Abstract

The propagation properties of strip waveguides are analyzed by a mode-matching technique. Mode coupling, which causes leakage effects, is taken into account in the analysis. The numerical results for the attenuation constants of the first three leaky modes are presented as a function of the strip width. The numerical results obtained by the present method are compared with other theoretical and experimental results. It is found that the higherorder leaky modes have varied and interesting properties, as does the fundamental leaky mode.

© 1983 Optical Society of America

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References

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  1. H. Furuta, H. Noda, and A. Ihaya, “Novel optical waveguide for integrated optics,” Appl. Opt. 13, 322–326 (1974).
    [Crossref] [PubMed]
  2. V. Ramaswamy, “Strip-loaded film waveguide,” Bell Syst. Tech. J. 53, 697–704 (1974).
    [Crossref]
  3. J. E. Goell, “Rib waveguide for integrated optical circuits,” Appl. Opt. 12, 2797–2798 (1973).
    [Crossref] [PubMed]
  4. S. T. Peng and A. A. Oliner, “Leakage and resonance effects on strip waveguides for integrated optics,” Trans. Inst. Electron. Jpn. E61, 151–154 (1978).
  5. K. Ogusu, S. Kawakami, and S. Nishida, “Optical strip waveguide: an analysis,” Appl. Opt. 18, 908–914 (1979);erratum Appl. Opt. 18, 3725 (1979).
    [Crossref] [PubMed]
  6. K. Ogusu and I. Tanaka, “Optical strip waveguide: an experiment,” Appl. Opt. 19, 3322–3325 (1980).
    [Crossref] [PubMed]
  7. S. T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I—mathematical formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 843–855 (1981).
    [Crossref]
  8. A. A. Oliner, S. T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
    [Crossref]
  9. R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric waveguides using mode-matching technique and variational methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36–43 (1980).
    [Crossref]
  10. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 153.

1981 (2)

S. T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I—mathematical formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 843–855 (1981).
[Crossref]

A. A. Oliner, S. T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[Crossref]

1980 (2)

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric waveguides using mode-matching technique and variational methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36–43 (1980).
[Crossref]

K. Ogusu and I. Tanaka, “Optical strip waveguide: an experiment,” Appl. Opt. 19, 3322–3325 (1980).
[Crossref] [PubMed]

1979 (1)

1978 (1)

S. T. Peng and A. A. Oliner, “Leakage and resonance effects on strip waveguides for integrated optics,” Trans. Inst. Electron. Jpn. E61, 151–154 (1978).

1974 (2)

1973 (1)

Furuta, H.

Goell, J. E.

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 153.

Hou, Y. L.

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric waveguides using mode-matching technique and variational methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36–43 (1980).
[Crossref]

Hsu, T. I.

A. A. Oliner, S. T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[Crossref]

Ihaya, A.

Jamnejad, V.

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric waveguides using mode-matching technique and variational methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36–43 (1980).
[Crossref]

Kawakami, S.

Mittra, R.

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric waveguides using mode-matching technique and variational methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36–43 (1980).
[Crossref]

Nishida, S.

Noda, H.

Ogusu, K.

Oliner, A. A.

S. T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I—mathematical formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 843–855 (1981).
[Crossref]

A. A. Oliner, S. T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[Crossref]

S. T. Peng and A. A. Oliner, “Leakage and resonance effects on strip waveguides for integrated optics,” Trans. Inst. Electron. Jpn. E61, 151–154 (1978).

Peng, S. T.

S. T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I—mathematical formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 843–855 (1981).
[Crossref]

A. A. Oliner, S. T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[Crossref]

S. T. Peng and A. A. Oliner, “Leakage and resonance effects on strip waveguides for integrated optics,” Trans. Inst. Electron. Jpn. E61, 151–154 (1978).

Ramaswamy, V.

V. Ramaswamy, “Strip-loaded film waveguide,” Bell Syst. Tech. J. 53, 697–704 (1974).
[Crossref]

Sanchez, A.

A. A. Oliner, S. T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[Crossref]

Tanaka, I.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

V. Ramaswamy, “Strip-loaded film waveguide,” Bell Syst. Tech. J. 53, 697–704 (1974).
[Crossref]

IEEE Trans. Microwave Theory Tech. (3)

S. T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: part I—mathematical formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 843–855 (1981).
[Crossref]

A. A. Oliner, S. T. Peng, T. I. Hsu, and A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides: part II—new physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[Crossref]

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric waveguides using mode-matching technique and variational methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36–43 (1980).
[Crossref]

Trans. Inst. Electron. Jpn. (1)

S. T. Peng and A. A. Oliner, “Leakage and resonance effects on strip waveguides for integrated optics,” Trans. Inst. Electron. Jpn. E61, 151–154 (1978).

Other (1)

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 153.

