Abstract

The origin of the instabilities of the Waterman method was studied previously and an improvement in the method was developed for one-dimensional gratings and s polarization [J. Opt. Soc. Am. 15, 1566 (1998)]. Later, the same kind of regularization was used to improve Rayleigh’s expansion method. We show that the same well-adapted regularization process can be generalized to two-dimensional (2D) gratings. Numerical implementations show that the convergence domain of the Waterman method is extended by a factor of ∼40% in the range of groove depth. In the same way, the convergence domain of the Rayleigh expansion method is extended by a factor of ∼35% for 2D sinusoidal gratings. As a consequence, the new versions of Waterman and Rayleigh methods become simple and efficient tools for use in investigating the properties of 2D gratings that have ratios of groove depth to period up to unity.

© 1999 Optical Society of America

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References

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  1. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
  2. N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
    [CrossRef]
  3. G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
    [CrossRef]
  4. J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
    [CrossRef]
  5. Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–415 (1907).
    [CrossRef]
  6. Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.
  7. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI, pp. 1–67.
  8. M. Bagieu, D. Maystre, “Waterman and Rayleigh methods for diffraction problems: extension of the convergence domain,” J. Opt. Soc. Am. A 15, 1566–1576 (1998).
    [CrossRef]
  9. J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
    [CrossRef]
  10. R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
    [CrossRef]
  11. 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]
  12. M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 53–62.
  13. R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
  14. R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
    [CrossRef]
  15. M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

1998

1994

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

1982

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

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, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (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]

1965

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 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]

Bagieu, M.

Botten, L. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Cabrera, N.

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

Cadilhac, M.

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 53–62.

Celli, V.

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

Chesneaux, J. M.

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

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Garcia, N.

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

Hill, N. R.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned 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.

M. Bagieu, D. Maystre, “Waterman and Rayleigh methods for diffraction problems: extension of the convergence domain,” J. Opt. Soc. Am. A 15, 1566–1576 (1998).
[CrossRef]

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[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, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI, pp. 1–67.

McPhedran, R. C.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[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]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Nevière, M.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

Petit, R.

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

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.

Stegun, A.

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

Uretsky, J. L.

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 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]

Ann. Phys.

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

J. Acoust. Soc. Am.

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

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

J. Opt. (Paris)

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

J. Opt. Soc. Am. A

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, “Ill-conditioned 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, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI, pp. 1–67.

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 53–62.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

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

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Figures (6)

Fig. 1
Fig. 1

Schematic representation of the 2D diffraction grating.

Fig. 2
Fig. 2

Definitions of incidence angles θ and ϕ and polarization angle δ. A00=cos δts+sin δtp.

Fig. 3
Fig. 3

Comparison of the numerical results obtained from the Waterman method with and without adapted regularization for a 1D grating and p polarization. Solid curve, ϵ=10-10; dotted line, curve, ϵ=0, N=20, λ=0.6328 µm, dx=1 µm, and θ=0°; the incident wave is p polarized.

Fig. 4
Fig. 4

2D grating. A comparison of the sum of the efficiencies obtained from the Waterman method with and without the adapted regularization. Solid curve, ϵ=10-8, ϵ=10-9; dotted curve, ϵ=0, ϵ=0.

Fig. 5
Fig. 5

Regularized Waterman method for a 2D grating. ϵ=10-8, ϵ=10-9, N=10, dx=1 µm, dz=5 µm, and λ=0.6328 µm; Hz is in micrometers. The grating is used at normal incidence with the electric field parallel to the z axis.

Fig. 6
Fig. 6

Rayleigh method for a 2D grating. A comparison of the sum of the efficiencies obtained from the Rayleigh method with and without adapted regularization. Solid curve, ϵ=9.10-11, ϵ=3.10-11; dotted curve, ϵ=0, ϵ=0.

Tables (2)

Tables Icon

Table 1 Regularization Parameters for a 2D Grating in s Polarization (δ=0°)a

Tables Icon

Table 2 Regularization Parameters for a 2D Grating in p Polarization (δ=90°)a

Equations (97)

