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A division of wavefrontpolarimeter and optical
analysis of red blood cells
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• A Doctoral Thesis. Submitted in partial fulfilment of the requirementsfor the award of Doctor of Philosophy at Loughborough University.
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LOUGHBOROUGH UNIVERSITY OF TECHNOLOGY
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A DIVISION OF WAVEFRONT POLARIMETER AND OPTICAL ANALYSIS
OF RED BLOOD CELLS
by
Gabriela Maria Ruiz de Marquez
A Doctoral Thesis
Submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy of Loughborough University
May 1996
© by G. M. Ruiz de Marquez, 1996.
Dat,
ClaSf I--~"-'::": > •• ,,-:-.---:.""
-
A Isabel, Clemente, Bruno y Ruben.
ABSTRACT
This research project is dedicated to obtain vital information about blood components, using optical engineering techniques. The theory and apparatus developed within the project are the preliminary steps for the future development of non-invasive and low cost instrumentation for blood analysis that could be used in the clinical practice.
The measurement of biological suspensions by modulation of polarised light required . the development of a Stokes' parameters polarimeter. For this purpose a theoretical description of a new polarisation state sensor, a Division of Wavefront Polarimeter, has been developed and constructed, together with the appropriate considerations to calibrate the instrument.
Optical techniques are implemented to determine variations in the concentration of red blood cells in a given sample and modifications in the morphology of the cells. For this purpose, measurements of red blood cell suspensions were performed, combining polarised light measurements with an imaging technique. The light traversing the sample is absorbed and multiply scattered by the contents of the sample, thus the detected intensity can provide information about the nature of the scatterers. The amountof depolarisation of the incident light by the blood sample is an indicator of the concentration of the scattering particles. It. was found that polarised light measurements are useful to discriminate absorbance from scattering at certain concentration ranges of red blood cells.
ii
ACKNOWLEDGEMENTS
I would like to thank Dr. Peter Smith, my supervisor, for introducing me to this area, for his guidance and advice on all the aspects of this work and for his continuous support.
I would also like to thank Prof. Harry Thomason, my external supervisor, for his support and encouragement. He told me many times that if my research topic was not difficult, someone would had already done it.
I would like to acknowledge the financial support from Universidad Nacional Autonoma de Mexico, which made my studies in England possible.
I am grateful to Prof. C. Williams for letting me use the Sport Sciences Research Laboratory, at Loughborough University, to perform some of the experiments on blood. I am also grateful to Prof. H. Thurston for making available to me the facilities to conduct experimental work at the Clinical Sciences Building, Leicester Royal Infirmary. I would like to thank too Dr. M Bennett for providing me with blood samples and invaluable technical assistance.
I would like to extend my special thanks to R. Marquez for his helpful discussions in all technical and non-technical matters, and his advice on the software and hardware aspects of this work.
I very much appreciate the assistance and patience of Dr. A. Prado Barragan for teaching me the essential chemistry laboratory techniques and explaining to me all the biochemistry I could not understand.
I thank Amarat, Dave, Doug, Naimi, Hans and Matt, my colleagues in the Optical Engineering Group, for their assistance, support and for answering all my queries. In particular I would like to thank Doug for many interesting discussions during the course of this research, and to Amarat for her help and friendship during the long hours that the experiments on blood used to take.
I thank Mr. D. Smith, at the Dept. of Chemical Engineering in Loughborough University, for helping me with various aspects of my research which benefited from his chemistry skills. I also want to thank Mr. P. Barrington, at the mechanical workshop in the Dept. of Electronic and Electrical Eng., for making good technical drawings from my sketches and supervising the prompt manufacturing of the pieces I used in my experimental rigs.
Finally, thanks to all my family and friends in England and Mexico.
iii
Glossary of Terms
ADC
CD
DAB
DAS
DOAP
DOP
DOWP
Analogue to Digital Converter
Circular Dichroism
Data Acquisition Board
Data Acquisition System
Division of Amplitude Polarimeter
Degree of Polarisation
Division of Wave front Polarimeter
Haemolysis: Diffusion of haemoglobin out of the erythrocytes, leaving an empty
membrane, a "ghost".
Haeparin:
Lysis:
LP
MCHC
OD
ORD
PiN diode
An anticoagulant for blood.
The membrane of an erythrocyte is destroyed and haemoglobin leaks
out of the cell.
Linear Polariser
Mean Cell Haemoglobin Concentration.
Optical Density
Optical Rotatory Dispersion
P-N junction by injection, refering to a photodiode.
Polaras: Software application used to drive the DAB to obtain polarimetric
measurements from the DOWP. This name was formed from the words
POLARisation AnalysiS.
Polycythemia: An excessive number of RBC in the blood.
QWR Quarter Wave Retarder
RBC Red Blood Cells
Serum: When a blood sample has clotted, the clot is suspended in a fluid free
of fibrinogen called serum.
SMA stands for "SMArt", referring to a type of connector.
iv
Contents Page No.
Certificate of Originality
Abstract ii
Acknowledgements iii
Glossary of Terms iv
I. Introduction. I
2. Polarimetry. 5 2.1 Introduction. 5 2.2 Division of Wave front Polarimeters. 6
2.2.1 Division of Wavefront Polarimeters requiring of Specific Orientation ofPolarisers and Quarter Wave Retarder. 2.2.2 An improvement to the Division of Wave front Polarimeters reported in the literature.
3. Derivation of Stokes Parameters using a DOWP 11 with arbitrary settings of three Polarisers and one Retarder.
3.1 Transmission of an Electric Field through a linear Polariser. 13 3.2 An expression for the Normalised Transmission Coefficient. 14 3.3 Normalised Transmission Coefficient for Non-polarised Light. 16 3.4 Transmission of an Electric Field through a Quarter Wave Retarder. 17 3.5 Transmission of an Electric Field through a Quarter Wave Retarder followed .. by a linear Polariser. 19 3.6 Equations describing the four Intensities measured by the DOWP. 21 3.7 Derivation of the Stokes Parameters from four measured Intensities. 23 3.8 The Poincare Sphere. 27
4. Description of the Polarimeter Experimental Equipment 30 4.1 Description of the Sensor Head. 30 4.2 Software Description. 33 4.3 Description of the Data Acquisition System. 37
5. Calibration of the Division of Wave front Polarimeter. 39 5.1 Offsets Measurement. 39 5.2 Calibration. 40 5.3 Measurement of Angles and Parameters Estimation. 42
6. Performance Evaluation of the Polarimeter. 52 6.1 Theoretical Analysis. 52
6.1.1 NO converter with 12 bit resolution. 6.1.2 Adding an error of one degree to the position of a linear polariser. 6.1.3 NO converter with 10 bit resolution.
6.2 Experimental Analysis. 67 6.3 Additional Sources of Error. 72 6.4 Summary and Discussion. 75
6.4.1 Problems Encountered while using the DOWP in biomedical meas. 78
v
7. Traditional Techniques for Blood Analysis 7.1 Methods for Blood Analysis currently in use.
7.1.1 Blood Composition. 7.1.2 Blood Indices. 7.1.3 Automatic Cell Counters. 7.1.4 Measurements of Haemoglobin.
7.1.4.1 Spectroscopic Measurements - An Application in Blood Oximetry.
7.1.4.2 Colorimetry. 7.1.5 Examination of a Stained Blood Film.
79 79
7.2 Research in Optical Techniques for Blood Analysis. 86 7.2.1 Motivation for Investigating Optical Methods 7.2.2 Light Transmission through Whole Blood
- Whole Blood Modelled as an Absorbing and Scattering Medium 7.2.2.1 Adding a Scattering term to the Beer-Lambert law.
- Twersky's Model. 7.2.2.2 Diffusion Theory. 7.2.2.3 Kubelka-Munk Theory. 7.2.2.4 Time Resolved Spectroscopy for investigations in Tissue and Blood Oximetry. 7.7.7.5 The path lenght dependency in Transmittancec measurements.
7.2.3 Summary.
8. Imaging Technique for Absorbance and Scattering Measurements on Blood 99 8.1 Introduction. 99 8.2 Description of the Experiment. 100 8.3 Materials and Methods. 101 8.4 Discussion of Results. 103
8.4.1 Data Processing. 8.4.2 Semi-Empirical Model.
8.5 Conclusions. 119
9. Polarised Light and Imaging Measurements of suspensions of Erythrocytes 120 9.1 Introduction. 120
9.1.1 Measurements on Blood using Polarised Light. 9.1.2 An Improved Imaging Measurement Technique complemented by Polarised Light Measurements.
9.2 Description of the Experiment. 123 9.3 Materials and Methods. 124 9.4 Discussion of Results. 125
9.4.1 Imaging Technique. 9.4.2 Measurements with Polarised Light.
9.5 Red Blood Cell Morphology. 135 9.5 Conclusions. 139
10. Conclusions and Suggestions for Further Work. 141 10.1 Conclusions. 141 10.2 Suggestions for Further Work. 142
10.2.1 Suggested Modifications to the Mechanical Design of the Sensor Head. 10.2.2 Improvements to other components of the DOWP. 10.2.3 Suggested Modifications to the glass containers used in Blood Experiments.
Bibliography 148
vi
Appendices
AppendixA. Appendix B. AppendixC.
Diagram of the Sensor Head. Non-Linear Fitting to the parameters, the Quasi-Newton Method. Normal Haematology Values. - Recipe for Phosphate-buffered Saline. - Recipe for Ringer Solution.
153 154 158
vii
Chapter 1
Introduction
Since the beginning of the twentieth century, polarised light has been used as an aide to
learn about the nature of various organic compounds. Some of the best known
applications include measuring concentrations of sugars in solution and determining the
molecular structure of some proteins. More recently, in the food and pharmaceutical
industries, polarised studies have been employed for distinguishing between pairs of
enantiomers. Sometimes one of the enantiomers can have a harmful effect on the human
body, while the other one can help to heal it.
One of the most complete instruments that can be used to fully characterise the
polarisation state of a given beam of light is a polarimeter. This device can provide
information about the direction and amount of rotation of the polarisation ellipse that
describes -a particular polarisation state. Also it can determine the dimensions of the
ellipse, quantify how much of the light detected by the polarimeter is completely
polarised and how much is non-polarised. From all these data, the four Stokes parameters -
are extracted.
Although some optical diagnostic instruments are currently being developed to be applied
in the biomedical field, the full potential of polarised light has not yet been exploited. One
of the areas that could benefit from incorporating a polarimeter to the tool bench is blood
analysis. The reason is that some of the blood constituents can modify the polarisation
state of the light and suspensions of blood cells can depolarise an incident completely
polarised beam. Another point of interest is that a polarimeter, as many other optical
engineering tools, has the potential to evolve into a non-invasive piece of instrumentation
for the medical practice, because the measuring principle is based on a light beam, which
1
is modified after traversing biological tissue.
In this research project a Polarimeter of general purpose was designed, built and tested.
One of the most important features of this polarimeter is that only one set of four intensity
measurements is required to calculate simultaneously all the Stokes parameters. The
polarimeter is provided with four optical channels, in one of the channels there is only a
photodiode, in two of the other channels there is a linear polariser followed by a
photodiode, and in the remaining channel there is a quarter wave retarder followed by a
linear polariser and then a photodiode. The azimuth angles of the three linear polarisers
and retarder are all different, so each of the four photodiodes measures a different
intensity value when the polarimetric sensor is illuminated with polarised light. By
algebraic manipulation of the four intensity readings from each channel, polarisation
information of a sample tested by the polarimeter is obtained.
The main difference between this "Division of Wavefront Polarimeter" and other
polarimeters of similar type, reported in the literature, is that the polarimeter reported here
- does not require to set the polarisers and retarder in the sensor head at unique and
predetermined azimuth angles, as all the other models, that we are aware of, do.
The recommendations that will appear in the following chapters of this text, to improve
the polarimeter and additional equipment, should be followed if the equipment is going to
be applied· for biomedical applications. However, in order to test the principle of
measuring blood components with polarised light, some less ambitious experiments were
successfully conducted. In these experiments linearly polarised light was used to
illuminate a blood sample, then it was studied with an analyser parallel and perpendicular
to the incident polarisation state. These studies were combined with an imaging
technique. Samples with various concentrations of whole blood and haemolysed blood
were studied.
This document is divided into ten chapters, including this introduction as the first one.
2
Chapter 2 is a literature review on the subject of Polarimetry. The Division of Wavefront
Polarimeter will be revised more carefully than other types, because the polarimeter
developed within this· research project falls into this category. The advantages of
developing a Division of Wave front Polarimeter, with arbitrary orientation of polarisers
and quarter wave retarder, are mentioned in this chapter.
Chapter 3 is dedicated to obtain, in some detail, the equations corresponding to the Stokes
Parameters. These parameters are derived from four intensity readings, measured by the
Polarimeter with arbitrary orientation of its polarising elements. The theory developed in
this chapter is a generalisation of the algorithms on which older versions of Division of
Wavefront polarimeters were based.
Chapter 4 is a complete description of the polarimeter that was built. Not only the sensor
head is mentioned here, which is the polarimeter basis, but also the additional software
and hardware required for using it as a measuring instrument.
In Chapter 5 a calibration routine is described. One of the procedures included was
designed to measure and eliminate some offsets present in the intensity measurements.
Another one to artificially correct the problem of uneven illumination of the sensor head,
and a final one dedicated to determine the orientation of three polarisers and one quarter
wave retarder comprised in the sensor head, plus their absorbance parameters.
Chapter 6 consists of a performance evaluation of the Polarimeter. Is included a
theoretical analysis of the errors introduced in the polarimetric measurements by
quantisation of the measured intensities, while being converted from analogue into digital
signals, and of the errors caused by an incorrect determination of the azimuth angles of
the polarising optics in the sensor head and their absorbance parameters. The effect of
these sources of error was corroborated experimentally.
Chapter 7 is a literature review on laboratory techniques for blood analysis. Special
3
consideration is given to automatic cell counters because of tbeir importance in
haematology laboratories. Also tbere is a section dedicated to tbe research on optical
techniques for blood analysis. This chapter includes a review of some of best known
tbeoretical models describing tbe interaction of light with a blood sample. Those models
take into account absorbance and scattering measurements.
A first set of experiments is reported in Chapter 8, in which an imaging technique was
developed to measure tbe absorbance and scattering of light by a sample of blood in
motion. These measurements are intended to provide information about tbe concentration
and type of red blood cells contained in a given sample. The experiments were perfected
and the experimental technique was complemented by adding a polarised light
measurement of suspensions of red blood cells. The combined technique is described in
Chapter 9, showing tbe potential tbat polarised light measurements have to distinguish the
effects of absorbance from tbe scattering of light produced by tbe blood sample.
Additionally, some comments are made relative to how tbe morphology of tbe cells is
affected. while being tested by most optical methods. Some suggestions. are made to
correct this problem in the future.
Finally. suggestions for future work can be found in Chapter 10, together with conclusive
remarks about tbe contributions of this project to tbe fields of polarimetry and optical
analysis of blood.
4
-----------------------------_ ..
Chapter 2
Polarimetry
2.1 Introduction
Polarimeters have been used since the early days of . this century, mainly in the sugar
industry. Their purpose was to quantify amounts of sugar in solution and to
discriminate among different types of sugars. Sugars, like many other organic and
inorganic compounds, are optically active. They have the ability to rotate the plane of
polarisation of linearly polarised light, because its molecules or crystals lack a plane
or centre of symmetry. Polarimeters that were calibrated to quantify the amount of
rotation in "sugar degrees" (percentage of sucrose by weight), were known as
saccharimeters. The early polarimeters consisted basically of a tungsten lamp used to
illuminate a train of a linearly polarising element, a sample chamber and a second
linearly polarising element, or analyser. All the information they could provide was
related to the optical rotation of light [Fluegge, 1., 19651, whereas modern polarisers
are designed to extract the full Stokes parameters from the detected light.
The most recent generation of polarimeters can be classified into two broad
categories: I) Division of Amplitude Polarimeters (DOAP) and IT) Division of
Wavefront Polarimeters (DOWP).
In the division of amplitude polarimeter designed by Azzam, R.M.A. [19821, a beam
splitter divides an incident beam under measurement into a reflected beam and a
transmitted beam, travelling in orthogonal directions. Each of these two beams are
incident on Wollaston prisms, which divide each beam again into another two beams,
and photodetectors measure each of the four resulting beams. Let F be a 4 x 4 Mueller
matrix of the polarimeter, which is determined by the reflection and transmission
matrices of the beam splitter, the azimuth angles of the prisms and by the detectors
sensitivities. For a given DOAP it is necessary to know F in order to detennine the
Stokes vectors of the incident light. F can be measured directly by calibration. The
procedure is to deliberately polarise the incident light into four known different states,
described by four linearly independent Stokes vectors and to detennine the associated
signal vector from the outputs of the linear photodetectors.
Azzam [ibid.] indicates that the advantages of the DOAP include the absence of
moving parts (as required by some DOAP and also by some DOWP), the absence of
modulation, its fast response limited only by the photodetectors, and that the
calibration makes it unnecessary to know the properties of the individual components
of the polarimeter. But the main disadvantage is that the calibration procedure requires
to invert the F matrix to obtain the Stokes parameters.
The same author designed another DOAP photopolarimeter, based on conical
diffraction from a metallic grating [Azzam, R.M., 1992]. In this device, a metallic
grating splits the incident beam in at least four different orders, modifying the
polarisation state of the incident beam, ~hen each of the diffracted beams are captured
by a photodetector. The four Stokes parameters are detennined from the four
photodiode outputs by means of a matrix that is obtained by calibration. In general the
advantages and limitations of this design with respect to a modulation based DOAP
are the same as those stated in the previous example.
2.2 Division of Wavefront Polarimeters
In a division of wavefront polarimeter, or DOWP, the incident beam is divided at least
into four segments that evenly illuminate the test sample, then a polarising device,
used as an analyser, is located in each of the beam paths before they are detected. The
limitations of this technique are that the incident light beam must be uniformly
polarised over its cross section, the light transmitted from the sample must illuminate
equally all photodetectors and the absolute responses from all the photodetectors must
be the same, or the system must be calibrated. From the (normally) four transmitted
intensity readings, the Stokes parameters are calculated.
6
Azzam et al. constructed a DOWP polarimeter requiring four Si photodiodes [Azzam,
R.M., et al., 1988]. The input light has to be reflected from the ftrst one to the second
one, from the second one to the third one and so on. Each detector is at a different
plane of incidence, with angles of approximately 45° between them. With an
optimum set of calibration states, the instrument matrix is determined, and the Stokes
parameters are obtained from this matrix.
2.2.1 Division of Wavefront Polarimeters requiring Specific Orientations
of Polarisers and Quarter Wave Retarder.
The main difference among various designs of DOWP polarimeters is the form in
which the polarisation state of each of the four signals are modifted. The polarimeters
reported in the literature during the past decade, required one linear polariser (LP) in
two of the channels, one quarter wave retarder (QWR) followed by a linear polariser
in another one of the channels, and one clear channel. Different designs diverge in the
. - orientations of the transmission axis of the polarising elements that they use, and in
consequence, in the form in which the Stokes parameters are extracted and the
polarimeter is calibrated. Until 1995 all polarimeters of this type, patented or reported
in the literature, had to localise the orientation of the polarisers and retarder at very
speciftc angles. By doing this, relations between intensity readings from the four
channels, to extract the Stokes parameters, were relatively simple, but on the other
hand they required successive measurements involving the rotation of the polariser
elements to various ftxed orientations. This produced, in consequence, inaccurate
results derived from errors caused by non positioning the polarisers and retarder at the
specifted angles. Some of these instruments are described next.
Collett [1980] designed one of these types of polarimeter, using a calcite prism in each
of the four channels and also a QWR in only one of them. The intensity detected in
each channel was given by I(9,cp), where 9 is the polariser angle and cp the phase shift
introduced by the retarder. The combinations of polariser-retarder were set to give :
1(0,0), I(rrl2,O), I(rrl4,O) and I(rrl4,rrl2).
7
A slightly different design of a DOWP required to have one LP in one channel,
another LP (orthogonal to the first one) in a second channel, a circular polarisation
state in the third one, and also a circular polarisation state in the fourth one, but
orthogonal to the previous one. Various positions of the polarising optics had to be
measured to calibrate the instrument [Abramov, V.!. and Tagunov, B.B., 1989].
One of the polarimeters which is nowadays in the market (Hewlett Packard Co.,
system HP8509 A) requires to have one polariser and one retarder in channel I, one
polariser in each of the channels 2 and 3, and nothing in channel 4. Apparently the
polarisers must be oriented at fixed angles -45°, 0° and 90° respectively. Also an extra
polariser and a polarisation adjuster must be provided [Cross, R. et al., 1991]. One of
the authors in this reference patented a method to calibrate the instrument using
various different polarisation states [Heffner, B.L., 1994].
Another instrument which has been patented, consists of the same elements as the
__ _ _ previous reference and performs four intensity measurements at various settings· of LP
and QWR. One setting that was suggested had two of the polarisers at 45° from each
other, a circular polariser (right handed) in the third channel, and in the fourth channel
it had a neutral density filter with 50% transmission and which is insensitive to
polarisation effects, [Siddiqui, A.S., 1992].
Scholl, B et al., [1995], reported another polarimeter that required to perform four
different measurements at different positions of a QWR, and an external polarisation
detector was used to calibrate the device.
Arnbirajan, A. and Look, D.C. [1995 a & 1995b] tried to determine which
combination of azimuth angles of a QWR, followed by a linear polariser, gave smaller
errors when calculating, by four successive intensity measurements, the Mueller
matrix of the combination. They found that if the LP had an azimuth angle of 0°, then
the set of four optimum values for the azimuth angle of the QWR were (-45°, 0°, 30°
and 60°), or (-90°, -45°, 30° and 60°). If they moved both parts, the set of four
8
optimum combinations of LP and QWR were [(0°,90°), (O°,-arcsine (113», (120°,
arcsine (113» and (240°,-arcsine (1/3»].
2.2.2 An improvement to the Division of Wavefront polarimeters reported
in the literature.
All the Division of Wavefront polarimeters mentioned in the previous section require
to have LPs and QWRs positioned at specific azimuth angles. In all cases except for
Ambibarajan A. and Look, D.e. [1955a & 1955b], who calculated optimum fixed
orientations, these angles were multiples of ~ . Presumably these values of the
angles were chosen in order to simplify the expressions developed to recover the
Stokes parameters from the transmitted light intensity readings. However the main
inconvenience caused by this choice of values is the difficulty in positioning the optics
at the required orientation, and small deviations from the predetermined orientations .
can render useless the simplified light transmission models.
The simplified models used to extract the Stokes parameters can be inaccurate also
because the polarisers and retarder are assumed to be ideal, i.e. as if they were non
absorbing components. This is the case in the models developed by Scholl B. et al.
[1955], Ambibarajan and Look [1995b], Abramov, V.I. and Tagunov, B.B., [1989]
and Siddiqui, A. [1992].
Also some devices such as those by Scholl B. et al. [1955] and Ambibarajan and Look
[1995b], must perform four successive measurements at various predetermined
positions of the same optical elements. By doing this not only the total measuring time
is long, but also they are likely to position inaccurately the polarisers and QWR, and
. that could happen four times during one total measurement.
9
For the reasons stated before, the main objectives motivating the design of a new
Division of Wave front Polarimeter included the following points:
I) Elimination of moving parts. This has the purpose of reducing the total measuring
time and also of fixing each of the polarisers and retarder at any arbitrary position.
IT) Generalisation of the model that describes the transmission of light through the
combination of non-ideal QWR and LP, to accommodate any arbitrary settings of
azimuth angles. This generalisation has the advantage that any orientation of the
polarising elements is useful. The only restriction, which is common to the other
existing designs, is that the azimuth angles of all the elements must be different from
each other. In our novel design, predetermined positions of the elements are not
sought, instead they are located arbitrarily and the orientation of each azimuth angle is
measured afterwards.
III) Consideration of the absorption coefficients of non-ideal polarisers and retarders
--in the equations generated to obtain the Stokes parameters .
. A Division of Wavefront Polarimeter, that incorporates into its design the conditions
mentioned above, was constructed and tested. Chapters 3,4,5 and 6 of this dissertation
are dedicated to it.
10
,------------------ - -- - -- -- --
Chapter 3
Derivation of the Stokes Parameters using a DOWP
with Arbitrary Settings of three Polarisers and one
Retarder
The intention of this chapter is to derive the expressions used to calculate the Stokes
Parameters. We will see that to accomplish this we need four intensity readings. The
readings are provided by the sensor head incorporated into the Division of Wavefront
Polarimeter (DOWP).
Any polarisation state of light can be represented by its polarisation ellipse (in two
dimensions, observing only a cross section of the ellipse). Figure 3.1 is just an
example of an infinite number of polarisation states. An arbitrary polarisation state is
- - characterised by the azimuth, ellipticity and handedness of the polarisation ellipse.
