Abstract

A generalized beam matrix method is used to investigate the mode structure of astigmatic misaligned optical systems with loss or gain. In these optical systems the usual real-argument polynomial-Gaussian beams are not eigenfunctions, and off-axis complex-argument polynomial beams must be used. New beam transformations for these complex-argument modes are reported. Stability criteria are developed, and mode selection in laser resonators that contain tilted, displaced, or curved complex optical elements is discussed.

© 1996 Optical Society of America

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  1. N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).
  2. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
  3. J. A. Arnaud, “Optical resonators in the approximation of Gauss,” Proc. IEEE 62, 1561–1570 (1974).
    [CrossRef]
  4. L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
    [CrossRef] [PubMed]
  5. A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
    [CrossRef]
  6. U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
    [CrossRef]
  7. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
    [CrossRef]
  8. A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. 43, 1857–1862 (1995).
    [CrossRef]
  9. L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
    [CrossRef]
  10. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  11. L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
    [CrossRef]
  12. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  13. N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
    [CrossRef] [PubMed]
  14. P. Lavigne, A. Parent, D. Pascale, N. McCarthy, “A compact wide-aperature single-mode TE-CO2laser with a low chirp rate,” IEEE J. Quantum Electron. QE-22, 2200–2203 (1986).
    [CrossRef]
  15. D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with a Gaussian reflectivity mirror,” IEEE J. Quantum Electron. 24, 849–855 (1988).
    [CrossRef]
  16. P. Lavigne, N. McCarthy, A. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–895 (1988).
    [CrossRef]
  17. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
    [CrossRef] [PubMed]
  18. K. J. Snell, N. McCarthy, M. Piche, P. Lavigne, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
    [CrossRef]
  19. S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
    [CrossRef]
  20. S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
    [CrossRef]
  21. J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
    [CrossRef]
  22. D. M. Tratt, “Optimizing coherent lidar performance with graded-reflectance laser resonator optics,” Appl. Opt. 31, 4233–4239 (1992).
    [CrossRef] [PubMed]
  23. K. L. Webster, C. C. Sung, “Mode-medium instability and its correction with a Gaussian-reflectivity mirror,” Appl. Opt. 31, 319–328 (1992).
    [CrossRef] [PubMed]
  24. See, for example, P. Lavigne, N. McCarthy, J.-G. Demers, “Design and characterization of complementary Gaussian reflectivity mirrors,” Appl. Opt. 24, 2581–2586 (1985).
    [CrossRef] [PubMed]
  25. W. R. Bennett, “Inversion mechanisms in gas lasers,” Appl. Opt. Suppl. 2, Chemical Lasers, pp. 3–33 (1965).
  26. B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
    [CrossRef]
  27. See the introduction of S. Zaidi, D. L. MacFarlane, “Mode evolution of optical resonators with a radial gain profile,” Phys. Rev. A 47, 588–596 (1993), and references therein.
    [CrossRef] [PubMed]
  28. F. R. Nash, “Mode guidance parallel to the junction plane of double-heterostructure GaAs lasers,” J. Appl. Phys. 44, 4696–4707 (1973).
    [CrossRef]
  29. N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: misalignment sensitivity,” Appl. Opt. 22, 2704–2708 (1983).
    [CrossRef] [PubMed]
  30. N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
    [CrossRef] [PubMed]
  31. N. McCarthy, M. Morin, “High-order transverse modes of misaligned laser resonators with Gaussian reflectivity mirrors,” Appl. Opt. 28, 2189–2191 (1989).
    [CrossRef] [PubMed]
  32. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  33. M. Nazarathy, A. Hardy, J. Shamir, “Generalized mode propagation in first-order optical systems with loss or gain,” J. Opt. Soc. Am. 72, 1409–1420 (1982).
    [CrossRef]
  34. L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
    [CrossRef]
  35. A. A. Tovar, L. W. Casperson, “Off-axis complex-argument polynomial-Gaussian beams in optical systems,” J. Opt. Soc. Am. A 8, 60–68 (1991).
    [CrossRef]
  36. L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [CrossRef]
  37. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 837.
  38. Ref. 37, pp. 798–801.

