Abstract

This paper is a review of the theory-of laser beams and resonators. It is meant to be tutorial in nature and useful in scope. No attempt is made to be exhaustive in the treatment. Rather, emphasis is placed on formulations and derivations which lead to basic understanding and on results which bear practical significance.

© 1966 Optical Society of America

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References

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  1. R. H. Dicke, “Molecular amplification and generation systems and methods,” U. S. Patent2 851 652, September9, 1958.
  2. A. M. Prokhorov, “Molecular amplifier and generator for sub-millimeter waves,” JETP (USSR), vol. 34, pp. 1658–1659, June1958; Sov. Phys. JETP, vol. 7, pp. 1140–1141, December1958.
  3. A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev., vol. 29, pp. 1940–1949, December1958.
    [CrossRef]
  4. A. G. Fox, T. Li, “Resonant modes in an optical maser,” Proc. IRE(Correspondence), vol. 48, pp. 1904–1905, November1960; “Resonant modes in a maser interferometer,” Bell Sys. Tech. J., vol. 40, pp. 453–488, March1961.
  5. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J., vol. 40, pp. 489–508, March1961.
  6. G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Sys. Tech. J., vol. 41, pp. 1347–1369, July1962.
  7. G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. on Antennas and Propagation, vol. AP-9, pp. 248–256, May1961.
    [CrossRef]
  8. J. R. Pierce, “Modes in sequences of lenses,” Proc. Nat’l Acad. Sci., vol. 47, pp. 1808–1813, November1961.
  9. G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas. New York: Macmillan, 1963, pp. 907–918.
  10. H. Kogelnik, “Imaging of optical mode—Resonators with internal lenses,” Bell Sys. Tech. J., vol. 44, pp. 455–494, March1965.
  11. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt., vol. 4, pp. 1562–1569, December1965.
    [CrossRef]
  12. A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE, vol. 53, pp. 277–287, March1965.
    [CrossRef]
  13. W. Brower, Matrix Methods in Optical Instrument Design. New York: Benjamin, 1964. E. L. O’Neill, Introduction to Statistical Optics. Reading, Mass.: Addison-Wesley, 1963.
  14. M. Bertolotti, “Matrix representation of geometrical properties of laser cavities,” Nuovo Cimento, vol. 32, pp. 1242–1257, June1964.V. P. Bykov, L. A. Vainshtein, “Geometrical optics of open resonators,” JETP (USSR), vol. 47, pp. 508–517, August1964. B. Macke, “Laser cavities in geometrical optics approximation,” J. Pys. (Paris), vol. 26, pp. 104A–112A, March1965. W. K. Kahn, “Geometric optical derivation of formula for the variation of the spot size in a spherical mirror resonator,” Appl. Opt., vol. 4, pp. 758–759, June1965.
    [CrossRef]
  15. J. R. Pierce, Theory and Design of Electron Beams. New York: Van Nostrand, 1954, p. 194.
  16. H. Kogelnik, W. W. Rigrod, “Visual display of isolated optical-resonator modes,” Proc. IRE (Correspondence), vol. 50, p. 220, February1962.
  17. G. A. Deschamps, P. E. Mast, “Beam tracing and applications,” in Proc. Symposium on Quasi-Optics. New York: Polytechnic Press, 1964, pp. 379–395.
  18. S. A. Collins, “Analysis of optical resonators involving focusing elements,” Appl. Opt., vol. 3, pp. 1263–1275, November1964.
    [CrossRef]
  19. T. Li, “Dual forms of the Gaussian beam chart,” Appl. Opt., vol. 3, pp. 1315–1317, November1964.
    [CrossRef]
  20. T. S. Chu, “Geometrical representation of Gaussian beam propagation,” Bell Sys. Tech. J., vol. 45, pp. 287–299, February1966.
  21. J. P. Gordon, “A circle diagram for optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 1826–1827, July1964. M. J. Offerhaus, “Geometry of the radiation field for a laser interferometer,” Philips Res. Rept., vol. 19, pp. 520–523, December1964.
  22. H. Statz, C. L. Tang, “Problem of mode deformation in optical masers,” J. Appl. Phys., vol. 36, pp. 1816–1819, June1965.
    [CrossRef]
  23. A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. of Quantum Electronics, vol. QE-2, p. lxii, April1966.
  24. A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE, vol. 51, pp. 80–89, January1963.
    [CrossRef]
  25. F. B. Hildebrand, Methods of Applied Mathematics. Englewood Cliffs, N. J.: Prentice Hall, 1952, pp. 412–413.
  26. D. J. Newman, S. P. Morgan, “Existence of eigenvalues of a class of integral equations arising in laser theory,” Bell Sys. Tech. J., vol. 43, pp. 113–126, January1964.
  27. J. A. Cochran, “The existence of eigenvalues for the integral equations of laser theory,” Bell Sys. Tech. J., vol. 44, pp. 77–88, January1965.
  28. H. Hochstadt, “On the eigenvalue of a class of integral equations arising in laser theory,” SIAM Rev., vol. 8, pp. 62–65, January1966.
    [CrossRef]
  29. D. Gloge, “Calculations of Fabry-Perot laser resonators by scattering matrices,” Arch. Elect. Ubertrag., vol. 18, pp. 197–203, March1964.
  30. W. Streifer, “Optical resonator modes—rectangular reflectors of spherical curvature,” J. Opt. Soc. Am., vol. 55, pp. 868–877, July1965
    [CrossRef]
  31. T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Sys. Tech. J., vol. 44, pp. 917–932, May–June, 1965.
  32. J. C. Heurtley, W. Streifer, “Optical resonator modes—circular reflectors of spherical curvature,” J. Opt. Soc. Am., vol. 55, pp. 1472–1479, November1965.
    [CrossRef]
  33. J. P. Gordon, H. Kogelnik, “Equivalence relations among spherical mirror optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 2873–2886, November1964.
  34. F. Schwering, “Reiterative wave beams of rectangular symmetry,” Arch. Elect. Übertrag., vol. 15, pp. 555–564, December1961
  35. A. G. Fox, T. Li, to be published.
  36. L. A. Vainshtein, “Open resonators for lasers,” JETP (USSR), vol. 44, pp. 1050–1067, March1963; Sov. Phys. JETP, vol. 17, pp. 709–719, September1963.
  37. S. R. Barone, “Resonances of the Fabry-Perot laser,” J. Appl. Phys., vol. 34, pp. 831–843, April1963.
    [CrossRef]
  38. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Sys. Tech. J., vol. 40, pp. 43–64, January1961.
  39. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Svs. Tech. J., vol. 43, pp. 3009–3057, November1964.
  40. J. C. Heurtley, “Hyperspheroidal functions—optical resonators with circular mirrors,” in Proc. Symposium on Quasi-Optics. New York: Polytechnic Press, 1964, pp. 367–375.
  41. S. R. Barone, M. C. Newstein, “Fabry-Perot resonances at small Fresnel numbers,” Appl. Opt., vol. 3, p. 1194, October1964.
    [CrossRef]
  42. L. Bergstein, H. Schachter, “Resonant modes of optic cavities of small Fresnel numbers,” J. Opt. Soc. Am., vol. 55, pp. 1226–1233, October1965.
    [CrossRef]
  43. A. G. Fox, T. Li, “Modes in a maser interferometer with curved mirrors,” in Proc. Third International Congress on Quantum Electronics. New York: Columbia University Press, 1964, pp. 1263–1270.
  44. H. Kogelnik, “Modes in optical resonators,” in Lasers, A. K. Levine, Ed. New York: Dekker, 1966.

