Abstract

Many modern applications of lasers involve understanding the transport of radiation through thin layers. The interactions of continuous wave and pulsed lasers with skin in dermatological use related to surgery and cosmetic procedures are examples of such. These highly scattering thin layers in skin are best modeled by the Monte Carlo method. However, most traditional Monte Carlo models may inaccurately account for the presence of thin layers. As an example, the very thin epidermis, with its highly absorbing melanin, is known to influence the laser penetration significantly. If the Monte Carlo model is implemented without special features, then the results of the simulation will show incorrect effects of thin layers because the path length of most photons would be significantly larger than the layer thickness. As a result, the computed photon travel path length would simply not feel the presence of the layer. In this paper, we present numerical and algorithmic features for computation of radiation transport through thin layers. It is noted that, while Monte Carlo without special features smears the radiative effect of the layers, the refined technique indicates that layers have a great impact on the absorption of energy, especially if the layer properties are distinctly different from those of the adjacent layers. The results have significant implications in the study of diagnostic and therapeutic applications of lasers in biomedicine and surgery.

© 2010 Optical Society of America

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2006

A. S. Glaich, P. M. Friedman, M. H. Jih, and L. H. Goldberg, “Treatment of inflammatory facial acne vulgaris with combination 595 nm pulsed-dye laser with dynamic-cooling-device and 1,450 nm diode laser,” Lasers Surg. Med. 38, 177-180(2006).
[CrossRef]

2005

K. Mariwalla and T. E. Rohrer, “Use of lasers and light-based therapies for treatment of acne vulgaris,” Lasers Surg. Med. 37, 333-342 (2005).
[CrossRef] [PubMed]

2004

D. Manstein, G. S. Herron, R. K. Sink, H. Tanner, and R. R. Anderson, “Fractional photothermolysis: a new concept for cutaneous remodeling using microscopic patterns of thermal injury,” Lasers Surg. Med. 34, 426-438 (2004).
[CrossRef] [PubMed]

2003

2002

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 73, 159-168 (2002).
[CrossRef]

S. Kimel, L. O. Svaasand, D. Cao, M. J. Hammer-Wilson, and J. S. Nelson, “Vascular response to laser photothermolysis as a function of pulse duration, vessel type, and diameter: implications for port wine stain laser therapy,” Lasers Surg. Med. 30, 160-169 (2002).
[CrossRef] [PubMed]

2000

J. S. Dover and K. A. Arndt, “New approaches to the treatment of vascular lesions,” Lasers Surg. Med. 26, 158-163 (2000).
[CrossRef] [PubMed]

J. Q. Lu, X. H. Hu, and K. Dong, “Modeling of the rough-interface effect on a converging light beam propagating in a skin tissue phantom,” Appl. Opt. 39, 5890-5897 (2000).
[CrossRef]

Z. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short pulse laser transport in scattering media,” J. Thermophys. Heat Transfer 14, pp. 504-511 (2000).
[CrossRef]

1997

G. Lask, M. Elman, P. Noren, P. Lee, and M. Nowfar-Rad, “Hair removal with the Epitouch ruby laser--a multicenter study,” Lasers Surg. Med. 9(suppl.), 32(1997).

G. Teikemeier and D. J. Goldberg, “Skin resurfacing with the erbium-YAG laser,” Dermatol. Surg. 23, 685-687 (1997).
[CrossRef] [PubMed]

1996

1995

L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34-40 (1995).
[CrossRef]

M. J. C. van Gemert, G. H. Gijsbers, J. W. Pickering, O. T. Tan, and A. J. Welch, “Wavelengths for laser treatment of portwine stains and telangiectasia,” Lasers Surg. Med. 16, 147-155(1995).
[CrossRef] [PubMed]

R. G. Wheeland, “Clinical uses of lasers in dermatology,” Lasers Surg. Med. 16, 2-23 (1995).
[CrossRef] [PubMed]

1993

M. P. Goldman, R. E. Fitzpatrick, and J. Ruiz-Esparza, “Treatment of port-wine stains (capillary malformation) with the flash lamp-pumped pulsed dye laser,” J. Pediatr. 122, 71-77(1993).
[CrossRef] [PubMed]

S. L. Kilmer and R. R. Anderson, “Clinical use of the Q-switched ruby and the Q-switched Nd:YAG (1064 nm and 532 nm) lasers for treatment of tattoos,” J. Dermatol. Surg. Oncol. 19, 330-338 (1993).
[PubMed]

R. E. Fitzpatrick, M. P. Goldman, and J. Ruiz-Esparza, “Use of the alexandrite laser (755 nm, 100 nsec) for tattoo pigment removal in an animal model,” J. Am. Acad. Dermatol. 28, 745-750 (1993).
[CrossRef] [PubMed]

