Abstract

A rigorous analysis of wave guidance by dielectric grating waveguides is presented for the general case that both the grating vector and the direction of the guided wave have an arbitrary orientation in three dimensions. Guided modes then become TE-like and TM-like modes owing to the coupling between TE and TM waves. The analysis is formulated in a unified matrix form so that guidance properties can be obtained accurately in any level by systematic matrix calculations. The calculated results indicate that peculiar stop bands and leaky waves appear as a result of coupling between TE and TM waves.

© 1993 Optical Society of America

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References

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  1. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chaps. 3 and 4.
  2. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  3. K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
    [CrossRef]
  4. S. L. Chuang, J. A. Kong, “Wave scattering and guidance by dielectric waveguides with periodic surfaces,” J. Opt. Soc. Am. 73, 669–679 (1983).
    [CrossRef]
  5. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  6. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  7. T. Yamasaki, T. Hinata, T. Hosono, J. Inagawa, “Propagation characteristics of modulated dielectric gratings,” Tech. Rep. Electromagn. Theory EMT-87-54 (Institute of Electrical Engineers of Japan, Tokyo, 1987), pp. 137–146.
  8. K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
    [CrossRef]
  9. K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
    [CrossRef]
  10. R. Petit, G. Tayeb, “About the electromagnetic theory of gratings made with anisotropic materials,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 11–16 (1987).
    [CrossRef]
  11. K. Rokushima, J. Yamakita, S. Mori, “Wave diffraction and guidance by anisotropic dielectric waveguides with space–time periodic index modulation,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 40–45 (1987).
    [CrossRef]
  12. E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080 (1987).
    [CrossRef]
  13. J. Yamakita, K. Matsumoto, S. Mori, K. Rokushima, “Analysis of anisotropic dielectric gratings using differential method,” Trans. Inst. Electr. Inf. Commun. Eng. J73-C-I, 1–8 (1990) [Electron. Commun. Jpn. 2, 73, 25–34 (1990)].
  14. K. Wagatsuma, H. Sasaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
    [CrossRef]
  15. G. I. Stegeman, D. Sarid, J. J. Burke, D. G. Hall, “Scattering of guided waves by surface periodic gratings for arbitrary angles of incidence: perturbation field theory and implications to normal-mode analysis,” J. Opt. Soc. Am. 71, 1497–1507 (1981).
    [CrossRef]
  16. L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: a comparison of the results of Rouard’s method and coupled-mode theory,” J. Opt. Soc. Am. A 4, 60–65 (1987).
    [CrossRef]
  17. S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
    [CrossRef]
  18. R. E. Collin, F. J. Zucker, eds., Antenna Theory (McGraw-Hill, New York, 1969), Chap. 19.

1990 (1)

J. Yamakita, K. Matsumoto, S. Mori, K. Rokushima, “Analysis of anisotropic dielectric gratings using differential method,” Trans. Inst. Electr. Inf. Commun. Eng. J73-C-I, 1–8 (1990) [Electron. Commun. Jpn. 2, 73, 25–34 (1990)].

1989 (1)

1987 (3)

1985 (1)

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

1983 (3)

1981 (1)

1980 (1)

1979 (1)

K. Wagatsuma, H. Sasaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

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]

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]

Burke, J. J.

Chang, K. C.

Chuang, S. L.

Gaylord, T. K.

Glytsis, E. N.

Hall, D. G.

Hinata, T.

T. Yamasaki, T. Hinata, T. Hosono, J. Inagawa, “Propagation characteristics of modulated dielectric gratings,” Tech. Rep. Electromagn. Theory EMT-87-54 (Institute of Electrical Engineers of Japan, Tokyo, 1987), pp. 137–146.

Hosono, T.

T. Yamasaki, T. Hinata, T. Hosono, J. Inagawa, “Propagation characteristics of modulated dielectric gratings,” Tech. Rep. Electromagn. Theory EMT-87-54 (Institute of Electrical Engineers of Japan, Tokyo, 1987), pp. 137–146.

Inagawa, J.

T. Yamasaki, T. Hinata, T. Hosono, J. Inagawa, “Propagation characteristics of modulated dielectric gratings,” Tech. Rep. Electromagn. Theory EMT-87-54 (Institute of Electrical Engineers of Japan, Tokyo, 1987), pp. 137–146.

Kong, J. A.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chaps. 3 and 4.

Matsumoto, K.

J. Yamakita, K. Matsumoto, S. Mori, K. Rokushima, “Analysis of anisotropic dielectric gratings using differential method,” Trans. Inst. Electr. Inf. Commun. Eng. J73-C-I, 1–8 (1990) [Electron. Commun. Jpn. 2, 73, 25–34 (1990)].

Moharam, M. G.

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, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

Mori, S.

J. Yamakita, K. Matsumoto, S. Mori, K. Rokushima, “Analysis of anisotropic dielectric gratings using differential method,” Trans. Inst. Electr. Inf. Commun. Eng. J73-C-I, 1–8 (1990) [Electron. Commun. Jpn. 2, 73, 25–34 (1990)].

