Abstract

The variational-moment method is used to demonstrate the calculation of propagation constants for optical waveguides with arbitrary index profiles. For simplicity, and for comparison with exact numerical solutions, only a few one-dimensional structures are considered. Examples include symmetric and nonsymmetric waveguides. Good agreement with the exact numerical solutions is obtained in most cases, even for the nonsymmetric waveguides.

© 1994 Optical Society of America

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References

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  1. R. M. Knox, P. P. Toulies, “Integrate circuits for the millimeter through optical frequency range,” in Technical Digest of the Symposium on Submillimeter Waves (Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1970), pp. 497–516.
  2. S. Ruschin, “Approximate formula for the propagation constant of the basic mode in slab waveguides of arbitrary index profiles,” Appl. Opt. 24, 4189–4191 (1985).
    [CrossRef] [PubMed]
  3. H. Matsumura, T. Suganuma, “Normalization of single-mode fibers having an arbitrary index profile,” Appl. Opt. 19, 3151–31581980).
    [CrossRef] [PubMed]
  4. C. D. Hussey, C. Pask, “Theory of the profile-moments description of single mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).
  5. R. J. Black, C. Pask, “Slab waveguides characterized by moments of the index profile,” IEEE J. Quantum Electron. QE-20, 996–999 (1984).
    [CrossRef]
  6. H. Kogelnik, “Theory of optical waveguides,” in Guided-Waves Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), pp. 7–88.
    [CrossRef]
  7. A. Sharma, P. Bindal, “An accurate variational analysis of single mode channel wave-guides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
    [CrossRef]
  8. A. Sharma, P. K. Mishra, A. K. Ghatak, “Single mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
    [CrossRef]
  9. P. K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204–212 (1986).
    [CrossRef]
  10. A. Hardy, M. Itzkowitz, G. Griffel, “Use of a variational moment method in calculating propagation constants for waveguides with an arbitrary index profile,” Appl. Opt. 28, 1910–1913 (1989).
    [CrossRef] [PubMed]
  11. I. S. Gradshteyn, I. M. Ryghik, Table of Integrals Series and Products (Academic, New York, 1980).
  12. Q. Li, J. Wang, “A modified ray-optic method for arbitrary dielectric waveguides,” IEEE J. Quantum Electron. 28, 2721–2727 (1992).
    [CrossRef]

1992 (2)

A. Sharma, P. Bindal, “An accurate variational analysis of single mode channel wave-guides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

Q. Li, J. Wang, “A modified ray-optic method for arbitrary dielectric waveguides,” IEEE J. Quantum Electron. 28, 2721–2727 (1992).
[CrossRef]

1989 (1)

1988 (1)

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

1986 (1)

P. K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204–212 (1986).
[CrossRef]

1985 (1)

1984 (1)

R. J. Black, C. Pask, “Slab waveguides characterized by moments of the index profile,” IEEE J. Quantum Electron. QE-20, 996–999 (1984).
[CrossRef]

1982 (1)

C. D. Hussey, C. Pask, “Theory of the profile-moments description of single mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).

1980 (1)

Bindal, P.

A. Sharma, P. Bindal, “An accurate variational analysis of single mode channel wave-guides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

Black, R. J.

R. J. Black, C. Pask, “Slab waveguides characterized by moments of the index profile,” IEEE J. Quantum Electron. QE-20, 996–999 (1984).
[CrossRef]

Ghatak, A. K.

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryghik, Table of Integrals Series and Products (Academic, New York, 1980).

Griffel, G.

Hardy, A.

Hussey, C. D.

C. D. Hussey, C. Pask, “Theory of the profile-moments description of single mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).

Itzkowitz, M.

Knox, R. M.

R. M. Knox, P. P. Toulies, “Integrate circuits for the millimeter through optical frequency range,” in Technical Digest of the Symposium on Submillimeter Waves (Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1970), pp. 497–516.

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Waves Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), pp. 7–88.
[CrossRef]

Li, Q.

Q. Li, J. Wang, “A modified ray-optic method for arbitrary dielectric waveguides,” IEEE J. Quantum Electron. 28, 2721–2727 (1992).
[CrossRef]

Matsumura, H.

Mishra, P. K.

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

P. K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204–212 (1986).
[CrossRef]

Pask, C.

R. J. Black, C. Pask, “Slab waveguides characterized by moments of the index profile,” IEEE J. Quantum Electron. QE-20, 996–999 (1984).
[CrossRef]

C. D. Hussey, C. Pask, “Theory of the profile-moments description of single mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).

