Abstract

We present an analytic variational scheme which predicts the modes and losses for empty resonators with large Fresnel number N. The method is most accurate for stable cavities, with decreasing effectiveness as the cavity becomes unstable. We consider mainly the boundary area between the stable and unstable regimes described by neither Gaussian beams nor approximate asymptotic solutions. Numerical results in this region for circular mirrors are presented. The method may be extended to stable cavities.

© 1980 Optical Society of America

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References

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  1. See references in the following reviews: H. Kogelnik and T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312–1329, (1966);A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353 (1974).
    [Crossref] [PubMed]
  2. A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [Crossref]
  3. R. L. Sanderson and W. Streifer, “Comparison of laser mode calculations,” Appl. Opt. 8, 131–140 (1969).
    [Crossref] [PubMed]
  4. A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2736 (1970).
    [Crossref] [PubMed]
  5. D. Rensch and A. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. 12, 997–1010 (1972).
    [Crossref]
  6. W. Murphy and M. L. Bernabe, “Numerical procedures for solving nonsymmetric eigenvalue problems associated with optical resonators,” Appl. Opt. 15, 2358–2365 (1978).
    [Crossref]
  7. M. Lax, G. Agrawal, and W. Louisell, “Continuous Fourier transform spline solution of unstable resonator field distribution,” Opt. Lett. 4, 303–305 (1979).
    [Crossref] [PubMed]
  8. P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [Crossref]
  9. R. R. Butts and P. V. Avizonis, “Asymptotic analysis of unstable laser resonators with circular mirrors,” J. Opt. Soc. Am. 68, 1072–1078 (1978).
    [Crossref]
  10. G. C. Dente, “Polarization effects in resonators,” Appl. Opt. 18, 2911 (1979).
    [Crossref] [PubMed]
  11. W. H. Louisell, in Physics of Quantum Electronics, edited by S. F. Jacobs, M. O. Scully, M. Sargent, and C. D. Cantrell (Addison-Wesley, Reading, Massachusetts, 1976), Vol. 3, p. 369.
  12. P. Morse and H. Feshbach, Methods in Mathematical Physics (McGraw-Hill, New York, 1953).
  13. G. Boyd and H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
    [Crossref]
  14. L. Weinstein, Open Resonators and Waveguides (Golem, Boulder, 1969).
  15. A. E. Siegman, “Unstable Optical Resonators for Laser Applications,” Proc. IEEE 53, 277–287 (1965).
    [Crossref]
  16. W. Kahn, “Unstable optical resonators,” Appl. Opt. 4, 407–417 (1966).
    [Crossref]
  17. L. Bergstein, “Modes of stable and unstable optical resonators,” Appl. Opt. 7, 495–504 (1968).
    [Crossref] [PubMed]
  18. L. Chen and L. Felsen, “Coupled Mode Theory of Unstable Resonators,” IEEE J. Quantum Electron. QE9, 1102–1113 (1973).
    [Crossref]
  19. C. Santana and L. Felsen, “Unstable open resonators: Two dimensional and three-dimensional losses by a waveguide analysis,” Appl. Opt. 15, 1470–1478 (1976).
    [Crossref] [PubMed]
  20. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York1964).
  21. G. Watson, Theory of Bessel Functions (Cambridge University, Cambridge, 1966).
  22. G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, “Investigation of the Selective Properties of Open Unstable Cavities,” Opt. Spectrosc. 34, 437–432 (1973).
  23. L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Claredon, Oxford, 1974).
  24. L. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).
  25. S. Morgan, “On the Integral Equations of Laser Theory,” IEEE Trans. Microwave Theory Tech. 8, 191–193 (1963).
    [Crossref]
  26. D. Newman and S. Morgan, “Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory,” Bell Syst. Tech. J. 43, 113–126 (1964).
    [Crossref]
  27. J. Cochran, “The Existence of Eigenvalues for the Integral Equations in Laser Theory,” Bell Syst. Tech. J. 44, 77–88 (1965).
    [Crossref]
  28. H. Hochstadt, “Integral Equations in Laser Theory,” SAIM Rev. 8, 62–65 (1966).
    [Crossref]
  29. J. Nagel, D. Rogovin, P. Avizonis, and R. Butts, “Asymptotic Approaches to Marginally Stable Resonators,” Opt. Lett. 4, 300–302 (1979).
    [Crossref] [PubMed]
  30. The condition is M > 1 + N−1 and not M > 1 + N−1/2 as stated in Ref. 29.
  31. A. Siegman and R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE3, 156–163 (1967).
    [Crossref]
  32. R. Sanderson and W. Streifer, “Unstable laser resonator modes,” Appl. Opt. 8, 2129–2136 (1969).
    [Crossref] [PubMed]
  33. R. Butts and A. Paxton (private communication).

