Abstract

The characteristics of two types of leaky modes on the asymmetric three-layered dielectric slab waveguide are fully investigated by solving the eigenvalue equation numerically. Loci of propagation constants and field distributions are presented and the effects of asymmetry on the propagation characteristics are discussed. It is found that the characteristics of substrate leaky modes are quite different from those of air leaky modes, and that the attenuation constants of substrate leaky modes become extremely small as the asymmetry becomes small.

© 1980 Optical Society of America

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References

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  1. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971).
    [Crossref] [PubMed]
  2. S. Kawakami and S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-11, 130–138 (1975).
    [Crossref]
  3. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.
  4. D. Marcuse, Theory of Dielectric Optical Waveguides (AcademicNew York, 1974), Sec. 1.5.
  5. S. E. Miller, “A survey of integrated optics,” IEEE J. Quantum Electron. QE-8, 199–205 (1972).
    [Crossref]
  6. M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics,” Opt. Quantum Electron. 10, 425–433 (1978).
    [Crossref]

1978 (1)

M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics,” Opt. Quantum Electron. 10, 425–433 (1978).
[Crossref]

1975 (1)

S. Kawakami and S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-11, 130–138 (1975).
[Crossref]

1972 (1)

S. E. Miller, “A survey of integrated optics,” IEEE J. Quantum Electron. QE-8, 199–205 (1972).
[Crossref]

1971 (1)

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.

Kawakami, S.

S. Kawakami and S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-11, 130–138 (1975).
[Crossref]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (AcademicNew York, 1974), Sec. 1.5.

Miller, S. E.

S. E. Miller, “A survey of integrated optics,” IEEE J. Quantum Electron. QE-8, 199–205 (1972).
[Crossref]

Miyagi, M.

M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics,” Opt. Quantum Electron. 10, 425–433 (1978).
[Crossref]

Nishida, S.

S. Kawakami and S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-11, 130–138 (1975).
[Crossref]

Tien, P. K.

Yip, G. L.

M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics,” Opt. Quantum Electron. 10, 425–433 (1978).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

S. Kawakami and S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-11, 130–138 (1975).
[Crossref]

S. E. Miller, “A survey of integrated optics,” IEEE J. Quantum Electron. QE-8, 199–205 (1972).
[Crossref]

Opt. Quantum Electron. (1)

M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics,” Opt. Quantum Electron. 10, 425–433 (1978).
[Crossref]

Other (2)

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.

D. Marcuse, Theory of Dielectric Optical Waveguides (AcademicNew York, 1974), Sec. 1.5.

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Figures (7)

FIG. 1
FIG. 1

Geometry of the asymmetric slab waveguide.

FIG. 2
FIG. 2

Loci of the normalized transverse propagation constant w of the leaky TE2 mode in the substrate as a function of the normalized frequency v. The solid and dashed lines correspond to the substrate and air leaky modes, respectively.

FIG. 3
FIG. 3

Loci of the normalized phase constant β/n0k0 of the leaky TE2 mode as a function of the normalized frequency v. The solid and dashed lines correspond to the substrate and air leaky modes, respectively.

FIG. 4
FIG. 4

Frequency dependence of the normalized attenuation constant −Im(β/n0k0) of the leaky TE2 mode. The solid and dashed lines correspond to the substrate and air leaky modes, respectively.

FIG. 5
FIG. 5

Amplitude distribution (|Ey|) of the leaky TE2 mode. (a) Substrate leaky mode; (b) Air leaky mode

FIG. 6
FIG. 6

Loci of the normalized transverse propagation constant w of the leaky TE1 mode in the substrate as a function of the normalized frequency v.

FIG. 7
FIG. 7

Frequency dependence of the normalized attenuation constant −Im(β/n0k0) of the leaky TE1 mode.

Tables (1)

Tables Icon

TABLE I Normalized cutoff frequencies vc and transit frequencies vs and va to the substrate and air leaky modes for several values of b (a = 1.1).

Equations (7)

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E y = { e w ˆ x / d e j ( w t β z ) for x > 0 , [ cos u ( x / d ) ( w ˆ / u ) sin u ( x / d ) ] × e j ( w t β z ) for 0 > x > d , [ cos u + ( w ˆ / u ) sin u ] e w [ ( x / d ) + 1 ] × e j ( w t β z ) for d > x ,
u 2 + w 2 = ( a 2 1 ) ( n 0 k 0 d ) 2 υ 2 ,
w ˆ 2 = w 2 + υ 2 ( 1 b 2 ) / ( a 2 1 ) ,
tan u = u ( w + w ˆ ) / ( u 2 w w ˆ ) ,
( β / n 0 k 0 ) 2 = 1 + ( w / υ ) 2 ( a 2 1 ) .
υ c = n π + tan 1 ( 1 b 2 a 2 1 ) 1 / 2 ,
E y exp [ ( w r / d ) x + β i z ] exp j { w t [ ( w i / d ) x + β r z ] } ,