Abstract

We show that the Waterman method, a classical and rigorous method of electromagnetics for scattering by surfaces or objects, can be significantly improved. In a first step, it is shown, in the case of scattering by gratings, that the origin of the instabilities encountered in the numerical implementation of the method must be found in the ill conditioning of the equations. A well-adapted regularization process allows us to extend the domain of convergence of the method by a factor of approximately 40% in the range of groove depth for one-dimensional gratings and s polarization. Finally, we show that the same kind of regularization can extend the domain of convergence of the Rayleigh method.

© 1998 Optical Society of America

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  1. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
  2. A. Wirgin, “Sur trois variantes de la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C.R. Acad. Sci. Ser. A 289, 259–262 (1979).
  3. A. Wirgin, “Aspects numeriques du problème de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. Ser. B 289, 273–276 (1979).
  4. Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–415 (1907).
    [CrossRef]
  5. Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.
  6. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, Amsterdam, 1984), pp. 1–67.
  7. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.
  8. J. Hadamard, Le problème de Cauchy (Hermann, Paris, 1932).
  9. K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1989).
  10. A. Tikhonov, V. Arsenine, Méthode de Résolution de Problèmes Mal Posés (Editions de Moscou, Moscow, 1976).
  11. N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Illconditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
    [CrossRef]
  12. G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
    [CrossRef]
  13. J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. (New York) 33, 400–427 (1965).
    [CrossRef]
  14. R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
    [CrossRef]
  15. D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
    [CrossRef]
  16. M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
  17. M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).
  18. R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. 262, 468–471 (1966).
  19. A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. (Paris) 9, 83–90 (1978).
    [CrossRef]
  20. J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
    [CrossRef]

1994

J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
[CrossRef]

1979

A. Wirgin, “Sur trois variantes de la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C.R. Acad. Sci. Ser. A 289, 259–262 (1979).

A. Wirgin, “Aspects numeriques du problème de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. Ser. B 289, 273–276 (1979).

G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
[CrossRef]

1978

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Illconditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

1975

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

1974

D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
[CrossRef]

1967

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[CrossRef]

1966

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. 262, 468–471 (1966).

1965

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. (New York) 33, 400–427 (1965).
[CrossRef]

1907

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–415 (1907).
[CrossRef]

Abramowitz, M.

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

Armand, G.

G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
[CrossRef]

Arsenine, V.

A. Tikhonov, V. Arsenine, Méthode de Résolution de Problèmes Mal Posés (Editions de Moscou, Moscow, 1976).

Cabrera, N.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Illconditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Cadilhac, M.

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. 262, 468–471 (1966).

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).

Celli, V.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Illconditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Chadan, K.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1989).

Chesneaux, J. M.

J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
[CrossRef]

Garcia, N.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Illconditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Hadamard, J.

J. Hadamard, Le problème de Cauchy (Hermann, Paris, 1932).

Hill, N. R.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Illconditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Manson, J. R.

G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
[CrossRef]

Maystre, D.

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, Amsterdam, 1984), pp. 1–67.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.

McPhedran, R. C.

D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
[CrossRef]

Petit, R.

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[CrossRef]

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. 262, 468–471 (1966).

Rayleigh, Lord

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–415 (1907).
[CrossRef]

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

Roger, A.

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

Sabatier, P. C.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1989).

Stegun, A.

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

Tikhonov, A.

A. Tikhonov, V. Arsenine, Méthode de Résolution de Problèmes Mal Posés (Editions de Moscou, Moscow, 1976).

Uretsky, J. L.

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. (New York) 33, 400–427 (1965).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

Wirgin, A.

J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
[CrossRef]

A. Wirgin, “Sur trois variantes de la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C.R. Acad. Sci. Ser. A 289, 259–262 (1979).

A. Wirgin, “Aspects numeriques du problème de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. Ser. B 289, 273–276 (1979).

Ann. Phys. (New York)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. (New York) 33, 400–427 (1965).
[CrossRef]

C. R. Acad. Sci.

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. 262, 468–471 (1966).

C. R. Acad. Sci. Ser. B

A. Wirgin, “Aspects numeriques du problème de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. Ser. B 289, 273–276 (1979).

C.R. Acad. Sci. Ser. A

A. Wirgin, “Sur trois variantes de la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C.R. Acad. Sci. Ser. A 289, 259–262 (1979).

J. Acoust. Soc. Am.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
[CrossRef]

J. Opt. (Paris)

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. (Paris) 9, 83–90 (1978).
[CrossRef]

Opt. Acta

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[CrossRef]

Opt. Commun.

D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
[CrossRef]

Phys. Rev. B

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Illconditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
[CrossRef]

Proc. R. Soc. London Ser. A

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–415 (1907).
[CrossRef]

Other

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, Amsterdam, 1984), pp. 1–67.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.

J. Hadamard, Le problème de Cauchy (Hermann, Paris, 1932).

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1989).

A. Tikhonov, V. Arsenine, Méthode de Résolution de Problèmes Mal Posés (Editions de Moscou, Moscow, 1976).

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

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Figures (10)

Fig. 1
Fig. 1

Notation. The angle of incidence θ is an algebraic number measured clockwise, and yM-ym=H.

Fig. 2
Fig. 2

Relative error on the results given by the Waterman method. Solid curves, relative error on efficiencies; dashed curves, relative error on surface current density. The wavelength λ=0.6328 µm, the grating period is equal to 1 µm, the angle of incidence θ=0°, and the incident wave is s polarized.

Fig. 3
Fig. 3

Convergence domain of the function δEi(x, y).

Fig. 4
Fig. 4

Comparison of the numerical results obtained from the Waterman method with and without Tikhonov’s regularization, with N=20, λ=0.6328 µm, d=1 µm, θ=0°, and an s-polarized incident wave. Solid curve, q0=10-16; dotted curve, q0=0.