1979 (3)

1978 (2)

1976 (1)

1973 (3)

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, “Investigation of the Selective Properties of Open Unstable Cavities,” Opt. Spectrosc. 34, 437–432 (1973).

P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
[Crossref]

L. Chen and L. Felsen, “Coupled Mode Theory of Unstable Resonators,” IEEE J. Quantum Electron. QE9, 1102–1113 (1973).
[Crossref]

1972 (1)

1970 (1)

1969 (2)

1968 (1)

1967 (1)

A. Siegman and R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE3, 156–163 (1967).
[Crossref]

1966 (3)

H. Hochstadt, “Integral Equations in Laser Theory,” SAIM Rev. 8, 62–65 (1966).
[Crossref]

W. Kahn, “Unstable optical resonators,” Appl. Opt. 4, 407–417 (1966).
[Crossref]

See references in the following reviews: H. Kogelnik and T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312–1329, (1966);A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353 (1974).
[Crossref] [PubMed]

1965 (2)

A. E. Siegman, “Unstable Optical Resonators for Laser Applications,” Proc. IEEE 53, 277–287 (1965).
[Crossref]

J. Cochran, “The Existence of Eigenvalues for the Integral Equations in Laser Theory,” Bell Syst. Tech. J. 44, 77–88 (1965).
[Crossref]

1964 (1)

D. Newman and S. Morgan, “Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory,” Bell Syst. Tech. J. 43, 113–126 (1964).
[Crossref]

1963 (1)

S. Morgan, “On the Integral Equations of Laser Theory,” IEEE Trans. Microwave Theory Tech. 8, 191–193 (1963).
[Crossref]

1962 (1)

G. Boyd and H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
[Crossref]

1961 (1)

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York1964).

Agrawal, G.

Arrathoon, R.

A. Siegman and R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE3, 156–163 (1967).
[Crossref]

Avizonis, P.

Avizonis, P. V.

Bergstein, L.

Bernabe, M. L.

Boyd, G.

G. Boyd and H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
[Crossref]

Butts, R.

Butts, R. R.

Chen, L.

L. Chen and L. Felsen, “Coupled Mode Theory of Unstable Resonators,” IEEE J. Quantum Electron. QE9, 1102–1113 (1973).
[Crossref]

Chester, A.

Cochran, J.

J. Cochran, “The Existence of Eigenvalues for the Integral Equations in Laser Theory,” Bell Syst. Tech. J. 44, 77–88 (1965).
[Crossref]

Delves, L. M.

L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Claredon, Oxford, 1974).

Dente, G. C.

Felsen, L.

Feshbach, H.

P. Morse and H. Feshbach, Methods in Mathematical Physics (McGraw-Hill, New York, 1953).

Fox, A. G.

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[Crossref]

Hochstadt, H.

H. Hochstadt, “Integral Equations in Laser Theory,” SAIM Rev. 8, 62–65 (1966).
[Crossref]

Horwitz, P.

Kahn, W.

W. Kahn, “Unstable optical resonators,” Appl. Opt. 4, 407–417 (1966).
[Crossref]

Kogelnik, H.

See references in the following reviews: H. Kogelnik and T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312–1329, (1966);A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353 (1974).
[Crossref] [PubMed]

G. Boyd and H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
[Crossref]

Lax, M.

