Abstract

Because they are not proper spectral modes, leaky modes do not obey the usual power orthogonality relations for an ideal waveguide. Two methods for circumventing this problem are presented in this paper. One approach is to use the fact that weakly leaky modes are approximately power orthogonal within a restricted region of space to define a new set of slightly coupled modes. The other is to abandon power orthogonality in favor of a more general, mathematically exact orthogonality relation. The relative merits of these two approaches are discussed.

© 1976 Optical Society of America

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Corrections

Rowland Sammut and Allan W. Snyder, "Leaky modes on a dielectric waveguide: orthogonality and excitation; erratum," Appl. Opt. 15, 2953-2953 (1976)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-15-12-2953

References

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  1. A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-19, 720 (1971).
    [CrossRef]
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  3. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1973).
  4. R. Sammut, C. Pask, A. W. Snyder, Proc. IEEE (Lond.) 122, 25 (1975).
  5. R. Sammut, A. W. Snyder, J. Opt. Soc. Am. 64, 1171 (1974).
    [CrossRef]
  6. A. W. Snyder, Appl. Phys. 4, 273 (1974).
    [CrossRef]
  7. A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
    [CrossRef]
  8. A. W. Snyder, D. J. Mitchell, Opto-Electron. 6, 287 (1974).
    [CrossRef]
  9. R. Sammut, A. W. Snyder, Appl. Opt.14, in press.
  10. A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
    [CrossRef]
  11. V. V. Shevchenko, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1242 (1971); [Radiophys. Quantum Electron. 14, 972 (1974)].
  12. K. G. Budden, The Wave-guide Mode Theory of Wave Propagation (Logos, London, 1961), pp. 219–221.
  13. A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).
    [CrossRef]
  14. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

1975 (1)

R. Sammut, C. Pask, A. W. Snyder, Proc. IEEE (Lond.) 122, 25 (1975).

1974 (4)

R. Sammut, A. W. Snyder, J. Opt. Soc. Am. 64, 1171 (1974).
[CrossRef]

A. W. Snyder, Appl. Phys. 4, 273 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, Opto-Electron. 6, 287 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
[CrossRef]

1972 (1)

1971 (2)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-19, 720 (1971).
[CrossRef]

V. V. Shevchenko, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1242 (1971); [Radiophys. Quantum Electron. 14, 972 (1974)].

1969 (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).
[CrossRef]

Budden, K. G.

K. G. Budden, The Wave-guide Mode Theory of Wave Propagation (Logos, London, 1961), pp. 219–221.

Collin, R. E.

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

Marcuse, D.

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

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1973).

Mitchell, D. J.

A. W. Snyder, D. J. Mitchell, Opto-Electron. 6, 287 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
[CrossRef]

Pask, C.

R. Sammut, C. Pask, A. W. Snyder, Proc. IEEE (Lond.) 122, 25 (1975).

Sammut, R.

R. Sammut, C. Pask, A. W. Snyder, Proc. IEEE (Lond.) 122, 25 (1975).

R. Sammut, A. W. Snyder, J. Opt. Soc. Am. 64, 1171 (1974).
[CrossRef]

R. Sammut, A. W. Snyder, Appl. Opt.14, in press.

Shevchenko, V. V.

V. V. Shevchenko, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1242 (1971); [Radiophys. Quantum Electron. 14, 972 (1974)].

Snyder, A. W.

R. Sammut, C. Pask, A. W. Snyder, Proc. IEEE (Lond.) 122, 25 (1975).

R. Sammut, A. W. Snyder, J. Opt. Soc. Am. 64, 1171 (1974).
[CrossRef]

A. W. Snyder, Appl. Phys. 4, 273 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, Opto-Electron. 6, 287 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
[CrossRef]

A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-19, 720 (1971).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).
[CrossRef]

R. Sammut, A. W. Snyder, Appl. Opt.14, in press.

Appl. Phys. (1)

A. W. Snyder, Appl. Phys. 4, 273 (1974).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-19, 720 (1971).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

V. V. Shevchenko, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1242 (1971); [Radiophys. Quantum Electron. 14, 972 (1974)].

J. Opt. Soc. Am. (3)

Opto-Electron. (1)

A. W. Snyder, D. J. Mitchell, Opto-Electron. 6, 287 (1974).
[CrossRef]

Proc. IEEE (Lond.) (1)

R. Sammut, C. Pask, A. W. Snyder, Proc. IEEE (Lond.) 122, 25 (1975).

Other (5)

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

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1973).

R. Sammut, A. W. Snyder, Appl. Opt.14, in press.

K. G. Budden, The Wave-guide Mode Theory of Wave Propagation (Logos, London, 1961), pp. 219–221.

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

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Figures (1)

Fig. 1
Fig. 1

Contour of integration to be used in Eq. (11). r′ and r″ are the real and imaginary parts of r, respectively.

