Abstract

A novel approach for the simulation of the field back-scattered from a rough surface is presented. It takes into account polarization and multiple scattering events on the surface, as well as diffraction effects. The validity and usefulness of this simulation is demonstrated in the case of surface topology measurement.

© 2012 Optical Society of America

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    [CrossRef]
  3. E. R. Freniere, G. G. Gregory, and R. A. Hassler, “Edge diffraction in Monte Carlo ray tracing,” Proc. SPIE 3780, 151–157 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  9. D. Didascalou, M. Dottling, N. Geng, and W. Wiesbeck, “An approach to include stochastic rough surface scattering into deterministic ray-optical wave propagation modeling,” IEEE Trans. Antennas Propag. 51, 1508–1515 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (Roberts, 1988), pp. 50–53.

2010

A. Vernes, J. Bohm, and G. Vorlaufer, “Ab initio optical properties of tribological/engineering surfaces,” Tribol. Lett. 39, 39–47 (2010).
[CrossRef]

O. Vasseur, I. Bergond, and X. Orlik, “A Gaussian transition of an optical speckle field studied by the minimal spanning tree method,” J. Eur. Opt. Soc. 5 (2010).
[CrossRef]

2009

2007

2005

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J Acoust. Soc. Am. 117, 1911–1921 (2005).

2003

D. Didascalou, M. Dottling, N. Geng, and W. Wiesbeck, “An approach to include stochastic rough surface scattering into deterministic ray-optical wave propagation modeling,” IEEE Trans. Antennas Propag. 51, 1508–1515 (2003).
[CrossRef]

2001

V. Serikov and S. Kawamoto, “Numerical experiments in Monte Carlo modeling of polarization, diffraction, and interference phenomena,” Proc. SPIE 4436, 80–88 (2001).
[CrossRef]

2000

A. Ettemeyer, “Combination of 3-D deformation and shape measurement by electronic speckle-pattern interferometry for quantitative strain-stress analysis,” Opt. Eng. 39, 212–215 (2000).
[CrossRef]

1999

1998

K. Coffey, K. Priestley, J. Mahan, and M. Sanchez, “Diffraction models of radiation entering an aperture for use in a Monte Carlo ray-trace environment,” Proc. SPIE 3429, 213–219 (1998).
[CrossRef]

1995

F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183–1191 (1995).

1987

H. Kadono, T. Asakura, and N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

Almoro, P.

Anand, A.

Asakura, T.

H. Kadono, T. Asakura, and N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

Bergond, I.

O. Vasseur, I. Bergond, and X. Orlik, “A Gaussian transition of an optical speckle field studied by the minimal spanning tree method,” J. Eur. Opt. Soc. 5 (2010).
[CrossRef]

Bhaduri, B.

U. Kumar, B. Bhaduri, M. Kothiyal, and N. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47, 223–229 (2009).
[CrossRef]

Boas, D.

Bohm, J.

A. Vernes, J. Bohm, and G. Vorlaufer, “Ab initio optical properties of tribological/engineering surfaces,” Tribol. Lett. 39, 39–47 (2010).
[CrossRef]

Broschat, S. L.

F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183–1191 (1995).

Charrière, F.

Chhaniwal, V.

Coffey, K.

K. Coffey, K. Priestley, J. Mahan, and M. Sanchez, “Diffraction models of radiation entering an aperture for use in a Monte Carlo ray-trace environment,” Proc. SPIE 3429, 213–219 (1998).
[CrossRef]

Colomb, T.

Cuche, E.

Depeursinge, C.

Didascalou, D.

D. Didascalou, M. Dottling, N. Geng, and W. Wiesbeck, “An approach to include stochastic rough surface scattering into deterministic ray-optical wave propagation modeling,” IEEE Trans. Antennas Propag. 51, 1508–1515 (2003).
[CrossRef]

Dottling, M.

D. Didascalou, M. Dottling, N. Geng, and W. Wiesbeck, “An approach to include stochastic rough surface scattering into deterministic ray-optical wave propagation modeling,” IEEE Trans. Antennas Propag. 51, 1508–1515 (2003).
[CrossRef]

Duncan, D.

Emery, Y.

Ettemeyer, A.

A. Ettemeyer, “Combination of 3-D deformation and shape measurement by electronic speckle-pattern interferometry for quantitative strain-stress analysis,” Opt. Eng. 39, 212–215 (2000).
[CrossRef]

Fang, Q.

Fischer, D.

Freniere, E. R.