1995 (2)

A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. 43, 1857–1862 (1995).
[CrossRef]

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
[CrossRef]

1993 (1)

See the introduction of S. Zaidi, D. L. MacFarlane, “Mode evolution of optical resonators with a radial gain profile,” Phys. Rev. A 47, 588–596 (1993), and references therein.
[CrossRef] [PubMed]

1992 (3)

1991 (1)

1990 (2)

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

1989 (1)

1988 (4)

D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with a Gaussian reflectivity mirror,” IEEE J. Quantum Electron. 24, 849–855 (1988).
[CrossRef]

P. Lavigne, N. McCarthy, A. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–895 (1988).
[CrossRef]

K. J. Snell, N. McCarthy, M. Piche, P. Lavigne, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

1986 (1)

P. Lavigne, A. Parent, D. Pascale, N. McCarthy, “A compact wide-aperature single-mode TE-CO2laser with a low chirp rate,” IEEE J. Quantum Electron. QE-22, 2200–2203 (1986).
[CrossRef]

1985 (2)

1984 (1)

1983 (2)

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: misalignment sensitivity,” Appl. Opt. 22, 2704–2708 (1983).
[CrossRef] [PubMed]

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

1982 (1)

1976 (1)

1975 (3)

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

1974 (3)

J. A. Arnaud, “Optical resonators in the approximation of Gauss,” Proc. IEEE 62, 1561–1570 (1974).
[CrossRef]

L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
[CrossRef]

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

1973 (2)

L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
[CrossRef] [PubMed]

F. R. Nash, “Mode guidance parallel to the junction plane of double-heterostructure GaAs lasers,” J. Appl. Phys. 44, 4696–4707 (1973).
[CrossRef]

1970 (1)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).

1968 (1)

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

1965 (4)

W. R. Bennett, “Inversion mechanisms in gas lasers,” Appl. Opt. Suppl. 2, Chemical Lasers, pp. 3–33 (1965).

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, “Optical resonators in the approximation of Gauss,” Proc. IEEE 62, 1561–1570 (1974).
[CrossRef]

Bennett, W. R.

W. R. Bennett, “Inversion mechanisms in gas lasers,” Appl. Opt. Suppl. 2, Chemical Lasers, pp. 3–33 (1965).

Bournes, P.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Burnham, R.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Casperson, L. W.

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
[CrossRef]

A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. 43, 1857–1862 (1995).
[CrossRef]

A. A. Tovar, L. W. Casperson, “Off-axis complex-argument polynomial-Gaussian beams in optical systems,” J. Opt. Soc. Am. A 8, 60–68 (1991).
[CrossRef]

L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
[CrossRef]

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
[CrossRef]

L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
[CrossRef] [PubMed]

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Cerullo, G.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

De Silvestri, S.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

Demers, J.-G.

DiBiase, D.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Ganiel, U.

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Hardy, A.

M. Nazarathy, A. Hardy, J. Shamir, “Generalized mode propagation in first-order optical systems with loss or gain,” J. Opt. Soc. Am. 72, 1409–1420 (1982).
[CrossRef]

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Harris, M. R.

D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with a Gaussian reflectivity mirror,” IEEE J. Quantum Electron. 24, 849–855 (1988).
[CrossRef]

Hughes, W.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Kasinski, J. J.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[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. 4, 1562–1569 (1965).
[CrossRef]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

Lavigne, P.

P. Lavigne, N. McCarthy, A. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–895 (1988).
[CrossRef]

K. J. Snell, N. McCarthy, M. Piche, P. Lavigne, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

P. Lavigne, A. Parent, D. Pascale, N. McCarthy, “A compact wide-aperature single-mode TE-CO2laser with a low chirp rate,” IEEE J. Quantum Electron. QE-22, 2200–2203 (1986).
[CrossRef]

See, for example, P. Lavigne, N. McCarthy, J.-G. Demers, “Design and characterization of complementary Gaussian reflectivity mirrors,” Appl. Opt. 24, 2581–2586 (1985).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: misalignment sensitivity,” Appl. Opt. 22, 2704–2708 (1983).
[CrossRef] [PubMed]

Lunnam, S. D.

MacFarlane, D. L.

See the introduction of S. Zaidi, D. L. MacFarlane, “Mode evolution of optical resonators with a radial gain profile,” Phys. Rev. A 47, 588–596 (1993), and references therein.
[CrossRef] [PubMed]

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

McCarthy, N.

Morin, M.

Nash, F. R.

F. R. Nash, “Mode guidance parallel to the junction plane of double-heterostructure GaAs lasers,” J. Appl. Phys. 44, 4696–4707 (1973).
[CrossRef]

Nazarathy, M.

Newstein, M.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Parent, A.

P. Lavigne, N. McCarthy, A. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–895 (1988).
[CrossRef]

P. Lavigne, A. Parent, D. Pascale, N. McCarthy, “A compact wide-aperature single-mode TE-CO2laser with a low chirp rate,” IEEE J. Quantum Electron. QE-22, 2200–2203 (1986).
[CrossRef]

Pascale, D.

