Abstract

In this paper, a method of analysis of dielectric waveguides using a variational method is presented. This method can be applied to dielectric waveguides with any general permittivity distribution. As examples, numerical calculations for dielectric waveguides with rectangular cross-sectional shape and quadratic inhomogeneous permittivity distribution are given.

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References

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  1. W. Schlosser and H. G. Unger, Advances in Microwaves (Academic, New York, 1966), p. 319.
  2. J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
    [CrossRef]
  3. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
    [CrossRef]
  4. E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
    [CrossRef]
  5. M. Otaka, M. Matsuhara, and N. Kumagai, Trans. IECE Jap.55B, 332 (1972).
  6. M. Matsuhara, J. Opt. Soc. Am. 63, 135 (1973).
    [CrossRef]
  7. S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).
    [CrossRef]
  8. J. P. Gordon, Bell Syst. Tech. J. 45, 321 (1966).
    [CrossRef]
  9. C. N. Kurtz and W. Striefer, IEEE Trans. Microwave Theory Tech. 17, 11 (1969); IEEE Trans. Microwave Theory Tech. 17, 250 (1969); IEEE Trans. Microwave Theory Tech. 17, 360(1969).
    [CrossRef]
  10. P. J. B. Clarricoats and K. B. Chan, Electron. Lett. 6, 694 (1970).
    [CrossRef]
  11. A. Stratton, Electromagnetic Theory (McGraw—Hill, NewYork, 1941), p. 343.
  12. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw—Hill, New York, 1953), p. 1106.

1973 (1)

M. Matsuhara, J. Opt. Soc. Am. 63, 135 (1973).
[CrossRef]

1972 (1)

M. Otaka, M. Matsuhara, and N. Kumagai, Trans. IECE Jap.55B, 332 (1972).

1970 (1)

P. J. B. Clarricoats and K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

1969 (3)

C. N. Kurtz and W. Striefer, IEEE Trans. Microwave Theory Tech. 17, 11 (1969); IEEE Trans. Microwave Theory Tech. 17, 250 (1969); IEEE Trans. Microwave Theory Tech. 17, 360(1969).
[CrossRef]

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
[CrossRef]

1966 (1)

J. P. Gordon, Bell Syst. Tech. J. 45, 321 (1966).
[CrossRef]

1965 (1)

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).
[CrossRef]

1964 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
[CrossRef]

Chan, K. B.

P. J. B. Clarricoats and K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

Clarricoats, P. J. B.

P. J. B. Clarricoats and K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw—Hill, New York, 1953), p. 1106.

Goell, J. E.

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[CrossRef]

Gordon, J. P.

J. P. Gordon, Bell Syst. Tech. J. 45, 321 (1966).
[CrossRef]

Kumagai, N.

M. Otaka, M. Matsuhara, and N. Kumagai, Trans. IECE Jap.55B, 332 (1972).

Kurtz, C. N.

C. N. Kurtz and W. Striefer, IEEE Trans. Microwave Theory Tech. 17, 11 (1969); IEEE Trans. Microwave Theory Tech. 17, 250 (1969); IEEE Trans. Microwave Theory Tech. 17, 360(1969).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
[CrossRef]

Matsuhara, M.

M. Matsuhara, J. Opt. Soc. Am. 63, 135 (1973).
[CrossRef]

M. Otaka, M. Matsuhara, and N. Kumagai, Trans. IECE Jap.55B, 332 (1972).

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw—Hill, New York, 1953), p. 1106.

Otaka, M.

M. Otaka, M. Matsuhara, and N. Kumagai, Trans. IECE Jap.55B, 332 (1972).

Schlosser, W.

W. Schlosser and H. G. Unger, Advances in Microwaves (Academic, New York, 1966), p. 319.

Stratton, J. A.

A. Stratton, Electromagnetic Theory (McGraw—Hill, NewYork, 1941), p. 343.

Striefer, W.

C. N. Kurtz and W. Striefer, IEEE Trans. Microwave Theory Tech. 17, 11 (1969); IEEE Trans. Microwave Theory Tech. 17, 250 (1969); IEEE Trans. Microwave Theory Tech. 17, 360(1969).
[CrossRef]

Unger, H. G.

