## Comparison of discrete ordinate and Monte Carlo simulations of polarized radiative transfer in two coupled slabs with different refractive indices |

Optics Express, Vol. 21, Issue 8, pp. 9592-9614 (2013)

http://dx.doi.org/10.1364/OE.21.009592

Acrobat PDF (1811 KB)

### Abstract

A comparison is presented of two different methods for polarized radiative transfer in coupled media consisting of two adjacent slabs with different refractive indices, each slab being a stratified medium with no change in optical properties except in the direction of stratification. One of the methods is based on solving the integro-differential radiative transfer equation for the two coupled slabs using the discrete ordinate approximation. The other method is based on probabilistic and statistical concepts and simulates the propagation of polarized light using the Monte Carlo approach. The emphasis is on non-Rayleigh scattering for particles in the Mie regime. Comparisons with benchmark results available for a slab with constant refractive index show that both methods reproduce these benchmark results when the refractive index is set to be the same in the two slabs. Computed results for test cases with coupling (different refractive indices in the two slabs) show that the two methods produce essentially identical results for identical input in terms of absorption and scattering coefficients and scattering phase matrices.

© 2013 OSA

## 1. Introduction

1. H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observation System era,” J. Geophys. Res. **102**, 17081–17106 (1997) [CrossRef] .

2. M. Mishchenko and L. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. **102**, 16989–7013 (1997) [CrossRef] .

4. W. Li, K. Stamnes, R. Spurr, and J. J. Stamnes, “Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. II. SeaWiFS case study for the Santa Barbara channel,” Int. J. Rem. Sens. **29**, 5689–5698 (2008). [CrossRef]

5. Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. **33**, 431–442 (1994) [CrossRef] [PubMed] .

6. R. Spurr, K. Stamnes, H. Eide, W. Li, K. Zhang, and J. J. Stamnes, “Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach: I. Analytic Jacobians from the linearized CAO-DISORT model,” J. Quant. Spectrosc. Radiat. Transfer **104**, 428–449 (2007) [CrossRef] .

4. W. Li, K. Stamnes, R. Spurr, and J. J. Stamnes, “Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. II. SeaWiFS case study for the Santa Barbara channel,” Int. J. Rem. Sens. **29**, 5689–5698 (2008). [CrossRef]

7. K. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra: A feasibility study,” J. Photochem. Photobiol. B **93**, 23–31 (2008) [CrossRef] [PubMed] .

9. D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg. **36**, 1–8 (2010). [CrossRef]

10. C. N. Adams and G. W. Kattawar, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: Effect of interface refractive index on radiance and polarization,” Limm. Ocean. **34**, 1453–1472 (1989) [CrossRef] .

12. A. A. Lacis, J. Chowdhary, M. I. Mishchenko, and B. Cairns, “Modeling errors in diffuse-sky radiation: vector vs. scalar treatment,” Geophys. Res. Lett. **25**, 135–138 (1998) [CrossRef] .

14. K. N. Liou, “A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer,” J. Atmos. Sci. **30**, 1303–1326 (1973) [CrossRef] .

15. K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. **38**, 387–399 (1981) [CrossRef] .

17. K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer **31**, 273–282 (1984) [CrossRef] .

18. K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. **27**2502–2509 (1988) [CrossRef] [PubMed] .

5. Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. **33**, 431–442 (1994) [CrossRef] [PubMed] .

19. K. P. Nielsen, L. Zhao, P. Juzenas, K. Stamnes, J. J. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol. **80**, 450–455 (2004) [PubMed] .

20. K. I. Gjerstad, J. J. Samnes, B. Hamre, J. K. Lotsberg, B. Yan, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of irradiances in the coupled atmosphere-ocean system,” Appl. Opt. **42**, 2609–2622 (2003) [CrossRef] [PubMed] .

21. K. Hestenes, K. P. Nielsen, L. Zhao, J. J. Stamnes, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of spectral radiances in the coupled air-tissue system,” Appl. Opt. **46**, 2333–2350 (2007) [CrossRef] [PubMed] .

22. F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - I. Theory,” J. Quant. Spectrosc. Radiat. Transfer , **47**, 19–33 (1992) [CrossRef] .

25. F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane parallel, vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transfer **65**, 609–620 (2000) [CrossRef] .