A. H. Gandbakche, R. Nossal, and R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504-516 (1993).
[CrossRef]

S. L. Jacques, “Role of tissue optics and pulse duration on tissue effects during high-power laser irradiation,” Appl. Opt. 32, 2447-2454 (1993).
[CrossRef] [PubMed]

1992

1991

Y. Hasegawa, Y. Yamada, M. Tamura, and Y. Nomura, “Monte Carlo simulation of light transmission through living tissues,” Appl. Opt. 30, 4515-4520 (1991).
[CrossRef] [PubMed]

R. Ashinoff and R. G. Geronemus, “Flashlamp-pumped pulsed dye laser for port-wine stains in infancy: earlier versus later treatment,” J. Am. Acad. Dermatol. 24, 467-472 (1991).
[CrossRef] [PubMed]

1990

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

1989

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. II: Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

M. J. C. van Gemert and A. J. Welch, “Clinical use of laser-tissue interactions,” IEEE Eng. Med. Biol. Mag. 8, 10-13(1989).
[CrossRef] [PubMed]

A. Ishimaru, “Diffusion of light in turbid materials,” Appl. Opt. 28, 2210-2215 (1989).
[CrossRef] [PubMed]

E. A. Profio, “Light transport in tissue,” Appl. Opt. 28, 2216-2222 (1989).
[CrossRef] [PubMed]

1988

J. R. Howell, “Thermal radiation in participating media: the past, the present, and some possible futures,” J. Heat Transfer 110, 1220-1229 (1988).
[CrossRef]

1983

R. R. Anderson and J. A. Parish, “Selective photothermolysis: precise microsurgery by selective absorption of pulsed radiation,” Science 220, 524-527 (1983).
[CrossRef] [PubMed]

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

1981

R. Anderson and J. A. Parish, “The optics of human skin,” J. Invest. Dermatol. 77, 13-19 (1981).
[CrossRef] [PubMed]

1977

L. I. Grossweiner, “Application of diffusion theory to photodynamic damage in large targets,” J. Photochem. Photobiol. 26, 309-311 (1977).
[CrossRef]

Aber, J.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 73, 159-168 (2002).
[CrossRef]

Adam, G.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

Anderson, R.

R. Anderson and J. A. Parish, “The optics of human skin,” J. Invest. Dermatol. 77, 13-19 (1981).
[CrossRef] [PubMed]

Anderson, R. R.

D. Manstein, G. S. Herron, R. K. Sink, H. Tanner, and R. R. Anderson, “Fractional photothermolysis: a new concept for cutaneous remodeling using microscopic patterns of thermal injury,” Lasers Surg. Med. 34, 426-438 (2004).
[CrossRef] [PubMed]

S. L. Kilmer and R. R. Anderson, “Clinical use of the Q-switched ruby and the Q-switched Nd:YAG (1064 nm and 532 nm) lasers for treatment of tattoos,” J. Dermatol. Surg. Oncol. 19, 330-338 (1993).
[PubMed]

S. D. DeCoste, W. Farinelli, T. Flotte, and R. R. Anderson, “Dye-enhanced laser welding for skin closure,” Lasers Surg. Med. 12, 25-32 (1992).
[CrossRef] [PubMed]

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

R. R. Anderson and J. A. Parish, “Selective photothermolysis: precise microsurgery by selective absorption of pulsed radiation,” Science 220, 524-527 (1983).
[CrossRef] [PubMed]

Arndt, K. A.

J. S. Dover and K. A. Arndt, “New approaches to the treatment of vascular lesions,” Lasers Surg. Med. 26, 158-163 (2000).
[CrossRef] [PubMed]

Ashinoff, R.

R. Ashinoff and R. G. Geronemus, “Flashlamp-pumped pulsed dye laser for port-wine stains in infancy: earlier versus later treatment,” J. Am. Acad. Dermatol. 24, 467-472 (1991).
[CrossRef] [PubMed]

Beard, P. C.

Bonner, R. F.

Brewster, M. Q.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley Interscience, 1992).

Cao, D.

S. Kimel, L. O. Svaasand, D. Cao, M. J. Hammer-Wilson, and J. S. Nelson, “Vascular response to laser photothermolysis as a function of pulse duration, vessel type, and diameter: implications for port wine stain laser therapy,” Lasers Surg. Med. 30, 160-169 (2002).
[CrossRef] [PubMed]

Chance, B.

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34-40 (1995).
[CrossRef]

DeCoste, S. D.

S. D. DeCoste, W. Farinelli, T. Flotte, and R. R. Anderson, “Dye-enhanced laser welding for skin closure,” Lasers Surg. Med. 12, 25-32 (1992).
[CrossRef] [PubMed]

Dong, K.

Dover, J. S.

J. S. Dover and K. A. Arndt, “New approaches to the treatment of vascular lesions,” Lasers Surg. Med. 26, 158-163 (2000).
[CrossRef] [PubMed]

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

Elman, M.

