Abstract

The accuracy of optical simulations including bulk diffusors is heavily dependent on the accuracy of the bulk scattering properties. If no knowledge on the physical scattering effects is available, an iterative procedure is usually used to obtain the scattering properties, such as the inverse Monte Carlo method or the inverse adding-doubling (AD) method. In these methods, a predefined phase function with one free parameter is usually used to limit the number of free parameters. In this work, three predefined phase functions (Henyey–Greenstein, two-term Henyey–Greenstein, and Gegenbauer kernel (GK) phase function) are implemented in the inverse AD method to determine the optical properties of two strongly diffusing materials: low-density polyethylene and TiO2 particles. Using the presented approach, an estimation of the effective phase function was made. It was found that the use of the GK phase function resulted in the best agreement between calculated and experimental transmittance, reflectance, and scattered radiant intensity distribution for the LDPE sample. For the TiO2 sample, a good agreement was obtained with both the two-term Henyey–Greenstein and the GK phase function.

© 2014 Optical Society of America

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2013

T. Kim, S. Kim, D. Y. Lim, and S.-W. Choi, “A novel diffuser sheet comprising nanosized birefringent fibers embedded within an isotropic polymer matrix,” Opt. Commun. 295, 125–128 (2013).
[CrossRef]

S. Leyre, F. B. Leloup, J. Audenaert, G. Durinck, J. Hofkens, G. Deconinck, and P. Hanselaer, “Determination of the bulk scattering parameters of diffusing materials,” Appl. Opt. 52, 4083–4090 (2013).
[CrossRef]

2012

2011

2010

B.-T. Liu and Y.-T. Teng, “A novel method to control inner and outer haze of an anti-glare film by surface modification of light-scattering particles,” J. Colloid Interf. Sci. 350, 421–426 (2010).

C. C. Sun, W. T. Chien, I. Moreno, C. T. Hsieh, M. C. Lin, S. L. Hsiao, and X. H. Lee, “Calculating model of light transmission efficiency of diffusers attached to a lighting cavity,” Opt. Express 18, 6137–6148 (2010).
[CrossRef]

2008

2006

2001

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, “A scattering phase function for blood with physiological haematocrit,” Phys. Med. Biol. 46, N65–N69 (2001).
[CrossRef]

1998

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Control Optim. 9, 112–147 (1998).

M. D. Schweitzer, B. Michel, E. Thamm, and A. Kolb, “Single scattering by red blood cells,” Appl. Opt. 37, 7410–7418 (1998).
[CrossRef]

1993

1991

R. S. Stein, “Some scattering studies of polymer microstructures, surfaces, and interfaces,” Pure Appl. Chem. 63, 941–950 (1991).
[CrossRef]

1990

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Elect. 26, 2166–2185 (1990).

1980

1975

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).

1969

J. E. Hansen, “Exact and approximate solutions for Multiple scattering by cloudy and hazy planetary atmospheres,” J. Atmos. Sci. 26, 478–487 (1969).
[CrossRef]

1941

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Akarçay, H. G.

Audenaert, J.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Chapman, G. H.

Cheong, W.-F.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Elect. 26, 2166–2185 (1990).

Chien, W. T.

Choi, S.-W.

T. Kim, S. Kim, D. Y. Lim, and S.-W. Choi, “A novel diffuser sheet comprising nanosized birefringent fibers embedded within an isotropic polymer matrix,” Opt. Commun. 295, 125–128 (2013).
[CrossRef]

Deconinck, G.

Durinck, G.

Dutré, P.

Forment, S.

Fredriksson, I.

Frenz, M.

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hammer, M.

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, “A scattering phase function for blood with physiological haematocrit,” Phys. Med. Biol. 46, N65–N69 (2001).
[CrossRef]

Hanselaer, P.

Hansen, J. E.

J. E. Hansen, “Exact and approximate solutions for Multiple scattering by cloudy and hazy planetary atmospheres,” J. Atmos. Sci. 26, 478–487 (1969).
[CrossRef]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hofkens, J.

Hsiao, S. L.

Hsieh, C. T.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Karlsson, H.

Kattawar, G. W.

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).

Kim, S.