1966

T. S. Chu, “Geometrical representation of Gaussian beam propagation,” Bell Sys. Tech. J., vol. 45, pp. 287–299, February1966.

A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. of Quantum Electronics, vol. QE-2, p. lxii, April1966.

H. Hochstadt, “On the eigenvalue of a class of integral equations arising in laser theory,” SIAM Rev., vol. 8, pp. 62–65, January1966.
[CrossRef]

1965

J. A. Cochran, “The existence of eigenvalues for the integral equations of laser theory,” Bell Sys. Tech. J., vol. 44, pp. 77–88, January1965.

H. Kogelnik, “Imaging of optical mode—Resonators with internal lenses,” Bell Sys. Tech. J., vol. 44, pp. 455–494, March1965.

H. Statz, C. L. Tang, “Problem of mode deformation in optical masers,” J. Appl. Phys., vol. 36, pp. 1816–1819, June1965.
[CrossRef]

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE, vol. 53, pp. 277–287, March1965.
[CrossRef]

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Sys. Tech. J., vol. 44, pp. 917–932, May–June, 1965.

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt., vol. 4, pp. 1562–1569, December1965.
[CrossRef]

W. Streifer, “Optical resonator modes—rectangular reflectors of spherical curvature,” J. Opt. Soc. Am., vol. 55, pp. 868–877, July1965
[CrossRef]

L. Bergstein, H. Schachter, “Resonant modes of optic cavities of small Fresnel numbers,” J. Opt. Soc. Am., vol. 55, pp. 1226–1233, October1965.
[CrossRef]

J. C. Heurtley, W. Streifer, “Optical resonator modes—circular reflectors of spherical curvature,” J. Opt. Soc. Am., vol. 55, pp. 1472–1479, November1965.
[CrossRef]

1964

S. R. Barone, M. C. Newstein, “Fabry-Perot resonances at small Fresnel numbers,” Appl. Opt., vol. 3, p. 1194, October1964.
[CrossRef]

T. Li, “Dual forms of the Gaussian beam chart,” Appl. Opt., vol. 3, pp. 1315–1317, November1964.
[CrossRef]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Svs. Tech. J., vol. 43, pp. 3009–3057, November1964.

