Abstract

We make a generalization of the integral method in the electromagnetic theory of gratings to study diffraction by echelles covered with dielectric lossless or absorbing layers. Numerical examples are given that show that, as in the resonance domain, the diffraction efficiency is more complicated than being a simple product of lossless diffraction efficiency curves and plane surface reflectivity.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
    [CrossRef]
  2. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  3. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity via the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 12, 2672–2678 (1995).
    [CrossRef]
  4. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  5. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  6. P. Lalanne, G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  7. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  8. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  9. J. P. Plumey, B. Guizal, J. Chandezon, “The coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
    [CrossRef]
  10. G. Granet, J. Chandezon, O. Coudert, “Extension of the C method to nonhomogeneous media: application to nonhomogeneous layers with parallel modulated faces and to inclined lamellar gratings,” J. Opt. Soc. Am. A 14, 1576–1582 (1997).
    [CrossRef]
  11. D. Maystre, “Integral method,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.
    [CrossRef]
  12. E. Loewen, D. Maystre, E. Popov, L. Tsonev, “Echelles: scalar, electromagnetic, and real-groove properties,” Appl. Opt. 34, 1707–1727 (1995).
    [CrossRef] [PubMed]
  13. E. Loewen, D. Maystre, E. Popov, L. Tsonev, “Diffraction efficiency of echelles working in extremely high orders,” Appl. Opt. 35, 1700–1704 (1996).
    [CrossRef] [PubMed]
  14. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  15. D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
    [CrossRef]
  16. A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109–120 (1991).
    [CrossRef]
  17. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  18. Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  19. R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973); M. Nevière, M. Cadilhac, “Sur la validite du developpement de Rayleigh,” Opt. Commun. 2, 235–238 (1970).
    [CrossRef]
  20. A. Wirgin, “Sur la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. Ser. B 288, 179–182 (1979).
  21. E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings: nonsinusoidal profile,” J. Mod. Opt. 34, 155–158 (1987).
    [CrossRef]
  22. P. M. Van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. 71, 1224–1229 (1981).
    [CrossRef]
  23. See E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 4.

1997

1996

1995

1994

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

1993

1991

A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

1987

E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings: nonsinusoidal profile,” J. Mod. Opt. 34, 155–158 (1987).
[CrossRef]

1982

1981

1979

A. Wirgin, “Sur la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. Ser. B 288, 179–182 (1979).

1978

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
[CrossRef]

1973

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973); M. Nevière, M. Cadilhac, “Sur la validite du developpement de Rayleigh,” Opt. Commun. 2, 235–238 (1970).
[CrossRef]

1971

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

1907

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Cadilhac, M.

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Cerutti-Maori, G.

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Chandezon, J.

Cornet, G.

Coudert, O.

Dupuis, M. T.

Gaylord, T. K.

Granet, G.

Guizal, B.

Lalanne, P.

Li, L.

Loewen, E.

Mashev, L.

E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings: nonsinusoidal profile,” J. Mod. Opt. 34, 155–158 (1987).
[CrossRef]

Maystre, D.

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973); M. Nevière, M. Cadilhac, “Sur la validite du developpement de Rayleigh,” Opt. Commun. 2, 235–238 (1970).
[CrossRef]

Moharam, M. G.

Montiel, F.

Morris, G.

Nevière, M.

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity via the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 12, 2672–2678 (1995).
[CrossRef]

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Plumey, J. P.

Pomp, A.

A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

Popov, E.

E. Loewen, D. Maystre, E. Popov, L. Tsonev, “Diffraction efficiency of echelles working in extremely high orders,” Appl. Opt. 35, 1700–1704 (1996).
[CrossRef] [PubMed]

E. Loewen, D. Maystre, E. Popov, L. Tsonev, “Echelles: scalar, electromagnetic, and real-groove properties,” Appl. Opt. 34, 1707–1727 (1995).
[CrossRef] [PubMed]

E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings: nonsinusoidal profile,” J. Mod. Opt. 34, 155–158 (1987).
[CrossRef]

See E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 4.