G. Lask, M. Elman, P. Noren, P. Lee, and M. Nowfar-Rad, “Hair removal with the Epitouch ruby laser--a multicenter study,” Lasers Surg. Med. 9(suppl.), 32(1997).

Farinelli, W.

S. D. DeCoste, W. Farinelli, T. Flotte, and R. R. Anderson, “Dye-enhanced laser welding for skin closure,” Lasers Surg. Med. 12, 25-32 (1992).
[CrossRef] [PubMed]

Farmer, J. T.

J. T. Farmer and J. R. Howell, “Comparison of Monte Carlo strategies for radiative transfer in participating media,” in Advances in Heat Transfer, J. P. Hartnett and T. Irvine, eds. (Academic, 1996), Vol. 31, pp. 1-97.

Fitzpatrick, R. E.

R. E. Fitzpatrick, M. P. Goldman, and J. Ruiz-Esparza, “Use of the alexandrite laser (755 nm, 100 nsec) for tattoo pigment removal in an animal model,” J. Am. Acad. Dermatol. 28, 745-750 (1993).
[CrossRef] [PubMed]

M. P. Goldman, R. E. Fitzpatrick, and J. Ruiz-Esparza, “Treatment of port-wine stains (capillary malformation) with the flash lamp-pumped pulsed dye laser,” J. Pediatr. 122, 71-77(1993).
[CrossRef] [PubMed]

Flock, S. T.

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. II: Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

Flotte, T.

S. D. DeCoste, W. Farinelli, T. Flotte, and R. R. Anderson, “Dye-enhanced laser welding for skin closure,” Lasers Surg. Med. 12, 25-32 (1992).
[CrossRef] [PubMed]

Flotte, T. J.

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

Friedman, P. M.

A. S. Glaich, P. M. Friedman, M. H. Jih, and L. H. Goldberg, “Treatment of inflammatory facial acne vulgaris with combination 595 nm pulsed-dye laser with dynamic-cooling-device and 1,450 nm diode laser,” Lasers Surg. Med. 38, 177-180(2006).
[CrossRef]

Gandbakche, A. H.

Gange, R. W.

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

Garetz, B. A.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 73, 159-168 (2002).
[CrossRef]

Geronemus, R. G.

R. Ashinoff and R. G. Geronemus, “Flashlamp-pumped pulsed dye laser for port-wine stains in infancy: earlier versus later treatment,” J. Am. Acad. Dermatol. 24, 467-472 (1991).
[CrossRef] [PubMed]

Gijsbers, G. H.

M. J. C. van Gemert, G. H. Gijsbers, J. W. Pickering, O. T. Tan, and A. J. Welch, “Wavelengths for laser treatment of portwine stains and telangiectasia,” Lasers Surg. Med. 16, 147-155(1995).
[CrossRef] [PubMed]

Glaich, A. S.

A. S. Glaich, P. M. Friedman, M. H. Jih, and L. H. Goldberg, “Treatment of inflammatory facial acne vulgaris with combination 595 nm pulsed-dye laser with dynamic-cooling-device and 1,450 nm diode laser,” Lasers Surg. Med. 38, 177-180(2006).
[CrossRef]

Goldberg, D. J.

G. Teikemeier and D. J. Goldberg, “Skin resurfacing with the erbium-YAG laser,” Dermatol. Surg. 23, 685-687 (1997).
[CrossRef] [PubMed]

Goldberg, L. H.

A. S. Glaich, P. M. Friedman, M. H. Jih, and L. H. Goldberg, “Treatment of inflammatory facial acne vulgaris with combination 595 nm pulsed-dye laser with dynamic-cooling-device and 1,450 nm diode laser,” Lasers Surg. Med. 38, 177-180(2006).
[CrossRef]

Goldman, M. P.

M. P. Goldman, R. E. Fitzpatrick, and J. Ruiz-Esparza, “Treatment of port-wine stains (capillary malformation) with the flash lamp-pumped pulsed dye laser,” J. Pediatr. 122, 71-77(1993).
[CrossRef] [PubMed]

R. E. Fitzpatrick, M. P. Goldman, and J. Ruiz-Esparza, “Use of the alexandrite laser (755 nm, 100 nsec) for tattoo pigment removal in an animal model,” J. Am. Acad. Dermatol. 28, 745-750 (1993).
[CrossRef] [PubMed]

Gonzalez, E.

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

Grossweiner, L. I.

L. I. Grossweiner, “Application of diffusion theory to photodynamic damage in large targets,” J. Photochem. Photobiol. 26, 309-311 (1977).
[CrossRef]

Guo, Z.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 73, 159-168 (2002).
[CrossRef]

Z. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short pulse laser transport in scattering media,” J. Thermophys. Heat Transfer 14, pp. 504-511 (2000).
[CrossRef]

Hammer-Wilson, M. J.