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Figures (6)

Fig. 1
Fig. 1

Cross section of a rib waveguide.

Fig. 2
Fig. 2

Convergence of the normalized attenuation constant α/k of the E11x mode with the increase in the number of expansion modes. A: f = 34 GHz, a = 4 mm, b = 3 mm, w = 7 mm, nf = 1.5, ns = nc = 1.0; B: ka = 1.6π, b/a = 0.875, w/λ = 1.84, nf = 1.68, ns = 1.48, nc = 1.0.

Fig. 3
Fig. 3

Comparison of numerical and experimental results.

Fig. 4
Fig. 4

Normalized attenuation constant α/k of the E11x mode as a function of the normalized rib width w/λ and comparison with other numerical results.

Fig. 5
Fig. 5

Normalized attenuation constants α/k oí the fírst three leaky modes as a function of the normalized rib width w/λ.

Fig. 6
Fig. 6

Dispersion curves of the first three leaky modes.

Equations (22)

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H y I = i A i ϕ i I ( x ) { cos ( k y i I y ) cos ( k y i I w 2 ) sin ( k y i I y ) sin ( k y i I w 2 ) } exp [ j ( w t k z z ) ] in region I ,
H y II = i B i ϕ i II ( x ) exp ( j k y i II | y w 2 | ) exp [ j ( w t k z z ) ] in region II ;
E y I = i C i ϕ i I ( x ) { sin ( k y i I y ) sin ( k y i I w 2 ) cos ( k y i I y ) cos ( k y i I w 2 ) } exp [ j ( w t k z z ) ] in region I ,
E y II = i D i ϕ i II ( x ) exp ( j k y i II | y w 2 | ) exp [ j ( w t k z z ) ] in region II ,
ϕ i ( x ) = { cos ( k xfi d + φ i ) cos k xci ( d x U ) cos k xci ( x x U ) x U > x > d , cos ( k xfi x + φ i ) d > x > 0 , cos φ i cos k xsi x L cos k xsi ( x + x L ) 0 > x > x L ,
φ i = tan 1 [ ( n f n s ) 2 k xsi k xfi tan k xsi x L ] .
tan k x f d = ( n f n c ) 2 k x c k x f tan k x c ( d x U ) ( n f n s ) 2 k x s k x f tan k x s x L 1 + ( n f n c ) 2 k x c k x f ( n f n s ) 2 k x s k x f tan k x c ( d x U ) tan k x s x L ,
k x c 2 = ( n c k ) 2 ( k y 2 + k z 2 ) = ( n c 2 n f 2 ) k 2 + k x f 2 ,
k x s 2 = ( n s k ) 2 ( k y 2 + k z 2 ) = ( n s 2 n f 2 ) k 2 + k x f 2 .
e i = ( k y i k ) 2 + ( k z k ) 2
ϕ i ( x ) = { sin ( k xfi d + φ i ) sin k xci ( d x U ) sin k xci ( x x U ) x U > x > d , sin ( k xfi x + φ i ) d > x > 0 , sin φ i sin k xsi x L sin k xsi ( x + x L ) 0 > x > x L ,
φ i = tan 1 ( k xfi k xsi tan k xsi x L ) .
tan k x f d = k x f k x c tan k x c ( d x U ) k x f k x s tan k x s x L 1 + k x f k x c k x f k x s tan k x c ( d x U ) tan k x s x L
k x ν = [ ( n ν k ) 2 ( k y 2 + k z 2 ) ] 1 / 2 .
i e i I ϕ i I ( x ) r I ( x ) A i i e i II ϕ i II ( x ) r II ( x ) C i = 0 ,
i e i I ϕ i I ( x ) B i i e i II ϕ i II ( x ) D i = 0 ,
i k z k ( μ 0 0 ) 1 / 2 1 r I ( x ) δ ϕ i I ( x ) δ x A i + i k y i I { cot ( k y i I w 2 ) tan ( k y i I w 2 ) } ϕ i I ( x ) B i + i k z k ( μ 0 0 ) 2 1 r II ( x ) δ ϕ i II ( x ) δ x C i + i j k y i II ϕ i II ( x ) D i = 0 ,
i k y i I { tan ( k y i I w 2 ) cot ( k y i I w 2 ) } ϕ i I ( x ) A i i k z k ( 0 μ 0 ) 1 / 2 δ ϕ i I ( x ) δ x B i i j k y i II ϕ i II ( x ) C i + i k z k ( 0 μ 0 ) 1 / 2 δ ϕ i II ( x ) δ x D i = 0 ,
ϕ i I ( x ) , ϕ i I ( x ) , ϕ i I ( x ) , 1 r I ( x ) ϕ i I ( x ) ,
η e p 1 I > β k > η e p 1 II for the E p q x mode ,
η e p 1 I > β k > η e p 1 II for the E p q y mode ,
k y 0 I w = 2 m π ,