Abstract

The propagating beam method utilizes discrete Fourier transforms for generating configuration-space solutions to optical waveguide problems without reference to modes. The propagating beam method can also give a complete description of the field in terms of modes by a Fourier analysis with respect to axial distance of the computed fields. Earlier work dealt with the accurate determination of mode propagation constants and group delays. In this paper the method is extended to the computation of mode eigenfunctions. The method is efficient, allowing generation of a large number of eigenfunctions from a single propagation run. Computations for parabolic-index profiles show excellent agreement between analytic and numerically generated eigenfunctions.

© 1980 Optical Society of America

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References

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  1. M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
    [Crossref] [PubMed]
  2. M. D. Feit, J. A. Fleck, Appl. Opt. 18, 2843 (1979).
    [Crossref] [PubMed]
  3. M. D. Feit, J. A. Fleck, Appl. Opt. 19, 1154 (1980).
    [Crossref] [PubMed]
  4. C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 250 (1969).
    [Crossref]
  5. R. Yamada, Y. Inabe, IEEE Trans. Microwave Theory Tech. MTT-22, 813 (1974).
    [Crossref]
  6. J. E. Midwinter, Opt. Quantum Electron. 7, 289 (1975).
    [Crossref]
  7. J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973).
    [Crossref]
  8. A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
    [Crossref]
  9. M. O. Vassell, Opto-electronics 6, 271 (1974).
    [Crossref]
  10. J. A. Arnaud, W. Mammel, Electron Lett. 12, 6 (1976).
    [Crossref]
  11. P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
    [Crossref]
  12. C. Yeh, G. Lindgren, Appl. Opt. 16, 483 (1977).
    [Crossref] [PubMed]
  13. E. Bianciardi, V. Rizzole, Opt. Quantum Electron 9, 121 (1977).
    [Crossref]
  14. T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun Eng Jpn. E59, 11 (1976).
  15. P. Di Vita, V. Rossi, Opt. Acta, to be published.
  16. D. Gloge, E. A. J. Marcatili, Bell. Syst. Tech. J. 52, 1563 (1973).
  17. R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [Crossref] [PubMed]
  18. J. J. Ramskov Hansen, E. Nicolaisen, Appl. Opt. 17, 2831 (1978).
    [Crossref]
  19. D. Marcuse, Appl. Opt. 18, 2073 (1979).
    [Crossref] [PubMed]
  20. S. Choudhary, L. B. Felsen, J. Opt. Soc. Am. 67, 1192 (1977).
    [Crossref]
  21. J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 216–218.
  22. J. M. Arnold, Electron. Lett. 13, 660 (1977).
    [Crossref]
  23. J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [Crossref]
  24. See, for example, D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 270.
  25. See, for example, Ref. 21, p. 106.
  26. The function V(x,y) and its integral defined in the text have been so labeled because they correspond closely with the potential energy function in the Schroedinger equation. They should not be confused with the normalized frequency ak(n1/n0)(2Δ)1/2 often called V in the literature.

1980 (1)

1979 (2)

1978 (3)

1977 (4)

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

E. Bianciardi, V. Rizzole, Opt. Quantum Electron 9, 121 (1977).
[Crossref]

S. Choudhary, L. B. Felsen, J. Opt. Soc. Am. 67, 1192 (1977).
[Crossref]

J. M. Arnold, Electron. Lett. 13, 660 (1977).
[Crossref]

1976 (4)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[Crossref]

T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun Eng Jpn. E59, 11 (1976).

J. A. Arnaud, W. Mammel, Electron Lett. 12, 6 (1976).
[Crossref]

R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[Crossref] [PubMed]

1975 (1)

J. E. Midwinter, Opt. Quantum Electron. 7, 289 (1975).
[Crossref]

1974 (2)

M. O. Vassell, Opto-electronics 6, 271 (1974).
[Crossref]

R. Yamada, Y. Inabe, IEEE Trans. Microwave Theory Tech. MTT-22, 813 (1974).
[Crossref]

1973 (2)

D. Gloge, E. A. J. Marcatili, Bell. Syst. Tech. J. 52, 1563 (1973).

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

1970 (1)

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

1969 (1)

C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 250 (1969).
[Crossref]

Arnaud, J.

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 216–218.

Arnaud, J. A.

J. A. Arnaud, W. Mammel, Electron Lett. 12, 6 (1976).
[Crossref]

Arnold, J. M.

J. M. Arnold, Electron. Lett. 13, 660 (1977).
[Crossref]

Bianciardi, E.

E. Bianciardi, V. Rizzole, Opt. Quantum Electron 9, 121 (1977).
[Crossref]

Blok, H.

A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
[Crossref]

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

Chan, K. B.

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

Choudhary, S.

Clarricoats, P. J. B.

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

Di Vita, P.

P. Di Vita, V. Rossi, Opt. Acta, to be published.

Dil, J. G.

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

Feit, M. D.