The azimuth ex. is the angle that the major axis of the ellipse makes with the X axis.
p
Figure 3.1: The polarisation ellipse.
Angles are positive when measured anticlockwise from the X axis.
Ellipticity is a term commonly used in optics for the inverse of the eccentricity of the
ellipse and is definedas the ratio of the length of the semi-minor axis (b) to that of the
semi-major axis (a). The ellipticity is defined to be always positive and to range from
o for linearly polarised light to I for circularly polarised light.
According to Figure (3.1) the electric field components (Ex,E,) of an arbitrary fully
polarised source are given by
Ex =acosa sin(rot+cp.)-bsina cos(rot+cpo)
Ey =asina sin(rot+CPo)+bcosacos(rot+cpo)
(3.1)
(3.2)
where ro is the wave frequency, a and b are the semi-axes of the ellipse, a is the
polarisation azimuth and cp 0 is a phase constant.
Since the sourCe is fully polarised, the corresponding average source intensity can be
obtained as the average in time of the sum of both squared components of the electric
field, having defined the time average as
1fT (J)~- j(t)dt TO'
and integrating over a time period T = 2n / ro, then
Using (3.1) and (3.2)
(3.3)
(3.4)
12
So the average source intensity is
3.1 Transmission of an Electric Field through a Linear Polariser.
(3.5)
When the field (3.1, 3.2) , traverses a linear polariser (LP) inclined at an angle e , the
same electric field components are projected over the polariser parallel and
perpendicular axis (called p and s respectively). The electric field components
emerging from the polariser are given by _
E p = F. (Ex cose + Ey sine)
(3.6)
E, = F, (Ex sine - Ey cose)
Where constant factors 't p and 't s represent the intensity transmission coefficients for
the polariser in transmission mode and extinction mode respectively.
Replacing Ex and E y
[ (acosasin(rot+<!>o)-bsinacos(rot+<!>o)) cose]
Ep =..[t; +( a sina sin(rot + <!>o) + bcosacos(rot + <!>.J ) sine
[ (acosasin(rot+<!>o)-bsinacos(rot+<!>.J) Sine]
E, = F. _( a sinasin(rot + <!>o) +bcosacos(rot+<!>o) )cose
(3.7)
13
3.2 An expression for the Normalised Transmission Coefficient.
The normalised transmission coefficient, T(a), for the LP is obtained by averaging in
time the sum of the square of the above components and dividing by the averaged
intensity, i.e.
(3.8)
Replacing I p from ( 3.5) gives
(3.9)
. After substitution in (3.9), the above integral can be evaluated easily using the
following well known integrals
I JTI JT I - sin2(u) du =- cos2(u) du =-coT 0 coT 0 2
1 JT - sin(u)cos(u) du = 0 coT 0
(3.10)
where u=co t+'I>o and T is one period. Then after integration,
T(e) = a2 !b2 ((t, sin2e +t p cos2 e )(a2 cos2 a. +b2 sin2 a.)
. +(t, cos2 e +t p sin2 e )(a 2 sin2 a. +b2 coS2 a.) (3.11)
+2(t p -t ,)cose sine (a 2 sina. cosa. _b2 sina. cos a. ))
14
From Figure (3.1) the ellipticity is given by
b e=
a
then Equation (3.11) can be rewritten in terms of the ellipticity as follows
T(9) = e2 ~1 ((-c, sin2 9 +t p cos2 9 )(cos2 a +e2 sin2 a)
+(t, cos2 9 +t p sin2 9 )(sin2 a +e2 cos2 a)
+2(t p -t ,)cos9 sin9 (sin a cos a _e2 sin a cos a ) )
Making use of the following trigonometric identities
cos(e -a) = cos9 cosa +sine sina
sin(e -a) = sine cosa -cose sina
Equation (3.13 ) can be stated in a shorter form:
(3.12)
(3.13)
(3.14)
(3.15)
If in terms of absorption the polariser was an ideal one, t p would be equal to unity,
and t, equal to zero. Also if circularly polarised light was transmitted through the
polariser(e=l), then T(e)=~, i.e., only half of the incident intensity would be
transmitted. However, if linearly fully polarised light is transmitted through an ideal
LP, (e=O) and T(e)=cos2(e-a), which has the form of Malus' law for the
transmission through a train of linear polarisers [Shurciiff, W.A., 1964].
15
3.3 NormalisedTransmission Coefficient for Non-polarised Light.
The above transmission coefficient is applicable to any fully polarised light source
illuminating the LP, however if the source is randomly polarised, another coefficient
't 0 must be used. The latter can be obtained by integrating T(a) over all possible
azimuth angles and ellipticities, multiplied by the normalised probability function of
the polarisation states. That is
IYz J 1 J _//T(a,a,e)j(a,e)da de o /2
'to=---~~--~------~.--L-
J 1 JYz _,// j(a,e)da de o /2
(3.16)
, )
Where . j(a ,e) = P, i.e. a constant. Then
... J'f:i j(~,e) da de =~ P o /2
(3.17)
(3.18)
Which can be reduced to
16
(3.19)
For an ideal LP 't 0 = 05. It means that all the light with a linear polarisation state
parallel to the polariser axis would be transmitted (a=OO), but none of the light with a
linear polarisation state perpendicular to the polariser axis (a=900), and only half of
the initial intensity of a randomly polarised beam traversing the polariser would be
transmitted.
3.4 Transmission of an Electric Field through a Quarter Wave Retarder.
We now consider the same electric field described by equations (3.1 & 3.2) to be
transmitted through a quarter wave retarder (QWR). The retarder is orientated at an
azimuth angle p and has absorption coefficients 't r and 't I along its fast and slow axis
respectively. The components of the electric field given by equations (3.1 & 3.2)
projected over the retarder fast and slow axis are given by Er and El respectively, thus
Er =F, a sin (rot + <1>.) cos {a - p)-F,bcos(rot+<I>o}sin{a - p) (3.20)
It is observed that there is a phase shift of rrJ2 between the fast and slow components
of the field. If the beam entering the retarder was linearly polarised (i.e. b=O) at an
angle of 45° to the retarder fast axis (i.e. a-p=rr14), the two components of the electric
vector describing the emerging beam would be identical, thus the emerging beam
would be circularly polarised.
17
A transmission coefficient T( p), nonualised with respect to the completely polarised
source intensity, can be constructed as follows
After integration
The above equation can be expressed in tenus of the ellipticity as
T(p) 't1sin2(a - p)+'t,cos2(a - p)+e2[t1cos2(a - p)+'t,sin2(a - p)]
e2 +1 .
(3.22)
(3.23)
When this equation is describing an ideal retarder, 't I = t, = 1, and T( p) = 1, meaning
as it is well known, that an ideal QWR only shifts the phase between the two
components of a beam passing through it, but the amplitude of the beam remains the
same.
Also one can find an algorithm for the transmission coefficient of randomly polarised
light through a QWR, using an expression equivalent to (3.16), obtaining
(3.24)
18
3.5 Transmission of an Electric Field through a Quarter Wave Retarder followed by a Linear Polariser.
The design of this DOWP requires that at least in one of its channels, both amplitude
and ellipticity of the incident light is modulated. Since the QWR can only introduce a
modulation in the ellipticity, it is necessary to add a polarising element that can
modulate the intensity in a predictable fashion and the obvious selection is a LP. The
order in which both elements are located within the sensor head is important, because
if a polariser is followed by a QWR, the emerging beam will have a constant
amplitude independent of the orientation of the elements. However if a QWR is
followed by a LP the ellipticity and amplitude of the beam leaving the set will depend
on the orientation of the LP relative to the QWR.
To obtain the electric field components of the beam transmitted through the
combination described above, the components E, and E, ' equations (3.20 & 3.21) of
_. the beam exiting the QWR have to be projected over the polariser's parallel and
perpendicular axes. The beam emerging from the combination has components Ep.
and E,. given by the following equations:
(3.25)
E,. =F, (-E,sin(a - p)+E,cos(a - p))
and after replacing the values of E, and E" the square of the transformed components
will be given by the following equations
E / =t p {cos2 ~ (t ,a2 sin2 y cos2 Tl +t ,b2 sin2 Tl cos2 y -2t ,absiny cosy sinTl cosTl)
+{sin2 ~ (t ,b2 sin2 y cos2 Tl +t ,a2 sin2 Tl cos2 y - 2t, absiny cosy sinTl cosTl)
+2sin~ cos~ (F,Ft a2 siny cosy sinTl cosTl +F, Ft b2 siny cosy sinTl cosTl
-F, Ft absin2y cos2 Tl -F, Ft abcos2y sin2 Tl)}
(3.26)
19
E/ ='t, {sin 2 ~ ('t fa 2 sin 2 y cos211 +'t fb 2 COS211 sin2 y - 2't fab siny cosy sinl1 cosl1)
+{cos2 ~ ('t ,b2 sin2 y cos211 +'t ,a2 cos211 sin2 y - 2't, ab siny cosy sinl1 cosl1)
-2sin~ cos~ (..jtj Fs a2 siny cosy sinl1 cosl1 +..jtj Fsb2 siny cosy sinl1 cosl1
-..jtj Fs absin2y cos211-..jtj Fs abcos2y sin211)}
(3.27)
where y =rot+cjlo' 11 =a - p and ~ =a - p.
The transmission coefficient r( 9 • p) for completely polarised light of a QWR
followed by a LP can be obtained by integrating the sum of equations (3.26 and 3.27)
with respect to t over one cycle and normalising it with respect to the polarised source
intensity.
r(a.p)=~{'tr['tpCos2(a - p)+'t s sin2(a - p)][cos2(a - p)+e2 sin2(a - p)] l+e
;'t r ['t p sin2(a - p)+'t s cos2(a ... p )][sin 2(a -p)+ e2 cos2(a "::p)]
+2..jt; Fz ('tp -'ts)esin(a - p)cos(a- p)}
(3.28 )
The previous equation can be written in a shorter form as follows
where
k2 = 't ,'t, sin2 (9 k .,.. p) +'t ,'t p cos2 (a k .:. p)
k3 =~'tl't, ('tp -'t~)sin2(ak - p)
(3.29)
(3.30)
20
Similarly, the transmission coefficient to £ for a QWR foIlowed by a LP when non
polarised light is transmitted through them is obtained by integration of (3.29) over all
possible azimuth angles and ellipticities, multiplied by the probability function of the
polarisation states and normalising with respect to that probability function. This
procedure gives
(3.31)
3.6 Equations describing the four Intensities measured by the DOWP.
Having obtained the equations describing the transmission for completely polarised
light and randomly polarised light through either a LP (equations 3.15 and 3.19
respectively) or a QWR followed by a LP (equations 3.29 and 3.31 respectively), it is
possible to write an expression for. the intensity I transmitted through the LP and
measured by a photodetector, assuming that the source is partially polarised. The total
intensity is given by a sum of polarised and unpolarised light intensities Ip + 10
, so
(3.32)
Equivalently the total intensity transmitted through the combination of QWR and LP
measured by a photodetector is
(3.33)
It is assumed that when an optically active sample is located in front of the DOWP, to
test the polarisation state of the sample, a collimated beam will illuminate it, and the
21
light transmitted through it will be received by each of the four channels of the DOWP
sensor head.
To fully characterise the polarisation state of a given sample, it is required to know the
corresponding Stokes Parameters or equivalently, the azimuth angle, ellipticity,
amount of polarised light and randomly polarised light transmitted through the
sample. All this information must be extracted from the DOWP intensity readings.
Then it is necessary to build a set of at least four equations describing the intensity
transmitted through the optics on each channel as they are measured by their
respective photodetector. Because a reading of the total intensity transmitted through
the sample is required, one of the channels should be free of any polarising optics. At
least one of the channels must contain a LP, so the azimuth angle of the polarisation
state of the sample can be found using an approach based on Malus' law. Finally one
of the sensors must include a QWR additional to a LP, so various amounts of
ellipticity can be measured.
It was found that the most convenient design for the DOWP consisted in using two
channels containing a LP each, another one containing a QWR followed by a LP, and
an extra clear channel.
Each of the two channels with a LP in the sensor head, admits a different intensity li
and I j modelled by equation (3.32) and labelled by the orientations of the polarisers
a I and a 1" The intensity I. transmitted through the channel comprising a QWR
followed by a LP with orientations p and a. respectively, is described by (3.33).
Finally the clear channel is assumed to measure the total partially polarised light
intensity emerging from the sample. The set of intensities describing the four channels
in the sensor head is therefore the following:
(3.34)
22
(3.35)
(3.36)
(3.37)
with kJ, k2 and k3 defined by the set of equations (3.30).
3.7 Derivation of the Stokes Parameters from four measured intensities.
As can be seen in fig. (3.1), any polarisation state measured by the DOWP can be
characterised by a polarisation ellipse, with a corresponding azimuth angle, ellipticity
and degree of polarisation. Using these. three quantities the complete Stokes
parameters can be obtained. Through some algebraic manipulations with equations
(3.34) to (3.37), these parameters will now be derived.
Replacing 10 from (3.37) in equations (3.34) and (3.35), and taking the ratio of the
resulting equations
Ii -'to I[
Ij-'t o I[
cosZ( Si -a) cosZ(Sra )
Using the trigonometric identity given by (3.14), (3.38) gives
(3.38)
23
coS2a(Ii cos29 j -to 1/ cos 29 j - I j cos29i +to It cos29i ) .
= sin 2a ( I j sin29i -to 1/ sin29i - Ii sin 29 j +to It sin 29 j)
Or
(3.39)
From this equation the azimuth angle a is easily obtained. To derive the ellipticity,
Equation (3.37) must be substituted in Equations (3.34 & 3.36 ), and after taking the
ratio of these two equations and rearranging terms
Ik -t/I/ 2e2[k\ cos2(a - p)+ k2 -to l]+2ek3 +2[k\ sin2(a - p)+ k2 -to l ]
Ii-t o 1/ (tp-ts)cos2(9i-a)(e2 -1)
Letting
Zl = kl cos2
(a - p)+k2 -'to l
z2 =k1 sin2
(a - P)+k2 -to l
Z 3 = ('t P - 't s )cos 2 (e i-a)
. l I k - to 1/
Z4 = . 2{Ii - 'to 1/)
(3.40)
(3.41)
Where k], kz and k3 are defined as above, (3.40) can be rewritten in terms of Z), Zz, Z3.
and Z4 as follows
24·
(3.42)
Leading to the following second degree equation in e
(3.43)
Solving this equation for the ellipticity, two solutions are obtained
(3.44)
Each of the solutions of the ellipticity have a physical meaning, the solution taking the
positive sign corresponds to the definition of the ellipticity used in Figure (3.1), with
e';' Ya ,but the solution-using the negative sign corresponds to the inverse definition
e = % . While the first definition gives a zero ellipticity for a measured linearly
polarised state (b=O), the second definition gives an infinite ellipticity for the same
polarisation state, for this reason a convention of using the more standard definition,
[e.g. Kiiger, S.D, et al. , 1990] with the positive sign will be adopted.
Having determined the azimuth (J, and ellipticity e, the completely polarised intensity
measured by the polarimeter, lp, can be obtained. This is derived from the subtraction
of (3.35) from (3.34),yielding
(3.45)
25
Finally the value of 10 for the amount of non-polarised intensity incident on the sensor
head is obtained directly from (3.37) as
(3.46)
Another useful quantity is the Degree of Polarisation (DOP), defined as
(3.47)
A very useful representation of an arbitrary polarisation state is in terms of the four
Stokes Parameters [ibid.], which are only functions of intensity. This mathematical
representation is particularly important, not only because it is very simple, but also
because any beam of light is described by a four element vector and any optical
element "acting" on the light is described by a 4x4 matrix (Mueller matrices). Then
the complicated optical problem of light propagation through any medium can be
reduced to an algebraic problem (if the Mueller matrix is known).
The most common notation for the Stokes parameters is {I, Q, U, V}. I is interpreted
as the total intensity, Q as the difference in intensities between horizontally and
vertically linearly polarised components, U as the difference in intensities between
linearly polarised components oriented at +450 and _450, and V as the difference in
intensities between right and left circularly polarised components.
Equations (3.39 and 3.44 - 3.46) can now be used to write the Stokes parameters by
the well known relationships for normalised intensities [ibid.],
1=1 (3.48)
Q = cos(2arctan(e»)cos(2a) (3.49)
26
U = cos(2 arctan(e)) sin(2a) (3.50)
v = sin(2arctan(e») (3.51)
It is possible to represent unpolarised or partially polarised light with a Stokes vector,
because unpolarised light can be described by an electric vector that, at any moment in
time, corresponds to a well-defined polarisation state, but that fluctuates randomly
between different polarisation states on a time scale that is small compared with the
frequency of light. Thus, over a relatively long period of time, all the rapidly varying
polarisation states are averaged and the beam appears unpolarised. For unpolarised
light, the polarisation dependent terms Q, U and V will disappear, whereas for
partially polarised light
(3.52)
The relationship given by equation (3.52) can be understood by considering partially
polarised light as made up of two beams, one of which is completely polarised and
the other one is unpolarised. The magnitude of the contribution of each of these beams
to the total beam determines the DOP of polarisation of the total beam.
The DOP can be obtained either using Equation (3.47) or also from the Stokes
parameters
DOP (3.53)
3.8 The Poincare Sphere.
A useful representation of completely polarised light is by the Poincare Sphere, a
graphical equivalent to the Stokes parameters. This sphere (see Fig 3.2) has a unit
radius spherical surface, and each point on the surface describes a different fully
27
polarised state. Any modulation involving the effect of a retarder on a monochromatic,
polarised beam is represented by "moving" along an arc of circle over the sphere.
The top and bottom of the sphere stand for left and right circular polarisation,
respectively. Every point on the equator represents a linear polarisation form. Points
between the equator and bottom (or south pole) of the sphere represent right elliptical
polarisation. The point in the equator, marked in Figure (3.2) by the word
"horizontal", represents light that is linearly polarised horizontally, and the point
located 180 degrees apart from the first one along the equator, represents vertically
linearly polarised light. Any two diametrically opposite points represent an orthogonal
pair of polarisation forms.
The Poincare Sphere has been traditionally used to determine the effect of retarders
over any monochromatic beam of completely polarised light.
Icp
7 ___ t---l-_ vertical I{------t-----t----i horizontal
rcp
Figure 3.2: Poincare Sphere. The top of the sphere corresponds to a right handed
circular polarisation state, while the bottom corresponds to a left handed circular
polarisation state
28
For a completely polarised beam of unit intensity, tbe Stokes parameters define a
sphere of unit radius, since d+u2+v=1. Points on tbe sphere have tbe cartesian
coordinates (Q, U, V) and hence correspond to specific states of polarisation.
The tbeory developed in this chapter made clear that only four simultaneous intensity
measurements of a beam of light are necessary to fully characterise tbe polarisation
state of tbe beam, with tbe only requirements of having tbe light beam illuminating
uniformly tbe whole surface of tbe sensor head, and of having tbe four polarising
elements in tbe sensor head positioned at different angles from each otber. In Chapter
4 is presented a complete description of the DOWP experimental equipment.
29
Chapter 4
Description of the Polarimeter Experimental Equipment
The principal components integrating the Division of Wave front Polarimeter (DOWP)
are the sensor head, the data acquisition board linked to a personal computer and the
software developed to acquire, manipulate and display the experimental data.
Although the software and data acquisition board were developed in the Electronics
and Electrical Eng. Dept. at Loughborough University, with the purpose of coupling
them to the sensor head for polarimetric measurements, their design is such that they
can work integrated to different systems to perform a variety of tasks involving the
acquisition and handling of data; the sensor head on the contrary, is the element that
uniquely characterises the DOWP as such.
4.1 Description of the Sensor Head
The sensor head consists of four plastic optical fibres arranged in a triangular
geometry, with one of the fibres located on the centroid of the triangle. A linear
polariser rests on top of each of the three fibre tips located on the vertices of the
triangle and a quarter wave retarder is placed on top of only one of the linear
polarisers. The central fibre has nothing on top. All the fibres are inserted in a PVC
cylinder and are held together with epoxy resin.
To maintain the polarising optics in place on top of the fibres, a brass casing was
designed, see Figure (4.1). This is a hollow brass cylinder made of two sections, the
frontal section holds first an ordered set of elements, then the cylinder with the optical
fibres is inserted and finally the rear section (not shown in Fig. 4.1) is screwed to the
frontal one to keep all the pieces fastened in place. The frontal piece clamps first a
clear optical glass flat or window, used to ease the removal of dust, then is clamped in
a holder containing the quarter wave retarder, followed by another glass plate and by a
holder with the three linear polarisers orientated at different angles. The complete
. brass casing is mounted inside an aluminium rig that can be secured to the optical
table. All the pieces integrating the sensor head can only be mounted in a single
position that guarantees the perfect alignment of all the pieces. A technical drawing of
the sensor can be found in Appendix A.
Outer Casing
Glass Plate
Section containing 4 optical fibres
linear polarisers
Figure 4.1: Diagram of the sensor head construction.
The other end of each of the optical fibres is fitted with an SMA connector which
attaches the fibre to a photodiode in the data acquisition system (DAS). The fibres can
be as long as desired because all intensity attenuation that could have occurred in the
fibres is accounted for in the calibration stage.
31
An intense, 633 nm wavelength, non-polarised light source must be used with this
polarimeter. It can be a LED, a filtered white light source, or a non-polarised laser
diode. The requirement of using a non-polarised source is imposed by the polarimeter
calibration routine and a strong light source is preferred in case the test sample is a
strongly absorbing material. The wavelength of the source is of particular importance
since the quarter wave retarder in the sensor head is designed to be used within a
narrow spectral window.
A LED was chosen as the light source when the DOWP was tested. This selection was
motivated by various factors [Fantini, S., et al., 1994]: i) Ease of modulation: A LED
can be easily intensity modulated using a signal generator, the efficiency of
modulation is limited by the electrical response time of the LED. ii) Stable output:
The intensity emitted by a LED is more stable than the output from a laser or an arc
lamp. iii) Safety: The low optical power of the emission and the wide angular
distribution are ideal for a portable instrument that will not induce any damage of the
samples. iv) Cost: The low cost of the DOWP light source makes more favourable the
possibility of commercialising the product.
The calibration routine can cope with a slightly uneven illumination of the optical
fibres, but an equal illumination is most desirable. The particular geometry of each
test sample will modify the illumination of the fibres and for a reference to be
established an equal illumination is ideal while the system is being calibrated. When
the performance of the DOWP was tested, a lens was used to maximise the
illumination of the whole sensor head, see Figure (5.3).
The design and construction of the prototype reported here had the light source
separated from the sensor head, but they were aligned one in front of the other by
securing them to a portable optical bench. Any necessary additional optics and the test
sample can be mounted on the bench between the source and the sensor head. By this
arrangement, a test sample such as a polariser is affixed to the bench by a mount and
post, but a liquid test sample requires a non-polarising clear cuvette and a cuvette
holder attached to the bench. Preferentially the cuvette should be made of clear glass
32
with flat walls, large enough to avoid reflection and diffraction effects of the light
beam from the cuvette surfaces.
Figure 4.2: Photograph of the Sensor Head
4.2 Software Description
The software used to link the computer to the DAS, to compute and to display the
measured polarisation parameters, was originally written having the DOWP in mind,
as a final year undergraduate project [Colquitt, D.l., 1994]. Later the algorithms used
to calculate the Stokes Parameters were modified and the calibration and offsets
determination procedure were added.
33
-, ;\-" ... '"
\ " \ .. \ ,
~ .. '
.'
Figure 4.3: Example of a "POLARAS" window.
The software application labelled as "POLARAS" was written within the Microsoft
Visual C++ environment and it provides a Windows type of interface with the user.
Figure (4.3) is an example of the computer screen when the application is initialised
and no data has been sampled. The. main functions of POLARAS could be
summarised as follows:
i) Communication between the DAS and the PC to acquire data.
ii) Calibration of the DOWP and pre-processing of data.
Hi) Calculation of the Stokes Parameters from measured data.
iv) Display of the results.
34
The communication between the computer and the DAS is performed using the
standard protocol of an RS232 interface. When the PC requests data, the DAB sends
one intensity reading per operating channel (in binary form), then POLARAS
transforms all received readings into decimal form and it can average any requested
number of readings.
POLARAS allows the modification of some of the settings, such as the frequency with
which· data is requested (sampling period), total duration of a sequence of
measurements, number of· measurements to be averaged, details of the
communications protocol, the number of operating channels or sensors (it offers a
selection of anything between one and eight channels), and it also offers the choice of
saving into a file only the raw intensity (or voltage) readings from the sensors, the
estimated polarisation parameters, or both.