28. E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transfer **111**, 616–633 (2010) [CrossRef] .

28. E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transfer **111**, 616–633 (2010) [CrossRef] .

28. E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transfer **111**, 616–633 (2010) [CrossRef] .

*δ*-M technique [29

29. W. J. Wiscombe, “The delta-M method: Rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. **34**, 1408–1422 (1977) [CrossRef] .

37. P. W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Expr. **17**, 2057–2079 (2009) [CrossRef] .

**111**, 616–633 (2010) [CrossRef] .

## 2. Radiative transfer theory

### 2.1. General radiative transfer

*dτ*= (

*α*+

*σ*)

*dz*is the differential optical depth,

*α*and

*σ*are the absorption and scattering coefficients, respectively,

*dz*is the differential vertical path length,

*u*′ = cos

*θ*′,

*θ*′ and

*ϕ*′ are the polar angle and the azimuth angle, respectively, of the direction of an incident parallel beam,

*u*= cos

*θ*,

*θ*and

*ϕ*are the polar angle and the azimuth angle, respectively, of the observation direction,

*a*(

*τ*) =

*σ*/(

*α*+

*σ*) is the single-scattering albedo,

**M**is the phase matrix describing the scattering properties of the medium [38

38. C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. **245**, 1080–1086 (1981) [CrossRef] .

*Q⃗*is the source term, which has different expressions in the two media of different refractive indices comprising a coupled system [5

5. Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. **33**, 431–442 (1994) [CrossRef] [PubMed] .

**111**, 616–633 (2010) [CrossRef] .

*Q⃗*and

_{atm}*Q⃗*are given by Eqs. (10) and (11) in [28

_{w}**111**, 616–633 (2010) [CrossRef] .

### 2.2. Types of scattering & input/output

*r*is the radius of the particle and

*λ*is the wavelength of the incident light [41,42

42. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley, 1998) [CrossRef] .

*x*<< 1, the scattering is said to be in the Rayleigh regime, which corresponds to scattering by an electric dipole. When 0.1 <

*x*< 50, the scattering is called intermediate or non-Rayleigh scattering. For spherical particles, this regime is referred to as Mie scattering, because solutions to the scattering problem applicable to dielectric spheres published by Gustav Mie in 1908 can be applied [42

42. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley, 1998) [CrossRef] .

42. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley, 1998) [CrossRef] .

43. X. Zhou, S. Li, and K. Stamnes, “Geometrical-optics code for computing optical properties of large dielectric spheres,” Appl. Opt. **42**, 4295–4306 (2003) [CrossRef] [PubMed] .

*n*(

*r*), the number of particles with radius

*r*in a volume

^{−1}, given by [44

44. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. **16**, 527–610 (1974) [CrossRef] .

*σ*, the dimensionless standard deviation of the size distribution, is given by The effective radius

_{g}*r*and the effective variance

_{eff}*ν*are defined as (

_{eff}*r*

_{1}and

*r*

_{2}are respectively the radii of the smallest and largest particles in the distribution, and Eqs. (3)–(5) have been used in the last steps of Eqs. (6) and (7) [44

44. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. **16**, 527–610 (1974) [CrossRef] .

#### Aerosol particles

45. A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer **111**, 1931–1946 (2010). [CrossRef] .

*r*= 0.3

_{g}*μ*m, and

*σ*= 0.92, and the smallest and largest particle radii were selected to be

_{g}*r*

_{1}= 0.005

*μ*m and

*r*

_{2}= 30

*μ*m. The refractive index of the aerosol particles was set to

*m*= 1.385, which yields a single-scattering albedo of 1.0, and an asymmetry factor

*g*= 0.79275 [45

45. A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer **111**, 1931–1946 (2010). [CrossRef] .

#### Cloud particles

45. A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer **111**, 1931–1946 (2010). [CrossRef] .

*r*= 5

_{g}*μ*m, and

*σ*= 0.4, and the smallest and largest particle radii were selected to be

_{g}*r*

_{1}= 0.005

*μ*m and

*r*

_{2}= 100

*μ*m. The refractive index was set to

*m*= 1.339, which yields a single-scattering albedo of 1.0, and an asymmetry factor

*g*= 0.86114 for the cloud particles.