Ruschin, S.

Ryghik, I. M.

I. S. Gradshteyn, I. M. Ryghik, Table of Integrals Series and Products (Academic, New York, 1980).

Sharma, A.

A. Sharma, P. Bindal, “An accurate variational analysis of single mode channel wave-guides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

P. K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204–212 (1986).
[CrossRef]

Suganuma, T.

Toulies, P. P.

R. M. Knox, P. P. Toulies, “Integrate circuits for the millimeter through optical frequency range,” in Technical Digest of the Symposium on Submillimeter Waves (Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1970), pp. 497–516.

Wang, J.

Q. Li, J. Wang, “A modified ray-optic method for arbitrary dielectric waveguides,” IEEE J. Quantum Electron. 28, 2721–2727 (1992).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (2)

Q. Li, J. Wang, “A modified ray-optic method for arbitrary dielectric waveguides,” IEEE J. Quantum Electron. 28, 2721–2727 (1992).
[CrossRef]

R. J. Black, C. Pask, “Slab waveguides characterized by moments of the index profile,” IEEE J. Quantum Electron. QE-20, 996–999 (1984).
[CrossRef]

J. Lightwave Technol. (2)

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

P. K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204–212 (1986).
[CrossRef]

Opt. Quantum Electron. (1)

A. Sharma, P. Bindal, “An accurate variational analysis of single mode channel wave-guides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

Proc. Inst. Electr. Eng. Part H (1)

C. D. Hussey, C. Pask, “Theory of the profile-moments description of single mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).

Other (3)

H. Kogelnik, “Theory of optical waveguides,” in Guided-Waves Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), pp. 7–88.
[CrossRef]

R. M. Knox, P. P. Toulies, “Integrate circuits for the millimeter through optical frequency range,” in Technical Digest of the Symposium on Submillimeter Waves (Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1970), pp. 497–516.

I. S. Gradshteyn, I. M. Ryghik, Table of Integrals Series and Products (Academic, New York, 1980).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the index profile distribution.

Fig. 2
Fig. 2

Relative error Δb/b for the lowest-order m = 0 mode for the symmetric cases as a function of the normalized frequency ν [see Eq. (3b)]. The dashed curves represent the zero-order approximation of the moment method [see Eq. (20a)]. The solid curves represent the variational correction [see Eq. (20b)].

Fig. 3
Fig. 3

Relative error Δb/b for the m = 1 mode for the symmetric cases as a function of the normalized frequency ν [see Eq. (3b)]. The dashed curves represent the moment method [see. Eq. (20a)]. The solid curves represent the variational correction [see Eq. (20b)].

Fig. 4
Fig. 4

Relative error Δb/b for the m = 2 mode for the symmetric cases as a function of ν [see Eq. (3b)]. The dashed curves represent the moment method [See Eq. (20a)]. The solid curve represents the variational correction [see Eq. (20b)].

Fig. 5
Fig. 5

Relative error Δb/b for the m = 3 mode for the symmetric cases as a function of ν [see Eq. (3b)]. The dashed curves represent the moment method [see Eq. (20a)]. The solid curves represent the variational correction [see Eq. (20b)].

Fig. 6
Fig. 6

Relative error Δb/b of the step-index profile for the m = 0 mode for the asymmetric cases as a function of ν [see Eq. (3b)]. The dashed curves represent the moment method [see Eq. (20a)]. The solid curves represent the variational correction [see Eq. (20b)].

Fig. 7
Fig. 7

Relative error Δb/b of the cladded parabolic profile for the m = 0 mode for the asymmetric cases as a function of ν [see. Eq. (3b)]. The dashed curves represent the moment method [see Eq. (20a)]. The solid curves represent the variational correction [See Eq. (20b)].

Fig. 8
Fig. 8

Relative error Δb/b of the exponential profile for the m = 0 mode for the asymmetric cases as a function of ν [see Eq. (3b)]. The dashed curves represent the moment method [see Eq. (20a)]. The solid curves represent the variational correction [see Eq. (20b)].

Fig. 9
Fig. 9

Relative error Δb/b of the Gaussian profile for the m = 0 mode for the asymmetric cases as a function of ν [see Eq. (3b)]. The dashed curves represent the moment method [see Eq. (20a)]. The solid curves represent the variational correction [see Eq. (20b)].

