Polarization acquisition using a commercial Fourier transform spectrometer in the MWIR

Polarization acquisition using a commercial Fourier transform spectrometer in the MWIR

Michael W. Kudenov, Nathan A. Hagen, Haitao Luo, Eustace L. Dereniak, Shawn Robertson,

Leonardo G. Montilla, Tom B. Vo, Justina Tam, Julia D. Nichols, College of Optical Sciences / The University of Arizona.

Grant R. Gerhart, U.S. Army Tank-Automotive Research, Development and Engineering Ctr.

ABSTRACT A spectropolarimeter utilizing an Oriel MIR8000 Fourier Transform Spectrometer in the MWIR is demonstrated. The use of the channeled spectral technique, originally developed by K. Oka, is created with the use of two AR coated Yttrium Vanadate (YVO4) crystal retarders with a 2:1 thickness ratio. A basic mathematical model for the system is presented, showing that the Stokes parameters are directly present in the interferogram. Theoretical results are then compared with real data from the system, an improved model is provided to simulate the effects of absorption within the crystal, and error between reconstructions with phase-corrected and raw interferograms is analyzed. Keywords: Yttrium Vanadate, spectropolarimeter, Fourier transform spectrometer, channeled spectrum.

1. INTRODUCTION The use of a Fourier Transform Spectrometer (FTS) has many advantages over other forms of spectroscopic measurement, such as the use of a diffraction grating, in the MWIR. Predominantly, the FTS maintains the throughput (Jacquinot) and multiplex (Fellgett) advantages.1,2 Yet aside from this, the FTS has another unique benefit for working with the channeled spectropolarimetric technique implemented by K. Oka3; the FTS operates directly Fourier space through its interferometer. This is important because it provides a more direct method of acquiring the polarization data directly from the interferogram. Consequently, this provides another advantage for the FTS in terms of data processing, although it may not be so reliable to do without phase correcting the interferogram. Here, we describe some aspects of our proof of concept experiments. First, we will describe the theoretical model behind the spectropolarimetric technique with an FTS. This will be followed with our laboratory results by means of a commercial FTS in the MWIR using Yttrium Vanadate (YVO4) crystal retarders. Detailed error analyses will be provided, and an improved system model will be discussed and simulated. Lastly, the effect of phase-correction on the reconstructions will be analyzed and compared to non phase-corrected results.

2. SYSTEM MODEL The basic setup of the Fourier Transform Spectropolarimeter (FTSP) can be seen in figure 1. It consists of two high-order retarders R1 and R2 with thicknesses d1 and d2 followed by an analyzer. The orientation of the retarders, with respect to their fast axes relative to the transmission axis of the analyzer, is 0° and 45° for R1 and R2 respectively. The output is then fed into an interferometer, where it is read and reconstructed on a computer.

Figure 1. Basic FTSP block diagram.

Infrared Detectors and Focal Plane Arrays VIII, edited by Eustace L. Dereniak, Robert E. Sampson,Proc. of SPIE Vol. 6295, 62950A, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.688401

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. ..

H1 R2 A

+: *1T

Detector

Fixed Mirror

To create a theoretical model of the aforementioned system, we use plane-wave propagation in a typical Twyman-Green interferometer to observe the functional form of the interferogram. The system in figure 2 was utilized for this task, where the interferometer consists of the beam splitter, moving and fixed mirrors, and the detector. The two retarders and analyzer are then placed at the input, and a spectrally dependent polarized electric field is incident upon them.

Figure 2. Twyman-Green interferometer used for the theoretical derivation of the interferogram.