Before the polarimeter is used to sample anything, the calibration procedure must be
accessed from the "operations" menu in POLARAS. By running this routine not only
is the polarimeter being calibrated to yield correct measurements, but also POLARAS
is being "initialised" to display correct figures. Within the calibration procedure, the
offsets are calculated following the same steps that will be described in section 5.1,
and they are subtracted from all further intensity readings. The relative angles between
the linear polarisers and also of the QWR must be determined beforehand (in the
unlikely case of the polarisers having moved physically in their mounts during
transportation) and their values typed as an input before the calibration procedure can
be performed. Because the transmittance parameters are unique for the set of linear
polarisers and QWR within the sensor head, changing their values is not an option
offered to the user.
There are three basic modes in which POLARAS obtains data:
. i) A single measurement. .
ii) Continuous measurements
iii) Detector Test.
35
In the first two modes, a calibration factor is applied to all intensity readings before
data are processed. The single measurement mode was designed for' experiments in
which the experimental settings were modified between measurements, and the
continuous measurement mode was intended to be used for those experiments in
which temporal changes were important. The Detector Test mode was used for
monitoring the system itself, like verifying the alignment and intensity levels of the
light source. In all three cases, the data were stored in a fIle that can be retrieved later
by a spreadsheet software package for further processing.
;:. "'
r , '.
\ '" \ \ , \ , , \ ,J ',,-
j'
, i
j. ': i
Figure 4.4: Example of a POLARAS window when elliptically polarised light has
been measured. The four traces in the Intensity Chart correspond to the raw
voltages received from the four optical fibres in the sensor head.
36
In the first two acquisition modes, the calibrated intensities are used to estimate the
Stokes Parameters relative to the sample in question. When these are calculated, the
information on the polarisation state of the sample, i.e. azimuth angle, ellipticity, lp, 1o,
and the DOP are displayed in the "Polarisation Data" window, see Figure (4.4). The
ellipticity is displayed in the "Ellipticity Trace" window, the Stokes Parameters are
displayed in the "Stokes Parameters" window (not shown in the previous figure), the
raw intensities can be seen in the "Intensity Chart" displayed as voltage levels, and
also the values of the Stokes Parameters can be plotted as a point in the Poincare
Sphere. All these quantities and traces can be displayed in real time while a
experiment is taking place.
4.3 Description of the Data Acquisition System
The inputs to the Data Acquisition System (DAS) consist of a set of six PiN diodes
and two analogue inputs connected to a Data Acquisition Board (DAB). The DAB,
built in. the Dept. of Electronic and Electrical Eng. at Loughborough University, has
the functions of transducing the optical signals into electrical ones, of amplifying the
analogue signal generated by each of the PiN diodes and analogue inputs, converting
them into digital signals and sending them multiplexed by the serial port of a personal
computer when they are requested. The DAS is also provided with its own power
supply.
All eight inputs are amplified in the DAS by two amplification stages. The first stages
consist of linear amplifiers for all the optical inputs. In both cases the gain in the
amplifiers can be· set to any of the following values
{1,2,3,4,5,1O,20,30,40,50,100,200,300,400,500) by changing the bracket settings in
the DAB.
37
The amplified analogue signals are time-multiplexed with an analogue switch and
then converted into digital signals by an Analogue to Digital Converter (ADC) with a
resolution of 12 bits. Both elements are under control of a 80c32 microcontroller. The
micro-controller reads the digitised signals and stores them in external RAM. The
microcontroller communicates with a Personal Computer (PC) via an RS 232 serial
port with a maximum transmission baud rate of 9600.
Not all the eight input channels have to be operational at the same time, only the
required channels are activated by a set of eight on/off switches, and the
microcontroller switches on a number of LEDs corresponding to the active channels,
in the D AS front panel.
AD·bus
Control & Program Data Switch Status r-- decoding E-PROM RAM Config. Signals
micro-logic Array Array
data bus controller
J J J control bus
I I address bus
ADC + Analogue
+-2nd.
~ 1st.
+- Photodetect. . Multiplexer Amplifier Amplifier Array ~
(MUX) Array Array Optical
I Fibres
IRS.232 I
I IPC I
Figure 4.5: Block Diagram of the Data Acquisition System.
38
Chapter 5
Calibration of the Division of Wavefront Polarimeter.
The DOWP consists of the optics integrating the sensor head, a bundle of optical
fibres conducting the received light from the sensor head to the photodiodes mounted
on the data acquisition system (DAS), a 12 bit DAB, and a personal computer linked
to the DAB via a serial port (a detailed description of the system can be found in
. Chapter 4). The polarimeter is ready to operate when the Windows based software
program POLARAS is executed, a high intensity non-polarised light source is used to
illuminate the sample, and the light transmitted (or reflected) by the sample evenly
illuminates the sensor head.
Before performing any polarisation measurement using the DOWP, a calibration
routine has to be followed. First of all, the values of the azimuth angles, ai, aj and ah of the three linear polarisers and p of the quarter wave retarder have to be known and
entered into POLARAS. Then the calibration routine integrated within the software
must be accessed. This calibration routine asks the user to blackout the photodiodes
and "press return" on the computer keyboard, then to uncover the photodiodes and
illuminate the sensor with non-polarised light only (no sample in place) and press
return again; having done this, the calibration is completed and the DOWP is ready to
operate. The calibration routine records the photodiodes offsets and equalises the
intensities in all four channels.
5.1 Offsets Measurement
The light received by the DOWP sensor head travels along optical fibres to each of the
four PiN diodes. The analogue output of each PiN diode is amplified before being
converted to a digital level and the offset on each amplifier can be adjusted, although
it is very difficult to set the same offset in the four amplifiers. An offset will appear
added to the actual intensity measurement on each channel, but because the offsets
may be different in all channels and it is practically impossible to set them to zero,
they must be subtracted from the measured intensity before the data can be
manipulated to extract the polarisation parameters.
If an offset results to be negative it is not possible to measure it and discriminate it
from the actual measurement, because the ADC can convert only positive voltages
between 0 and 10 volts. For this reason the offset on each of the PiN diode amplifiers
was set to be a small positive quantity (about 15 quantisation levels). A hundred
measurements with the unilluminated photodiodes were taken and averaged to obtain
a more accurate offset. The offsets measured in this .way were subtracted from data
used for calibration and from all subsequent intensity readings. Because the offsets
may drift with temperature fluctuations, this procedure has to be repeated at time
intervals to keep the offsets updated.
5.2 Calibration to Equalise the Intensities in all Channels.
When a collimated, randomly polarised, beam illuminates the DOWP sensor head, it
is expected that if the linear polarisers and QWR are perfect, all three polarised
channels will yield identical intensity readings, and the bare channel will receive twice
the intensity of any of the others. However, this is not the case, because the polarisers
and QWR are non-ideal, and because the alignment of the light in all the channels may
be different. Furthermore each PiN diode could output a different voltage for the same
input illumination. Also the losses on each of the fibres, taking the light received by
the sensor head to the PiN diodes in the DAB, could be different. Having different
intensities on each of the channels, when there is no sample present to test, is a serious
problem, because the theory used to derive the expressions for the polarisation
parameters is based on the assumption that all variations in the measured intensities
should be due only to polarisation effects in the sample.
40
The fact that the polarising elements are non-ideal is not a real problem, because if
their absorbance or transmittance are known, these values can be taken into account in
the algorithms describing the light transmitted through each channel (see Chapter 3) ..
However it is important to know them accurately.
The problem of differences in the intensities, due to all the other mentioned factors,
was removed by a calibration procedure. When non-polarised or randomly polarised
light is illuminating the sensor head the two channels i and j, with only a polariser on
top of their respective optical fibre, should measure intensities Ij and Ij> and these two
values should be identical but for a calibration factor, i.e.
(5.1)
Because there is a linear polariser in front of their respective photodiodes, both should
measure the total intensity (received by channel I) times the transmission coefficient
of a linear polariser, 'to, for randomly polarised light (given by Equation 3.19). Then
the intensity measured by channel i is given by
(5.2)
and an expression very similar to this one exists for channelj.
Equalising the intensities in all the channels to the one measured by sensor I, factor It should be equal to unity, i.e.
It=I (5.3)
so replacing Equation (5.3) and its equivalent for channel j in Equation (5.2),
calibration factorsji andjj are given by
41
•. = to 11 J I I.
I
j.= toll J I.
J
(5.4)
By a similar approach, calibration factor fk for channel k, comprising a QWR followed
by a LP is given by the following expression
(5.5)
where to l is given by Equation (3.31). After these calibration factors were calculated,
all subsequent intensity measurements were normalised with respect to the
corresponding factor prior to the calculation of the polarisation parameters. All
intensity readings that remain unnormalised are referred to as "raw data".
These procedures to calculate the offsets and calibration factors were incorporated
into the software POLARAS used to acquire and manipulate intensity data. In practice
the following procedure was carried out:
i) Switch the light source off or cover the PiN diodes and take the average of one
hundred sets of measurements, to obtain the offsets.
ii) Switch on the light source or uncover the PiN diodes, take another set of one
hundred measurements, subtract offsets and obtain calibration factors according to
equations (5.4 and 5.5).
5.3 Measurement of Angles and Parameter Determination
An accurate determination of the angles and parameters of the LP and QWR is crucial
for the polarimeter to yield accurate measurements. Most of the time the values of
parameters t p, t" tr and t/ are provided by the manufacturer, but in many
42
circumstances the real experimental conditions are different from those at which the
parameters were measured, so they had to be obtained again under operational
conditions.
For a prior measurement of 9 i and 9 j' it was useful to illuminate the sensor head with
a non-polarised source and to rotate a external LP mounted on a rotary mount with a
one degree resolution scale (a diagram of the set up is found in Figure (5.3)). Then, by
Malus' law, when a photodiode detects a maximum in intensity, it means that the
azimuth angle of the external polariser is identical· to the azimuth angle of the
polariser in the sensor head. By doing this, the angles of the polarisers in the sensor
head are measured using the external polariser as the reference. Measuring 9 k and p
was a more difficult task due to the fact that they had to be measured simultaneously
. and p cannot be found following the Malus' law approach, because an ideal QWR
should not absorb any of the incident light. For all the above reasons, a data fitting
procedure seemed a good approach to find the values of all the angles and parameters.
With respect to the transmittance parameters, the easiest procedure for measuring the
transmission and extinction coefficients of a polariser, involves the use of a second
identical polariser and a very sensitive photodetector. If I j is the incident intensity on
the combination of polarisers, and It the transmitted intensity measured by the
detector, the transmission coefficient will be given by the ratio of It over Ij. When the
relative angle between the axis of both polarisers is zero (when the polarisers'
transmission axis are parallel to each other), the transmission coefficient will be 'tp ,
but if the relative angle is 1tI2 (when the polarisers' transmission axis are
perpendicular to each other) the measured coefficient will be 'ts•
, 10
't =_t_ P I· t
. 90' It
'ts =-I j
(5.6)
(5.7)
43
A similar approach must be followed to find tr and tl, corresponding to the fast and
slow axis of the QWR, with the additional inconvenience of having to identify these
axes. However, this approach could not be followed to find the parameters of the
components in the DOWP, because the small dimensions of the polarisers and
retardation plate (2mm diameter each) prevent them from being clamped adequately,
and made the intensity measurements just described extremely difficult.
The technique applied to find the remaining parameters consisted of modelling the
equations for the intensities (3.34-3.37), and then comparing the model to
experimentally measured intensities. Parameters used within the model would give a
worse or better fit to the experimental intensities. A plot of intensity vs. rotation angle
was produced when a LP external to the sensor, used as the sample, was rotated
manually in the interval 0° to 180°, in steps of 5°. Some assumptions had to be made
for modelling the intensities (3.34-3.37). First, since the linear polariser should
produce a linear polarisation state, the ellipticity e was set to zero, and second, the
light emerging from the external LP should be fully polarised, so Ip»Io.
The intensity 1/ measured in the central channel I, is the sum of the polarised and non
polarised intensities reaching the sensor head,
(3.37)
Then assuming that the LP is perfect and the light transmitted through the polariser is
completely polarised, the non-polarised intensity contribution is zero, so I p= 1/.
Putting the ellipticity to zero in Equations (3.34-3.36), the modelled intensities in the
. three polarised channels are given by
(5.8)
44
(5.9)
(5.10)
The rotation IX in this case corresponds to the azimuth angle of the external polariser,
but the angles 9" 9 l' 9k and p still have to be found. Equations (5.7 - 5.10) were
modelled using a spreadsheet in a software package. The theoretical curves were
matched to the experimental ones by changing the values of the angles and the
parameters, the best fit was obtained according to the least mean square error.
o· 20 40 60 80 100 120 140 160 160
Rotation Angle (degrees)
Figure 5.1: Preliminary measurement of angles and parameters.
Experimental and predicted intensity measurements obtained when a linear
polariser was rotated from 0° to 180°, in steps of 5°. Solid symbols represent
experimental data, diamonds stand for sensor 4 squares for sensor j, triangles for
sensor k and crosses for sensor 1.
45
Figure (5.1) shows an example of both experimental data and fitted data obtained
from transmitted intensity measurements when an external polariser was rotated from
0° to 180°. Ip was chosen to be about the same value as It. so in the model Ip was
arbitrarily set to the value 1168 and 10 to 5, all the other values were found using the
least RMS value between the model and the data. The fitted angles and parameters for
this case were
Table 5.1: Erroneous Values of the angles and parameters of the polarising optics
'tp 'ts to t/ 'tr 't/ p Sj S· J Sk
0.96 0.002 0.481 0.481 1 1 159 44.5 167 15.7
The values in Table (5.1) were erroneous and this was proved when the rotation ex. was
calculated with the fitted values. Figure (5.2) shows the difference between the actual
rotation of the external polariser and the measured rotation calculated using the
incorrectly fitted angles and parameters.
~r---------------------------------------------, .- .-40 .- .-.- .-30 .- .--.- .-i: ::- .::-
t of-I __ -+ ____ ~--~~--_r~.~-_+----+_--~----~--__. 20 40 60 .- 100 120 140 160 11 0
] -10 .- , .- , -20 .- I .- I -30 .- .1 .- .--40 .- .-
. .
-~
Rotated angle (degrees)
Figure 5.2: Rotation of a linear polariser from 0° to 180° in steps of 5°. Circles
represent experimental data and squares predicted data.
46
The accuracy in the determination of the angles relies on the accuracy with which the
value of the azimuth angles are measured on the mount's scale, the sensitivity of the
photodiodes and the precision of the DAB used to process the signals. The DAB
linked to the DOWP is provided with a 12 bit analogue to digital converter, for an
input voltage of 0 to 10 volts.
An attempt has been made to fit the three curves simultaneously. This approach
required of making some assumptions about the values of Ip and 10 , These two values
can influence greatly the determination of the parameters (erroneous guesses of 10 and
Ip could be responsible for the difference between observed and predicted data in
Figure (5.2», so instead of fitting the intensities, the algorithm for the ellipticity was
, fitted. This option was suitable not only because the lack of dependence on the Ip and
10 values, but also because all the required parameters appeared in one single equation.
The algorithm used to calculate the ellipticity (3.44) involves calculating the rotation
previously (3.39), and the equation for the rotation depends on two of the parameters
and two of the angles.
If the ellipticity is going to be fitted by any of the gradient optimisation methods [e.g.
Massara, R.E., 19911, the derivative of the ellipticity has to be calculated with respect
to each of the parameters to be fitted and this leads to extremely cumbersome
expressions. A numerical method already implemented within a software package was
therefore preferred.
When the software package ST A TISTICA became available, the ellipticity was fitted
by the Quasi-Newton method (see Appendix B), which is the default method for non
linear fitting in that package.
47
Red LED
Lens
Rotating Linear Polariser
Figure 5.3 Schematic of the experimental set-up used to measure the parameters
and angles of the polarising optics in the sensor head.
Figure (5.4) is an example of experimental and artificial intensity data generated
using non-linear fitted parameters .
90
80
S 70
:!. .~ 80 .. c:
i 50
~ 40 ~ .. .. .. :E 30
20
10
-20
.
XXXXXXXXXXXXXXXXXXXXXXXXX xxxxxXxxx . ~~~~~~ .. ~ ..
$ $ []
~
~ ~. ~. ~ ~ .
~ ~ .. ~ . ~
• •
•
+ sensor i • sensor j
,,"sensor k IJ • *
~ .. . $~ ....
20 80 100 140
Rotation angle (degrees)
180
Figure 5.4: Definitive angles and parameters. Experimental and predicted data of
intensity measurements obtained when a linear polariser was rotated from 0° to
180°, in steps of 5°. Solid symbols represent experimental data. The crosses stand
for sensor I.
48
Although the four transmission coefficients and four angles were obtained by fitting
the ellipticity equation by the quasi-Newton method, the reconstruction of the
intensities in Figure (5.4) still uses guessed values for Ip and I", so Ip was set to 1 and
10 to 0 (i.e, the total intensity was assumed to be fully polarised). The difference in the
magnitude of the intensities in Figure (5.4) with respect to Figure (5.1) is due' to a
difference in the light source power. The fitted angles and parameters for this case are
shown in the following table:
Table 5.2: Values olthe angles and parameters olthe polarising optics in the
DOWP sensor head
'1:p '1:, to t/ '1:, '1:/ P Si Sj Sk
0.965 0.0014 0.4839 0.4839 1 1 68 41 167.5 197
The RMSE values corresponding to the three fitted intensities Ii, Ij and h in Figure
(5.4), were calculated as the percentage of the experimental values, as follows:
Rl!JSE= (5.1I)
n
where in are the theoretical intensity values and Xn the measured values. The RMS
errors for the three intensity curves i, j, k resulted:
RMSEi= 1.5224%, RMSEF 2.2997%, RMSEF 1.3394%.
The rotation Cl (Figure 5.5), calculated with non-linear fitted parameters, produced
very similar values to those recorded by rotating the external polariser.
The curves for rotation shown in Figure (5.5) are wrapped at intervals of nl4, because
Equation (3.39) used to extract the value of Cl involves the calculation of the
49
arctangent of a ratio of intensities, and the arctangent function has the same
periodicity as the curves shown next.
The reader may also notice that the theoretical calculation of ex. is based on
experimental intensities, but the values of the "rotated angle", plotted as the x values
in Figure (5.5) are estimated values of how much the external polariser had rotated.
The real physical rotation of the external polariser could have been different to the
exact values used in Figures (5.2 and 5.5).
50 [J
40 [J B 0 0
30 [J 0 ~ 9 0 ~
~ 20 0 [J
'" 0 [J
"Cl 10 0 0 ~
'" [J El "6lJ ~ 0
"Cl 3 5 7 9 11 13 15 17 019 21 23 25 27 29 31 33 35 [J
~ -10 [J 0
B 9
~ -20 0 0
El [J
-30 0 [J
0 8 .4Q g 0
[J 0 .
-50
Rotated angle (degrees)
Figure 5.5: Rotation of a linear polariser from 0° to 180° in steps of S°. Circles
represent experimental data and squares predicted data. RMS error is 3.0572 as a
percentage of the experimental value.
50
0.025
.
0.Q2 • 0.015
om • •
Q • o.OOS •• • • • '0 •• • 'l:I • • i • •
0 Y
I 20 40 • 60 80 100 120 140. 160
·(l.OOS • • • •• • • • .0.01 •• •
.0.015 • • •• .o.Q2
Azimuth angle (degrees)
Figure 5.6: Predicted ellipticity versus azimuth angle of a linear polariser. The
RMS error is 0.0064%, as a percentage of the experimental value.
~
.
The ellipticity, Figure (5.6), was calculated from experimental intensities, the fitted
parameters shown in Table (5.1) and the rotation shown in the previous figure. In
theory, e should be zero for any value of IX, however in this example it was as large as
0.02. This could be due to errors incurred during the acquisition of the parameters, to
imperfections of the polarising optics, such as non-uniformity along their surface, and
to the light source not being perfectly monochromatic, affecting the performance of
theQWR.
The parameters in Table (5.2), found by the procedure described in this chapter were
incorporated into a header file in the software application POLARAS and used to
calibrate the polarimeter.
51
Chapter 6
Performance Evaluation of the Polarimeter
6.1 Theoretical Analysis.
6.1.6 AID Converter with 12 bit Resolution.
The factors most likely to affect the performance of the Division of Wavefront
Polarimeter are:
i) The error in the assessment of the absorption parameters (tp , t s, t" tl) and the
azimuth angles of the polarising optics in the sensor head (9i, 9j. 9k and p).
ii) The resolution of the analogue to digital converter (ADC) used by the DAB.
iii) The performance of the four photodiodes and their respective linear amplifiers.
From the three factors listed above, only the first one is intrinsic to the DOWP, while
the other two very much depend on the particular DAB used to convert the light
intensity signals from the DOWP into amplified digital signals that a computer can
read. However, if the performance of the photodiodes or the amplifiers in the DAB is
poor, or if the accuracy of the ADC is low, the accuracy in the determination of the
azimuth angles and absorption parameters will also be low.
To ascertain the magnitude of the absolute error introduced into the Stokes parameters
. measured by the DOWP, due to quantisation and incorrect measurement of the
azimuth angles, the two effects were simulated theoretically.
The computer simulation examined the situation in which the DOWP measured
changes in the polarisation state of a linear polariser rotating from 0° to 180°. This
condition was chosen to be analysed theoretically, because it is simple and easy to
reproduce experimentally in a controlled way. Initially the simulation routine assumed
that the angles and the parameters were exact, but the four intensities were quantised
into 4096 digital levels, simulating the accuracy of a 12 bit ADC.
If 11 is the intensity measured in channel i, then the corresponding estimated quantised
intensity Ilq in the same channel will be given by
(Ip+Io) . {Ii X4096 \ Iiq = 4096 xRoun I +1
po) (6.1)
where Ip is the amount of completely polarised light, and 10 is the amount of randomly
polarised light. The quantised intensity given by Equation (6.1) will give the minimum
error as it uses the full dynamic range in each channel. There is a similar equation for
the quantised intensities in Channels j, and k, but the error in the quantised intensity in
channell was not calculated since
(6.2)
The error in each of the channels was estimated as the difference between the exact
and the estimated intensities. Figures (6.1- 6.3) show the absolute quantisation error in
the intensities from channels i, j and k. The ordinate in each of the plots is the
polarisation azimuth varying from 0° to 180°, while the abscissa is the absolute error
with respect to 1/ set equal to 1. The size of the absolute quantisation errors are the
same for the three intensities i, j and k, in all of them the error varies at random.
53
Quantisation Error Intensity i
0.0001
0.00005
o
-0.00005
-0.0001
Polarisation Azimuth (deg)
Figure 6.1: Absolute quantisation error in the estimation of Intensity I.
RMS error of 0.00007.
Quantisation Error Intensity j
0.0001
0.00005
o
-0.00005
-0.0001
Polarisation Azimuth (deg') '.-
Figure 6.2: Absolute quantisation error in the estimation of Intensity j.
RMS error of 0.00007.
54
Quantisation Error Intensity k
0.0001
0.00005
o
-0.00005
-0.0001
Polarisation Azimuth (deg)
Figure 6.3: Absolute quantisation error in the estimation of Intensity k.
RMS error of 0.00007.
Statistically, the three plots shown before (Figures 6.1-6.3) should be identical, but
there are some differences in the shape of each plot, because the different values of
azimuth angles of the linear polarisers produce different effects on the polarisation
azimuth and the ellipticity. For this reason, some settings of the polarisers produce
larger errors than others, but still any value can be used to extract the Stokes
parameters, if the positions of the polarisers and retarder are different from each other.
55
Absolute Error in the Polarisation Azimuth (deg)
0.04
0.02
o
-0.02
-0.04
o 0.2 0.4 0.6 0.8 1
Ellipticity
Figure 6.4: Absolute Quantisation Error in the Polarisation Azimuth.
RMS error of 0.0263.
Figure (6.4) shows the absolute quantisation error in the measurement of a
polarisation azimuth exact value of zero, when the ellipticity varies from 0 to 1. The
ordinate is the ellipticity (defined as the ratio of the two semi-axes of an ellipse) and
the abscissa is the absolute error (in degrees) in the reconstructed polarisation azimuth
lx. From the figure it is very clear that the absolute error due to quantisation in the
polarisation azimuth is smaller for values of ellipticity closer to zero than closer to
one. Thus measurements of the polarisation azimuth of linear polarisation states will
be obtained with better accuracy than those corresponding to circular polarisation
states. But still one can notice from Figure (6.4) that if the AID converter has a 12 bit
resolution, the minimum theoretical error that can be expected for the measurement of
a linear polarisation state is of ± 0.008 of a degree.
Figure (6.5) shows the absolute quantisation error in the reconstructed ellipticity,
simulating the case when a linear polariser (e=O), varies from 00 to 1800• The abscissa
in this plot is the absolute error in the ellipticity (dimensionless).
56
Reconstructed Ellipticity
0.0002
0.0001
o
-0.0001
-0.0002LO~--~2~5~--~5~0----~~--71~0~0--~1~2~5~~1~5~0--~1~7~5~
Polarisation Azimuth (degrees)
Figure 6.5: Absolute quantisation error in the Ellipticity. RMS error of 0.00009.
