Abstract

We simulated light scattering from a particle located on a smooth surface. We developed a new approach utilizing the discrete sources method based on a strict mathematical model for this scattering problem. The main features of the corresponding numerical algorithm are presented. The results of modeling and comparisons with other theoretical results and experimental data are shown as well.

© 1996 Optical Society of America

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References

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  1. H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.
  2. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991).
    [CrossRef]
  3. M. A. Taubenblatt, T. K. Tran, “Calculation of light scattering from particles and structures on a surface by the coupled-dipole method,” J. Opt. Soc. Am. A 10, 912–919 (1993).
    [CrossRef]
  4. Yu. Eremin, A. Sveshnikov, “Discrete sources method in electromagnetic scatterer problems,” Electromagnetics 13, 203–216 (1993).
    [CrossRef]
  5. A. Sveshnikov, Yu. Yeremin, N. Orlov, “Study of some mathematical models in the theory of diffraction by the method of non-orthogonal series,” Sov. J. Commun. Technol. Electron. 30(7), 73–81 (1985).
  6. Yu. Eremin, N. Orlov, A. Sveshnikov, “Electromagnetic scattering analysis based on discrete sources method,” ACES J. 9(3), 46–56 (1994).
  7. H. Lee, S. Chae, Y. Ye, D. Pui, G. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol Sci. Technol. 14, 177–192 (1991).
    [CrossRef]
  8. G. Wojcik, D. Vaughan, L. Galbraith, “Calculation of light scattering from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE 774, 21–31 (1987).
  9. L. Clementy, M. Fossey, A. Rakhmanov, “The experimental study of the angular scattering distribution of the PSL particle placed on silicone substrate,” in Report of ADE and ADE Optical System Research Laboratory (ADE Optical System Corporation, Charlotte, N.C., 1994).

1994 (1)

Yu. Eremin, N. Orlov, A. Sveshnikov, “Electromagnetic scattering analysis based on discrete sources method,” ACES J. 9(3), 46–56 (1994).

1993 (2)

Yu. Eremin, A. Sveshnikov, “Discrete sources method in electromagnetic scatterer problems,” Electromagnetics 13, 203–216 (1993).
[CrossRef]

M. A. Taubenblatt, T. K. Tran, “Calculation of light scattering from particles and structures on a surface by the coupled-dipole method,” J. Opt. Soc. Am. A 10, 912–919 (1993).
[CrossRef]

1991 (2)

H. Lee, S. Chae, Y. Ye, D. Pui, G. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol Sci. Technol. 14, 177–192 (1991).
[CrossRef]

G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991).
[CrossRef]

1985 (1)

A. Sveshnikov, Yu. Yeremin, N. Orlov, “Study of some mathematical models in the theory of diffraction by the method of non-orthogonal series,” Sov. J. Commun. Technol. Electron. 30(7), 73–81 (1985).

Chae, S.

H. Lee, S. Chae, Y. Ye, D. Pui, G. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Clementy, L.

L. Clementy, M. Fossey, A. Rakhmanov, “The experimental study of the angular scattering distribution of the PSL particle placed on silicone substrate,” in Report of ADE and ADE Optical System Research Laboratory (ADE Optical System Corporation, Charlotte, N.C., 1994).

Eremin, Yu.

Yu. Eremin, N. Orlov, A. Sveshnikov, “Electromagnetic scattering analysis based on discrete sources method,” ACES J. 9(3), 46–56 (1994).

Yu. Eremin, A. Sveshnikov, “Discrete sources method in electromagnetic scatterer problems,” Electromagnetics 13, 203–216 (1993).
[CrossRef]

Fossey, M.

L. Clementy, M. Fossey, A. Rakhmanov, “The experimental study of the angular scattering distribution of the PSL particle placed on silicone substrate,” in Report of ADE and ADE Optical System Research Laboratory (ADE Optical System Corporation, Charlotte, N.C., 1994).

Galbraith, L.

G. Wojcik, D. Vaughan, L. Galbraith, “Calculation of light scattering from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE 774, 21–31 (1987).

Gildersleeve, K.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Goodall, R.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Hirleman, D.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Huff, H.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Lee, H.

H. Lee, S. Chae, Y. Ye, D. Pui, G. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Liu, B.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Murray, W.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Orlov, N.

