Abstract

We present a solution based on the Rayleigh hypothesis. The calculation leads to a matrix system that can be computed to yield the efficiencies of a two-dimensional dielectric grating. The results are compared with previously reported numerical results obtained both with exact theories and with perturbation methods. We show that our method can treat rougher surfaces than can classic perturbation methods.

© 1999 Optical Society of America

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References

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  1. B. W. Shore, M. D. Perry, J. A. Britten, R. D. Boyd, M. D. Feit, H. T. Nguyen, R. Chow, G. E. Loomis, Lifeng Li, “Design of high efficiency dielectric reflection gratings,” J. Opt. Soc. Am. A 14, 1124–1135 (1997).
    [CrossRef]
  2. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1981).
  3. D. Maystre, ed., Selected Papers on Diffraction Gratings, Vol. MS83 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1993).
  4. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  5. D. Maystre, M. Neviére, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
    [CrossRef]
  6. P. Vincent, “A finite difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  7. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981).
    [CrossRef]
  8. R. Bräeuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  9. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  10. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  11. J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
    [CrossRef]
  12. J. J. Greffet, C. Baylard, P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a series solution,” Opt. Lett. 17, 1740–1742 (1992).
    [CrossRef] [PubMed]
  13. J. J. Greffet, Z. Maassarani, “Scattering of electromagnetic waves by a grating: a numerical evaluation of the iterative-series solution,” J. Opt. Soc. Am. A 7, 1483–1493 (1990).
    [CrossRef]
  14. T. Delort, D. Maystre, “Finite-element method for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
    [CrossRef]
  15. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  16. G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
    [CrossRef]
  17. J. W. S. Rayleigh, The Theory of Sound (Macmillan, London, 1896).
  18. A. A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
    [CrossRef]
  19. S. Mainguy, J. J. Greffet, “A numerical evaluation of Rayleigh’s theory applied to scattering by randomly rough dielectric surfaces,” Waves Random Media 8, 79–101 (1998).
    [CrossRef]
  20. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 4–7.
  21. V. I. Tatarskii, “Relation between the Rayleigh equation in diffraction theory and the equation based on Green’s formula,” J. Opt. Soc. Am. A 12, 1254–1260 (1995).
    [CrossRef]

1998

S. Mainguy, J. J. Greffet, “A numerical evaluation of Rayleigh’s theory applied to scattering by randomly rough dielectric surfaces,” Waves Random Media 8, 79–101 (1998).
[CrossRef]

1997

1996

1995

V. I. Tatarskii, “Relation between the Rayleigh equation in diffraction theory and the equation based on Green’s formula,” J. Opt. Soc. Am. A 12, 1254–1260 (1995).
[CrossRef]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

1993

1992

1990

1988

J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
[CrossRef]

1981

A. A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981).
[CrossRef]

1979

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978

D. Maystre, M. Neviére, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

P. Vincent, “A finite difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Baylard, C.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 4–7.

Boyd, R. D.

Bräeuer, R.

R. Bräeuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Britten, J. A.

Bruno, O. P.

Bryngdahl, O.

R. Bräeuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Chow, R.

Delort, T.

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Feit, M. D.

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981).
[CrossRef]

Granet, G.

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Greffet, J. J.

S. Mainguy, J. J. Greffet, “A numerical evaluation of Rayleigh’s theory applied to scattering by randomly rough dielectric surfaces,” Waves Random Media 8, 79–101 (1998).
[CrossRef]

J. J. Greffet, C. Baylard, P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a series solution,” Opt. Lett. 17, 1740–1742 (1992).
[CrossRef] [PubMed]

J. J. Greffet, Z. Maassarani, “Scattering of electromagnetic waves by a grating: a numerical evaluation of the iterative-series solution,” J. Opt. Soc. Am. A 7, 1483–1493 (1990).
[CrossRef]

J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
[CrossRef]

Lalanne, P.

Li, L.

Li, Lifeng

Loomis, G. E.

Maassarani, Z.

Mainguy, S.

S. Mainguy, J. J. Greffet, “A numerical evaluation of Rayleigh’s theory applied to scattering by randomly rough dielectric surfaces,” Waves Random Media 8, 79–101 (1998).
[CrossRef]

Maystre, D.

T. Delort, D. Maystre, “Finite-element method for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

D. Maystre, M. Neviére, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981).
[CrossRef]

Morris, G. M.

Neviére, M.

