Abstract

We describe a numerical algorithm for the evaluation of the electromagnetic-field distribution in a loaded unstable resonator. The storage requirements are minimized so that the resulting code can be used for large Fresnel numbers. Edge diffraction is accounted for by a recently developed continuous Fourier-transform algorithm. Use is made of a new gain formula that incorporates the effects of interference between the forward and backward waves. The present method yields improved accuracy over previous methods and enables one to perform calculations for systems with large Fresnel numbers on a medium-sized computer. Numerical results are presented for a loaded confocal unstable resonator to study the effect of the saturated gain on the mode profile. An important conclusion is that the saturated gain does not alter the number of peaks and their relative positions in the intensity distribution. This supports the simplified view that these features arise from edge diffraction and that the saturated gain amplifies each peak by a different amount depending on the peak intensities.

© 1985 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  22. V. S. Rogov, M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 7, 18–21 (1977).
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    [PubMed]
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    [CrossRef]
  25. W. H. Louisell, M. Lax, G. P. Agrawal, H. W. Gatzke, “Simultaneous forward and backward integration for standing waves in a resonator,” Appl. Opt. 18, 2730–2731 (1979).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  34. M. Lax, G. P. Agrawal, “Evaluation of Fourier integrals using B-splines,” Math. Comput. 39, 535–548 (1982).
  35. B. Coffey, M. Lax, “Two efficient continuous Fourier transform algorithms for unstable resonator simulation,” Appl. Opt. (to be published).
  36. M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 8.
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    [CrossRef]
  38. D. O. Riska, S. Stenholm, “The influence of the mode structure on the quantum theory of the laser,” Phys. Lett. 30A, 16–17 (1969).
  39. H. J. Carmichael, “The mean-field approximation and validity of a truncated Bloch hierarchy in an absorptive bistability,” Opt. Acta 27, 147–158 (1980).
    [CrossRef]
  40. G. P. Agrawal, H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta 27, 651–660 (1980).
    [CrossRef]
  41. H. J. Carmichael, J. A. Hermann, “Analytic description of optical bistability including spatial effects,”Z. Phys. B 38, 365–380 (1980).
    [CrossRef]
  42. G. P. Agrawal, M. Lax, “Analytic evaluation of interference effects on laser output in a Fabry–Perot resonator,”J. Opt. Soc. Am. 71, 515–519 (1981).
    [CrossRef]
  43. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976), App. A.
    [CrossRef]
  44. M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
    [CrossRef]
  45. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  46. L. R. Rabiner, C. M. Radar, eds., Digital Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1972).
  47. M. J. Marsden, G. D. Taylor, “Numerical evaluation of Fourier integrals,” in Numerische Methoden der Approximations Theorie (Birkhauser, Basel, 1972), Band 16, pp. 61–76.
  48. B. Einarsson, “Numerical calculation of Fourier integrals with cubic splines,” Nord. Tidsskr. Informationsbehand. 8, 279–286 (1968).
  49. B. K. Swartz, R. S. Varga, “Error bounds for spline and L-spline interpolations,”J. Approx. Theory 6, 6–49 (1972).
    [CrossRef]

1984 (1)

1983 (1)

1982 (2)

W. P. Latham, M. E. Smithers, “Diffractive effect of a scraper in an unstable resonator,”J. Opt. Soc. Am. 72, 1321–1327 (1982).
[CrossRef]

M. Lax, G. P. Agrawal, “Evaluation of Fourier integrals using B-splines,” Math. Comput. 39, 535–548 (1982).

1981 (4)

1980 (3)

H. J. Carmichael, “The mean-field approximation and validity of a truncated Bloch hierarchy in an absorptive bistability,” Opt. Acta 27, 147–158 (1980).
[CrossRef]

G. P. Agrawal, H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta 27, 651–660 (1980).
[CrossRef]

H. J. Carmichael, J. A. Hermann, “Analytic description of optical bistability including spatial effects,”Z. Phys. B 38, 365–380 (1980).
[CrossRef]

1979 (6)

1978 (3)

1977 (5)

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, A. M. Shalagin, “Shaping of the output beam in a pulsed gas laser with an unstable resonator,” Sov. J. Quantum Electron. 7, 746–752 (1977).
[CrossRef]