Yu. Eremin, N. Orlov, A. Sveshnikov, “Electromagnetic scattering analysis based on discrete sources method,” ACES J. 9(3), 46–56 (1994).

A. Sveshnikov, Yu. Yeremin, N. Orlov, “Study of some mathematical models in the theory of diffraction by the method of non-orthogonal series,” Sov. J. Commun. Technol. Electron. 30(7), 73–81 (1985).

Pui, D.

H. Lee, S. Chae, Y. Ye, D. Pui, G. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Rakhmanov, A.

L. Clementy, M. Fossey, A. Rakhmanov, “The experimental study of the angular scattering distribution of the PSL particle placed on silicone substrate,” in Report of ADE and ADE Optical System Research Laboratory (ADE Optical System Corporation, Charlotte, N.C., 1994).

Scheer, B.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Sveshnikov, A.

Yu. Eremin, N. Orlov, A. Sveshnikov, “Electromagnetic scattering analysis based on discrete sources method,” ACES J. 9(3), 46–56 (1994).

Yu. Eremin, A. Sveshnikov, “Discrete sources method in electromagnetic scatterer problems,” Electromagnetics 13, 203–216 (1993).
[CrossRef]

A. Sveshnikov, Yu. Yeremin, N. Orlov, “Study of some mathematical models in the theory of diffraction by the method of non-orthogonal series,” Sov. J. Commun. Technol. Electron. 30(7), 73–81 (1985).

Taubenblatt, M. A.

Tran, T. K.

Vaughan, D.

G. Wojcik, D. Vaughan, L. Galbraith, “Calculation of light scattering from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE 774, 21–31 (1987).

Videen, G.

Warner, T.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Williams, E.

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

Wojcik, G.

H. Lee, S. Chae, Y. Ye, D. Pui, G. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol Sci. Technol. 14, 177–192 (1991).
[CrossRef]

G. Wojcik, D. Vaughan, L. Galbraith, “Calculation of light scattering from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE 774, 21–31 (1987).

Ye, Y.

H. Lee, S. Chae, Y. Ye, D. Pui, G. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Yeremin, Yu.

A. Sveshnikov, Yu. Yeremin, N. Orlov, “Study of some mathematical models in the theory of diffraction by the method of non-orthogonal series,” Sov. J. Commun. Technol. Electron. 30(7), 73–81 (1985).

ACES J. (1)

Yu. Eremin, N. Orlov, A. Sveshnikov, “Electromagnetic scattering analysis based on discrete sources method,” ACES J. 9(3), 46–56 (1994).

Aerosol Sci. Technol. (1)

H. Lee, S. Chae, Y. Ye, D. Pui, G. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Electromagnetics (1)

Yu. Eremin, A. Sveshnikov, “Discrete sources method in electromagnetic scatterer problems,” Electromagnetics 13, 203–216 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

Sov. J. Commun. Technol. Electron. (1)

A. Sveshnikov, Yu. Yeremin, N. Orlov, “Study of some mathematical models in the theory of diffraction by the method of non-orthogonal series,” Sov. J. Commun. Technol. Electron. 30(7), 73–81 (1985).

Other (3)

G. Wojcik, D. Vaughan, L. Galbraith, “Calculation of light scattering from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE 774, 21–31 (1987).

L. Clementy, M. Fossey, A. Rakhmanov, “The experimental study of the angular scattering distribution of the PSL particle placed on silicone substrate,” in Report of ADE and ADE Optical System Research Laboratory (ADE Optical System Corporation, Charlotte, N.C., 1994).

H. Huff, R. Goodall, E. Williams, B. Liu, T. Warner, D. Hirleman, K. Gildersleeve, W. Murray, B. Scheer, “Measurement of silicon particles by laser surface scanning and angle-resolved light scattering,” in SEMATECH Meeting: Particle Counting/Microroughness Task Force (Minneapolis, 23–24 June 1994), p. 5.

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Figures (4)

Fig. 1
Fig. 1

Geometry of the scattering problem.

Fig. 2
Fig. 2

Relative intensity versus scattering angle. Excitation wavelength is 0.633 μm at normal incidence. The PSL sphere diameter is 0.76 μm; the refractive indices are 1.59 (PSL) and 3.8 (silicon). Curves 1 and 2 correspond to DSM results of P and S polarization, respectively. The circles and squares (3 and 4) correspond to the Wojcik et al.8 numerical results.

