Abstract

The angular dependence and the polarization of light scattered by a small particle a distance d inside and outside a reflecting surface is calculated in the Rayleigh limit. This calculation yields expressions for the polarized bidirectional reflectance distribution function matrices for in-plane and out-of-plane scattering. The results are compared with those obtained from microroughness-induced scattering. For the p-in/p-out configuration with oblique incidence, there exist out-of-plane angles for which scattering that is due to one of the mechanisms vanishes, whereas that from the others does not. By exploiting this knowledge, we can make improvements in the detection of small particles or subsurface defects. It is also shown that one must take care when differentiating subsurface-defect-induced scattering from microroughness-induced scattering using in-plane scattering and wavelength scaling laws.

© 1997 Optical Society of America

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References

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  1. P. Burggraaf, “Pursuing advanced metrology solutions,” Semicond. Int. 17, 62–64 (1994).
  2. R. S. Howland, “Detecting killer particles on rough surfaces,” Semicond. Int. 17, 164–170 (1994).
  3. E. Morita, H. Okuda, F. Inoue, “Distinguishing COPs from real particles,” Semicond. Int. 17, 156–162 (1994).
  4. K. Moriya, A. Yazaki, K. Hirai, “Detection and identification of near-surface microprecipitates in silicon wafers by laser scattering tomography,” Jpn. J. Appl. Phys. Pt. 1 34, 5721–5728 (1995).
    [CrossRef]
  5. T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
    [CrossRef]
  6. T. A. Germer, C. C. Asmail, “Bidirectional ellipsometry and its application to the characterization of surfaces,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 173–182 (1997).
  7. G. Videen, W. L. Wolfe, W. S. Bickel, “Light scattering Mueller matrix for a surface contaminated by a single particle in the Rayleigh limit,” Opt. Eng. 31, 341–349 (1992).
    [CrossRef]
  8. G. Videen, M. G. Turner, V. J. Iafelice, W. S. Bickel, W. L. Wolfe, “Scattering from a small sphere near a surface,” J. Opt. Soc. Am. A 10, 118–126 (1993).
    [CrossRef]
  9. G. Videen, “Light scattering from a sphere behind a surface,” J. Opt. Soc. Am. A 10, 110–117 (1993).
    [CrossRef]
  10. D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1980).
  12. G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
    [CrossRef]
  13. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  14. D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970).
  15. W. S. Bickel, A. J. Watkins, G. Videen, “The light-scattering Mueller matrix elements for Rayleigh, Rayleigh-Gans, and Mie spheres,” Am. J. Phys. 55, 559–561 (1987).
    [CrossRef]
  16. P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica A 137, 243–257 (1986).
    [CrossRef]
  17. K. B. Nahm, W. L. Wolfe, “Light-scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987).
    [CrossRef] [PubMed]
  18. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991).
    [CrossRef]
  19. G. Videen, “Light scattering from a sphere on or near a surface: errata,” J. Opt. Soc. Am. A 9, 844–845 (1992).
    [CrossRef]
  20. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
    [CrossRef]
  21. D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 27, 4019–4026 (1988).
    [CrossRef] [PubMed]
  22. J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).

1997 (1)

1995 (2)

K. Moriya, A. Yazaki, K. Hirai, “Detection and identification of near-surface microprecipitates in silicon wafers by laser scattering tomography,” Jpn. J. Appl. Phys. Pt. 1 34, 5721–5728 (1995).
[CrossRef]

D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
[CrossRef]

1994 (3)

P. Burggraaf, “Pursuing advanced metrology solutions,” Semicond. Int. 17, 62–64 (1994).

R. S. Howland, “Detecting killer particles on rough surfaces,” Semicond. Int. 17, 164–170 (1994).

E. Morita, H. Okuda, F. Inoue, “Distinguishing COPs from real particles,” Semicond. Int. 17, 156–162 (1994).

1993 (2)

1992 (2)

G. Videen, W. L. Wolfe, W. S. Bickel, “Light scattering Mueller matrix for a surface contaminated by a single particle in the Rayleigh limit,” Opt. Eng. 31, 341–349 (1992).
[CrossRef]

G. Videen, “Light scattering from a sphere on or near a surface: errata,” J. Opt. Soc. Am. A 9, 844–845 (1992).
[CrossRef]

1991 (1)

1988 (1)

1987 (2)

K. B. Nahm, W. L. Wolfe, “Light-scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987).
[CrossRef] [PubMed]

W. S. Bickel, A. J. Watkins, G. Videen, “The light-scattering Mueller matrix elements for Rayleigh, Rayleigh-Gans, and Mie spheres,” Am. J. Phys. 55, 559–561 (1987).
[CrossRef]

1986 (2)

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

1967 (1)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Alexander, C.