V. S. Rogov, M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 7, 18–21 (1977).
[CrossRef]

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]

C. Santana, L. B. Felson, “Effects of medium and gain inhomogeneities in unstable optical resonators,” Appl. Opt. 16, 1058–1066 (1977).
[PubMed]

P. W. Milonni, “Criteria for the thin-sheet gain approximation,” Appl. Opt. 16, 2794–2795 (1977).
[CrossRef] [PubMed]

1976 (4)

1975 (3)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Yu. N. Karamzin, Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
[CrossRef]

E. A. Sziklas, A. E. Siegman, “Mode calculation in unstable resonators with flowing saturable gain. II: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[CrossRef] [PubMed]

1974 (5)

A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974); we refer to this paper for a review and for an extensive bibliography of the work on unstable resonators before 1974.
[CrossRef] [PubMed]

A. E. Siegman, E. A. Sziklas, “Mode calculation in unstable resonators with flowing saturable gain. I: Hermite–Gaussian expansion,” Appl. Opt. 13, 2775–2792 (1974).
[CrossRef] [PubMed]

I. M. Bel’dyugin, E. M. Zemskov, A. Kh. Mamyan, V. N. Seminogov, “Theory of open resonators with cylindrical mirrors,” Sov. J. Quantum Electron. 4, 484–490 (1974).

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, “Converging beams in unstable telescopic resonators,” Sov. J. Quantum Electron. 4, 761–766 (1974).
[CrossRef]

Yu. A. Anan’ev, L. V. Koval’chuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

1973 (3)

1972 (1)

B. K. Swartz, R. S. Varga, “Error bounds for spline and L-spline interpolations,”J. Approx. Theory 6, 6–49 (1972).
[CrossRef]

1969 (1)

D. O. Riska, S. Stenholm, “The influence of the mode structure on the quantum theory of the laser,” Phys. Lett. 30A, 16–17 (1969).

1968 (1)

B. Einarsson, “Numerical calculation of Fourier integrals with cubic splines,” Nord. Tidsskr. Informationsbehand. 8, 279–286 (1968).

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Agrawal, G. P.

Anan’ev, Yu. A.

Yu. A. Anan’ev, L. V. Koval’chuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Avizonis, P. V.

Batteh, J. H.

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
[CrossRef]

Bel’dyugin, I. M.

I. M. Bel’dyugin, E. M. Zemskov, A. Kh. Mamyan, V. N. Seminogov, “Theory of open resonators with cylindrical mirrors,” Sov. J. Quantum Electron. 4, 484–490 (1974).

Butts, R. R.

Carmichael, H. J.

G. P. Agrawal, H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta 27, 651–660 (1980).
[CrossRef]

H. J. Carmichael, “The mean-field approximation and validity of a truncated Bloch hierarchy in an absorptive bistability,” Opt. Acta 27, 147–158 (1980).
[CrossRef]

H. J. Carmichael, J. A. Hermann, “Analytic description of optical bistability including spatial effects,”Z. Phys. B 38, 365–380 (1980).
[CrossRef]

Chen, L. W.

L. W. Chen, L. B. Felson, “Coupled-mode-theory of unstable resonators,” IEEE J. Quantum Electron. QE-9, 1102–1113 (1973).
[CrossRef]

Chester, A. N.

Cho, S. H.

Chodzko, R.

Coffey, B.

B. Coffey, M. Lax, “Two efficient continuous Fourier transform algorithms for unstable resonator simulation,” Appl. Opt. (to be published).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Craig, J. M.

Cross, E. F.

Dewhurst, R. J.

R. J. Dewhurst, D. Jacoby, “A mode-locked unstable Nd:YAG laser,” Opt. Commun. 28, 107–110 (1979).
[CrossRef]

Einarsson, B.

B. Einarsson, “Numerical calculation of Fourier integrals with cubic splines,” Nord. Tidsskr. Informationsbehand. 8, 279–286 (1968).

Ewanizky, T. F.

Feit, M. D.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976), App. A.
[CrossRef]

Felson, L. B.

Ferguson, T. R.

Fleck, J. A.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976), App. A.
[CrossRef]

Gatzke, H. W.

Hermann, J. A.