Beginning by defining the terms used for the analysis, we have for the phase retardances φ1 and φ2 in R1 and R2,

1 1

2 2

( ) 2 ( )

( ) 2 ( )

B d

B d

φ σ π σ σφ σ π σ σ

==

(1)

where σ is the wavenumber, B(σ) is the birefringence, and d1, d2 are the aforementioned retarder thicknesses. Assuming arbitrarily polarized light at the input, defined as,

( )

0

( )

0

( ) ( )

( ) ( )

x

y

j jx x

j jy y

E E e e

E E e e

δ σ τ

δ σ τ

σ σ

σ σ

=

= (2)

where Ex and Ey are the field components along the x and y axis, respectively; Ex0 and Ey0 are the field amplitudes, δx and δy are the initial phase differences, and τ is the propagator, t kzτ ω= − (3) In the rest of this analysis, the propagator is neglected but implied; yet it is included here for completeness. Propagating these fields through the two retarders and analyzer produces the following electric field parallel to the transmission axis of the analyzer,

' 0 1 2 0 1 2cos( 2 )cos( 2) sin( 2 ) sin( 2)x x x y yE E Eδ φ τ φ δ φ τ φ= + + − − + (4)

where each term is implicitly dependent on σ. This is the effective input to the Twyman-Green interferometer. We now define a new term for the phase imparted by the moving mirror, ( ) 2z zφ σ π σ= ∆ (5)

where ∆z is the optical path difference (OPD). Ignoring non-ideal effects from the beamsplitter and mirrors, interfering equation 4 with a phase-delayed version of itself yields the electric field at the detector,

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OPD

-cas-)s-4-ô,)1 c:) r -S(-q-)-Sin(

' '

0 1 2 0 1 2

0 1 2 0 1 2

cos( 2 )cos( 2) sin( 2 )sin( 2)

cos( 2 ) cos( 2) sin( 2 )sin( 2)

zjd x x

d x x y y

x x x y y x

E E E e

E E E

E E

φ

δ φ τ φ δ φ τ φδ φ φ τ φ δ φ φ τ φ

∝ + →∝ + + − − +

+ + + + − − + +

(6)

Integrating over time to find the effective intensity yields,

( )( )2 2 2 2

0 0 0 0 2

0 0 1 2

0 0 1 2

1 1( ) ( ) cos

2 2( ) 1 cos( ) cos( )sin( )sin( )

sin( )cos( )sin( )

x y x y

x x y y x

x y y x

E E E E

I E E

E E

φ

σ φ δ δ φ φδ δ φ φ

⎡ ⎤+ + − +⎢ ⎥⎢ ⎥

∝ + − −⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

(7)

Now, reference to the definition of the Stokes vectors4 reveals that equation 7 can be rewritten in terms of S0, S1, S2, and S3 as,

[ ]0 1 2 2 1 2 3 1 2

(1 cos( ))( ) cos( ) sin( ) sin( ) cos( )sin( )

2xI S S S S

φσ φ φ φ φ φ+∝ + + − (8a)

Further expansion of this equation reveals the presence of seven channels, each shifted in the interferogram by an amount proportional to the magnitude of the retardances φ1 and φ2,

( ) ( )

[ ]

[ ]

0 12 2

21 2 1 2 1 2 1 2

31 2 1 2 1 2 1 2

cos( ) cos( ) cos( )2 4

cos( ) cos( ) cos( ) cos( )8

sin( ) sin( ) sin( ) sin( )8

x x x

x x x x

x x x x

S SI

S

S

σ φ φ φ φ φ

φ φ φ φ φ φ φ φ φ φ φ φ

φ φ φ φ φ φ φ φ φ φ φ φ

= + + + − +

+ + + − − − + − − − + +

+ + − − − − + − + − +

(8b)

This can be seen schematically in figure 3. Demodulation of the spectral Stokes vectors can now be done by filtering the desired channels, followed by a Fourier transformation. Performing this on the channels labeled C0, C2, and C3 per figure 3 yields,

Figure 3. The 7 channels in the interferogram are separated in OPD space by the retardances φ1 and φ2 for a 2:1 thickness ratio (d2:d1).

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( ) ( )

( ) ( )

( ) ( ) ( )( )

2

1 2

0 0

22 1

2 23 2 3

1

21

41

8

j

j j

F C S

F C S e

F C S jS e e

πφ σ

πφ σ πφ σ

σ

σ

σ σ

−

−

=

=

= − +

(9)

Hence, by measuring a known state of polarization (SOP), the modulating phase factors in C2 and C3 can be calibrated out and the unknown values reconstructed.