Fig. 5
Fig. 5

Solid curve, function |Wn(x)|; dashed curve, function |W˜n(x)|. The parameters are n=10, H=1 µm, d=1 µm, λ=0.6328 µm, and θ=0°.

Fig. 6
Fig. 6

Relative error on the results given by the Waterman method, with N=20, H=1 µm, d=1 µm, λ=0.6328 µm, and θ=0°. Solid curve, relative error on efficiencies; dashed curve, relative error on surface current density.

Fig. 7
Fig. 7

Energy balance criterion on the results given by the Waterman method, with the same parameters as those in Fig. 6.

Fig. 8
Fig. 8

Norm of the function ϕ, with the same parameters as those in Fig. 6.

Fig. 9
Fig. 9

Relative error on the results given by the Waterman method with adapted regularization. Solid curves, relative error on efficiencies; dashed curves, relative error on surface current density. The regularization coefficient =6×10-8, the period d=1 µm, the wavelength λ=0.6328 µm, the angle of incidence θ=0°, and the incident wave is s polarized.

Fig. 10
Fig. 10

Comparison of the numerical results obtained from the Rayleigh expansion method with and without adapted regularization. Solid curves: =9×10-9 for N=5, =9×10-9 for N=10, =3×10-9 for N=20, and =2×10-9 for N=30; dotted curves: =0. The other parameters are λ=0.6328 µm, d=1 µm, and θ=0°, and the incident wave is s polarized.

Tables (2)

Tables Icon

Table 1 Normalized Eigenvalues of the Matrix M*M in the Case of a Sinusoidal Profile f(x)=(H/2)[cos(Kx)+1], with N=20, H=0.2 µm, d=1 µm, λ=0.6328 µm, and θ=0°

Tables Icon

Table 2 Values of the Complex Amplitudes Bn of the Diffracted Wave, with N=20, H=0.5 µm, d=1 µm, λ=0.6328 µm, and θ=0°

Equations (61)

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

Ei=Eizˆ,
Ei=exp(iαx-iβy),
α=k sin θ,β=k cos θ.
Ed=nBn exp(iαnx+iβny)ify>yM,
αn=α+nλ/d,
βn=(k2-αn2)1/2if nU,i(αn2-k2)1/2else,
E(x+d, y)=E(x, y)exp(iαd).
u(x, y)=nAn exp(iαnx-iβny)+nBn exp(iαnx+iβny),
v(x, y)=nAn exp(iαnx-iβny)+nBn exp(iαnx+iβny).
12idPu dv¯dn-v¯dudnds=nUβn(AnAn¯-BnBn¯),
v(x, y)=exp(iαnx-iβny).
δn,0=-12idβn0d exp[-iαnx+iβnf(x)]ϕ(x)dx,
ϕ(x)={1+[f(x)]2}1/2(dE/dn),
=-iωμ0{1+[f(x)]2}1/2jP(x),
v(x, y)=exp(iαnx+iβny),
Bn=12idβn0d exp[-iαnx-iβnf(x)]ϕ(x)dx.
ϕˆ(x)=mϕm exp(imKx),
ϕ(x)=mϕm exp(iαmx).
m=-N+NMn,mϕm=Snn  (-N,+N),
Mn,m=-12idβn0d exp[iβnf(x)+i(m-n)Kx]dx,
Sn=δn,0.
f(x)=H2cos(Kx),
Mn,m=-12iβnin-mJn-mβn H2,
ρ=nU(e˜n-en)21/2,
ρ=ϕ˜-ϕ/ϕ,
-12idβn0d exp[-iαnx+iβnf(x)]ϕ˜(x)dx=δn,0+sn
n[-N,+N].
δϕ(x)=ϕ˜(x)-ϕ(x),
-12idβn0d exp[-iαnx+iβnf(x)]δϕ(x)dx=sn
n[-N,+N].
δEi(x, y)=n=-N+Nsn exp(iαnx-iβny).
|sn|=|βn||In|,
In=0d exp[-iαnx+iβnf(x)]ϕ(x)dx,
βni|n|Kifn,
Incnϕ(xc)exp[-iαnxc-|n|Kf(xc)]ifn,
f(xc)=-iifn>0iifn<0,
sin(Kxc)=2iHK,
xc=-d2-iKarcsinh2HK.
f(xc)=-H21+4H2K2,
In(-1)n cn1/2ϕ(xc)exp{n[1+h2-arcsinh(1/h)]},
h=KH/2.
|sn||n|3/2exp{n[1+h2-arcsinh(1/h)]},
=|c| |ϕ(xc)|.
(δEi)n|n|3/2exp{n[1+h2-arcsinh(1/h)+Ky]}.
1+h2-arcsinh(1/h)+Ky<0;
y<1K[arcsinh(1/h)-1+h2].
hc+1+hc2-arcsinh(1/hc)=0.
hc=0.447743,
Mϕ=S,
M*Mϕ=M*S
M*M(ϕ+χVn)=M*S+τnχVnM*S.
(M*M+q0I)ϕ=M*S.
Wn(x)=exp[-iαnx+iβnf(x)].
|W10(x)|=exp[-β˜10f(x)].
W˜n(x)=exp(-iαnx){exp[iβnf(x)]+n},
n=d0d exp[iβnf(x)]dx,
E=exp(iαx-iβy)+nBn exp(iαnx+iβny),
exp[iαx-iβf(x)]+nBn
×exp[iαnx+iβnf(x)]=0x.
Wn(x)=exp[iαnx+iβnf(x)].
W˜n=exp(iαnx){exp[iβnf(x)]+n},

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