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

K. Rokushima, J. Yamakita, S. Mori, “Wave diffraction and guidance by anisotropic dielectric waveguides with space–time periodic index modulation,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 40–45 (1987).
[CrossRef]

Peng, S. T.

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

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

Petit, R.

R. Petit, G. Tayeb, “About the electromagnetic theory of gratings made with anisotropic materials,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 11–16 (1987).
[CrossRef]

Rokushima, K.

J. Yamakita, K. Matsumoto, S. Mori, K. Rokushima, “Analysis of anisotropic dielectric gratings using differential method,” Trans. Inst. Electr. Inf. Commun. Eng. J73-C-I, 1–8 (1990) [Electron. Commun. Jpn. 2, 73, 25–34 (1990)].

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
[CrossRef]

K. Rokushima, J. Yamakita, S. Mori, “Wave diffraction and guidance by anisotropic dielectric waveguides with space–time periodic index modulation,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 40–45 (1987).
[CrossRef]

Saito, S.

K. Wagatsuma, H. Sasaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Sarid, D.

Sasaki, H.

K. Wagatsuma, H. Sasaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Shah, V.

Stegeman, G. I.

Tamir, T.

K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
[CrossRef]

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

Tayeb, G.

R. Petit, G. Tayeb, “About the electromagnetic theory of gratings made with anisotropic materials,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 11–16 (1987).
[CrossRef]

Tominaga, K.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

Wagatsuma, K.

K. Wagatsuma, H. Sasaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Weller-Brophy, L. A.

Yamakita, J.

J. Yamakita, K. Matsumoto, S. Mori, K. Rokushima, “Analysis of anisotropic dielectric gratings using differential method,” Trans. Inst. Electr. Inf. Commun. Eng. J73-C-I, 1–8 (1990) [Electron. Commun. Jpn. 2, 73, 25–34 (1990)].

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
[CrossRef]

K. Rokushima, J. Yamakita, S. Mori, “Wave diffraction and guidance by anisotropic dielectric waveguides with space–time periodic index modulation,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 40–45 (1987).
[CrossRef]

Yamasaki, T.

T. Yamasaki, T. Hinata, T. Hosono, J. Inagawa, “Propagation characteristics of modulated dielectric gratings,” Tech. Rep. Electromagn. Theory EMT-87-54 (Institute of Electrical Engineers of Japan, Tokyo, 1987), pp. 137–146.

IEEE J. Quantum Electron. (1)

K. Wagatsuma, H. Sasaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

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

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

Proc. IEEE (1)

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

Trans. Inst. Electr. Inf. Commun. Eng. (1)

J. Yamakita, K. Matsumoto, S. Mori, K. Rokushima, “Analysis of anisotropic dielectric gratings using differential method,” Trans. Inst. Electr. Inf. Commun. Eng. J73-C-I, 1–8 (1990) [Electron. Commun. Jpn. 2, 73, 25–34 (1990)].

Other (5)

R. E. Collin, F. J. Zucker, eds., Antenna Theory (McGraw-Hill, New York, 1969), Chap. 19.

T. Yamasaki, T. Hinata, T. Hosono, J. Inagawa, “Propagation characteristics of modulated dielectric gratings,” Tech. Rep. Electromagn. Theory EMT-87-54 (Institute of Electrical Engineers of Japan, Tokyo, 1987), pp. 137–146.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chaps. 3 and 4.

R. Petit, G. Tayeb, “About the electromagnetic theory of gratings made with anisotropic materials,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 11–16 (1987).
[CrossRef]

K. Rokushima, J. Yamakita, S. Mori, “Wave diffraction and guidance by anisotropic dielectric waveguides with space–time periodic index modulation,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 40–45 (1987).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Three-dimensional geometry of a dielectric grating waveguide.

Fig. 2
Fig. 2

Guidance of waves by dielectric grating waveguide.

Fig. 3
Fig. 3

Schematic configuration of three-dimensional waveguiding process, where nt−1 = nt0nK and n I = I ( I = 1 , 2 , 3 ).

Fig. 4
Fig. 4

Cross-sectional view of Fig. 3.

Fig. 5
Fig. 5

Variation of propagation constant j(β) with normalized frequency 1/nK = Λ/λ. (a) Phase factor (βΛ cos θ)/2π versus grating factor (Λ cos θ)/λ, (b) attenuation factor αΛ versus (Λ cos θ)/λ, (c) attenuation factor αΛ versus Λ/λ for θ = 0°.

Fig. 6
Fig. 6

Brillouin diagrams for the first-order Bragg region for θ = 30°.

Fig. 7
Fig. 7

Brillouin diagrams for the first-order Bragg region for TE0-like mode with θ as a parameter. (a) Phase factor (βΛ cos θ)/2π, (b) attenuation factor αΛ.

Fig. 8
Fig. 8

Brillouin diagrams for the first-order Bragg region for TM0-like mode with θ as a parameter. (a) Phase factor (βΛ cos θ)/2π, (b) attenuation factor αΛ.