3. METHODOLOGY

The theoretical model indicates that channeled spectropolarimetry can be done with an FTS through the interferogram, so we configured the MWIR FTS in our lab for this task. The FTS equipment that is used for our experiments is an Oriel MIR8000 with an 80026 MCT detector and 80007 MWIR source. The crystal retarders are made from YVO4 and, according to the vendor, are AR coated for 4 µm. The thickness ratio used is 2:1 for d2:d1 where d2 = 4 mm and d1 = 2 mm. The experimental setup can be seen in figure 4. It consists of a wire-grid generating polarizer (G) to allow known SOP input for calibration and sample data measurements. This is followed by the two retarders (R1 and R2), wire-grid analyzer (A), and finally the FTS. 3.1 Data collection The data collection process involves placing G at known orientations from 0° to 180° in 5° increments relative to the transmission axis of A and measuring the interferogram. After reconstruction, the data are then normalized to the S0 component in order to analyze the accuracy of the output compared to the theoretical input states. The primary limitation to this setup, however, is that it does not allow us to perform a detailed error analysis on the S3 component, other than that it should be equal to zero. Hence for S3, we analyze its reconstruction accuracy based on the RMS error. Lastly, for all data collection, the FTS was internally purged with Nitrogen gas to reduce the CO2 absorption line, and the external source was purged to the best of our ability.

Figure 4. (Left) The experimental setup consists of the MWIR source, generating polarizer G, the two retarders R1 and R2,

the Analyzer A, and the FTS with the MCT detector. (Right) An image of the experimental setup.

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120

IOU

80

80

3.8 4 4.8 8 3.8 4 4.8 8Wavenumher (gym) Wavelength (gym)

Wavelength (gym) Wavelength (gym)

4. RESULTS & ERROR The reconstructions obtained for the Stokes parameters from the aforementioned procedure of rotating G from 0° to 180° in 5° increments can be seen in figure 5 for S0, S1, S2, and S3, along with an interferogram obtained with G oriented at 22.5° to show verification of the seven channels per figure 6. In these data, the S1, S2, and S3 components have been normalized by the S0 component. Additionally, it should be noted that the S0 component has been normalized to the calibration spectrum to give an impression of how S0 changes with respect to it. Lastly, all reconstructions have a spectral resolution of approximately 30.5 cm-1 (FWHM). 4.1 Data trends If we begin by analyzing the trends in the data from figure 5, we notice a slight disturbance at ~ 4.25 µm primarily in the S2 reconstruction. This is being caused by aliasing from the CO2 absorption line, which is changing during the course of the experiment. Consequently, since the calibration reference angle is fixed at 22.5°, the reconstruction suffers more as one progresses further away from this location. It is therefore apparent that better and more consistent nitrogen purging outside the FTS system around the sample would further reduce the error associated with this phenomena. Additionally, further examination of the data yields a second set of trends located in the reconstruction from 4.9-5 µm and 3.75-4.2 µm for rotation angles of 45° to 135°. To highlight these areas, the percent error in S1 and S2 can be seen in figure 7, where higher values are present around the angles previously indicated. After significant testing, this error in the data was pinned down to a defect in the AR coating of the retarders, which is enhanced by the crystal’s absorption.

Figure 5. Reconstruction of the data from the FTS after phase-correction of the interferogram. This phase correction was achieved by the Mertz phase correction method, and the interferogram was reconstructed by taking the real part of the Fourier transform of the double-sided phase corrected spectrum.

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Phase corrfld Inthrf&on, Oat 22.t0.06

0.05 - -

0.04 - -

0.03 - -

0.02 - -

0.01 - -

0 --—-0.01 - -

-0.02 - -

-003-2.5 -2 -1.5 -I -0.5 0 0.5 I 1.5 2 2.5

OPO (mm)

WveIength (tm) WveIength (tm)

Figure 6. Mertz phase corrected interferogram obtained from the experiment for G at 22.5°. All seven channels are present. The reasoning behind the phase correction is discussed in section 4.3.4.

Figure 7. Percent error for the reconstruction of S1 and S2 in 10° increments. The error is high for regions where S1 and S2 are small, especially around 4 µm. In the case of the above data, the percent error for S2 at 0°, 90°, and 180° is undefined and has therefore been set to zero.