*p*(cosΘ) ≡

*a*

_{1}(Θ), which is the only element that matters if polarization effects are ignored. The first moment of the scattering phase function: is called the asymmetry factor of the scattering phase function. It is positive for scattering in the forward direction, negative for scattering in the backward direction, and zero for scattering phase functions that are symmetric about Θ =

*π*/2.

46. M. I. Mishchenko, “Light scattering by randomly oriented rotationally symmetric particles,” J. Opt. Soc. Am. A **8**, 871–882 (1991) [CrossRef] .

**111**, 1931–1946 (2010). [CrossRef] .

### 2.3. The interface between the two media

*θ*=

_{r}*θ*, where

_{i}*θ*and

_{i}*θ*are the angles of incidence and reflection, respectively, applies at the interface. The angle of refraction (or transmission)

_{r}*θ*is determined by Snell’s law

_{t}*n*

_{1}sin

*θ*=

_{i}*n*

_{2}sin

*θ*, where

_{t}*n*

_{1}and

*n*

_{2}are the real parts of the refractive indices in medium 1 (containing the incident beam) and medium 2 (containing the transmitted beam), respectively. The reflection and transmission coefficients for parallel and perpendicular polarization follow from Fresnel’s equations [42

*Absorption and Scattering of Light by Small Particles* (John Wiley, 1998) [CrossRef] .

*m*

_{1}=

*n*

_{1}+

*in*′

_{1}and

*m*

_{2}=

*n*

_{2}+

*in*′

_{2}are the complex refractive indices,

*E*

_{||}is the component of electric field that is parallel to the plane of incidence (the plane spanned by the unit vector in the direction of the incident beam and the interface normal), while

*E*

_{⊥}is the component perpendicular to the plane of incidence,

*R*

_{||}and

*T*

_{||}are the reflection and transmission coefficients for the parallel polarization component, while

*R*

_{⊥}and

*T*

_{⊥}are the reflection and transmission coefficients for the perpendicular polarization component.

*I⃗*= [

_{inc}*I*

_{||i},

*I*

_{⊥i},

*U*,

_{i}*V*]

_{i}*by the reflection matrix to obtain the Stokes vector of the reflected beam [42*

^{T}*Absorption and Scattering of Light by Small Particles* (John Wiley, 1998) [CrossRef] .

*Absorption and Scattering of Light by Small Particles* (John Wiley, 1998) [CrossRef] .

*m*

_{1}=

*n*

_{1}and

*m*

_{2}=

*n*

_{2}are real numbers (

*n*′

_{1}=

*n*′

_{2}= 0). Then the absolute value of each of the reflection coefficients is 1, as expected for total internal reflection. Total internal reflection couples the two Stokes vector elements

*U*and

*V*since the off-diagonal elements

## 3. Radiative transfer models

*θ*

_{0},

*ϕ*

_{0}) for a coupled system, were provided as input to the polarized RT model, which calculated the Stokes vector at user-specified optical depths and observation directions, each given by its polar angle

*θ*and azimuth angle

*ϕ*.

### 3.1. C-VDISORT

**M**, which can be calculated as explained in Appendix A.

### 3.2. C-PMC

*ρ*between 0 and 1 is generated to determine its initial optical path length

*τ*. To compute the new path length

*τ*one generates a new random number

*ρ*between 0 and 1, and

*τ*is computed from the formula where

*d*is the geometric path length and

_{j}*k*is the attenuation coefficient. The summation is carried out until the sum becomes larger than ln

_{e,j}*ρ*, and then

*d*is adjusted in layer

_{N}*N*so that the sum in Eq. (25) becomes equal to ln

*ρ*. This procedure determines the new vertical position of the beam. Once the new vertical position of the beam has been calculated, the attenuation coefficient, given by

*k*=

_{e}*α*+

*σ*, where

*α*and

*σ*are the absorption and scattering coefficients, respectively, determines whether or not the beam is scattered or absorbed. If

*ρ*between 0 and 1 is generated and used to determine whether the scattering is of the Rayleigh or non-Rayleigh type, each defined by a user-provided Mueller matrix.

*ρ*is used to determine whether the beam is reflected or refracted by comparing it to the reflectance

*R*for unpolarized light given by If

*ρ*<

*R*the beam is reflected; otherwise the beam is refracted according to Snell’s law, for details see [30, 31

31. J. K. Lotsberg and J. J. Stamnes, “Impact of particulate oceanic composition on the radiance and polarization of underwater and backscattered light,” Opt. Expr. **18**, 10432–10445 (2010) [CrossRef] .