Fig. 9
Fig. 9

Variation of propagation constant j(β) with oblique angle θ for 1/nK = Λ/λ = 0.65. (a) Phase factor βΛ/2π versus θ, (b) attenuation factor αΛ versus θ

Tables (1)

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Table 1 Accuracy of Solution by Truncation of the Order of Space Harmonicsa

Equations (45)

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n K = K / k 0 = i x p + i y q + i z s ,
| n K | = n K = λ / Λ ,
̂ 2 ( r ) = m b m exp ( j m n K r ) ,
n m = n 0 + m n K ,
n 0 = i x p 0 + i y q 0 + i z s 0
β / k 0 = | n t 0 | = q 0 2 + [ Re ( s 0 ) ] 2 , n t 0 = i y q 0 + i z Re ( s 0 ) ,
α / k 0 = Im ( s 0 ) ,
Y 0 E = m e m ( x ) exp ( j n m r ) ,
Z 0 H = m h m ( x ) exp ( j n m r ) ,
e m ( x ) = i x e x m ( x ) + i y e y m ( x ) + i z e z m ( x ) ,
h m ( x ) = i x h x m ( x ) + i y h y m ( x ) + i z h z m ( x ) ,
n m = i x p m + i y q m + i z s m ,
p m = p 0 + m p , q m = q 0 + m q , s m = s 0 + m s .
curl Y 0 E = j Z 0 H , curl Z 0 H = j ̂ 2 ( r ) Y 0 E ,
d f t d x = j C f t ,
f t = [ e y h z e z h y ] .
C = [ p q 1 q 1 0 q 1 s + s 2 p s q 0 0 s 1 q p s 1 s + 1 q s 0 q 2 p ] ,
= [ m n ] = [ b n m ] ,
p = [ δ m n p m ] , q = [ δ m n q m ] , s = [ δ m n s m ] , 1 = [ 1 ] , 0 = [ 0 ] .
f t = T g ,
d g d x = j κ g .
C m u = [ 0 q m 2 1 0 q m s m + s m 2 s m q m 0 0 s m q m 0 s m 2 + 1 q m s m 0 q m 2 0 ] .
κ E m ± = κ M m ± = ξ m ,
u E m u ± = [ ± s ˙ m s ˙ m ξ m ± q ˙ m q ˙ m ξ m ] t ,
u M m u ± = [ q ˙ m ξ m / ± q ˙ m s ˙ m ξ m / s ˙ m ] t ,
ξ m = q m 2 s m 2 , q ˙ m = q m q m 2 + s m 2 , s ˙ m = s m q m 2 + s m 2 .
T = [ s ˙ ξ q ˙ s ˙ ξ q ˙ ξ s ˙ q ˙ ξ s ˙ q ˙ q ˙ ξ s ˙ q ˙ ξ s ˙ ξ q ˙ s ˙ ξ q ˙ s ˙ ] = [ T u + T u ] ,
g = [ g + g ] = [ g + E g + M g E g M ] ,
g ± E = ( g M ± E g 0 ± E g + M ± E ) t ,
g ± M = ( g M ± M g 0 ± M g + M ± M ) t .
[ g 2 + ( x 1 ) g 2 ( x 1 ) ] = [ U [ j κ 2 + ( x 1 x 2 ) ] 0 0 U [ j κ 2 ( x 1 x 2 ) ] ] [ g 2 + ( x 1 ) g 2 ( x 1 ) ] ,
f t 1 ( x 1 ) = P 2 ( x 1 ) f t 2 ( x 1 ) , P 2 ( x 2 ) f t 2 ( x 2 ) = f t 3 ( x 2 ) ,
P 2 ( x ) = [ p ( x ) 0 p ( x ) p ( x ) 0 p ( x ) ]
g 1 ( x 1 ) = g 3 + ( x 2 ) = 0 .
[ T 1 u + B 12 + 0 0 B 23 T 3 u ] [ g 1 + ( x 1 ) g 2 + ( x 2 ) g 2 ( x 1 ) g 3 ( x 2 ) ] = 0 ,
B 12 + = P 2 ( x 1 ) T 2 [ U [ j κ 2 + ( x 2 x 2 ) ] 0 0 1 ] ,
B 23 = P 2 ( x 2 ) T 2 [ 1 0 0 U [ j κ 2 ( x 1 x 2 ) ] ] .
det W = 0 , W = [ T 1 u + B 12 + 0 0 B 23 T 3 u ] .
q 0 = Re ( s 0 ) tan θ .
n K = i z s = i z ( λ / Λ ) ,
̂ 2 ( z ) = 2 [ 1 + δ cos ( n K z ) ] .
cos θ = m n K 2 n E t 0 ( TE TE ) ,
cos θ = ( m n K ) 2 + n E t 0 2 n M t 0 2 2 m n K n E t 0 ( TE TM ) ,
cos θ = m n K 2 n M t 0 ( TM TM ) ,
cos θ = ( m n K ) 2 + n M t 0 2 n E t 0 2 2 m n K n M t 0 ( TM TE ) ,

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