H. J. Carmichael, J. A. Hermann, “Analytic description of optical bistability including spatial effects,”Z. Phys. B 38, 365–380 (1980).
[CrossRef]

Horwitz, P.

Isaev, A. A.

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, A. M. Shalagin, “Shaping of the output beam in a pulsed gas laser with an unstable resonator,” Sov. J. Quantum Electron. 7, 746–752 (1977).
[CrossRef]

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, “Converging beams in unstable telescopic resonators,” Sov. J. Quantum Electron. 4, 761–766 (1974).
[CrossRef]

Jacoby, D.

R. J. Dewhurst, D. Jacoby, “A mode-locked unstable Nd:YAG laser,” Opt. Commun. 28, 107–110 (1979).
[CrossRef]

Karamzin, Yu. N.

Yu. N. Karamzin, Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
[CrossRef]

Kazaryan, M. A.

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, A. M. Shalagin, “Shaping of the output beam in a pulsed gas laser with an unstable resonator,” Sov. J. Quantum Electron. 7, 746–752 (1977).
[CrossRef]

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, “Converging beams in unstable telescopic resonators,” Sov. J. Quantum Electron. 4, 761–766 (1974).
[CrossRef]

Konev, Yu. B.

Yu. N. Karamzin, Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
[CrossRef]

Koval’chuk, L. V.

Yu. A. Anan’ev, L. V. Koval’chuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Kreuzer, J. L.

Lamb, W. E.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 8.

Latham, W. P.

Lax, M.

M. Lax, G. P. Agrawal, “Evaluation of Fourier integrals using B-splines,” Math. Comput. 39, 535–548 (1982).

G. P. Agrawal, M. Lax, “Analytic evaluation of interference effects on laser output in a Fabry–Perot resonator,”J. Opt. Soc. Am. 71, 515–519 (1981).
[CrossRef]

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
[CrossRef]

W. H. Louisell, M. Lax, G. P. Agrawal, H. W. Gatzke, “Simultaneous forward and backward integration for standing waves in a resonator,” Appl. Opt. 18, 2730–2731 (1979).
[CrossRef] [PubMed]

G. P. Agrawal, M. Lax, “Effects of interference on gain saturation in laser resonators,”J. Opt. Soc. Am. 69, 1717–1719 (1979).
[CrossRef]

M. Lax, G. P. Agrawal, W. H. Louisell, “Continuous Fourier-transform spline solution of unstable resonator field distribution,” Opt. Lett. 9, 303–305 (1979).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

B. Coffey, M. Lax, “Two efficient continuous Fourier transform algorithms for unstable resonator simulation,” Appl. Opt. (to be published).

Louisell, W. H.

Mamyan, A. Kh.

I. M. Bel’dyugin, E. M. Zemskov, A. Kh. Mamyan, V. N. Seminogov, “Theory of open resonators with cylindrical mirrors,” Sov. J. Quantum Electron. 4, 484–490 (1974).

Marsden, M. J.

M. J. Marsden, G. D. Taylor, “Numerical evaluation of Fourier integrals,” in Numerische Methoden der Approximations Theorie (Birkhauser, Basel, 1972), Band 16, pp. 61–76.

Mason, S. B.

McCarthy, R. J.

McCullough, A. W.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Milonni, P. W.

P. W. Milonni, A. H. Paxton, “Model for the unstable resonator carbon monoxide electric-discharge laser,” J. Appl. Phys. 42, 1012–1026 (1978).
[CrossRef]

P. W. Milonni, “Criteria for the thin-sheet gain approximation,” Appl. Opt. 16, 2794–2795 (1977).
[CrossRef] [PubMed]

Moore, G. T.

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976), App. A.
[CrossRef]

Mumola, P. B.

Oughstun, K. E.

Paxton, A. H.

P. W. Milonni, A. H. Paxton, “Model for the unstable resonator carbon monoxide electric-discharge laser,” J. Appl. Phys. 42, 1012–1026 (1978).
[CrossRef]

Petrash, G. G.

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, A. M. Shalagin, “Shaping of the output beam in a pulsed gas laser with an unstable resonator,” Sov. J. Quantum Electron. 7, 746–752 (1977).
[CrossRef]

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, “Converging beams in unstable telescopic resonators,” Sov. J. Quantum Electron. 4, 761–766 (1974).
[CrossRef]

Rautian, S. G.

