Abstract

An efficient and powerful technique has been developed to treat the problem of wave propagation along arbitrarily shaped single-mode dielectric waveguides with inhomogeneous index variations in the cross-sectional plane. This technique is based on a modified finite-element method. Illustrative examples were given for the following guides: (a) the triangular fiber guide; (b) the elliptical fiber guide; (c) the single material fiber guide; (d) the rectangular fiber guide; (e) the embossed integrated optics guide; (f) the diffused channel guide; (g) the optical stripline guide.

© 1979 Optical Society of America

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References

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  1. J. Carson, S. P. Mead, S. A. Schelkunoff, Bell Syst. Tech. J. 15, 310 (1936).
  2. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [Crossref]
  3. C. Yeh, J. Appl. Phys. 33, 3235 (1962).
    [Crossref]
  4. J. Goell, Bell Syst. Techn. J. 48, 2133 (1969).
  5. E. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  6. C. Yeh, G. Lindgren, Appl. Opt. 16, 483 (1977).
    [Crossref] [PubMed]
  7. C. Yeh, S. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
    [Crossref]
  8. S. Dong et al., Int. J. Numer. Methods Eng. 4, 155 (1972).
    [Crossref]
  9. J. Dil, H. Blok, Opto-Electronics 5, 415 (1973).
    [Crossref]
  10. F. Blum, D. Shaw, W. C. Holton, Appl. Phys. Lett. 25, 116 (1974).
    [Crossref]
  11. H. Furuta et al., Appl. Opt. 13, 322 (1974).
    [Crossref] [PubMed]
  12. E. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
  13. M. Ohtaka et al., IEEE J. Quantum Electron. QE-12, 378 (1976).
    [Crossref]
  14. G. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
    [Crossref]
  15. H. Taylor, IEEE J. Quantum Electron. QE-12, 748 (1976).
    [Crossref]

1977 (1)

1976 (2)

M. Ohtaka et al., IEEE J. Quantum Electron. QE-12, 378 (1976).
[Crossref]

H. Taylor, IEEE J. Quantum Electron. QE-12, 748 (1976).
[Crossref]

1975 (2)

G. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[Crossref]

C. Yeh, S. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[Crossref]

1974 (3)

F. Blum, D. Shaw, W. C. Holton, Appl. Phys. Lett. 25, 116 (1974).
[Crossref]

H. Furuta et al., Appl. Opt. 13, 322 (1974).
[Crossref] [PubMed]

E. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).

1973 (1)

J. Dil, H. Blok, Opto-Electronics 5, 415 (1973).
[Crossref]

1972 (1)

S. Dong et al., Int. J. Numer. Methods Eng. 4, 155 (1972).
[Crossref]

1969 (2)

J. Goell, Bell Syst. Techn. J. 48, 2133 (1969).

E. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

1962 (1)

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[Crossref]

1961 (1)

1936 (1)

J. Carson, S. P. Mead, S. A. Schelkunoff, Bell Syst. Tech. J. 15, 310 (1936).

Blok, H.

J. Dil, H. Blok, Opto-Electronics 5, 415 (1973).
[Crossref]

Blum, F.

F. Blum, D. Shaw, W. C. Holton, Appl. Phys. Lett. 25, 116 (1974).
[Crossref]

Burns, W. K.

G. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[Crossref]

Carson, J.

J. Carson, S. P. Mead, S. A. Schelkunoff, Bell Syst. Tech. J. 15, 310 (1936).

Dil, J.

J. Dil, H. Blok, Opto-Electronics 5, 415 (1973).
[Crossref]

Dong, S.

C. Yeh, S. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[Crossref]

S. Dong et al., Int. J. Numer. Methods Eng. 4, 155 (1972).
[Crossref]

Furuta, H.

Goell, J.

J. Goell, Bell Syst. Techn. J. 48, 2133 (1969).

Hocker, G.

G. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[Crossref]

Holton, W. C.