T. Kim, S. Kim, D. Y. Lim, and S.-W. Choi, “A novel diffuser sheet comprising nanosized birefringent fibers embedded within an isotropic polymer matrix,” Opt. Commun. 295, 125–128 (2013).
[CrossRef]

Kim, T.

T. Kim, S. Kim, D. Y. Lim, and S.-W. Choi, “A novel diffuser sheet comprising nanosized birefringent fibers embedded within an isotropic polymer matrix,” Opt. Commun. 295, 125–128 (2013).
[CrossRef]

Kolb, A.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Control Optim. 9, 112–147 (1998).

Larsson, M.

Lee, X. H.

Leloup, F. B.

Leyre, S.

Lim, D. Y.

T. Kim, S. Kim, D. Y. Lim, and S.-W. Choi, “A novel diffuser sheet comprising nanosized birefringent fibers embedded within an isotropic polymer matrix,” Opt. Commun. 295, 125–128 (2013).
[CrossRef]

Lin, M. C.

Liu, B.-T.

B.-T. Liu and Y.-T. Teng, “A novel method to control inner and outer haze of an anti-glare film by surface modification of light-scattering particles,” J. Colloid Interf. Sci. 350, 421–426 (2010).

McKee, D.

Michel, B.

Moreno, I.

Palmer, G. M.

Pfeiffer, N.

Piskozub, J.

Pointer, M.

Prahl, S. A.

S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef]

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Elect. 26, 2166–2185 (1990).

Preisser, S.

Ramanujam, N.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Control Optim. 9, 112–147 (1998).

Ricka, J.

Saeys, W.

Schweitzer, D.

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, “A scattering phase function for blood with physiological haematocrit,” Phys. Med. Biol. 46, N65–N69 (2001).
[CrossRef]

Schweitzer, M. D.

Stein, R. S.

R. S. Stein, “Some scattering studies of polymer microstructures, surfaces, and interfaces,” Pure Appl. Chem. 63, 941–950 (1991).
[CrossRef]

Strömberg, T.

Sun, C. C.

Sun, C.-C.

Teng, Y.-T.

B.-T. Liu and Y.-T. Teng, “A novel method to control inner and outer haze of an anti-glare film by surface modification of light-scattering particles,” J. Colloid Interf. Sci. 350, 421–426 (2010).

Thamm, E.

van Gemert, M. J. C.

Van Giel, B.

Welch, A. J.

S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef]

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Elect. 26, 2166–2185 (1990).

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Control Optim. 9, 112–147 (1998).

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Control Optim. 9, 112–147 (1998).

Yaroslavsky, A. N.

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, “A scattering phase function for blood with physiological haematocrit,” Phys. Med. Biol. 46, N65–N69 (2001).
[CrossRef]

Appl. Opt.

Astrophys. J.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Biomed. Opt. Express

IEEE J. Quantum Elect.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Elect. 26, 2166–2185 (1990).

J. Atmos. Sci.

J. E. Hansen, “Exact and approximate solutions for Multiple scattering by cloudy and hazy planetary atmospheres,” J. Atmos. Sci. 26, 478–487 (1969).
[CrossRef]

J. Colloid Interf. Sci.

B.-T. Liu and Y.-T. Teng, “A novel method to control inner and outer haze of an anti-glare film by surface modification of light-scattering particles,” J. Colloid Interf. Sci. 350, 421–426 (2010).

J. Opt. Soc. Am.

J. Quant. Spectrosc. Radiat. Transfer

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).

Opt. Commun.

T. Kim, S. Kim, D. Y. Lim, and S.-W. Choi, “A novel diffuser sheet comprising nanosized birefringent fibers embedded within an isotropic polymer matrix,” Opt. Commun. 295, 125–128 (2013).
[CrossRef]

Opt. Express

Phys. Med. Biol.

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, “A scattering phase function for blood with physiological haematocrit,” Phys. Med. Biol. 46, N65–N69 (2001).
[CrossRef]

Pure Appl. Chem.

R. S. Stein, “Some scattering studies of polymer microstructures, surfaces, and interfaces,” Pure Appl. Chem. 63, 941–950 (1991).
[CrossRef]

SIAM J. Control Optim.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Control Optim. 9, 112–147 (1998).