Rayleigh, Lord

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Tsonev, L.

Van den Berg, P. M.

Wirgin, A.

A. Wirgin, “Sur la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. Ser. B 288, 179–182 (1979).

Appl. Opt.

C. R. Acad. Sci. Ser. B

A. Wirgin, “Sur la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. Ser. B 288, 179–182 (1979).

J. Mod. Opt.

E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings: nonsinusoidal profile,” J. Mod. Opt. 34, 155–158 (1987).
[CrossRef]

A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
[CrossRef]

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Proc. R. Soc. London Ser. A

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Pure Appl. Opt.

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Radio Sci.

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973); M. Nevière, M. Cadilhac, “Sur la validite du developpement de Rayleigh,” Opt. Commun. 2, 235–238 (1970).
[CrossRef]

Other

See E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 4.

D. Maystre, “Integral method,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Schematic representation of a grating covered by a dielectric layer, together with the notation used in the text.

Fig. 2
Fig. 2

Schematic representation of a reflective echelle. Only a few diffracted orders are shown.

Fig. 3
Fig. 3

Total reflectivity of an aluminum surface covered by 25-nm-thick layers of Al2O3 or MgF2 as a function of wavelength. Solid curves, TE polarization; dashed curves, TM polarization; thinner curves, plane mirror at an 80° angle of incidence; heavier curves, echelle with 83 grooves/mm, 80° groove angle, and incidence of 80°; dotted curves, normal incidence upon a plane mirror.

Fig. 4
Fig. 4

(a) Detailed view of Fig. 3(a) in the vicinity of the absorption edge of Al2O3. (b) Diffraction efficiency (thinner curves) in order 153 and the total diffracted energy (heavier curves) of the echelle described in Fig. 3.

Fig. 5
Fig. 5

Same as in Fig. 4 but for a MgF2 covering layer.

Fig. 6
Fig. 6

Spectral dependence of diffraction efficiency in order -1 of an aluminum sinusoidal grating with period d = 0.3 µm and groove depth h = 0.12 µm in a -1-order Littrow mount. The grating is covered with layers of MgF2 with the thicknesses (in nanometers) shown. (a) TE polarization, (b) TM polarization. Dashed curves, bare aluminum grating.

Fig. 7
Fig. 7

(a) Same as in Fig. 6(a), but here only three layer thicknesses are presented to permit the anomalies in the spectral region 0.10–0.12 µm to be seen. (b) Comparison of the results of the integral method and the formalism of Chandezon et al.8 for the sinusoidal grating presented in Fig. 6 for a layer 50 nm thick.

Equations (48)