F. Blum, D. Shaw, W. C. Holton, Appl. Phys. Lett. 25, 116 (1974).
[Crossref]

Lindgren, G.

Marcatili, E.

E. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).

E. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Mead, S. P.

J. Carson, S. P. Mead, S. A. Schelkunoff, Bell Syst. Tech. J. 15, 310 (1936).

Ohtaka, M.

M. Ohtaka et al., IEEE J. Quantum Electron. QE-12, 378 (1976).
[Crossref]

Oliver, W.

C. Yeh, S. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[Crossref]

Schelkunoff, S. A.

J. Carson, S. P. Mead, S. A. Schelkunoff, Bell Syst. Tech. J. 15, 310 (1936).

Shaw, D.

F. Blum, D. Shaw, W. C. Holton, Appl. Phys. Lett. 25, 116 (1974).
[Crossref]

Snitzer, E.

Taylor, H.

H. Taylor, IEEE J. Quantum Electron. QE-12, 748 (1976).
[Crossref]

Yeh, C.

C. Yeh, G. Lindgren, Appl. Opt. 16, 483 (1977).
[Crossref] [PubMed]

C. Yeh, S. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[Crossref]

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

F. Blum, D. Shaw, W. C. Holton, Appl. Phys. Lett. 25, 116 (1974).
[Crossref]

Bell Syst. Tech. J. (3)

E. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

E. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).

J. Carson, S. P. Mead, S. A. Schelkunoff, Bell Syst. Tech. J. 15, 310 (1936).

Bell Syst. Techn. J. (1)

J. Goell, Bell Syst. Techn. J. 48, 2133 (1969).

IEEE J. Quantum Electron. (3)

M. Ohtaka et al., IEEE J. Quantum Electron. QE-12, 378 (1976).
[Crossref]

G. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[Crossref]

H. Taylor, IEEE J. Quantum Electron. QE-12, 748 (1976).
[Crossref]

Int. J. Numer. Methods Eng. (1)

S. Dong et al., Int. J. Numer. Methods Eng. 4, 155 (1972).
[Crossref]

J. Appl. Phys. (2)

C. Yeh, S. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[Crossref]

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

Opto-Electronics (1)

J. Dil, H. Blok, Opto-Electronics 5, 415 (1973).
[Crossref]

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Figures (28)

Fig. 1
Fig. 1

The triangular coordinates.

Fig. 2
Fig. 2

A generalized special element.

Fig. 3
Fig. 3

Variation of eigenfrequency as a function of α for β ¯ = 1.0004.

Fig. 4
Fig. 4

Comparison of finite element approach results with the exact results for the dispersion characteristics of the HE11 mode in a step-index fiber.

Fig. 5
Fig. 5

Comparison of the finite element results for the normalized field component |Ez| of the HE11 mode for a step-index fiber with the exact Bessel function J1(μr). The solid line represents finite element results. The circles are calculated according to the Bessel function.

Fig. 6
Fig. 6

Comparison of the finite element approach results with the exact Yeh and Lindgren’s results for the dispersion characteristics of the HE11 mode in a graded-index fiber. Data are finite element results.

Fig. 7
Fig. 7

Comparison of finite element approach results with the exact Dil and Blok’s results for the dispersion characteristics of the HE11 mode in a graded-index fiber. Data are finite element results.

Fig. 8
Fig. 8

Comparison of the finite element results for the normalized field component |Ez| of the HE11 mode for a graded-index fiber with the exact modified Bessel function K1(γr). The solid line represents the finite element results. The circles are calculated according to the modified Bessel function.

Fig. 9
Fig. 9

Comparison of the finite element results for the HE11 mode with Goell’s results and with Marcatili’s approximate results for the rectangular fiber guide.

Fig. 10
Fig. 10

Comparison of the finite element results for the lowest order mode with Marcatili’s approximate results for the channel waveguide.

Fig. 11
Fig. 11

Dispersion curves for the dominant E 11 y mode in an optical stripline waveguide. The solid line represents results found according to the finite-element method, while the dashed line represents results found according to the vector variational method.

