1 / 52

Modeling of polarized light transfer in scattering media

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.

aysel
Télécharger la présentation

Modeling of polarized light transfer in scattering media

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Modeling of polarized light transfer in scattering media Monte Carlo

  2. Introduction The Monte Carlo method simulates the “random walk” of photons in a medium that contains absorption and scattering.

  3. 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.

  4. 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)

  5. Monte Carlo flow chart

  6. 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

  7. 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.

  8. 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

  9. Meridian plane Monte Carlo

  10. 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.

  11. 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]

  12. Meridian plane • The unit vector I1and the Z-axis determine a plane COA. • This plane is the meridian plane

  13. 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].

  14. 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.

  15. 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.

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

  17. Move cnt. • The trajectory of the photon is specified by the unit vector I characterized by the direction cosines [ux, uy, uz]. • where xy and z are the unit vectors in the laboratory frame XYZ.

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

  19. 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

  20. 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

  21. 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

  22. 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.

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

  24. 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.

  25. 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 • *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).

  26. 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

  27. Rejection method • For unpolarized light [1 0 0 0] • The rejection method is used to generate random variables with a particular distribution.

  28. Parallel incidence- parallel detection F B D=2.0µm D=0.01µm D=1.0µm

  29. Parallel incidence- perpendicular detection D=2.0µm D=0.01µm D=1.0µm

  30. Rejection method-step 1 • Generate Prand - uniform random number between 0 and 1. • arand is generated uniformly between 0 and π. • The angle arandis accepted as the new scattering angle if Prand ≤ P(arand). • If not, a new Prandand arand are generated

  31. 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 Prandand arand , brandare generated

  32. Scattering from spherical particles Single scattering

  33. Update of direction cosines • q and f are obtained at in the scattering step of the Monte Carlo program If |uz|≈ 1

  34. Update of direction cosines If |uz|≠ 1

  35. 3. Return to a new meridian plane • The Stokes vector is rotated so that it is referenced to the new meridian plane COB

  36. 3. Return to a new meridian plane • The Stokes vector is multiplied by the rotational matrix R(-i2) so that it is referenced to the meridian plane COB. uare the direction cosines before scattering and û are the direction cosines after scattering.

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

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

  39. 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).

  40. 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.

  41. 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

  42. 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

  43. Results • Average Stokes vector values • HI, HQ, HU, HV, VI, VQ (…) IMAGES • All you need to build an image of the Mueller matrix

  44. Monte Carlo testing

  45. 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).

  46. Mueller Matrix -microspheres solutions Monte Carlo Experimental* *Cameron et al.

  47. Mueller Matrix-asymmetric illumination

  48. Mueller Matrix - microspheres solutions Monte Carlo Experimental

  49. Side View Experiments

  50. Results -1 Parallel experimental Parallel Monte Carlo

More Related