Abstract

We performed Mueller matrix Monte Carlo simulations of the propagation of optical radiation in diffusely scattering media for collimated incidence and report the results as a function of thickness and the angle subtended by the detector. For sufficiently small thickness, a fraction of the radiation does not undergo any scattering events and is emitted at zero angle. Thus, for a very small detector angle, the measured signal will indicate mostly the attenuation of the coherent contribution, while for larger angles, the diffuse scattering radiation will contribute significantly more. The degree to which the radiation is depolarized thus depends on the angle subtended by the detector. A three-stream model—where the coherent radiation, the forward diffusely scattered radiation, and the backward scattered radiation are propagated according to the differential Mueller matrix formalism—is introduced and describes the results from the Monte Carlo simulations and the results of measurements well. This scatter-based model for depolarization in diffusely scattering media is an alternative to that based upon elementary fluctuation theory applied to a single propagation stream. Results for average photon path length, determined from the Monte Carlo simulations, suggest that applying fluctuation theory to photon path length may unify the two approaches.

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2020 (1)

2018 (2)

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

B. Gompf, M. Gill, M. Dressel, and A. Berrier, “On the depolarization in granular thin films: a Mueller-matrix approach,” J. Opt. Soc. Am. A 35, 301–308 (2018).
[Crossref]

2017 (1)

S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
[Crossref]

2016 (1)

2015 (3)

2014 (1)

2011 (4)

2002 (1)

1997 (1)

1996 (1)

M. Dogariu and T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[Crossref]

1994 (1)

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

1993 (1)

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[Crossref]

1989 (1)

G. H. Weiss, R. Nossal, and R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[Crossref]

1986 (1)

1983 (1)

W. Burns, R. Moeller, and C.-L. Chen, “Depolarization in a single-mode optical fiber,” J. Lightw. Technol. 1, 44–50 (1983).
[Crossref]

1980 (1)

1978 (1)

1948 (1)

1931 (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Agarwal, N.

Alfano, R. R.

Antonelli, M.-R.

Arce-Diego, J. L.

Arteaga, O.

Asakura, T.

M. Dogariu and T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[Crossref]

Azzam, R. M. A.

Benali, A.

Berrier, A.

Bicout, D.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Bonner, R. F.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[Crossref]

G. H. Weiss, R. Nossal, and R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[Crossref]

Brosseau, C.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Burns, W.

W. Burns, R. Moeller, and C.-L. Chen, “Depolarization in a single-mode optical fiber,” J. Lightw. Technol. 1, 44–50 (1983).
[Crossref]

Bykov, A.

Charbois, J. M.

Chen, C.-L.

W. Burns, R. Moeller, and C.-L. Chen, “Depolarization in a single-mode optical fiber,” J. Lightw. Technol. 1, 44–50 (1983).
[Crossref]

Chen, W.-N.

Chiang, C.-W.

Dembski, S.

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

Devlaminck, V.

Dogariu, M.

M. Dogariu and T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[Crossref]

Dolne, J.

Dressel, M.

Gandjbakhche, A. H.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[Crossref]

Garcia-Caurel, E.

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
[Crossref]

N. Agarwal, J. Yoon, E. Garcia-Caurel, T. Novikova, J.-C. Vanel, A. Pierangelo, A. Bykov, A. Popov, I. Meglinski, and R. Ossikovski, “Spatial evolution of depolarization in homogeneous turbid media within the differential Mueller matrix formalism,” Opt. Lett. 40, 5634–5637 (2015).
[Crossref]

Gayet, B.

Germer, T. A.

Gill, M.

Gompf, B.

Groeber-Becker, F. K.

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

Hapke, B.

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (Cambridge University, 1993).

Jones, R. C.

Kubelka, P.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Lee, H. R.

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

Li, P.

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

Liu, F.

Lotz, C.

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

Martinez, A. S.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Martino, A. D.

Meglinski, I.

Moeller, R.

W. Burns, R. Moeller, and C.-L. Chen, “Depolarization in a single-mode optical fiber,” J. Lightw. Technol. 1, 44–50 (1983).
[Crossref]

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Nee, J.-B.

Nossal, R.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[Crossref]

G. H. Weiss, R. Nossal, and R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[Crossref]

Novikova, T.

Ortega-Quijano, N.