*R*

_{||}and

*R*

_{⊥}are the Fresnel reflection coefficients given by Eqs. (9) and (11).

## 4. Comparison of C-VDISORT and C-PMC RT simulations

*I*,

*Q*, and

*U*, and the degree of linear polarization

### 4.1. Mie Scattering in the Rayleigh limit for a coupled system

**111**, 616–633 (2010) [CrossRef] .

*i.e.*

46. M. I. Mishchenko, “Light scattering by randomly oriented rotationally symmetric particles,” J. Opt. Soc. Am. A **8**, 871–882 (1991) [CrossRef] .

*r*

_{1}= 14 nm and

*r*

_{2}= 15 nm [see Eqs. (3)–(7)]. We found that the C-VDISORT and C-PMC RT models produced results in very good agreement with one another, and also with the results obtained for pure Rayleigh scattering [28

**111**, 616–633 (2010) [CrossRef] .

### 4.2. C-VDISORT and C-PMC vs benchmark – aerosol layer – reflection

**111**, 1931–1946 (2010). [CrossRef] .

*r*= 0.3

_{g}*μ*m, standard deviation

*σ*= 0.92, and refractive index

_{g}*m*= 1.385, which yields a single-scattering albedo of 1.0, and an asymmetry factor

*g*= 0.79275 as explained in §2.2. In the simulations using the C-VDISORT and C-PMC RT models, we set the index of refraction to 1.0 in both slabs, and put half of the aerosol particles (optical depth = 0.1631) in each slab. The C-VDISORT results are seen to be indistinguishable from the benchmark results; in fact the dotted and solid curves lie on top of each other in the left panels of Fig. 1. The right four panels of Fig. 1 show comparisons between reflected polarized radiation components computed by the C-VDISORT model (same results as in the four left panels) and the C-PMC RT model for the benchmark case [45

**111**, 1931–1946 (2010). [CrossRef] .

### 4.3. C-VDISORT and C-PMC vs benchmark – aerosol layer – transmission

**111**, 1931–1946 (2010). [CrossRef] .

*I*,

*Q*, and

*U*parameters around the forward direction (cos

*θ*= −0.5), the agreement between C-VDISORT and the benchmark results [45

**111**, 1931–1946 (2010). [CrossRef] .

*δ*-M approximation [29

29. W. J. Wiscombe, “The delta-M method: Rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. **34**, 1408–1422 (1977) [CrossRef] .

*δ*-M approximation in C-VDISORT.

*S*stands for the

*I*,

*Q*, or

*U*component of the Stokes vector. Note that we are using the absolute value of the maximum magnitude of the Stokes parameter [max{

*S*

_{BENCH}}] in the interval

*μ*≡ cos

*θ*∈ [−1, 1] as the point of reference to avoid dividing by values close to zero for the

*Q*and

*U*components. This difference between the C-VDISORT and the benchmark results [Eq. (27)] is provided in Fig. 3 for the aerosol case. The relative difference for the

*I*parameter shown in the upper left panel of Fig. 3 is a fraction of one percent for all viewing angles. The difference in the

*Q*parameter is of similar magnitude except around the forward direction where it lies between 1 and 2%. The lower left panel shows that the difference in the

*U*parameter is a fraction of one percent.

### 4.4. C-VDISORT and C-PMC vs benchmark – cloud layer – reflection

**111**, 1931–1946 (2010). [CrossRef] .

*r*= 5

_{g}*μ*m, and a standard deviation of

*σ*= 0.4, and the smallest and largest particle radii were selected to be

_{g}*r*

_{1}= 0.005

*μ*m and

*r*

_{2}= 100

*μ*m. The refractive index was set to

*m*= 1.339, which yields a single-scattering albedo of 1.0, and an asymmetry factor

*g*= 0.86114 for the cloud particles. In the simulations with C-VDISORT and C-PMC, we set the refractive index equal to 1.0 in both slabs, and put half of the cloud particles (optical depth = 2.5) in each slab. The left four panels of Fig. 4 show comparisons of C-VDISORT and benchmark results [45

**111**, 1931–1946 (2010). [CrossRef] .

*I*and

*Q*parameters for Δ

*ϕ*= 180° around the forward scattering direction due to the use of the

*δ*-M approximation. A quantitative assessment of these results is provided in §4.5.