E. R. Freniere, G. G. Gregory, and R. A. Hassler, “Edge diffraction in Monte Carlo ray tracing,” Proc. SPIE 3780, 151–157 (1999).
[CrossRef]

Geng, N.

D. Didascalou, M. Dottling, N. Geng, and W. Wiesbeck, “An approach to include stochastic rough surface scattering into deterministic ray-optical wave propagation modeling,” IEEE Trans. Antennas Propag. 51, 1508–1515 (2003).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 1988), pp. 50–53.

Gordon, G.

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J Acoust. Soc. Am. 117, 1911–1921 (2005).

Gregory, G. G.

E. R. Freniere, G. G. Gregory, and R. A. Hassler, “Edge diffraction in Monte Carlo ray tracing,” Proc. SPIE 3780, 151–157 (1999).
[CrossRef]

Hassler, R. A.

E. R. Freniere, G. G. Gregory, and R. A. Hassler, “Edge diffraction in Monte Carlo ray tracing,” Proc. SPIE 3780, 151–157 (1999).
[CrossRef]

Hastings, F. D.

F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183–1191 (1995).

Heyman, E.

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J Acoust. Soc. Am. 117, 1911–1921 (2005).

Kadono, H.

H. Kadono, T. Asakura, and N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

Kawamoto, S.

V. Serikov and S. Kawamoto, “Numerical experiments in Monte Carlo modeling of polarization, diffraction, and interference phenomena,” Proc. SPIE 4436, 80–88 (2001).
[CrossRef]

Kothiyal, M.

U. Kumar, B. Bhaduri, M. Kothiyal, and N. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47, 223–229 (2009).
[CrossRef]

Kuhn, J.

Kumar, U.

U. Kumar, B. Bhaduri, M. Kothiyal, and N. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47, 223–229 (2009).
[CrossRef]

Mahan, J.

K. Coffey, K. Priestley, J. Mahan, and M. Sanchez, “Diffraction models of radiation entering an aperture for use in a Monte Carlo ray-trace environment,” Proc. SPIE 3429, 213–219 (1998).
[CrossRef]

Marquet, P.

Mazar, R.

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J Acoust. Soc. Am. 117, 1911–1921 (2005).

Mohan, N.

U. Kumar, B. Bhaduri, M. Kothiyal, and N. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47, 223–229 (2009).
[CrossRef]

Montfort, F.

Orlik, X.

O. Vasseur, I. Bergond, and X. Orlik, “A Gaussian transition of an optical speckle field studied by the minimal spanning tree method,” J. Eur. Opt. Soc. 5 (2010).
[CrossRef]

Osten, W.

Pedrini, G.

Prahl, S.

Priestley, K.

K. Coffey, K. Priestley, J. Mahan, and M. Sanchez, “Diffraction models of radiation entering an aperture for use in a Monte Carlo ray-trace environment,” Proc. SPIE 3429, 213–219 (1998).
[CrossRef]

Sanchez, M.

K. Coffey, K. Priestley, J. Mahan, and M. Sanchez, “Diffraction models of radiation entering an aperture for use in a Monte Carlo ray-trace environment,” Proc. SPIE 3429, 213–219 (1998).
[CrossRef]

Schneider, J. B.

F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183–1191 (1995).

Serikov, V.

V. Serikov and S. Kawamoto, “Numerical experiments in Monte Carlo modeling of polarization, diffraction, and interference phenomena,” Proc. SPIE 4436, 80–88 (2001).
[CrossRef]

Takai, N.

H. Kadono, T. Asakura, and N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

Vasseur, O.

O. Vasseur, I. Bergond, and X. Orlik, “A Gaussian transition of an optical speckle field studied by the minimal spanning tree method,” J. Eur. Opt. Soc. 5 (2010).
[CrossRef]

Vernes, A.

A. Vernes, J. Bohm, and G. Vorlaufer, “Ab initio optical properties of tribological/engineering surfaces,” Tribol. Lett. 39, 39–47 (2010).
[CrossRef]

Vorlaufer, G.

A. Vernes, J. Bohm, and G. Vorlaufer, “Ab initio optical properties of tribological/engineering surfaces,” Tribol. Lett. 39, 39–47 (2010).
[CrossRef]

Wiesbeck, W.

D. Didascalou, M. Dottling, N. Geng, and W. Wiesbeck, “An approach to include stochastic rough surface scattering into deterministic ray-optical wave propagation modeling,” IEEE Trans. Antennas Propag. 51, 1508–1515 (2003).
[CrossRef]

Appl. Opt.

Appl. Phys. B

H. Kadono, T. Asakura, and N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

IEEE Trans. Antennas Propag.