Ossikovski, R.

Pierangelo, A.

Polishchuk, A. Y.

Popov, A.

Schmitt, J. M.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Strang, G.

G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, 1980), p. 205.

Validire, P.

Vanel, J.-C.

Weiss, G. H.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[Crossref]

G. H. Weiss, R. Nossal, and R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[Crossref]

Williams, M. W.

Yoo, S. H.

S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
[Crossref]

Yoo, T. S. H.

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

Yoon, J.

Young, A. T.

Appl. Opt. (3)

Appl. Surf. Sci. (1)

S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
[Crossref]

J. Lightw. Technol. (1)

W. Burns, R. Moeller, and C.-L. Chen, “Depolarization in a single-mode optical fiber,” J. Lightw. Technol. 1, 44–50 (1983).
[Crossref]

J. Mod. Opt. (1)

G. H. Weiss, R. Nossal, and R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

M. Dogariu and T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. E (2)

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[Crossref]

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[Crossref]

Proc. SPIE (1)

H. R. Lee, T. S. H. Yoo, P. Li, C. Lotz, F. K. Groeber-Becker, S. Dembski, E. Garcia-Caurel, R. Ossikovski, and T. Novikova, “Mueller microscopy of anisotropic scattering media: theory and experiments,” Proc. SPIE 10677, 222–229 (2018).
[Crossref]

Z. Tech. Phys. (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (2)

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (Cambridge University, 1993).

G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, 1980), p. 205.

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Figures (6)

Fig. 1.
Fig. 1. Virtual measurement setup. Collimated radiation strikes material S of thickness $\Delta z$, and radiation emitted from the material with angles within $2\beta$ is collected by detector D.
Fig. 2.
Fig. 2. Unique, non-zero elements of the Mueller matrix phase function ${{\textbf{M}}_{{\rm pf}}}$ for the three different particle size distributions used in the simulations.
Fig. 3.
Fig. 3. Schematics of the (a) two-stream approach and (b) three-stream approach. Horizontal arrows indicate Mueller matrix streams, and dashed curves indicate scattering elements coupling the streams.
Fig. 4.
Fig. 4. Bidirectional transmittance distribution function ${f_{\rm t}}$ calculated using the MC method with ${D_0} = 250 \;{\rm nm}$ distribution for five different sample thicknesses. The markings at the top indicate the virtual detector collection angles used in this study.
Fig. 5.
Fig. 5. Results of the MC simulations (symbols) as a function of normalized sample thickness. The top row shows the effective transmittance ${T_{00}}$. The bottom two rows show the non-zero depolarizing logarithmic decomposition elements ${L_{11}} = {L_{22}}$ and ${L_{33}}$. The phase functions are for (left) Rayleigh, (middle) ${D_0} = 250 \;{\rm nm}$, and (right) ${D_0} = 700 \;{\rm nm}$. The curves show the best fit to the three-stream model with values given in Table 1.
Fig. 6.
Fig. 6. Mean path length $\langle \Lambda \rangle$ of transmitted rays calculated from the MC simulations for the three phase functions and for $\beta {= 20^ \circ}$ as a function of thickness $\Delta z$. The dashed line shows $\langle \Lambda \rangle = \Delta z$, corresponding to the unscattered path.

Tables (1)

Tables Icon

Table 1. Best Fit Three-Stream Parameters Shown as Curves in Fig. 5

Equations (46)

