Abstract

Standard formulations of rigorous Fourier-expansion analysis methods of lamellar gratings reduce, irrespective of the state of polarization, to the TE result of the lowest-order theory of form birefringence when only the zeroth-order terms are retained in the field and permittivity expansions. A reformulation is presented that reduces to the correct form-birefringence result also in the TM case. Expressions are given for the effective relative permittivities eff of a subwavelength-period grating with an arbitrary relative-permittivity profile r(x): the lowest-order approximation for eff is given by the zeroth-order Fourier coefficient of r(x) in TE polarization and of 1/r(x) in TM polarization.

© 1996 Optical Society of America

Full Article  |  PDF Article

Errata

Jari Turunen, "Form-birefringence limits of Fourier-expansion methods in grating theory: errata," J. Opt. Soc. Am. A 14, 2317-2322 (1997)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-14-9-2317

References

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  1. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
    [CrossRef]
  2. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  3. M. G. Moharam, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  4. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  5. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  6. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  7. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  8. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]
  9. M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
    [CrossRef]
  10. M. Nevière, E. Popov, “Analysis of dielectric gratings of arbitrary profiles and thicknesses: comment,” J. Opt. Soc. Am. A 9, 2095–2096 (1992).
    [CrossRef]
  11. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]
  12. D. H. Raguin, G. M. Morris, “Analysis of antireflection structured surfaces with continuous one-dimensional profiles,” Appl. Opt. 32, 2582–2598 (1993).
    [CrossRef] [PubMed]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 705–708 and 61–66.
  14. G. Campbell, R. K. Kostuk, “Effective-medium theory of sinusoidally modulated volume holograms,” J. Opt. Soc. Am. A 12, 1113–1117 (1995).
    [CrossRef]
  15. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  16. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  17. The choice of the reference plane z= hin the terms exp[−iγ(z− h)] that appear in Eqs. (9) and (10) ensures that the physical nature of evanescent waves will be properly accounted for in the solution of the boundary value problem: the amplitudes of evanescent waves are appreciable only in the vicinity of the particular boundary, either z= 0 or z= h, at which they are generated. Therefore the numerical stability problems resulting from growing exponentials9,10 are eliminated.
  18. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

1995 (3)

1993 (2)

1992 (1)

1988 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983 (1)

1982 (1)

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

1978 (1)

1975 (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1966 (1)

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 705–708 and 61–66.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Burckhardt, C. B.

Campbell, G.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Haggans, C. W.

Knop, K.

Kostuk, R. K.

Li, L.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Morf, R. H.

Morris, G. M.

Nevière, M.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Popov, E.

Raguin, D. H.

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Vincent, P.

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 705–708 and 61–66.

Appl. Opt. (1)

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (7)

Opt. Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 705–708 and 61–66.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

The choice of the reference plane z= hin the terms exp[−iγ(z− h)] that appear in Eqs. (9) and (10) ensures that the physical nature of evanescent waves will be properly accounted for in the solution of the boundary value problem: the amplitudes of evanescent waves are appreciable only in the vicinity of the particular boundary, either z= 0 or z= h, at which they are generated. Therefore the numerical stability problems resulting from growing exponentials9,10 are eliminated.

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Figures (3)

Fig. 1
Fig. 1

Diffraction of a plane wave by a grating. If the period d is sufficiently small, only the zeroth backward- and forward-diffracted orders with complex amplitudes R0 and T0, respectively, can propagate.

Fig. 2
Fig. 2

Convergence comparison for a subwavelength-period grating antireflection layer: (a) convergence of the lowest-order eigenvalue when the number N of eigenmodes and Rayleigh coefficients retained in the analysis is increased; (b) convergence of the zeroth-order reflectance. Filled circles, modified formulation; open circles, standard formulation; horizontal lines, exact results.

Fig. 3
Fig. 3

Convergence comparison for a beam-splitter grating of period d = 2.5λ. Convergence of the eigenvalues (a) γ0 and (b) γ1 and the efficiencies of transmitted orders (c) l = 0 and (d) l = ±1. Filled circles, modified formulation; open circles, standard formulation; horizontal lines, exact results.

Equations (56)

