Abstract

We examine the properties of the transverse eigenmodes of optical resonators containing a hard aperture. We show that for orthogonal optics and arbitrary aperture shape, the trace of the ray matrix and the scaled aperture size determine the major properties of interest: loss, frequency, and distortion that is due to diffraction. We discuss three different methods of reducing the Huygen–Fresnel integral equation to a matrix equation whose eigenvalues and eigenvectors can be easily found. The dependence of loss, frequency, distortion, and required matrix order on cavity parameters is presented for cylindrically symmetrical resonators. We show that undiffracted modes inadequately approximate the actual modes in nearly unstable cavities.

© 1999 Optical Society of America

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  1. N. B. Abraham, W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B 7, 951–961 (1990) and references therein.
    [Crossref]
  2. A. W. Yu, G. P. Agrawal, R. Roy, “Power spectra and spatial pattern dynamics of a ring laser,” J. Stat. Phys. 54, 1223–1241 (1989).
    [Crossref]
  3. N. N. Rasonov, V. E. Semenov, “The kinetics of the hysteresis change of the beam profile in nonlinear interferometers,” Opt. Commun. 38, 435–438 (1981).
    [Crossref]
  4. F. Hollinger, C. Jung, H. Weber, “Simple mathematical model describing multi-transversal solid-state lasers,” J. Opt. Soc. Am. B 7, 1013–1018 (1990).
    [Crossref]
  5. L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
    [Crossref] [PubMed]
  6. L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, M. A. Pernigo, “Instabilities and spatial complexity in a laser,” J. Opt. Soc. Am. B 7, 1019–1033 (1990).
    [Crossref]
  7. L. A. Lugiato, “Spatio-temporal structures. Part I,” Phys. Rep. 219, 293–310 (1992).
    [Crossref]
  8. H. Adachihara, O. Hess, E. Abraham, P. Ru, J. V. Moloney, “Spatiotemporal chaos in broad-area semiconductor lasers,” J. Opt. Soc. Am. B 10, 658–665 (1993).
    [Crossref]
  9. W. Kaige, N. B. Abraham, L. A. Lugiato, “Leading role of optical phase instabilities in the formation of certain laser transverse patterns,” Phys. Rev. A 47, 1263–1273 (1993).
    [Crossref]
  10. A. M. Dunlop, W. J. Firth, E. M. Wright, “Master equation for spatio-temporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
    [Crossref]
  11. J. Martin-Regalado, S. Balle, M. San Miguel, “Polarization and transverse mode dynamics of gain-guided vertical-cavity surface emitting lasers,” Opt. Lett. 22, 460–462 (1997).
    [Crossref] [PubMed]
  12. D. J. Jones, D. K. Bandy, “Hysteresis effects attributed to free-space diffraction in a passive optical system,” Opt. Commun. 92, 376–384 (1992).
    [Crossref]
  13. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
    [Crossref] [PubMed]
  14. T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
    [Crossref]
  15. T. Burkhard, M. O. Ziegler, I. Fischer, W. Elsaber, “Spatio-temporal dynamics of broad area semiconductor lasers and its characterization,” Chaos, Solitons and Fractals 10, 845–850 (1999).
  16. M. A. Rippin, G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
    [Crossref]
  17. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
    [Crossref] [PubMed]
  18. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
    [Crossref] [PubMed]
  19. M. Brunel, G. Ropars, A. L. Floch, F. Bretenaker, “Transverse excess noise factor in geometrically stable resonators,” Phys. Rev. A 55, 4563–4567 (1997).
    [Crossref]
  20. K. Joosten, G. Nienhuis, “Combined longitudinal and transverse noise enhancement in lasers,” Phys. Rev. A 58, 4937–4945 (1998).
    [Crossref]
  21. Yuh-Jen Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
    [Crossref] [PubMed]
  22. M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
    [Crossref] [PubMed]
  23. M. A. van Eijkelenborg, M. P. van Exter, J. P. Woerdman, “Threshold characteristics and intensity fluctuations of lasers with excess quantum noise,” Phys. Rev. A 57, 571–579 (1998).
    [Crossref]
  24. Å. M. Lindberg, M. A. van Eijkelenborg, K. Joosten, G. Nienhuis, J. P. Woerdman, “Observation of excess quantum noise in a geometrically stable laser,” Phys. Rev. A 57, 3036–3039 (1998).
    [Crossref]
  25. O. Emile, M. Brunel, F. Bretenaker, A. Le Floch, “Direct measurement of the transverse excess noise factor in a geometrically stable resonator,” Phys. Rev. A 57, 4889–4893 (1998).
    [Crossref]
  26. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 626–652.
  27. O. Haderka, “Properties of the transverse eigenmode set in optical resonators with apertures,” J. Opt. Soc. Am. A 12, 340–345 (1995).
    [Crossref]
  28. S. Ruschin, T. Hurvits, M. Keselbrener, “Properties of the transverse eigenmode set in optical resonators with apertures: comment,” J. Opt. Soc. Am. A 13, 1287–1288 (1996).
    [Crossref]
  29. O. Haderka, “Properties of the transverse eigenmode set in optical resonators with apertures: reply to comment,” J. Opt. Soc. Am. A 13, 1289–1290 (1996).
    [Crossref]
  30. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 904–913.
  31. A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
    [Crossref]
  32. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [Crossref]
  33. A. E. Siegmann, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2735 (1970); see also pages 570–577 of Ref. 12.
    [Crossref]
  34. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), pp. 887–890.
  35. G. D’Alessandro, G. L. Oppo, “Gauss–Laguerre modes: a ‘sensible’ basis for laser dynamics,” Opt. Commun. 88, 130–136 (1992).
    [Crossref]