D. Maystre, M. Neviére, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

Nevière, M.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Nguyen, H. T.

Perry, M. D.

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1981).

Rayleigh, J. W. S.

J. W. S. Rayleigh, The Theory of Sound (Macmillan, London, 1896).

Reitich, F.

Shore, B. W.

Tatarskii, V. I.

Versaevel, P.

Vincent, P.

P. Vincent, “A finite difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Wirgin, A. A.

A. A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 4–7.

Appl. Phys.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

J. Opt. (Paris)

D. Maystre, M. Neviére, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

A. A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
[CrossRef]

Opt. Commun.

R. Bräeuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

P. Vincent, “A finite difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Opt. Lett.

Phys. Rev. B

J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
[CrossRef]

Pure Appl. Opt.

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Waves Random Media

S. Mainguy, J. J. Greffet, “A numerical evaluation of Rayleigh’s theory applied to scattering by randomly rough dielectric surfaces,” Waves Random Media 8, 79–101 (1998).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 4–7.

J. W. S. Rayleigh, The Theory of Sound (Macmillan, London, 1896).

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1981).

D. Maystre, ed., Selected Papers on Diffraction Gratings, Vol. MS83 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1993).

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Figures (7)

Fig. 1
Fig. 1

Electromagnetic field incident on a two-dimensional rough dielectric surface.

Fig. 2
Fig. 2

Reflection, absorption, and conservation of energy for a bisinusoidal grating: n=2.25+i0.75, Dx=Dy=36 µm, h=12 µm, incidence parameters θi=20°, ψi=45°, and λ=10.6 µm, and p polarization.

Fig. 3
Fig. 3

Reflection, absorption and conservation of energy for a bisinusoidal grating: n=2.25+i0.75, Dx=Dy=36 µm, h=12 µm, incidence parameters θi=20°, ψi=45°, and λ=10.6 µm, and s polarization.

Fig. 4
Fig. 4

Absorptivity T, reflectivity R, and total energy for a sinusoidal gold grating: n=0.31898+i2.31986, Dx=1 µm, λ/Dx=0.5, p polarization.

Fig. 5
Fig. 5

Absorptivity T and total energy for two bisinusoidal gold gratings: n=0.31898+i2.31986, D=1 µm, λ/D=0.5, p polarization.

Fig. 6
Fig. 6

Energy conservation as a function of the height-to-period ratio (h/D) for four methods, the RFS and those of the three references indicated. The grating is a bisinusoidal surface with λ/D=0.83, θi=0°, ψi=0°, and =4.

Fig. 7
Fig. 7

Absorptivity of a bisinusoidal gold grating illuminated at 0.5 µm. The parameters are D=0.445 µm, h=0.1 µm, and s polarization.

Tables (3)

Tables Icon

Table 1 Comparison of Different Methods

Tables Icon

Table 2 Comparison of the RFS and the Method of Ref. 16a

Tables Icon

Table 3 Comparison of the RFS and the Method of Ref. 16a

Equations (60)