Li, T.

See references in the following reviews: H. Kogelnik and T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312–1329, (1966);A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353 (1974).
[Crossref] [PubMed]

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[Crossref]

Louisell, W.

Louisell, W. H.

W. H. Louisell, in Physics of Quantum Electronics, edited by S. F. Jacobs, M. O. Scully, M. Sargent, and C. D. Cantrell (Addison-Wesley, Reading, Massachusetts, 1976), Vol. 3, p. 369.

Lyubimov, V. V.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, “Investigation of the Selective Properties of Open Unstable Cavities,” Opt. Spectrosc. 34, 437–432 (1973).

Miller, H. Y.

Morgan, S.

D. Newman and S. Morgan, “Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory,” Bell Syst. Tech. J. 43, 113–126 (1964).
[Crossref]

S. Morgan, “On the Integral Equations of Laser Theory,” IEEE Trans. Microwave Theory Tech. 8, 191–193 (1963).
[Crossref]

Morse, P.

P. Morse and H. Feshbach, Methods in Mathematical Physics (McGraw-Hill, New York, 1953).

Murphy, W.

Nagel, J.

Newman, D.

D. Newman and S. Morgan, “Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory,” Bell Syst. Tech. J. 43, 113–126 (1964).
[Crossref]

Orlova, I. B.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, “Investigation of the Selective Properties of Open Unstable Cavities,” Opt. Spectrosc. 34, 437–432 (1973).

Paxton, A.

R. Butts and A. Paxton (private communication).

Rall, L.

L. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).

Rensch, D.

Rogovin, D.

Sanderson, R.

Sanderson, R. L.

Santana, C.

Siegman, A.

A. Siegman and R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE3, 156–163 (1967).
[Crossref]

Siegman, A. E.

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York1964).

Streifer, W.

Vinokurov, G. N.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, “Investigation of the Selective Properties of Open Unstable Cavities,” Opt. Spectrosc. 34, 437–432 (1973).

Walsh, J.

L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Claredon, Oxford, 1974).

Watson, G.

G. Watson, Theory of Bessel Functions (Cambridge University, Cambridge, 1966).

Weinstein, L.

L. Weinstein, Open Resonators and Waveguides (Golem, Boulder, 1969).

Appl. Opt. (9)

Bell Syst. Tech. J. (4)

D. Newman and S. Morgan, “Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory,” Bell Syst. Tech. J. 43, 113–126 (1964).
[Crossref]

J. Cochran, “The Existence of Eigenvalues for the Integral Equations in Laser Theory,” Bell Syst. Tech. J. 44, 77–88 (1965).
[Crossref]

G. Boyd and H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
[Crossref]

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[Crossref]

IEEE J. Quantum Electron. (2)

L. Chen and L. Felsen, “Coupled Mode Theory of Unstable Resonators,” IEEE J. Quantum Electron. QE9, 1102–1113 (1973).
[Crossref]

A. Siegman and R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE3, 156–163 (1967).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

S. Morgan, “On the Integral Equations of Laser Theory,” IEEE Trans. Microwave Theory Tech. 8, 191–193 (1963).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Lett. (2)

Opt. Spectrosc. (1)

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, “Investigation of the Selective Properties of Open Unstable Cavities,” Opt. Spectrosc. 34, 437–432 (1973).

Proc. IEEE (2)

A. E. Siegman, “Unstable Optical Resonators for Laser Applications,” Proc. IEEE 53, 277–287 (1965).
[Crossref]

See references in the following reviews: H. Kogelnik and T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312–1329, (1966);A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353 (1974).
[Crossref] [PubMed]

SAIM Rev. (1)

H. Hochstadt, “Integral Equations in Laser Theory,” SAIM Rev. 8, 62–65 (1966).
[Crossref]

Other (9)

The condition is M > 1 + N−1 and not M > 1 + N−1/2 as stated in Ref. 29.

L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Claredon, Oxford, 1974).

L. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).