4.2 Retarder defects To test the retarders, an experiment was done to test the percent transmission of the fast and slow axes. First, the linear polarizers G and A were aligned so that their transmission axes were parallel, and a reference spectrum was obtained. Each retarder was then independently inserted in-between G and A, initially with the fast axis parallel to the transmission axes. The spectrum was measured, the percent transmission calculated, and the procedure was repeated for the slow axis. This test was also completed on a non-AR coated YVO4 retarder to provide a baseline to reference the results to. The percent transmission from this experiment for retarder R1, including the percent differences in transmission between each axis, can be seen in figure 8. The transmission for R2 is not shown, but it is essentially the same as figure 8 from 3-3.75 µm and has the same trends that correspond roughly to the square of the R1 transmission from 3.75 – 5 µm.

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F..t •nd 'low •xI• trn•mI••Ion p,rc,rg.g,, Pircint dlIILinc• bitwi•n thi HI •nd ilow •xI• trn•mI••Ion•

Fast-Axis (AR)lI ——Fast-Axis(NoAR)

—-—Slow-Axis(AR)Slow-Axis (No AR)

3.2 3.4 3.8 3.8 4 4.2 4.4 4.8 4.8 8

WveIen2th (tm) WveIen2th (tm)

Interferometer

so() R1 p1 R2. P2 D

Lf÷ sXf ___ +

3 Computer

Figure 8. (Left) Percent transmission of R1 and the non-AR coated YVO4 retarder. (Right) Percent difference between the fast and slow axis transmission for the AR coated and non-AR coated samples, as well as a scaled overlay of the S1 error at 140° to show how well its trends correspond to the percent difference’s lineshape. As can be seen, the AR coating is creating a larger percent difference between the fast and slow axes from 3.75-4.2 µm and 3-3.1 µm compared to the non-AR coated sample. Additionally, the AR coating shows little to no effect on the difference from 4.9-5 µm.

This defect is causing both retarders to act as weak polarizers, contributing error predominantly to S1 due to retarder R1’s angular orientation with respect to R2 at 45° (R2 is solely responsible for measuring S1), meaning the defect is given maximum modulation by R2 into the S1 channel. Error is then introduced into the S2 and S3 reconstruction due to its required normalization by S0, from which the defect extracts its energy. Hence, one needs to be careful about the quality of the AR coating. Aside from that, even if the AR coating were made ideal, there may still be error located in these spectral regions due to the significant percent transmission difference seen in the non AR coated sample (most likely due to unavoidable absorption inside the crystal). Hence, we are currently looking into methods of improving the quality of the reconstruction with a more thorough system model and different calibration techniques. 4.3 Improved system model Although our improved system model is still in its infancy, we provide it here as a reference to our starting point. In the meantime, it allows for a more thorough simulation of the system when computing the theoretically measured output. We begin by expanding the system model by including two partial polarizers behind R1 and R2 with fast axes transmissions px1, px2 and slow axes transmissions py1, py2, respectively. This improved model can be seen in figure 9. Now, from the theoretical model, we can ascertain that the interferometer simply measures the S0 output from the retarders and modulates it by (1/2)(1 + cos(φz)). Performing the Muller calculus on R1, P1, R2, P2, and A, then extracting the S0 component and multiplying it by our phase factor from the moving mirror yields the effective output of the interferometer per equation 10,

Figure 9. Improved system model layout. Partial polarizers with transmissions px1, px2, py1, and py2 have been included to simulate the effects of the varying transmission on the reconstructed results.

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( ) ( ) ( )

( ) ( ) ( )

( )

2 2 2 2 2 201 1 2 2 2 2 1 1 2

2 2 2 2 2 211 1 2 2 2 2 1 1 2

2 221 1 2 2 1 1 1 2 2 1 2

231 1 2

1cos( )

4 2

1cos( )

4 2(1 cos( ))( )

2 1cos( ) sin( ) sin( )

2 2

1

2 2

x y x y x y x y

x y x y x y x y

x

x y x y x y x y

x y x

Sp p p p p p p p

Sp p p p p p p p

IS

p p p p p p p p

Sp p p

φ

φφσ

φ φ φ

⎡ ⎤+ + + − +⎢ ⎥⎣ ⎦

⎡ ⎤− + + + +⎢ ⎥+ ⎣ ⎦∝⎡ ⎤− + +⎢ ⎥⎣ ⎦

−( )22 1 1 1 2 2 1 2sin( ) cos( ) sin( )y x y x yp p p p pφ φ φ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪

⎡ ⎤⎪ ⎪−⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

(10)

Analysis of this equation yields three key results:

1. The S0 and S1 components are transferring energy between each other in spectral locations where there are differences between px1 and py1, as can be seen from the difference terms in the first and second lines.

