II Escuela de Optica Biomedica, Puebla, 2011 Modeling of polarized light transfer in scattering media Monte Carlo

II Escuela de Optica Biomedica, Puebla, 2011

Modeling of polarized light transfer in scattering media

Monte Carlo


Introduction

The Monte Carlo method simulates the “random walk” of photons in a medium that contains

absorption and scattering.


Introduction

• The method is based on a set of rules that governs the movement of a photon in tissue.

• The two key decisions are:• The mean free path for a scattering or

absorption event.• The scattering angle.


Introduction

• Monte Carlo programs estimates ensamble average quantities

• The Monte Carlo method is essentially a technique for sampling a probability density function based on a computer generated random number.

• Need large number of photons – Error 1/√(Number of photons)


Monte Carlo flow chart


Polarized light Monte Carlo

• Main issues– Define a polarization reference plane– Propagate a Stokes vector & reference plane– Handle scattering – No Heyney Greenstain phase

function– Handle scattering – Rejection method/Mie

scatterers– Handle interface and exit


J.C. Ramella-Roman, S.A. Prahl , S.L. Jacques, 'Three Monte Carlo programs of polarized light transport into scattering media: part I,' Optics Express, 13, pp. 4420-4438, 2005.

J.C. Ramella-Roman, S.A. Prahl, S.L. Jacques, 'Three Monte Carlo programs of polarized light transport into scattering media: part II,' Optics Express, 13, pp.10392-10405, 2005.


Three methods

• Meridian Plane Monte Carlo – Today• Euler and Quaternion Monte Carlo –

Tomorrow• The difference among three programs is the

tracking of the reference plane


Meridian plane Monte Carlo


Meridian plane

• Chandrasekhar was the first to envision the scattering of a photon from one location to the next in the form of meridian and scattering planes.

• The photon directions, before and after scattering, are represented as points on the unit sphere and each photon direction is uniquely described by two angles q and f.


Meridian plane

• q is the angle between the initial photon direction and the Z-axis. – Scattering angle

• f is the angle between the meridian plane and the X-Z plane.– Azimuth angle

• The photon direction is specified by a unit vector I1

• Elements of I1 are the direction cosines [ux, uy, uz]


Meridian plane

• The unit vector I1 and the Z-axis determine a plane COA.

• This plane is the meridian plane


Launch

• A photon is launched in a specific meridian/reference frame and status of polarization.

• Slab geometry, perpendicular illumination, pencil beam

• Direction cosines – [ux, uy, uz] = [0, 0, 1].


Launch-STEP A

• Initially the meridian plane f =0,• The reference frame is initially equal to the x-z

plane. • The initial Stokes vector is relative to this

meridian plane.


Launch-STEP B

• The status of polarization as an appropriate Stokes vector, S = [I Q U V].

• Example S=[1 1 0 0] – launched photon will be linearly polarized parallel

to the x-z plane.


Move

• The photon is moved as in a standard Monte Carlo program.

• The photon is moved a propagation distance Ds that is calculated based on pseudo random number generated in the interval (0,1].

€

Δs=−(ln ζ )μt

µt = µa +µs, µa is the absorption coefficient of the media. µs is the scattering coefficient of the media. €

ζ


Move cnt.

• The trajectory of the photon is specified by the unit vector I characterized by the direction cosines [ux, uy, uz].

• where x y and z are the unit vectors in the laboratory frame XYZ.

€

ux =I ⋅ x

uy = I ⋅ y

uz = I ⋅ z


Move cnt.

• The photon position is updated to a new position [x',y',z'] with the following equations:

€

′ x =x+uxΔs=I xΔs

′ y =y+uyΔs=I yΔs

′ z =z+uzΔs=I zΔs


Scatter

• Three matrix multiplications are necessary to handle the scattering of a photon and track its polarization

• 1. Stokes vector rotation in the scattering plane

• 2. Scattering -updating the Stokes vector• 3. Return to a new meridian plane


Rotation to a new reference plane

• The E field is originally defined respect to a meridian plane COA.

