Abstract

We describe a spectral method for solving the paraxial wave equation in cylindrical geometry that is based on expansion of the exponential evolution operator in a Taylor series and use of fast Fourier transforms to evaluate derivatives. A fourth-order expansion gives excellent agreement with a two-transverse-dimensional split-operator calculation at a fraction of the cost in computation time per z step and at a considerable savings in storage.

© 1989 Optical Society of America

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References

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  1. J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  2. A. E. Siegman, Opt. Lett. 1, 13 (1977).
    [CrossRef] [PubMed]
  3. D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, J. Chem. Phys. 31, 987 (1959).
    [CrossRef]
  4. S.-C. Sheng, “Studies of laser resonators and beam propagation using fast transform methods,” Ph.D. dissertation (Department of Applied Physics, Stanford University, 1980);G. P. Agrawal, M. Lax, Opt. Lett. 6, 171 (1981).
    [CrossRef] [PubMed]
  5. M. D. Feit, J. A. Fleck, Appl. Opt. 19, 1154, 3140 (1980).
    [CrossRef] [PubMed]
  6. M. D. Feit, J. A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982).
    [CrossRef]

1982 (1)

M. D. Feit, J. A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982).
[CrossRef]

1980 (1)

1977 (1)

1976 (1)

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

1959 (1)

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, J. Chem. Phys. 31, 987 (1959).
[CrossRef]

Feit, M. D.

M. D. Feit, J. A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982).
[CrossRef]

M. D. Feit, J. A. Fleck, Appl. Opt. 19, 1154, 3140 (1980).
[CrossRef] [PubMed]

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

Fleck, J. A.

M. D. Feit, J. A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982).
[CrossRef]

M. D. Feit, J. A. Fleck, Appl. Opt. 19, 1154, 3140 (1980).
[CrossRef] [PubMed]

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

Gardner, D. G.

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, J. Chem. Phys. 31, 987 (1959).
[CrossRef]

Gardner, J. C.

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, J. Chem. Phys. 31, 987 (1959).
[CrossRef]

Lausch, G.

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, J. Chem. Phys. 31, 987 (1959).
[CrossRef]

Meinke, W. W.

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, J. Chem. Phys. 31, 987 (1959).
[CrossRef]

Morris, J. R.

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

Sheng, S.-C.

S.-C. Sheng, “Studies of laser resonators and beam propagation using fast transform methods,” Ph.D. dissertation (Department of Applied Physics, Stanford University, 1980);G. P. Agrawal, M. Lax, Opt. Lett. 6, 171 (1981).
[CrossRef] [PubMed]

Siegman, A. E.

Steiger, A.

M. D. Feit, J. A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

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

J. Chem. Phys. (1)

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, J. Chem. Phys. 31, 987 (1959).
[CrossRef]

J. Comput. Phys. (1)

M. D. Feit, J. A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982).
[CrossRef]

Opt. Lett. (1)

Other (1)

S.-C. Sheng, “Studies of laser resonators and beam propagation using fast transform methods,” Ph.D. dissertation (Department of Applied Physics, Stanford University, 1980);G. P. Agrawal, M. Lax, Opt. Lett. 6, 171 (1981).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

Mode spectra generated for a Gaussian beam, E(r) = rν exp(−r2/2σ2), propagating in a highly multimode quadratic-index-profile optical fiber, (a) ν = 0 mode, (b) ν = 2 mode.

Fig. 2
Fig. 2

Comparison between the computed and the analytic mode eigenfunction for the μ = 9, ν = 0 mode. The radial units are such that 1 unit =12 μm.

Tables (1)

Tables Icon

Table 1 Comparison of Calculated and Analytic Propagation Constants for Modes of Quadratic Index Fiber Obtained from the Paraxial Wave Equationa

Equations (9)

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2 i k E z = 2 E r 2 + 1 r E r ν 2 r 2 E 2 + k 2 [ ( n n 0 ) 2 1 ] E ,
E ( Δ z ) = exp ( i Δ z 2 k H ) E ( 0 ) = [ 1 i Δ z 2 k H + 1 2 ( i Δ z 2 k H ) 2 1 3 ! ( i Δ z 2 k H ) 3 + 1 4 ! ( i Δ z 2 k H ) 4 + O ( Δ z 5 ) ] E ( 0 ) .
E ( r , z ) = n = ( N / 2 ) + 1 N / 2 E n ( z ) exp ( 2 π inr / R ) ,
lim r 0 1 r E r = ( 2 E r 2 ) r = 0 , lim r 0 E r 2 = 1 2 ( 2 E r 2 ) r = 0
n 2 = n 1 2 [ 1 2 Δ ( r r 0 ) 2 ] r < r 0 = n 0 2 = n 1 2 ( 1 2 Δ ) r r 0 .
P ( z ) = E * ( r , 0 ) E ( r , z ) r d r
β = Δ n 1 2 n 0 ω c n 1 n 0 ( 2 Δ ) 1 / 2 r 0 ( 2 μ + ν + 1 ) ,
u μ ν = const . × 0 Z E ( r , z ) w ( z ) exp ( i β μ ν z ) d z ,
Δ z < 1 10 min ( k ( Δ r ) 2 , { k [ ( n n 0 ) 2 1 ] } 1 )

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