Absolute Error, Reconstructed Stokes Parameter Q
0.0002
o
-0.0002
-0.0004 LO~----2~5~~~5~0~---7~5~--~1~00~--~1~2~5~~~1~5~0----1-7~5-J
Polarisation Azimuth (degrees)
Figure 6.6: Reconstructed Stokes Parameter Q. RMS error of 0.00013.
57
Absolute Error, Reconstructed Stokes Parameter U
0.0003
0.0002
0.0001
o
-0.0001
-0.0002
-0.0003
-0.0004
25 50 100 125 150 1
polari~ationl\~\m~th (degrees)
Figure 6.7: Reconstructed Stokes Parameter U. RMS error 0/0.00010.
Absolute Error, Reconstructed Stokes Parameter V
0.0006
0.0004
0.0002
o
-0.0002
Polarisation Azimuth (degrees)
Figure 6.8: Reconstructed Stokes Parameter V. RMS error 0/0.00034.
58
Figures (6.6 to 6.8) show the reconstructed Stokes parameters Q, U and V for the case
of linearly polarised light (e=O), and the polarisation azimuth varying from 0° to 180°.
The reconstructed parameter 1 is not shown because it is identical to the exact
parameter r.
When e=O, Equations (3.48 to 3.51) for the Stokes parameters in terms of the
polarisation azimuth and ellipticity, are reduced to:
1=1 (6.4) .
Q=cos(2a) (6.5)
U =sin(2a) (6.6)
V=O (6.7)
The shape of the plots shown in Figures (6.6 and 6.7) are modulated by the
reconstructed polarisation azimuth times two, where 2cx (given by Equation 3.39) is a
wrapped function of the reconstructed intensities. Figure (6.8) shows random
oscillations close to the zero value, and the two well defined oscillation bands are due
to the estimated polarisation azimuth being a wrapped function of a , which although
it does not appear explicitly in the calculation of the Stokes parameter V, it is involved
in the estimation of the ellipticity.
6.1.2 Adding an error of one degree to the azimuth angle of a linear
polariser.
To study the repercussion that, an additional error in the angles and absorbance
parameters of the polarising elements in the sensor head, has on the measurement of
the Stokes parameters, a deviation of one degree from the exact value was added to
one of the angles, ai, leaving all the other angles and absorbance parameters intact.
59
The reconstruction of the Polarisation Azimuth (considering the additional deviation
in one of the angles), as a function of a varying ellipticity from 0 to I, can be seen in
Figure (6.9), where the abscissa is the absolute error in degrees. The exact polarisation
azimuth has a value of 0.0.
Absolute Error in the Polarisation Azimuth (deg)
O. 78 ...... ---~-~-.----~----.,-.
0.76
0.74
0.72
0.7
o 0.6
Ellipticity
0.8 1
Figure 6.9: Absolute error in the reconstructed Polarisation Azimuth, due to an
error localising the orientation of one of the polarisers in the sensor head,
additional to quantisation noise. RMS error of 0.0606.
The shape of the curve for the reconstruction of the Polarisation Azimuth (Fig. 6.9) is
identical to the shape of the curve shown in Figure (6.4), but with an added offset of
0.74, which means that the quantisation noise was unaffected by an error of one
degree in the determination of one of the angles. But the reconstructed rotation
differed from the exact value by at least 0.7 of a degree, for the case of linearly
polarised light.
Figure (6.10) shows the reconstructed ellipticity when an error of one degree in one of
the polarisers settings has been added to the quantisation noise. In this case, the shape
60
of the curve suffered a noticeable change from the one shown in Figure (6.5). In
Figure (6.10) random noise introduced by quantisation is almost negligible compared
to the contribution by the added deviation in the polariser angle.
Reconstructed E11ipticity
0.001
O~------~-r---------------.f-~--------~
-0.001
-0.002
-0.003
o 25 50 75 100 125
Polarisation Azimuth (degrees)
Figure 6.10: Absolute error in the reconstructed Ellipticity, due to an error in
localising the orientation of one of the polarisers in the sensor head, additional to
quantisation noise. RMS error of 0.0024.
This additional error also affected the shape of the plot of the reconstructed Stokes
parameters Q (Figure 6.11) and U (not shown because it is very similar to the one for
parameter Q), when they are varied with respect to the polarisation azimuth; the most
noticeable effect is the modulation produced by the wrapped polarisation azimuth.
The shape of the plot of the Stokes parameter V (not shown), is the mirror image of
the ellipticity shown in Figure (6.10).
61
Absolute Error, Reconstructed Stokes Parameter Q
o
-0.02
-0.04
-0.06
-0.08
o 25 50 75 100 125· 150 175
Polarisation Azimuth (degrees)
Figure 6.11: Absolute Error in the Stokes Parameter Q, due to an error in
localising the orientation of one of the polarisers in the sensor head, additional to
quantisation noise.
The RMS errors in the reconstruction of the Stokes parameters under the conditions
just described are:. QRMS =0.046, U RMS =0.079, VRMS =0.005.
Additional errors, in the settings of other angles and absorbance parameters of the
polarising optics, will increase the error in the detennination of the Stokes parameters,
in both the simulated and the experimental measurements.
6.1.3 AID Converter with 10 bit resolution.
Although the NO converter has a resolution of 12 bit, the experimental conditions
prevented the use of the whole dynamic range, so the experiments conducted to test
the performance of the polarimeter made use of a resolution of about 10 bit.
Additionally, the DAS introduced analogue noise (from digital switching circuitry) to
the measured data. Then in reality the experiments were conducted as if using a
resolution smaller than that provided by a 10 bit NO converter. To simulate this
situation, the theoretical calculations performed in section 6.1.1 were repeated, but
62
assuming now a quantisation with a resolution of 10 bit.
The intensities Ij. Ij and h. the polarisation azimuth. the ellipticity and the four Stokes
parameters were recalculated. The following are the plots of the difference between
the exact and the estimated intensities as functions of the polarisation azimuth. varied
from 0 to 180°.
Quantisation Error Intensity i
0.0004
0.0002
o
-0.0002
-0.0004
Polarisation Azimuth (deg)
Figure 6.12: Absolute quantisation error in the estimation of Intensity I.
RMS error of 0.00027.
63
Quantisation Error Intensity j
0.0004
0.0002
o
-0.0002
-0.0004
Polarisation Azimuth (deg)
Figure 6.13: Absolute quantisation error in the estimation of Intensity 1.
RMS error of 0.00026.
Quantisation Error Intensity k
0.0004
0.0002
o
-0.0002
-0.0004
o 25 50 75 100 125 150 175
Polarisation Azimuth (deg)
Figure 6.14: Absolute quantisation error in the estimation of Intensity K.
RMS error of 0.00027.
64
The three plots of the error in the quantised intensities (Figures 6.12-6.14) show the
error varying randomly for different values of the polarisation azimuth, in a similar
way than those using a resolution of 12 bit, but the maximum size of the error has
increased four times after decreasing the resolution. This was expected since the
resolution is 2 bit smaller. Also Figure (6.14) shows some modulation in the absolute
error for values of the polarisation azimuth close to ltA' and 3'X, because on these
points the wrapped curve for the polarisation azimuth changes sign.
Absolute Error in the Polarisation Azimuth (deg)
0.1
Of--~'"
-0.1
Ellipticity
Figure 6.15: Absolute quantisation error in the reconstructed polarisation
azimuth. RMS error of 0.0397.
When the resolution of the NO converter is reduced, the values that the reconstructed
polarisation azimuth can have are spread over a smaller number of digital levels. This
is confirmed when Figure (6.15) is compared with its 12 bit resolution counterpart
(Figure 6.4). Also one can notice that the error between the exact and the
reconstructed polarisation azimuth (both functions of the ellipticity) is increased by a
factor of almost five.
65
When the ellipticity was reconstructed using a resolution of 10 bit, a plot of the
difference between this and the exact value (not shown) was not too different to the
one for 12 bit, but the absolute error increased by a factor of about 4.0. The same
effect occurred to the plots of the absolute error of Stokes parameters Q and U. Only
the plot of the absolute error of the Stokes parameter V suffered a change of shape,
the different quantisation levels can not be differentiated clearly in the plot using only
10 bit of resolution (Figure 6.16).
0.0015
0.001
0.0005
o
-0.0005
-0.001
Absolute Error, Reconstructed Stokes Parameter V
Polarisation Azimuth (degrees)
Figure 6.16: Absolute quantisation error in the estimation
of Stokes parameter V. RMS error of 0.00049.
66
6.2 Experimental Analysis.
The conditions analysed in the simulation were reproduced experimentally. A linear
polariser was rotated from 0° to 178.2° in 33 steps of 5.4° each, using a stepper motor
with a resolution of 1.8°± 5% (quoted by the manufacturer and without considering
the accuracy of the driving circuit).
The experimental set up was practically identical to the one shown previously in
Figure (5.3), but the LED used in this application changed from the 660 nm LED used
for the initial calibration, reported in Chapter 5, to a 685 nm device.
After the polarisers and QWR in the sensor head were adjusted, the· system was
recalibrated. The new values of the azimuth angles, which were used for the
theoretical caIculations, are the following:
Table 6.1: Values of the angles and parameters of the polarising optics in the
DOWP's sensor head, with a 685 nm light source.
1:p 1:, to to' 1:r 1:/ p 9j 9j 9k
0.965 0.0014 0.4832 0.4832 1 1 88.2 123.9 70 51.9
The DOWP performed 500 polarimetric measurements at each of the 33 different
orientations of the polariser. The four intensities measured by Channels. i, j, k and 1
were plotted versus the orientation of the rotating polariser, see Figure (6.12).
67
1000
900 M~"E HKH ". MHMHHHKM ' HHKHMt$~HH o OIannell D /I ....
800 <> "'o. D ChannelJ D DD
~ o. '" D 700 DO.
.. .. AQannelK D
-R o. D 600 "'D xOlannelL
.~ & .... ·u .... D .. SOO
D .... .. .... .!; D .... .. ..
f D ....
400 .... .. .... .. 1:" .. D .... 6 300 6666 0 ....
Z 0 .. D .. ~ 200
D 0 .. 100 0 D .. o.
DD D '" .. D <> '" 0
0 20 40 60 80 100 120 140 160 180
Azimuth Angle (deg)
Figure 6.12: Intensities measured when a linear polariser rotated through 180
degrees.
In this plot the abscissa is the value of intensity in digital levels. The error bars plotted
in each of the four intensities are the standard deviation of the 500 measurements at
each of the 33 azimuth angles of the polariser. Although the whole dynamic range of
the photodiodes is of 4096 levels, less than 25% of this range is being used. This is
because the DOWP has to. be calibrated with non-polarised light, and to avoid
saturation, the gain in Channel 1 was set to provide an amplified output just below
4000 digital levels. When the linear polariser is inserted, the intensity in Channel 1 is
absorbed by about 60% (the transmittance coefficient of the polariser is 0.41). The
intensities in the other three channels are sized proportionally to this channel by the
corresponding calibration factor.
Figure (6.12) shows that the light source illuminating the sample is partially polarised,
as it is observed from the intensity modulation in Channel 1. Because this channel is
not polarised, when illuminated by non polarised light it should provide a constant
intensity reading throughout the complete rotation of the motorised polariser. An
examination of the experimental set-up made evident that the initially non-polarised
light source was partially polarised by the expanding lens.
The calibration stage described in Chapter 5 requires a non-polarised light source. If
68
the source is partially polarised, the calibration factors are calculated incorrectly and
the difficulty in detennining the exact position of the polarisers and QWR in the
sensor head is increased. However, this added complication is not inherent to the
DOWP itself and can be removed by using a different type of lens (e.g. one with an
anti-reflective coating) and ensuring that the light source is truly randomly polarised.
The partial polarisation of the source does not affect the measurement of the
polarisation azimuth, as it is observed in Figure (6.13), but it does affect the
measurement of the ellipticity (Figure 6.14) and the measurement of the Degree of
Polarisation (OaP) (Figure 6.15) .
. ~r---------~--~======~-------;",--------~ 40 I!t l.6.fuperirrental value I *
lit + Theoretical value I dr
30 '" '"
'" '" ~ 20 '" '" '" . ~ '" :8 10 iD" •
~ '" ... § O'---~~---+-----r----+-~L-~---4----~C----4---~
] -10 20 ~. 60 " I!t 100 120 140 160 '" ~ 0
~ ;!; !It • ::. -20 ;!; !It ;!;
-30 ;!; '" '" ~;!; '" ;!;
-50 .L-~ ____________________ --,--______________________ ---'
Azimuth Angle (dog)
Figure 6.13: Polarisation Azimuth measurement.
In Figure (6.13), the ordinate is the angle in degrees by which the polariser has been
rotated by the motor, and the abscissa is the measured rotation in degrees. Both
theoretical and experimental values have been plotted together, and the standard
deviation of the experimental values is plotted as the measurement error. The average
of the standard deviation of the 500 samples resulted in 0.103 of a degree. Both curves
69
in Figure (6.13) show a very good correlation, a linear regression of the unwrapped
experimental curve yielded a correlation coefficient of 0.9999.
0.025
0.02
0.015 I
b 0.01 .L!;r ." a .LI ;;; 0.005 "' .LI ]
• 0 I. ~
140 • :s 20 .L 60 80 100 120 140 160 1 0 -0.005 I
·0.01 .LI
.L ·0.015
Azimuth Angle (deg)
Figure 6.14: Ellipticity measurement.
The experimental measurement of the ellipticity is shown in Figure (6.14), where the
errors are more apparent than in the plot for the polarisation azimuth. Again this
measurement should have been a constant zero in an ideal situation, but as was
observed in the theoretical simulation, the accuracy is a non-linear function of the
polarisation azimuth. In this plot the modulation in the experimental ellipticity is
believed to be due to incorrect values of the parameters and azimuth angles. In the
theoretical simulation (Figure 6.10) where a deviation added to one of the angles was
simulated together with an error introduced by 12 bit quantisation, the plot showed
also a similar type of modulation, but out of phase from Figure (6.14). Which suggests
an error in one or more of the parameters or azimuth angles in the polarimeter.
Figure (6.15) shows an experimental measurement of the degree of polarisation,
derived from the ellipticity measurement. At some points the DOP is greater than
unity, and this is evidently wrong. From Equation (3.47) the degree of polarisation is
70
obtained from the ratio of the non-polarised light and the completely polarised light,
but these two quantities are sensitive in different degree to the partial polarisation of
the source, which in some instances causes Ip to be greater than It.
1.S.,.-------------------, Cl 1 ++++++++ i '5 0.5
l
+++++++++
~ O+--------+--------+--------+------~
k .0.5 50 100 150 2 0
I ·1 ++++++++++++++++
.1.S .L.. __ ;..,;... ________________ --'-__________ ---'
Azimuth Angle (de g)
Figure 6.15: Measurement of the Degree of Polarisation •
r"'. •• . " " . • ." 0.8 • • • • • • • • • • • 0.6 • • • • • • 0.4 • • • • • • • • • 0.2 • • •
l •• ·t .................. • 0 30 80. 130 I
·0.2 • ~Q
.u o .. v
·0.4 • • ·0.6 • • • ·0.8 • •• • _ .. •
Azimuth Angle (deg)
Figure 6.16: Measurement of normalised Stokes Parameters Q, U and V.
Figure (6.16) shows the normalised Stokes parameters Q, U and V plotted versus the
71
position of the rotating polariser. The three curves follow the behaviour predicted by
Equations (6.2-6.4). Parameter V is different from zero, as is the ellipticity from which
it was derived. Parameter I is not plotted since it is normalised and has a constant
value of one. The same type of plots were found theoretically for the Stokes
parameters (Figure 6.11 for parameter Q is the only one shown), but the values had the
oposite sign, indicating that the theoretical error was also modulated by cos(2a ) (in
the case of parameter Q) and by sin(:n) (in the case of parameter U).
6.3 Additional Sources of Error.
At each azimuthal settings of the polariser, the DOWP performed 500 measurements
consecutively. Each set of 500 measurements was produced after 10 minutes, even
when the DAB was sending data to the computer at its fastest baud rate. If for a
particular fixed position of the linear polariser, the intensity measurements in each of
the channels are plotted versus acquisition time or versus sample number, different
quantisation levels are observed on each of the channels.
In figure (6.17), the plots for Channels i, j and k show about the same number of
quantisation levels, but in the case of Channel i, the intensity values are spread over
almost all of the levels, while in Channels j and k the majority of the measurements
are concentrated in' a smaller number of levels. Channel 1 shows only half of the
number of levels shown in the other channels but the intensity readings are
concentrated basically in only two of them. The cause of this difference among the
channels is the second stage of amplification of the photodiodes signals in the DAB.
When the analogue signals, from each of the photodiodes attached to the DAB, were
followed with an oscilloscope through the different electronic components in the
board, it was found that at the output of the first amplification stage, all four signals
from the four photodiodes have the same shape with small variations in amplitude, but
72
at the output of the second stage of amplification they present some degree of
modulation. The modulation is largest in Channel i and almost negligible in Channell.
This modulation could be suppressed designing the adequate electronic filter for each
channel, or linking the DOWP sensor head to a different DAB with less noise
problems, since the board used in this application was not designed specifically to be
used for polarimetric measurements.
5Mr-------------------, S ~564
1562 ~-: ----••• - :.:. .... -. -~ - -(J 560 .. - _.- - ---_ .... _ ••• ~ ~558
~ 558 - -----
o 100 200 300 400 500
Sample Number
289,---------------------,
[288 - _-:- _--: - - --:M:: 287 -i 286 ---==:':: .. -=---_ .. :,- - -~-..:-:
! 285 r ...... _ .... -:'-:-.. -=---=-= ';! 284 _ .... ___ .. __ .. _
.. 283 - ----- :---: 1ii 282 - .. - _ ••••• -
li 281 -I----I'---_~_4~__'t----I o 100 200 300 400 500
Sample Number
~Or--------------------, S --,,7191---~ 718 - ...
§ 717 ... _ ... --lJ 716 - - --_-_--_-_-_-_-_-____ -+ f ~~: .---::~_-. .. -___ -.5 713 -1----+----+----+----<-----"
o 100 200 300 400 500
Sample Number
846r-~~--------------_.
.. 847
~ 846 ~ 8451----_______ _ 6... __ _ ~ 843 __________ -+ Iii 842 li 841 _ _ ______ _
84O-I---+---+----+---~--~ o 100 200 300 400 500
Sample Number
Figure 6.17: Observation of quantisation levels in intensity readings from the four
Channels in the sensor head.
Experimental errors caused by quantisation noise are propagated to the calculation of
the polarisation azimuth and ellipticity as seen in Figure (6.18). The maximum
absolute experimental error in the measurement of the polarisation azimuth is 0.8 of a
degree, while the theoretical simulation for linearly polarised light (Figure· 6.4)
predicted only 0.02 of a degree, i.e. 40 times smaller. However the quantised
intensities in the simulation are making use of the whole 4096 quantisation levels,
while the experimental intensities used less than 25% of the total number of levels.
73
0.022 0r----+~--~--~----~--~
'Si -0.1 100 2(f).. 300 .. 4~ .. "1>0 ~ ,: "'" .-.. . ... . i :.: .-:: -:-.- .-. -: .. - .. ::_ : ...... -.: -... li '.. " ... ~~ :.-: .... -.... -~~....:....,"""' .. "! o "()4 ~"_ ... - .. .r,!\oo:.,~ .. -...... ..""" .. _ .... a: . ~--. ....... ..,- ...... .. -_~ ..
1 ::! : :':..--= .:.~:::i..:-; .. : ... i .. :~il~ i .(J,7 ~ .. -:.~":':::-~ .. =':._.,;:~:'o~~.:=..~-..
.
0.01.\----+----+----<----+-----l -0.' '------------------------' o 100 200 300 400 500
Simple Number Semple Number
Figure 6.18: Observation of quantisation levels in the Polarisation Azimuth and
EUipticity
In an effort to improve the precision of the measurement, the experimental intensities
were compared to the theoretical intensities from the simulation, then in the
simulation the absorbance parameters were modified until both sets of intensities
matched, according to the minimum RMS error, and using the recalculated parameters
(Table 6.3), the ellipticity was calculated again using the experimental intensities.
Table 6.3: Values of the angles and recalculated parameters of the polarising optics
in the DOWP's sensor head, compensating for the effects from a polarised light
source.
'tp 'ts to t/ 'tr 't/ p Si Sj Sk
0.97 0.0014 0.4857 0.4579 0.95 0.93 88.2 123.9 70 51.9
The result is plotted in Figure (6.\9). In this plot. the ellipticity calculated from
measured intensities is ten times smaller than in previous calculations (Figure 5.6).
74
0.004 • T • 0.003 ••• • •• b
;g 0.002 • • ~ 0.001 • • • • • ] 0
.~ -0.001 • 100 •• 4150 210 ~ • 2 .().002 •• ••• -= ... • .().003 • .().004 •
Azimuth Angle (<leg)
Figure 6.19: Theoretical Ellipticity generatedfrom measured intensifies and
modified absorbance coefficients.
The reason why the recalculated absorbance parameters yielded a better result is that
they are compensating for polarisation effects of the source which generated erroneous
calibration factors to size the intensities. It is not recommended to use the new
calibrated parameters, shown in Table (6.3), because "the old parameters" quoted in
Table (6.1) will perform well if a non-polarised light source is used; however this
parameter fitting exercise shows that a fine tuning of the absorbance parameters can
affect dramatically the accuracy of the measurements.
6.4 Summary and Discussion.
A comparison between the theoretical and the experimental errors cannot be made in a
straight forward manner because only a small number of sources of error have been
modelled. Additionally an assumption, that is not necessarily correct, has been made
when comparing the experimental and theoretical results, and this is that the motor is
assumed capable of repeating with perfect accuracy 500 rotations of a linear polariser,
in 34 steps each, and that the size of all steps is the same:
75
,-------------------------------------------------------------
Ii
Ij
Ik
Ellipticity
Q
U
V
Table 6.2: Summary of the Theoretical RMSerrors and
Experimental Standard Deviation.
12 bit resolution 12 bit resolution + 10 bit resolution Experimental data.
RMS error. deviation of 10. RMS error. Standard Deviation.
RMS error.
0.00007 0.00007 0.00027 0.0025
0.00007 0.00007 0.00026 0.0011
0.00007 0.00007 0.00027 0.0013
0.0001 0.0024 0.00035 0.0017
0.00013 0.0463 0.00049 . 0.0014
0.00010 0.0794 0.00045 0.0044 .
0.00033 0.0050 0.00079 0.0045
In Table (6.3), all theoretical data referring to intensity measurements was modelled in
such a way that it was only affected when the resolution of the AID converter was
modified, but not when additional deviations to the parameters were added. This
distinction was made to facilitate the discrimination of the consequences that
different sources of error have on the recovery of the Stokes parameters. However in
reality incorrect settings of azimuth angles and parameters can influence the
calibration procedure, thus affecting the amplitude of the intensity curves.
In the. same Table (6.3) the standard deviation of the experimental intensities is one
order of magnitude greater than the 10 bit resolution RMS error. This could mean that
the experiment was performed using an even smaller dynamic range, or that incorrect .
values of the experimental parameters an angles affected the intensities through
incorrect calibration factors. Considering that a resolution of 12 bit means that the
four intensities can use 4096 quantisation levels, and that in reality only about 25% of
the number of available levels are used (as it is shown in Figure 6.12), one cannot
expect to observe experimental errors smaller than those produced by a 10 bit
quantisation.
Nevertheless, another factor that could account for the difference between the
76
experimental errors in the intensities and the 10 bits quantisation noise could well be
analogue noise. An analysis of the four plots shown in Figure (6.17) indicates that
after performing 500 consecutive measurements with each channel, in exactly the
. same conditions" the measurements are spread, for channels i, j and k, over a
surprisingly large number of quantisation levels. As was explained before in Section
(6.3), this additional noise was identified even before the light intensity signals
reached the NO converter, so this problem is not quantisation noise alone. Thus
variations in intensity values over at least 15 different levels ( as is the case of channel
i), could increase then times the quantisation noise. Any source of noise in the
intensities will be propagated into the measurements of the polarisation azimuth,
ellipticity, degree of polarisation and Stokes parameters.
Data referent to the polarization azimuth values were not recorded in Table (6.3)
because in the theoretical simulation the polarisation azimuth is plotted for various
values of ellipticity, while in the experiment only linearly polarised light (with an
ideal ellipticity value of zero) was tested. Still it is interesting to see how the different
theoretical sources of error are likely to influence the polarisation azimuth results. The
theoretical absolute error in the azimuth determination for linearly polarised light, due
to quantisation effects, was of 0.008.
In the simulation the RMS error in the ellipticity, due to only 12 bit quantisation, i.e.
using the full dynamic range, is 0.001 (see Figure 6.10). The averaged standard
deviation of the experimental measurements is 0.0017, and this is the averaged size of
the error bars in Figure (6.14). So the error in the experimental data is one order of
magnitude larger than the theoretical results. This could be due to additional errors in
the settings of the parameters, to the experimental work being done using only a
portion of the available dynamic range, to analogue noise (as is shown in Figure 6.18),
or to a combination of all these factors.