S. A. Collins, “Analysis of optical resonators involving focusing elements,” Appl. Opt., vol. 3, pp. 1263–1275, November1964.
[CrossRef]

M. Bertolotti, “Matrix representation of geometrical properties of laser cavities,” Nuovo Cimento, vol. 32, pp. 1242–1257, June1964.V. P. Bykov, L. A. Vainshtein, “Geometrical optics of open resonators,” JETP (USSR), vol. 47, pp. 508–517, August1964. B. Macke, “Laser cavities in geometrical optics approximation,” J. Pys. (Paris), vol. 26, pp. 104A–112A, March1965. W. K. Kahn, “Geometric optical derivation of formula for the variation of the spot size in a spherical mirror resonator,” Appl. Opt., vol. 4, pp. 758–759, June1965.
[CrossRef]

J. P. Gordon, H. Kogelnik, “Equivalence relations among spherical mirror optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 2873–2886, November1964.

D. J. Newman, S. P. Morgan, “Existence of eigenvalues of a class of integral equations arising in laser theory,” Bell Sys. Tech. J., vol. 43, pp. 113–126, January1964.

D. Gloge, “Calculations of Fabry-Perot laser resonators by scattering matrices,” Arch. Elect. Ubertrag., vol. 18, pp. 197–203, March1964.

J. P. Gordon, “A circle diagram for optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 1826–1827, July1964. M. J. Offerhaus, “Geometry of the radiation field for a laser interferometer,” Philips Res. Rept., vol. 19, pp. 520–523, December1964.

1963

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE, vol. 51, pp. 80–89, January1963.
[CrossRef]

L. A. Vainshtein, “Open resonators for lasers,” JETP (USSR), vol. 44, pp. 1050–1067, March1963; Sov. Phys. JETP, vol. 17, pp. 709–719, September1963.

S. R. Barone, “Resonances of the Fabry-Perot laser,” J. Appl. Phys., vol. 34, pp. 831–843, April1963.
[CrossRef]

1962

H. Kogelnik, W. W. Rigrod, “Visual display of isolated optical-resonator modes,” Proc. IRE (Correspondence), vol. 50, p. 220, February1962.

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Sys. Tech. J., vol. 41, pp. 1347–1369, July1962.

1961

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. on Antennas and Propagation, vol. AP-9, pp. 248–256, May1961.
[CrossRef]

J. R. Pierce, “Modes in sequences of lenses,” Proc. Nat’l Acad. Sci., vol. 47, pp. 1808–1813, November1961.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J., vol. 40, pp. 489–508, March1961.

F. Schwering, “Reiterative wave beams of rectangular symmetry,” Arch. Elect. Übertrag., vol. 15, pp. 555–564, December1961

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Sys. Tech. J., vol. 40, pp. 43–64, January1961.

1960

A. G. Fox, T. Li, “Resonant modes in an optical maser,” Proc. IRE(Correspondence), vol. 48, pp. 1904–1905, November1960; “Resonant modes in a maser interferometer,” Bell Sys. Tech. J., vol. 40, pp. 453–488, March1961.

1958

A. M. Prokhorov, “Molecular amplifier and generator for sub-millimeter waves,” JETP (USSR), vol. 34, pp. 1658–1659, June1958; Sov. Phys. JETP, vol. 7, pp. 1140–1141, December1958.

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev., vol. 29, pp. 1940–1949, December1958.
[CrossRef]

Barone, S. R.

S. R. Barone, M. C. Newstein, “Fabry-Perot resonances at small Fresnel numbers,” Appl. Opt., vol. 3, p. 1194, October1964.
[CrossRef]

S. R. Barone, “Resonances of the Fabry-Perot laser,” J. Appl. Phys., vol. 34, pp. 831–843, April1963.
[CrossRef]

Bergstein, L.

Bertolotti, M.

M. Bertolotti, “Matrix representation of geometrical properties of laser cavities,” Nuovo Cimento, vol. 32, pp. 1242–1257, June1964.V. P. Bykov, L. A. Vainshtein, “Geometrical optics of open resonators,” JETP (USSR), vol. 47, pp. 508–517, August1964. B. Macke, “Laser cavities in geometrical optics approximation,” J. Pys. (Paris), vol. 26, pp. 104A–112A, March1965. W. K. Kahn, “Geometric optical derivation of formula for the variation of the spot size in a spherical mirror resonator,” Appl. Opt., vol. 4, pp. 758–759, June1965.
[CrossRef]

Boyd, G. D.

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Sys. Tech. J., vol. 41, pp. 1347–1369, July1962.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J., vol. 40, pp. 489–508, March1961.

Brower, W.

W. Brower, Matrix Methods in Optical Instrument Design. New York: Benjamin, 1964. E. L. O’Neill, Introduction to Statistical Optics. Reading, Mass.: Addison-Wesley, 1963.

Chu, T. S.

T. S. Chu, “Geometrical representation of Gaussian beam propagation,” Bell Sys. Tech. J., vol. 45, pp. 287–299, February1966.

Cochran, J. A.

J. A. Cochran, “The existence of eigenvalues for the integral equations of laser theory,” Bell Sys. Tech. J., vol. 44, pp. 77–88, January1965.

Collins, S. A.

Deschamps, G. A.

G. A. Deschamps, P. E. Mast, “Beam tracing and applications,” in Proc. Symposium on Quasi-Optics. New York: Polytechnic Press, 1964, pp. 379–395.

Dicke, R. H.

R. H. Dicke, “Molecular amplification and generation systems and methods,” U. S. Patent2 851 652, September9, 1958.

Fox, A. G.

