Abstract

We report a wave-optical analysis of light propagation along leaky waveguides using a symmetric hollow dielectric slab as a model. The field deformation due to leakage into the surrounding regions for a guided wave that penetrated into the hollow structure is studied by using the eigenmode expansion method, where the propagation is described by an integral over the modes of the continuum. The numerical evaluation of the integral expansion enables us to estimate the radiation loss for the incident wave. Results are compared with those based on the simplified zigzag ray-optics approach and complex leaky waves.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  2. T. Tamir, “Beam and waveguide couplers,” in Topics in Applied Physics, Vol. 7: Integrated Optics (Springer-Verlag, New York, 1975).
    [Crossref]
  3. D. B. Hall and C. Yeh, “Leaky waves in a heteroepitaxial film,” J. Appl. Phys. 44, 2271–2274 (1973).
    [Crossref]
  4. R. Ulrich and W. Prettl, “Planar leaky light-guides and couplers,” Appl. Phys. 1, 55–68 (1973).
    [Crossref]
  5. T. P. Sosnowski, “Polarization mode filters for integrated optics,” Opt. Commun. 4, 408–412 (1972).
    [Crossref]
  6. I. J. Kurland and H. L. Bertoni, “Birefringent prism couplers for thin-film optical waveguides,” Appl. Opt. 17, 1030–1037 (1978).
    [Crossref] [PubMed]
  7. S. Yamamoto and T. Makimoto, “Semileaky-type thin-film magneto-optic waveguide for modular application,” J. Appl. Phys. 48, 1680–1682 (1977).
    [Crossref]
  8. S. Yamamoto, Y. Okamura, and T. Makimoto, “Analysis and design of semileaky-type thin-film optical waveguide isolator,” IEEE J. Quantum Electron. QE-12, 764–770 (1976).
    [Crossref]
  9. J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
    [Crossref]
  10. E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
    [Crossref]
  11. P. W. Smith, “A waveguide gas laser,” Appl. Phys. Lett. 19, 132–134 (1971).
    [Crossref]
  12. D. Marcuse, “Hollow dielectric waveguide for distributed feedback lasers,” IEEE J. Quantum Electron. QE-8, 661–669 (1972).
    [Crossref]
  13. D. Marcuse, “Wave propagation along a dielectric interface,” J. Opt. Soc. Am. 64, 794–797 (1974).
    [Crossref]
  14. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971).
    [Crossref] [PubMed]
  15. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  16. H. Kogelnik and H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. 64, 174–185 (1974).
    [Crossref]

1978 (1)

1977 (1)

S. Yamamoto and T. Makimoto, “Semileaky-type thin-film magneto-optic waveguide for modular application,” J. Appl. Phys. 48, 1680–1682 (1977).
[Crossref]

1976 (1)

S. Yamamoto, Y. Okamura, and T. Makimoto, “Analysis and design of semileaky-type thin-film optical waveguide isolator,” IEEE J. Quantum Electron. QE-12, 764–770 (1976).
[Crossref]

1974 (2)

1973 (3)

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[Crossref]

D. B. Hall and C. Yeh, “Leaky waves in a heteroepitaxial film,” J. Appl. Phys. 44, 2271–2274 (1973).
[Crossref]

R. Ulrich and W. Prettl, “Planar leaky light-guides and couplers,” Appl. Phys. 1, 55–68 (1973).
[Crossref]

1972 (2)

T. P. Sosnowski, “Polarization mode filters for integrated optics,” Opt. Commun. 4, 408–412 (1972).
[Crossref]

D. Marcuse, “Hollow dielectric waveguide for distributed feedback lasers,” IEEE J. Quantum Electron. QE-8, 661–669 (1972).
[Crossref]

1971 (2)

1964 (1)

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Bertoni, H. L.

Bjorkholm, J. E.

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[Crossref]

Hall, D. B.

D. B. Hall and C. Yeh, “Leaky waves in a heteroepitaxial film,” J. Appl. Phys. 44, 2271–2274 (1973).
[Crossref]

Kogelnik, H.

Kurland, I. J.

Makimoto, T.

S. Yamamoto and T. Makimoto, “Semileaky-type thin-film magneto-optic waveguide for modular application,” J. Appl. Phys. 48, 1680–1682 (1977).
[Crossref]

S. Yamamoto, Y. Okamura, and T. Makimoto, “Analysis and design of semileaky-type thin-film optical waveguide isolator,” IEEE J. Quantum Electron. QE-12, 764–770 (1976).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Marcuse, D.

D. Marcuse, “Wave propagation along a dielectric interface,” J. Opt. Soc. Am. 64, 794–797 (1974).
[Crossref]

D. Marcuse, “Hollow dielectric waveguide for distributed feedback lasers,” IEEE J. Quantum Electron. QE-8, 661–669 (1972).
[Crossref]

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Okamura, Y.