From the data recorded in Table (6.2), one can observe that the theoretical RMS error
in the measurement of the ellipticity, when a deviation of one degree has been added
to the exact value of one of the azimuth angles, is of 0.0024. So one could presume
77
that the experimental errors in the measurement of the ellipticity are a combination of
quantisation related errors and incorrect parameters.
Also the ellipticity measurement reflects errors in the calculation of the polarisation
azimuth, and the errors incurred in the measurement of the ellipticity will be
propagated to the determination of the Stokes parameters.
In conclusion, the DOWP designed here would benefit from using the full dynamic
range available, if this was possible, and from estimating accurately the parameters
and azimuth angles of polarisers and retarder. In addition, the polarimeter requires a
dedicated electronic receiver system designed for this purpose only.
6.4.1 Problems encountered while using the DOWP in biomedical
measurements.
One of the main reasons behind the design and construction of the DOWP discussed
in the previous chapters, was to use it investigating the polarisation, absorption and
scattering properties of blood. The development of the DOWP ran in parallel with
basic research in various fields such as biochemistry, medicine, physiology and optics,
to understand and provide an optical model of blood. The experiments described in
Chapters 8 and 9 are the results of such investigations.
Further experiments attempting to measure some blood pathologies using the DOWP,
shown that blood absorbs light very strongly, and the large path lengths required by
the experimental conditions prevented any light from reaching the DOWP. A stronger
light source was then required, but it must have been unpolarised (by the reasons
explained in Chapter 5) and high intensity unpolarised sources are not easily available.
Since a suitable light source could not be found, the DOWP was not used to measure
blood. Instead some experiments on blood with polarised light were still conducted
using a laser source (Chapter 9). Suggested modifications to the DOWP for using it in
biomedical applications can be found in Chapter 10.
78
Chapter 7
Techniques for Blood Analysis
7.1 Methods for blood analysis currently in use.
7.1.1 Blood Composition.
Blood and the circulatory system constitute a transport system for exchanging
chemical products between the specialised cells of various organs. Blood consists
basically of plasma, a fluid rich in fibrinogen protein, in which white blood cells, red
blood· cells (RBC) and platelets are suspended. All these cells are generated in the
bone marrow.
Red blood cells, or erythrocytes, are the most abundant cells in blood. They are
biconcave discs of an average diameter of 8 microns. These cells have no nucleus, but
they have a membrane and the cells are filled with haemoglobin, an iron containing
protein. RBC transport oxygen by chemically binding the oxygen molecules to
haemoglobin. Arterial blood carries oxygen saturated haemoglobin (oxy
haemoglobin), which has a bright red colour, while venous blood carries oxygen
depleted haemoglobin (deoxy-haemoglobin), of a dark red colour. The average
number ofRBC is 5x1012 per litre of blood.
White blood cells, or leukocytes, are nucleated cells with normally a round shape, of
an average diameter of 15 microns. These cells are much less numerous than RBC and
they constitute the defence mechanism of the body to fight disease. White blood cells
can change shape. They attack foreign organisms folding themselves around and
digesting them. The total average leukocyte number is 8x109 per litre of blood.
Blood platelets, or thrombocytes, are non-nucleated, colourless cells of an average
diameter of 3 microns and they can have various shapes. Their main function is in the
blood clotting mechanism. The total average platelet concentration is 300x 109 per litre
of blood.
7.1.2 Blood Indices.
A count of the blood cells, an inspection of their size and shape, or a chemical
analysis of the blood serum, provide important information for the diagnosis of
diseases .. The red blood cell indices, which are of importance in diagnosis, are the
packed cell volume (PCV), mean cell haemoglobin concentration (MCHC), the mean
cell haemoglobin (MCH) and the mean cell volume (MCV). The normal values for an
adult, corresponding to these indices, can be found in Appendix C.
There are two methods of obtaining these indices, either by manual means or by
automatic counters. Indices derived by manual methods are calculated using three
basic measurements: haemoglobin concentration, PCV and RBC count. The
haemoglobin can be measured by spectroscopy and the PCV is obtained by
centrifugation of whole blood; both these measurements can be made accurately, but
the cell count is very inaccurate when done manually, because a counting chamber
must be used and then it is viewed with the aid of a microscope, and so it is of little
use to the clinical practice.
Serological tests, or bacteriological tests, have the purpose to determine the type of
bacteria that have invaded the body. Serological tests are based on the fact that the
organism, when invaded by an infectious disease, develops antibodies in the blood.
These antibodies are selective to certain strains of organisms, and their action can be
observed in vitro by various methods. For example in some methods, agglutination
becomes visible under the microscope when a test serum containing the antigen of the
organism is added.
80
Many diseases cause characteristic variations in the blood indices. These variations
can be a change in the number, size or shape of certain blood cells (in anaemia, for
example, the RBC count is reduced). But other diseases cause changes in the chemical
composition of the blood serum or other body fluids. In diabetes mellitus, for instance,
glucose levels are more elevated than normal.
Blood counts and chemical blood tests are often performed routinely to monitor the
process of an illness. Therefore even in a small clinical practice, the use of automatic
methods of blood analysis is widely spread.
7.1.3 Automatic Cell Counters.
A number of automatic counters have been developed and several books have chapters
devoted to it [e.g. Rowan, R.M. and England, J.M., eds. ,1986]. The most popular type
of automatic blood counter is the Coulter counter.
The first step in obtaining an automatic blood count is to aspirate a fixed volume of
blood from the sample (from 100 to 250 Ill) and dilute it, adding reagents. Each
automatic counting instrument uses a dilution of whole blood in a saline based
diluent.
For haemoglobinometry, various modifications of Drabkin type of reagent are added
to lyse the erythrocytes, and convert the haemoglobin into forms containing cyanide.
For white cells studies, blood must be diluted and red cells lysed. One dilution (1120)
is treated with a red cell lytic agent, fixed with formalin and stained for peroxidase
activity. This dilution is used to determine the total white cells count and to
differentiate the leukocytes. Another dilution (1120) is treated with a red lytic agent,
and stained with Astra blue to allow the basophil count to be determined.
81
RBC counting can be performed in two fundamentally different ways. The first of
them involves the detection of a change in the aperture-impedance when a cell passes
through an orifice (automatic counter Coulter model S Plus IV). There are two
electrodes in the orifice and as the cell passes between them, the impedance is reduced
because the cell acts as an insulator. This change in impedance can be detected and
used to count the cells. A sweep flow arrangement removes cells from the rear of the
orifice as soon as they have been detected.
The second method for RBC counting (automatic counter Ortho ELT 800) involves
detecting light scattered when a cell passes through a sensing zone. In this instrument
light from a He-Ne laser is focused onto the cells as they pass through the flow
channel. Because a "stop" is placed in the light path, light can not pass directly into
the forward scattering collecting lens. If, however, a cell is in the light beam, light is
scattered, passes around the "stop" and then it is detected. Light scattered at right
angles to the light beam is also detected for use in the white cell measurements.
In the counter Technicon H601O, only forward non-laser light scattering is detected.
Diluted blood enters the flow chamber and is made to pass through a very narrow
stream by the use of another fluid, which is directed through the flow chamber and
. surrounds the diluted blood. Because this diluted blood is in a very narrow stream, the
light can be focused precisely onto the cells as they pass through the detector. This
counter requires to make the cells spherical, so their volume can be predicted by the
Mie theory for spherical scattering particles.
All the instruments of the scattering type, count the rate at which the cells pass
through the sensing zone (impulses! unit time), rather than counting the number of
cells per unit volume of dilution. The instruments must therefore be calibrated with
blood samples of known red cell and platelet count.
Light scattering systems are sensitive to changes in internal refractive index, as well as
to changes in volume. For red cells, it is the mean cell haemoglobin concentration
(MCHC) which is primarily affected by changes in the internal refractive index.
82
If the blood is not very diluted, there is a possibility that more than one cell may be in
the sensing zone and can not be resolved by the detector. This problem is called
"coincidence" and a correction is applied by an algorithm built into the instrument.
Another correction factor used by automatic counters is the shape factor, defined as
the ratio between the apparent volume and the true volume of a particle. This factor
varies from particle to particle, depending upon the shape the particle adopts when
passing through the orifice in a fast flowing fluid stream.
With red cells, flexibility is a very important parameter, and the Mean Corpuscular
Haemoglobin Concentration (MCHC) is probably the most important determinant of
flexibility. Cells with a high MCHC are less flexible and appear enlarged.
For white cell counting it is necessary to lyse the RBC before the white cells can be
counted. With aperture impedance systems the RBC' membranes must be broken
down to very small fragments, since "ghosts" cause impulses which would interfere
with the measurements. The presence of "ghosts" is less important with light
scattering systems. When the RBC' membranes are destroyed very rapidly, some
damage to the leukocytes is inevitable and their apparent cell volume is markedly
reduced.
The haemoglobin derivatives are measured by their absorbance at various
wavelengths.
7.1.4 Measurements of Haemoglobin
The haemoglobin content of blood may be determined by physical properties such as
specific gravity, chemical composition, gas analysis or by spectroscopic
measurements. The estimation of haemoglobin is dependent on its property of
absorbing light in the yellow-green region of the visible spectrum. Since various types
83
of haemoglobin (oxy-haemoglobin, reduced haemoglobin, metha-haemoglobin and
carboxy-haemoglobin) absorb light to a different extent, the blood is diluted with a
cyanide solution, which converts all the types of haemoglobin to a more stable
derivative. By exposure to potassium ferricyanide and potassium cyanide,
haemoglobin is converted to the extremely stable cyanmetha-haemoglobin [Hughes
Jones, N.C., 1984]. The absorbance of this derivative at 530-550 nm is proportional to
the haemoglobin content in blood [Biochemical Organic Compounds (Sigma), 1993).
A spectrophotometer allows the measurement of the absorption of samples at different
wavelengths. It uses a light source of a wide spectral band and a monochromator to
spread the light into its spectral components. A measurement of the light transmitted
intensity, as a function of wavelength, is performed and compared with the initial
intensity. The concentration of haemoglobin is obtained using the Beer-Lambert Law.
7.1.4.1 Spectroscopic Measurements.
When a spectroscopic study of a pure absorber takes place, the simple Beer-Lambert
law [Parikh, V.M., 1974] will govern the absorption process. This law states that
1=10 exp (-edc) ( 7.1 )
where 10 is the intensity of the incident radiation and I the intensity of the transmitted
radiation. The thickness of the cell d is given in centimetres, and the concentration of
the solution c is given in moles per litre. The quantity e is the extinction coefficient,
with the units [It Imol x cm].
In spectroscopy the quantity named "Optical Density" (OD) provides important
information about a test sample. The OD is defined as
OD = log (Io/l) ( 7.2)
84
An Application in Blood Oxymetry.
Using Equation (7.1), the concentration of a haemolised blood sample (Hb) and its
oxygenated aliquot (Hb02) can be detennined from the light transmission through the
sample. Haemoglobin solutions follow the Beer-Lambert law and the fractional
oxygen saturation of haemoglobin in solution can be detennined by measuring the OD
at a wavelength where the oxy and deoxy forms of the molecule have the same molar
extinction coefficient (805 nm), and at a wavelength where they have different
extinction coefficients (550 nm). If this is done, a unique linear relation between the
fractional oxygen saturation of the haemoglobin and the ratio of the OD of the
solution, at the two different wavelengths, can be obtained [Takatani, S., 1994].
7.1.4.2 Colorimetry.
The haemoglobin concentration can be detennined by lysing the RBC to release the
haemoglobin, and chemically converting the haemoglobin into another coloured
compound (acid-haematin or Cyanmethaemoglobin). The colour concentration of
these components does not depend on the oxygenation of the blood. Following the
reaction, the concentration of the new components can be detennined by colorimetry.
A colorimetric detennination of the concentration of a substance uses the fact that
many chemical compounds in solution appear coloured, with the saturation of the
colour depending on the concentration of the compound.
In a colorimeter the absorbance of the solution is measured ideally at a wavelength of
540 nm. Traditionally it consists of a light source, a yellow-green filter to select the
appropriate wavelength, and a collimating lens used to evenly illuminate a glass
cuvette. When a solution of a given concentration is placed in the cuvette, it will
absorb a portion of the incident light. Then Beer-Lambert law is used to detennine the
concentration of the substance, having previously detennined the extinction
coefficient of the substance.
85
7.1.5 Examination of a stained blood film.
All haematological diagnostics include the examination of a blood film. A small drop
of blood is spread as a film on a slide and stained with a dye containing methylene
blue and eosin. The film is observed under the microscope to examine if cells show
abnormal shapes or sizes.
7.2 Research in Optical Techniques for Blood Analysis.
7.2.1 Motivation for Investigating Optical Methods.
During the last two decades, optical methods dedicated to the solution of problems in
biomedicine have gained increasing popularity. The most important reasons for which
so much research effort is focused into the field are: the high accuracy of the
measurements and the potential of optical methods to evolve into non-invasive
technology. These advantages are in particular very attractive for research in human
blood, because traditional blood analyses are performed in very large quantities world
wide, they are expensive and carry intrinsic health risks for both the donor/patient and
the analyst. Optical techniques for blood analysis are traditionally spectroscopic,
however, instrumentation is being developed using other technologies, mainly applied
in the subjects of blood cells ablation and coagulation of vessels, blood flow through
capillaries and vessels, blood oxymetry, ektacytometry, and estimation of the
concentration of blood constituents.
The work carried out within this research project, in the development of alternative
techniques for studying blood components, makes use of measurements of light
transmitted through a sample of blood. For this reason the theoretical models
86
elaborated to describe the interaction between the blood and the incident light, and the
relationship between the light transmission and the absorbing and scattering
coefficients of blood are of particular importance. Some of the models most
commonly used, and which are a direct antecedent to this work, will be revised in the
following sections.
7.2.2 Light Transmission through Whole blood.
A plot of the transmission of light through a sample of whole blood, as a function of
the haematocrit, does not follow Beer-Lambert law because whole blood not only
absorbs light, but the RBC suspended in the plasma also scatter the incident light.
Kramer, K. et al. [1955] explained that samples of erythrocytes refract light and cause
a greater absorption, by the intracellular haemoglobin pigments, than haemolised
aliquots. Also the extinction coefficients of oxygenated and reduced whole blood, are
greater than their haemolised equivalents (Hb02 and Hb) by factors ranging from 7 to
20, for certain wavelengths.
However, when scattering from RBC in whole blood is eliminated by suspending
erythrocytes in concentrated protein solutions, the resulting non-scattering RBC
suspensions absorb light according to the Beer-Lambert law [Barer, R, 1955]. Barer
also showed that haemoglobin in RBC absorbs similarly as free haemoglobin, if the
scattering is eliminated by suspending the cells in a medium with a refractive index
equal to the cells' cytoplasm refractive index. A sample of very tightly packed cells,
with almost no plasma, absorbs light as if it was a haemoglobin solution. This
observation was confirmed by Kramer, K. et al. [1955].
The main scattering particles in whole blood are Red Blood Cells (RBC). Scattering
of light by RBC takes place at interfaces where there is a change in the refractive
index. The angles through which the light is scattered and the scattering parameter
87
depend on the size and shape of the scatterers relative to the source wavelength, and to
differences in the refractive index of the scatterers and the surrounding medium.
Scattering from molecules or structures whose size is much smaller than the
wavelength (Rayleigh Scattering), is relatively weak, nearly isotropic, and strongly
decreases with wavelength. When the wavelength and scattering particles are about
the same size, the scattering is stronger and more forward-directed, but decreases
roughly inversely with the wavelength [Bohren, C.F. and Huffman, D.R., 1983].
Whole Blood Modelled as an Absorbing and Scattering Medium.
7.2.2.1 Adding a Scattering term to the Beer-Lambert law.
The spectrophotometric behaviour of whole blood follows a pattern predictable as the
sum of a linear absorption term (i.e. following Beer-Lambert law) and a non-linear
scattering term [Anderson, N.M. and Sekelj, P., 1967a and 1967b].
Anderson and Sekelj also developed an expression for OD, which depends on the
usual variables: the extinction coefficient (E), the concentration (C), the path length
(P), and on an additional term (b):
OD =cpE+b (7.3 )
where b depends on the fraction of the total scattering cross section of one scatterer in
free space received by the detector (for non-absorbing scatterers), on the density of
scatterers and on the dimensions of the scattering particles. This new equation means
that when the concentration and path length are constant, the extinction coefficient of
the absorber within the scattering particle is the only independent variable, as it is
linearly related to the OD. This hypothesis indicates that absorption is dominant at
wavelengths with high extinction coefficients, whereas scattering predominates at
wavelengths with low extinction coefficients.
88
Twersky's Model.
Anderson and Sekelj [1967 a, b] showed that light propagated through whole blood
follows the form predicted by Twersky [Twersky, V., 1962]. The following is
Twersky's transmission equation:
T(y)=exp(-w(y)) x [exp(-~p(y)+q(1-exp(-~p(y))] (7.4)
The terms in this equation will be explained in Chapter 8, but in summary, the first
factor in Equation (7.4) is a purely absorbing term, while the second one involves
scattering effects.
This theory implies that it is possible to separate the effects of absorption from the
effects of scattering in the OD of blood and it demonstrates the following points:
i) The linear relationship postulated by Beer-Lambert law between OD and the
extinction coefficient of haemoglobin is valid for haemoglobin even when it is
contained within scattering particles, and contrary to Kramer's theory, absorption is
not increased as a result of an increase in the length of the light path induced by
scattering.
ii) It makes possible to evaluate separately the effects of light scattering and
absorption on the light transmission of thin films.
Twersky's theory is applicable when the medium thickness is very thin, or the
concentration of the particles very low, so the coherence of the incident beam is not
lost when transmitted through the sample. If the coherence is lost, the light becomes
diffused.
Although Lipowsky H., et al. [1980] reported successful results using Twersky's
model, Steinke and Shepherd [1988] explained that Twersky's theory n ••• does not
describe the· spatial distribution of the reflected and transmitted light... and that
89
Twersky's model does not lend itself to simulating the optical effects of variables such
as mean corpuscular haemoglobin concentration and red blood cell volume, nor does
it accommodate light detectors and sources that do not share a common optical axis".
7.2.2.2 Diffusion Theory.
The diffusion equation was derived from the transport equation. The transport
equation assumes that each scattering particle within the medium is sufficiently distant
from its neighbours to prevent interactions between successive scattering effects. In
theory these scatterers and absorbers must be uniformly distributed throughout the
medium. Polarisation effects are neglected [Cheong, W., 1990]. Diffusion theory
requires the absorption effects on the incident irradiance to be much smaller than the
scattering effects [Zdrojkowski, R.I. and Pisharoty, N.R., 1970].
Diffusion takes place when the particle concentration increases above 5% [Takatani,
S. and Ling, J, 1994], or the medium thickness is large. Diffusion theory is based in
the assumption that the interaction of the radiance of light with the media can be
separated into unscattered and scattered components. Unscattered light is attenuated
exponentially following Beer-Lambert's law. For light passing through a slab of an
absorbing material with thickness (or effective path length) p and having no
reflections at the surface, the transmission coefficient is given by
(7.5)
where J.Lu is the extinction coefficient.
In regions within the scattering medium, far from light sources and boundaries, the
fluence rate decays exponentially. The rate of decay is called the effective attenuation
coefficient (J.L f) or the diffusion exponent. This coefficient is known as (l( ) if the
scattering parameter is greater than the absorption parameter.
90
The transmission coefficient, in presence of scattering, through a slab of thickness p
with matched boundaries is given by the following equation [Cheong, W., 1990]:
(7.6)
where (Ila) is the absorption coefficient, (11.) is the scattering coefficient and (g) is the
mean scattering cosine of the scattering angle or the parameter of anisotropy. The
parameters a], a2 and aJ are part of the solution of the diffusion equation for the total
fluence rate in a parallel slab [Cheong, W., ibid.], their values depend on boundary
conditions.
The total transmission coefficient is T = Tu + Ts (7.7)
And the diffuse reflection coefficient is
(7.8)
The values of the three optical coefficients (Ila' Ils' g) are determined when the
unscattered transmission, the scattered transmission and the diffuse reflection are
measured. However if only the total transmission coefficient and the diffuse reflection
coefficient are measured, then only the absorption parameter and the reduced
scattering parameter ( 11 s ') can be measured.
(7.9)
The value of g varies between -1 and 1; g=0 corresponds to the case of isotropic
scattering, g= 1 corresponds to total forward scattering and g=-1 corresponds to total
backward scattering. The g ofRBC ranges from 0.7 to 0.967, and is therefore strongly
forward scattering. The back scattering section of the RBC in the visible and near
91
infrared regions is approximately constant [Takatani, S. and Ling, J, 1994]. These·
parameters were measured for a whole blood sample of 6.38 mm thickness at a
wavelength A = 633 nm, yielding Ita = 25 cm·!, Its = 400 cm·!, g = 0.98 and refractive
index n = 1.35 [Tuchin, V.V., 1993]. The same parameters were also measured under
different conditions, as quoted by [Cheong, W. et al., 1990], at 1..=960 nm, resulting in
Ita = 2.84 cm-I, Its = 505 cm-! and g = 0.992 for oxygenated and haeparinised whole
blood (haematocrit = 0.41).
Some authors [e.g. Reynolds, L. et aI., 1976] instead of using directly the above
coefficients, prefer to quote two parameters involving a combination of the scattering
and absorbing coefficients. One of this parameters is the albedo, defined as
albedo = Jls!( ). The other one is the photon penetration depth, or the distance /(Jls + Jla
light can penetrate a given medium without loosing coherence, this parameter is
proportional to the albedo.
7.2.2.3 Kubelka·Munk Theory.
One of the first models developed to describe optical propagation through diffuse
media is the two flux Kubelka·Munk model [van Gemert, M.J.C. and Star, W.M.,
1987]. This model is based on the concept of forward and backward travelling fluxes,
but because of its phenomenological and heuristic nature, as well as its one
dimensionality, the theory has serious limitations when applied to whole blood
[Takatani, S. and Ling, J, 1994].
The Kubelka-Munk equations for the reflection and transmission coefficients of light
through an absorbing and scattering medium of thickness p, when the medium has a
finite thickness [Kubelka, K., 1948] are:
92
b T =-a-si-nh'( b-S-p )'+-b-c-os-h'( b-Sp')
sinh(bSp) R=----__ ~~~,-~ a sinh(bSp)+b cosh(bSp)
Where,
b=~a2-1 S+K
a=--S
(7.10)
(7.11)
(7.12)
(7.13)
S is proportional to the scattering coefficient and K to the absorption coefficient.
Using the hyperbolic functions
sinh(a.) exp(a)-exp(-a)· and 2
exp( a )+ exp( -a ) cosh(a)
2
Equation (7.1 0) for the transmission coefficient through a scattering and absorbing
medium can be rewritten as follows:
2b T
(a+b )exp(bSp) -(a -b )exp( -bSp)
After some algebraic manipulation, Equation (7.14) results
2b 1 .( (b-a} J1 T=-b -x ( ) 1+ -b - xp(-2bSp) +a exp bSp +a
This equation can be expanded as a sum of exponentiaIs as follows:
T=~exp(_bSP[I_(b-a}xp(_2bSp)+(b-a y exp(-4bSp)+ ... J b+a \ b+a b+a)
(7.14)
(7.15)
(7.16)
93
Equation (7.12) ensures the convergence of Equation (7.16).
The Kubelka-Munk absorption and scattering coefficients are dependent on the
scattering anisotropy of the medium. These parameters can be found by measurements
of the transmitted and reflected fluxes in the medium.
7.2.2.4 Time resolved spectroscopy for investigations in tissue and blood
oxymetry.
Time resolved spectroscopy has been studied for oxymetry applications, when the
optical path length is unknown. A very narrow duration pulse at the near-infrared
region is applied to the tissue of interest, where absorption due to haemoglobin is low,
and the transit time of the photon through the tissue is measured. One can estimate
transit times as well as tissue scattering absorption properties from this wave form.
This technique involves counting photons rather than measuring the intensity of light
transmitted through the sample of interest [Chance, B., et aI., 1988].
7.2.2.5 The path length dependency in light transmission measurements.
For whole blood, Zdrojkowski and Pisharoty [1970], calculated the mean total optical
path travelled by a photon within the sample before absorption, using optical data
taken from samples of haemolised blood. This mean path was found to be equal for
both whole and haemolised blood with the same relative oxygen saturation and
haemoglobin concentration, provided that the absorption per erythrocyte is small.
Blood meets this condition for wavelengths between 600 nm and 850 nm. The· mean
optical path varies inversely with the haemoglobin concentration.