R. Butts and A. Paxton (private communication).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York1964).

G. Watson, Theory of Bessel Functions (Cambridge University, Cambridge, 1966).

L. Weinstein, Open Resonators and Waveguides (Golem, Boulder, 1969).

W. H. Louisell, in Physics of Quantum Electronics, edited by S. F. Jacobs, M. O. Scully, M. Sargent, and C. D. Cantrell (Addison-Wesley, Reading, Massachusetts, 1976), Vol. 3, p. 369.

P. Morse and H. Feshbach, Methods in Mathematical Physics (McGraw-Hill, New York, 1953).

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Figures (12)

FIG. 1
FIG. 1

Diagram of an open resonator with curved mirrors showing the coordinates used. Symmetric mirrors are assumed, with R < 0 for the unstable case pictured.

FIG. 2
FIG. 2

Plot of the complex magnification M for stable and unstable resonators with equal mirrors. The curved portion corresponds to stable resonators, for which the core term is a good solution. The M > 1 segment represents Type I unstable, and the M < − 1 segment represents Type II unstable resonators. Point A is a Fabry-Perot resonator, B is the confocal point, and C is the concentric point. The marginally stable region for which numerical results are given is indicated by hash marks.

FIG. 3
FIG. 3

Plot of |F1(x)| for x = r/a where 0 ≤ r ≤ 2a, with N = 10 and M = 1. The small oscillations have period N−1 and amplitude ∼(πN)−1/2. The Poisson spot at x = 0 and the diffuse shadow boundary near x = 1 are evident as well as the diffraction rings immediately outside x = 1. The infinite Fresnel number limit is |F1| = 0 for x < 1, |F1| = 1 for x > 1, and |F1| = ½ for x = 1.

FIG. 4
FIG. 4

Plot of |ϕ1|2 and |ϕ0|2 vs x = r/a for the lowest loss N = 10, M = 1 mode. These are compared with the numerical results,33 which are indicated by a solid line.

FIG. 5
FIG. 5

Plot of |ϕ0|2 vs r/a for the lowest loss N = 10, M = 1 mode. Also plotted for comparison are the core term |ϕ0| and the numerical result.33

FIG. 6
FIG. 6

Phase of ϕ2(x) vs x = r/a for the lowest loss N = 10, M = 1 mode, compared with the core-term phase.

FIG. 7
FIG. 7

Losses in percent for N = 10 cavities of various magnification are plotted. The six lowest loss modes are given. The x axis is indexed by the mirror curvature given by g − 1 = −L/R. The stable region is on the left, unstable on the right. The geometric-optics losses are indicated by solid lines. Note that they are discontinuous at M = 1, and zero for stable resonators.

FIG. 8
FIG. 8

Phase shifts for the six lowest loss modes of the N = 10 cavity are plotted for various mirror curvatures. The phase shifts are given in degrees, and the x axis is the same as Fig. 7. The geometric-optics phase shifts are indicated by solid lines. Note that the geometric-optics phase shifts are discontinuous at M = 1 and zero for unstable resonators.

FIG. 9
FIG. 9

|ϕ1| and |ϕ0| for the nl = 00 loss mode of the R/L = 220, N = 10 resonator are given and compared with the Gaussian beam predictions. The Laguerre polynomials are indicated by solid lines, |ϕ1| by dotted lines, and |ϕ0| by dashed lines. The modes are labeled by n and l, where n labels the radial excitations and l the azimuthal symmetry.

FIG. 10
FIG. 10

|ϕ1| and |ϕ0| for the nl = 01 loss mode of the R/L = 220, N = 10 resonator are given and compared with the Gaussian beam predictions. The Laguerre polynomials are indicated by solid lines, |ϕ1| by dotted lines, and |ϕ0| by dashed lines. The modes are labeled by n and l, where n labels the radial excitations and l the azimuthal symmetry.

FIG. 11
FIG. 11

|ϕ| and |ϕ0| for the nl = 10 loss mode of the R/L = 220, N = 10 resonator are given and compared with the Gaussian beam predictions. The Laguerre polynomials are indicated by solid lines, |ϕ1| by dotted lines, and |ϕ0| by dashed lines. The modes are labeled by n and l where n labels the radial excitations and l the azimuthal symmetry.