D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
[CrossRef]

Asmail, C. C.

T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
[CrossRef]

T. A. Germer, C. C. Asmail, “Bidirectional ellipsometry and its application to the characterization of surfaces,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 173–182 (1997).

Barrick, D. E.

D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970).

Bickel, W. S.

G. Videen, M. G. Turner, V. J. Iafelice, W. S. Bickel, W. L. Wolfe, “Scattering from a small sphere near a surface,” J. Opt. Soc. Am. A 10, 118–126 (1993).
[CrossRef]

G. Videen, W. L. Wolfe, W. S. Bickel, “Light scattering Mueller matrix for a surface contaminated by a single particle in the Rayleigh limit,” Opt. Eng. 31, 341–349 (1992).
[CrossRef]

W. S. Bickel, A. J. Watkins, G. Videen, “The light-scattering Mueller matrix elements for Rayleigh, Rayleigh-Gans, and Mie spheres,” Am. J. Phys. 55, 559–561 (1987).
[CrossRef]

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1980).

Burggraaf, P.

P. Burggraaf, “Pursuing advanced metrology solutions,” Semicond. Int. 17, 62–64 (1994).

Flynn, D. S.

D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
[CrossRef]

Germer, T. A.

T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
[CrossRef]

T. A. Germer, C. C. Asmail, “Bidirectional ellipsometry and its application to the characterization of surfaces,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 173–182 (1997).

Greef, R.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

Hirai, K.

K. Moriya, A. Yazaki, K. Hirai, “Detection and identification of near-surface microprecipitates in silicon wafers by laser scattering tomography,” Jpn. J. Appl. Phys. Pt. 1 34, 5721–5728 (1995).
[CrossRef]

Hirleman, E. D.

Howland, R. S.

R. S. Howland, “Detecting killer particles on rough surfaces,” Semicond. Int. 17, 164–170 (1994).

Iafelice, V. J.

Inoue, F.

E. Morita, H. Okuda, F. Inoue, “Distinguishing COPs from real particles,” Semicond. Int. 17, 156–162 (1994).

Morita, E.

E. Morita, H. Okuda, F. Inoue, “Distinguishing COPs from real particles,” Semicond. Int. 17, 156–162 (1994).

Moriya, K.

K. Moriya, A. Yazaki, K. Hirai, “Detection and identification of near-surface microprecipitates in silicon wafers by laser scattering tomography,” Jpn. J. Appl. Phys. Pt. 1 34, 5721–5728 (1995).
[CrossRef]

Nahm, K. B.

Okuda, H.

E. Morita, H. Okuda, F. Inoue, “Distinguishing COPs from real particles,” Semicond. Int. 17, 156–162 (1994).

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Scheer, B. W.

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).

Turner, M. G.

Valenzuela, G. R.

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

Videen, G.

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

Watkins, A. J.

W. S. Bickel, A. J. Watkins, G. Videen, “The light-scattering Mueller matrix elements for Rayleigh, Rayleigh-Gans, and Mie spheres,” Am. J. Phys. 55, 559–561 (1987).
[CrossRef]

Weber, D. C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1980).

Wolfe, W. L.

Yazaki, A.