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, A. M. Shalagin, “Shaping of the output beam in a pulsed gas laser with an unstable resonator,” Sov. J. Quantum Electron. 7, 746–752 (1977).
[CrossRef]

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, “Converging beams in unstable telescopic resonators,” Sov. J. Quantum Electron. 4, 761–766 (1974).
[CrossRef]

Rensch, D. B.

Rikenglaz, M. M.

V. S. Rogov, M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 7, 18–21 (1977).
[CrossRef]

Riska, D. O.

D. O. Riska, S. Stenholm, “The influence of the mode structure on the quantum theory of the laser,” Phys. Lett. 30A, 16–17 (1969).

Robertson, H. J.

Rogov, V. S.

V. S. Rogov, M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 7, 18–21 (1977).
[CrossRef]

Santana, C.

Sargent, M.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 8.

Scully, M. O.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 8.

Seminogov, V. N.

I. M. Bel’dyugin, E. M. Zemskov, A. Kh. Mamyan, V. N. Seminogov, “Theory of open resonators with cylindrical mirrors,” Sov. J. Quantum Electron. 4, 484–490 (1974).

Shalagin, A. M.

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, A. M. Shalagin, “Shaping of the output beam in a pulsed gas laser with an unstable resonator,” Sov. J. Quantum Electron. 7, 746–752 (1977).
[CrossRef]

Sherstobitov, V. E.

Yu. A. Anan’ev, L. V. Koval’chuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

Smith, M. J.

Smithers, M. E.

Steier, W. H.

W. H. Steier, “Unstable Resonators,” in Laser Handbook, M. L. Stitch, ed. (North-Holland, Amsterdam, 1979), Vol 3, pp. 3–39.

Steinberg, G. N.

Stenholm, S.

D. O. Riska, S. Stenholm, “The influence of the mode structure on the quantum theory of the laser,” Phys. Lett. 30A, 16–17 (1969).

Swartz, B. K.

B. K. Swartz, R. S. Varga, “Error bounds for spline and L-spline interpolations,”J. Approx. Theory 6, 6–49 (1972).
[CrossRef]

Sziklas, E. A.

Taylor, G. D.

M. J. Marsden, G. D. Taylor, “Numerical evaluation of Fourier integrals,” in Numerische Methoden der Approximations Theorie (Birkhauser, Basel, 1972), Band 16, pp. 61–76.

Trusov, V. P.

Yu. A. Anan’ev, L. V. Koval’chuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Varga, R. S.

B. K. Swartz, R. S. Varga, “Error bounds for spline and L-spline interpolations,”J. Approx. Theory 6, 6–49 (1972).
[CrossRef]

Zemskov, E. M.

I. M. Bel’dyugin, E. M. Zemskov, A. Kh. Mamyan, V. N. Seminogov, “Theory of open resonators with cylindrical mirrors,” Sov. J. Quantum Electron. 4, 484–490 (1974).

Appl. Opt. (13)

D. B. Rensch, A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. 12, 997–1010 (1973).
[CrossRef] [PubMed]

A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974); we refer to this paper for a review and for an extensive bibliography of the work on unstable resonators before 1974.
[CrossRef] [PubMed]

A. E. Siegman, E. A. Sziklas, “Mode calculation in unstable resonators with flowing saturable gain. I: Hermite–Gaussian expansion,” Appl. Opt. 13, 2775–2792 (1974).
[CrossRef] [PubMed]

E. A. Sziklas, A. E. Siegman, “Mode calculation in unstable resonators with flowing saturable gain. II: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[CrossRef] [PubMed]

T. F. Ewanizky, J. M. Craig, “Negative-branch unstable resonator Nd:YAG laser,” Appl. Opt. 15, 1465–1469 (1976).
[CrossRef] [PubMed]

C. Santana, L. B. Felson, “Unstable open resonators: two-dimensional and three-dimensional losses by a waveguide analysis,” Appl. Opt. 15, 1470–1478 (1976).
[CrossRef] [PubMed]

R. Chodzko, S. B. Mason, E. F. Cross, “Annular converging wave cavity,” Appl. Opt. 15, 2137–2144 (1976).
[CrossRef] [PubMed]