Following the work made by Loewinger et al. [Loewinger et al;, 1964], Zdrojkowski
and Pisharoty [ibid.], arrived at the conclusion that the mean total optical path (P)
94
travelled by a photon before being randomly scattered, is related to the haematocrit
through the following equation:
P (l-H)H for O~H~1 (7.17)
where Jl s is the scattering coefficient and H the haematocrit Equation (7.17) indicates
that no scattering takes place when blood cells are not present and when the cells are
tightly packed. The scattering is maximum when the haematocrit has the value of
0.50. They found good agreement with the results of Anderson and Sekelj [1967].
Lipowsky H. et al. [1980] studied variations in the optical density of whole blood
samples as function of haematocrit and path length. For this they circulated blood
through tubes of different sizes and modelled the light transmission through the tube
using Twersky's [1970] theory. They found good agreement between the theory and
the experiment when they described the attenuation of a coherent light beam by cell to
cell scattering using equation (7.17), where p is the total diameter of the tube and H
the haematocrit.
Those authors that studied the light transmission through whole blood samples, as
function of the path length and haematocrit, used containers with fixed path length.
They did a sequence of measurements at different haematocrit values, then in order to
modify the path length they had to use a different container, of a different size, and
vary the haematocrit again.
Changing the position of the cuvette or changing cuvettes in order to modify the path
length is not ,an optimum experimental solution, because external sources of error are
brought into the measurements, such as different reflection and diffraction effects by
the cuvette walls, different wall thickness and refractive index. The incident beam can
also reach the cuvette at different angles of incidence. Not all these extra variables can
be eliminated by calibration alone.
95
Trying to overcome the problems stated above, the experimental techniques that will
be described in Chapters 8 and 9 make use of a blood container which allows
simultaneous measurements of single haematocrit values at different path lengths, in
other words, a container of varying path length is being used.
7.2.3 Summary
In the previous sections were reported the most popular and best known theoretical
models used to describe the transmission of light through a blood sample. Although
the Beer-Lambert law, the Twersky's equation, Diffusion theory and the Kubelka
Munk treatment (see Table 7.1 for a summary) are all used to describe the interaction
of blood with the radiation traversing the sample, they all describe different problems
in the sense that each of them is useful for particular boundary conditions (e.g.,
matched or mismatched interfaces), type of radiation (diffuse or coherent), thickness
of the sample and the type of interaction between blood and light occurring within the
sample (absorbing vs. scattering).
However, all the mentioned transmission equations (Equations 7.4 to 7.7, 7.15 and .
7.17) can be generalised in one dimension into an expression made of a sum of
exponentials such as:
(7.18) n
where an and bn are the particular coefficients indicating the absorbing and scattering
characteristics of the sample, and p is the path length.
96
In the work that will be described in the next two chapters, the general equation (7.18)
was used to model the light transmission measurements. In particular, when the
experimental transmission data for whole blood samples was fitted through a non
linear procedure, the best fit was given by the equation at the bottom of Table (7.1)
[Ruiz de Marquez, G., et al., 1996], the details can be found in Chapter 9.
Table 7.1 Summary of Theoretical models describing the transmission of light
through a sample of path length p.
Transmission Model Conditions
Beer- T = exp( -/luP ) Absorbing but non-
Lambert scattering medium.
Twersky's T = exp( -'¥P)[exp( -~p +q(l-exp( -~p)] Coherent illumination, thin absorbing and scattering medium.
Diffusion T=ex (_ )+ /lsg exp(-/lup)
Depends on illumination and boundary conditions.
p /luP /la + /l.(I- g) This case is for index
-{alK exp(Kp )-a2K exp( -Kp )-a3/lu exp( -/lup)] matched boundaties, isotropic scattering and uniform optical properties.
Kubelka- 2b Uniform, diffuse irradiance T
(a +b )exp(bSp )-(a -b )exp( -bSp) or through a one-dimensional
Munk isotropic slab, with no
T=(~ }XP(-bSP) reflection at the boundaties.
b+a Two-flux (forward and
1_(b-a }XP(-2bSP)
, backward) model. Non-useful for laser
b+a irradiance. X
+(::: r exp( -4bSp )+ ...
Ruizde T = Aexp(-ax)(I+ Bexp(-bx)) Empirical model, white Marquezet light illumination, varying al., 1996. path lengthmedium.
The blood samples used in our own experiments were stationary diluted whole blood
samples, placed in a glass chamber of varying path length and illuminated with diffuse
light. Although an attempt was not made to fit the experimental results to diffusion
97
theory, we expected the results from the empirical model to be not very different from
those of diffusion, because a diffuse white light source was used, and although the
concentration of the particles was relatively low, the path length was of the order of
millimetres. The terms of the simple transmission equation, that was empirically
obtained, were written in the Twersky's notation but just for the sake of comparison,
since Twersky's equation is both simple and relatively easy to be interpreted.
However the comparison among the two models is not a direct one, because
Twersky's assumes laser light illumination. In our last set of experiments we used also
a laser light source for polarisation measurements, over a large and constant path
length.
Another important experimental variation introduced here, to the optical analysis of
blood, is the use of polarised light. The full capabilities of this useful tool have not yet
been exploited in the context of blood measurements, but the results that will be
reported at the end of Chapter 9 are encouraging and motivate the pursuit of this line
of research.
98
Chapter 8
Imaging Technique for Absorbance and Scattering
Measurements on Blood
8.1 Introduction
The motivation for performing the set of experiments labelled as .. imaging technique
for absorbance and scattering measurements on blood" is to study the morphology and
concentration of RBC suspensions using an uncomplicated sensor. Complete physical
and chemical analysis of blood is not always necessarily required to detect, screen or
monitor a single pathology and the development of a simple sensor for specific
investigations could be valuable for future patient-centred medical care.
Traditionally, the optical analysis of blood, as was illustrated in Chapter 7, has
involved the measurement of the light absorbed by a sample of haemoglobin in order
to determine the concentration of the protein, but recently some authors have shown
that scattering methods can be used to determine the characteristics of whole and
haemolysed blood [e.g. Lee, V.S. and Tarassenko, L., 1991]. These techniques are
based on transport models or multiple scattering theories which do not readily provide
a direct and unambiguous interpretation of the measurements.· This is not surprising
given the large number of variables governing the scattering process, the complexity
of the interaction and the relatively narrow distribution of observations which are
usually made.
In order to obtain more information from the scattered field, we have developed a
simple technique to discriminate absorption and scattering within the framework of
Twersky's multiple scattering theory [Twersky, V., 1970). But equally we could have
used the empirical model given by Equation (7.19), Diffusion theory or the Kubelka
Munkmodel.
8.2 Description of the Experiment
A white light source was used to illuminate the entrance aperture of an integrating
sphere and diffuse light from the sphere was incident on the sample chamber, which
was situated at 90° relative to the entrance aperture. The light transmitted through the
sample was detected at normal incidence by a linear array of 1024 photodiodes and
data were captured using a PC based data acquisition system, manufactured by Oriel
Corporation, with a 16-bit NO converter.
The size of the photodiode array is 2.5 mm width and each element measures 25 J.Un.
The detector head is provided with thermoelectric coolers to maintain the temperature
of the array.
The sample chamber was a flexible latex tube, or vessel, of 2.5 mm internal diameter.
Because the detector array was located perpendicular to the length of the tube, the
vessel acted as a varying path length chamber.
The vessel was imaged on the detector array by a combination of two biconvex lenses.
The first lens created a magnified and inverted image of the vessel, and because the
two lenses were separated by a distance smaller than the focal distance of the second
lens, the second lens focused on the detector a slightly magnified image.
The high intensity white light source used for this experiment was manufactured by
Dolan- Jener Industries, and the integrating sphere used to diffuse the white light was
lOO
designed and produced "at home". The sphere is about 20 cm diameter with entrance
and exit circular apertures of 2.54 cm diameter. To cover the inside of the mild steel
sphere we used an air drying matt acrylic white paint (barium sulphate based). When
light is incident on a layer of this paint, the reflected light is diffused due to the Iow
refractive index of the barium sulphate.
LIGHT SOURCE A
B x
VESSEL
Figure 8.1: Diagram of the Experimental Set Up.
8.3 Materials and Methods
For each complete set of experiments, heparinised whole blood from a single healthy
human donor was obtained.· One portion of the whole blood sample was haemolysed
and diluted with distilled water and the remaining portion diluted at different
concentrations with Ringer solution (see recipe for Ringer in Appendix C). Distilled
water and some dyes diluted with distilled water and filtered through a 0.7 !Lm pore
filter were also prepared to be used as reference solutions ..
101
Each of the profiles reported here, corresponding to the image of the vessel, is the
average of some 150 profiles; the acquisition time for each profile was approximately
3.5 ms. The main cause of error in the measurements arises from fluctuations in the
light source itself, but since they occur on time scales greater than the scan time, they
can be compensated for by the appropriate offset.
Blood will suffer haemolysis if a strange substance with a pH different to 7.4 is added
to it, or if by some means the balance in the osmotic pressure between the inside and
the outside of the RBC is lost, causing the cell membrane to disrupt. In this
experiment, the haemolysed blood samples were prepared by adding 3 rnI of
haeparinised whole blood to each of the following amounts of distilled water:
Table 8.1: Concentration o/haemolysed blood samples.
Sample Whole Blood Distilled Concentration Type of
Number. (mI) Water (ml) Sample
1 3 4.5 0.667 haemolysed
2 3 5.5 0.545 haemolysed
3 3 6.7 0.461 haemolysed
4 . 3 7.5 0.40 haemolysed
5 3 10 0.30 haemolysed .
Ringer solution, at the correct pH, is one of the substances commonly used to dilute
whole blood without altering its properties, so it was used in this experiment to dilute
the whole blood samples:
102
Table 8.2: Concentration of whole blood samples
Sample Whole Blood Ringer Concentration Type of
Number (ml) Solution (ml) Sample
6 3 4.5 0.667 whole blood
7 3' 5.5 0.545 whole blood , .
8 3 7.5 0.40 whole blood
9 3 10.0 0.30 whole blood .
A peristaltic pump was used to circulate the samples at a slow flow rate of about 12
mlImin. After each measurement, Ringer solution was circulated through the system to
rinse it before the next sample was pumped in.
During the image acquisition time, the whole measuring system was covered by a
black plastic sheet to eliminate ambient light.
8.4 Discussion of Results
8.4.1 Data Processing.
Raw data obtained are averaged and normalised with respect t() a profile obtained
from a pure absorber (filtered dye). In Figure (8.2), the normalised intensity profiles
derived from a sample of distilled water and an extra filtered dye are shown. The dye,
whose profile is shown in the figure, is a stronger absorber than the dye used for
normalising, so the transmittance for the more absorbing dye has a value between 0
and I as a function of path length, while the water, on the contrary, is less absorbing
than the dye used for normalising and its transmittance is therefore greater than I for
most of the path length.
103
1 1===:::::::~
I i .!;l O.B E :g El ~ 1 0.6
z
Distilled Water
Fittered Dye
0.4+----+----~~~--~~--~--~----+_--_+----+_--~
o 20 40 80 80 100 120 140 180 180 200
X 8xls across the vessel
Figure 8.2: Normalised transmitted intensity profiles for samples of distilled water
. and a filtered dye, imaging the profiles across the shortest length of the vessel.
The resolution of the linear array of photodiodes is 1024 pixels, but restrictions in the
physical distance between the elements in the set up and the availability of lenses,
forced the image of the vessel on the array to be relatively small. The vessel is a tube
with an approximately circular cross section, so the study of only half of the data is
enough to obtain a full analysis of the light transmission across the complete vessel.
The initial section of the profile, between pixels 0 and 30 approximately, falls outside
the vessel, the following 30 pixels correspond to the wall of the vessel and the
remainding points are all located between the internal wall and the centre of the vessel.
In the next two figures (8.3 and 8.4) only one in every four pixels are plotted, so that
pixel number 'n' actually corresponds to pixel number 4n+ 1 on the photodiode array.
Average standard deviations in these data are -0.01 and therefore are not visible on
the scale.
In Figure (8.3), a series of normalised transmission profiles are shown, the
concentrations of all samples are found in Tables (8.1 and 8.2).
104
1.1 -r--------------------------,
•• •••••••••••••••••••••••••••••••••• f 1 ..... ""II;;llIlllx _-Clii • .X
0.9 • ",X 'tI • &"X
~C11 • • X • Aiixx E 0.8 • A.XXX
'" • .x X a • A .xxxx f:. 0.7 .bluedye • A ·.XXXXXX .
'tI • water • A • .XXx xxxxx
•5:_ • A ••• xxxxx XXXXXXXXXX
Awb(7) •.• xxx ~ 0.6 Xhae(2) .~A •••••• XXXXXXXX
- xhae(4) •• • •••••••• o lit. z .wb(8) .... M A.~ AAA ••••••••••••
AAA AAA A
0.4 +---+---+----+---+----I---+---f---+=.....u, ....... &.a.~ o 5 10 15 20 25 30 35 40 45 50
Pixel Number (across the vessel)
Figure 8.3: Transmission profiles across a flexible tube containing dye, water,
whole blood and haemolysed blood (the numbers in brackets indicate the sample
number, See Tables 8.1 & 8.2). All profiles are normalised with respect to a profile
obtained/or a pure absorber.
The most concentrated samples: number 7 ( whole blood, 0.545 m1 blood in 1 m1
Ringer) and number 2 (haemolysed blood, 0.545 m1 blood in 1 m1 water) absorb more
light and their transmittances are low, with respect to samples of the same type but
less concentrated: number 8 (whole blood, 0.4 ml blood in 1 m1 Ringer) and number 4
(haemolysed blood, 0.4 m1 blood in 1 m1 water). Also according to the figure, the
normalised transmitted intensity for whole blood samples is lower than for
haemolysed samples. This result coincides with Kramer K. et al. [1955].
All the samples are clearly differentiated in the central region of the vessel, but their
relative positions can be reversed close to the wall. Figure (8.4) shows a series of
normalised transmission profiles near the internal edge of the vessel for three whole
blood concentrations. A dye is also shown for the sake of comparison.
105
1
i 0.95
0.9 ! ] 0.85 ] le 0.8 e ...
1 0.75
0.7 0 z
0.65
I iI Q i Q
[J [J
A A [J
• • A
~ • • i • ~ • • It . • [J •
• blue dye A • • A • [J
[J wb (7) • A • • • [J A • A wb (8)
A • • [J A • • • A • • • wb (5) [J A • • A • [J
A A A
0.6 • [J
10 15 20 25 30 35
Pixel Number (across the vesseQ
Figure 8.4: Transmission profiles near the edge of a flexible tube containing dye or
whole blood (the numbers in brackets indicate the type of sample, see Tables 8.1&
8.2). All profiles are normalised with respect to a profile obtained from a pure
absorber.
It may be noted that the relative levels of the three whole blood samples are reversed
near the edge of the vessel (pixei number 16), suggesting a strong depletion of the
apparent absorption. Many factors could be responsible for such an effect including:
RBC scattering, scattering at the wall and fluid interface and formation of a boundary
layer due to RBC rouleaux. This occurs because the tendency which the cells have to
align, one over the other, in a reversible fashion when they are in plasma; The extent
of rouleaux formation determines the sedimentation rate of blood, [Bessis, M, 1973].
A highly complex physical model is required to asses the relative magnitude of these
phenomena since it would need to incorporate the fluid mechanics of RBC
suspensions with a multiple scattering theory in at least two dimensions. Nevertheless
an empirical interpretation of profiles such as those in Figures (8.3 and 8.4) may prove
useful.
106
8.4.2 Semi-Empirical Model.
It is known that when RBC exhibit laminar flow in tubes they tend to migrate toward
the axis. The axial accumulation of RBC may increase the haematocrit in the middle
of the tube, thereby increasing the formation of aggregates, which then can increase
the light transmitted through the gaps [Lindberg, L. G. And Oberg, P.A., 1993]. We
attempted a preliminary interpretation of the data presented in the previous section via
a zeroth order empirical model of the phenomenon just described.
An empirical model for the transmittance along the path length p(y) can be expressed
in the general form:
T(y)= IaiexP(hi p(y») i=!
(8.1)
Where the coefficients ai are related to the scatterers' properties and the coefficients hi
are related to the absorbance by the scatterers and suspending medium.
The one dimensional empirical scattering model (Equation 8.1) can be written in terms
of Twersky's notation [Twersky, V., 1970] (Equation 8.4), to facilitate the
interpretation and comparison of the results with those from other authors. Twersky's
. theory is based on fundamental physical assumptions, and was reported to be
successful by [Lipowsky, H.H. et al., 1980].
The approximate multiple scattering theory developed by Twersky predicts the total
transmitted intensity T through a suspension of spheroids to be
T(y)=exp(--1 p(y»[exp(-~ p(y»+q(1-exp(-~ p(y»)] (8.2)
where pry) is the path length at position y. In the case of a suspension of red blood
cells (RBC), "( is an absorption coefficient proportional to the haematocrit volume
fraction (v); ~ is a coefficient dependent on the haematocrit (proportional to v (l-v2»,
107
, .
I
the size and total scattering cross section of the particles, and q is a dimension-less
constant dependent upon particle size, wavelength, photodetector aperture angle and
refractive index of the particles and suspending medium [ibid.].
The interaction with the vessel was assumed to be one dimensional as scale lengths
transverse to the propagation direction are two orders of magnitude greater than the
wavelength. Using the geometry of the blood vessel model shown in Figure (S.l), the
path length profiles p(y) for the respective regions A (inside the vessel) and B (region
between the internal and external walls of the vessel) are given by:
PA =29t.j(a2_y2) (S.3)
. PH =29t~(b2 -i) - 29t.j(a2 -i) (S.4)
where a and b are the internal and external radii respectively and 9\ indicates the
positive solution of the square root.
Within the one dimensional approximation and ignoring Fresnel losses, the
transmission coefficient through the blood vessel can be written
(S.S)
where it may be noted that the physical parameters are assumed constant in the vessel
wall (region B). In the experiments these transmission profiles were compared with
that of a pure absorber. The relative transmission coefficient t is determined by the
ratio between Equation (S.4) for the blood vessel and that one for the same vessel
filled with an absorber.
try) T(PA( y))scatkrer T(PB( y))
T(PA ( y ))absoroer T(PB( y)) (S.6)
lOS
Assuming that the transmittance through the walls of the vessel is independent of the
nature of the sample inside the vessel, the terms in region B cancel, then
'try) T(PA ( y ))""""''''' T(PA ( y ))_ro.r (8.7)
For the case of a pure absorber, there are no scattering particles inside the vessel, thus
the expression for the transmittance in region A reduces to Beer-Lambert Law [Parikh, .
V.M,1974]:
T(y)absorber = exp( -a AP A (y» (8.8)
where we have renamed Cl the absorbance coefficient of the pure absorbing substance
held inside of the vessel. Then by substitution of Equations (8.4 and 8.8) in Equation
(8.7)
't( ) = exp(-y p( y))[exp( -~p( y)+q(l-exp(-~p( y))] y exp( -ap( y))
(8.9)
rearranging terms in this equation
't (y) = exp( -y p(y)+Cl p(y»[exp(-~ p(y»+q(l ~exp(-~ p(y)))]
(8.10)
Following Lipowsky's et al. [1980] interpretation of Twersky's coefficients for the
transmittance of a white light source through a suspension of RBC, for a uniform
distribution of the cells inside the vessel,
(8.11)
where D is the total internal diameter of the vessel and v is the fractional haematocrit.
Then replacing Equation (8.11) in Equation (8.10)
109
t (y)= exp( --'Yp(y)+cx p(y»[exp(-Dv(l-v2)p(y»+Q(1-exp( -Dv(1-v2)p(y»)]
(8.12)
However if the distribution of the scatterers is non-uniform, Equation (8.12) can be
generalised to an effective path length proportional to the projected optical density
along the line of sight (i.e. x direction):
t (y) = exp(--'Y V(y)+cx p(y»[exp(-~l (V(y)-W(Y»)+Q(1-exp(-~l (V(y)-W(y»)))]
~a1._y2
where V(y)= J v(x,y)dx, (8.13) -Jal _ y2
and ~l is a proportional constant.
The functions V and W project the volume fraction haematocrit along the optical path
and can take many forms. A simple choice is to take a Gaussian in the radial distance r
v(x,y)= vo 2 exp(-<x2 + i)/2r2) 21tr
r<a (8.14)
which allows for some control of the size and extent of the boundary layer between the
blood and the vessel wall, although physical modelling of the process will ideally
replace this assumption. Replacing Equation (8.14) in V(y), Equation (8.13), the
projection integral is then
V(y)= 4a: ex{- x22 trf(~a2 - i /r..fi) 21tr 2r r
. .
(8.15)
where a normalisation factor is introduced to ensure that the total haematocrit is
conserved irrespective of the values of a and r, as long as the Gaussian distribution
110
does not extend significantly beyond the vessel diameter. Using (S.15) and a similar
expression for W(y), the transmission profile (S.13) can be evaluated.
Parameters chosen for this study are taken from experimental studies of whole blood
under white light illumination, ~1=7.5 mm·t, ,),=0.037mm·1, q=O.4 [Lipowsky et al.,
19S0]. A fully quantitative comparison is not made here as that would only be
advisable with a monochromatic source. Nevertheless the values used are
representative of what may be expected and are therefore sufficient to reproduce the
main features of the profiles.
Evaluations of the relative transmission profiles using Equation (S.13) are shown in
Figures (S.5 and S.6). These theoretical profiles were generated using a spread sheet.
1.1
1
0.9
0.8
0.7
L-~~-+---+--:::::;:::;;;;;:;:===l 0.6
1.2 1 0.8 0.6 0.4 0.2 o Radial Distance (mm)
Figure 8.5: Theoretical normalised transmission profiles. Curves are respectively A:
absorber with optical density less than normalising absorber, B: absorber with
optical density greater than normalising absorber, C: whole blood uniformly
distributed, D: whole blood Gaussian distributed (r=O.5 mm), E: whole blood
Gaussian distributed (r=O.6 mm).
111
The theoretical profiles shown in figure (8.5) are transmission profiles through a
vessel containing a whole blood sample or a dye, relative to the same vessel
containing a pure absorber and having unifonn concentration across the vessel. Curves
labelled C, D and E correspond to whole blood samples with a fractional haematocrit
volume of 0.1. The internal diameter of the vessel is 2.5 mm.
All curves are unity at the edge of the vessel cavity, since each profile is nonnalised to
that of a pure absorber. Thus excursions above (or below) unity indicate decreased or
increased absorption with respect to that of a pure absorber.
A comparison with the data shown in Figures (8.3 and 8.4) should therefore be made
recalling that the edge of the vessel cavity correspond to pixel number 16. In Figure
(8.5) curve A is for distilled water and therefore lies above unity and increases
relatively little with path length, indicating a relatively small optical density. Curve B
however results from an absorber with much greater optical density. The transmission
profile for uniformly distributed whole blood (curve C) has a fairly large apparent
absorption with path length, experimental evidence [Lipowsky et al., 1980] strongly
suggests that the optical density can be fitted to the expression (8.2) for which the
scattering tenns, parametrised by ~ and q, are very much more significant than the
pure absorption tenn. When the whole blood is distributed non-uniformly ( curves D
and E) it is possible for the fluid in the boundary layer to be less absorbing than the
pure absorber and therefore the nonnalised profile can exceed unity. Since the total
volume of whole blood is held constant this effect is balanced by an increased
absorption in the longer path length region. The effect of changing the boundary layer
dimensions can be seen by comparing curves D and E.
Figure (8.6) shows the effect of changing the whole blood concentration and the
standard deviation of the radial concentration distribution. Studies of blood in motion
indicate that RBC aggregation is sensitive to the blood volume and conditions of the
flow [Lindberg, L.G, and Oberg, P.A., 1993]. We might expect axial aggregation to
increase with the blood volume [ibid.], which in the above model would imply that r
would decrease with increasing v. The effect of this phenomenon can be seen by
112
comparing say curve D with curves B or C. The haematocrits used in this simulation
are chosen to correspond with those used in the experimental work (Le. v=0.1 is the
same as sample 9: 0.3 ml of whole blood diluted in 1 rn1 of Ringer). It may be noted
that the redistribution of the blood volume causes the relative absorption levels of the
curves to reverse near the edge of the vessel wall as was observed in Figure (8.4).
1.1
't 1
0.9
0.8
0.7
0.6
1.2 1 0.8 0.6 0.4 0.2 0
Radial Di<>tance (mm)
Figure 8.6: Theoretical normalised transmission profiles o/whole blood distributed
radially according to a Gaussian with radial distribution r. Labelled CUT'l'es are
respectil'ely A: (1'=0.10, r=0.s mm), B: (1'=0.13, r=0.5 mm), C: (v=0.16, r=0.s mm),
D: (1'=0.10, r=0.6 mm), E: (1'=0.13, r=0.6 mm), F: (1'=0.16, r=0.6 mm).