S. Kimel, L. O. Svaasand, D. Cao, M. J. Hammer-Wilson, and J. S. Nelson, “Vascular response to laser photothermolysis as a function of pulse duration, vessel type, and diameter: implications for port wine stain laser therapy,” Lasers Surg. Med. 30, 160-169 (2002).
[CrossRef] [PubMed]

Hasegawa, Y.

Hefetz, Y.

Herron, G. S.

D. Manstein, G. S. Herron, R. K. Sink, H. Tanner, and R. R. Anderson, “Fractional photothermolysis: a new concept for cutaneous remodeling using microscopic patterns of thermal injury,” Lasers Surg. Med. 34, 426-438 (2004).
[CrossRef] [PubMed]

Howell, J. R.

J. R. Howell, “Thermal radiation in participating media: the past, the present, and some possible futures,” J. Heat Transfer 110, 1220-1229 (1988).
[CrossRef]

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (Taylor and Francis, 2002).

J. T. Farmer and J. R. Howell, “Comparison of Monte Carlo strategies for radiative transfer in participating media,” in Advances in Heat Transfer, J. P. Hartnett and T. Irvine, eds. (Academic, 1996), Vol. 31, pp. 1-97.

Hu, X. H.

Ishimaru, A.

Jacques, S. J.

Jacques, S. L.

L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

S. L. Jacques, “Role of tissue optics and pulse duration on tissue effects during high-power laser irradiation,” Appl. Opt. 32, 2447-2454 (1993).
[CrossRef] [PubMed]

S. L. Jacques and L. H. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 73-100.

Jih, M. H.

A. S. Glaich, P. M. Friedman, M. H. Jih, and L. H. Goldberg, “Treatment of inflammatory facial acne vulgaris with combination 595 nm pulsed-dye laser with dynamic-cooling-device and 1,450 nm diode laser,” Lasers Surg. Med. 38, 177-180(2006).
[CrossRef]

Judy, M. M.

M. M. Judy, “Biomedical lasers,” Medical Devices and Systems (CRC, 1995), pp. 1333-1345.

Kilmer, S. L.

S. L. Kilmer and R. R. Anderson, “Clinical use of the Q-switched ruby and the Q-switched Nd:YAG (1064 nm and 532 nm) lasers for treatment of tattoos,” J. Dermatol. Surg. Oncol. 19, 330-338 (1993).
[PubMed]

Kimel, S.

S. Kimel, L. O. Svaasand, D. Cao, M. J. Hammer-Wilson, and J. S. Nelson, “Vascular response to laser photothermolysis as a function of pulse duration, vessel type, and diameter: implications for port wine stain laser therapy,” Lasers Surg. Med. 30, 160-169 (2002).
[CrossRef] [PubMed]

Kostli, K. P.

Kudo, K.

W. J. Yang, H. Taniguchi, and K. Kudo, “Radiative heat transfer by the Monte Carlo method,” in Advances in Heat Transfer (Academic, 1995), Vol. 27, pp. 1-215.
[CrossRef]

Kumar, S.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 73, 159-168 (2002).
[CrossRef]

Z. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short pulse laser transport in scattering media,” J. Thermophys. Heat Transfer 14, pp. 504-511 (2000).
[CrossRef]

S. Kumar, K. Mitra, and Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scattering media,” Appl. Opt. 35, 3372-3378 (1996).
[CrossRef] [PubMed]

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Advances in Heat Transfer (Academic, 1999), Vol. 33, pp. 187-294.
[CrossRef]

Lask, G.

G. Lask, M. Elman, P. Noren, P. Lee, and M. Nowfar-Rad, “Hair removal with the Epitouch ruby laser--a multicenter study,” Lasers Surg. Med. 9(suppl.), 32(1997).

Lee, P.

G. Lask, M. Elman, P. Noren, P. Lee, and M. Nowfar-Rad, “Hair removal with the Epitouch ruby laser--a multicenter study,” Lasers Surg. Med. 9(suppl.), 32(1997).

Lu, J. Q.

Madsen, S. J.

Manstein, D.

D. Manstein, G. S. Herron, R. K. Sink, H. Tanner, and R. R. Anderson, “Fractional photothermolysis: a new concept for cutaneous remodeling using microscopic patterns of thermal injury,” Lasers Surg. Med. 34, 426-438 (2004).
[CrossRef] [PubMed]

Mariwalla, K.

K. Mariwalla and T. E. Rohrer, “Use of lasers and light-based therapies for treatment of acne vulgaris,” Lasers Surg. Med. 37, 333-342 (2005).
[CrossRef] [PubMed]

Michaud, N.

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

Mitra, K.

S. Kumar, K. Mitra, and Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scattering media,” Appl. Opt. 35, 3372-3378 (1996).
[CrossRef] [PubMed]

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Advances in Heat Transfer (Academic, 1999), Vol. 33, pp. 187-294.
[CrossRef]

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (Academic, 2003).