Felsen, L. B.

Fleck, J. A.

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell. Syst. Tech. J. 52, 1563 (1973).

Gronthoud, A. G.

A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
[Crossref]

Inabe, Y.

R. Yamada, Y. Inabe, IEEE Trans. Microwave Theory Tech. MTT-22, 813 (1974).
[Crossref]

Keck, D. B.

Kurtz, C. N.

C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 250 (1969).
[Crossref]

Lindgren, G.

Mammel, W.

J. A. Arnaud, W. Mammel, Electron Lett. 12, 6 (1976).
[Crossref]

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, Bell. Syst. Tech. J. 52, 1563 (1973).

Marcuse, D.

D. Marcuse, Appl. Opt. 18, 2073 (1979).
[Crossref] [PubMed]

See, for example, D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 270.

Midwinter, J. E.

J. E. Midwinter, Opt. Quantum Electron. 7, 289 (1975).
[Crossref]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[Crossref]

Nicolaisen, E.

Olshansky, R.

Ramskov Hansen, J. J.

Rizzole, V.

E. Bianciardi, V. Rizzole, Opt. Quantum Electron 9, 121 (1977).
[Crossref]

Rossi, V.

P. Di Vita, V. Rossi, Opt. Acta, to be published.

Streifer, W.

C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 250 (1969).
[Crossref]

Suematsu, Y.

T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun Eng Jpn. E59, 11 (1976).

Tanaka, T.

T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun Eng Jpn. E59, 11 (1976).

Vassell, M. O.

M. O. Vassell, Opto-electronics 6, 271 (1974).
[Crossref]

Yamada, R.

R. Yamada, Y. Inabe, IEEE Trans. Microwave Theory Tech. MTT-22, 813 (1974).
[Crossref]

Yeh, C.

Appl. Opt. (7)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[Crossref]

Bell. Syst. Tech. J. (1)

D. Gloge, E. A. J. Marcatili, Bell. Syst. Tech. J. 52, 1563 (1973).

Electron Lett. (1)

J. A. Arnaud, W. Mammel, Electron Lett. 12, 6 (1976).
[Crossref]

Electron. Lett. (2)

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

J. M. Arnold, Electron. Lett. 13, 660 (1977).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 250 (1969).
[Crossref]

R. Yamada, Y. Inabe, IEEE Trans. Microwave Theory Tech. MTT-22, 813 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Quantum Electron (1)

E. Bianciardi, V. Rizzole, Opt. Quantum Electron 9, 121 (1977).
[Crossref]

Opt. Quantum Electron. (2)

J. E. Midwinter, Opt. Quantum Electron. 7, 289 (1975).
[Crossref]

A. G. Gronthoud, H. Blok, Opt. Quantum Electron. 10, 95 (1978).
[Crossref]

Opto-electronics (2)

M. O. Vassell, Opto-electronics 6, 271 (1974).
[Crossref]

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

Trans. Inst. Electron. Commun Eng Jpn. (1)

T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun Eng Jpn. E59, 11 (1976).

Other (5)

P. Di Vita, V. Rossi, Opt. Acta, to be published.

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 216–218.

See, for example, D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 270.

See, for example, Ref. 21, p. 106.

The function V(x,y) and its integral defined in the text have been so labeled because they correspond closely with the potential energy function in the Schroedinger equation. They should not be confused with the normalized frequency ak(n1/n0)(2Δ)1/2 often called V in the literature.

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Figures (4)

Fig. 1
Fig. 1

Numerical and analytic eigenfunction comparison for 1-D parabolic profile. Absolute value of eigenfunction is plotted for the following mode numbers: (a) n = 0; (b) n = 9; (c) n = 19.

Fig. 2
Fig. 2

Numerical and analytic eigenfunction comparison for axisymmetric parabolic profile. Absolute value of radial eigenfunction is plotted for the following mode numbers: (a) ν = 5, μ = 0; (b) ν = 5, μ = 1; (c) ν = 4, μ = 7; (d) ν = 5, μ = 7.

Fig. 3
Fig. 3

One-dimensional power-law profile with α = 1.85 illuminated by incoherent source. Cladding begins 31.25 μm from center. (a) Field spectrum vs −β showing guided and leaky modes; (b) group delays vs −β.

Fig. 4
Fig. 4

Eigenfunctions for 1-D power-law profile with α = 1.85 (Fig. 3); absolute values of eigenfunctions plotted for following mode numbers: (a) n = 19; (b) n = 18; (c) n = 17.