C. Santana, L. B. Felson, “Effects of medium and gain inhomogeneities in unstable optical resonators,” Appl. Opt. 16, 1058–1066 (1977).
[PubMed]

P. B. Mumola, H. J. Robertson, G. N. Steinberg, J. L. Kreuzer, A. W. McCullough, “Unstable resonator for annular gain volume lasers,” Appl. Opt. 17, 936–943 (1978).
[CrossRef] [PubMed]

M. J. Smith, “Mode properties of strip confocal unstable resonators with saturable gain,” Appl. Opt. 20, 1611–1620 (1981).
[CrossRef] [PubMed]

M. E. Smithers, T. R. Ferguson, “Unstable optical resonators with linear magnification,” Appl. Opt. 23, 3718–3724 (1984).
[CrossRef] [PubMed]

P. W. Milonni, “Criteria for the thin-sheet gain approximation,” Appl. Opt. 16, 2794–2795 (1977).
[CrossRef] [PubMed]

W. H. Louisell, M. Lax, G. P. Agrawal, H. W. Gatzke, “Simultaneous forward and backward integration for standing waves in a resonator,” Appl. Opt. 18, 2730–2731 (1979).
[CrossRef] [PubMed]

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976), App. A.
[CrossRef]

IEEE J. Quantum Electron. (1)

L. W. Chen, L. B. Felson, “Coupled-mode-theory of unstable resonators,” IEEE J. Quantum Electron. QE-9, 1102–1113 (1973).
[CrossRef]

J. Appl. Phys. (2)

P. W. Milonni, A. H. Paxton, “Model for the unstable resonator carbon monoxide electric-discharge laser,” J. Appl. Phys. 42, 1012–1026 (1978).
[CrossRef]

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
[CrossRef]

J. Approx. Theory (1)

B. K. Swartz, R. S. Varga, “Error bounds for spline and L-spline interpolations,”J. Approx. Theory 6, 6–49 (1972).
[CrossRef]

J. Opt. Soc. Am. (10)

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

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]

R. R. Butts, P. V. Avizonis, “Asymptotic analysis of unstable laser resonators with circular mirrors,”J. Opt. Soc. Am. 68, 1072–1078 (1978).
[CrossRef]

G. P. Agrawal, M. Lax, “Analytic evaluation of interference effects on laser output in a Fabry–Perot resonator,”J. Opt. Soc. Am. 71, 515–519 (1981).
[CrossRef]

K. E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations. I: Analysis,”J. Opt. Soc. Am. 71, 862–872 (1981); Part II, J. Opt. Soc. Am. 71, 1180–1192 (1981).
[CrossRef]

W. P. Latham, M. E. Smithers, “Diffractive effect of a scraper in an unstable resonator,”J. Opt. Soc. Am. 72, 1321–1327 (1982).
[CrossRef]

K. E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations. III: Passive and active cavity study for a large Neqresonator,”J. Opt. Soc. Am. 73, 282–302 (1983).
[CrossRef]

G. P. Agrawal, M. Lax, “Effects of interference on gain saturation in laser resonators,”J. Opt. Soc. Am. 69, 1717–1719 (1979).
[CrossRef]

S. H. Cho, L. B. Felson, “Ray-optical analysis of unstable resonators with spherical mirrors,”J. Opt. Soc. Am. 69, 1377–1384 (1979).
[CrossRef]

S. H. Cho, S. Y. Shin, L. B. Felson, “Ray-optical analysis of cylindrical unstable resonators,”J. Opt. Soc. Am. 69, 563–574 (1979).
[CrossRef]

Math. Comput. (2)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

M. Lax, G. P. Agrawal, “Evaluation of Fourier integrals using B-splines,” Math. Comput. 39, 535–548 (1982).

Nord. Tidsskr. Informationsbehand. (1)

B. Einarsson, “Numerical calculation of Fourier integrals with cubic splines,” Nord. Tidsskr. Informationsbehand. 8, 279–286 (1968).

