Abstract

To deal with sharply cut off fields at mirror edges, a continuous Fourier integration procedure is described. A spline fit is made to the discrete data, followed by analytic integration of the spline functions. End corrections associated with the difference between spline functions near the edges and the remaining uniform splines are made. This procedure permits an accurate integration of the paraxial equation in the thin-gain-sheet approximation.

© 1979 Optical Society of America

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References

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  1. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974).
    [CrossRef] [PubMed]
  2. P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [CrossRef]
  3. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453 (1961).
  4. D. B. Rensh, A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. 12, 997–1000 (1973).
    [CrossRef]
  5. G. T. Moore, R. J. McCarthy, “Theory of modes in a loaded strip confocal unstable resonator,” J. Opt. Soc. Am. 67, 228–241 (1977).
    [CrossRef]
  6. E. A. Sziklas, A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain, 2: fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef] [PubMed]
  7. I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions,” Q. Appl. Math. 4, 45–49 (1946).I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions,” Q. Appl. Math. 4, 121–141 (1946).
  8. C. D. Boor, “On calculating with B-splines,” J. Approx. Theory, 6, 50–62 (1972); “Package for calculating with B-splines,” SIAM J. Numerical Anal. 14, 441–472 (1977).
    [CrossRef]
  9. Algebraic details and a program will be supplied to a computer-science-oriented journal.
  10. H. Jeffreys, B. S. Jeffreys, Methods of Mathematical Physics (Cambridge U. Press, Oxford, England, 1978), p. 262.

1977 (1)

G. T. Moore, R. J. McCarthy, “Theory of modes in a loaded strip confocal unstable resonator,” J. Opt. Soc. Am. 67, 228–241 (1977).
[CrossRef]

1975 (1)

1974 (1)

1973 (2)

1972 (1)

C. D. Boor, “On calculating with B-splines,” J. Approx. Theory, 6, 50–62 (1972); “Package for calculating with B-splines,” SIAM J. Numerical Anal. 14, 441–472 (1977).
[CrossRef]

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453 (1961).

1946 (1)

I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions,” Q. Appl. Math. 4, 45–49 (1946).I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions,” Q. Appl. Math. 4, 121–141 (1946).

Boor, C. D.

C. D. Boor, “On calculating with B-splines,” J. Approx. Theory, 6, 50–62 (1972); “Package for calculating with B-splines,” SIAM J. Numerical Anal. 14, 441–472 (1977).
[CrossRef]

Chester, A. N.

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453 (1961).

Horwitz, P.

Jeffreys, B. S.

H. Jeffreys, B. S. Jeffreys, Methods of Mathematical Physics (Cambridge U. Press, Oxford, England, 1978), p. 262.

Jeffreys, H.

H. Jeffreys, B. S. Jeffreys, Methods of Mathematical Physics (Cambridge U. Press, Oxford, England, 1978), p. 262.

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453 (1961).

McCarthy, R. J.

G. T. Moore, R. J. McCarthy, “Theory of modes in a loaded strip confocal unstable resonator,” J. Opt. Soc. Am. 67, 228–241 (1977).
[CrossRef]

Moore, G. T.

G. T. Moore, R. J. McCarthy, “Theory of modes in a loaded strip confocal unstable resonator,” J. Opt. Soc. Am. 67, 228–241 (1977).
[CrossRef]

Rensh, D. B.

Schoenberg, I. J.

I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions,” Q. Appl. Math. 4, 45–49 (1946).I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions,” Q. Appl. Math. 4, 121–141 (1946).

Siegman, A. E.

Sziklas, E. A.

Appl. Opt. (3)

Bell. Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453 (1961).

J. Approx. Theory (1)

C. D. Boor, “On calculating with B-splines,” J. Approx. Theory, 6, 50–62 (1972); “Package for calculating with B-splines,” SIAM J. Numerical Anal. 14, 441–472 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

G. T. Moore, R. J. McCarthy, “Theory of modes in a loaded strip confocal unstable resonator,” J. Opt. Soc. Am. 67, 228–241 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

Q. Appl. Math. (1)

I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions,” Q. Appl. Math. 4, 45–49 (1946).I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions,” Q. Appl. Math. 4, 121–141 (1946).

Other (2)

Algebraic details and a program will be supplied to a computer-science-oriented journal.

H. Jeffreys, B. S. Jeffreys, Methods of Mathematical Physics (Cambridge U. Press, Oxford, England, 1978), p. 262.

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Tables (1)

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Table 1 Comparison of FFT and CFT for a Truncated Cornu Spiral

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

g ( μ ) = - f ( x ) e i μ x d x .
f ( x ) = j = 1 N - k a j B j , k ( x ) ,
B j , k ( x ) = x - x j x j + k - 1 - x j B j , k - 1 ( x ) + x j + k - x x j + k - x j + l B j + 1 , k - 1 ( x ) ,
g ( μ ) = j a j x j x j + k B j , k ( x ) e i μ x d x ,
g ( μ ) = j a j ( - 1 ) k - 1 ( i μ ) k - 1 x j x j + k B j , k ( k - 1 ) ( x ) e i μ x d x ,
B j , k ( k - 1 ) ( x ) = k - 1 x j + k - 1 - x j B j , k - 1 ( k - 2 ) ( x ) - k - 1 x j + k - x j + 1 B j + 1 , k - 1 ( k - 2 ) ( x ) .
g ( μ ) = ( k - 1 ) ! ( i μ ) k j = 1 N - k a j ( x j + k - x j ) × ( x j , x j + 1 , x j + k ) ,
g ( μ ) = h j = 1 N - k a j e i μ j h Φ j ( μ ) ,
Φ j ( μ ) = ( e i μ h - 1 i μ h ) k Φ ( μ )
Φ 1 = 2 λ 3 [ e λ - ( 1 + λ + 1 2 λ 2 ) ] ,
Φ 2 = 1 λ 3 [ e 2 λ - 4 e λ + 3 + 2 λ ] ,
Φ N - 4 = - 1 λ 3 [ 1 - 4 e λ + ( 3 - 2 λ ) e 2 λ ] ,
Φ N - 3 = - 2 λ 3 [ 1 - ( 1 - λ + 1 2 λ 2 ) e λ ] ,
f ( x ) = { 1 for x < 1 0 otherwise ,

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