K. Moriya, A. Yazaki, K. Hirai, “Detection and identification of near-surface microprecipitates in silicon wafers by laser scattering tomography,” Jpn. J. Appl. Phys. Pt. 1 34, 5721–5728 (1995).
[CrossRef]

Am. J. Phys. (1)

W. S. Bickel, A. J. Watkins, G. Videen, “The light-scattering Mueller matrix elements for Rayleigh, Rayleigh-Gans, and Mie spheres,” Am. J. Phys. 55, 559–561 (1987).
[CrossRef]

Appl. Opt. (2)

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

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

Jpn. J. Appl. Phys. Pt. 1 (1)

K. Moriya, A. Yazaki, K. Hirai, “Detection and identification of near-surface microprecipitates in silicon wafers by laser scattering tomography,” Jpn. J. Appl. Phys. Pt. 1 34, 5721–5728 (1995).
[CrossRef]

Opt. Eng. (2)

G. Videen, W. L. Wolfe, W. S. Bickel, “Light scattering Mueller matrix for a surface contaminated by a single particle in the Rayleigh limit,” Opt. Eng. 31, 341–349 (1992).
[CrossRef]

D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
[CrossRef]

Opt. Lett. (1)

Physica A (2)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

Semicond. Int. (3)

P. Burggraaf, “Pursuing advanced metrology solutions,” Semicond. Int. 17, 62–64 (1994).

R. S. Howland, “Detecting killer particles on rough surfaces,” Semicond. Int. 17, 164–170 (1994).

E. Morita, H. Okuda, F. Inoue, “Distinguishing COPs from real particles,” Semicond. Int. 17, 156–162 (1994).

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1980).

D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970).

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).

T. A. Germer, C. C. Asmail, “Bidirectional ellipsometry and its application to the characterization of surfaces,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 173–182 (1997).

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Figures (7)

Fig. 1
Fig. 1

Measuring geometry defining the angles θi, θs, and ϕs.

Fig. 2
Fig. 2

First-order interactions between light and a sphere (a–d) above a surface and (e) below a surface.

Fig. 3
Fig. 3

|qpp|2 factors for a sphere above a surface (d = 0), below a surface, and for microroughness as functions of the azimuthal angle ϕs. The incident angle θi and viewing angle θs are both 45°. The substrate material is assumed to be silicon (nmat = 3.882 + 0.012i) at λ = 633 nm.

Fig. 4
Fig. 4

|qpppart|2 factors for a sphere above a silicon surface (d = 0, nmat = 3.882 + 0.012i) as a function of θs and ϕs for three incident angles: (a) θi = 70°, (b) θi = 45°, and (c) θi = 20°.

Fig. 5
Fig. 5

|qppsub|2 factors for a sphere below a silicon surface (nmat = 3.882 + 0.012i) as a function of θs and ϕs for three incident angles: (a) θi = 70°, (b) θi = 45°, and (c) θi = 20°.

Fig. 6
Fig. 6

|qpptopo|2 factors for a microrough silicon surface (nmat = 3.882 + 0.012i) as a function of θs and ϕs for three incident angles: (a) θi = 70°, (b) θi = 45°, and (c) θi = 20°.

Fig. 7
Fig. 7

Curves of constant azimuthal angle ϕs for which zeros exist in qpp, plotted in the θi–θs plane for scattering from particles, subsurface defects, and microroughness. The functions are evaluated with the refractive indices for glass (nmat = 1.4) and silicon (nmat = 3.882 + 0.012i). For (θi, θs) to the upper right of each ϕs = 0 curve there exist no zeros in the qpp. The ϕs = 90° curve is identical to the θi and θs axes. For defects on silicon, the angles ϕs are always greater than 80°, so the contours are not shown. For scattering from particles, d = 0.

Equations (31)