D. Didascalou, M. Dottling, N. Geng, and W. Wiesbeck, “An approach to include stochastic rough surface scattering into deterministic ray-optical wave propagation modeling,” IEEE Trans. Antennas Propag. 51, 1508–1515 (2003).
[CrossRef]

F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183–1191 (1995).

J Acoust. Soc. Am.

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J Acoust. Soc. Am. 117, 1911–1921 (2005).

J. Eur. Opt. Soc.

O. Vasseur, I. Bergond, and X. Orlik, “A Gaussian transition of an optical speckle field studied by the minimal spanning tree method,” J. Eur. Opt. Soc. 5 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

A. Ettemeyer, “Combination of 3-D deformation and shape measurement by electronic speckle-pattern interferometry for quantitative strain-stress analysis,” Opt. Eng. 39, 212–215 (2000).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

U. Kumar, B. Bhaduri, M. Kothiyal, and N. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47, 223–229 (2009).
[CrossRef]

Opt. Lett.

Proc. SPIE

K. Coffey, K. Priestley, J. Mahan, and M. Sanchez, “Diffraction models of radiation entering an aperture for use in a Monte Carlo ray-trace environment,” Proc. SPIE 3429, 213–219 (1998).
[CrossRef]

E. R. Freniere, G. G. Gregory, and R. A. Hassler, “Edge diffraction in Monte Carlo ray tracing,” Proc. SPIE 3780, 151–157 (1999).
[CrossRef]

V. Serikov and S. Kawamoto, “Numerical experiments in Monte Carlo modeling of polarization, diffraction, and interference phenomena,” Proc. SPIE 4436, 80–88 (2001).
[CrossRef]

Tribol. Lett.

A. Vernes, J. Bohm, and G. Vorlaufer, “Ab initio optical properties of tribological/engineering surfaces,” Tribol. Lett. 39, 39–47 (2010).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (Roberts, 1988), pp. 50–53.

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Figures (12)

Fig. 1.
Fig. 1.

Schematic of the simulation process.

Fig. 2.
Fig. 2.

Polarization vector decomposition during a reflection on a face of the surface mesh.

Fig. 10.
Fig. 10.

Input triangle patch used to define a rough object with Sa=120nm. Inset: Fourier spectrum of the height variations.

Fig. 3.
Fig. 3.

Flowchart of the simulation process.

Fig. 4.
Fig. 4.

Computing speed assessment for Rayleigh–Sommerfeld diffraction integral (Eq. (5)) in millions of diffracted rays per second (MDiffRay/s).

Fig. 5.
Fig. 5.

Diffracted amplitude at z=5mm for a square flat object of a×a=50μm×50μm, composed of two right triangles. (a) Amplitude map obtained with simulation. (b) Comparison of the simulated result with sinc(afx).

Fig. 6.
Fig. 6.

Input triangle patch used to define a 1.5 μm high and 50 μm large pyramid made of 596 triangles.

Fig. 7.
Fig. 7.

(a) Diffracted amplitude and (b) phase at z=1.25mm due to the input pyramid-shaped model presented in Fig. 6.

Fig. 8.
Fig. 8.

(a) Amplitude (top) and phase (bottom) at z=0 recovered after back propagation of the field shown in Fig. 7 using the Fresnel approximation. (b) Height profile recovered along the dashed line shown on (a).

Fig. 9.
Fig. 9.

Validation of the polarization computation. Linearly polarized illumination: diffracted field amplitude along (a) ex, along (c) ey, and along (e) ez. Circularly polarized illumination: diffracted field amplitude along (b) ex, along (d) ey, and along (f) ez.

Fig. 11.
Fig. 11.

Simulation of the field back-scattered at zd=1.25mm from the surface described with the mesh of Fig. 10, for a wavelength of λ1=685nm. (a) Amplitude and (b) phase.

Fig. 12.
Fig. 12.

Retrieved modulo-2π phase map. (a) Phase map. (b) Profile plot along white dashed line specified on (a). “Plain”: simulated phase profile; “dashed”: phase profile from original mesh.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Er=A0Pexp(iϕ0).
{rp=(n2cosθrn1cosθt)/(n2cosθr+n1cosθt)rs=(n1cosθrn2cosθt)/(n1cosθr+n2cosθt)
Er=epEprp+esEsrs.
Ex,y,z=iλSErx,y,zexp(ik·rd)rdcos(n,rd)dS,
Eijx,y,z=iλrRErx,y,zexp(ik·rij)rijcos(n,rij)
NrNdiffr=Ng12Ng22,

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