Abstract

An optimization algorithm for the reconstruction of a surface profile is presented. This algorithm uses the far-field transmitted intensity to retrieve the surface profile of a thin film. We approach the inverse transmission problem as a constrained optimization problem. A mathematical representation of the surface based on B-spline curves and an optimum profile search with the self-adaptation evolutionary strategy (ES) are adopted. As the input data for the ES algorithm for surface inversion, the transmitted intensity has been measured by both a laser bidirectional reflectometer instrument and an in-line digital speckle camera system. The reconstructed 1D surfaces are compared with the profile measured by an atomic force microscope.

© 2008 Optical Society of America

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  1. A. J. Sant, J. C. Dainty, and M. J. Kim, “Comparison of surface scattering between identical randomly rough metal and dielectric diffuser,” Opt. Lett. 14, 1183-1185 (1989).
    [CrossRef] [PubMed]
  2. A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255-307 (1990).
    [CrossRef]
  3. Z.-H. Gu, J. Q. Lu, and M. M. Tehrani, “Enhanced backscattering of polarized light at vacuum/dielectric interface,” Opt. Eng. (Bellingham) 34, 1611-1623 (1995).
    [CrossRef]
  4. W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269-287 (1977).
    [CrossRef]
  5. J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31-47 (1979).
    [CrossRef]
  6. J. C. Stover, S. A. Serati, and C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. (Bellingham) 23, 406-412 (1984).
  7. V. Malyshkin, S. Simeonov, A. R. McGurn, and A. A. Maradudin, “Determination of surface profile statistics from electromagnetic scattering data,” Opt. Lett. 22, 58-60 (1997).
    [CrossRef] [PubMed]
  8. D. Macías, G. Olague, and E. R. Méndez, “Surface profile reconstruction from scattered intensity data using evolutionary strategies,” in EvoWorkshops 2002, Vol. 2279 of Lecture Notes in Computer Science (Springer, 2002), pp. 233.
  9. D. Macías, G. Olague, and E. R. Méndez, “Hybrid evolution strategy--downhill simplex algorithm for inverse light scattering problems,” in EvoWorkshops 2003, Vol. 2611 of Lecture Notes in Computer Science (Springer, 2003), pp. 399.
  10. D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: Random surfaces that belong to a well-defined statistical class,” Waves Random Complex Media 16, 545-560 (2006).
    [CrossRef]
  11. A. Wang and Z.-H. Gu, “Inverse scattering simulation for a 1-D surface reconstruction,” Proc. SPIE 6672, 66720E (2007).
    [CrossRef]
  12. T. Back, U. Hammel, and H.-P. Schwefel, “Evolutionary computation: Comments on the history and current state,” IEEE Trans. Evol. Comput. 1, 3-17 (1997).
    [CrossRef]
  13. B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A, Pure Appl. Opt. 8, S91-S207 (2006).
    [CrossRef]
  14. A. A. Maradudin, E. R. Méndez, and T. A. Leskova, Designer Surface (Elsevier, in press).
  15. Z.-H. Gu and A. Wang, “Experimental reconstruction for inverse scattering of one-dimensional surfaces,” Proc. SPIE 6672, 66720H (2007).
    [CrossRef]
  16. H. G. Beyer, The Theory of Evolution Strategies (Springer-Verlag, 2001), p. 380.
  17. Z.-H. Gu, “Angular spectrum redistribution from rough surface scattering,” Proc. SPIE 4447, 109-114 (2001).
    [CrossRef]
  18. Z.-H. Gu, I. M. Fuks, and M. Ciftan, “Grazing angle enhanced backscattering from a dielectric film on a reflecting metal substrate,” Opt. Eng. (Bellingham) 43, 559-567 (2004).
    [CrossRef]
  19. Z.-H. Gu and M. Josse, “Fabrication of 1-D gratings on photoresist for light-scattering and memory effect measurements,” Proc. SPIE 2726, 158-163 (1996).
    [CrossRef]

2007

A. Wang and Z.-H. Gu, “Inverse scattering simulation for a 1-D surface reconstruction,” Proc. SPIE 6672, 66720E (2007).
[CrossRef]