Nelson, J. S.

S. Kimel, L. O. Svaasand, D. Cao, M. J. Hammer-Wilson, and J. S. Nelson, “Vascular response to laser photothermolysis as a function of pulse duration, vessel type, and diameter: implications for port wine stain laser therapy,” Lasers Surg. Med. 30, 160-169 (2002).
[CrossRef] [PubMed]

Nomura, Y.

Noren, P.

G. Lask, M. Elman, P. Noren, P. Lee, and M. Nowfar-Rad, “Hair removal with the Epitouch ruby laser--a multicenter study,” Lasers Surg. Med. 9(suppl.), 32(1997).

Nossal, R.

Nowfar-Rad, M.

G. Lask, M. Elman, P. Noren, P. Lee, and M. Nowfar-Rad, “Hair removal with the Epitouch ruby laser--a multicenter study,” Lasers Surg. Med. 9(suppl.), 32(1997).

Parish, J. A.

R. R. Anderson and J. A. Parish, “Selective photothermolysis: precise microsurgery by selective absorption of pulsed radiation,” Science 220, 524-527 (1983).
[CrossRef] [PubMed]

R. Anderson and J. A. Parish, “The optics of human skin,” J. Invest. Dermatol. 77, 13-19 (1981).
[CrossRef] [PubMed]

Park, Y. D.

Patterson, M. S.

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. J. Jacques, and Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffusion reflectance measurements,” Appl. Opt. 31, 3509-3517(1992).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. II: Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

Pickering, J. W.

M. J. C. van Gemert, G. H. Gijsbers, J. W. Pickering, O. T. Tan, and A. J. Welch, “Wavelengths for laser treatment of portwine stains and telangiectasia,” Lasers Surg. Med. 16, 147-155(1995).
[CrossRef] [PubMed]

M. J. C. van Gemert, J. W. Pickering, and A. J. Welch, “Modeling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea and Febiger, 1992), pp. 24-47.

Profio, E. A.

Rohrer, T. E.

K. Mariwalla and T. E. Rohrer, “Use of lasers and light-based therapies for treatment of acne vulgaris,” Lasers Surg. Med. 37, 333-342 (2005).
[CrossRef] [PubMed]

Ruiz-Esparza, J.

M. P. Goldman, R. E. Fitzpatrick, and J. Ruiz-Esparza, “Treatment of port-wine stains (capillary malformation) with the flash lamp-pumped pulsed dye laser,” J. Pediatr. 122, 71-77(1993).
[CrossRef] [PubMed]

R. E. Fitzpatrick, M. P. Goldman, and J. Ruiz-Esparza, “Use of the alexandrite laser (755 nm, 100 nsec) for tattoo pigment removal in an animal model,” J. Am. Acad. Dermatol. 28, 745-750 (1993).
[CrossRef] [PubMed]

San, K. C.

Z. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short pulse laser transport in scattering media,” J. Thermophys. Heat Transfer 14, pp. 504-511 (2000).
[CrossRef]

Siegel, R.

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (Taylor and Francis, 2002).

Sink, R. K.

D. Manstein, G. S. Herron, R. K. Sink, H. Tanner, and R. R. Anderson, “Fractional photothermolysis: a new concept for cutaneous remodeling using microscopic patterns of thermal injury,” Lasers Surg. Med. 34, 426-438 (2004).
[CrossRef] [PubMed]

Svaasand, L. O.

S. Kimel, L. O. Svaasand, D. Cao, M. J. Hammer-Wilson, and J. S. Nelson, “Vascular response to laser photothermolysis as a function of pulse duration, vessel type, and diameter: implications for port wine stain laser therapy,” Lasers Surg. Med. 30, 160-169 (2002).
[CrossRef] [PubMed]

Tamura, M.

Tan, O. T.

M. J. C. van Gemert, G. H. Gijsbers, J. W. Pickering, O. T. Tan, and A. J. Welch, “Wavelengths for laser treatment of portwine stains and telangiectasia,” Lasers Surg. Med. 16, 147-155(1995).
[CrossRef] [PubMed]

Taniguchi, H.

W. J. Yang, H. Taniguchi, and K. Kudo, “Radiative heat transfer by the Monte Carlo method,” in Advances in Heat Transfer (Academic, 1995), Vol. 27, pp. 1-215.
[CrossRef]

Tanner, H.

D. Manstein, G. S. Herron, R. K. Sink, H. Tanner, and R. R. Anderson, “Fractional photothermolysis: a new concept for cutaneous remodeling using microscopic patterns of thermal injury,” Lasers Surg. Med. 34, 426-438 (2004).
[CrossRef] [PubMed]

Taylor, C. R.

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

Teikemeier, G.