A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. of Quantum Electronics, vol. QE-2, p. lxii, April1966.

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE, vol. 51, pp. 80–89, January1963.
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in an optical maser,” Proc. IRE(Correspondence), vol. 48, pp. 1904–1905, November1960; “Resonant modes in a maser interferometer,” Bell Sys. Tech. J., vol. 40, pp. 453–488, March1961.

A. G. Fox, T. Li, “Modes in a maser interferometer with curved mirrors,” in Proc. Third International Congress on Quantum Electronics. New York: Columbia University Press, 1964, pp. 1263–1270.

A. G. Fox, T. Li, to be published.

Gloge, D.

D. Gloge, “Calculations of Fabry-Perot laser resonators by scattering matrices,” Arch. Elect. Ubertrag., vol. 18, pp. 197–203, March1964.

Gordon, J. P.

J. P. Gordon, H. Kogelnik, “Equivalence relations among spherical mirror optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 2873–2886, November1964.

J. P. Gordon, “A circle diagram for optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 1826–1827, July1964. M. J. Offerhaus, “Geometry of the radiation field for a laser interferometer,” Philips Res. Rept., vol. 19, pp. 520–523, December1964.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J., vol. 40, pp. 489–508, March1961.

Goubau, G.

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. on Antennas and Propagation, vol. AP-9, pp. 248–256, May1961.
[CrossRef]

G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas. New York: Macmillan, 1963, pp. 907–918.

Heurtley, J. C.

J. C. Heurtley, W. Streifer, “Optical resonator modes—circular reflectors of spherical curvature,” J. Opt. Soc. Am., vol. 55, pp. 1472–1479, November1965.
[CrossRef]

J. C. Heurtley, “Hyperspheroidal functions—optical resonators with circular mirrors,” in Proc. Symposium on Quasi-Optics. New York: Polytechnic Press, 1964, pp. 367–375.

Hildebrand, F. B.

F. B. Hildebrand, Methods of Applied Mathematics. Englewood Cliffs, N. J.: Prentice Hall, 1952, pp. 412–413.

Hochstadt, H.

H. Hochstadt, “On the eigenvalue of a class of integral equations arising in laser theory,” SIAM Rev., vol. 8, pp. 62–65, January1966.
[CrossRef]

Kogelnik, H.

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt., vol. 4, pp. 1562–1569, December1965.
[CrossRef]

H. Kogelnik, “Imaging of optical mode—Resonators with internal lenses,” Bell Sys. Tech. J., vol. 44, pp. 455–494, March1965.

J. P. Gordon, H. Kogelnik, “Equivalence relations among spherical mirror optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 2873–2886, November1964.

H. Kogelnik, W. W. Rigrod, “Visual display of isolated optical-resonator modes,” Proc. IRE (Correspondence), vol. 50, p. 220, February1962.

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Sys. Tech. J., vol. 41, pp. 1347–1369, July1962.

H. Kogelnik, “Modes in optical resonators,” in Lasers, A. K. Levine, Ed. New York: Dekker, 1966.

Li, T.

A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. of Quantum Electronics, vol. QE-2, p. lxii, April1966.

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Sys. Tech. J., vol. 44, pp. 917–932, May–June, 1965.

T. Li, “Dual forms of the Gaussian beam chart,” Appl. Opt., vol. 3, pp. 1315–1317, November1964.
[CrossRef]

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE, vol. 51, pp. 80–89, January1963.
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in an optical maser,” Proc. IRE(Correspondence), vol. 48, pp. 1904–1905, November1960; “Resonant modes in a maser interferometer,” Bell Sys. Tech. J., vol. 40, pp. 453–488, March1961.

A. G. Fox, T. Li, “Modes in a maser interferometer with curved mirrors,” in Proc. Third International Congress on Quantum Electronics. New York: Columbia University Press, 1964, pp. 1263–1270.

A. G. Fox, T. Li, to be published.

Mast, P. E.

G. A. Deschamps, P. E. Mast, “Beam tracing and applications,” in Proc. Symposium on Quasi-Optics. New York: Polytechnic Press, 1964, pp. 379–395.

Morgan, S. P.

D. J. Newman, S. P. Morgan, “Existence of eigenvalues of a class of integral equations arising in laser theory,” Bell Sys. Tech. J., vol. 43, pp. 113–126, January1964.

Newman, D. J.

D. J. Newman, S. P. Morgan, “Existence of eigenvalues of a class of integral equations arising in laser theory,” Bell Sys. Tech. J., vol. 43, pp. 113–126, January1964.

Newstein, M. C.

Pierce, J. R.

J. R. Pierce, “Modes in sequences of lenses,” Proc. Nat’l Acad. Sci., vol. 47, pp. 1808–1813, November1961.

J. R. Pierce, Theory and Design of Electron Beams. New York: Van Nostrand, 1954, p. 194.

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Sys. Tech. J., vol. 40, pp. 43–64, January1961.

Prokhorov, A. M.

A. M. Prokhorov, “Molecular amplifier and generator for sub-millimeter waves,” JETP (USSR), vol. 34, pp. 1658–1659, June1958; Sov. Phys. JETP, vol. 7, pp. 1140–1141, December1958.