Z.-H. Gu and A. Wang, “Experimental reconstruction for inverse scattering of one-dimensional surfaces,” Proc. SPIE 6672, 66720H (2007).
[CrossRef]

2006

D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: Random surfaces that belong to a well-defined statistical class,” Waves Random Complex Media 16, 545-560 (2006).
[CrossRef]

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A, Pure Appl. Opt. 8, S91-S207 (2006).
[CrossRef]

2004

Z.-H. Gu, I. M. Fuks, and M. Ciftan, “Grazing angle enhanced backscattering from a dielectric film on a reflecting metal substrate,” Opt. Eng. (Bellingham) 43, 559-567 (2004).
[CrossRef]

2003

D. Macías, G. Olague, and E. R. Méndez, “Hybrid evolution strategy--downhill simplex algorithm for inverse light scattering problems,” in EvoWorkshops 2003, Vol. 2611 of Lecture Notes in Computer Science (Springer, 2003), pp. 399.

2002

D. Macías, G. Olague, and E. R. Méndez, “Surface profile reconstruction from scattered intensity data using evolutionary strategies,” in EvoWorkshops 2002, Vol. 2279 of Lecture Notes in Computer Science (Springer, 2002), pp. 233.

2001

H. G. Beyer, The Theory of Evolution Strategies (Springer-Verlag, 2001), p. 380.

Z.-H. Gu, “Angular spectrum redistribution from rough surface scattering,” Proc. SPIE 4447, 109-114 (2001).
[CrossRef]

1997

T. Back, U. Hammel, and H.-P. Schwefel, “Evolutionary computation: Comments on the history and current state,” IEEE Trans. Evol. Comput. 1, 3-17 (1997).
[CrossRef]

V. Malyshkin, S. Simeonov, A. R. McGurn, and A. A. Maradudin, “Determination of surface profile statistics from electromagnetic scattering data,” Opt. Lett. 22, 58-60 (1997).
[CrossRef] [PubMed]

1996

Z.-H. Gu and M. Josse, “Fabrication of 1-D gratings on photoresist for light-scattering and memory effect measurements,” Proc. SPIE 2726, 158-163 (1996).
[CrossRef]

1995

Z.-H. Gu, J. Q. Lu, and M. M. Tehrani, “Enhanced backscattering of polarized light at vacuum/dielectric interface,” Opt. Eng. (Bellingham) 34, 1611-1623 (1995).
[CrossRef]

1990

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

1989

1984

J. C. Stover, S. A. Serati, and C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. (Bellingham) 23, 406-412 (1984).

1979

1977

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269-287 (1977).
[CrossRef]

Back, T.

T. Back, U. Hammel, and H.-P. Schwefel, “Evolutionary computation: Comments on the history and current state,” IEEE Trans. Evol. Comput. 1, 3-17 (1997).
[CrossRef]

Baumeier, B.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A, Pure Appl. Opt. 8, S91-S207 (2006).
[CrossRef]

Bennett, J. M.

Beyer, H. G.

H. G. Beyer, The Theory of Evolution Strategies (Springer-Verlag, 2001), p. 380.

Ciftan, M.

Z.-H. Gu, I. M. Fuks, and M. Ciftan, “Grazing angle enhanced backscattering from a dielectric film on a reflecting metal substrate,” Opt. Eng. (Bellingham) 43, 559-567 (2004).
[CrossRef]

Dainty, J. C.

Elson, J. M.

Fuks, I. M.

Z.-H. Gu, I. M. Fuks, and M. Ciftan, “Grazing angle enhanced backscattering from a dielectric film on a reflecting metal substrate,” Opt. Eng. (Bellingham) 43, 559-567 (2004).
[CrossRef]

Gu, Z.-H.