FIG. 12
FIG. 12

|ϕ1| and |ϕ0| for the nl = 11 loss mode of the R/L = 220, N = 10 resonator are given and compared with the Gaussian beam predictions. The Laguerre polynomials are indicated by solid lines, |ϕ1| by dotted lines, and |ϕ0| by dashed lines. The modes are labeled by n and l, where n labels the radial excitations and l the azimuthal symmetry.

Tables (6)

Tables Icon

TABLE I M = 1 eigenvalues. The magnitudes and phases of the eigenvalues for the six lowest loss modes with N = 10, M = 1 are calculated and compared with numerical results and predictions from Weinstein. The numerical results are taken from a graph in Ref. 2. |λ|,θ = present results; |λI|,θI = numerical integration1,2; |λw|,θw = Weinstein predictions.14

Tables Icon

TABLE II Magnitudes of eigenvalues for cavities with curved mirrors. The magnitudes of the full eigenvalue γ of the six lowest loss modes for the N = 10 resonator with various marror curvatures are given. The curvature is given in terms of g − 1 = −L/R. The stable region corresponds to g − 1 = 0.

Tables Icon

TABLE III Phases of eigenvalues for cavities with curved mirrors. The phases in degrees of the eigenvalue γ for the modes in Table II are given.

Tables Icon

TABLE IV s parameter for marginally stable cavities. The magnitude of the complex parameter |s| is given for each of the modes in Table II. The geometric-optics result is |s| = 2n + l, and is approximated by the stable cavities. Near M = 1, |s| becomes large as the core term approaches a Bessel function.

Tables Icon

TABLE V Magnitudes of eigenvalues for M = 1.02 cavities. The magnitudes of the full eigenvalue γ for six low order modes are given for various Fresnel numbers.

Tables Icon

TABLE VI Phases of eigenvalues for M = 1.02 cavities. The phase shifts for the modes in Table V are given.

Equations (67)