Opt. Acta (2)

H. J. Carmichael, “The mean-field approximation and validity of a truncated Bloch hierarchy in an absorptive bistability,” Opt. Acta 27, 147–158 (1980).
[CrossRef]

G. P. Agrawal, H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta 27, 651–660 (1980).
[CrossRef]

Opt. Commun. (1)

R. J. Dewhurst, D. Jacoby, “A mode-locked unstable Nd:YAG laser,” Opt. Commun. 28, 107–110 (1979).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. (1)

D. O. Riska, S. Stenholm, “The influence of the mode structure on the quantum theory of the laser,” Phys. Lett. 30A, 16–17 (1969).

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Sov. J. Quantum Electron. (6)

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, A. M. Shalagin, “Shaping of the output beam in a pulsed gas laser with an unstable resonator,” Sov. J. Quantum Electron. 7, 746–752 (1977).
[CrossRef]

I. M. Bel’dyugin, E. M. Zemskov, A. Kh. Mamyan, V. N. Seminogov, “Theory of open resonators with cylindrical mirrors,” Sov. J. Quantum Electron. 4, 484–490 (1974).

A. A. Isaev, M. A. Kazaryan, G. G. Petrash, S. G. Rautian, “Converging beams in unstable telescopic resonators,” Sov. J. Quantum Electron. 4, 761–766 (1974).
[CrossRef]

Yu. A. Anan’ev, L. V. Koval’chuk, V. P. Trusov, V. E. Sherstobitov, “Method for calculating the efficiency of lasers with unstable resonators,” Sov. J. Quantum Electron. 4, 659–664 (1974).
[CrossRef]

Yu. N. Karamzin, Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
[CrossRef]

V. S. Rogov, M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 7, 18–21 (1977).
[CrossRef]

Z. Phys. B (1)

H. J. Carmichael, J. A. Hermann, “Analytic description of optical bistability including spatial effects,”Z. Phys. B 38, 365–380 (1980).
[CrossRef]

Other (5)

L. R. Rabiner, C. M. Radar, eds., Digital Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1972).

M. J. Marsden, G. D. Taylor, “Numerical evaluation of Fourier integrals,” in Numerische Methoden der Approximations Theorie (Birkhauser, Basel, 1972), Band 16, pp. 61–76.

W. H. Steier, “Unstable Resonators,” in Laser Handbook, M. L. Stitch, ed. (North-Holland, Amsterdam, 1979), Vol 3, pp. 3–39.

B. Coffey, M. Lax, “Two efficient continuous Fourier transform algorithms for unstable resonator simulation,” Appl. Opt. (to be published).

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 8.

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Figures (8)

Fig. 1
Fig. 1

Illustrating the coordinates and geometry of a positive-branch confocal unstable resonator. A cross-section view of the y = 0 plane is shown. Two mirrors at z = 0 and z = d are square-shaped spherical mirrors with radii of curvature b1 and b2, respectively. The gain medium between planes z = d1 and z = d2 is divided into a number of equispaced segments, and the gain sheets are situated midway on each segment. In the geometrical-optics approximation the light beam expands to a width of Ma, where M is the round-trip magnification. The light that spills over the mirror edges at z = 0 is the resonator output.

Fig. 2
Fig. 2

Normalized intensity distribution at z = 0 of the lowest-loss mode of a strip bare confocal unstable resonator with the Fresnel number F = 5 and round-trip magnification M = 2.25.

Fig. 3
Fig. 3

Normalized intensity profile of a strip bare confocal unstable resonator with F = 50. The other parameters are identical to those of Fig. 2. To resolve the rapid transverse variations, (b) shows an enlarged view covering only the small-mirror region xa.

Fig. 4
Fig. 4

Comparison of the intensity profiles obtained for a strip bare unstable resonator with F = 10 using CFT (solid curve) and FFT (dotted curve) algorithms.

Fig. 5
Fig. 5

Comparison of the phase profiles obtained for a strip bare unstable resonator with F = 10 using CFT and FFT algorithms. The rapid oscillations in phase beyond x = Ma merely reflect the fact that the phase is plotted modulo 2π.

Fig. 6
Fig. 6

Illustrating numerical instability in a loaded unstable resonator arising from the amplification of wing fluctuations. The intensity profile is shown (a) with uniform small-signal gain and (b) with the Fermi profile using Eq. (4.3) with x0 = 4a. In both cases g0d = 10 and the intensity is normalized to the saturation intensity.