• The field can be decomposed into its parallel and perpendicular components E|| and E.

• The choice of f and q is done with the rejection method


Rotation to a new reference plane

• First the Stokes vector is rotated so that its reference plane is ABO

• This rotation is necessary because the scattering matrix, that defines the elastic interaction of a photon with a sphere, is specified with respect to the frame of reference of the scattering plane


1. Stokes vector rotation in the scattering plane

• The Stokes vector S1 defined relative to the scattering plane (ABO) is found by multiplying by a rotational matrix R(i1)

• This action corresponds to a counterclockwise rotation of an angle i1 about the direction of propagation.

€

R i1( ) =

1 0 0 0

0 cos( 2i1 ) sin( 2i1 ) 0

0 − sin( 2i1 ) cos( 2i1 ) 0

0 0 0 1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥


Scattering

• The Stokes vector is multiplied by the scattering matrix that accounts for scattering of the photon at an angle a


2. Scattering -updating the Stokes vector

• The Stokes vector is multiplied by a scattering matrix, M(a).

• a is the scattering angle between the direction of the photon before and after scattering.

€

M α( ) =

s11( α ) s12( α ) 0 0

s12( α ) s11( α ) 0 0

0 0 s33( α ) s34( α )

0 0 −s34( α ) s33( α )

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥


Updating the Stokes vector

• The parameters s11 , s12 , s33 , s34 are calculated with Rayleigh or Mie theory.

• These terms are expressed as

• S1, S2* – Mie scattering

€

s11(α )=12[ S2(α )

2+ S1(α )

2]

s12(α )=12[ S2(α )

2−S1(α )

2]

s33(α )=12[ S2

* (α )S1(α )+S2(α )S1* (α )]

s34(α )= i2[ S2

* (α )S1(α )−S2(α )S1* (α )]

*http://omlc.ogi.edu/calc/mie_calc.html

C. Bohren and D. R. Huffman, Absorption and scattering of light by small particles, (Wiley Science Paperback Series,1998).


Rejection method

• The phase function P(α,β) for incident light with a Stokes vector So= [Io, Qo, Uo, Vo] is

• α angle of scattering • b angle of rotation into the scattering plane

€

(P α ,β ) = 11 (s α ) Io+ 12 (s α ) [ (2Qo cosβ ) + (2Uo sin β )]


Rejection method

• For unpolarized light [1 0 0 0]

• The rejection method is used to generate randomvariables with a particular distribution.

€

(P α ,β ) = 11 (s α ) Io


Parallel incidence- parallel detection

D=0.01µm D=1.0µm D=2.0µm

F

B


Parallel incidence- perpendicular detection

D=0.01µm D=1.0µm D=2.0µm


Rejection method-step 1

• Generate Prand - uniform random number between 0 and 1.

• arand is generated uniformly between 0 and π.

• The angle arand is accepted as the new scattering angle if Prand ≤ P(arand).

• If not, a new Prand and arand are generated


Rejection method-step 2

• Generate Prand - uniform random number between 0 and 1.

• brand is generated uniformly between 0 and 2π.

• The angle brand is accepted as the new scattering angle if Prand ≤ P(arand, brand).

• If not, a new Prand and arand , brand are generated

€

(P α ,β ) = 11 (s α ) Io+ 12 (s α ) [ (2Qo cosβ ) + (2Uo sin β )]


Scattering from spherical particlesSingle scattering


Update of direction cosines

• q and f are obtained at in the scattering step of the Monte Carlo program

€

u^

x = (sin θ ) (cos φ)

u^

y = (sin θ ) (sin φ)

u^

z = (cosθ ) uz

uz

If |uz| ≈ 1


Update of direction cosines

€

u^

x =1

1−uz

2(sinθ )(uxuy (cosφ)−uy (sinφ))+ux (cosθ )

u^

y =1

1−uz

2(sinθ )(uxuz (cosφ)−ux (sinφ))+uy (cosθ )

u^

z = 1−uz

2 (sinθ ) (cosφ)(uyuz (cosφ)−ux (sinφ))+uz (cosθ )

If |uz| ≠ 1


3. Return to a new meridian plane

• The Stokes vector is rotated so that it is referenced to the new meridian plane COB



• The Stokes vector is multiplied by the rotational matrix R(-i2) so that it is referenced to the meridian plane COB.