Before fitting Equation (8.4) to the experimental profiles, the path length pry) had to
be determined and replaced in this equation. The simplest way to find the path length
is using the transmission normalised profile for a dye. The transmission coefficient for
a pure absorber is given by Beer-Lambert law (Equation 8.8):
T(y)absorber =exp(-<lAP A(Y» then,
113
PA(Y)= -l ln (T(y)absorber) (lA
(8.16)
since the path length is proportional to the transmitted intensity for an absorber, we
can use the profile for a filtered dye to obtainpA(Y). Usually the absorbance coefficient
(lA is found by spectroscopy, after several spectroscopic measurements are performed
of dilutions with different concentrations of the dye. The relationship. between
transmission and concentration is almost linear, and the slope of the plot of these two
quantities is the absorbance coefficient. Unfortunately, the absorbance coefficients of
the dyes used in these experiments could not be found in the way just described,
because their concentration was unknown and very high for spectroscopic
measurements.
The path length of the inner section of the vessel was determined according to
Equation (8.16), by calculating the natural logarithm of the normalised transmittance
of a dye, relative to the absorbance coefficient of this dye and relative to the
absorbance coefficient of the dye used to normalise all the profiles.
Equation (8.4) depends on the path length, which has been shown in pixels until this
point. However, in order to determine the parameters '"(, ~ and q with the correct units,
the path length must be converted to the vessel dimensions (mm). Using the
information that the internal diameter of the vessel is 2.5 mm, a suitable conversion
factor was found.
Figure (8.7) shows a plot of the path length in (mm) vs. pixel number. From this plot
one can observe that the path length is not shown exactly as a section of a circle, as
expected from the cross section of a perfect tube, instead it is rather shallow and
elongated towards the edge. This effect is due to using different scales for the axes in
the figure, making a circular cross section rather elliptical.
114
3,-------------------------------------------------, 2.5
~
E E 2 .....
..c: -Cl 1.5 C
~ ..c: "la Q, 0.5
centre of the vessel
O~~~=-~----~--_+----~----+_--~--~--~ edge of the vessel
o 20 40 60 80 100 120 140
Plxel Number
Figure 8.7: Plot of path length, PA(I), versus pixel number, as obtained from
equation (8.15), and scaled to dimensions in mm.
i!' 1
.iij c 0.95 GI -.E 0.9 "C
!J 0.85 E UI
0.8 c f!! I- 0.75 "C GI .!!l 0.7 "ii E
0.85 .. 0 Z
0.6 0 0.5 1 1.5 2 2.5
Path length (mm)
Figure 8.8: Normalised transmitted intensity profiles versus path length. Three
concentrations of haemolysed blood and a dye (curve D) are plotted. Curves A (the
less concentrated sample), B, and C (the most concentrated sample) correspond to
samples 5,4 and 2 respectively.
115
Figure (8.8) shows the profiles for three different concentrations of haemolysed blood
and a dye, having converted the path length to dimensions in (mm). After the
dimensional transformation, the relationship between absorbance and concentration is
maintained. The dye plotted for comparison is seen to absorb more uniformly along
the varying path length than the blood samples. It is probable that scattering from the
empty cells' membranes accounts for this effect.
Each of the experimental normalised transmitted intensity profiles were fitted with a
non-linear fitting software package based on the quasi-Newton algorithm (see
Appendix B). The function used to fit the profiles was the empirical model given by
Equation (8.1), used on normalised data such as that in Figure (8.2). We were
interested in fitting only the section of the profile corresponding to the inner side of
the vessel because only this section contained information about the scattering
suspension, and so the first 60 pixels in Figure (8.2) were discarded.
Data for pACy) were substituted in Equation (8.1) to fit the profiles. Equivalently, an
expression obtained by non-linear fitting of PA(Y) could also have been used:
b PA(Y)= +d
c+exp(ay) (8.17)
where y is in (mm), a= 0.0533, b= -16.0770, c= 5.3462 and d= 2.5347. The correlation
coefficient between the observed and the calculated data is R= 0.99986.
An example of an experimental profile of whole blood, fitted using the transmitted
intensity model given by Equation (8.1), andpA(y) given by Equation (8.17) is shown
in Figure (8.9).
116
~ 11)
·S .5
" CD -:t:: E 11) C ca -I-" CD .la ia E -0 Z
0.9
0.8
0.7
0.6
0.5
r
X Experimental Data .RUed Model
0.4 +-----f------+-----+------/------"":::" ---- ------------o 0.5 -----------1- ----_.- ~- 1.5 -----_ .. - -- 2 ----_. - --._._. 2.5
Path length (mm)
Figure 8.9: Normalised experimental data of the transmitted intensity for whole
blood (sample 9) and an empirical model used to fit the data, plotted versus path
length. _
Some of the profiles for whole blood and haemolysed blood were fitted as described
above, obtaining on average a correlation coefficient of 0.99. The parameters obtained
to fit the transmitted intensity, according to the empirical model, but using Twersky's
notation to facilitate a comparison, are shown in Table (8.3).
Table 8:3 CoeffICients for the non linear fitting model
haemolysed whole blood
blood
concentration "( ~ q r' po q'
0.3 0.165933 0 6.909406 0.233694 -0.75561 1.04677
0.4 0.22992 0.010768 6.921025 0.219472 -8.60723 1
0.545 0.194969 -0.00033 6.867281 0.233694 -0.75561 1.046772
117
A dye with different colour and concentration than those used for nonnalisation and
for obtaining PA(Y) was also fitted, yielding the following parameters: y"= 0.0058,
/3"=2.7736 and q"=0.7918.
From the predicted parameters is very clear that a change in concentration cannot be
quantified using Twersky's model, however this disagreement is not very surprising
since Twersky's theory is a model generated for a suspension of spheroids illuminated
with coherent light, whereas the experiment was perfonned with white light.
Furthennore, the samples of RBC in whole blood quite probably had a relatively
_______ unifonn. discoi~ s~ap~ but the h~erIlolysed samples could be better described as
suspensions of multifonn ghosts.
However the parameters in Table (8.3) can still indicate which samples are made of
whole blood and which of haemolysed blood, because parameter J3' for whole blood
is at least two orders of magnitude larger than parameter /3 for haemolysed blood and
parameter q is about seven times greater than q'. This last result may indicate that
haemolysed blood is a stronger scatterer than whole blood. Also "(' is on average larger
than y, indicating that whole blood is more absorbing than haemolysed blood, as was
expected.
Comparing now the results for the dye with respect to the blood samples, parameter "("
was found to be much smaller than either y or "(' ,. indicating that the dye is a weak
absorber. The parameter /3" was expected to be very small because the filtered dye
was regarded as a pure absorber, however it was found to be about the averaged size
of its counterpart J3' for whole blood, but parameter q" was smaller indeed than the
equivalent for the two types of blood, indicating that in general, the dye was a weaker
absorber and scatterer than the blood samples.
The experimental results just described here were presented at two conferences
[Smith, P.R. et al., (1994) and Smith, P.R. et al., (1995)]. The experimental technique
for discriminating absorbance from multiple scattering was further improved and some
118
more experiments were conducted on blood samples, the results of which can be find
in Chapter 9.
8.5 Conclusions
The technique reported here can discriminate, theoretically, whole blood
concentrations and can classify properties of the RBC suspensions via a simple and
--------- fast measurement on untreated samples. A non-linear curve fit to the model described ______ _
was capable of extracting the absorption and scattering parameters of experimental
data, clearly differentiating whole blood samples from haemolysed samples.
Twersky's model was regarded as an empirical model which can be used with partial
success to fit data from experiments such as those described in this chapter, however
other expressions involving a sum of exponentials could equally yield a good fit to the
data (as it will be shown in Chapter 9).
The lack of precision in differentiating samples with varying concentrations could be
improved if the absorbance coefficients of the dyes used for normalisation, and for
obtaining an expression for the path length were known; and also if the number of
samples with different concentration is increased. A careful manipulation and
preparation of the blood samples is a key for the success of the experiments.
119
Chapter 9
Polarised Light and Imaging Measurements of
Suspensions of Erythrocytes
9.1 Introduction
9.1.1 Measurements on Blood using Polarised Light
The first studies made on blood using polarised light, were applied in crystallography
to discriminate the molecular conformation of the blood protein haemoglobin [e.g.
Beychok, S., (1964); Sugita, Y., et al. (1968); Einterz, C.M. et al. (1985)]. Polarised
absorption or optical rotatory dispersion (the measurement of wavelength dependence
of the ability of an optically active chromophore to rotate linearly polarised light) and
circular dichroism (the difference of absorption coefficients for left and right
circularly polarised light) are techniques that are used to study the optical properties of
oriented systems, such as blood proteins. i
Unlike solutions in which light polarised in any direction is absorbed equally, because
the molecules are randomly positioned, the absorption of plane polarised light by
oriented molecules is dependent on the polarisation direction of the incident beam.
Anisotropic absorption occurs because molecules fixed in space exhibit maximum
absorption when the electric vector of the light is parallel to well defined directions in
the molecule. Measurements of optical rotatory dispersion (ORD) and circular
dichroism (CD), can provide equivalent information about molecular orientation,
absorption bands and structural conformation of a protein, and they have been used to
study the haemoglobin protein [Antonini, E. et al., 1981].
The most abUli.dant components in blood, the erythrocytes, have a particular polarising
property. When RBC have been observed under the microscope they have shown
some birefringence. A birefringent cell shows two different refractive indices in two
orthogonal directions, and the two polarised components of a polarised light beam
incident on a haemoglobin sample will travel at different speeds when traversing each
cell [Bessis, M. and Mohandas, N., 1977].
In more recent studies, propagation of light through whole blood is characterised not
only by absorption and scattering coefficients, but also by changes of the polarisation
properties of the scattered light, depending on the size, refractive index, morphology,
internal structure, and optical activity of the scatterers. Elastic light scattering can be
described by a light scattering matrix (the "S" matrix), a Mueller matrix of 16
elements. Each of the elements is a function of the wavelength, scatterers sizes, their
form and material [Tuchin, V.V., 1993]. Measurements of the polarisation scattering
matrix of RBC in terms of their diffraction characteristics has identified the relative
importance of the off diagonal matrix elements [Korolevich, A.N., 1990] and a
specific study of depolarisation orthogonal to the light propagation direction [de
Grooth, B.G., et al., 1987] has demonstrated the capability to distinguish among
different features in blood cells without having to stain them. Also the polarisation
state of photons exiting a diffuse medium can allow to distinguish between photons
which travelled a short path from those that travelled a long path within [Schrnitt, J .M.
et al., 1992] and [Morgan, S.P. et al., 1996]. This information can be used to
distinguish the scatterers from the densely absorbing background [Schnorrenberg,
HJ., et al. 1995].
121
9.1.2 An Improved Imaging Measurement Technique complemented by
Polarised Light Measurements.
The focus of this section of the work is placed on the onset and nature of polarisation
modulation, including depolarisation, as a function of RBC concentration. The
polarised light measurements are combined with an imaging approach, which studies
the absorption and scattering of light from a blood sample. When a linearly polarised
collimated beam is incident on the sample, it will re-polarise or depolarise the light
according to the amount and type of erythrocytes contained in it. Simultaneously, the
linear imaging array will detect a diffused light beam transmitted through the sample,
from which the absorbance and scattering characteristics of the sample will be
extracted.
The set of experiments described in this chapter are a development of those reported
in Chapter 8. In this case a new design of cuvette is tested and the performance of the
technique is extended by the introduction of polarised light measurements to
complement the imaging measurements. The use of polarised light adds further
degrees of freedom to these investigations. One important variation, with respect to
the experiment described in the previous chapter, is that for all the experiments that
will be described here the sample was not in motion. This happened as a restriction
imposed by the quantity of. blood required to prepare several dilutions, and
unfortunately it was not enough to make it circulate through the experimental system.
RBC were chosen as the subject of investigation using this new set up, because they
are the most abundant type of scatterers in blood and they are prone to suffer very
interesting shape changes.
Since the cylindrical cuvette is made of glass, . it is expected that at small
concentrations most of the washed healthy red blood cells will crenate even when they
are suspended in an isotonic buffer at the correct pH. However because the vast
122
majority of the cells in a single sample will take this abnormal shape, each sample can
be assumed to consist of a single type of scatterer. Also since the total volume of the
cell is maintained when the discocyte cell is transformed into an echinocyte, a
reference to an analysis made of normal cells by standard blood laboratory techniques
is still applicable [Bessis, M., 1973].
9.2 Description of the experiment
For the combined polarisation and profile measurement, as indicated in Fig (9.1), the
blood sample was contained within a cylindrical flow through glass cuvette. The path
length within the cuvette is 20 mm, measured along its largest dimension, whilst the
internal diameter of the circular cross section is 6 mm. The internal volume of the
glass cuvette is about 1.5 ml. Both ends of the cuvette have flat faces of optical
quality, but the cylindrical section was glass blown and the glass shows a few small
irregularities.
A 2 mW He-Ne laser light source was transmitted through the cuvette, having the flat
ends of the chamber parallel to the electric field of the laser. After traversing the
sample, the laser beam was also transmitted through a linear polariser, used as an
analyser, before being measured by a digital power meter. Two consecutive
measurements of a single sample were performed, one with the analyser parallel to the
polarisation state of the laser, providing a measurement of the co-polarised transmitted
intensity, and another one perpendicular to the laser azimuth, for a measurement of the
cross-polarised transmitted intensity. To avoid white light interfering with the
detection of polarised light, the white light source was blocked during this
measurement.
To obtain a profile of the chamber, the same experimental equipment as the one
described in Chapter 8 was utilised. A diffused white light source was used to
illuminate the entrance aperture of an integrating sphere and diffuse light from the
sphere was incident on the sample chamber, perpendicular to the entrance aperture of
123
the sphere. The transmitted intensity was detected by a 1024 element linear detector
array, provided with a l6-bit resolution data acquisition board and software driving
routines.
Lascr
Line scan Camem
Dillusll 1N11ite Light Source
---17 .d-..... D Powcr MetcI
1---------1 Computcr
Figure 9.1: Experimental system for simultaneous measurement of the
transmitted image and linear polarisation modulation.
9.3 Materials and Methods.
Whole blood was obtained from 8 different healthy donors. Immediately after
collection, the heparinised samples were slowly centrifuged, for about 30 minutes, to
separate the Red Blood Cells (RBC) from the plasma. The supernatant layer was
removed with a pipette and the packed cells were washed in fresh phosphate-buffered
124
saline (PBS) at a pH 7 (see the recipe for PBS in Appendix C). To wash the cells,
some PBS was added to the packed RBC and slowly centrifuged again for about 10
minutes, then the PBS and remaining plasma were removed with a pipette, but the
heparin balls were not removed from the RBC samples.
The washed cells were diluted with PBS at concentrations ranging from 0.0008 to
0.0143 ml RBC in 1 ml ofPBS. Measurements were carried out at a room temperature
of 27°C in a black-out environment, but the washed RBC were kept in ice until the
different dilutions were prepared. The complete measurement time for each dilution
was about 120 seconds.
The samples were pumped from a beaker holding the sample into the chamber and
then out from the chamber into the drain, with a slow peristaltic pump (the flow rate
was about 12 mlI min), taking care that no air bubbles were trapped inside the
chamber. The pump was switched off while the measurements took place. After
measuring a series of different RBC dilutions for a single donor, the cuvette was
decontaminated and then thoroughly rinsed with PBS.
9.4 Discussion of Results
9.4.1 Imaging Technique.
Different concentrations of RBC (Table 9.1) from 8 different donors were tested as
described in the previous section. Simultaneously, additional whole blood samples
from the same donors were tested on a Coulter Counter. The reason for having made a
set of measurements using one of the traditional instruments for blood analysis is to
provide an external reference for the results obtained by the imaging and polarisation
studies. The results obtained from the Coulter Counter analysis are shown in Table
(9.2).
125
Table 9.1: Range of concentrations used to dilute RBCfrom all donors.
Sample 1 2 3 4 5 6 7 8
No.
mlRBC 0.0008 0.0017 0.0026 0.0034 0.0043 0.0057 0.0086 0.0143
Ilml
PBS
Table 9.2: Summary of results obtainedfrom a Coulter Counter Test of each
donor's blood. HCT= Haematocrit, MCV= Mean Cell Volume, Hb=Haemoglobin.
HCT=MCV x RBC.
Donor No. RBC HCT MCV Hb
1
2
3
4
5
6
7
8
x lOe121I fl g/ dl
4.9 0.402 82 14.7
5.49 0.448 81.6 15.3
4.49 0.405 90.2 14.3
4.13 0.345 83.5 12.2
4.97 0.435 87.5 15.1
3.94 0.361 91.6 12.3
4.62 0.372 80.5 12.6
4.4 0.375 85.2 12.9 ..
An example of a sequence of line scan images for donor 8 is shown in Figure (9.2),
normalised with respect to a line scan image of PBS. It can be observed that the
absorbance of the blood sample increases with concentration, although certain
variations are apparent across the chamber, due to changes in scattering as a function
126
of path length as well as inaccuracies in the physical construction of the vessel itself.
Nevertheless the amplitude of the transmittance is a good indicator of the optical
density (OD) of the sample.
The normalised profile of a non-scattering substance (distilled water) is also shown in
Figure (9.2, curve A) and it may be noted that the absorption is less than PBS, so the
trend broadly "mirrors" the blood samples. In so far as the line scans derived from the
suspensions exceed unity it is likely that a scattering process is responsible and this
may involve the vessel as well as the RBC. There are a number of features in the line
scans of RBC, particularly at the edges of the vessel, whose shapes are modulated by
the sample concentration and which do not appear in the line scan of a pure absorber.
Sedimentation of the RBC was considered a possible cause of this effect, but the
samples remained static inside the cuvette for less than one minute, and the
measurement time was very uniform for all samples, thus the contribution of
sedimentation to the shape of the profiles should not be significant.
1.4
1.2
~
1 0.8
~ 0.6 ...
~
:§ ~ 0.4
Z 0.2
0 0 100 200 300 400 500
X Axis across Cylindrical Cuveite (A U)
Figure 9.2: Linear Scans of eight different RBC concentrations (B-1) are plotted for
Donor No. 8. All the profiles have been normalised with respect to a profile for
pure PBS. The range of concentrations listed in Table 1 correspond to the curves B·
I, whilst curve A is for a sample of distilled water. B corresponds to the least
127
concentrated blood sample, while I corresponds to the most concentrated sample.
To estimate the effective absorption characteristics of the RBC suspensions, data from
the centre of the line scan images for all donors have been extracted and are plotted in
Figure (9.3). A number of models for the light transmission characteristics have been
tested and though they are based on disparate physical assumptions (EM wave
[Twersky, V., 19701, diffusion [Ishimaru, A., 19781, flux [Kubelka, P., 1948]) their
functionality can be reduced to that of a sum of exponentials either analytically or by
approximation.
A series of three exponentials is normally quoted but the data averages considered
here are well described (correlation coefficient 0.99) by the following expression.
T = Aexp(-ax)(I+ Bexp(-bx)) (9.1)
where x is the concentration measured as a fraction by volume, A=O.6, a=57 conc·I ,
B=I, b=482 conc· l• This model was obtained by fitting empirically an equation to the
experimental data plotted in Figure (9.3). The non-linear fitting algorithm used for this
purpose was available through the non-linear fitting module of a commercial software
package, the algorithm is based on the quasi-Newton method (see Appendix B).
Following Lipowsky [Lipowsky, H., 19801 the first factor in (9.1) can be interpreted
as an absorption process and the second derives from scattering. The same author
finds the corresponding OD due to absorption and scattering to be different by a factor
of -10, consistent with the data presented here.
128
1.4
" ~ 1.2
'8 1 '" ~ 0.8
'2 0.6 '" :a 0.4 e 0
0.2 Z
0
:0:
x
0 0.005 0.01
Concentration ofRBC in 1 ml PBS
0.015
o Donor!
[] Donor2
'" Donor3 X Donor4
x DonorS
o Donor6
+ Donor7
• DonorS
-Mode!
Figure 9.3: Normalised Transmittance vs. Concentration for white light
illumination. Eight sets of data are plotted together, one for each donor. The
concentrations of all dilutions are the same as those in Table1. The path length for
all data is 0.6 cm, corresponding to the centre of the chamber. The model is an
empiricalfit to equation (9.1).
It is clear from Table (9.2) that the donors have a range of RBC concentrations and
sizes, therefore we might expect some ordering of the donor data in the experimental
results which reflect those same trends.
When linear scans of samples of identical dilutions from different donors were plotted
together, differences in OD for each are noticeable as can be observed in Figure (9.4).
The variations in the amplitudes of the curves in Figure (9.4) are consistent with the
variations in RBC and Haemoglobin (Hb) obtained by a Coulter Counter, as is
reported in Table (9.2). In this figure only the results of samples from donors 1-4 are
shown, because the alignment of the camera changed for the linear scans from donors
5-8 and the two sets cannot be compared directly. The unusual shape of curve B may
. be due to blood clots formed when the sample was prepared. Variations in the Mean
Cell Volume (MCV) of blood from different donors may produce a different
scattering signature for each sample and a consequent effect on the shape of the linear
scans however no such correlations were found in the above data.
129
1
~ 0.9
0.8 '8 '" 0.7
~ 0.6 ." 1;l
0.5 ~ § 0.4 Z
0.3
0.2
0 100 200 300 400 500
X Axis across Cylindrical Cuvette (AU)
Figure 9.4: Normalised Linear Scans of samples with same concentration (Sample
8), from four different donors. A: Donor 1, B: Donor 2, C: Donor 3, D: Donor 4.
9.4.2 Measurements with polarised light.
A more sensitive indicator of the scattering process taking place in samples with
different concentrations might be expected from measurements using polarised light.
We now consider results from the laser transmission experiments performed on the
same samples as were considered above. The co-polarised and cross-polarised
transmitted intensities from all donors are shown in Figures (9.5 and 9.6) respectively .
. 130
60
50 0 Donor!
;; [] Donor 2 ~ 40 + "- Donor3 o~ j;I.,\O
Donor 4 '"d ,., X BO 30 X DonorS --'8 ><
Donor 6 §~ 0
~ 20 + Donor 7
• DonorS "-
10 ~ --Model
0
0 0.005 0.01 0.015
Concentration ofRBC in 1 ml PBS
Figure 9.5: Co-polarised Transmitted Intensity. Eight curves are plotted together,
one for each donor. The concentrations of all dilutions are the same as in Table
(9. I}.
The averaged data in Figure (9.5) have been fitted (correlation coefficient 0.99) to the
model expression (9.1) with parameters A=8.6, a=251 cone' I , B=8, b=1721 conc· l •.
Since the cross-polarised levels are much smaller than the. co-polarised, for the
majority of the concentration range, the co-polarised level is nearly identical to the
total intensity and the data fit is therefore the same within experimental error. The
data fit is consistent with the domination of scattering phenomena over absorption.
The cross-polarised transmission characteristic exhibits a distinct maximum at ..,0.002
fractional volume of RBC for the particular path length used (20 mm). The cross
polarised levels are generally small but at the highest concentrations considered they
are about 113 the co-polarised levels, from which it may be supposed that the light is
significantly depolarised, although strictly re-polarisation is an unlikely possibility. If
one calculates the cross section for interaction between a photon and the RBC (by
131
estimating the average number of RBC in a cylindrical element of volume of the
cuvette having its diameter containing a RBC), then because of the relatively long
path length, each photon will encounter between 6 and 180 RBC across the range of
concentrations considered. Thus even though the concentrations are very low a
regime of multiple scattering exists.
0.7 [J ~ Donor!
[J Donor2
\0' 0.6 '" Donor3
, 01)
0 X X Donor4
..... 0.5 ><
~ + ....
0.4 01)
~
[J X DonorS 9 0 Donor 6 +
:I 0 + Donor7
~ "0 0.3 ~
• Donor 8 <>
--Model
'8 '" 0.2 a ~
0.1
0
0 0.005 O.oJ 0.015
Concentration ofRBC in Iml PBS
Figure 9.6: Cross-polarised Transmitted Intensity. Eight curves are plotted
together, one for each donor. The concentrations of all dilutions are the same as
those in Table (9.1).
The simplest empirical data fit (correlation coefficient 0.99) for the cross-polarised
intensity is also given, coincidentally, by equation (9.1) with parameters A=0.7, a=11O
conc'l, B=-I, b=200 1 conc -1_
From a qualitative point of view it is likely that two distinct process are taking place;
polarisation modulation due to scattering, dominant at the lower concentrations, and
re-absorption of the scattered light, dominant at the higher concentrations. This
132
significant effect is only noticed if very diluted concentrations of RBC in the buffer
solution are studied, specifically if the concentrations are varied around the region of
0.002 ml RBC in 1 ml PBS.