Rigrod, W. W.

H. Kogelnik, W. W. Rigrod, “Visual display of isolated optical-resonator modes,” Proc. IRE (Correspondence), vol. 50, p. 220, February1962.

Schachter, H.

Schawlow, A. L.

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev., vol. 29, pp. 1940–1949, December1958.
[CrossRef]

Schwering, F.

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. on Antennas and Propagation, vol. AP-9, pp. 248–256, May1961.
[CrossRef]

F. Schwering, “Reiterative wave beams of rectangular symmetry,” Arch. Elect. Übertrag., vol. 15, pp. 555–564, December1961

Siegman, A. E.

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE, vol. 53, pp. 277–287, March1965.
[CrossRef]

Slepian, D.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Svs. Tech. J., vol. 43, pp. 3009–3057, November1964.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Sys. Tech. J., vol. 40, pp. 43–64, January1961.

Statz, H.

H. Statz, C. L. Tang, “Problem of mode deformation in optical masers,” J. Appl. Phys., vol. 36, pp. 1816–1819, June1965.
[CrossRef]

Streifer, W.

Tang, C. L.

H. Statz, C. L. Tang, “Problem of mode deformation in optical masers,” J. Appl. Phys., vol. 36, pp. 1816–1819, June1965.
[CrossRef]

Townes, C. H.

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev., vol. 29, pp. 1940–1949, December1958.
[CrossRef]

Vainshtein, L. A.

L. A. Vainshtein, “Open resonators for lasers,” JETP (USSR), vol. 44, pp. 1050–1067, March1963; Sov. Phys. JETP, vol. 17, pp. 709–719, September1963.

Appl. Opt.

Arch. Elect. Ubertrag.

D. Gloge, “Calculations of Fabry-Perot laser resonators by scattering matrices,” Arch. Elect. Ubertrag., vol. 18, pp. 197–203, March1964.

Arch. Elect. Übertrag.

F. Schwering, “Reiterative wave beams of rectangular symmetry,” Arch. Elect. Übertrag., vol. 15, pp. 555–564, December1961

Bell Svs. Tech. J.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Svs. Tech. J., vol. 43, pp. 3009–3057, November1964.

Bell Sys. Tech. J.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Sys. Tech. J., vol. 40, pp. 43–64, January1961.

J. P. Gordon, H. Kogelnik, “Equivalence relations among spherical mirror optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 2873–2886, November1964.

D. J. Newman, S. P. Morgan, “Existence of eigenvalues of a class of integral equations arising in laser theory,” Bell Sys. Tech. J., vol. 43, pp. 113–126, January1964.

J. A. Cochran, “The existence of eigenvalues for the integral equations of laser theory,” Bell Sys. Tech. J., vol. 44, pp. 77–88, January1965.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J., vol. 40, pp. 489–508, March1961.

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Sys. Tech. J., vol. 41, pp. 1347–1369, July1962.

H. Kogelnik, “Imaging of optical mode—Resonators with internal lenses,” Bell Sys. Tech. J., vol. 44, pp. 455–494, March1965.

T. S. Chu, “Geometrical representation of Gaussian beam propagation,” Bell Sys. Tech. J., vol. 45, pp. 287–299, February1966.

J. P. Gordon, “A circle diagram for optical resonators,” Bell Sys. Tech. J., vol. 43, pp. 1826–1827, July1964. M. J. Offerhaus, “Geometry of the radiation field for a laser interferometer,” Philips Res. Rept., vol. 19, pp. 520–523, December1964.

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Sys. Tech. J., vol. 44, pp. 917–932, May–June, 1965.

IEEE J. of Quantum Electronics

A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. of Quantum Electronics, vol. QE-2, p. lxii, April1966.

IRE Trans. on Antennas and Propagation

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. on Antennas and Propagation, vol. AP-9, pp. 248–256, May1961.
[CrossRef]

J. Appl. Phys.

H. Statz, C. L. Tang, “Problem of mode deformation in optical masers,” J. Appl. Phys., vol. 36, pp. 1816–1819, June1965.
[CrossRef]

S. R. Barone, “Resonances of the Fabry-Perot laser,” J. Appl. Phys., vol. 34, pp. 831–843, April1963.
[CrossRef]

J. Opt. Soc. Am.

JETP (USSR)

L. A. Vainshtein, “Open resonators for lasers,” JETP (USSR), vol. 44, pp. 1050–1067, March1963; Sov. Phys. JETP, vol. 17, pp. 709–719, September1963.

A. M. Prokhorov, “Molecular amplifier and generator for sub-millimeter waves,” JETP (USSR), vol. 34, pp. 1658–1659, June1958; Sov. Phys. JETP, vol. 7, pp. 1140–1141, December1958.

