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

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II Escuela de Optica Biomedica, Puebla, 2011 Introduction The Monte Carlo method simulates the “random walk” of photons in a medium that contains absorption and scattering.

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II Escuela de Optica Biomedica, Puebla, 2011 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.

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II Escuela de Optica Biomedica, Puebla, 2011 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)

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II Escuela de Optica Biomedica, Puebla, 2011 Monte Carlo flow chart

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II Escuela de Optica Biomedica, Puebla, 2011 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

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II Escuela de Optica Biomedica, Puebla, 2011 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.

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II Escuela de Optica Biomedica, Puebla, 2011 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

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II Escuela de Optica Biomedica, Puebla, 2011 Meridian plane Monte Carlo

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II Escuela de Optica Biomedica, Puebla, 2011 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 and .

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II Escuela de Optica Biomedica, Puebla, 2011 Meridian plane is the angle between the initial photon direction and the Z-axis. – Scattering angle is the angle between the meridian plane and the X-Z plane. – Azimuth angle The photon direction is specified by a unit vector I 1 Elements of I 1 are the direction cosines [u x, u y, u z ]

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II Escuela de Optica Biomedica, Puebla, 2011 Meridian plane The unit vector I 1 and the Z-axis determine a plane COA. This plane is the meridian plane

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II Escuela de Optica Biomedica, Puebla, 2011 Launch A photon is launched in a specific meridian/reference frame and status of polarization. Slab geometry, perpendicular illumination, pencil beam Direction cosines – [u x, u y, u z ] = [0, 0, 1].

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II Escuela de Optica Biomedica, Puebla, 2011 Launch-STEP A Initially the meridian plane =0, The reference frame is initially equal to the x-z plane. The initial Stokes vector is relative to this meridian plane.

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II Escuela de Optica Biomedica, Puebla, 2011 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.

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II Escuela de Optica Biomedica, Puebla, 2011 Move The photon is moved as in a standard Monte Carlo program. The photon is moved a propagation distance s that is calculated based on pseudo random number generated in the interval (0,1]. µ t = µ a +µ s, µ a is the absorption coefficient of the media. µ s is the scattering coefficient of the media.

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II Escuela de Optica Biomedica, Puebla, 2011 Move cnt. The trajectory of the photon is specified by the unit vector I characterized by the direction cosines [u x, u y, u z ]. where x y and z are the unit vectors in the laboratory frame XYZ.

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II Escuela de Optica Biomedica, Puebla, 2011 Move cnt. The photon position is updated to a new position [x',y',z'] with the following equations:

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II Escuela de Optica Biomedica, Puebla, 2011 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

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II Escuela de Optica Biomedica, Puebla, 2011 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 and is done with the rejection method

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II Escuela de Optica Biomedica, Puebla, 2011 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

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II Escuela de Optica Biomedica, Puebla, 2011 1. Stokes vector rotation in the scattering plane The Stokes vector S 1 defined relative to the scattering plane (ABO) is found by multiplying by a rotational matrix R(i 1 ) This action corresponds to a counterclockwise rotation of an angle i 1 about the direction of propagation.

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II Escuela de Optica Biomedica, Puebla, 2011 Scattering The Stokes vector is multiplied by the scattering matrix that accounts for scattering of the photon at an angle

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II Escuela de Optica Biomedica, Puebla, 2011 2. Scattering -updating the Stokes vector The Stokes vector is multiplied by a scattering matrix, M( ). is the scattering angle between the direction of the photon before and after scattering.

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II Escuela de Optica Biomedica, Puebla, 2011 Updating the Stokes vector The parameters s 11, s 12, s 33, s 34 are calculated with Rayleigh or Mie theory. These terms are expressed as S1, S2* – Mie scattering *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).

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II Escuela de Optica Biomedica, Puebla, 2011 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 angle of rotation into the scattering plane

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II Escuela de Optica Biomedica, Puebla, 2011 Rejection method For unpolarized light [1 0 0 0] The rejection method is used to generate random variables with a particular distribution.