The relationship between the polarisation state of the incident and scattered wave at
any angle (J in the scattering plane can be described by a 4 x 4 intensity scattering
matrix S, for optically inactive scatterers. Let So ~ be the Stokes vector representing
the intensity measured by the photodetector, where the upper sign indicates the co
polarised intensity and the lower sign the cross-polarised intensity. Making no
assumptions as to the form of the scattering matrix, the co and cross-polarised Stokes
intensities are
1 ±I 0 0yl
Sl2 s13 SI4 1
I ±1 1 0 o S21 S22 S23 S24 1 S =- (9.2) -± 2 0 0 0 °t31 S32 S33 S34 0
0 0 0 o S41 S42 S43 S44 0
where the vector on the right hand side of Equation (9.2) represents a linearly
horizontally polarised beam, the matrix in the middle is the scattering matrix
representing the RBC sample and the matrix to the left of the scattering matrix is
representing a linear polariser. The choice of sign determines the orientation of the
polariser. If none of the amplitude-scattering matrix elements is zero, the scattered
wave is elliptically polarised [Schmitt, I.M., et al., 1992].
Performing the matrix multiplication and observing the first term of each resulting .
vector, the total detected intensity in each case will be given by
(9.3)
Adding the two previous equations
133
---------------------------_._----- ----
(9.4)
and subtracting them
(9.5)
Similar information to the one provided by equation (9.4) could have been obtained if
the analyser had been rotated by 1tI4 in any direction from either the cross-polarised or
the co-polarised directions to perform one extra measurement, in that case the total
measured intensity would have been
(9.6)
Many other possibilities can be explored to obtain the terms of the scattering matrix,
including changing the polarisation state of the light source but maintaining the same
alignment of the optical elements, which is not a simple procedure.
For a single concentration, the transmitted intensity measurements vary from one
donor to another, and these variations are larger for smaller concentrations. These
fluctuations can be due to changes in mean blood cell size and type or result from
experimental errors. Errors arising from sample preparation are too low to be plotted
but the effects of air bubbles trapped inside the cuvette when the sample is pumped
into the vessel can be more significant, though not measurable with the system used.
The speed at which the measurement took place would indicate that the distribution of
particles inside the container can be regarded as uniform. Instrumentation errors are
negligible on the scale of the fluctuations observed.
9.5 Red Blood Cell Morphology
134
The objectives of the present study were to observe changes in· shape and
concentration of blood samples by optical techniques, involving the measurement of
absorbance, scattering and polarisation parameters of light interacting with a blood
sample. The size and shape of the scatterers and the relative refractive index between
the scatterers and the suspending medium, in addition to the light source wavelength,
will determine the type of scattering regime. For this reason, changes in the mentioned
optical parameters will reflect changes in the morphology of blood cells.
Figure 9.7: Image of crenated red blood cells, suspended in PBS and held on a
glass microscope slide.
In normal in vivo conditions, a healthy human Red Blood Cell (RBC), or erythrocyte,
is a biconcave disc with an average diameter of 7.5 to 8.3 microns, an average
135
thickness of 1.7 IlIIl , a volume of 83 11m3 and a surface area of approximately 140 sq.
microns. Under normal conditions, the variations about the mean do not exceed more
than 5% [Bessis, M., 1973].
Erythrocytes are very flexible, they can change their shape to allow passage through
very narrow vessels, and then regain their former shape. They can also suffer
irreversible shape changes due to disease such as spherocytosis and autoimmune
haemolytic anaemia, by which they adopt a spherical shape, or in sickle cell anaemia,
where cells acquire the less flexible sickle shape [Plasek, I. and Marik,T., 1982]. A
change in size is also likely to occur through disease, such as microcytosis, but RBC
with low mean haemoglobin count (MCHC) can look smaller too, because low mean
MCHC leads to changes both in the internal refractive index and the deformability of
the cell [Hinchliffe, R.F. and Lilleyman, J.S. eds., 1987].
Other morphological changes can happen to a RBC in vitro, some of them due to the
tonicity of the saline suspending medium or to the pH of the solution (the normal pH
is 7.4). A low pH can induce a ring like appearance while a high one can provoke an
echinocyte shape (similar to a sea urchin), the last one can be also the consequence of
a hypertonic saline solution.
For the experiments described in this chapter, with samples of RBC, the erythrocytes
were centrifuged and separated from the plasma, and then re-suspended in PBS at
normal pH and correct tonicity. In order to verify that the cells had not lysed by
centrifugation, samples of various concentrations of RBC were observed under the
microscope. It was observed that the cells crenated, i. e. they were transformed from a
discocyte into an echinocyte, see Figure (9.7). While just a number of cells crenated
for high concentrations of RBC, the effect was enhanced for lower concentrations.
Only if the RBC were not separated from the plasma and if they were kept at high
concentrations, did their shape look normal when observed between plastic slides,
(Figure 9.9).
Because the coupling between the microscope and video camera used to take the
images shown in Figures (9.7 and 9.9) was not perfect, the resolution of the images is
136
very poor. Figure (9.8) is a clearer photograph showing the two types of cell.
Figure 9.8: Crenated red blood cell and two normal cells in the background.
Bessis indicated [Bessis, M., 1973 and references therein] that the basic principle
causing this change of shape is still not clearly understood, although Ponder had
identified already in 1948 the problem of a morphology change by RBC suspended in
saline when they are in contact with glass. He noticed that the effect was not
attributable to the saline itself but to the removal of a protective layer of plasma
protein when the cells are washed in the saline, because when the cells are returned to
fresh plasma they regain their normal shape. If the cells are washed several times, they
can crenate even when placed in contact with plastic. Bessis also mentioned that close
contact with glass or a high ratio of glass to red cell surface area is necessary to
137
produce the change, since exposure to the glass in a test tube is insufficient. He adds
that the "glass effect" is probably due to the elevated pH( > 9) between glass slides
and coverslips. On the other hand, George et al. [1971] presented evidence that
adhesion of erythrocytes to glass depends upon non-polar forces but that the
diminution of adhesiveness by serum or plasma depends upon electrostatic forces
acting on the serum and plasma factors.
Figure 9.9: Image of red blood cells, suspended in PBS and held on a plastic
microscope slide.
We observed that cells washed only once in PBS undergo crenation when they are
placed between plastic slide and coverslip if the concentration of the cells is about
0.015%.
Most of the optical methods for blood analysis make use of glass containers, which
normally have short path lengths, and in many caSes whole blood or blood cells are
138
diluted down to small concentrations, so RBC are very likely to crenate. For the
purpose of light scattering studies, this change of shape is highly undesirable although
the total volume of the cell is maintained, because the scattering signature from the
two types of cells is very different, and the theoretical solution to the echinocyte
scattering problem has yet to be found.
9.5 Conclusions
We have found that the study of low concentrations of RBC with polarised light is of
significant importance, because it allows the observation of the relative weight of the
absorbance process in the sample versus the scattering modulation, which otherwise
cannot be distinguished from each other in an imaging measurement of the suspension
alone. We have identified a peak in the cross-polarised intensity response with
concentration, at -0.2 % by volume, for a path length of 20 mm. However the
polarised light measurement made here is unable to determine what proportions of the
effect of the sample on the polarisation state of the incident light are due to
depolarisation, modulation due to scattering or optical activity. For this reason it is
necessary in future experiments to incorporate a polarimeter in the experimental array,
that can provide information about the Stokes' parameters describing the beam
transmitted through the sample and hence further details of the scattering matrix.
An abnormal morphology of red blood cells was identified when red blood cells free
of plasma are placed in contact with a glass container. Although this is not the first
time that this anomaly has been reported in the literature, this is the first time that the
problem has been addressed in the context of studies assessing the absorption and
scattering properties of blood.
Some attention must be paid to the selection of an appropriate cuvette, which requires
a very short path length to allow for high concentrations of RBC (Le. undiluted whole
blood) but at the same time ·its cross section must be large enough to prevent
undesired reflections from its walls. The choice of material is also critical, because it
139
has to be optically inactive and at the same time should not affect the normal
appearance of the cells; an important condition for tests leading to potentially non
invasive blood measurements.
140
Chapter 10
Conclusions and Suggestions for Further Work.
10.1. Conclusions
A polarimeter of general purpose has been designed, constructed and· tested. The
novelty of its design lies in the fact that, the theory developed to extract
simultaneously the complete Stokes parameters from only one set of intensity
measurements, is a generalisation to the algorithms used by all the other authors. This
polarimeter allows the polarisers and quarter wave retarder in the sensor head to be
positioned at arbitrary azimuth angles, while all other polarimeters of the same kind
require fixed predetermined angles. This feature makes the polarimeter presented here
more flexible; and because there are no moving parts, the angles can be found with
better accuracy. Furthermore, because only one set of intensity measurements is
required, results using this polarimeter can be produced faster than in those requiring
to re-calibrate at various different settings of the optical elements involved.
A calibration routine was designed to improve the accuracy of the measurements and
some suggestions were made to enhance the performance of the software and
hardware tools accompanying the polarimeter sensor head, so it can be used for
applications in various fields, such as optical measurements of blood.
The cost of producing a sensor head for the Division of Wavefront Polarimeter is
relatively low, because the most expensive parts are the three polarisers and the
quarter wave retarder. Also all mechanical parts can be easily manufactured.
To test the applicability of polarised light techniques for the analysis of blood
samples, some experiments were conducted by varying the concentration of red blood
cells in the samples and also by lysing the cells. It was found that the techniques
developed here can distinguish among different concentrations of the sample.
Polarised light measurements are a useful tool to discriminate among those
concentrations of the sample where the light absorbing process is weaker than the
scattering process. These observations of the polarisation scattering properties of red
blood cells have established the principle for future invasive and non-invasive sensor
technologies but the most effective configurations of such sensors are yet to be
established.
Another important contribution of these thesis is that while performing the
experiments on blood using polarised light, I observed that washed RBC suffered a
distortion of their normal appearance when they are placed in contact with a glass
container, but to this date no other author had pointed out this problem, or attempted
to avoid it, within the context of the optical analysis of red blood cells. I also observed
that cells washed only once will crenate if their concentration is about 0.015% or less,
even when they are placed in contact with a plastic surface.
Suggestions have been included to improve the performance of the developed
polarimeter and of the experimental techniques for optical analysis of blood.
10.2 Suggestions for Further Work.
10.2.1 Suggested Modifications to the Mechanical Design of the Sensor Head.
When the DOWP is used to test a sample with a low transmittance, the light source
has to be very intense, so it can be detected by the photodiodes in the DAS after it has
traversed the sample and the polarising optics in the sensor head. It may be possible to
reduce the thickness of the sample, however this is not always feasible and for this
reason a new probe with a smaller cross section should be considered.
142
small PIN diode __ ~'V'--'
Figure 10.1: Proposed Design for an Improved Sensor Head.
If the cross section of the sensor head is small, when the light source is expanded and
collimated, less power is necessary to evenly illuminate the sensor head. Also, because
achieving a uniform illumination of the sensor head is very important, a number of
smaller photodiodes could be added to the polarimeter to be located at strategic
positions (see Figure 5.6); then from the readings of these extra photodiodes a map of
intensity variations across the sensor could be constructed and incorporated into the
algorithms that calculate the Stokes Parameters. In this way variations of intensity
across the sensor can be compensated for and the polarimeter will be immune to a
non-uniform illumination of the sensor head.
An additional advantage of this improved design is that optical fibres are not
necessary, because the receivers are positioned directly in the sensor head. This
improvement makes the sensor more robust because there is no danger of disturbing
the fibres when the sensor head or other pieces of the experimental set-up are moved
around.
143
10.2.2 Improvements to other components of the DOWP.
The DOWP described in the previous chapter can be improved by upgrading some of
the equipment that is external to the polarimeter itself, but that is also needed to use
the instrument as a measuring tool. Because some of these pieces of equipment were
not constructed or acquired having necessarily a polarimetric application in mind, they
could be replaced by others with specifications matching more closely the
requirements of the polarimeter.
One of the pieces which demands close attention is the data acquisition system. This
system must be provided with photodiodes optimised to work at the same wavelengths
as the light source and the quarter wave retarder. At present the calibration procedure
of the DOWP (Section 5.1) assumes that because the offsets in the photodiode
readings remain constant throughout a complete experiment, it is enough to measure
them just before an experiments starts. In reality these offsets vary by a small amount,
depending on the time the equipment has been functioning. Then in order to provide
even more accurate measurements of a sample, the offsets should be measured also
during the experiment, so they can be removed from the actual measurement of the
sample. The analogue signals from the photodiodes should be linearly amplified and
any source of noise should be filtered before the signals reach the multiplexer.
The accuracy of the analogue to digital converter is also of great importance, as was
mentioned previously, and the more accurate it can be, the better. If the DOWP is
intended to be used for example, in biomedical applications, at least an ADC of 16
bits is necessary. For example, one of the optically active substances present in the
human body requiring careful monitoring is glucose, so a polarimeter should be able
to measure variations in the concentration of this substance. The specific rotation of
any substance is defined as:
[(XlI = (100!x) I Le (10.1)
where", is the wavelength, T the temperature, a is the observed rotation in degrees, L
the optical path length in decimetres and C the concentration of the sample in grams
144
per 100 ml of solution. The specific rotation of glucose is 41.8, at a temperature of
20°C and illuminated with a source of 633 nm wavelength [Cote, G.L. et al., 1990].
Therefore the polarimeter must have enough accuracy to detect variations in the
polarisation azimuth of 0.004°, if the concentration of glucose is 100 mg/100 ml and
the optical path length is 1 cm.
At present, using an ADC with only 12 bits, together with the noise problems from the
amplifier, the accuracy of the DOWP is at least two orders of magnitude away from
the required accuracy to sample substances such as glucose. Although the polarisation
measurement shown in Figure (6.13) seems to have a small error, it is achieved by
averaging 500 measurements of a single sample, and the time required to obtain the
500 measurements and produce the averaged result is at least of 10 minutes, which is
an unrealistic acquisition time for testing biomedical substances with polarisation
properties rapidly varying in time. Real time acquisition is another issue that must be
taken into consideration. The data acquisition system must sample data in real time, to
be able to store in RAM a large number of samples and then to communicate
efficiently with the personal computer.
The light source must be non-polarised and the optics taking the light to the sample
and sensor head must not modify the polarisation state of the source. If this can not be
accomplished, a theoretical model must be produced of the polarising effects external
to the sample which must be incorporated into the calibration stage. Furthermore, to
distinguish light source fluctuations from intensity variations due to the sample,
stabilising the light source power supply and keeping track of source fluctuations may
prove useful.
All the possible improvements mentioned above are not applied to the polarimeter
itself, however another source of error directly related to the polarimeter and that
could be improved, is the measurement of the angles and absorbance parameters of the
optics in the sensor head. Up to now, the angles were measured by rotating a linear
polariser and doing a non-linear fitting to the intensity curves, resulting from plotting
intensity levels versus azimuth angle of rotation. The accuracy of this measurement is
145
closely related to the resolution of the motor rotating the polariser, so the size of the
step by which the polariser is rotated should be minimised.
Another possibility that could be explored, to reduce the sensitivity of the Stokes
parameters to quantisation noise, is to find which set of azimuth angles of the
polarisers and retarder minimise the error in the determination of the polarisation
azimuth and ellipticity. Any set of angles can be used, because the algorithms
developed for this polarimeter to derive the Stokes parameters from four intensity
readings, allow arbitrary azimuth angles. The only condition is that they all are
different between them. Solving this theoretical problem may be cumbersome,
because the derivative of the ellipticity must be calculated and it is a very long
expression, although once it is solved, the real difficulty would be to set the polarisers
and retarder at the required positions.
If all the suggestions made above can be accomplished successfully, the performance
of the polarimeter may be enhanced greatly and it could be used then to perform
polarimetric measurements of highly absorbing samples, based on principles such as
those that were discussed in Chapters 8 and 9.
An extra line of research that could be explored, as an academic exercise, is the
. derivation of the Mueller matrices that describe non-ideal LP and QWR. If this is
done, the theoretical model for the intensity transmitted through a QWR followed by a
LP. (Equation 3.36), could be obtained very easily from the multiplication of both
matrices times a Stokes vector describing a general polarisation state.
10.2.3 Suggested modifications to the glass containers used in blood experiments.
For the reasons mentioned in Chapter 9, one of the aspects of this research project that
would benefit from further improvement is the usage of an appropriate biopolymer
coating the interior of the glass cuvette, intended to hold the RBC samples during
146
measurements. The biopolymer must be layered over the glass in a very uniform film,
because any imperfections will cause scattering and diffractive problems.
Unfortunately, not many people have worked in the study of this solution, apart from
Ponder [Ponder, 1964], who tried to use gelatine at minimum concentrations to
reduce the adhesiveness of the cells to the glass surface. Most of the authors trying to
solve problems of biocompatibility focus their research on blood coagulation as a
result of blood interacting with artificial surfaces. The fundamental causes of damage
in blood cells induced by foreign surfaces [Keller, K. in Salzman, E.w. ed., 1981], are
still not clearly understood
Not every choice of biopolymer is suitable for experiments involving polarised light,
because polymers normally are some form of polarising elements whose polarising
properties are not very uniform along its surface, and that will affect the
measurements. An ideal choice of polymer would be one producing a linear uniform
polarisation state, so a distinction between the polarisation state introduced by the
sample and the polarisation state introduced by the container could be attempted. The
solution of this problem is very important for optical studies of RBC morphology
because not only healthy RBC suffer a change of shape, also abnormal cells like sickle
cells show a dramatic change into an echinocyte form, making difficult the
observation of different natural occurring cell pathologies.
147
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152
Appendix A
Technical Drawing of the DOWP Sensor Head
Drawing made by P. Barrington at Loughborough University of Technology.
153
AppendixB
Non-Linear Fitting to the parameters, the QuasiNewton method.
In general terms, the requirement for using a non linear fitting method was aroused by
the need to fit a function of n number of variables and m number of parameters to a set
of experimental· data. The best fit to the data will be obtained when the set of
parameters is such that the error between the predicted values and the experimentally
observed values is minimum. The fitting problem is basically a problem of finding the
minimum of a surface generated by plotting the errors in parameter space.
The quasi-Newton method is a variation of the Newton method. If the surface just
described has a minimum point, in the close neighbourhood to that point the surface
can by approximated by a quadratic surface. The Newton method works on the
assumption that if the first and second derivatives exist and the function is quadratic,
then the position of the minimum can be predicted in a single step. The description of
the method can be found in [Massara, R.E., 1991].
Let <!i{,!) be a function in the space of parameters,! = xl, x2, ... , xn. If the point,!. * is a stationary point, then <!i{,! *) on that point must satisfy the condition
(5.11)
and if this stationary point is a minimum
(5.12)
Moving from the stationary point an increment Ll,!, the new point can be written as
(5.13)
The change in the function introduced by this increment can be studied expanding in
Taylor series the function «ll(,!) evaluated in this new point ,!. For the case of one
variable
(5.14)
but for a function of n variables, the above expansion in vector form results:
(5.15)
where g is a gradient vector of the first n partial derivatives of «ll(,!) with respect to Xi,
T indicates transpose, and H is the n x n Hessian matrix of the second partial
derivatives. H is symmetric if «ll(,!) is continuos. In this case,
a 2«ll( X) a 2«ll( X) axjJXj aXjJXi
(5.16)
assuming the increment .1.,! is small, is then justified to discard terms of higher order
than 2 in equation (5.15).
If the stationary point at ,! * is a strong local minimum «ll(,!) «ll(,! *) , then
(5.17)
and
(5.18)
inequality (5.18) is satisfied if the Hessian H is positive definite. The practical
significance of a positive definite H matrix is that the local form of the function «ll(,!)
155
is that of a quadratic surface. A Hessian is positive definite if all n principal minors
are positive. i.e. if Di > O. for all i. The principal minors are the following:
D - ~l ~2 D. = IHI 2 - ~1~2 •••••••••• (5.19)
Let .J.' be a minimum point and l£Oa point in the neighbourhood of l£ *. The first
derivative of <lJ{l£) in matrix form is
(5.20)
This equation provides a prediction of the gradient at .J.. given the gradient and
Hessian at l£O. Since dl£ is required for a shift to the minimum .J.' therefore g(l£ *) is
null. hence
(5.21)
so that
(5.22)
using
(5.23)
This equation forms the basis of Newton method: the predicted stationary point l£ * will be a minimum if H is positive definite. For a general non-quadratic function. l£ * can only be viewed as an approximation to the actual minimum of <lJ{l£) and the
following iterative algorithm. derived from Equation (5.23) is applied
(5.24)
156
-- - - -----------
The tenn in brackets in this equation is regarded as a search direction. A scalar a 'is
detennined according to some strategy defined by the user. This scalar controls the
possibility of divergence that can often occur.
The quasi-Newton method is based on the Newton technique, but avoids the explicit
evaluation of the Hessian and its inversion. Instead it uses an approximation to the
inverse Hessian. Thus HI is replaced by H' representing an approximation to HI
after r iterations. This approximation is updated at each iteration in such way that:
(i) H'+I is positive definite providing H' was positive definite.
(ii) The sequence of matrices H', r=0,1,2, ... tends to the value of HI (the actual
inverse Hessian matrix).
An efficient optimisation process should only use the function and the first-derivative
infonnation, and guarantee convergence. Thus to evaluate the current search direction
d', a positive definite matrix H' must be used, equivalently to equation (5.22).
d'=-H'g', (5.25) .
and the next point, "r+1, is detennined as
xr+1=xr+a rd r - - , (5.26)
d' in equation (5.25) is guaranteed to be a locally "down hill" direction if H' is
positive definite. Convergence is assured if the scalar a ' is chosen to at least reduce
CP, which simply requires that a ' is not too large.
When the set of parameters that minimise the function are found, predicted data are
calculated and compared to observed data. Any deviation means some loss in the
accuracy of the prediction, so many non-linear regression models use Least-Square
Estimation procedures to minimise a loss function. Specifically, the loss function used
in our problem was defined as the sum of the squared deviations about the predicted
values.
157
- AppendixC
Normal Haematology Values
Normal haematology values for adults, quoted by the Leicestershire Haematology
Service (UK) are the following:
Hb 13.5-18.0 gldl male 11.5-16.5 gldl female
Total Haemoglobin Concentration: This test measures the grams of haemoglobin in
lOOm! of blood, which can help to diagnose the severity of anaemia or polycythemia.
RBC 4.5-6.5 x 101211 male 3.9-5.6 x 101211 female
RBC (Red Blood Cell Count): Counts the number of red blood cells in a single drop (a
microliter) of blood. A low RBC count may indicate anaemia, excess body fluid, or
haemorrhaging. A high RBC count may indicate polycythemia or dehydration.
HCT 0.4-0.54 male 0.37-0.47 female
Haematocrit: Measures the percentage of red blood cells in the sample.
Erythrocyte (RBC) indices:
MCV 80-99 fl
MCV: Mean Corpuscular Volume measures the volume of red blood cells.
MCH 27-32pg
MCH: Mean Corpuscular Haemoglobin measures the amount of haemoglobin
in an average cell.
MCHC: Mean Corpuscular Haemoglobin Concentration measures the
concentration of haemoglobin ill red blood cells. Normal is 30% to 36%.
WBC 4-11 X 109/1
WBC: White Blood Cell Count measures the number of leukocytes in a microliter
(drop) of blood. Normal values range from 4,100 to 10,900, but can be altered greatly
by factors such as exercise, stress and disease. A low WBC may indicate viral
infection or toxic reactions. A high WBC count may indicate infection, leukaemia, or
tissue damage. An increased risk of infection occurs once the WBC drops below
I,OOO/rnl.
WBC Differential: Determines the percentage of each type of white blood cell in the
sample. Multiplying the percentage by the total count of white blood cells indicates
the actual number of each type of white blood cell in the sample. Normal values are:
Neutrophil 50-60%
Eosinophils 1-4%
Basophils 0.5-2%
Lymphocytes 20-40%
Monocytes 2-9%
Reticulocytes 0-2 %
Platelets 150-400 x 109/1
Platelet Count: Measures. the number of platelets in a drop (microliter) of blood.
Platelet counts increase during strenuous activity and in certain conditions leading
infections, inflarnmations, malignancies and when the spleen has been removed.
Platelet count decreases just before menstruation. A count below 50,000 can result in
spontaneous bleeding.
P.V. 1.5-1.72 cp
159
Recipe For Phosphate-Buffered Saline
Equal volumes must be used of iso-osmotic phosphate buffer and 9 gIl NaCl. For
serological tests, a pH 7.0 buffer is recommended [Dacie, J.V. and Lewis, S.M.,
. 1984].
To prepare iso-osmotic phosphate buffer, pH 7.0, use 32 ml of solution A and 68 ml
of solution B.
(A) NaH2P04.2H20 (150 mmoIlI)
(B) NaH2P04 (150 mmoIlI)
23.4 gIl
21.3 gIl
Recipe for Ringer Solution - 1/4 strength
One tablet makes 500 ml of 114 strenght Ringer solution. To prepare it dissolve 1
tablet (available from major chemical products manufacturers) in 500 ml of distilled
water and sterilize by autoclaving at 121 °C for 15 minutes (optional).
160