Nuovo Cimento

M. Bertolotti, “Matrix representation of geometrical properties of laser cavities,” Nuovo Cimento, vol. 32, pp. 1242–1257, June1964.V. P. Bykov, L. A. Vainshtein, “Geometrical optics of open resonators,” JETP (USSR), vol. 47, pp. 508–517, August1964. B. Macke, “Laser cavities in geometrical optics approximation,” J. Pys. (Paris), vol. 26, pp. 104A–112A, March1965. W. K. Kahn, “Geometric optical derivation of formula for the variation of the spot size in a spherical mirror resonator,” Appl. Opt., vol. 4, pp. 758–759, June1965.
[CrossRef]

Phys. Rev.

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev., vol. 29, pp. 1940–1949, December1958.
[CrossRef]

Proc. IEEE

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE, vol. 53, pp. 277–287, March1965.
[CrossRef]

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE, vol. 51, pp. 80–89, January1963.
[CrossRef]

Proc. IRE (Correspondence)

H. Kogelnik, W. W. Rigrod, “Visual display of isolated optical-resonator modes,” Proc. IRE (Correspondence), vol. 50, p. 220, February1962.

Proc. IRE(Correspondence)

A. G. Fox, T. Li, “Resonant modes in an optical maser,” Proc. IRE(Correspondence), vol. 48, pp. 1904–1905, November1960; “Resonant modes in a maser interferometer,” Bell Sys. Tech. J., vol. 40, pp. 453–488, March1961.

Proc. Nat’l Acad. Sci.

J. R. Pierce, “Modes in sequences of lenses,” Proc. Nat’l Acad. Sci., vol. 47, pp. 1808–1813, November1961.

SIAM Rev.

H. Hochstadt, “On the eigenvalue of a class of integral equations arising in laser theory,” SIAM Rev., vol. 8, pp. 62–65, January1966.
[CrossRef]

Other

R. H. Dicke, “Molecular amplification and generation systems and methods,” U. S. Patent2 851 652, September9, 1958.

A. G. Fox, T. Li, to be published.

J. C. Heurtley, “Hyperspheroidal functions—optical resonators with circular mirrors,” in Proc. Symposium on Quasi-Optics. New York: Polytechnic Press, 1964, pp. 367–375.

G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas. New York: Macmillan, 1963, pp. 907–918.

W. Brower, Matrix Methods in Optical Instrument Design. New York: Benjamin, 1964. E. L. O’Neill, Introduction to Statistical Optics. Reading, Mass.: Addison-Wesley, 1963.

G. A. Deschamps, P. E. Mast, “Beam tracing and applications,” in Proc. Symposium on Quasi-Optics. New York: Polytechnic Press, 1964, pp. 379–395.

J. R. Pierce, Theory and Design of Electron Beams. New York: Van Nostrand, 1954, p. 194.

F. B. Hildebrand, Methods of Applied Mathematics. Englewood Cliffs, N. J.: Prentice Hall, 1952, pp. 412–413.

A. G. Fox, T. Li, “Modes in a maser interferometer with curved mirrors,” in Proc. Third International Congress on Quantum Electronics. New York: Columbia University Press, 1964, pp. 1263–1270.

H. Kogelnik, “Modes in optical resonators,” in Lasers, A. K. Levine, Ed. New York: Dekker, 1966.

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Figures (25)

Fig. 1
Fig. 1

Reference planes of an optical system. A typical ray path is indicated.

Fig. 2
Fig. 2

Periodic sequence of identical systems, each characterized by its ABCD matrix.

Fig. 3
Fig. 3

Spherical-mirror resonator and the equivalent sequence of lenses.

Fig. 4
Fig. 4

Stability diagram. Unstable resonator systems lie in shaded regions.

Fig. 5
Fig. 5

Amplitude distribution of the fundamental beam.

Fig. 6
Fig. 6

Contour of a Gaussian beam.

Fig. 7
Fig. 7

Mode patterns of a gas laser oscillator (rectangular symmetry).

Fig. 8
Fig. 8

Transformation of wavefronts by a thin lens.

Fig. 9
Fig. 9

Distances and parameters for a beam transformed by a thin lens.

Fig. 10
Fig. 10

Symmetrical laser resonator and the equivalent sequence of lenses. The beam parameters, q1 and q2, are indicated.

Fig. 11
Fig. 11

Mode parameters of interest for a resonator with mirrors of unequal curvature.

Fig. 12
Fig. 12

The confocal parameter b2 as a function of the lens-waist spacing d1.

Fig. 13
Fig. 13

The waist spacing d2 as a function of the lens-waist spacing d1.

Fig. 14
Fig. 14

Geometry for the W-plane circle diagram.

Fig. 15
Fig. 15

The Gaussian beam chart. Both W-plane and Z-plane circle diagram are combined into one.

Fig. 16
Fig. 16

Geometry of a spherical-mirror resonator with finite mirror apertures and the equivalent sequence of lenses set in opaque absorbing screens.

Fig. 17
Fig. 17

Linearly polarized resonator mode configurations for square and circular mirrors.

Fig. 18
Fig. 18

Synthesis of different polarization configurations from the linearly polarized TEM01 mode.

Fig. 19
Fig. 19

Relative field distributions of the TEM00 mode for a resonator with circular mirrors (N = 1).

Fig. 20
Fig. 20

Relative field distributions of the TEM01 mode for a resonator with circular mirrors (N = 1).