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E ( r , z ) = l ( α r l ( r , z ) e ikz + α 1 l ( r , z ) e ikz ) e i l θ .
[ 2 ± 2 i k d d z ] α r , 1 l ( r , z ) = 0 .
α 1 l ( r , z ) = i l + 1 k z 0 a r exp [ i k ( r 2 + r 2 ) 2 z ] × J l ( k r r z ) α 1 l ( r , 0 ) d r ,
( 1 ) q + 1 α 1 l ( r , 0 ) e ikL e i k r 2 / R r = i l + 1 k L 0 a r exp [ i k ( r 2 + r 2 ) / 2 L ] × J l ( k r r / L ) α 1 l ( r , 0 ) d r .
γ = e i [ k L ( q + 1 ) π ] ,
f ( r ) = α 1 l ( r , 0 ) e i k r 2 / 2 R ,
g = 1 L / R ,
γ f ( r ) = i l + 1 k / L 0 a r exp [ ikg ( r 2 + r 2 ) / 2 L ] × J l ( k r r / L ) f ( r ) d r ,
γ λ / M
f ( x ) exp ( i π N e q x 2 ) ϕ ( x ) ,
λ ϕ ( x ) = 2 i l + 1 π N M 0 1 ydy exp [ i π N ( M y 2 + x 2 / M ) ] × J l ( 2 π Nxy ) ϕ ( y ) .
λ ϕ ( k ) = M 2 exp ( i M k 2 / 4 π N ) ϕ ( k M ) .
ϕ ( k ) = 1 k s + 2 exp { M k 2 / [ 4 π N ( M 2 1 ) ] } ,
λ = ( 1 / M ) s ,
ϕ l s ( x ) = Γ ( ½ l + ½ s + 1 ) i l / 2 s / 2 η l / 2 s / 2 x l × F 1 ( ½ l ½ s , l + 1 , i η x 2 ) ,
λ = M s ,
η = π N ( M 2 1 ) M ,
ϕ l s ( x ) = x s [ j = 0 ( ½ l ½ s ) j ( ½ l ½ s ) j j ! ( 1 i η x 2 ) j + Γ ( ½ l + ½ s + 1 ) e i η x 2 Γ ( ½ l ½ s ) ( i η x 2 ) s + 1 e i π ( l / 2 s / 2 ) j = 0 ( ½ l + ½ s + 1 ) j ( 1 ½ l + ½ s ) j j ! ( 1 i η x 2 ) j ] ,
( α ) j Γ ( α + j ) Γ ( α ) .
s = i α 2 / 2 η ,
ϕ l α ( x ) = k = 0 c k x k J l + k ( α x ) ,
λ α = exp [ iln ( M ) α 2 M / 2 π N ( M 2 1 ) ] .
c 0 = 1 ,
c 1 = l η / α ,
c k + 1 = η [ ( l + 2 k ) c k / α c k 1 ] / ( k + 1 ) .
α 1 l ( r , 0 ) = g ( L ) e + i k r 2 / 2 R e i η x 2 / 2 ( η x 2 ) l / 2 × F 1 ( ½ l ½ s , l + 1 , i η x 2 ) ,
η = k a 2 g 2 1 / L .
exp [ i η x 2 / 2 ] exp [ k r 2 L ( 2 | R | L ) / 2 | R | L ] .
k L 2 L R + L 2 R 2 i k L 2 L | R | L 2 | R | 2 .
ω 2 ( z = 0 ) = 2 | R | L / k L ( 2 | R | L ) ,
R ( z = 0 ) = | R | .
α 1 l ( r , 0 ) = g ( L ) e i k r 2 / 2 R ( 0 ) e r 2 / ω 2 ( 0 ) × [ 2 r / ω ( 0 ) ] l L n l [ 2 r 2 / ω 2 ( 0 ) ] .
ϕ M = ϕ M * ,
λ M = ( 1 ) l λ M * ,
ϕ n + 1 = λ 1 K [ ϕ n ] ,
K [ f ] K [ g ] κ f g ,
ϕ n + 1 ϕ n ( κ / λ ) n ϕ 1 ϕ 0 .
K 2 ( s , t ) dsdt 0 ,
ϕ 0 = c 0 + i = 1 n c i F i ( x ) ,
ϕ 1 ϕ 0 | ϕ 1 ( 1 ) ϕ 0 ( 1 ) | ,
0 1 0 1 .
K 1 [ ϕ 0 ] = i l exp [ i π N M ( 1 + x 2 / M 2 ) ] υ , k , s c υ ( i x M ) k ( i α 2 π N ) s × ( s + k s ) R s + k l + υ ( i π N M ) J l + k ( 2 π N x ) J l + υ + s ( α )
R j i ( x ) = k = 0 i ( i k ) ( j + k ) ! j ! ( 1 x ) k .
ϕ 1 ϕ 0 = λ 0 1 K 1 [ ϕ 0 ] .
E ( α ) = 0 1 x l + 1 d x [ ϕ 1 ( x ) ϕ 0 ( x ) ] = 0 .
E ( α ) = i l exp [ i π N M ( 1 + 1 / M 2 ) ] 2 π N υ , k , s , j c υ ( i M ) k + j × ( i α 2 π N ) s ( s + k s ) R s + k l + υ ( i π N M ) J l + k + j + 1 ( 2 π N ) J l + υ + s ( α ) .