Fig. 3
Fig. 3

Relative intensity versus scattering angle. Plane wave incidence is −60° from normal; the wavelength is the same as in Fig. 2. The silicon refractive index is 3.88–0.02i, the particle is a 0.45-μm PSL sphere. Curve 1, the DSM; circles 2, Taubenblatt and Tran.3

Fig. 4
Fig. 4

Integral cross section (response in μm2) as a function of the PSL sphere diameter (μm). The DSM theoretical prediction (filled circles) compared with the real world surface scanner WIS-8500II experimental data9 (open boxes). The excitation wavelength is 0.488 μm; the plane wave incidence is 215° from normal.

Equations (29)

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× H t = i k ɛ t E t ;             × E t = - i k μ t H t in D t , t = 0 , 1 , i ,
n p × E i ( p ) - E 0 ( p ) = n p × E 0 ( p ) , n p × H i ( p ) - H 0 ( p ) = n p × H 0 ( p ) , p D ,
e z × E 0 ( p ) - E 1 ( p ) = 0 , e z × H 0 ( p ) - H 1 ( p ) = 0 , p Ξ ,
attenuation / radiation conditions at infinity ( z 0 ) .
A m x e = e x g m e + e z ( e x , ) f m , A m y h = e y g m h + e z ( e y , ) f m ,
g m e , h ( q , w n ) = Y m ( q , w n ) - Y m ( q , - w n ) + 0 J m ( λ ρ ) κ e , h exp [ - η 0 ( z + w n ) ] λ 1 + m d λ ,
f m ( q , w n ) = 0 J m ( λ ρ ) ζ exp [ - η 0 ( z + w n ) ] λ 1 + m d λ , σ m e , h ( q , w n ) = g m h , e ( q , w n ) ,
Y m ( q , w n ) = Y n m 0 ( q ) = k 0 m + 1 / i × h m ( 2 ) ( k 0 R ) × ( ρ / R ) m
κ e = 2 μ 1 / ( μ 1 η 0 + μ 0 η 1 ) ,             κ h = 2 ɛ 1 / ( ɛ 1 η 0 + ɛ 0 η 1 ) , ζ = 2 ( μ 1 ɛ 0 - μ 0 ɛ 1 ) / [ ( ɛ 1 η 0 + ɛ 0 η 1 ) ( μ 1 η 0 + μ 0 η 1 ) ] , η 0 , 1 2 = λ 2 - k 0 , 1 2 ,             k 0 , 1 2 = k 2 ɛ 0 , 1 μ 0 , 1 .
A m x e cos m φ - A m y e sin m φ A m e , A m x h sin m φ - A m y h cos m φ A m h .
( E t s H t s ) = s S { p s t ( E s t e H s t e ) + q s t ( E s t h H s t h ) } + Σ dip .
( E v t e H v t e ) = ( i / k ɛ t μ × × - 1 / μ t × ) A v t e , ( E v t h H v t h ) = ( 1 / ɛ t × i / k ɛ t μ t × × ) A v t h .
A m x e ( q ) = e x Y n m i ( q ) ,             A m y h ( q ) = e y Y m n i ( q ) , Y n m i ( q ) = Y n m i ( q , w n ) , = j m ( k i R ) ( ρ / R ) m ,
A n , 0 O z ( q ) = σ e ( q , w n ) e z ,             A n , i O z ( q ) = Y 0 i ( q , w n ) e z .
E 0 = e x cos γ ( χ 1 - R p χ 2 ) + e z sin γ ( χ 1 + R p χ 2 ) , H 0 = e y cos γ ( χ 1 + R p χ 2 ) , χ 1 = exp [ - i k 0 ( x sin γ - z cos γ ) ] , χ 2 = exp [ - i k 0 ( x sin γ + z cos γ ) ] ,