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( E p scat E s scat ) = exp ( i k R ) R ( S p p S s p S p s S s s ) ( E p inc E s inc ) ,
BRDF = N A S · e ^ 2 cos θ s cos θ i F ,
P sphere = 4 π 0 n sph 2 - n 0 2 n sph 2 + 2 n 0 2 a 3 E ,
E scat = - n 0 2 k 2 exp ( i n 0 k R ) 4 π 0 R k ^ × ( k ^ × P sphere ) ,
E scat = n 0 2 k 2 exp ( i n 0 k R ) 4 π 0 R ( p ^ p ^ + s ^ s ^ ) · P sphere .
t p i j ( θ i ) p ^ j p ^ i + t s i j ( θ i ) s ^ j s ^ i ,
t s i j ( θ i ) = 2 cos θ i cos θ i + [ ( n j / n i ) 2 - sin 2 θ i ] 1 / 2 , t p i j ( θ i ) = 2 ( n j / n i ) cos θ i ( n j / n i ) 2 cos θ i + [ ( n j / n i ) 2 - sin 2 θ i ] 1 / 2 .
n i n j ( cos θ i cos θ j ) 1 / 2 [ t p i j ( θ i ) p ^ j p ^ i + t s i j ( θ i ) s ^ j s ^ i ] ,
r p i j ( θ i ) p ^ j p ^ i + r s i j ( θ i ) s ^ j s ^ i ,
r p i j ( θ ) = ( n j / n i ) 2 cos θ - [ ( n j / n i ) 2 - sin 2 θ ] 1 / 2 ( n j / n i ) 2 cos θ + [ ( n j / n i ) 2 - sin 2 θ ] 1 / 2 , r s i j ( θ ) = cos θ - [ ( n j / n i ) 2 - sin 2 θ ] 1 / 2 cos + [ ( n j / n i ) 2 - sin 2 θ ] 1 / 2 .
E = E p inc [ 1 - α r p 12 ( θ i ) ] cos θ i x ^ + E s inc [ 1 + α r s 12 ( θ i ) ] y ^ + E p inc [ 1 + α r p 12 ( θ i ) ] sin θ i z ^ ,
s ^ scat = - sin ϕ s x ^ + cos ϕ s y ^ , p ^ scat = - cos θ s cos ϕ s x ^ - cos θ s sin ϕ s y ^ + sin θ s z ^ , k ^ scat = sin θ s cos ϕ s x ^ + sin θ s sin ϕ s y ^ + cos θ s z ^ .
s ^ rfl = - sin ϕ s x ^ + cos ϕ s y ^ , p ^ rfl = cos θ s cos ϕ s x ^ + cos θ s sin ϕ s y ^ + sin θ s z ^ , k ^ rfl = sin θ s cos ϕ s x ^ + sin θ s sin ϕ s y ^ - cos θ s z ^ .
E scat = k 2 exp ( i k R ) 4 π 0 R [ p ^ scat p ^ scat + s ^ scat s ^ scat + β r p 12 ( θ s ) p ^ scat p ^ rfl + β r s 12 ( θ s ) s ^ scat s ^ rfl ] · P sphere ,
S 0 part = ( n sph 2 - 1 n sph 2 + 2 ) a 3 k 2 ,
q s s part = [ 1 + β r s 12 ( θ s ) ] [ 1 + α r s 12 ( θ i ) ] cos ϕ s , q s p part = - [ 1 + β r p 12 ( θ s ) ] [ 1 + α r s 12 ( θ i ) ] cos θ s sin ϕ s , q p s part = - [ 1 + β r s 12 ( θ s ) ] [ 1 - α r p 12 ( θ i ) ] cos θ i sin ϕ s , q p p part = [ 1 + β r p 12 ( θ s ) ] [ 1 + α r p 12 ( θ i ) ] sin θ i sin θ s - [ 1 - β r p 12 ( θ s ) ] [ 1 - α r p 12 ( θ i ) ] cos θ s cos θ i cos ϕ s .
BRDF part = 16 π 4 λ 4 ( n sph 2 - 1 n sph 2 + 2 ) 2 a 6 cos θ s cos θ i N F A × q i j part · e ^ 2 .
E = E p inc γ t p 12 ( θ i ) cos θ i x ^ + E s inc γ t s 12 ( θ i ) y ^ + E p i n c γ t p 12 ( θ i ) sin θ i z ^ ,
sin θ i = 1 n mat sin θ i , cos θ i = 1 n mat ( n mat 2 - sin 2 θ i ) 1 / 2 ,