### 4.5. C-VDISORT and C-PMC vs benchmark – cloud layer – transmission

**111**, 1931–1946 (2010). [CrossRef] .

### 4.6. C-VDISORT vs C-PMC – aerosol particles – coupled case

*Q*and

*U*components for directions close to the horizon at Δ

*ϕ*= 45°. The corresponding results at the TOA are shown in Fig. 9 (upper panel); there is generally very good agreement also at the TOA.

### 4.7. C-VDISORT vs C-PMC – aerosol/cloud particles – coupled case

## 5. Conclusions

**111**, 616–633 (2010) [CrossRef] .

**111**, 616–633 (2010) [CrossRef] .

## 6. Appendix A

## 6.1. Stokes vector representation I⃗_{S} = [I, Q, U, V]^{T} used in C-PMC

**F**

*. If any of the following conditions are fulfilled [51]:*

_{S}- each particle in the volume element has a plane of symmetry, and the particles are randomly oriented,
- each volume element contains an equal number of particles and their mirror particles in random orientation,
- the particles are much smaller than the wavelength of the incident light,

*I⃗*= [

_{S}*I*,

*Q*,

*U*,

*V*]

*has the following form Each of the six independent matrix elements in Eq. (30) depends on the scattering angle Θ, but are independent of the optical depth within a given horizontal layer of the medium. For spherical particles, the matrix in Eq. (30) simplifies, since*

^{T}*a*

_{1}=

*a*

_{2}and

*a*

_{3}=

*a*

_{4}, so that only four independent elements remain.

**F**

*(Θ) in Eq. (30) by where*

_{S}**R**

*is the rotation matrix described in Eq. (29) with*

_{S}*ω*replaced by Ψ or Φ where Ψ and Φ are angles describing the rotation of the incident and scattered parallel beams from the frame of the scattering plane to the laboratory frame [13].

**R**

*(Φ) before it is multiplied by the Stokes scattering matrix*

_{S}**F**

*(Θ), whereafter it must be multiplied by the rotation matrix*

_{S}**R**

*(Ψ). These matrix multiplications are carried out explicitly in C-PMC, while they are automatically taken care of in C-VDISORT.*

_{S}## 6.2. Stokes vector representation I⃗ = [I_{||}, I_{⊥}, U, V]^{T} used in C-VDISORT

## 6.3. Generalized Spherical Functions

**M**in Eq. (1) is expanded in generalized spherical functions with particular coefficients. To that end, the phase matrix

**M**is first expanded in a Fourier series in the azimuth angles using 2

*N*coefficients as shown in the following equation: Here

*u*= cos

*θ*,

*θ*being the polar angle after scattering, while

*u*′ = cos

*θ*′,

*θ*′ being the polar angle prior to scattering,

*ϕ*is the azimuth angle after scattering, and

*ϕ*′ is the azimuth angle prior to scattering. An addition theorem for the generalized functions can be used to express the Fourier expansion coefficients directly in terms of the expansion coefficients of the Stokes scattering matrix [38

38. C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. **245**, 1080–1086 (1981) [CrossRef] .

46. M. I. Mishchenko, “Light scattering by randomly oriented rotationally symmetric particles,” J. Opt. Soc. Am. A **8**, 871–882 (1991) [CrossRef] .

**D̃**= diag{1, 1, −1, −1}. The matrix

**A**

*(*

^{m}*u,u*′) is given by: where and Here

*a*(Θ) and

_{j}*b*(Θ) are the elements of the Stokes scattering matrix in Eq. (30).