G. Teikemeier and D. J. Goldberg, “Skin resurfacing with the erbium-YAG laser,” Dermatol. Surg. 23, 685-687 (1997).
[CrossRef] [PubMed]

van Gemert, M. J. C.

M. J. C. van Gemert, G. H. Gijsbers, J. W. Pickering, O. T. Tan, and A. J. Welch, “Wavelengths for laser treatment of portwine stains and telangiectasia,” Lasers Surg. Med. 16, 147-155(1995).
[CrossRef] [PubMed]

M. J. C. van Gemert and A. J. Welch, “Clinical use of laser-tissue interactions,” IEEE Eng. Med. Biol. Mag. 8, 10-13(1989).
[CrossRef] [PubMed]

M. J. C. van Gemert, J. W. Pickering, and A. J. Welch, “Modeling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea and Febiger, 1992), pp. 24-47.

Wang, L. H.

S. L. Jacques and L. H. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 73-100.

Wang, L.-H.

L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

Welch, A. J.

M. J. C. van Gemert, G. H. Gijsbers, J. W. Pickering, O. T. Tan, and A. J. Welch, “Wavelengths for laser treatment of portwine stains and telangiectasia,” Lasers Surg. Med. 16, 147-155(1995).
[CrossRef] [PubMed]

M. J. C. van Gemert and A. J. Welch, “Clinical use of laser-tissue interactions,” IEEE Eng. Med. Biol. Mag. 8, 10-13(1989).
[CrossRef] [PubMed]

M. J. C. van Gemert, J. W. Pickering, and A. J. Welch, “Modeling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea and Febiger, 1992), pp. 24-47.

Wheeland, R. G.

R. G. Wheeland, “Clinical uses of lasers in dermatology,” Lasers Surg. Med. 16, 2-23 (1995).
[CrossRef] [PubMed]

Wilson, B. C.

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. J. Jacques, and Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffusion reflectance measurements,” Appl. Opt. 31, 3509-3517(1992).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. II: Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

Wyman, D. R.

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. II: Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

Yamada, Y.

Yang, W. J.

W. J. Yang, H. Taniguchi, and K. Kudo, “Radiative heat transfer by the Monte Carlo method,” in Advances in Heat Transfer (Academic, 1995), Vol. 27, pp. 1-215.
[CrossRef]

Yodh, A.

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34-40 (1995).
[CrossRef]

Zheng, L.-Q.

L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

Appl. Opt.

A. Ishimaru, “Diffusion of light in turbid materials,” Appl. Opt. 28, 2210-2215 (1989).
[CrossRef] [PubMed]

E. A. Profio, “Light transport in tissue,” Appl. Opt. 28, 2216-2222 (1989).
[CrossRef] [PubMed]

Y. Hasegawa, Y. Yamada, M. Tamura, and Y. Nomura, “Monte Carlo simulation of light transmission through living tissues,” Appl. Opt. 30, 4515-4520 (1991).
[CrossRef] [PubMed]

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. J. Jacques, and Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffusion reflectance measurements,” Appl. Opt. 31, 3509-3517(1992).
[CrossRef] [PubMed]

A. H. Gandbakche, R. Nossal, and R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504-516 (1993).
[CrossRef]

S. L. Jacques, “Role of tissue optics and pulse duration on tissue effects during high-power laser irradiation,” Appl. Opt. 32, 2447-2454 (1993).
[CrossRef] [PubMed]

S. Kumar, K. Mitra, and Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scattering media,” Appl. Opt. 35, 3372-3378 (1996).
[CrossRef] [PubMed]

J. Q. Lu, X. H. Hu, and K. Dong, “Modeling of the rough-interface effect on a converging light beam propagating in a skin tissue phantom,” Appl. Opt. 39, 5890-5897 (2000).
[CrossRef]

K. P. Kostli and P. C. Beard, “Two-dimensional photoacoustic imaging by use of Fourier-transform image reconstruction and a detector with an anisotropic response,” Appl. Opt. 42, 1899-1908 (2003).
[CrossRef] [PubMed]

Arch. Dermatol.

C. R. Taylor, R. W. Gange, J. S. Dover, T. J. Flotte, E. Gonzalez, N. Michaud, and R. R. Anderson, “Treatment of tattoos by Q-switched ruby laser: a dose-response study,” Arch. Dermatol. 126, 893-899 (1990).
[CrossRef] [PubMed]

Comp. Methods Programs Biomed.

L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

Dermatol. Surg.

G. Teikemeier and D. J. Goldberg, “Skin resurfacing with the erbium-YAG laser,” Dermatol. Surg. 23, 685-687 (1997).
[CrossRef] [PubMed]

IEEE Eng. Med. Biol. Mag.

M. J. C. van Gemert and A. J. Welch, “Clinical use of laser-tissue interactions,” IEEE Eng. Med. Biol. Mag. 8, 10-13(1989).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng.

