Abstract

We demonstrate that relaxation algorithms for the determination of the lowest-order modes of a refractive-index profile are identical to imaginary-distance propagation procedures.

© 1992 Optical Society of America

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References

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  1. R. Hawkins, R. Deri, in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 110.
  2. M. Robertson, S. Ritchie, P. Dayan, IEE Proc. Pt. J 132, 336 (1985).
  3. A. Delage, Can. J. Phys. 69, 512 (1991).
    [CrossRef]
  4. R. Deri, R. Hawkins, E. Kapon, Appl. Phys. Lett. 53, 1483 (1988).
    [CrossRef]
  5. R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
    [CrossRef]
  6. R. Deri, R. Thurston, R. Hawkins, Electron. Lett. 27, 483 (1991).
    [CrossRef]
  7. D. Yevick, B. Hermansson, Electron. Lett. 21, 1029 (1985).
    [CrossRef]
  8. B. Hermansson, D. Yevick, J. Opt. Soc. Am. A 4, 379 (1987).
    [CrossRef]
  9. N. Sehmi, Computational Physics (Benjamin Cummings, Menlo Park, Calif., 1986).
  10. D. Yevick, B. Hermansson, IEEE J. Quantum Electron. 26, 109 (1990).
    [CrossRef]

1991 (2)

A. Delage, Can. J. Phys. 69, 512 (1991).
[CrossRef]

R. Deri, R. Thurston, R. Hawkins, Electron. Lett. 27, 483 (1991).
[CrossRef]

1990 (2)

D. Yevick, B. Hermansson, IEEE J. Quantum Electron. 26, 109 (1990).
[CrossRef]

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

1988 (1)

R. Deri, R. Hawkins, E. Kapon, Appl. Phys. Lett. 53, 1483 (1988).
[CrossRef]

1987 (1)

1985 (2)

D. Yevick, B. Hermansson, Electron. Lett. 21, 1029 (1985).
[CrossRef]

M. Robertson, S. Ritchie, P. Dayan, IEE Proc. Pt. J 132, 336 (1985).

Colas, E.

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

Dayan, P.

M. Robertson, S. Ritchie, P. Dayan, IEE Proc. Pt. J 132, 336 (1985).

Delage, A.

A. Delage, Can. J. Phys. 69, 512 (1991).
[CrossRef]

Deri, R.

R. Deri, R. Thurston, R. Hawkins, Electron. Lett. 27, 483 (1991).
[CrossRef]

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

R. Deri, R. Hawkins, E. Kapon, Appl. Phys. Lett. 53, 1483 (1988).
[CrossRef]

R. Hawkins, R. Deri, in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 110.

Hawkins, R.

R. Deri, R. Thurston, R. Hawkins, Electron. Lett. 27, 483 (1991).
[CrossRef]

R. Deri, R. Hawkins, E. Kapon, Appl. Phys. Lett. 53, 1483 (1988).
[CrossRef]

R. Hawkins, R. Deri, in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 110.

Hermansson, B.

D. Yevick, B. Hermansson, IEEE J. Quantum Electron. 26, 109 (1990).
[CrossRef]

B. Hermansson, D. Yevick, J. Opt. Soc. Am. A 4, 379 (1987).
[CrossRef]

D. Yevick, B. Hermansson, Electron. Lett. 21, 1029 (1985).
[CrossRef]

Kapon, E.

R. Deri, R. Hawkins, E. Kapon, Appl. Phys. Lett. 53, 1483 (1988).
[CrossRef]

Ritchie, S.

M. Robertson, S. Ritchie, P. Dayan, IEE Proc. Pt. J 132, 336 (1985).

Robertson, M.

M. Robertson, S. Ritchie, P. Dayan, IEE Proc. Pt. J 132, 336 (1985).

Sehmi, N.

N. Sehmi, Computational Physics (Benjamin Cummings, Menlo Park, Calif., 1986).

Seto, M.

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

Shahar, A.

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

Thurston, R.

R. Deri, R. Thurston, R. Hawkins, Electron. Lett. 27, 483 (1991).
[CrossRef]

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

Tomlinson, W.

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

Yevick, D.

D. Yevick, B. Hermansson, IEEE J. Quantum Electron. 26, 109 (1990).
[CrossRef]

B. Hermansson, D. Yevick, J. Opt. Soc. Am. A 4, 379 (1987).
[CrossRef]

D. Yevick, B. Hermansson, Electron. Lett. 21, 1029 (1985).
[CrossRef]

Yi-Yan, A.