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z H y ( x , z ) = i ω D x ( x , z ) ,
x H y ( x , z ) = - i ω D z ( x , z ) ,
z E x ( x , z ) - x E z ( x , z ) = i ω μ 0 H y ( x , z ) ,
z H y ( x , z ) = i ω 0 r ( x ) E x ( x , z ) ,
z E x ( x , z ) = i ω μ 0 H y ( x , z ) - 1 i ω 0 x [ 1 r ( x ) x H y ( x , z ) ] .
x [ 1 r ( x ) x H y ( x , y ) ] + z [ 1 r ( x ) z H y ( x , y ) ] + k 2 H y ( x , z ) = 0 ,
r ( x ) = p = - ɛ p exp ( i 2 π p x / d ) ,
1 r ( x ) = p = - ξ p exp ( i 2 π p x / d ) ,
H y ( x , z ) = { a exp ( i γ z ) + b exp [ - i γ ( z - h ) ] } × m = - H m exp ( i α m x ) ,
E x ( x , z ) = γ ω 0 { a exp ( i γ z ) - b exp [ - i γ ( z - h ) ] } × m = - E m exp ( i α m x ) ,
H l = m = - ɛ l - m E m ,
m = - ( k 2 δ l m - α l ξ l - m α m ) H m = γ 2 E l ,
MLE = γ 2 E ,
1 r ( x ) z D x ( x , z ) = i ( ω / c 2 ) H y ( x , z ) - 1 i ω x [ 1 r ( x ) x H y ( x , z ) ] ,
D x ( x , z ) = γ ω { a exp ( i γ z ) - b exp [ - i γ ( z - h ) ] } × m = - D m exp ( i α m x ) .
m = - ( k 2 δ l m - α l ξ l - m α m ) H m = γ 2 m = - ξ l - m H m .
P - 1 MH = γ 2 H ,
H y I ( x , z ) = exp [ i ( α 0 x + r 0 z ) ] + l = - R l exp [ i ( α l x + r l z ) ] ,
H y III ( x , z ) = l = - T l exp { i [ α l x + t l ( z - h ) ] } ,
H y II ( x , z ) = n = 0 { a n exp ( i γ n z ) + b n exp [ - i γ n ( z - h ) ] } × l = - H l n exp ( i α l x ) ,
E x II ( x , z ) = 1 ω 0 n = 0 γ n { a n exp ( i γ n z ) - b n exp [ - i γ n ( z - h ) ] } × l = - E l n exp ( i α l x ) ,
E l n = m = - ξ l - m D m n
R l = n = 0 H l n [ a n + b n exp ( i γ n h ) ] - δ l 0 ,
n 1 - 2 r l ( δ l 0 - R l ) = n = 0 γ n E l n [ a n - b n exp ( i γ n h ) ] ,
T l = n = 0 H l n [ a n exp ( i γ n h ) + b n ] ,
n 3 - 2 t l T l = n = 0 γ n E l n [ a n exp ( i γ n h ) - b n ] .
n = 0 ( n 1 - 2 r l H l n + γ n E l n ) a n + n = 0 ( n 1 - 2 r l H l n - γ n E l n ) × exp ( i γ n h ) b n = 2 n 1 - 2 r l δ l 0 ,
n = 0 ( n 3 - 2 t l H l n - γ n E l n ) exp ( i γ n h ) a n + n = 0 ( n 3 - 2 t l H l n + γ n E l n ) b n = 0 ,
m = - ( k 2 ɛ l - m - α m 2 δ l m ) E m = γ 2 E l .
n = 0 ( r l + γ n ) E l n a n + n = 0 ( r l - γ n ) exp ( i γ n h ) E l n b n = 2 r l δ l 0 ,
n = 0 ( t l - γ n ) exp ( i γ n h ) E l n a n + n = 0 ( t l + γ n ) E l n b n = 0.
γ 0 = k ( ɛ 0 - n 1 2 sin 2 θ 1 ) 1 / 2 .
R 0 = R 12 + R 23 exp ( i 2 γ 0 h ) 1 + R 12 R 23 exp ( i 2 γ 0 h ) ,
T 0 = T 12 T 23 exp ( i γ 0 h ) 1 + R 12 R 23 exp ( i 2 γ 0 h ) ,
R 12 = r 0 - γ 0 r 0 + γ 0 ,
R 23 = γ 0 - t 0 γ 0 + t 0 ,
T 12 = 2 r 0 r 0 + γ 0 ,
T 23 = 2 γ 0 γ 0 + t 0 .
= ɛ 0 = 1 d 0 d r ( x ) d x ,
γ 0 = k ( ɛ 0 - ɛ 0 ξ 0 n 1 2 sin 2 θ 1 ) 1 / 2 .
ɛ 0 ξ 0 = [ 1 d 0 d r ( x ) d x ] [ 1 d 0 d 1 r ( x ) d x ]
γ 0 = k ( ξ 0 - 1 - n 1 2 sin 2 θ ) 1 / 2 .
= ξ 0 - 1 = [ 1 d 0 d 1 r ( x ) d x ] - 1 .
R 12 = n 1 - 2 r 0 - ξ 0 γ 0 n 1 - 2 r 0 + ξ 0 γ 0 ,
R 23 = ξ 0 γ 0 - n 3 - 2 t 0 ξ 0 γ 0 + n 3 - 2 t 0 ,
T 12 = 2 n 1 - 2 r 0 n 1 - 2 r 0 + ξ 0 γ 0 ,
T 23 = 2 ξ 0 γ 0 ξ 0 γ 0 + n 3 - 2 t 0 .
r ( x ) = { n 1 2 when 0 x < c n 3 2 when c x < d .
= n 1 2 c / d + n 3 2 ( 1 - c / d ) ,
= n 1 2 n 3 2 n 3 2 c / d + n 1 2 ( 1 - c / d ) .
r ( x ) = ¯ + Δ sin ( 2 π x / d ) .
= ¯ ,
= ¯ 2 + Δ 2 .
r ( x ) = low + ( high - low ) x / d .
= 1 / 2 ( high + low ) ,
= high - low ln high - low .

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