Fig. 12
Fig. 12

Gray-scale plots for the power intensity distributions of the dominant E 11 y mode in an optical stripline waveguide at two different frequencies.

Fig. 13
Fig. 13

Dispersion curve for the dominant mode in a triangular fiber.

Fig. 14
Fig. 14

Gray-scale plots for the power intensity distributions of the dominant mode in a triangular fiber at three different frequencies.

Fig. 15
Fig. 15

Dispersion curve for the dominant eHE11 mode in an elliptical fiber.

Fig. 16
Fig. 16

Gray-scale plots for the power intensity distributions of the dominant eHE11 mode in an elliptical fiber at three different frequencies.

Fig. 17
Fig. 17

Cross-sectional views of single material fibers.

Fig. 18
Fig. 18

Dispersion curve for the dominant mode in a single material fiber.

Fig. 19
Fig. 19

Gray-scale plots for the power intensity distributions of the dominant mode in a single material fiber at three different frequencies.

Fig. 20
Fig. 20

Dispersion curves for the dominant eHE11 mode in rectangular fibers with different aspect ratios.

Fig. 21
Fig. 21

Gray-scale plots for the power intensity distributions of the dominant eHE11 mode in rectangular fibers with different aspect ratios.

Fig. 22
Fig. 22

Various configurations for integrated optics waveguides.

Fig. 23
Fig. 23

Dispersion curves for the dominant E 11 y mode in three embossed rectangular waveguides.

Fig. 24
Fig. 24

Gray-scale plots for the magnitude distribution of Ez and Hz for the dominant E 11 y mode in embossed rectangular waveguides at three different frequencies.

Fig. 25
Fig. 25

A display of the index distribution in the core region of a diffused channel waveguide.

Fig. 26
Fig. 26

Dispersion curves for the dominant E 11 y or E 11 x mode in a 1-D diffused channel waveguide and a uniform channel waveguide.

Fig. 27
Fig. 27

Dispersion curves for the dominant E 11 y, or E 11 x mode in a 2-D diffused channel waveguide and in a uniform channel waveguide.

Fig. 28
Fig. 28

Gray-scale plots for the magnitude distributions of Ez and Hz for the dominant E 11 y mode in a 2-D diffused channel waveguide at three different frequencies.

Tables (1)

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Table I Numerical Examples for the Optical Stripline

Equations (34)