S. Yamamoto, Y. Okamura, and T. Makimoto, “Analysis and design of semileaky-type thin-film optical waveguide isolator,” IEEE J. Quantum Electron. QE-12, 764–770 (1976).
[Crossref]

Prettl, W.

R. Ulrich and W. Prettl, “Planar leaky light-guides and couplers,” Appl. Phys. 1, 55–68 (1973).
[Crossref]

Schmeltzer, R. A.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Shank, C. V.

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[Crossref]

Smith, P. W.

P. W. Smith, “A waveguide gas laser,” Appl. Phys. Lett. 19, 132–134 (1971).
[Crossref]

Sosnowski, T. P.

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[Crossref]

T. P. Sosnowski, “Polarization mode filters for integrated optics,” Opt. Commun. 4, 408–412 (1972).
[Crossref]

Tamir, T.

T. Tamir, “Beam and waveguide couplers,” in Topics in Applied Physics, Vol. 7: Integrated Optics (Springer-Verlag, New York, 1975).
[Crossref]

Tien, P. K.

Ulrich, R.

R. Ulrich and W. Prettl, “Planar leaky light-guides and couplers,” Appl. Phys. 1, 55–68 (1973).
[Crossref]

Weber, H. P.

Yamamoto, S.

S. Yamamoto and T. Makimoto, “Semileaky-type thin-film magneto-optic waveguide for modular application,” J. Appl. Phys. 48, 1680–1682 (1977).
[Crossref]

S. Yamamoto, Y. Okamura, and T. Makimoto, “Analysis and design of semileaky-type thin-film optical waveguide isolator,” IEEE J. Quantum Electron. QE-12, 764–770 (1976).
[Crossref]

Yeh, C.

D. B. Hall and C. Yeh, “Leaky waves in a heteroepitaxial film,” J. Appl. Phys. 44, 2271–2274 (1973).
[Crossref]

Appl. Opt. (2)

Appl. Phys. (1)

R. Ulrich and W. Prettl, “Planar leaky light-guides and couplers,” Appl. Phys. 1, 55–68 (1973).
[Crossref]

Appl. Phys. Lett. (2)

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[Crossref]

P. W. Smith, “A waveguide gas laser,” Appl. Phys. Lett. 19, 132–134 (1971).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

IEEE J. Quantum Electron. (2)

D. Marcuse, “Hollow dielectric waveguide for distributed feedback lasers,” IEEE J. Quantum Electron. QE-8, 661–669 (1972).
[Crossref]

S. Yamamoto, Y. Okamura, and T. Makimoto, “Analysis and design of semileaky-type thin-film optical waveguide isolator,” IEEE J. Quantum Electron. QE-12, 764–770 (1976).
[Crossref]

J. Appl. Phys. (2)

D. B. Hall and C. Yeh, “Leaky waves in a heteroepitaxial film,” J. Appl. Phys. 44, 2271–2274 (1973).
[Crossref]

S. Yamamoto and T. Makimoto, “Semileaky-type thin-film magneto-optic waveguide for modular application,” J. Appl. Phys. 48, 1680–1682 (1977).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

T. P. Sosnowski, “Polarization mode filters for integrated optics,” Opt. Commun. 4, 408–412 (1972).
[Crossref]

Other (3)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

T. Tamir, “Beam and waveguide couplers,” in Topics in Applied Physics, Vol. 7: Integrated Optics (Springer-Verlag, New York, 1975).
[Crossref]

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

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

Geometry of the problem. A TE guided mode of an ordinary dielectric waveguide is incident on a hollow dielectric waveguide. The whole structure is uniform along the y axis

FIG. 2
FIG. 2

Transverse electric field distribution Ey of TE eigenmodes of a symmetric hollow dielectric slab waveguide: (a) propagating modes I (n2k0 > β > n1k0) and (b) propagating modes II (n1k0 > β > 0) and evanescent modes (β = −jξ, ∞ > ξ > 0).

FIG. 3
FIG. 3

Field patterns in the hollow structure for TE0 mode incidence at 2d = 2λ (λ = 0.63 μm).

FIG. 4
FIG. 4

Same as Fig. 3 for TE1 mode incidence.

FIG. 5
FIG. 5

Radiation loss as a function of the film thickness for TE0 mode incidence on a hollow structure with the refractive-index relation 1.55/1.50/1.55 from the input ordinary waveguide with 1.45/1.50/1.45. Wavelength is λ = 0.63 μm. Small circles indicate the losses evaluated from the numerical field patterns whereas the solid curve is based on the ray-optical formula [Eq. (5)]. Also, the dash-dotted curve shows a d−3 profile as a reference, which is an extended plot of the radiation loss formula for TE0 leaky wave [Eq. (6) for ν = 0] of the hollow structure valid at the low-loss region.