2. Portions of S2 and S3 are directly modulated by φ1, and consequently their energy appears as an additional modulation in the C1 channel (per figure 3). This may be useful in a 3:1 ratio for improved calibration, as these data would appear in its own isolated channel.

3. S2 and S3 can be reconstructed and calibrated in the traditional sense despite the defect if one uses channel C3 or its conjugate (e.g. (φ1 + φ2) or –(φ1 + φ2)). However, error will still be present in the reconstruction of these channels due to the normalization of S2 and S3 to S0 from the mixing of S0 with S1.

Therefore it can be presumed that if one knows px1, py1, px2, and py2 that this effect can be compensated for. However, calibrating the data with this model has not been accomplished at this time, but its effects have been verified in our simulated output of the system. 4.3.2 Model output To get an idea of the output from this model, we simulated our experiment for the output of polarizer G. This yielded the following results in figure 10, where the values of px1, py1, px2, and py2 were all measured using the procedure described previously in section 4.3. As can be seen, the major aforementioned trends are present in the output when compared to the actual reconstructions in figure 5, with large deviations around 4 µm and 4.95 µm. 4.3.3 Error of improved model vs. ideal output To inspect the model’s accuracy in the S1

and S2 reconstructions, we analyze the RMS error for the linear polarizer between the perfectly ideal output of polarizer G and the output predicted by our improved model. This yielded the following result in figure 11. In this figure, the RMS error is defined in the traditional sense as,

2

0 0

1 a aRMS

n Model Measured

S S

N S Sε

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠∑ (11)

where a indicates the number of the Stokes vector under analysis. As can be observed from figure 11, error is still present between the improved model and the measured results around 4 µm, although in general it is lower than that of the ideal model. This indicates that further improvement of the model is in order, or that there is an additional defect to quantify in one of the components. For now, it can be emphasized that careful design of the AR coating is crucial when using YVO4 in this form of spectropolarimetry. Even though we specified to our supplier that we wanted our retarders AR coated for 4 µm, we discovered that their AR coating made this spectral region worse, or had no effect on the percent transmission at all. This is likely due to absorption effects in the crystal for this spectral region that need to be taken into account. However, the reconstruction results outside of these troublesome regions are encouraging, and with further work it is anticipated that this effect can be significantly reduced with improved calibration routines.

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;;..A. ) — — ;; — —

0 120

IOU

80

80

03Wavenumher (km)

Wavelength (gym) Wavelength (gym)

02/St (im pnvved mvde

20

03 38 4 48 8Wavelength (gym)

RMB ErTor of BISQ 1 .1gm. RMB EITOr of Bi .. .1gm.

W88818fl2th (tm) W88818fl2th (tm)

8.88

888

8 84

Figure 10. Theoretical output from the improved system model. Similarities can be directly observed between this output and the measured output in figure 5.

Figure 11. RMS error of the models compared to the measured data. An improvement can be seen, however, there is still error present around 4 µm and 4.95 µm.