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

Appl. Phys. Lett. (2)

R. Deri, R. Hawkins, E. Kapon, Appl. Phys. Lett. 53, 1483 (1988).
[CrossRef]

R. Deri, A. Shahar, E. Colas, R. Thurston, W. Tomlinson, A. Yi-Yan, M. Seto, Appl. Phys. Lett. 57, 2396 (1990).
[CrossRef]

Can. J. Phys. (1)

A. Delage, Can. J. Phys. 69, 512 (1991).
[CrossRef]

Electron. Lett. (2)

R. Deri, R. Thurston, R. Hawkins, Electron. Lett. 27, 483 (1991).
[CrossRef]

D. Yevick, B. Hermansson, Electron. Lett. 21, 1029 (1985).
[CrossRef]

IEE Proc. Pt. J (1)

M. Robertson, S. Ritchie, P. Dayan, IEE Proc. Pt. J 132, 336 (1985).

IEEE J. Quantum Electron. (1)

D. Yevick, B. Hermansson, IEEE J. Quantum Electron. 26, 109 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

Other (2)

R. Hawkins, R. Deri, in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 110.

N. Sehmi, Computational Physics (Benjamin Cummings, Menlo Park, Calif., 1986).

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Equations (18)

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I = x a x a { 1 2 [ E ( x ) x ] 2 + k 0 2 n 2 ( x ) E 2 ( x ) } d x
x a x a E 2 ( x ) d x = 1 .
I = x a x a { 1 2 [ E ( x ) x ] 2 + [ k 0 2 n 2 ( x ) β 2 ] E 2 ( x ) } d x .
E i + 1 2 E i + E i 1 ( Δ x ) 2 + k 0 2 n 2 ( x i ) E i = 0 .
E i ( m + 1 ) = 1 2 [ E i 1 ( m ) + E i + 1 ( m ) + ( Δ x ) 2 k 0 2 n 2 ( x i ) E i ( m ) ] ,
E i ( m + 1 ) = ( 1 α ) E i ( m ) + α 2 × [ E i 1 ( m ) + E i + 1 ( m ) + ( Δ x ) 2 k 0 2 n 2 ( x i ) E i ( m ) ] .
E i ( m + 1 ) = ( 1 α ) E i ( m ) + α 2 × { E i 1 ( m ) + E i + 1 ( m ) + ( Δ x ) 2 k 0 2 [ n 2 ( x i ) n 0 2 ] E i ( m ) } .
k 0 2 n 0 2 = x a x a E ( x ) [ 2 x 2 + k 0 2 n 2 ( x ) ] E ( x ) d x x a x a E 2 ( x ) d x .
E i ( m + 1 ) = ( 1 α ˜ ) E i ( m ) + α ˜ [ E i 1 ( m ) + E i + 1 ( m ) ] 2 + ( Δ x ) 2 k 0 2 [ n 0 2 n 2 ( x i ) ] .
2 i k 0 n 0 E z = { 2 x 2 + k 0 2 [ n 2 ( x ) n 0 2 ] } E ( x ) .
E ( x , z ) = j E j ( x ) exp ( i β j z ) .
E ( x , z ) = j E j ( x ) exp ( β j z ) ,
E i ( m + 1 ) E i ( m ) = Δ z 2 k 0 n 0 { E i + 1 ( m ) 2 E i ( m ) + E i 1 ( m ) ( Δ x ) 2 + k 0 2 [ n 2 ( x i ) n 0 2 ] E i ( m ) } ,
α = Δ z / 2 k 0 n 0 ( Δ x ) 2 .
d E d z = ( O 1 + O 2 ) E ,
O 1 E i = α Δ z { E i 1 2 E i + E i + 1 + k 0 ( Δ x ) 2 [ n 2 ( x i ) n 0 2 ] E i } k 0 2 ( Δ x ) 2 2 Δ z [ n 2 ( x i ) n 0 2 ] E i ,
O 2 E i = k 0 2 ( Δ x ) 2 2 Δ z [ n 2 ( x i ) n 0 2 ] E i .
E i ( m + 1 ) = ( 1 γ ˆ Δ z O 2 ) 1 [ 1 + ( 1 γ ˆ ) Δ z O 2 ] × ( 1 γ Δ z O 1 ) 1 [ 1 + ( 1 γ ) Δ z O 1 ] E i ( m ) .

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