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k=αxˆ-βyˆ+γzˆ,
α=k sin θ cos ϕ,
β=k cos θ,
γ=k sin θ sin ϕ.
Ax,00=-cos δ sin ϕ+sin δ cos θ cos ϕ,
Ay,00=sin δ sin θ,
Az,00=cos δ cos ϕ+sin δ cos θ sin ϕ.
Ei=A00 exp(iαx-iβy+iγz).
Ed=nmBnm exp(iαnx+iβnmy+iγmz),
y>yM,
αn=α+2πn/dx,
γm=γ+2πm/dz,
βnm=(k2-αn2-γm2)1/2ifnandmUi(αn2+γm2-k2)1/2otherwise.
E(x+dx, y, z)=E(x, y, z)exp(iαdx),
E(x, y, z+dz)=E(x, y, z)exp(iγdz).
×uE=iωμ0uH,
×uH=-iωϵ0uE,
uE(x, y, z)=nmAnmE exp(iαnx-iβnmy+iγmz)+nmBnmE×exp(iαnx+iβnmy+iγmz),
1dxdzP(uE×vH¯-vE¯×uH)·nˆds
=nUmU(AnmE×AnmH¯+BnmE×BnmH¯-AnmE¯×AnmH-BnmE¯×BnmH).yˆ.
vE(x, y, z)=ts,nmi exp(iαnx-iβnmy+iγmz),
ts,nmi=knmi×yˆ|knmi×yˆ|,
knmi=αnxˆ-βnmyˆ+γmzˆ,
(γmAx,00-αnAz,00)δn,0δm,0
=-12idxdzβnm0dx0dz[γmϕx(x, z)-αnϕz(x, z)]×exp[-iαnx+iβnmf(x, z)-iγmz]dxdz,
Φ(x, z)=1+f(x, z)x2+f(x, z)z21/2dE/dn,=-iωμ01+f(x, z)x2+f(x, z)z21/2jP(x, z).
vE(x, y, z)=tp,nmi exp(iαnx-iβnmy+iγmz),
tp,nmi=ts,nmi×knmik.
(αnAx,00+γmAz,00)δn,0δm,0
=-12idxdzk20dx0dzαnβnm+(αn2+γm2) f(x, z)xϕx(x, z)+γmβnm+(αn2+γm2) f(x, z)zϕz(x, z)×exp(-iαnx+iβnmf(x, z)-iγmz)dxdz.
vE(x, y, z)=ts,nmd exp(iαnx+iβnmy+iγmz),
ts,nmd=knmd×yˆ|knmd×yˆ|,
knmd=αnxˆ+βnmyˆ+γmzˆ.
(γmBx,nm-αnBz,nm)
=12idxdzβnm0dx0dz[γmϕx(x, z)-αnϕz(x, z)]×exp(-iαnx-iβnmf(x, z)-iγmz)dxdz.
vE(x, y, z)=tp,nmd exp(iαnx+iβnmy+iγmz),
tp,nmd=ts,nmd×knmdk,
(αnBx,nm+γmBz,nm)
=-12idxdzk20dx0dzαnβnm-(αn2+γm2) f(x, z)x×ϕx(x, z)+γmβnm-(αn2+γm2) f(x, z)z×ϕz(x, z)exp[-iαnx-iβnmf(x, z)-iγmz]dxdz.
By,nm=-αnβnmBx,nm-γmβnmBz,nm,
enm=(|Bx,nm|2+|By,nm|2+|Bz,nm|2) βnmβ00.
δn,0=-βn2idxkβ00dx1+αnβnf(x)×exp[-iαnx+iβnf(x)]ϕx(x)dx,
Ed=nBn exp(iαnx+iβny),
Bx,n=-βn2idk20dx1-αnβnf(x)×exp[-iαnx-iβnf(x)]ϕx(x)dx,
By,n=-αnβnBx,n,
en=(|Bx,n|2+|By,n|2) βnβ0.
y=Hx2cos(Kxx).
W˜n(x)={exp[iβnf(x)]+ϵn}exp(-iαnx),
Tn(x)=1+αnβnf(x)exp[iβnf(x)-iαnx],
T˜n(x)=1+αnβnf(x)exp[iβnf(x)]+ϵn×exp(-iαnx),
ϵn=ϵdx0dx exp[iβnf(x)]dx.
ρ=nU(e˜n-en)21/2,
ρ=Φ˜-ΦΦ,
y=Hx2cos(Kxx)+Hz2cos(Kzz).
ϑnm(x, z)=exp[-iαnx+iβnmf(x, z)-iγmz]
ϑ˜nm(x, z)=exp(-iαnx-iγmz)×{exp[iβnmf(x, z)]+ϵnm},