P. Lavigne, A. Parent, D. Pascale, N. McCarthy, “A compact wide-aperature single-mode TE-CO2laser with a low chirp rate,” IEEE J. Quantum Electron. QE-22, 2200–2203 (1986).
[CrossRef]

Perry, B. N.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Piche, M.

K. J. Snell, N. McCarthy, M. Piche, P. Lavigne, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

Rabinowitz, P.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Shamir, J.

Sheldrake, S. J.

L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 837.

Silberberg, Y.

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Snell, K. J.

P. Lavigne, N. McCarthy, A. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–895 (1988).
[CrossRef]

K. J. Snell, N. McCarthy, M. Piche, P. Lavigne, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

Sung, C. C.

Svelto, O.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

Tovar, A. A.

Tratt, D. M.

Vakhimov, N. G.

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).

Valentini, G.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Webster, K. L.

Willetts, D. V.

D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with a Gaussian reflectivity mirror,” IEEE J. Quantum Electron. 24, 849–855 (1988).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Yeh, P.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Zaidi, S.

See the introduction of S. Zaidi, D. L. MacFarlane, “Mode evolution of optical resonators with a radial gain profile,” Phys. Rev. A 47, 588–596 (1993), and references therein.
[CrossRef] [PubMed]

Zucker, H.

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).

Appl. Opt. (9)

Appl. Opt. Suppl. 2 (1)

W. R. Bennett, “Inversion mechanisms in gas lasers,” Appl. Opt. Suppl. 2, Chemical Lasers, pp. 3–33 (1965).

Appl. Phys. Lett. (1)

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Bell Syst. Tech. J. (2)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Can. J. Phys. (1)

P. Lavigne, N. McCarthy, A. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–895 (1988).
[CrossRef]

IEEE J. Quantum Electron. (5)

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

P. Lavigne, A. Parent, D. Pascale, N. McCarthy, “A compact wide-aperature single-mode TE-CO2laser with a low chirp rate,” IEEE J. Quantum Electron. QE-22, 2200–2203 (1986).
[CrossRef]

D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with a Gaussian reflectivity mirror,” IEEE J. Quantum Electron. 24, 849–855 (1988).
[CrossRef]

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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J. Opt. Soc. Am. A (2)

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Equations (51)