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, D. R. Wyman, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues. II: Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

J. Am. Acad. Dermatol.

R. Ashinoff and R. G. Geronemus, “Flashlamp-pumped pulsed dye laser for port-wine stains in infancy: earlier versus later treatment,” J. Am. Acad. Dermatol. 24, 467-472 (1991).
[CrossRef] [PubMed]

R. E. Fitzpatrick, M. P. Goldman, and J. Ruiz-Esparza, “Use of the alexandrite laser (755 nm, 100 nsec) for tattoo pigment removal in an animal model,” J. Am. Acad. Dermatol. 28, 745-750 (1993).
[CrossRef] [PubMed]

J. Dermatol. Surg. Oncol.

S. L. Kilmer and R. R. Anderson, “Clinical use of the Q-switched ruby and the Q-switched Nd:YAG (1064 nm and 532 nm) lasers for treatment of tattoos,” J. Dermatol. Surg. Oncol. 19, 330-338 (1993).
[PubMed]

J. Heat Transfer

J. R. Howell, “Thermal radiation in participating media: the past, the present, and some possible futures,” J. Heat Transfer 110, 1220-1229 (1988).
[CrossRef]

J. Invest. Dermatol.

R. Anderson and J. A. Parish, “The optics of human skin,” J. Invest. Dermatol. 77, 13-19 (1981).
[CrossRef] [PubMed]

J. Pediatr.

M. P. Goldman, R. E. Fitzpatrick, and J. Ruiz-Esparza, “Treatment of port-wine stains (capillary malformation) with the flash lamp-pumped pulsed dye laser,” J. Pediatr. 122, 71-77(1993).
[CrossRef] [PubMed]

J. Photochem. Photobiol.

L. I. Grossweiner, “Application of diffusion theory to photodynamic damage in large targets,” J. Photochem. Photobiol. 26, 309-311 (1977).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 73, 159-168 (2002).
[CrossRef]

J. Thermophys. Heat Transfer

Z. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short pulse laser transport in scattering media,” J. Thermophys. Heat Transfer 14, pp. 504-511 (2000).
[CrossRef]

Lasers Surg. Med.

S. D. DeCoste, W. Farinelli, T. Flotte, and R. R. Anderson, “Dye-enhanced laser welding for skin closure,” Lasers Surg. Med. 12, 25-32 (1992).
[CrossRef] [PubMed]

S. Kimel, L. O. Svaasand, D. Cao, M. J. Hammer-Wilson, and J. S. Nelson, “Vascular response to laser photothermolysis as a function of pulse duration, vessel type, and diameter: implications for port wine stain laser therapy,” Lasers Surg. Med. 30, 160-169 (2002).
[CrossRef] [PubMed]

D. Manstein, G. S. Herron, R. K. Sink, H. Tanner, and R. R. Anderson, “Fractional photothermolysis: a new concept for cutaneous remodeling using microscopic patterns of thermal injury,” Lasers Surg. Med. 34, 426-438 (2004).
[CrossRef] [PubMed]

R. G. Wheeland, “Clinical uses of lasers in dermatology,” Lasers Surg. Med. 16, 2-23 (1995).
[CrossRef] [PubMed]

A. S. Glaich, P. M. Friedman, M. H. Jih, and L. H. Goldberg, “Treatment of inflammatory facial acne vulgaris with combination 595 nm pulsed-dye laser with dynamic-cooling-device and 1,450 nm diode laser,” Lasers Surg. Med. 38, 177-180(2006).
[CrossRef]

K. Mariwalla and T. E. Rohrer, “Use of lasers and light-based therapies for treatment of acne vulgaris,” Lasers Surg. Med. 37, 333-342 (2005).
[CrossRef] [PubMed]

J. S. Dover and K. A. Arndt, “New approaches to the treatment of vascular lesions,” Lasers Surg. Med. 26, 158-163 (2000).
[CrossRef] [PubMed]

M. J. C. van Gemert, G. H. Gijsbers, J. W. Pickering, O. T. Tan, and A. J. Welch, “Wavelengths for laser treatment of portwine stains and telangiectasia,” Lasers Surg. Med. 16, 147-155(1995).
[CrossRef] [PubMed]

G. Lask, M. Elman, P. Noren, P. Lee, and M. Nowfar-Rad, “Hair removal with the Epitouch ruby laser--a multicenter study,” Lasers Surg. Med. 9(suppl.), 32(1997).

Med. Phys.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

Phys. Today

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34-40 (1995).
[CrossRef]

Science

R. R. Anderson and J. A. Parish, “Selective photothermolysis: precise microsurgery by selective absorption of pulsed radiation,” Science 220, 524-527 (1983).
[CrossRef] [PubMed]

Other

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Advances in Heat Transfer (Academic, 1999), Vol. 33, pp. 187-294.
[CrossRef]

M. M. Judy, “Biomedical lasers,” Medical Devices and Systems (CRC, 1995), pp. 1333-1345.

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (Taylor and Francis, 2002).