ϵnm=ϵdxdz0dx0dx exp[iβnmf(x, z)]dxdz,
ξnm(x, z)=αnβnm+(αn2+γm2) f(x, z)x×exp[-iαnx+iβnmf(x, z)-iγmz]
ξ˜nm(x, z)=αnβnm+(αn2+γm2) f(x, z)x×exp[iβnmf(x, z)]+αnβnmϵnm×exp(-iαnx-iγmz),
χnm(x, z)=γmβnm+(αn2+γm2) f(x, z)z×exp[-iαnx+iβnmf(x, z)-iγmz]
χ˜nm(x, z)=γmβnm+(αn2+γm2) f(x, z)z×exp[iβnmf(x, z)]+γmβnmϵnm×exp(-iαnx-iγmz),
ϵnm=ϵdxdz0dx0dx exp[iβnmf(x, z)]dxdz.
ϕx(x, z)=pqϕˆx,pq exp(iαpx+iγqz),
ϕz(x, z)=pqϕˆz,pq exp(iαpx+iγqz),
M1Φˆx+M2Φˆz=S1,
M3Φˆx+M4Φˆz=S2,
pq(Mnmpq1ϕˆx,pq+Mnmpq2ϕˆz,pq)=Snm1,
pq(Mnmpq3ϕˆx,pq+Mnmpq3ϕˆz,pq)=Snm2,
Mnmpq1=-γm2iβnm(i)n-pJn-pβnm Hx2(i)m-q-Jm-qβnm Hz2+ϵnmδn,pδm,q,
Mnmpq2=αn2iβnm(i)n-pJn-pβnm Hx2(i)m-q×Jm-qβnm Hz2+ϵnmδn,pδm,q,
Mnmpq3=-12ik2αnβnm+(αn2+γm2) (n-p)Kxβnm×(i)n-pJn-pβnm Hx2(i)m-q×Jm-qβnm Hz2+ϵnmδn,pδm,q,
Mnmpq4=-12ik2γmβnm+(αn2+γm2) (m-q)Kzβnm×(i)n-pJn-pβnm Hx2(i)m-q×Jm-qβnm Hz2+ϵnmδn,pδm,q,
ϵnm=ϵJ0βnm Hx2J0βnm Hz2,
ϵnm=ϵϵϵnm,
Snm1=(γmAx,00-αnAz,00)δn,0δm,0,
Snm2=(αnAx,00+γmAz,00)δn,0δm,0.
Ein=a0+a1Hx2+a2Hz2+a3Hx4+a4Hz4+a5Hx2Hz2+.
Ein=1+a2Hz2+a4Hz4+a5Hx2Hz2+.
nˆ×E=nˆ×(Ei+Ed)=0.
nmnˆ×Bnm exp(iαnx+iβnmy+iγmz)
=-nˆ×A00 exp(iαx-iβy+iγz).
Bnm=bnmts,nmd+bnmtp,nmd.
nm 1αn2+γm20dx0dzαnbnm-γmβnmkbnm
+(αn2+γm2)kf(x, z)zbnmexp[i(n-p)Kxx
+iβnmf(x, z)+i(m-q)Kzz]dxdz
=-0dx0dzAz,00+Ay,00 f(x, z)z×exp(-ipKxx-iβf(x, z)-iqKzz)dxdz,
nm 1αn2+γm20dx0dzγmbnm+αnβnmkbnm
-(αn2+γm2)kf(x, z)xbnmexp[i(n-p)Kxx
+iβnmf(x, z)+i(m-q)Kzz]dxdz
=0dx0dzAx,00+Ay,00 f(x, z)x×exp[-ipKxx-iβf(x, z)-iqKzz]dxdz.
enm=(|bnm|2+|bnm|2) βnmβ00.
ζnm(x, z)=αnbnm-γmβnmkbnm+(αn2+γm2)kf(x, z)zbnm×exp[i(n-p)Kxx+iβnmf(x, z)+i(m-q)Kzz]
ζ˜nm(x, z)=αnbnm-γmβnmkbnm×exp[i(n-p)Kxx+i(m-q)Kzz]×{exp[iβnmf(x, z)]+ϵnm}+(αn2+γm2)kf(x, z)zbnm×exp[i(n-p)Kxx+iβnmf(x, z)+i(m-q)Kzz]
ηnm(x, z)
=γmbnm+αnβnmkbnm-(αn2+γm2)kf(x, z)xbnm×exp[i(n-p)Kxx+iβnmf(x, z)+i(m-q)kzz]
η˜nm(x, z)
=γmbnm+αnβnmkbnm×exp[i(n-p)Kxx+i(m-q)Kzz]×{exp[iβnmf(x, z)]+ϵnm}-(αn2+γm2)kf(x, z)xbnm exp[i(n-p)Kxx+iβnmf(x, z)+i(m-q)Kzz].

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