Abstract

We present a new approximate solution of the scalar-wave equation for planar optical waveguides with arbitrary refractive-index profiles. Test calculations are done for an index profile with a known solution. The comparison demonstrates the accuracy of our method. The method may also be applied to circularly symmetric optical fibers.

© 1991 Optical Society of America

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References

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  1. A. K. Ghatak, S. Lokanathan, Quantum Mechanics: Theory and Applications (Macmillan, New York, 1984).
  2. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C., 1964), Chap. 10.
  3. E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
    [Crossref]
  4. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).
  5. A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
    [Crossref]
  6. A. Gedeon, Opt. Commun. 12, 329 (1974).
    [Crossref]

1987 (1)

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

1974 (1)

A. Gedeon, Opt. Commun. 12, 329 (1974).
[Crossref]

1973 (1)

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C., 1964), Chap. 10.

Conwell, E. M.

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[Crossref]

Gedeon, A.

A. Gedeon, Opt. Commun. 12, 329 (1974).
[Crossref]

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

A. K. Ghatak, S. Lokanathan, Quantum Mechanics: Theory and Applications (Macmillan, New York, 1984).

Lokanathan, S.

A. K. Ghatak, S. Lokanathan, Quantum Mechanics: Theory and Applications (Macmillan, New York, 1984).

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

Shenoy, M. R.

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C., 1964), Chap. 10.

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

Appl. Phys. Lett. (1)

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[Crossref]

IEEE J. Lightwave Technol. (1)

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

Opt. Commun. (1)

A. Gedeon, Opt. Commun. 12, 329 (1974).
[Crossref]

Other (3)

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

A. K. Ghatak, S. Lokanathan, Quantum Mechanics: Theory and Applications (Macmillan, New York, 1984).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C., 1964), Chap. 10.

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Figures (1)

Fig. 1
Fig. 1

Electromagnetic field as a function of X for V = 4. The index profile is shown in the inset.

Tables (1)

Tables Icon

Table 1 Normalized Propagation Constants for Different Values of V

Equations (20)

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d 2 ψ d x 2 + κ 2 ( x ) ψ ( x ) = 0 ,
n 2 ( x ) = n 2 2 + ( n 1 2 - n 2 2 ) exp ( - x / d )             for x > 0 = n c 2             for x < 0
d 2 ψ d X 2 + V 2 [ exp ( - X ) - b ] ψ = 0             for X > 0 ,
d 2 ψ d X 2 - V 2 ( b + B ) ψ = 0             for X < 0 ,
V 2 = k 2 d 2 ( n 1 2 - n 2 2 ) ,             X = x / d ,             B = n 2 2 - n c 2 n 1 2 - n 2 2 ,
b = n e 2 - n 2 2 n 1 2 - n 2 2 ,             n e = β / k .
ψ ( X ) = { J ν [ 2 V exp ( X / 2 ) ] J ν ( 2 V )             for X > 0 exp [ V ( b + B ) 1 / 2 X ]             for X < 0 ,
ν = 2 V b .
J ν ( 2 V ) J ν ( 2 V ) = - ( n e 2 - n c 2 n 1 2 - n 2 2 ) 1 / 2 .
ψ ( x ) = { F ( x ) A i [ ξ ( x ) ] G ( x ) B i [ ξ ( x ) ] ,
ξ ( x ) [ ξ ( x ) ] 2 + κ 2 ( x ) = 0 ,
2 F ( x ) A i ( ξ ) ξ + F ( x ) A ( ξ ) ξ = 0.
F ( x ) = const . [ ξ ( x ) ] 1 / 2 ,             ξ ( x ) = { 3 2 x t x [ - κ 2 ( x ) ] 1 / 2 d x } 2 / 3 ,
ψ ( x ) [ C 1 A i ( ξ ) + C 2 B i ( ξ ) ] ( ξ ) - 1 / 2 .
ψ ( X ) = ( ξ 0 ξ ) 1 / 2 A i ( ξ ) A i ( ξ 0 )             for X > 0 ,
ξ = - { 3 V [ exp ( - X ) - b ] 1 / 2 - b 1 / 2 tan - 1 × [ exp ( - X ) b - 1 ] 1 / 2 } 2 / 3             for X < X t , = { 3 V - [ b - exp ( - X ) ] 1 / 2 + b 1 / 2 2 × ln b 1 / 2 + [ b - exp ( - X ) ] 1 / 2 b 1 / 2 - [ b - exp ( - X ) ] 1 / 2 } 2 / 3             for X > X t .
V ( B + b ) 1 / 2 ξ 0 = A i ( ξ 0 ) A i ( ξ 0 ) - 1 2 ξ ξ 0 2 .
d 2 ψ d X 2 + V 2 [ exp ( - X ) - b ] ψ + ( 1 2 ξ ξ - 3 4 ξ 2 ξ 2 ) ψ = 0.
δ b 0 ( 3 4 ξ 2 ξ 2 - 1 2 ξ 2 ξ ) ψ 2 d X V 2 - ψ 2 ( X ) d X .
Δ b b Δ β β n 1 n 1 - n 2 1 b .

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