Fig. 21
Fig. 21

Relative field distributions of four of the low order modes of a Fabry-Perot resonator with (parallel-plane) circular mirrors (N = 10).

Fig. 22
Fig. 22

Diffraction loss per transit (in decibels) for the TEM00 mode of a stable resonator with circular mirrors.

Fig. 23
Fig. 23

Diffraction loss per transit (in decibels) for the TEM01 mode of a stable resonator with circular mirrors.

Fig. 24
Fig. 24

Phase shift per transit for the TEM01 mode of a stable resonator with circular mirrors.

Fig. 25
Fig. 25

Phase shift per transit for the TEM01 mode of a stable resonator with circular mirrors.

Tables (2)

Tables Icon

TABLE I Ray Transfer Matrices of Six Elementary Optical Structures

Tables Icon

TABLE II Formulas for the Confocal Parameter and the Location of Beam Waist for Various Optical Structures

Equations (91)

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| x 2 x 2 | = | A B C D | | x 1 x 1 |
A D - B C = 1.
f = - 1 C h 1 = D - 1 C h 2 = A - 1 C
n = n 0 - 1 2 n 2 r 2 .
| A B C D | n = 1 sin Θ · | A sin n Θ - sin ( n - 1 ) Θ B sin n Θ C sin n Θ D sin n Θ - sin ( n - 1 ) Θ |
cos Θ = 1 2 ( A + D ) .
- 1 < 1 2 ( A + D ) < 1.
0 < ( 1 - d R 1 ) ( 1 - d R 2 ) < 1.
2 u + k 2 u = 0
u = ψ ( x , y , z ) exp ( - j k z )
2 ψ x 2 + 2 ψ y 2 - 2 j k ψ z = 0
ψ = exp { - j ( P + k 2 q r 2 ) }
r 2 = x 2 + y 2 .
q = 1
P = - j q
q 2 = q 1 + z
1 q = 1 R - j λ π w 2 .
q 0 = j π w 0 2 λ
q = q 0 + z = j π w 0 2 λ + z .
w 2 ( z ) = w 0 2 [ 1 + ( λ z π w 0 2 ) 2 ]
R ( z ) = z [ 1 + ( π w 0 2 λ z ) 2 ] .
θ = λ π w 0 .
λ z π w 0 2 = π w 2 λ R
w 0 2 = w 2 / [ 1 + ( π w 2 λ R ) 2 ]
z = R / [ 1 + ( λ R π w 2 ) 2 ] .
P = - j q = - j z + j ( π w 0 2 / λ ) .
j P ( z ) = ln [ 1 - j ( λ z / π w 0 2 ) ] = ln 1 + ( λ z / π w 0 2 ) 2 - j arc tan ( λ z / π w 0 2 ) .
u ( r , z ) = w 0 w · exp { - j ( k z - Φ ) - r 2 ( 1 w 2 + j k 2 R ) }
Φ = arc tan ( λ z / π w 0 2 ) .
ψ = g ( x w ) · h ( y w ) · exp { - j [ P + k 2 q ( x 2 + y 2 ) ] }
d 2 H m d x 2 - 2 x d H m d x + 2 m H m = 0.
g · h = H m ( 2 x w ) H n ( 2 y w )
H 0 ( x ) = 1 H 1 ( x ) = x H 2 ( x ) = 4 x 2 - 2 H 3 ( x ) = 8 x 3 - 12 x .
Φ ( m , n ; z ) = ( m + n + 1 ) arc tan ( λ z / π w 0 2 ) .
ψ = g ( r w ) · exp { - j ( P + k 2 q r 2 + l ϕ ) } .
g = ( 2 r w ) l · L p l ( 2 r 2 w 2 )
x d 2 L p l d x 2 + ( l + 1 - x ) d L p l d x + p L p l = 0.
L 0 l ( x ) = 1 L 1 l ( x ) = l + 1 - x L 2 l ( x ) = 1 2 ( l + 1 ) ( l + 2 ) - ( l + 2 ) x + 1 2 x 2 .
Φ ( p , l ; z ) = ( 2 p + l + 1 ) arc tan ( λ z / π w 0 2 ) .
1 R 2 = 1 R 1 - 1 f .
1 q 2 = 1 q 1 - 1 f ,
q 2 = ( 1 - d 2 / f ) q 1 + ( d 1 + d 2 - d 1 d 2 / f ) - ( q 1 / f ) + ( 1 - d 1 / f ) .
q 2 = A q 1 + B C q 1 + D .
1 q 2 = 1 q 1 + d - 1 f .
1 q 2 + 1 f q + 1 f d = 0.