W. Schlosser and H. G. Unger, Advances in Microwaves (Academic, New York, 1966), p. 319.

Other (12)

W. Schlosser and H. G. Unger, Advances in Microwaves (Academic, New York, 1966), p. 319.

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
[CrossRef]

M. Otaka, M. Matsuhara, and N. Kumagai, Trans. IECE Jap.55B, 332 (1972).

M. Matsuhara, J. Opt. Soc. Am. 63, 135 (1973).
[CrossRef]

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).
[CrossRef]

J. P. Gordon, Bell Syst. Tech. J. 45, 321 (1966).
[CrossRef]

C. N. Kurtz and W. Striefer, IEEE Trans. Microwave Theory Tech. 17, 11 (1969); IEEE Trans. Microwave Theory Tech. 17, 250 (1969); IEEE Trans. Microwave Theory Tech. 17, 360(1969).
[CrossRef]

P. J. B. Clarricoats and K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

A. Stratton, Electromagnetic Theory (McGraw—Hill, NewYork, 1941), p. 343.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw—Hill, New York, 1953), p. 1106.

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Figures (9)

Fig. 1
Fig. 1

Dispersion curves.

Fig. 2
Fig. 2

Irradiance distribution. (0,0) mode, Δ 1 2 u / λ = 1.0.

Fig. 3
Fig. 3

Irradiance distribution. (1,0) mode, Δ 1 2 u / λ = 1.0.

Fig. 4
Fig. 4

Dispersion curves.

Fig. 5
Fig. 5

Irradiance distribution. (0,0) mode, Δ 1 2 u / λ = 1.59.

Fig. 6
Fig. 6

Irradiance distribution. (0,1) mode, Δ 1 2 u / λ = 1.59.

Fig. 7
Fig. 7

Dispersion curves.

Fig. 8
Fig. 8

Irradiance distribution. (0,0) mode, Δ 1 2 u / λ = 1.59.

Fig. 9
Fig. 9

Irradiance distribution. (1,0) mode, Δ 1 2 u / λ = 1.59.

Equations (23)

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| 1 λ | 1 ,
2 Φ + ω 2 μ Φ = 0
| 1 δ | 1 ,
Φ ( u , υ , z ) = ϕ ( u , υ ) exp ( j β z ) ,
2 ϕ + ( k 2 β 2 ) ϕ = 0 ,
β 2 = [ { k 2 ϕ 2 ( ϕ ) 2 } d S ] / ϕ 2 d S .
k ( u , υ ) 2 = k 0 2 { 1 f ( u , υ ) } , β 2 = k 0 2 { 1 b } ,
b = [ { f ( u , υ ) ϕ 2 + 1 k 0 2 ( ϕ ) 2 } d S ] / ϕ 2 d S .
ϕ ( x , y ) = i , j a i j ( 2 π ξ η i ! j ! ) 1 2 D i ( x / ξ ) D j ( y / η ) ,
ϕ ( r , θ ) = p , l [ ( 2 p ! ) 1 2 { π ( l + p ) ! } 1 2 ζ 1 ] ( a p l cos l θ + b p l sin l θ ) ( r / ζ ) l L p l ( r 2 / ζ 2 ) exp ( r 2 2 ζ 2 ) ,
b = i , j k , l N i j k l a i j a k l / i , j k , l δ i k δ j l a i j a k l ,
N i j k l ( ξ , η ) = I i j k l ( ξ , η ) + 1 4 k 0 2 ( 1 ξ 2 D i k δ j l + 1 η 2 δ i k D j l ) ,
I i j k l ( ξ , η ) = { ( 2 π ) 2 i ! j ! k ! l ! } 1 2 f ( ξ x , η y ) D i ( x ) D j ( y ) · D k ( x ) D l ( y ) d x d y ,
D m n = { m ( n + 1 ) } 1 2 δ m ( n + 2 ) + [ { m n } 1 2 + { ( m + 1 ) ( n + 1 ) } 1 2 ] δ m n { ( m + 1 ) n } 1 2 δ m ( n 2 ) ,
δ m n = { 1 ( m = n ) 0 ( m n ) .
b / a k l = 0 ,
i , j ( N i j k l δ i k δ j l b ) a i j = 0.
b ( ξ , η ) / ξ = 0 , b ( ξ , η ) / η = 0.
b = b ( ξ , η ) ,
ϕ = i , j a i j ( ξ , η ) ( 2 π ξ η i ! j ! ) 1 2 D i ( x / ξ ) D j ( y / η ) .
f ( x , y ) = { 0 ( | x | u and | y | υ ) Δ ( other region ) .
f ( x , y ) = Δ [ 1 exp { 0.69315 ( x 2 / u 2 + y 2 / υ 2 ) } ] .
f ( x , y ) = { 0 ( | x | u and | y | υ ) 10 Δ ( y > υ ) Δ ( other region ) .