_{j}## Acknowledgments

## References and links

1. | H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observation System era,” J. Geophys. Res. |

2. | M. Mishchenko and L. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. |

3. | C. Rodgers, |

4. | W. Li, K. Stamnes, R. Spurr, and J. J. Stamnes, “Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. II. SeaWiFS case study for the Santa Barbara channel,” Int. J. Rem. Sens. |

5. | Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. |

6. | R. Spurr, K. Stamnes, H. Eide, W. Li, K. Zhang, and J. J. Stamnes, “Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach: I. Analytic Jacobians from the linearized CAO-DISORT model,” J. Quant. Spectrosc. Radiat. Transfer |

7. | K. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra: A feasibility study,” J. Photochem. Photobiol. B |

8. | D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol. |

9. | D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg. |

10. | C. N. Adams and G. W. Kattawar, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: Effect of interface refractive index on radiance and polarization,” Limm. Ocean. |

11. | M. I. Mishchenko, A.A. Lacis, and L. D. Travis, “Errors due to the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant Spectrosc. Radiat Transfer |

12. | A. A. Lacis, J. Chowdhary, M. I. Mishchenko, and B. Cairns, “Modeling errors in diffuse-sky radiation: vector vs. scalar treatment,” Geophys. Res. Lett. |

13. | S. Chandrasekhar, |

14. | K. N. Liou, “A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer,” J. Atmos. Sci. |

15. | K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. |

16. | K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II. Intensity computations,” J. Atmos. Sci. |

17. | K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer |

18. | K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. |

19. | K. P. Nielsen, L. Zhao, P. Juzenas, K. Stamnes, J. J. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol. |

20. | K. I. Gjerstad, J. J. Samnes, B. Hamre, J. K. Lotsberg, B. Yan, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of irradiances in the coupled atmosphere-ocean system,” Appl. Opt. |

21. | K. Hestenes, K. P. Nielsen, L. Zhao, J. J. Stamnes, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of spectral radiances in the coupled air-tissue system,” Appl. Opt. |

22. | F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - I. Theory,” J. Quant. Spectrosc. Radiat. Transfer , |

23. | F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - II. Applications,” J. Quant. Spectrosc. Radiat. Transfer |

24. | F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: An improved and generalized discrete ordinate method for polarized vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer |

25. | F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane parallel, vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transfer |

26. | S. Jiang, “Radiative Transfer in the Coupled Atmosphere-Sea Ice-Ocean System with Applications in Remote Sensing,” |

27. | E. R. Sommersten, “CAO-VDISORT: A discrete ordinate method for polarized (vector) radiative transfer in a coupled system consisting of two media with different indices of refraction,” |

28. | E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transfer |

29. | W. J. Wiscombe, “The delta-M method: Rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. |

30. | J. K. Lotsberg, “Impact of particulate oceanic composition on the radiance and polarization of the natural light field,” |

31. | J. K. Lotsberg and J. J. Stamnes, “Impact of particulate oceanic composition on the radiance and polarization of underwater and backscattered light,” Opt. Expr. |

32. | G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. |

33. | G. I. Marchuk, G. A Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, |

34. | H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. |

35. | H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light,” J. Quant. Spectrosc. Radiat. Transfer |

36. | D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer |

37. | P. W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Expr. |

38. | C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. |

39. | C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. |

40. | J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. |

41. | J. M. Wallace and P. V. Hobbs, |

42. | C. F. Bohren and D. R. Huffman, |

43. | X. Zhou, S. Li, and K. Stamnes, “Geometrical-optics code for computing optical properties of large dielectric spheres,” Appl. Opt. |

44. | J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. |

45. | A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer |

46. | M. I. Mishchenko, “Light scattering by randomly oriented rotationally symmetric particles,” J. Opt. Soc. Am. A |

47. | M. I. Mischenko and L. D. Travis, “Capabilities and Limitations of a Current FORTRAN Implementation of the T-Matrix Method for Randomly Oriented, Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transfer |

48. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

49. | M. Born and E. Wolf, |

50. | J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. |

51. | J. W. Hovenier and C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. |

**OCIS Codes**

(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(280.0280) Remote sensing and sensors : Remote sensing and sensors

(010.5620) Atmospheric and oceanic optics : Radiative transfer

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: January 24, 2013

Revised Manuscript: March 14, 2013

Manuscript Accepted: March 15, 2013

Published: April 10, 2013

**Virtual Issues**

Vol. 8, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

D. Cohen, S. Stamnes, T. Tanikawa, E. R. Sommersten, J. J. Stamnes, J. K. Lotsberg, and K. Stamnes, "Comparison of discrete ordinate and Monte Carlo simulations of polarized radiative transfer in two coupled slabs with different refractive indices," Opt. Express **21**, 9592-9614 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-8-9592

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