FIG. 6
FIG. 6

Same as Fig. 5 for TE1 mode incidence. The dash-dotted curve corresponds to TE1 leaky wave [Eq. (6) for ν = 1].

FIG. 7
FIG. 7

Ray model of the propagation along a composite ordinary/hollow dielectric slab waveguide.

Equations (19)

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

E y ( x , z ) = 0 V [ L 1 e ( ρ ) E y 1 e ( x ; ρ ) + L 1 o ( ρ ) E y 1 o ( x ; ρ ) ] exp ( j β z ) d ρ + V [ L 2 e ( ρ ) E y 2 e ( x ; ρ ) + L 2 o ( ρ ) E y 2 o ( x ; ρ ) ] exp ( j β z ) d ρ ,
L i ( ρ ) = ( | β | / 2 ω μ 0 P ) E y i * ( x ; ρ ) E y , in ( x , 0 ) d x ,
L 1 e ( ρ ) = A A 1 e ( β / ω μ 0 ) [ ( κ cosh Δ d sin κ d + Δ sinh Δ d cos κ d ) / ( κ 2 + Δ 2 ) + cos κ d ( γ cosh Δ d + Δ sinh Δ d ) / ( γ 2 + ρ 2 ) ] , L 2 e ( ρ ) = A A 2 e ( | β | / ω μ 0 ) [ ( κ cos σ d sin κ d σ sin σ d cos κ d ) / ( κ 2 σ 2 ) + cos κ d ( γ cos σ d σ sin σ d ) / ( γ 2 + ρ 2 ) ] , L 1 o ( ρ ) = L 2 o ( ρ ) = 0.
L 1 o ( ρ ) = A A 1 o ( β / ω μ 0 ) [ ( κ sinh Δ d cos κ d + Δ cosh Δ d sin κ d ) / ( κ 2 + Δ 2 ) + sin κ d ( γ sinh Δ d + Δ cosh Δ d ) / ( γ 2 + ρ 2 ) ] , L 2 o ( ρ ) = A A 2 o ( | β | / ω μ 0 ) [ ( κ sin σ d cos κ d + σ cos σ d sin κ d ) / ( κ 2 σ 2 ) + sin κ d ( γ sin σ d + σ cos σ d ) / ( γ 2 + ρ 2 ) ] , L 1 e ( ρ ) = L 2 e ( ρ ) = 0 ,
2 α = 2 κ 1 2 κ 2 / [ ( κ 1 + κ 2 ) 2 β g d ] ,
2 α v = ( v + 1 ) 2 π 2 / [ 2 ( n 2 2 n 1 2 ) 1 / 2 n 1 k 0 2 d 3 ] ,
E y 1 e ( x ; ρ ) = A 1 e cosh Δ x ( | x | < d ) , = A 1 e [ cosh Δ d cos ρ ( | x | d ) + ( Δ / ρ ) sinh Δ d sin ρ ( | x | d ) ] ( | x | > d ) ,
A 1 e = [ 2 ω μ 0 P / π β ( cosh 2 Δ d + ( Δ / ρ ) 2 sinh 2 Δ d ) 1 / 2 ,
Δ = ( β 2 n 1 2 k 0 2 ) 1 / 2 .
E y 1 o ( x ; ρ ) = A 1 o sinh Δ x ( | x | < d ) , = ( x / | x | ) A 1 o [ sinh Δ d cos ρ ( | x | d ) + ( Δ / ρ ) cosh Δ d sin ρ ( | x | d ) ] ( | x | > d ) ,
A 1 o = [ 2 ω μ 0 P / π β ( sinh 2 Δ d + ( Δ / ρ ) 2 cosh 2 Δ d ) ] 1 / 2 .
E y 2 e ( x ; ρ ) = A 2 e cos σ x ( | x | < d ) , = A 2 e [ cos σ d cos ρ ( | x | d ) ( σ / ρ ) sin σ d sin ρ ( | x | d ) ] ( | x | > d ) ,
A 2 e = [ 2 ω μ 0 P / π | β | ( cos 2 σ d + ( σ / ρ ) 2 sin 2 σ d ) ] 1 / 2 ,
σ = ( n 1 2 k 0 2 β 2 ) 1 / 2 .
E y 2 o ( x ; ρ ) = A 2 o sin σ x ( | x | < d ) , = ( x / | x | ) A 2 o [ sin σ d cos ρ ( | x | d ) + ( σ / ρ ) cos σ d sin ρ ( | x | d ) ] ( | x | > d ) ,
A 2 o = [ 2 ω μ 0 P / π | β | ( sin 2 σ d + ( σ / ρ ) 2 cos 2 σ d ) ] 1 / 2 .
H x = ( β / ω μ 0 ) E y ,
H z = ( j / ω μ 0 ) E y / x .
( 1 / 2 ) E y i ( x ; ρ ) H x j * ( x ; ρ ) d x = ( β * / | β | ) P δ ( ρ ρ ) δ i j ,