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Ei(r, t)=Ei(r)exp(-iωt)=ei(r)exp(ik·r)exp(-iωt).
Er(r)=er(κ)exp[i(k·r+γrz)]dκ
Et(r)=et(κ)exp[i(κ·r-γtz)]dκ,
κ2+γr2=ω2/c2,Im(γr)>0,
κ2+γt2=(ω)(ω2/c2),Im(γt)>0.
n  (Br+Bi)=n  Bt,
n  (Er+Ei)=n  Et.
n=1[(S)2+1]1/2(-S, 1).
as(k)=z κ/|κ|,
ap(k)=as  k/|k|.
et(κ)=ets(κ)as(kt)+etp(κ)ap(kt),
er(κ)=ers(κ)as(kr)+erp(κ)ap(kr).
Bt(r, t)=-iω  Et(r, t).
Bt(r)=-1ω[ket(κ)]exp[i(κr-γtz)]dκ.
Btx(r)=-1ω-etp(κ)kt2κy|kt||κ|+ets(κ)κxγt|κ|×exp[i(κ·r-γtz)]dκ,
Bty(r)=-1ωetp(κ)kt2κx|kt||κ|+ets(κ)·κyγt|κ|×exp[i(κ·r-γtz)]dκ,
Btz(r)=-1ω[ets(κ)|κ|]exp[i(κ·r-γtz)]dκ.
-ets(κ)Sy|κ|+κyγt|κ|-etp(κ)kt2κx|kt||κ|
×exp[i(κ·r-γtz)]dκ+ers(κ)Sy|κ|-κyγr|κ|
+erp(κ)kr2κx|kr||κ|exp[i(κ·r+γrz)]dκ
=-eis(κi)Sy|κi|+κiyγi|κi|-eip(κi)ki2κix|ki||κi|×exp[i(κi·r-γiz)],
-ets(κ)Sx|κ|+κxγt|κ|+etp(κ)kt2κy|kt||κ|
×exp[i(κ·r-γtz)]dκ+ers(κ)Sx|κ|-κxγr|κ|
-erp(κ)kr2κy|kr||κ|exp[i(κ·r+γrz)]dκ
=-eis(κi)Sx|κi|+κixγi|κi|+eip(κi)ki2κiy|ki||κi|×exp[i(κi·r-γiz)],
ets(κ)-κx|κ|+etp(κ)Sy|κ||kt|+κyγt|κ||kt|
×exp[i(κ·r-γtz)]dκ
+ers(κ)κx|κ|-erp(κ)Sy|κ||kr|-κyγr|κ||kr|
×exp[i(κ·r+γrz)]dκ
=-eis(κi)κix|ki|+eip(κi)Sy|κi||ki|+κiyγi|κi||ki|×exp[i(κi·r-γiz)],
ets(κ)-κy|κ|-etp(κ)Sx|κ||kt|+κxγt|κ||kt|
×exp[i(κ·r-γtz)]dκ
+ers(κ)κy|κ|+erp(κ)Sx|κ||kr|-κxγr|κ||kr|
×exp[i(κ·r+γrz)]dκ
=-eis(κi)κiy|κi|+eip(κi)Sx|κi||ki|+κixγi|κi||ki|×exp[i(κi·r-γiz)].
Er(r)=a=-Pa=Pb=-Pb=Per(κa,b)exp[i(κa,b·r+γra,bz)],
Et(r)=a=-Pa=Pb=-Pb=Pet(κa,b)exp[i(κa,b·r-γta,bz)],
κa,b=κi+2πaDxx+2πbDyy.
AX=Y,
Ts(a, b)=|ets(κa,b)|2 Re(γt)|ei|2(ω/c)cos(θi),
Tp(a, b)=|etp(κa,b)|2 Re(γt)|ei|2(ω/c)cos(θi),
Rs(a, b)=|ers(κa,b)|2 Re(γr)|ei|2(ω/c)cos(θi),
Rp(a, b)=|erp(κa,b)|2 Re(γr)|ei|2(ω/c)cos(θi).
α+ρ=1.
ϕr=-12µ0ωDxDy Re(a,b)γra,ber(κa,b)·er*(κa,b),
ϕt=12µ0ωRe(a,b)(a,b)I(κa,b, κa,b)×[er(κa,b)·er*(κa,b)]kt*·kt-kt*γta,b-γt*a,b,
I(κa,b, κa,b)=S exp[i(κa,b-κa,b)·ρ-(γta,b-γt*a,b)S(ρ)]dρ.
z(x, y)=h4cos2πxDx+cos2πyDy.
z(x, y)=h2cos2πxDx.
ksp=k0 Re+11/2.
ksp=k0 sin(θi)+n(2π/D)(ninteger).
ksp=n(2π/D).
π=12µ0ωRe(E  B*).
π=12µ0ωReet(κ) [kt*  et*(κ)]×exp[i(κ-κ)·ρ-i(γt-γt*)S(ρ)]dκdκ  .
 exp[i(κ-κ)·ρ-i(γt-γt*)S(ρ)]
=[i(κ-κ)·ρ-i(γt-γt*)S(ρ)]×exp[i(κ-κ)·ρ-i(γt-γt*)S(ρ)],
S exp[i(κ-κ)·ρ-i(γt-γt*)S(ρ)]dρ=0.
ϕt=-12µ0ωReI(κ, κ)[kt*·et(κ)][kt·et*(κ)]γt-γt*-[et(κ)·et*(κ)]kt*·kt-kt*γt-γt*dκdκ,
I(κ, κ)=S exp[i(κ-κ)·ρ-(γt-γt*)S(ρ)]dρ.
ϕt=12µ0ωReI(κ, κ)[et(κ)·et*(κ)]×kt*·kt-kt*γt-γt*dκdκ.

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