Abstract

The scattering of an electromagnetic plane wave incident upon an inhomogeneous multilayer structure is considered in symbolic form. In this framework a scattering-matrix propagation algorithm that decouples recurrences for backward- and forward-scattered wave amplitudes is developed. By construction the scattering-matrix solution procedure is stable against increase of truncation order and depths and number of layers, irrespective of numerical implementation. For grating structures a numerical study using Fourier-transform discretization is performed. In this implementation the convergence issue for TM polarization is recapitulated.

© 1996 Optical Society of America

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References

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1995

M. Auslender, S. Hava, Proc. SPIE 2399, 95 (1995); Infrared Phys. Technol 36, 1077 (1995).
[CrossRef]

R. H. Morf, J. Opt. Soc. Am. A 12, 1043 (1995).
[CrossRef]

1994

1993

L. Li, J. Opt. Soc. Am. A 10, 2581 (1993).
[CrossRef]

S. Hava, M. Auslender, D. Rabinovich, Bull. Isr. Phys. Soc. 39, 139 (1993); Appl. Opt. 33, 4807 (1994).
[PubMed]

1991

1986

1981

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, Opt. Acta 28, 413, 1087 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, Opt. Acta 28, 1103 (1981).
[CrossRef]

M. G. Moharam, T. K. Gaylord, J. Opt. Soc. Am. 71, 811 (1981); M. G. Moharam, T. K. Gaylord, J. Opt. Soc. Am. 72, 1385 (1982).
[CrossRef]

1978

1956

S. V. Rytov, Sov. Phys. JETP 2, 466 (1956).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, Opt. Acta 28, 413, 1087 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, Opt. Acta 28, 413, 1087 (1981).
[CrossRef]

Auslender, M.

M. Auslender, S. Hava, Proc. SPIE 2399, 95 (1995); Infrared Phys. Technol 36, 1077 (1995).
[CrossRef]

S. Hava, M. Auslender, D. Rabinovich, Bull. Isr. Phys. Soc. 39, 139 (1993); Appl. Opt. 33, 4807 (1994).
[PubMed]

Awada, K. A.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, Opt. Acta 28, 413, 1087 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, Opt. Acta 28, 1103 (1981).
[CrossRef]

Chateau, N.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, Opt. Acta 28, 1103 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, Opt. Acta 28, 413, 1087 (1981).
[CrossRef]

Gaylord, T. K.

Hava, S.

M. Auslender, S. Hava, Proc. SPIE 2399, 95 (1995); Infrared Phys. Technol 36, 1077 (1995).
[CrossRef]

S. Hava, M. Auslender, D. Rabinovich, Bull. Isr. Phys. Soc. 39, 139 (1993); Appl. Opt. 33, 4807 (1994).
[PubMed]

Hugonin, J.-P.

Hutley, M.

M. Hutley, Diffraction Gratings (Academic, London, 1982).

Knop, K.

Li, L.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, Opt. Acta 28, 413, 1087 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, Opt. Acta 28, 1103 (1981).
[CrossRef]

Moharam, M. G.

Morf, R. H.

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York, 1882).

Pai, D. M.

Rabinovich, D.

S. Hava, M. Auslender, D. Rabinovich, Bull. Isr. Phys. Soc. 39, 139 (1993); Appl. Opt. 33, 4807 (1994).
[PubMed]

Rytov, S. V.

S. V. Rytov, Sov. Phys. JETP 2, 466 (1956).

Bull. Isr. Phys. Soc.

S. Hava, M. Auslender, D. Rabinovich, Bull. Isr. Phys. Soc. 39, 139 (1993); Appl. Opt. 33, 4807 (1994).
[PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, Opt. Acta 28, 413, 1087 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, Opt. Acta 28, 1103 (1981).
[CrossRef]

Proc. SPIE

M. Auslender, S. Hava, Proc. SPIE 2399, 95 (1995); Infrared Phys. Technol 36, 1077 (1995).
[CrossRef]

Sov. Phys. JETP

S. V. Rytov, Sov. Phys. JETP 2, 466 (1956).

Other

M. Hutley, Diffraction Gratings (Academic, London, 1982).

R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York, 1882).

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Figures (3)

Fig. 1
Fig. 1

Convergence in TE polarization: truncation error of zero-order reflection efficiency at θ = 0° for gratings with Λ = 2 λ , d = 0.51, h = 0.96λ. ○, Si, ε = 3 . 442 ( λ = 2 . 50 μ m ) ; *, Ag, ε = 0 . 14 + i 4 . 15 ( λ = 0 . 653 μ m ) ; ×, Al, ε = 1 . 47 + i 7 . 79 ( λ = 0 . 650 μ m ) .

Fig. 2
Fig. 2

Convergence in TM polarization: truncation errors of zero-order diffraction efficiencies at θ = 0° for Si gratings. ○, TM0 transmission (Λ = 0.9λ, d = 0.36, h = 0.56λ); *, TM1 reflection ( Λ = 2 λ , d = 0.51, h = 1.76λ); ×, the first TM2 transmission (Λ = 0.9λ, d = 0.36, h = 0.96λ); solid curve, the second TM2 reflection ( Λ = 2 λ , d = 0.51, h = 1.36λ).

Fig. 3
Fig. 3

Efficiency of the ±1st diffracted order in TM2 at θ = 0° for a Ag grating (λ = 0.653 μm) with Λ = 1.4λ, d = 0.36, h = 0.16λ. Solid curve, the second TM2; dashed curve, the first TM2; dashed–dotted curve, TM1.

Tables (1)

Tables Icon

Table 1 Power Conservation Error δ = 1 − R T for Gratings with Λ= 1.9λ, d = 0.4, and h = 0.4λ at Normal Incidencea

Equations (11)

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( 2 / z 2 ) Ψ + k 0 2 A ˆ j Ψ = 0 , z j 1 > z > z j ,
A ˆ j = τ j ( x ) k 0 x τ j 1 ( x ) k 0 x + ε j ( x ) , τ j ( x ) = { 1 TE ε j ( x ) TM .
Ψ = exp [ i k 0 ( z z 0 ) Q ˆ 0 ] ξ 0 + exp [ i k 0 ( z z 0 ) Q ˆ 0 ] η 0 , z 0 z < .
Ψ = exp [ i k 0 ( z z j 1 ) Q ˆ j ] ξ j + exp [ i k 0 × ( z z j 1 ) Q ˆ j ] η j , z j z < z j 1 .
C ˆ j 1 ξ j + C ˆ j η j = ξ j + 1 + η j + 1 , Q ˆ j ( C ˆ j 1 ξ j C ˆ j η j ) = Q ˆ j + 1 ( ξ j + 1 η j + 1 ) ,
C ˆ 0 = I ˆ , C ˆ j = exp ( i k 0 h j Q ˆ j ) , Q ˆ j = τ ˆ j 1 Q ˆ j .
η 0 = exp ( i k 0 x sin θ ) ,
ξ L + 1 = 0 .
ξ j = S ˆ j η j ,
S ˆ j = C ˆ j [ Q ˆ j + ( I ˆ S ˆ j + 1 ) ( I ˆ + S ˆ j + 1 ) 1 Q ˆ j + 1 ] 1 × [ Q ˆ j ( I ˆ S ˆ j + 1 ) ( I ˆ + S ˆ j + 1 ) 1 Q ˆ j + 1 ] C ˆ j ,
η j + 1 = 2 [ Q ˆ j ( I ˆ + S ˆ j + 1 ) + Q ˆ j + 1 ( I ˆ S ˆ j + 1 ) ] 1 Q ˆ j C ˆ j η j ,

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