Equations (43)

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

E ( x , y , z ) = p = 1 N a p e p ( x , y ) exp ( i β p z ) + Σ 0 a ξ e ξ ( x , y ) exp [ i β ( ξ ) z ] d ξ ,
H ( x , y , z ) = p = 1 N a p h p ( x , y ) exp ( i β p z ) + Σ 0 a ξ h ξ ( x , y ) exp [ i β ( ξ ) z ] d ξ ;
A e p ^ h q * · z ^ d A = ψ p δ p q ,
a i = 1 ψ i A E ^ h i * · z ^ d A .
P = p = 1 N a p 2 + Σ 0 a ξ 2 d ξ .
E ( x , y , z ) p = 1 N a p ( z ) e p ( x , y ) + q = 1 M a q ( z ) e q ( x , y )
e ˜ p = e p r r t p p = 0 r > r t p p ,
E ( x , y , z ) p a p ( z ) e ˜ p ( x , y ) ,
a p ( 0 ) A E ^ h ˜ p * · z ^ d A A e ˜ p ^ h ˜ p * · z ^ d A .
r - 1 / 2 exp [ i ( β z - l ϕ + λ r ) ] ,
S e p ^ h q · z ^ d a = δ p q ,
a p = S E ^ h p · z ^ d a S e p ^ h p · z ^ d a
S d a lim R 0 2 π d ϕ ( 0 d r d r + L R r d r ) .
p a p 2 .
C p q = A t p e ˜ p ^ h ˜ q * · z ^ d A
L p q = A t p e ˜ p ^ h ˜ q * · r d l ,
q ( C p q + C q p * ) ( d a q * d z + i β p a q * ) = - q ( L p q + L q p * ) a q * .
A ˜ d a * d z = B ˜ a * ,
A p q = ( C p q + C q p * ) / 2 C p p
B p q = - [ L p q + L q p * + i β p ( C p q + C q p * ) ] / 2 C p p .
E q ( x , y , z ) = e ˜ q ( x , y ) exp ( i β q z ) ,
( β p - β q * ) ( C p q + C q p * ) = i ( L p q + L q p * ) .
( β p + β q * ) ( - C p q + C q p * ) = i ( L p , - q + L - q , p * ) .
2 C p q = i ( L p q + L q p * ) β p - β q * - i ( L p , - q + L - q , p * ) β p + β q * .
h ± s ± ( 1 μ ) 1 / 2 z ^ ^ e s .
C p q = i β p ( L p q + L q p * ) β p 2 - β q * 2 = 2 π i β p β p 2 - β q * 2 [ ( e ˜ p ^ h ˜ q * + e ˜ q * ^ h ˜ p ) · r ^ ] r = r t p p .
e s z ( R ) = ( μ 1 ) 1 / 4 i U s Q s k 1 V ( π ρ 2 ) 1 / 2 · H 1 ( Q s R ) [ H l ( Q s ) H l - 2 ( Q s ) ] 1 / 2
e s r ( R ) = ( μ 1 ) 1 / 4 U s V ( π ρ 2 ) 1 / 2 · H l - 1 ( Q s R ) [ H l ( Q s ) H l - 2 ( Q s ) ] 1 / 2 ,
V 2 = ρ 2 ( k 1 2 - k 2 2 ) = U s 2 - Q s 2 ,
C p q C p p = ( β p 2 - β p * 2 β p 2 - β q * 2 ) × U q * H l - 1 * ( Q q * R ) U p * H l - 1 * ( Q p * R ) [ H l * ( Q p * ) H l - 2 * ( Q p * ) H l ( Q q ) H l - 2 ( Q q ) ] 1 / 2 × [ Q p H l ( Q p R ) H l - 1 ( Q p R ) + Q q * H 1 * ( Q q * R ) H l - 1 * ( Q q * R ) Q p H l ( Q p R ) H l - 1 ( Q p R ) + Q p * H l * ( Q p * R ) H l - 1 * ( Q p * R ) ] | R = R t p p
| C p q C p p | | U q ( β p 2 - β p * 2 ) U p ( β p 2 - β q * 2 ) | · 3 2 a ( 2 π ) 1 / 2 × exp ( Q p 2 - Q q 2 ) exp ( - l 2 ) l 1 / 6 [ 1 + ( 1 - Q q 2 / Q p 2 ) 1 / 2 ] l ,
a = Γ ( 1 / 3 ) 2 2 / 3 3 1 / 6 .
| C p q C p p | A ( l ) β p 2 - β p * 2 β p 2 - β q * 2 ,
C p q / C p p 0 ,             when p q .
A ˜ = I ˜ + ˜ ,
( d a * ) / ( d z ) = ( I ˜ - ˜ ) B ˜ a * .
I ( k , m ; l , n ) = S R e k m ^ h l n · z ^ d a ,
e k m ^ h l n · z ^ = f k l m n ( r ) exp [ - i ( k + l ) ϕ ] .
I ( k , m ; l , n ) = Γ R 0 2 π f k l m n ( r ) exp [ - i ( k + l ) ϕ ] d ϕ r d r = 2 π I ( m , n ) δ k , - l ,
I ( m , n ) = Γ R f l l m n ( r ) r d r .
I ( m , n ) = 0 d f l l m n ( r ) r d r + L R f l l m n ( r ) r d r ,
( λ m + λ n ) + ( λ m + λ n ) tan γ > 0.
I ( m , n ) = 1 2 π δ m , n

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