Fig. 7
Fig. 7

Intensity profiles in a loaded unstable resonator with (solid curve) and without (dashed curve) inclusion of the standing-wave effects. The gain formulas, Eqs. (2.10) and (2.12), respectively, are used.

Fig. 8
Fig. 8

Same as in Fig. 7 except that the off-resonance case is considered. Note that the standing-wave effects are more pronounced.

Equations (71)

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E ( r , t ) = Re [ n ^ E ( r ) e - i ω t ] ,
( 2 + k 2 ) E ( r ) = i k g ( E 2 ) E ( r ) ,
g ( E 2 ) = g 0 ( 1 + i Ω ) 1 + Ω 2 + E ( r ) 2 / E s 2 ,
g 0 = N a p 2 D w ω κ L γ a b 0 c ;             D w = λ a γ a - λ b γ b ,
E ( r ) = E s odd n ψ n ( r ) e i n k z ,
n e i n k z [ k 2 ( 1 - n 2 ) ψ n + 2 i n k ψ n z + 2 ψ n z 2 + T 2 ψ n ] = i k g 0 ( 1 + i Ω ) m ψ m e i k m z ( 1 + Ω 2 + | p ψ p e i p k z | 2 ) - 1 ,
ψ n / ψ 1 ( g 0 / k ) ,             n 1.
( 2 i k z + T 2 ) ψ R = i k g R ψ R ,
( - 2 i k z + T 2 ) ψ L = i k g L ψ L ,
g μ = g 0 ( 1 + i Ω ) ( a 2 - b 2 ) 1 / 2 [ 1 - a - ( a 2 - b 2 ) 1 / 2 2 ψ μ 2 ]             ( μ = R , L ) ,
a = 1 + Ω 2 + ψ R 2 + ψ L 2 ,             b = 2 ψ R ψ L .
g R = g L = g 0 ( 1 + i Ω ) 1 + Ω 2 + ψ R 2 + ψ L 2 .
g R = g L = g 0 I T [ 1 - ( 1 + 2 I T ) - 1 / 2 ] .
ψ R z = [ A + B ] ψ R ,
A = i 2 k T 2 ,             B = 1 2 g R .
ψ R ( z ) = U ( z ) ψ R ( 0 ) ,             U ( z ) = exp [ ( A + B ) z ] ,
U ( z ) e A z e B z .
U ( z ) = exp ( A z + B z ) = exp ( ½ A z ) exp ( C ) exp ( ½ A z ) ,
C = B z - z 3 24 [ A , [ B , A ] ] - z 3 12 [ B , [ B , A ] ] + .
ψ R ( z ) exp ( ½ A z ) exp [ 0 z B ( z ) d z ] exp ( ½ A z ) ψ R ( 0 ) .
= [ A , B / z ] z 2 4 B = h 8 k B [ T 2 , Δ B ] ,
[ T 2 , Δ B ] 2 [ 2 x 2 , Δ B ] 2 2 x 2 Δ B + 4 z Δ B x 6 Δ B 2 ,
3 4 h k δ 2 Δ g R g R = 3 4 ( h d ) ( a δ ) 2 ( 1 2 π F ) Δ g R g R ,
ψ R ( - ) ( z + h / 2 ) = exp [ i h 4 k T 2 ] ψ R ( z ) .
ψ R ( + ) ( z + h / 2 ) = exp [ ½ z z + h g R ( z ) d z ] ψ R ( - ) ( z + h / 2 ) .
ψ R ( z + h ) = exp [ i h 4 k R 2 ] ψ R ( + ) ( z + h / 2 ) .
z z + h g R ( z ) d z h 2 [ g R ( z + h ) + g R ( z ) ] .
z z + h g R ( z ) d z = h g R ( z + h / 2 ) .
ψ ˜ R ( k z , k y , z ) = FT [ ψ R ( x , y , z ) ] = ψ R ( x , y , z ) exp [ i ( k x x + k y y ) ] d x d y .
ψ R ( - ) ( z + h / 2 ) = ( FT ) - 1 exp [ - i h 4 k ( k x 2 + k y 2 ) ] × ( FT ) ψ R ( z ) ,