Equations (38)

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d 2 E ( x ) d x 2 + [ k 0 2 n 2 ( x ) - β 2 ] E ( x ) = 0 ,
n 2 ( x ) = n s 2 + ( n 0 2 - n s 2 ) f ( x / a ) ,
ζ 2 = k 0 2 ( n 0 2 - n s 2 ) x 2 ,
ν = k 0 a ( n 0 2 - n s 2 ) 1 / 2 ,
b 2 = ( k 0 - 2 β 2 - n s 2 ) / ( n 0 2 - n s 2 ) ,
d 2 ψ ( ζ ) d ζ 2 + [ f ( ζ / ν ) - b 2 ] ψ ( ζ ) = 0 ,
n ¯ 2 ( x ) = n ¯ s 2 + ( n ¯ 0 2 - n ¯ s 2 ) f ¯ ( x / a ¯ ) ,
d 2 E ¯ ( x ) d x 2 + [ k 0 2 n ¯ 2 ( x ) - β ¯ 2 ] E ¯ ( x ) = 0.
n d 2 = ( n ¯ 0 2 - n ¯ s 2 ) / ( n 0 2 - n s 2 ) ,
ν ¯ = ( a ¯ / a ) ν = p ν ,
b ¯ 2 = ( k 0 - 2 β ¯ 2 - n ¯ s 2 ) / ( n 0 2 - n s 2 ) ,
d 2 ψ ¯ ( ζ ) d ζ 2 + [ n d 2 f ¯ ( ζ / ν ¯ ) - b ¯ 2 ] ψ ¯ ( ζ ) = 0 ,
M l = - ζ l f ( ζ / ν ) d ζ ,
M ¯ l = n d 2 - ζ l f ¯ ( ζ / ν ¯ ) d ζ ,
M ¯ 0 = M 0 ,
M ¯ 2 = M 2 ,
δ β = k 0 2 2 β ¯ - δ n 2 ( x ) E ¯ 2 ( x ) d x - E ¯ 2 ( x ) d x ,
δ b = 1 2 b ¯ - [ f ( ζ / ν ) - n d 2 f ¯ ( ζ / p ν ) ] [ ψ ¯ ( ζ / p ν ) ] 2 d ζ - [ ψ ¯ ( ζ / p ν ) ] 2 d ζ ,
k 0 2 δ n 2 = f ( ζ / ν ) - n d 2 f ¯ ( ζ / p ν ) .
f ¯ ( ζ / ν ¯ ) = cosh - 2 ( ζ / ν ¯ ) ,
b ¯ m = ( p ν ) - 1 [ ( p 2 ν 2 n d 2 + 1 4 ) 1 / 2 - m - 1 2 ] ,
ψ ¯ m ( ζ / p ν ) = u m ( ζ / p ν ) cosh - s ¯ ( ζ / p ν ) ,
s = 1 2 [ ( 1 + 4 p 2 ν 2 n d 2 ) 1 / 2 - 1 ] = p ν b ¯ m + m .
u 0 ( ζ / p ν ) = 1 ,
u 1 ( ζ / p ν ) = sinh ( ζ / p ν ) ,
u 2 ( ζ / p ν ) = 1 - 2 ( s - 1 ) sinh 2 ( ζ / p ν ) ,
u 3 ( ζ / p ν ) = [ 1 - 2 3 ( s - 2 ) sinh 2 ( ζ / p ν ) ] sinh ( ζ / p ν ) .
M ¯ 0 = 2 p ν n d 2 ,
M ¯ 2 = π 2 n d 2 ( p ν ) 3 .
n d 2 = ¼ π [ M 0 3 / 3 M 2 ] 1 / 2 ,
p ν = ( 2 / π ) [ 3 M 2 / M 0 ] 1 / 2 .
Δ b m b m = b ¯ m - b m b m ,
Δ b m b m = b ¯ m + δ b m - b m b m ,
f ( ζ / ν ) = { 1 0 ζ < ν c - ν < ζ < 0 0 ζ ν ,
f ( ζ / ν ) = { 1 - ( ζ / ν ) 2 0 ζ < ν 1 - ( ζ / c ν ) 2 - c ν < ζ < 0 0 else ,
f ( ζ / ν ) = { exp ( - ζ / ν ) ζ 0 exp ( ζ / c ν ) ζ < 0 ,
f ( ζ / ν ) = { exp ( - ζ 2 / ν 2 ) ζ 0 exp ( - ζ 2 / ( c ν ) 2 ) ζ < 0 ,
- cosh - 2 S ( ζ / p ν ) d ζ = p ν Γ ( 1 / 2 ) Γ ( s ) / Γ ( 1 / 2 + s ) ,

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