K 1 [ f ] = 2 i l + 1 π N M 1 ydy exp [ i π N ( M y 2 + x 2 / M ) ] × J l ( 2 π Nxy ) f ( y )
I k = 1 ydy exp [ i π N M y 2 ] J l ( 2 π Nxy ) y k J l + k ( α y ) .
J l ( α y ) = y l r = 0 ( 1 y 2 ) r ( α / 2 ) r r ! J l + r ( α )
I k = ½ r , p 1 d u e i π NMu u l + k ( 1 u ) r + p ( π N x ) r ( α / 2 ) p r ! p ! × J l + r ( 2 π N x ) J l + k + p ( α ) .
1 d u u l + k ( 1 u ) r + p e i π NMu = ( i π N M ) r p 1 j = 0 l + k ( l + k j ) ( r + p + j ) ! ( π N M ) j .
K 1 [ ϕ 0 ] = K [ ϕ 0 ] K 0 [ ϕ 0 ] ,
I k = 0 1 ydy e i π N M y 2 J l ( 2 π Nxy ) y k J l + k ( α y ) .
I k = e i π N M r = 0 0 π / 2 ( sin θ ) l + 1 cos θ d θ J l ( 2 π N x sin θ ) ( sin θ ) 2 k × e i π N M cos 2 θ cos 2 r θ r ! ( α / 2 ) r J l + k + r ( α ) .
( 1 u ) k e x u = p = 0 ( x u ) p p ! a k , p ( x ) ,
a k , p ( x ) = j = 0 Min ( k , p ) ( k j ) ( p j ) ( 1 ) j j ! x r ,
I k = e i π N M ( 2 π N ) r , p ( r + p r ) a k , p ( i π N M ) ( i x / M ) p × ( α 2 π N x ) r J l + r + p + 1 ( 2 π N x ) J l + k + r ( α ) .
K 1 [ ϕ 0 ] = i l + 1 ( x / M ) e i π N M ( 1 + x 2 / M 2 ) k , r , p c k ( i x / M ) p ( α 2 π N x ) r × ( r + p r ) a k , p ( i π N M ) J l + r + p + 1 ( 2 π N x ) J l + k + r ( α ) λ 0 ϕ 0 ( x ) ,
ϕ 0 ( y ) = ( i η ) l / 2 s / 2 y l i l / 2 s / 2 Γ ( ½ l ½ s ) × 0 1 u l / 2 s / 2 1 ( 1 u ) l / 2 + s / 2 e i η y 2 u d u .
e i η y 2 u = p ( y 2 1 ) p ( i η u ) p p ! e inu .
K 1 [ ϕ 0 ] = ( i η ) l / 2 s / 2 ( 2 i l + 1 π N M ) i l / 2 s / 2 Γ ( ½ l ½ s ) × e i π N x 2 / M r , p = 0 0 1 d u u l / 2 s / 2 1 × ( 1 u ) l / 2 + s / 2 ( i η u ) p p ! e i η u 1 y 2 l + 1 d y × e i π N M y 2 ( y 2 1 ) r + p r ! ( π N x ) r J l + r ( 2 π N x ) ,
0 1 d u u l / 2 s / 2 + p 1 ( 1 u ) l / 2 + s / 2 e i η u ,
K 1 [ ϕ l s ( x ) ] = i l e i π N M ( 1 + x 2 / M 2 ) p , r ( i x M ) r ( i π N ) p × ( r + p r ) R r + p l ( i π N M ) J l + r ( 2 π N x ) ( ½ l ½ s ) r ϕ l + p s p ( 1 ) ,
K 1 2 [ J l ( α x ) ] = 2 i l + 1 π N e i π N x 2 1 ydy J l ( 2 π Nxy ) × e i π N y 2 i l + 1 y e i π N ( 1 + y 2 ) r , p ( i y ) p ( α 2 π N y ) r ( r + p r ) × J l + r + p + 1 ( 2 π N x ) J l + r ( α ) λ 0 K 1 [ J l ] .
K 1 2 [ J l ( α x ) ] = ( 1 ) l + 1 i 2 e i π N ( 3 + x 2 ) × k , s , r , p ( i ) k ( α 2 π N ) s ( i 2 ) p ( i x 2 ) r ( s + k s ) ( r + p r ) × R r + p l ( 2 i π N ) J l + r ( 2 π N x ) J l + r + s + p + 1 ( 2 π N ) J l + s ( α ) λ o K 1 [ J l ( α x ) ] .
I k = 0 1 x l + k + 1 exp ( i π N x 2 / M ) J l + k ( 2 π N x ) .
I k = 1 2 π N e i π N / M j = 0 ( i M ) j J l + k + j + 1 ( 2 π N ) .