Tables (1)

Tables Icon

Table I Comparison Between Analytical and Numerical Values of the Integral |〈νμ|V|νμ〉 | for ν = 4 and 5 Modes in a Square-Law Fiber

Equations (40)

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2 E x 2 + 2 E y 2 + 2 E z 2 + ω 2 c 2 n 2 ( ω , x , y ) E = 0 ,
E ( ω , x , y , z ) = ( ω , x , y , z ) exp ( ikz ) ,
k = ( n 0 ω ) / c ,
2 z 2 + 2 ik z = 2 + k 2 { [ n ( x , y ) n 0 ] 2 1 } ,
2 ik z = 2 + k 2 { [ n ( x , y ) n 0 ] 2 1 } .
( x , y , z ) = u n ( x , y ) exp ( i β n z ) ,
( x , y , z ) = u n ( x , y ) exp ( i β n z ) ,
u n ( x , y ) = u n ( x , y ) ,
β n = ( β n 2 + 2 k β n ) / 2 k ,
β n = k [ 1 ( 1 + 2 β n / k ) 1 / 2 ] .
( x , y , z + Δ z ) = exp ( ic Δ z 4 n 0 ω 2 ) × exp ( in 0 ω Δ z 2 c { [ n ( x , y ) n 0 ] 2 1 } ) × exp ( ic Δ z 4 n 0 ω 2 ) ( x , y , z ) + 0 ( Δ z ) 3 ,
P 1 ( z ) = * ( x , y , 0 ) ( x , y , z ) dx d y = * ( x , y , 0 ) ( x , y , z ) ,
( x , y , z ) = n , j A nj u nj ( x , y ) exp ( i β n z ) ,
P 1 ( z ) = n , j | A nj | 2 exp ( i β n z ) ,
P 1 ( β ) = n , j | A nj | 2 δ ( β β n ) .
P 1 ( β ) = n W n 1 ( β β n ) ,
W n = j A nj 2 ,
1 ( β β n ) = 1 Z 0 Z exp [ i ( β β n ) z ] w ( z ) dz .
w ( z ) = 1 cos [ ( 2 π z ) / Z ] ,
1 ( β β n ) = exp [ i ( β β n ) Z ] 1 i ( β β n ) Z 1 2 ( exp { i [ ( β β n ) Z + 2 π ] } 1 i [ ( β β n ) Z + 2 π ] + exp { i [ ( β β n ) Z 2 π ] } 1 i [ ( β β n ) Z 2 π ] ) .
( x , y , β ) = 1 Z 0 Z ( x , y , z ) w ( z ) exp ( i β z ) dz = n , j A nj u nj ( x , y ) 1 ( β β n ) .
( x , y , β n ) = j A n j u n j ( x , y ) L 1 ( 0 ) + n , j A nj u nj ( x , y ) L 1 ( β n β n ) ,
( x , y , β n ) = j A nj u nj ( x , y ) 1 ( 0 ) .
u n ( x , y ) = const × 0 Z ( x , y , z ) w ( z ) exp ( i β n z ) dz = const × ( x , y , β n ) ,
u μ ν ( r , θ ) = exp ( i ν θ ) f μ ν ( r ) ,
f μ ν ( r ) = const × 0 Z ( x , 0 , z ) w ( z ) exp ( i β μ ν z ) dz = const × ( x , 0 , β μ ν ) .
( x , y , 0 ) = F ( r ) { r ν exp ( i ν θ ) + r ν + 1 exp [ i ( ν + 1 ) θ ] } ,
n 2 = { n 1 2 [ 1 2 Δ ( x / a ) α ] x a n 0 2 = ( 1 2 Δ ) n 1 2 x a ,
u n ( x ) = ( π 1 / 2 2 n n ! ) 1 / 2 H n ( x / σ a ) exp ( x 2 / 2 σ 2 ) ,
σ a = [ a k ( 2 Δ ) 1 / 2 n 1 / n 0 ] 1 / 2 .
β n = Δ n 1 2 n 0 ω c n 1 n 0 ( 2 Δ ) 1 / 2 a ( n + ½ ) .
( x , 0 ) = exp [ ( x x 0 ) 2 / 2 σ 2 ]
n 2 = { n 1 2 [ 1 2 Δ ( r / a ) 2 ] r a n 0 2 = ( 1 2 Δ ) n 1 2 r a ,
u μ ν ( r , θ ) = exp ( i ν θ ) exp ( r 2 / 2 σ a 2 ) ( r / σ a ) ν L μ ν ( r 2 / σ a 2 ) ,
L μ ν ( x ) = s = 0 μ ( μ + ν ) ! ( x ) s ( ν + s ) ! ( μ s ) ! s ! .
β μ ν = Δ n 1 2 n 0 ω c n 1 n 0 ( 2 Δ ) 1 / 2 a ( 2 μ + ν + 1 ) .
n | V | n = V ( x , y ) u n * ( x , y ) u n ( x , y ) dxdy ,
V = n 0 ω 2 c { 1 [ n ( x , y ) n 0 ] 2 } .
ν μ | V | ν μ = 2 π 0 R V ( r ) [ f μ ν ( r ) ] 2 rdr ,
ν μ | V | ν μ = Δ n 1 2 n 0 ω c + n 1 2 n 0 ( 2 Δ ) 1 / 2 a ( 2 μ + ν + 1 ) ,

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