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II Escuela de Optica Biomedica, Puebla, 2011 Parallel incidence- parallel detection D=0.01µmD=1.0µm D=2.0µm F B

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II Escuela de Optica Biomedica, Puebla, 2011 Parallel incidence- perpendicular detection D=0.01µmD=1.0µm D=2.0µm

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II Escuela de Optica Biomedica, Puebla, 2011 Rejection method-step 1 Generate P rand - uniform random number between 0 and 1. rand is generated uniformly between 0 and π. The angle rand is accepted as the new scattering angle if P rand ≤ P( rand ). If not, a new P rand and rand are generated

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II Escuela de Optica Biomedica, Puebla, 2011 Rejection method-step 2 Generate P rand - uniform random number between 0 and 1. rand is generated uniformly between 0 and 2π. The angle rand is accepted as the new scattering angle if P rand ≤ P( rand, rand ). If not, a new P rand and rand, rand are generated

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II Escuela de Optica Biomedica, Puebla, 2011 Scattering from spherical particles Single scattering

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II Escuela de Optica Biomedica, Puebla, 2011 Update of direction cosines and are obtained at in the scattering step of the Monte Carlo program If |u z | ≈ 1

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II Escuela de Optica Biomedica, Puebla, 2011 Update of direction cosines If |u z | ≠ 1

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II Escuela de Optica Biomedica, Puebla, 2011 3. Return to a new meridian plane The Stokes vector is rotated so that it is referenced to the new meridian plane COB

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II Escuela de Optica Biomedica, Puebla, 2011 3. Return to a new meridian plane The Stokes vector is multiplied by the rotational matrix R(-i 2 ) so that it is referenced to the meridian plane COB. u are the direction cosines before scattering and û are the direction cosines after scattering.

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II Escuela de Optica Biomedica, Puebla, 2011 3. Return to a new meridian plane The Stokes vector is multiplied by the rotational matrix R(-i 2 ) so that it is referenced to the meridian plane COB.

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II Escuela de Optica Biomedica, Puebla, 2011 Summary In summary the Stokes vector S new after a scattering event is obtained from the Stokes vector before the scattering event S

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II Escuela de Optica Biomedica, Puebla, 2011 Choice of and A fundamental problem in every Monte Carlo program with polarization information is the choice of the angles (angle of scattering) and (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).

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II Escuela de Optica Biomedica, Puebla, 2011 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 W th. – 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.

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II Escuela de Optica Biomedica, Puebla, 2011 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

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II Escuela de Optica Biomedica, Puebla, 2011 Boundaries For a photon backscattered from a slab, the last rotation to the meridian plane of the detector is of an angle The Stokes vector of the reflected photon is multiplied one final time by R(y). For a transmitted photon the angle

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II Escuela de Optica Biomedica, Puebla, 2011 Results Average Stokes vector values HI, HQ, HU, HV, VI, VQ (…) IMAGES – All you need to build an image of the Mueller matrix

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II Escuela de Optica Biomedica, Puebla, 2011 Monte Carlo testing

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II Escuela de Optica Biomedica, Puebla, 2011 Comparison with Evans’ adding- doubling * *K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat Transfer. 46, 413-423, (1991).

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II Escuela de Optica Biomedica, Puebla, 2011 Mueller Matrix -microspheres solutions Monte Carlo Experimental* *Cameron et al.

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II Escuela de Optica Biomedica, Puebla, 2011 Mueller Matrix-asymmetric illumination

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II Escuela de Optica Biomedica, Puebla, 2011 Mueller Matrix - microspheres solutions Monte Carlo Experimental

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II Escuela de Optica Biomedica, Puebla, 2011 Side View Experiments

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II Escuela de Optica Biomedica, Puebla, 2011 Results -1 Parallel experimental Parallel Monte Carlo

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II Escuela de Optica Biomedica, Puebla, 2011 Results -2 Parallel Max Parallel

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II Escuela de Optica Biomedica, Puebla, 2011 Tomorrow Euler and Quaternion based Monte Carlo How to run the codes Application

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