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( t 2 + k p 2 ) [ E z ( p ) H z ( p ) ] = 0 ,
k p 2 = ( ω / c ) 2 ( p / o ) - β 2 ,
E z ( p ) = E z ( q ) , H z ( p ) = H z ( q ) , τ p [ γ ( o μ o ) 1 / 2 E z ( p ) s - H z ( p ) n ] = τ q [ γ ( o μ o ) 1 / 2 E z ( q ) s - H z ( q ) n ] , τ p [ p o γ ( o μ o ) 1 / 2 E z ( p ) n + H z ( p ) s ] = τ q [ q o γ ( o μ o ) 1 / 2 E z ( q ) n + H z ( q ) s ] ,
τ i = ( γ 2 - 1 ) / ( γ 2 - i / o ) , γ = ( β c ) / ω .
E x ( p ) = i ω μ o k p 2 [ H z ( p ) y + ( o μ o ) 1 / 2 γ E z ( p ) x ] , E y ( p ) = i ω μ o k p 2 [ - H z ( p ) x + ( o μ o ) 1 / 2 γ E z ( p ) y ] , H x ( p ) = i ω o k p 2 [ - p o E z ( p ) y + ( μ o o ) 1 / 2 γ H z ( p ) x ] , H y ( p ) = i ω o k p 2 [ p o E z ( p ) x + ( μ o o ) 1 / 2 γ H z ( p ) y ] .
δ I = 0 ,
I = p = 1 I p = p = 1 ( τ p | H z ( p ) | 2 + γ 2 τ p p o | 1 γ ( o μ o ) 1 / 2 E z ( p ) | 2 + 2 τ p e ^ z · [ 1 γ ( o μ o ) 1 / 2 E z ( p ) × H z ( p ) ] - ( ω c ) 2 { [ H z ( p ) ] 2 + γ 2 p o [ 1 γ ( o μ o ) 1 / 2 E z ( p ) ] } ) d x d y
ϕ = H z ;             ψ = [ ( ω o ) / β ] E z .
ξ j = A j / A             ( j = r , s , t )
ξ r + ξ s + ξ t = 1.
{ ξ } = [ ξ r ξ s ξ t ] = 1 2 A [ 2 A s t b r a r 2 A t r b s a s 2 A r s b t a t ] [ 1 x y ] ,
A m n = x m y n - x n y m ( m , n = r , s , t ) , a r = x t - x s ; a s = x r - x t ; a t = x s - x r , b r = y s - y t ; b s = y t - y r ; b t = y r - y s ,
ϕ p ( ξ j ) = ϕ r , ϕ s , ϕ t [ ξ r ξ s ξ t ] = ϕ p { ξ } ,
ψ p ( ξ j ) = ψ r , ψ s , ψ t [ ξ r ξ s ξ t ] = ψ p { ξ } ,
x { ξ } = 1 2 A [ b r b s b t ] = { b } ,
y { ξ } = 1 2 A [ a r a s a t ] = { a } ,
ϕ / x = ϕ p T { b } ; ϕ / y = ϕ p T { a } , ψ / x = ψ p T { b } ; ψ / y = ψ p T { a } .
I p = θ p T [ A p ] { θ p } - Γ θ p T [ B p ] { θ p } ,
Γ = ( ω / c ) 2 ( 1 - β ¯ 2 ) , β ¯ = β c / ω ,
θ p T = { ϕ r , ψ r , ϕ s , ψ s , ϕ t , ψ t } .
ϕ p ( r , θ ) = exp ( - α r ) ( ϕ a + ϕ b - ϕ a Δ θ θ ) ; ψ p ( r , θ ) = exp ( - α r ) ( ψ a + ψ b - ψ a Δ θ θ ) ,
Δ θ = ( Δ r ) / r o r o = ( r a + r b ) / 2             Δ r = [ ( x a - x b ) 2 + ( y a - y b ) 2 ] 1 / 2 , r a = ( x a 2 + y a 2 ) 1 / 2             r b = ( x b 2 + y b 2 ) 1 / 2
I p b = θ p b T [ A p b ] { θ p b } - Γ θ p b T [ B p b ] { θ p b } ,
θ p b T = { ϕ a , ψ a , ϕ b , ψ b } .
E 1 = r o 1 r exp ( - 2 α r ) d r ,
I = θ T [ A ] { θ } - Γ θ T [ B ] { θ } ,
[ A ] { θ } = Γ [ B ] { θ } .
n ( r ) = n i ( 1 - a r 2 ) ,             n i = 1.53 ,             n 0 = 1.50 ,             a = ( n i - n 0 ) / n i ,
n 1 > n 2 n 3 > n 4 ,             n 1 > n 3 n 2 > n 4 .
n 1 ( x , y ) = n 2 + - ( n 1 m - n 2 ) b 2 ( y 2 - b 2 ) ,             - b y 0 ,
n 1 ( x , y ) = n 2 + - ( n 1 m - n 2 ) L 2 ( x 2 + y 2 - L 2 ) , - ( a / 2 ) x a / 2 - b y 0 ,
L = ( b 2 + x 2 ) 1 / 2 when y x , L = [ ( a / 2 ) 2 + y 2 ] 1 / 2 when x < y ,
n 1 ( x , y ) = n 2 + ( n 1 m - n 2 ) × [ N / ( 1000 ) ] ,
n ¯ 1 = 1 A 0 b a 2 a 2 n 1 ( x , y ) d x d y ,

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