ψ ˜ m n k = 1 N l = 1 N exp [ 2 π i ( k m + ln ) N ] ψ k l .
ψ ( x , 0 ) = exp ( i π C x 2 ) ,
ψ ( x , D ) = ( - i F ) 1 / 2 - 1 + 1 exp ( i π F x - y 2 ) ψ ( y , 0 ) d y ,
a b exp [ - i t f ( y ) ] d y = i t { exp [ - i t f ( b ) ] f ( b ) - exp [ - i t f ( a ) ] f ( a ) } ,
ψ ( x , D ) 2 = F 4 π 2 [ 1 ( F + C - F x ) 2 + 1 ( F + C + F x ) 2 + 2 cos ( 4 π F x ) ( F + C + F x ) ( F + C - F x ) ] .
1 ( G ) 1 2 π 2 F G G 2 - 1 - 2 D / B .
1 ( G ) 1 2 π 2 F 1 G - 1 ,
G 1 + 1 2 π 2 F 1
ϕ ( ν ) = - 1 + 1 exp ( 2 π i ν x ) exp ( i π C x 2 ) d x .
ϕ ( ν ) 2 = 1 4 π 2 C 2 [ 1 ( 1 - ν / C ) 2 + 1 ( 1 + ν / C ) 2 + 2 cos ( 4 π F ν ) ( 1 - ν 2 / C 2 ) ] .
2 = 1 2 π 2 ν max 1 1 - ( C / ν max ) 2 ,
N p G π 2 2 [ 1 + ( 1 + 16 C π 2 2 ) 1 / 2 ] .
N p 4 G ( G + 1 ) F = N p + 8 F G ,
Ψ R ( x , y , 0 ) = - R Ψ L ( x , y , 0 ) exp [ - i k ( x 2 + y 2 ) / b 1 ]
Ψ L ( x , y , d ) = - R Ψ R ( x , y , d ) × exp ( 2 i k d ) exp [ - i k ( x 2 + y 2 ) / b 2 ] ,
k d = m π .
M = M R M L ,
M R = g 1 [ 1 + ( 1 - 1 / g 1 g 2 ) 1 / 2 ] ,
M L = g 2 [ 1 + ( 1 - 1 / g 1 g 2 ) 1 / 2 ] ,
g i = [ 1 - d b i ] ,             i = 1 , 2 ,
F = a 2 / λ d = k a 2 / ( 2 π d ) ,
F eq = ½ ( M - 1 ) F ,             F T = M 2 F .
d I R d z + 2 [ ( M R - 1 ) d + ( M R - 1 ) z ] I R = Re ( g R ) I R ,
d I L d z + 2 [ ( M L - 1 ) d + ( M L - 1 ) ( d - z ) ] I L = - Re ( g L ) I L ,
P = Ψ L 2 d x d y ,
σ x 2 = 1 P Ψ L 2 x 2 d x d y .
g 0 ( x ) = { 1 + exp [ 20 ( x 0 - x ) ] } - 1 ,
g ( μ ) = a b exp ( i μ x ) f ( x ) d x
g ( μ ) = h A B [ j = 0 N exp ( i μ x j ) f ( x j ) - ½ f ( x 0 ) e i μ a - ½ f ( x N ) e i μ b ] - i h A × r = 0 k - 1 i r h r C r [ f ( r ) ( x N ) e i μ b - f ( r ) ( x 0 ) e i μ a ] + R f ,
A = 3 / ( 2 + cos u ) ,
B = [ ( 2 / u ) sin ( u / 2 ) ] 4 ,
C 0 = [ u 3 ( 2 + cos u ) - 12 sin 2 ( u / 2 ) sin u ] / 3 u 4 ,
C 1 = [ u 2 ( 2 + cos u ) - 12 sin 2 ( u / 2 ) ] / 3 u 4 ,
C 2 = [ u ( 2 + cos u ) - 3 sin ( u / 2 ) ] / 3 u 4 ,
C 3 = 0.
μ = μ 0 + m Δ μ ,             m = 0 , 1 , M .
s m = j = 0 N - 1 exp [ i ( μ 0 + Δ μ m ) h j ] f j
ϕ m = n = 0 N - 1 z m - n ψ n ,             m = 0 , 1 , , M - 1 ,
z m = z 0 W - m ,
L M + N - 1.
z 0 = exp [ - i μ 0 h ] ,             W = exp [ i Δ μ h ]

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