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

U0=F0 exp-iα0xin V00elsewhere,U1=F1 exp-iα0xin V10elsewhere,U2=F2-Fiexp-iα0xin V20elsewhere,
Fix, y=expiα0x-i2πλν2 cos θiyin V20elsewhere,
α0=2πλν2 sin θi,
Below S1:  U0x, y=-S1 A1-x, y, s1ϕ1-s1ds1-S1 1-x, y, s1ψ1-s1ds1,
Between S1 and S2: U1x, y=S1 A1+x, y, s1ϕ1+s1ds1+S1 1+x, y, s1ψ1+s1ds1-S2 A2-x, y, s2ϕ2-s2ds2-S2 2-x, y, s2ψ2-s2ds2.
Above S2:  U2x, y=S2 A2+x, y, s2ϕ2+s2ds2+S2 2+x, y, s2ψ2+s2ds2.
Aj±x, y, sj=12idm=-1βj,m±expIm Kx-xjsj+iβj,m±|y-yjsj|,
j±x, y, sj=12dm=-dxjdsjsgny-yjsj-αmβj,m±dyjdsj expIm Kx-xjsj+iβj,m±|y-yjsj|,
αm=α0+mK,
K=2πd,
β1,m-=2πλν02-αm21/2,
β1,m+=β2,m-=2πλν12-αm21/2,
β2,m+=2πλν22-αm21/2.
ψ1-=F0s1exp-iα0xs1limM0P1 U0x, y,
ψ1+=F1s1exp-iα0xs1limM1P1 U1x, y,
ψ2-=F1s2exp-iα0xs2limM1P2 U1x, y,
ψ2+=F2s2-Fis2exp-iα0xs2limM2P2 U2x, y,
ϕ1-=dF0x, ydnˆ1s1 exp-iα0xs1limM0P1dU0x, ydnˆ1+iα0n1,xU0x, y,
ϕ1+=dF1x, ydnˆ1s1 exp-iα0xs1limM1P1dU1x, ydnˆ1+iα0n1,xU1x, y,
ϕ2-=dF1x, ydnˆ2s2 exp-iα0xs2limM1P2dU1x, ydnˆ2+iα0n2,xU1x, y,
ϕ2+=dF2x, y-Fix, ydnˆ2s2 exp-iα0xs2limM2P2dU2x, ydnˆ2+iα0n2,xU2x, y,
U0s1=1/2 limM1P1 U0x, y+limM0P1 U0x, y=ψ1-s12.
U0M10.
ψ1-s12=-S1 A1-s1, s1ϕ1-s1ds1-S1 B1-s1, s1ψ1-s1ds1,
ψ1+s12=S1 A1+s1, s1ϕ1+s1ds1+S1 B1+s1, s1ψ1+s1ds1-S1 A2-s1, s2ϕ2-s2ds2-S1 2-s1, s2ψ2-s2ds2.
ψ2-s22=S1 A1+s2, s1ϕ1+s1ds1+S1 1+s2, s1ψ1+s1ds1-S1 A2-s2, s2ϕ2-s2ds2-S1 B2-s2, s2ψ2-s2ds2.
ψ2+s22=S2 A2+s2, s2ϕ2+s2ds2+S2 B2+s2, s2ψ2+s2ds2.
ψ1-=ψ1+,
ψ2-=ψ2++ψi,
q1-ϕ1-=q1+ϕ1+,
q2-ϕ2-=q2+ϕ2++ϕi,
q1-=1/0TM case1/μ0TE case,
q1+=q2-=1/1TM case1/μ1TE case,
q2+=1/2TM case1/μ2TE case,
ψi=exp-i 2πλν2 cos θiys2, ϕi=-i 2πλν2dys2ds2sin θi+dxs2ds2cos θi×exp-i 2πλν2 cos θiys2.
ϕ1-=C*ψ1-,
C=-A1--1B1-+I2,
A2-*ϕ2-+2-*ψ2-=W1*ψ1-, A2-*ϕ2-+B2-+I2*ψ2-=W2*ψ1-,
W1=B1+-I2+q1-q1+ A1+C, W2=1++q1-q1+ A1+C.
QQ11Q12Q21Q22=defA2-2-A2-B2-+I/2-1;
ϕ2-ψ2-=R1R2*ψ1-,
R1R2=QW1W2.
q2-q2+ A2+R1+B2+-I2R2*ψ1-=A2+*ϕi+B2+-I2*ψi.
Aj,pq±=1/2Aj±sjPj,p, sjPj,qsjPj,q+1-sjPj,q-1,  pq, Aj,pq±=1/2Aj±siPi,p, sjPj,qsiPi,q+1-sjPj,q-1,  ij,
U2x, y=m=- rn expinKx+iβ2,m+y.
rm=12idβ2,m+S2exp-Im Kx2s2-iβ2,m+yjsjϕ2+s2ds2+12dS2dx2ds2-αmβ2,m+dy2ds2exp-Im Kx2s2-iβ2,m+yjsjψ2+s2ds2.
ηm=β2,m+β2,0+ |rm|2.
Np,min  d/t.

Metrics