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d M ( z ) d z = m ( z ) M ( z ) ,
M ( Δ z ) = exp ( m Δ z ) ,
L = log ( M / M 00 ) .
L = m Δ z + 1 2 Δ m 2 Δ z 2 ,
L = m Δ z + 1 2 Δ m 2 ( π 1 / 2 z g Δ z z g 2 ) .
M = exp ( n m z 0 + n 2 Δ m 2 z 0 2 ) .
M n = R b M p f R a M n 1 ,
d d z ( M + ( z ) M ( z ) ) = m ( M + ( z ) M ( z ) ) ,
m = ( m a ( s ) r ( s ) s r [ a ( s ) m ] ) ,
a ( s ) = ( s 00 s 01 s 02 s 03 s 01 s 00 0 0 s 02 0 s 00 0 s 03 0 0 s 00 ) .
r ( m ) = ( m 00 m 01 m 02 m 03 m 10 m 11 m 12 m 13 m 20 m 21 m 22 m 23 m 30 m 31 m 32 m 33 ) .
( M + ( z ) M ( z ) ) = exp ( m z ) ( M + ( 0 ) M ( 0 ) ) ,
( M + ( 0 ) M ( 0 ) ) = ( I R d ) ,
( M + ( Δ z ) M ( Δ z ) ) = ( T d 0 ) ,
s = d i a g ( s 0 , s 1 , s 1 , s 3 ) .
( s 0 s j s j s 0 ) .
T d , 00 = 1 / ( 1 + s 0 Δ z ) ,
R d , 00 = s 0 Δ z / ( 1 + s 0 Δ z ) ,
T d , j j = P j P j cosh ( P j Δ z ) + s 0 sinh ( P j Δ z ) ,
R d , j j = s j s 0 + P j coth ( P j Δ z ) ,
L jj = 1 2 P j 2 ( Δ z ) 2 + 1 3 s 0 P j 2 ( Δ z ) 3 .
L jj = P j Δ z + log [ Δ z ( 2 s 0 2 s 0 3 s j 2 + 2 s 0 2 P j s j 2 ) ] .
d d z ( M c ( z ) M + ( z ) M ( z ) ) = ( m a ( s f + s b ) 0 0 s f m a ( s b + s f ) + s f r ( s b ) s b s b r [ m a ( s b + s f ) + s f ] ) ( M c ( z ) M + ( z ) M ( z ) ) ,
( M c ( 0 ) M + ( 0 ) M ( 0 ) ) = ( I 0 R d ) ,
( M c ( Δ z ) M + ( Δ z ) M ( Δ z ) ) = ( T c T d 0 ) ,
s f = d i a g ( s f , 00 , s f , 11 , s f , 11 , s f , 33 ) ,
s b = d i a g ( s b , 00 , s b , 11 , s b , 11 , s b , 33 ) .
( s f , 00 s b , 00 0 0 s f , j j C j s b , j j s b , j j s b , j j C j ) ,
C j = s b , 00 + s f , 00 s f , j j .
T d , 00 = 1 1 + s b , 00 Δ z T c , 00 ,
T c , j j = T c , 00 = exp [ ( s f , 00 + s b , 00 ) Δ z ] ,
T d , j j = cosh ( S j Δ z ) C j sinh ( S j Δ z ) S j + s b , j j 2 sinh ( S j Δ z ) S j 2 coth ( S j Δ z ) + S j C j T c , 00 ,
R d , j j = s b , j j C j + S j coth ( S j Δ z ) ,
S j = C j 2 s b , j j 2 .
T d , j j s f , j j Δ z + ( Δ z ) 2 2 { s b , j j + s f , j j [ s f , j j 2 ( s b , 00 + s f , 00 ) ] } .
T = T c + f T d .
L jj = log [ ( T c , j j + f T d , j j ) / ( T c , 00 + f T d , 00 ) ] .
M ( Δ z ) = I + 0 Δ z m ( z ) d z + 0 Δ z 0 z m ( z ) m ( z ) d z d z + .
Δ m ( z ) Δ m ( z ) = Δ m 2 exp [ ( z z ) 2 / z g 2 ] ,
M ( Δ z ) = I + m 0 Δ z + 1 2 m 0 2 ( Δ z ) 2 + Δ m 2 z g 2 { z g [ exp ( ( Δ z ) 2 z g 2 ) 1 ] + π 1 / 2 Δ z e r f ( Δ z z g ) } + .
L = m 0 Δ z + Δ m 2 z g 2 { z g [ exp ( ( Δ z ) 2 z g 2 ) 1 ] + π 1 / 2 Δ z e r f ( Δ z z g ) } + .
L = m Δ z + 1 2 Δ m 2 ( Δ z ) 2
L = m Δ z + 1 2 Δ m 2 ( π 1 / 2 z g Δ z z g 2 )
m ( z ) m ( z ) = Δ m 2 exp ( | z z | / z e 2 ) ,
L = m 0 Δ z + Δ m 2 z e { z e [ exp ( Δ z z e ) 1 ] + Δ z } + .
L = m Δ z + Δ m 2 ( z e Δ z z e 2 ) .