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Me4z Me4z Me4z- None - None - None

0.26

C______ 0__38 38442 44 46 48 8 Dm3838442444826Wooden gOh (km) Wooden gOh (km) Wooden gOh (km)

4.3.4 Phase correction The last issue we wanted to investigate in our proof of concept experiment is whether or not phase correction of the interferogram yields an improvement in the reconstruction, or if it is completely unnecessary. As has been stated before, theoretically we should be able to reconstruct the data without phase correcting the interferogram. However, it may turn out that improved reconstructions can be obtained with phase corrected interferograms extracted from Mertz corrected spectra. Mathematically there should be no difference considering the phase error in the interferogram should be constant with respect to wavenumber.5 This means that when we calibrate to a known SOP, it will also be compensated for. Essentially, this amounts to equation 9 with an extra phase term,

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( )

2

1 2

0 0

22 1

2 23 2 3

1

21

41

8

j

jj

jj j

F C S e

F C S e e

F C S jS e e e

χ σ

χ σπφ σ

χ σπφ σ πφ σ

σ

σ

σ σ

−

−−

−−

=

=

= − +

(12)

where χ is the wavenumber dependent phase error. To test this, we utilized our linear polarizer data from before, except we reconstructed the data with a raw interferogram and compared it to the Mertz corrected reconstructions. Comparing their RMS errors, again using equation 11, yielded the results in figure 12.

Figure 12. RMS error (between the measured data and the improved model) of the reconstruction results for phase and non-phase corrected interferograms. As can be seen, improved results are obtained with phase correction.

As a result of this analysis, all data previously mentioned was reconstructed using Mertz phase-corrected interferograms, as it provides a reduction in the overall error. The exact reasoning for this unknown, but it is speculated that small changes over time in the instrument (temperature changes induced by the N2 purging, mechanical expansion, etc.) during the course of the measurements could be affecting the phase error, thereby creating a deviation from the reference data. This means that we may lose our data processing advantage that we had from working in Fourier space directly. However, there are other methods of phase correction that do not involve additional Fourier transformations. For instance, phase correction could be accomplished using the method proposed by Robert Furstenberg and Jeffrey White,6 in which the interferogram is phase-corrected using a digital all-pass filter. The advantage of this is that it can be integrated into DSP hardware, from which a phase-corrected interferogram is directly obtained.

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5. CONCLUSIONS From the quality of the reconstructions, it is apparent that there are several issues with the present system that contribute to an increased error. These include:

1. Aliasing effects will always limit the maximum achievable accuracy of this technique. This was seen to some extent in the CO2 absorption line and can be a significant issue if a priori knowledge of aliasing effects cannot be obtained.

2. The AR coating on the YVO4 retarders must be carefully designed to assure that the percent transmission along both axes is close to identical. If it is not, the retarders will behave as partial polarizers, contributing error to the reconstructions.

3. Phase correction appears to improve the error in the reconstructions. This is thought to be due to small changes in the instrumental profile over the course of the measurements that cause a deviation in the phase error over time, which would be subsequently corrected in the Mertz algorithm.

Ultimately, aside from the error induced by the difference in transmission between the fast and slow axes of our YVO4 retarders, the system preformed as expected. It provided all seven channels and the Stokes parameters could be reconstructed directly from the interferogram without having to go to the spectrum first, albeit at the cost of additional error. In the future, it is envisioned that this technique could be implemented into an imaging MWIR FTS, which would then be able to provide reasonable spectral and Stokes vector resolution (30.5 cm-1 or better) with a fairly minimal amount of modification to the system. The retarders could then be removed at any time to revert back to high spectral resolution non-polarimetric measurements. Lastly, the YVO4 retarders proved to be a reasonably inexpensive option, especially if one already owns an MWIR FTS.

REFERENCES

1. MacDonald, Michael, et al. “Architectural Trades for an Advanced Geostationary Atmospheric Sounding Instrument,” Aerospace Conference, IEEE Proceedings, Vol. 4, pp. 1693-1711 (2001)

2. Griffiths, de Haseth. “Fourier Transform Infrared Spectrometry,” John Wiley & Sons, Inc., 1986. 3. Oka, Kazuhiko and Takayuki Kato. “Spectroscopic Polarimetry with a Channeled Spectrum,” Optics Letters,

1999, Vol. 24, No. 21, pp. 1475-1477. 4. Goldstein, Dennis. “Polarized Light,” Marcel Dekker, Inc. 2003. 5. Bell, R. J. “Introductory Fourier Transform Spectroscopy,” New York: Academic Press, 1972. 6. Furstenberg, Robert and Jeffrey White. “Phase Correction of Interferograms Using Digital All-Pass Filters,”

Applied Spectroscopy, Vol. 59, N.3, 2005.

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