A. Wang and Z.-H. Gu, “Inverse scattering simulation for a 1-D surface reconstruction,” Proc. SPIE 6672, 66720E (2007).
[CrossRef]

Z.-H. Gu and A. Wang, “Experimental reconstruction for inverse scattering of one-dimensional surfaces,” Proc. SPIE 6672, 66720H (2007).
[CrossRef]

Z.-H. Gu, I. M. Fuks, and M. Ciftan, “Grazing angle enhanced backscattering from a dielectric film on a reflecting metal substrate,” Opt. Eng. (Bellingham) 43, 559-567 (2004).
[CrossRef]

Z.-H. Gu, “Angular spectrum redistribution from rough surface scattering,” Proc. SPIE 4447, 109-114 (2001).
[CrossRef]

Z.-H. Gu and M. Josse, “Fabrication of 1-D gratings on photoresist for light-scattering and memory effect measurements,” Proc. SPIE 2726, 158-163 (1996).
[CrossRef]

Z.-H. Gu, J. Q. Lu, and M. M. Tehrani, “Enhanced backscattering of polarized light at vacuum/dielectric interface,” Opt. Eng. (Bellingham) 34, 1611-1623 (1995).
[CrossRef]

Guillespie, C. H.

J. C. Stover, S. A. Serati, and C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. (Bellingham) 23, 406-412 (1984).

Hammel, U.

T. Back, U. Hammel, and H.-P. Schwefel, “Evolutionary computation: Comments on the history and current state,” IEEE Trans. Evol. Comput. 1, 3-17 (1997).
[CrossRef]

Josse, M.

Z.-H. Gu and M. Josse, “Fabrication of 1-D gratings on photoresist for light-scattering and memory effect measurements,” Proc. SPIE 2726, 158-163 (1996).
[CrossRef]

Kim, M. J.

Leskova, T. A.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A, Pure Appl. Opt. 8, S91-S207 (2006).
[CrossRef]

A. A. Maradudin, E. R. Méndez, and T. A. Leskova, Designer Surface (Elsevier, in press).

Lu, J. Q.

Z.-H. Gu, J. Q. Lu, and M. M. Tehrani, “Enhanced backscattering of polarized light at vacuum/dielectric interface,” Opt. Eng. (Bellingham) 34, 1611-1623 (1995).
[CrossRef]

Macías, D.

D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: Random surfaces that belong to a well-defined statistical class,” Waves Random Complex Media 16, 545-560 (2006).
[CrossRef]

D. Macías, G. Olague, and E. R. Méndez, “Hybrid evolution strategy--downhill simplex algorithm for inverse light scattering problems,” in EvoWorkshops 2003, Vol. 2611 of Lecture Notes in Computer Science (Springer, 2003), pp. 399.

D. Macías, G. Olague, and E. R. Méndez, “Surface profile reconstruction from scattered intensity data using evolutionary strategies,” in EvoWorkshops 2002, Vol. 2279 of Lecture Notes in Computer Science (Springer, 2002), pp. 233.

Malyshkin, V.

Maradudin, A. A.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A, Pure Appl. Opt. 8, S91-S207 (2006).
[CrossRef]

V. Malyshkin, S. Simeonov, A. R. McGurn, and A. A. Maradudin, “Determination of surface profile statistics from electromagnetic scattering data,” Opt. Lett. 22, 58-60 (1997).
[CrossRef] [PubMed]

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, and T. A. Leskova, Designer Surface (Elsevier, in press).

McGurn, A. R.

V. Malyshkin, S. Simeonov, A. R. McGurn, and A. A. Maradudin, “Determination of surface profile statistics from electromagnetic scattering data,” Opt. Lett. 22, 58-60 (1997).
[CrossRef] [PubMed]

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Mendez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Méndez, E. R.

D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: Random surfaces that belong to a well-defined statistical class,” Waves Random Complex Media 16, 545-560 (2006).
[CrossRef]

D. Macías, G. Olague, and E. R. Méndez, “Hybrid evolution strategy--downhill simplex algorithm for inverse light scattering problems,” in EvoWorkshops 2003, Vol. 2611 of Lecture Notes in Computer Science (Springer, 2003), pp. 399.

D. Macías, G. Olague, and E. R. Méndez, “Surface profile reconstruction from scattered intensity data using evolutionary strategies,” in EvoWorkshops 2002, Vol. 2279 of Lecture Notes in Computer Science (Springer, 2002), pp. 233.