€

cosi2 =−uz + u

^

zcosα

± (1 −cos2α )(1 −u^

z2 )

u are the direction cosines before scattering and û are the direction cosines after scattering.



• The Stokes vector is multiplied by the rotational matrix R(-i2) so that it is referenced to the meridian plane COB.

€

R i2( ) =

1 0 0 0

0 cos(2i2 ) sin(2i2 ) 0

0 − sin(2i2 ) cos(2i2 ) 0

0 0 0 1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥


Summary

• In summary the Stokes vector Snew after a scattering event is obtained from the Stokes vector before the scattering event S

€

Snew = R( −i2 )M( α )R( i1 )S


Choice of f and a

• A fundamental problem in every Monte Carlo program with polarization information is the choice of the angles a (angle of scattering) and f (angle of rotation into the scattering plane).

• These angles are selected based on the phase function of the considered scatterers and a rejection method (**)

** W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the art of Scientific Computing, Cambridge University Press; (1992).


Photon life

• The life of a photon ends when the photon passes through a boundary or when its weight W value falls below a threshold.

• Roulette is used to terminate the photon packet when W £ Wth. – Gives the photon packet one chance of surviving– If the photon packet does not survive the roulette,

the photon weight is reduced to zero and the photon is terminated.


Boundaries

• One last rotation of the Stokes vector is necessary to put the photon polarization in the reference frame of the detector.

• For simplicity we are neglecting Fresnel reflectance


Boundaries

• For a photon backscattered from a slab, the last rotation to the meridian plane of the detector is of an angle y

• The Stokes vector of the reflected photon is multiplied one final time by R(y). For a transmitted photon the angle y

€

ψ =tan−1(uy

ux)

€

ψ =−tan−1(uy

ux)


Results

• Average Stokes vector values• HI, HQ, HU, HV, VI, VQ (…) IMAGES

– All you need to build an image of the Mueller matrix


Monte Carlo testing


Comparison with Evans’ adding-doubling*

Diameter

(nm)

I Evans I This code Q Evans Q this code

10 0.68833 0.68867 -0.10416 -0.10423

100 0.67698 0.67695 -0.10152 -0.10094

1000 0.44799 0.44848 0.04992 0.04990

2000 0.29301 0.29311 0.0089989 0.00887

*K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant.Spectrosc. Radiat Transfer. 46, 413-423, (1991).


Mueller Matrix -microspheres solutions

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Monte Carlo Experimental**Cameron et al.


Mueller Matrix-asymmetric illumination

Polarizer

Polarizer

Laser

Camera

Microsphere solution

θz

x

y


Mueller Matrix - microspheres solutions

Monte Carlo Experimental

m11

2 4

1

2

3

4

m12

2 4

1

2

3

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m13

2 4

1

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3

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m21

2 4

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m22

2 4

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m23

2 4

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m31

2 4

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2

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m32

2 4

1

2

3

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m33

2 4

1

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m11

1 2 3 4

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m12

1 2 3 4

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3

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m13

1 2 3 4

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m21

1 2 3 4

1

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m22

1 2 3 4

1

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m23

1 2 3 4

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m31

1 2 3 4

1

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m32

1 2 3 4

1

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m33

1 2 3 4

1

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4


Side View Experiments

He-NeLaser 633nm

200 ml glass container7x7x4 cm

G4 computer

7 cm

0.5 µm microspheressolution

0.25cm1cm

12 bit digital camera

AnalyzerPolarizer


Results -1

Parallelexperimental

ParallelMonte Carlo


Results -2

Parallel

Max Parallel


Tomorrow

• Euler and Quaternion based Monte Carlo

• How to run the codes

• Application

Documents

II Escuela de Optica Biomedica, Puebla, 2011 Modeling of polarized light transfer in scattering media Monte Carlo