s ^ sub = - sin ϕ s x ^ + cos ϕ s y ^ , p ^ sub = - cos θ s cos ϕ s x ^ - cos θ s sin ϕ s y ^ + sin θ s z ^ , k ^ sub = sin θ s cos ϕ s x ^ + sin θ s sin ϕ s y ^ + cos θ s z ^ ,
sin θ s = 1 n mat sin θ s , cos θ s = 1 n mat ( n mat 2 - sin 2 θ s ) 1 / 2 ,
E scat = n mat k 2 exp ( i k R ) 4 π 0 R ( cos θ s cos θ s ) 1 / 2 × [ δ t p 21 ( θ s ) p ^ scat p ^ sub + δ t s 21 ( θ s ) s ^ scat s ^ sub ] · P sphere ,
S 0 sub = 4 δ γ ( n sph 2 - n mat 2 n sph 2 + 2 n mat 2 ) cos θ s cos θ i a 3 k 2 n mat 3 / 2 [ ( n mat 2 - sin 2 θ s ) 1 / 2 cos θ s ] 1 / 2 ,
q s s sub = cos ϕ s [ cos θ i + ( n mat 2 - sin 2 θ i ) 1 / 2 ] [ cos θ s + ( n mat 2 - sin 2 θ s ) 1 / 2 ] , q s p sub = - sin ϕ s ( n mat 2 - sin 2 θ s ) 1 / 2 [ cos θ i + ( n mat 2 - sin 2 θ i ) 1 / 2 ] [ n mat 2 cos θ s + ( n mat 2 - sin 2 θ s ) 1 / 2 ] , q p s sub = - sin ϕ s ( n mat 2 - sin 2 θ i ) 1 / 2 [ n mat 2 cos θ i + ( n mat 2 - sin 2 θ i ) 1 / 2 ] [ cos θ s + ( n mat 2 - sin 2 θ s ) 1 / 2 ] , q p p sub = sin θ i sin θ s - ( n mat 2 - sin 2 θ i ) 1 / 2 ( n mat 2 - sin 2 θ s ) 1 / 2 cos ϕ s [ n mat 2 cos θ i + ( n mat 2 - sin 2 θ i ) 1 / 2 ] [ n mat 2 cos θ s + ( n mat 2 - sin 2 θ s ) 1 / 2 ] .
BRDF sub = 256 π 4 λ 4 ( n sph 2 - n mat 2 n sph 2 + 2 n mat 2 ) 2 a 6 cos θ i ( n mat 2 - sin 2 θ s ) 1 / 2 γ δ 2 n mat 3 N F A × q i j sub · e ^ 2 .
BRDF topo = 16 π 2 λ 4 cos θ i cos θ s S ( f ) × q i j topo · e ^ 2 ,
q s s topo = ( n mat 2 - 1 ) cos ϕ s [ cos θ i + ( n mat 2 - sin 2 θ i ) 1 / 2 ] [ cos θ s + ( n mat 2 - sin 2 θ s ) 1 / 2 ] , q s p topo = - ( n mat 2 - 1 ) sin ϕ s ( n mat 2 - sin 2 θ s ) 1 / 2 [ cos θ i + ( n mat 2 - sin 2 θ i ) 1 / 2 ] [ n mat 2 cos θ s + ( n mat 2 - sin 2 θ s ) 1 / 2 ] , q p s topo = - ( n mat 2 - 1 ) sin ϕ s ( n mat 2 - sin 2 θ i ) 1 / 2 [ n mat 2 cos θ i + ( n mat 2 - sin 2 θ i ) 1 / 2 ] [ cos θ s + ( n mat 2 - sin 2 θ s ) 1 / 2 ] , q p p topo = ( n mat 2 - 1 ) [ n mat 2 sin θ i sin θ s - ( n mat 2 - sin 2 θ i ) 1 / 2 ( n mat 2 - sin 2 θ s ) 1 / 2 cos ϕ s ] [ n mat 2 cos θ i + ( n mat 2 - sin 2 θ i ) 1 / 2 ] [ n mat 2 cos θ s + ( n mat 2 - sin 2 θ s ) 1 / 2 ] ,
λ f x = sin θ s cos ϕ s - sin θ i , λ f y = sin θ s sin ϕ s .
n 1 sin θ 1 = n 2 sin θ 2 .
n 1 cos θ 1 d θ 1 = n 2 cos θ 2 d θ 2 .
d Ω 2 = sin θ 2 d θ 2 d ϕ 2 = n 1 sin θ 1 n 2 n 1 cos θ 1 n 2 cos θ 2 d θ 1 d ϕ 1 = n 1 2 n 2 2 cos θ 1 cos θ 2 d Ω 1 .

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