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley Interscience, 1992).

M. F. Modest, Radiative Heat Transfer (Academic, 2003).

M. J. C. van Gemert, J. W. Pickering, and A. J. Welch, “Modeling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea and Febiger, 1992), pp. 24-47.

S. L. Jacques and L. H. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 73-100.

J. T. Farmer and J. R. Howell, “Comparison of Monte Carlo strategies for radiative transfer in participating media,” in Advances in Heat Transfer, J. P. Hartnett and T. Irvine, eds. (Academic, 1996), Vol. 31, pp. 1-97.

W. J. Yang, H. Taniguchi, and K. Kudo, “Radiative heat transfer by the Monte Carlo method,” in Advances in Heat Transfer (Academic, 1995), Vol. 27, pp. 1-215.
[CrossRef]

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” in Annual Review of Heat Transfer, C. L. Tien, ed. (Begell House, 1995), Vol. 6, pp. 1-59.

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Figures (6)

Fig. 1
Fig. 1

(a) Typical skin cross section (http://www.healthy-skin-guide.com/skin-diagram.html). © Wikimedia Commons (b) Multilayer equivalent model of the skin with the very thin layer corresponding to the epidermis. (c) Parametric multilayer model to examine the effect of placement of thin layer. (d) Schematic representation of path length estimation.

Fig. 2
Fig. 2

Flow chart of the multilayer variable step size Monte Carlo technique.

Fig. 3
Fig. 3

(a) Semi-log plot showing the absorption density profile for both the TMC model and the TLMC model for reflecting and nonreflecting topmost layer. (b) Log-log plot showing the absorption density profile for both the TMC model and the TLMC model for reflecting and nonreflecting topmost layers.

Fig. 4
Fig. 4

Semi-log plot showing the absorption density profile for various values of absorption coefficient in the thin layer using the model of Monte Carlo tailored for thin layers (TLMC) for a reflecting top layer.

Fig. 5
Fig. 5

(a) Plot showing the effect of placement of a thin layer between the top and bottom layers and its significance on the absorption density profile for both the TMC model and the TLMC model for reflecting and nonreflecting topmost layers. The region of interest is encased in boxed area A. (b) Magnification of the boxed section A of Fig. 5a detailing the effect of placement of a thin layer between the top and bottom layers and its significance on the absorption density profile for both the TMC model and the TLMC model for reflecting and nonreflecting topmost layers.

Fig. 6
Fig. 6

(a) Parametric study of the effect of placement of thin layer between the top and bottom layers and its significance on the absorption density profile for the TLMC model tailored for thin layers for reflecting topmost layer. The region of interest is enclosed in boxed area B. (b) Magnification of the boxed section B of Fig. 6a detailing the parametric assessment of the effect of effect of placement of a thin layer between the top and bottom layers and its significance on the absorption density profile for the TLMC model for a reflecting topmost layer.

Tables (1)

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Table 1 Optical Properties of Caucasian Type 1 Blonde Skin with 5% Melanin used in the Simulation

Equations (23)

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Δ W = W μ a μ a + μ s = W μ a μ t ,
W = W Δ W .
r source = r ξ ,
m f p = 1 μ t = 1 ( μ a + μ s ) .
d = ln ξ μ t .
x x + u d ,
y y + v d ,
z z + w d .
d new = { μ t a μ t b ( d z 1 z w ) + z 1 z w , if     w > 0 μ t a μ t b ( d + z 0 z w ) z 0 z w , if     w < 0 ,
R s ( θ i ) = 1 2 ( sin 2 ( θ i θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i θ t ) tan 2 ( θ i + θ t ) ) ,
n i sin θ i = n t sin θ t .
R s n = ( n i n t n i + n t ) 2 ,
I reflected = R d cos θ i π I incident .
ϕ = cos 1 ( 2 ξ 1 ) .
P h g ( cos θ 0 ) = 1 2 1 g 2 ( 1 + g 2 2 g cos θ 0 ) 3 / 2 .
cos θ 0 = { 1 2 { 1 + g 2 ( 1 g 2 1 g + 2 g ξ ) 2 } , if    g 0 2 ξ 1 , if    g = 0
cos θ 0 = cos θ cos θ + sin θ sin θ cos ( ϕ ϕ ) ,
u sin θ 1 w 2 ( u w cos ϕ v sin ϕ ) + u cos θ ,
v sin θ 1 w 2 ( v w cos ϕ + u sin ϕ ) + v cos θ ,
w sin θ cos ϕ 1 w 2 + u cos θ .
u sin θ cos ϕ ,
v sin θ sin θ ,
w sign ( w ) cos θ .

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