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k ( x , y , z ) 2 π n ( x , y , z ) λ + i α ( x , y , z )
= k 0 ( z ) - k 1 x ( z ) x / 2 - k 1 y ( z ) y / 2 - k 2 x ( z ) x 2 / 2 - k 2 y ( z ) y 2 / 2.
E ¯ m n ( x , y , z , t ) = Re ( E m n ( x , y , z ) × exp { i [ ω t - 0 z k 0 ( z ) d z ] } ) × [ cos ( ν ) i ¯ x + sin ( ν ) i ¯ y ] ,
E m n ( x , y , z ) = E m n , 0 exp { - i [ Q x ( z ) x 2 / 2 + Q y ( z ) y 2 / 2 + S x ( z ) x + S y ( z ) y + P ( z ) ] } × H m { 2 [ x - δ x ( z ) ] / W x ( z ) } × H n { 2 [ y - δ y ( z ) ] / W y ( z ) }
Q x β 0 1 q x = 1 R x - i 2 β 0 w x 2 ,
S x β 0 = - d x a q x + d x a = ( - d x a R x + d x a ) + i 2 d x a β 0 w x 2 .
d x a = S x i w x 2 / 2 ,
d x a = S x r / β 0 + S x i w x 2 2 R x .
( u x 2 ( 1 / q x 2 ) u x 2 S x 2 u x 2 ) = [ A x B x 0 C x D x 0 G x H x 1 ] ( u x 1 ( 1 / q x 1 ) u x 1 S x 1 u x 1 ) ,
1 q x 2 = C x + D x / q x 1 A x + B x / q x 1 .
S x 2 = S x 1 A x + B x / q x 1 + G x + H x / q x 1 A x + B x / q x 1 .
W x 2 2 = W x 1 2 ( A x + B x / q x 1 ) 2 + 4 i B x ( A x + B x / q x 1 ) / k 01 .
C x d A x / d z ,
D x d B x / d z ,
G x - 1 2 0 z k 1 ( z ) A x ( z ) d z ,
H x - 1 2 0 z k 1 ( z ) B x ( z ) d z ,
d δ x d z - 1 q x ( z ) δ x ( z ) = S x ( z ) k 0 ( z ) .
δ x 2 = δ x 1 ( A x + B x / q x 1 ) + S x 1 B x / k 01 + ( B x G x - A x H x ) / k 01 .
P 2 - P 1 = - i 2 ln ( A x + B x / q x 1 ) - i 2 ln ( A y + B y / q y 1 ) + i 2 [ m ln ( 1 + 4 i k 01 W x 1 2 B x A x + B x / q x 1 ) + n ln ( 1 + 4 i k 01 W y 1 2 B y A y + B y / q y 1 ) ] - 1 2 k 01 ( S x 1 2 B x A x + B x / q x 1 ) - 1 2 k 01 ( S y 1 2 B y A y + B y / q y 1 ) .
exp ( - i P 2 ) exp ( - i P 1 ) = ( 1 + 4 i k 01 W x 1 2 B x A x + B x / q x 1 ) m / 2 ( 1 + 4 i k 01 W y 1 2 B y A y + B y / q y 1 ) n / 2 ( A x + B x / q x 1 ) 1 / 2 ( A y + B y / q y 1 ) 1 / 2 exp [ i 2 k 01 ( S x 1 2 B x A x + B x / q x 1 + S y 1 2 B y A y + B y / q y 1 ) ] .
1 / q x = C x + D x / q x A x + B x / q x .
1 / q x = D x - A x 2 B x ± i B x [ 1 - ( A x + D x 2 ) 2 ] 1 / 2 ,
A x + B x / q x = exp ( ± i θ x ) ,
cos θ x A x + D x 2 .
S x = G x + H x / q x A x + B x / q x - 1 .
1 q x + δ q = C x + D x ( 1 / q x + δ q ) A x + B x ( 1 / q x + δ q )
= ( C x + D x / q x A x + B x / q x ) [ 1 + δ q D x / ( C x + D x / q x ) 1 + δ q B x ( A x + B x / q x ) ]
( C x + D x / q x A x + B x / q x ) ( 1 + δ q D x C x + D x / q x - δ q B x A x + B x / q x ) .
| δ q δ q | 1 A x + B x / q x 2 .
F x s A x + B x / q x > 1.
S x + δ S = S x + δ S A x + B x / q x + G x + H x / q x A x + B x / q x .
| δ S δ S | = 1 A x + B x / q x = F x s - 1 .
W x 2 = 4 i B x ( A x + B x / q x ) / k 0 1 - ( A x + B x / q x ) 2
= 2 B x / k 0 { 1 - [ ( A x + D x ) / 2 ] 2 } 1 / 2 ,
δ x = ( S x B x + B x G x - A x H x ) / k 0 1 - ( A x + B x / q x )
= [ ( A x - 1 ) H x - B x G x ] ( A x + B x / q x ) / k 0 ( A x + B x / q x - 1 ) 2
= [ B x G x - ( A x - 1 ) H x ] / k 0 2 [ 1 - ( A x + D x ) / 2 ] ,
W x 2 + δ W = ( W x 2 + δ W ) ( A x + B x / q x ) 2 + 4 i B x ( A x + B x / q x ) / k 0 ,
| δ W δ W | = A x + B x / q x 2 = F x s 2 ,
δ x + δ δ = ( δ x + δ δ ) ( A x + B x / q x ) + S x B x / k 0 + ( B x G x - A x H x ) / k 0 ,
| δ δ δ δ | = A x + B x / q x = F x s .
F x s = A x + B x / q x = | A x + D x 2 ± i [ 1 - ( A x + D x 2 ) 2 ] 1 / 2 | > 1
E 00 ( x , y , z ) = E 00 , 0 exp { - i [ Q x ( z ) x 2 / 2 + Q y ( z ) y 2 / 2 + S x ( z ) x + S y ( z ) y + P ( z ) ] } ,
E 10 ( x , y , z ) = E 00 E 10 , 0 E 00 , 0 2 ( x - δ x ) W x 1 ( A x + B x / q x 0 ) ,
G p , m n Power m n ( z ) Power m n ( 0 ) = - - E m n * ( x , y , z ) E m n ( x , y , z ) d x d y - - E m n * ( x , y , 0 ) E m n ( x , y , 0 ) d x d y .
G p , 00 G p , 10 = E 00 , 0 2 E 10 , 0 2 W x 1 2 w x 1 2 A x + B x / q x 0 2 .
G p , 00 G p , 10 = A x + B x / q x 2 = F x s 2 ,
δ x 2 = d x a 1 ( A x + B x / q x 1 ) + ( d x a 1 - d x a 1 / q x 1 ) B x + ( B x G x - A x H x ) / β 0
= A x d x a 1 + B x d x a 1 + ( B x G x - A x H x ) / β 0
= A x d x a 1 + B x d x a 1 + E x
= d x a 2 ,

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