R p = ɛ 1 cos γ - ( ɛ 1 / ɛ 0 - sin 2 γ ) 1 / 2 ɛ 1 cos γ + ( ɛ 1 / ɛ 0 - sin 2 γ ) 1 / 2 .
E 0 = e y cos γ ( χ 1 - R S χ 2 ) , H 0 = e x cos γ ( χ 1 + R S χ 2 ) + e z sin γ ( χ 1 - R S χ 2 ) ,
A m e : = A m x e sin m φ + A m y e cos m φ , A m h : = A m x h cos m φ - A m y h sin m φ .
R S = cos γ - ( ɛ 1 / ɛ 0 - sin 2 γ ) 1 / 2 cos γ + ( ɛ 1 / ɛ 0 - sin 2 γ ) 1 / 2 .
p = arg min { | | n p × E i s ( p ) - E 0 s ( p ) - E 0 ( p ) | | L 2 ( D ) + | | n p × H i S ( p ) - H 0 S ( p ) - H 0 ( p ) | | L 2 ( D ) } .
exp { - i ϖ cos φ } = m = 0 ( 2 - δ 0 m ) ( - i ) m J m ( ϖ ) cos m φ ,
B m p m = 0.5 × ( Q m + R m + 1 ) ,             m = 0 , M ¯ ,
Q m = { α cos γ ( J ˜ m - J m + 2 ) · e - , cos γ ( J ˜ m + J m + 2 ) · e - , - α ( J ˜ m + J m + 2 ) · e + , - ( J ˜ m - J m + 2 ) · e + } × ( - i ) m , R m + 1 = { - 2 i β sin γ J m + 1 · e + , 0 , 0 , 0 } × ( - i ) m , e - = exp ( i k 0 z cos γ ) - R p · exp ( - i k 0 z cos γ ) , e + = exp ( i k 0 z cos γ ) - R p · exp ( - i k 0 z cos γ ) , J ˜ = J m · ( 1 + δ 0 m ) ,             J m = J m ( ϖ ) ,
E ( r ) / E 0 = exp { - i k 0 r } r F ( θ , φ ) + O ( 1 / r 2 ) ,             r ,
0 J m ( λ ρ ) f ( λ ) exp { - η 0 ( z - z 0 ) } λ 1 + m d λ = ψ 0 i k 0 cos θ ( i k 0 sin θ ) m G n f ( k 0 sin θ ) + o r - 1 ,
F θ = i k 0 m = 0 M cos ( m + 1 ) φ ( i k 0 sin θ ) m × n = 1 N { p n m e cos θ [ G + ( κ ¯ e - sin 2 θ ζ ¯ ) G ] + q n m e ( G + κ ¯ h G ) } - i k 0 sin θ n = 1 N r n 0 e ( G + κ ¯ h G ) , F φ = - i k 0 m = 0 M sin ( m + 1 ) φ ( i k 0 sin θ ) m × n = 1 N { p n m e ( G + κ ¯ e G ) + q n m e cos θ [ G + ( k ¯ h - sin 2 θ ζ ¯ ) G ] } ,
κ ¯ e = i cos θ - ( sin 2 θ - ɛ 1 / ɛ 0 ) 1 / 2 i cos θ + ( sin 2 θ - ɛ 1 / ɛ 0 ) 1 / 2 ,             κ ¯ h = i ɛ 1 cos θ - ɛ 0 ( sin 2 θ - ɛ 1 / ɛ 0 ) 1 / 2 i ɛ 1 cos θ + ɛ 0 ( sin 2 θ - ɛ 1 / ɛ 0 ) 1 / 2 , ζ ¯ = 2 ( ɛ 1 - ɛ 0 ) [ i cos θ + ( sin 2 θ - ɛ 1 / ɛ 0 ) 1 / 2 ( i ɛ 1 cos θ + ɛ 0 ( sin 2 θ - ɛ 1 / ɛ 0 ) 1 / 2 ] , G n = exp { i k 0 z n cos θ } .
F θ = - i k 0 m = 0 M cos ( m + 1 ) φ ( i k e sin θ ) m n = 1 N { p n m e × ( G + κ ¯ e G ) - q n m e cos θ [ G + ( κ ¯ h - sin 2 θ ζ ¯ ) G ] } - i k 0 sin θ n = 1 N r n 0 e ( G + κ ¯ h G ) , F φ = i k 0 m = 0 M sin ( m + 1 ) φ ( i k 0 sin θ ) m n = 1 N { p n m e cos θ [ G + ( κ ¯ e - sin 2 θ ζ ¯ ) G ] - q n m e ( G + κ ¯ h G ) } .
I p = F θ p 2 + F φ p 2 ,             I S = F θ S 2 + F φ S 2 ,

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