A. A. Maradudin, E. R. Méndez, and T. A. Leskova, Designer Surface (Elsevier, in press).

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Olague, G.

D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: Random surfaces that belong to a well-defined statistical class,” Waves Random Complex Media 16, 545-560 (2006).
[CrossRef]

D. Macías, G. Olague, and E. R. Méndez, “Hybrid evolution strategy--downhill simplex algorithm for inverse light scattering problems,” in EvoWorkshops 2003, Vol. 2611 of Lecture Notes in Computer Science (Springer, 2003), pp. 399.

D. Macías, G. Olague, and E. R. Méndez, “Surface profile reconstruction from scattered intensity data using evolutionary strategies,” in EvoWorkshops 2002, Vol. 2279 of Lecture Notes in Computer Science (Springer, 2002), pp. 233.

Sant, A. J.

Schwefel, H.-P.

T. Back, U. Hammel, and H.-P. Schwefel, “Evolutionary computation: Comments on the history and current state,” IEEE Trans. Evol. Comput. 1, 3-17 (1997).
[CrossRef]

Serati, S. A.

J. C. Stover, S. A. Serati, and C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. (Bellingham) 23, 406-412 (1984).

Simeonov, S.

Stover, J. C.

J. C. Stover, S. A. Serati, and C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. (Bellingham) 23, 406-412 (1984).

Tehrani, M. M.

Z.-H. Gu, J. Q. Lu, and M. M. Tehrani, “Enhanced backscattering of polarized light at vacuum/dielectric interface,” Opt. Eng. (Bellingham) 34, 1611-1623 (1995).
[CrossRef]

Wang, A.

A. Wang and Z.-H. Gu, “Inverse scattering simulation for a 1-D surface reconstruction,” Proc. SPIE 6672, 66720E (2007).
[CrossRef]

Z.-H. Gu and A. Wang, “Experimental reconstruction for inverse scattering of one-dimensional surfaces,” Proc. SPIE 6672, 66720H (2007).
[CrossRef]

Welford, W. T.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269-287 (1977).
[CrossRef]

Ann. Phys. (N.Y.)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

IEEE Trans. Evol. Comput.

T. Back, U. Hammel, and H.-P. Schwefel, “Evolutionary computation: Comments on the history and current state,” IEEE Trans. Evol. Comput. 1, 3-17 (1997).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A, Pure Appl. Opt. 8, S91-S207 (2006).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng. (Bellingham)

J. C. Stover, S. A. Serati, and C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. (Bellingham) 23, 406-412 (1984).

Z.-H. Gu, J. Q. Lu, and M. M. Tehrani, “Enhanced backscattering of polarized light at vacuum/dielectric interface,” Opt. Eng. (Bellingham) 34, 1611-1623 (1995).
[CrossRef]

Z.-H. Gu, I. M. Fuks, and M. Ciftan, “Grazing angle enhanced backscattering from a dielectric film on a reflecting metal substrate,” Opt. Eng. (Bellingham) 43, 559-567 (2004).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269-287 (1977).
[CrossRef]

Proc. SPIE

Z.-H. Gu and M. Josse, “Fabrication of 1-D gratings on photoresist for light-scattering and memory effect measurements,” Proc. SPIE 2726, 158-163 (1996).
[CrossRef]

Z.-H. Gu, “Angular spectrum redistribution from rough surface scattering,” Proc. SPIE 4447, 109-114 (2001).
[CrossRef]

Z.-H. Gu and A. Wang, “Experimental reconstruction for inverse scattering of one-dimensional surfaces,” Proc. SPIE 6672, 66720H (2007).
[CrossRef]

A. Wang and Z.-H. Gu, “Inverse scattering simulation for a 1-D surface reconstruction,” Proc. SPIE 6672, 66720E (2007).
[CrossRef]

Waves Random Complex Media

D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: Random surfaces that belong to a well-defined statistical class,” Waves Random Complex Media 16, 545-560 (2006).
[CrossRef]

Other

D. Macías, G. Olague, and E. R. Méndez, “Surface profile reconstruction from scattered intensity data using evolutionary strategies,” in EvoWorkshops 2002, Vol. 2279 of Lecture Notes in Computer Science (Springer, 2002), pp. 233.