1 q = - 1 2 f ( ) j 1 f d - 1 4 f 2
w 2 = ( λ R π ) / 2 R d - 1 .
w 0 2 = λ 2 π d ( 2 R - d ) .
k d - 2 ( m + n + 1 ) arc tan ( λ d / 2 π w 0 2 ) = π ( q + 1 )
ν 0 = c / 2 d
ν / ν 0 = ( q + 1 ) + 1 π ( m + n + 1 ) arc cos ( 1 - d / R ) .
w 2 = λ b / π ,             w 0 2 = λ b / 2 π ; ν / ν 0 = ( q + 1 ) + 1 2 ( m + n + 1 ) .
w 1 4 = ( λ R 1 / π ) 2 R 2 - d R 1 - d d R 1 + R 2 - d w 2 4 = ( λ R 2 / π ) 2 R 1 - d R 2 - d d R 1 + R 2 - d .
w 0 4 = ( λ π ) 2 d ( R 1 - d ) ( R 2 - d ) ( R 1 + R 2 - d ) ( R 1 + R 2 - 2 d ) 2 .
t 1 = d ( R 2 - d ) R 1 + R 2 - 2 d t 2 = d ( R 1 - d ) R 1 + R 2 - 2 d .
ν / ν 0 = ( q + 1 ) + 1 π ( m + n + 1 ) arc cos ( 1 - d / R 1 ) ( 1 - d / R 2 )
1 q = D - A 2 B ( ) j 2 B 4 - ( A + D ) 2 ,
w 2 = ( 2 λ B / π ) / 4 - ( A + D ) 2 .
q 1 = j π w 1 2 / λ ,             q 2 = j π w 2 2 / λ
d 1 - f d 2 - f = w 1 2 w 2 2 .
( d 1 - f ) ( d 2 - f ) = f 2 - f 0 2
f 0 = π w 1 w 2 / λ .
d 1 = f ± w 1 w 2 f 2 - f 0 2 , d 2 = f ± w 2 w 1 f 2 - f 0 2 .
b 1 = 2 π w 1 2 / λ ,             b 2 = 2 π w 2 2 / λ .
f 0 2 = 1 4 b 1 b 2 ,
d 1 = f ± 1 2 b 1 ( f 2 / f 0 2 ) - 1 , d 2 = f ± 1 2 b 2 ( f 2 / f 0 2 ) - 1 .
b 2 / f = b 1 / f ( 1 - d 1 / f ) 2 + ( b 1 / 2 f ) 2 .
1 - d 2 / f = 1 - d 1 / f ( 1 - d 1 / f ) 2 + ( b 1 / 2 f ) 2
( λ π w 2 + j 1 R ) ( π w 0 2 λ - j z ) = 1.
W = λ π w 2 + j 1 R Z = π w 0 2 λ - j z = b / 2 - j z ,
W = 1 / Z .
γ ( 1 ) E ( 1 ) ( s 1 ) = S 2 K ( 2 ) ( s 1 , s 2 ) E ( 2 ) ( s 2 ) d S 2 γ ( 2 ) E ( 2 ) ( s 2 ) = S 1 K ( 1 ) ( s 2 , s 1 ) E ( 1 ) ( s 1 ) d S 1
S 1 E m ( 1 ) ( s 1 ) E n ( 1 ) ( s 1 ) d S 1 = 0 ,             m n S 2 E m ( 2 ) ( s 2 ) E n ( 2 ) ( s 2 ) d S 2 = 0 ,             m n
γ x ( 1 ) u ( 1 ) ( x 1 ) = - a 2 a 2 K ( x 1 , x 2 ) u ( 2 ) ( x 2 ) d x 2 γ x ( 2 ) u ( 2 ) ( x 2 ) = - a 1 a 1 K ( x 1 , x 2 ) u ( 1 ) ( x 1 ) d x 1
K ( x 1 , x 2 ) = j λ d · exp { - j k 2 d ( g 1 x 1 2 + g 2 x 2 2 - 2 x 1 x 2 ) } .
g 1 = 1 - d R 1 g 2 = 1 - d R 2 .
γ l ( 1 ) R l ( 1 ) ( r 1 ) r 1 = 0 a 2 K l ( r 1 , r 2 ) R l ( 2 ) ( r 2 ) r 2 d r 2 γ l ( 2 ) R l ( 2 ) ( r 2 ) r 2 = 0 a 1 K l ( r 1 , r 2 ) R l ( 1 ) ( r 1 ) r 1 d r 1
K l ( r 1 , r 2 ) = j l + 1 d J l ( k r 1 r 2 d ) r 1 r 2 · exp { - j k 2 d ( g 1 r 1 2 + g 2 r 2 2 ) }
N = a 1 a 2 λ d G 1 = g 1 a 1 a 2 G 2 = g 2 a 2 a 1 .
0 < G 1 G 2 < 1
0 < g 1 g 2 < 1.
α = 1 - γ 2
β = angle of γ
Q = 2 π d λ α t
ν / ν 0 = ( q + 1 ) + β / π
β = ( 2 p + l + 1 ) arc cos g 1 g 2 = ( 2 p + l + 1 ) arc cos g ,             for g 1 = g 2
α = 8 κ p l δ ( M + δ ) [ ( M + δ ) 2 + δ 2 ] 2
β = ( M 4 δ ) α
α = 2 π ( 8 π N ) 2 p + l + 1 e - 4 π N p ! ( p + l + 1 ) ! [ 1 + 0 ( 1 2 π N ) ]
β = ( 2 p + l - 1 ) π 2 .
α = 1 ± 1 - 1 - ( g 1 g 2 ) - 1 1 + 1 - ( g 1 g 2 ) - 1

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