Other

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic representation of the setup: Φi(λ) represents the incident beam, θ is the scatter angle, Ωrec is the solid angle from sample to the receiver, and Is(θ,λ) represents the spectral scattered intensity in scatter direction θ.

Fig. 2.
Fig. 2.

Spectral reflectance (red), transmittance (blue), and regular transmittance (black) of an LDPE layer with thickness 0.37 mm (dashed lines) and 0.1 wt. % TiO2 in silicone oil in a glass cuvette with path length 1.1 mm (solid lines).

Fig. 3.
Fig. 3.

Deviation between experimental and calculated spectral reflectance (|ΔR|) and transmittance (|ΔT|) when using the HG phase function (red solid lines), the TTHG phase function (blue long dashed lines) and the GK phase function (black short dashed lines).

Fig. 4.
Fig. 4.

NCC (solid lines) and ΔRMS (dashed lines) values for the experimental and calculated scattered radiant intensity patterns for the LDPE sample obtained when using the HG phase function (red), the TTHG phase function (blue), and the GK phase function (black).

Fig. 5.
Fig. 5.

Experimental (magenta solid line) and calculated scattered radiant intensity distribution patterns in transmission at 550 nm obtained when using the HG phase function (red long dashed line), the TTHG phase function (blue short dashed line), and the GK phase function (black dash dot line). 0° represents the normal direction on the LDPE sample.

Fig. 6.
Fig. 6.

Experimental (magenta solid line) and calculated scattered radiant intensity distribution patterns in reflection at 550 nm obtained when using the HG phase function (red long dashed line), the TTHG phase function (blue short dashed line), and the GK phase function (black dash dot line). 0° represents the normal direction on the LDPE sample.

Fig. 7.
Fig. 7.

Bulk scattering properties of the LDPE layer obtained with the IAD method using the GK phase function: τ (red), a (magenta), g (blue), α (black).

Fig. 8.
Fig. 8.

Phase functions for LDPE at 550 nm as obtained with the IAD method using the HG phase function (red solid line), the TTHG phase function (blue short dashed line), and the GK phase function (black long dashed line).

Fig. 9.
Fig. 9.

Experimental (magenta solid line) and calculated scattered radiant intensity distribution patterns in transmission at 550 nm obtained when using the HG phase function (red long dashed line), the TTHG phase function (blue short dashed line), and the GK phase function (black dash dot line). 0° represents the normal direction on the TiO2 sample.

Fig. 10.
Fig. 10.

Experimental (magenta solid line) and calculated scattered radiant intensity distribution patterns in reflection at 550 nm obtained when using the HG phase function (red long dashed line), the TTHG phase function (blue short dashed line), and the GK phase function (black dash dot line). 0° represents the normal direction on the TiO2 sample.

Fig. 11.
Fig. 11.

Phase functions for TiO2 in silicone oil at 550 nm as obtained with the IAD method using the HG phase function (red solid line), the TTHG phase function (blue short dashed line), and the GK phase function (black long dashed line).

Equations (11)

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T(λ)=2·πΦi(λ)0π/2Is(θ,λ)·sin(θ)·dθ+Treg(λ),
R(λ)=2·πΦi(λ)π/2πIs(θ,λ)·sin(θ)·dθ+Rreg(λ).
pHG(θ)=1g24π(1+g22gcos(θ))3/2.
Treg=(1r)2eτ1r2e2τ.
eHG=|RERCRE|+|TETCTE|.
pTTHG(θ)=α1g124π(1+g122g1cos(θ))3/2+(1α)1g224π(1+g222g2cos(θ))3/2.
eTTHG=C1|RERCRE|+C2|TETCTE|+C3(1NCC)+C4ΔRMS.
NCC=i=1N[IE(θi)I¯E][IC(θi)I¯C]i=1N[IE(θi)I¯E]2i=1N[IC(θi)I¯C]2.
ΔRMS=1Ni=1N[IC(θi)IE(θi)]2.
pGK(θ)=K·(1+g22gcos(θ))(α+1)
K=αg(1g2)2απ[(1+g)2α(1g2α)].

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