D. Macías, G. Olague, and E. R. Méndez, “Hybrid evolution strategy--downhill simplex algorithm for inverse light scattering problems,” in EvoWorkshops 2003, Vol. 2611 of Lecture Notes in Computer Science (Springer, 2003), pp. 399.

H. G. Beyer, The Theory of Evolution Strategies (Springer-Verlag, 2001), p. 380.

A. A. Maradudin, E. R. Méndez, and T. A. Leskova, Designer Surface (Elsevier, in press).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the scattering problem considered.

Fig. 2
Fig. 2

Flow diagram of the self-adaptation ES algorithm.

Fig. 3
Fig. 3

Schematic of the laser bidirectional reflectometer.

Fig. 4
Fig. 4

Measured intensity of the transmitted light for the case of (a) normal incidence and (b) incidence angle of 20 ° .

Fig. 5
Fig. 5

Reconstructed profiles are compared with those by an AFM measurement. (a) Reconstructed profile from the transmitted intensity at normal incidence, (b) reconstructed profile from the transmitted intensity for an incidence angle of 20 ° .

Fig. 6
Fig. 6

In-line digital speckle camera setup. P 1 , P 2 , polarizers; L 1 , L 2 , L 3 , lenses; AI, adjustable iris.

Fig. 7
Fig. 7

(a) Speckle distribution on the CCD plane, (b) the corresponding angle-resolved intensity.

Fig. 8
Fig. 8

Reconstructed profile is compared with that given by AFM.

Equations (21)

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ψ i n c ( x , z ) = exp [ i k x i α 1 ( k ) z ] ,
k = ( ω c ) n 1 sin θ 0 ,
α 1 ( k ) = ( ω c ) n 1 cos θ 0 ,
T 0 ( q k ) = γ ( q k ) ψ 0 ( k ) α 3 ( q ) [ γ ( q k ) + β ( q k ) ] { β ( q k ) α 3 ( q ) α 2 ( k ) + β ( q k ) α 3 ( q ) α 2 ( k ) } ,
ψ 0 ( k ) = 2 α 1 ( k ) exp [ i α 2 ( k ) H ] α 1 ( k ) + α 2 ( k ) ,
γ ( q k ) = q k + α 3 ( q ) α 2 ( k ) ε 2 ω 2 c 2 ,
β ( q k ) = ε 3 ω 2 c 2 q k α 3 ( q ) α 2 ( k ) .
q = ( ω c ) n 3 sin θ t ,
α 3 ( q ) = ( ω c ) n 3 cos θ t .
α 2 ( k ) = ( ω c ) [ n 2 2 n 1 2 sin 2 θ 0 ] 1 2 .
T ( q k ) = T 0 ( q k ) + d x exp [ i ( q k ) x ] exp { i [ ( α 3 ( q ) α 2 ( k ) ] ζ ( x ) } ,
I ( q k ) = c 2 n 3 16 π ω T ( q k ) 2 .
ζ ( x ) = i = 1 n C i B i g ( x ) x min < x < x max ,
B i 1 ( x ) = { 1 t i x < t i + 1 0 otherwise ,
B i g ( x ) = η i g ( x ) B i g 1 ( x ) + [ 1 η i + 1 g ( x ) B i + 1 g 1 ( x ) ] g > 1 ,
η i g ( x ) = { x t i t i + g 1 t i for t i t i + g 1 0 otherwise .
t i = { L 2 1 i g ( n + g + 2 2 i ) g L 2 ( n 1 ) ( g 1 ) g < i n L 2 n < i n + g .
f ( ζ c ( x ) ) = i = 1 N inc n 3 ( ω / c ) n 3 ( ω / c ) I m ( q k i ) I c ( q k i ) d q ,
ζ j ( x ) = i = 1 n C i j B i g ( x ) with j = 1 , 2 , , U + V .
C i j = 1 ρ j = 1 ρ C i j + σ mut N ( 0 , 1 ) ,
σ mut = σ 0 [ f c ( ζ ( x ) ) ] min .

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