1999 (1)

T. Burkhard, M. O. Ziegler, I. Fischer, W. Elsaber, “Spatio-temporal dynamics of broad area semiconductor lasers and its characterization,” Chaos, Solitons and Fractals 10, 845–850 (1999).

1998 (5)

T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
[Crossref]

K. Joosten, G. Nienhuis, “Combined longitudinal and transverse noise enhancement in lasers,” Phys. Rev. A 58, 4937–4945 (1998).
[Crossref]

M. A. van Eijkelenborg, M. P. van Exter, J. P. Woerdman, “Threshold characteristics and intensity fluctuations of lasers with excess quantum noise,” Phys. Rev. A 57, 571–579 (1998).
[Crossref]

Å. M. Lindberg, M. A. van Eijkelenborg, K. Joosten, G. Nienhuis, J. P. Woerdman, “Observation of excess quantum noise in a geometrically stable laser,” Phys. Rev. A 57, 3036–3039 (1998).
[Crossref]

O. Emile, M. Brunel, F. Bretenaker, A. Le Floch, “Direct measurement of the transverse excess noise factor in a geometrically stable resonator,” Phys. Rev. A 57, 4889–4893 (1998).
[Crossref]

1997 (3)

M. Brunel, G. Ropars, A. L. Floch, F. Bretenaker, “Transverse excess noise factor in geometrically stable resonators,” Phys. Rev. A 55, 4563–4567 (1997).
[Crossref]

A. M. Dunlop, W. J. Firth, E. M. Wright, “Master equation for spatio-temporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[Crossref]

J. Martin-Regalado, S. Balle, M. San Miguel, “Polarization and transverse mode dynamics of gain-guided vertical-cavity surface emitting lasers,” Opt. Lett. 22, 460–462 (1997).
[Crossref] [PubMed]

1996 (5)

M. A. Rippin, G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[Crossref]

Yuh-Jen Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[Crossref] [PubMed]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[Crossref] [PubMed]

S. Ruschin, T. Hurvits, M. Keselbrener, “Properties of the transverse eigenmode set in optical resonators with apertures: comment,” J. Opt. Soc. Am. A 13, 1287–1288 (1996).
[Crossref]

O. Haderka, “Properties of the transverse eigenmode set in optical resonators with apertures: reply to comment,” J. Opt. Soc. Am. A 13, 1289–1290 (1996).
[Crossref]

1995 (1)

1993 (2)

H. Adachihara, O. Hess, E. Abraham, P. Ru, J. V. Moloney, “Spatiotemporal chaos in broad-area semiconductor lasers,” J. Opt. Soc. Am. B 10, 658–665 (1993).
[Crossref]

W. Kaige, N. B. Abraham, L. A. Lugiato, “Leading role of optical phase instabilities in the formation of certain laser transverse patterns,” Phys. Rev. A 47, 1263–1273 (1993).
[Crossref]

1992 (3)

L. A. Lugiato, “Spatio-temporal structures. Part I,” Phys. Rep. 219, 293–310 (1992).
[Crossref]

D. J. Jones, D. K. Bandy, “Hysteresis effects attributed to free-space diffraction in a passive optical system,” Opt. Commun. 92, 376–384 (1992).
[Crossref]

G. D’Alessandro, G. L. Oppo, “Gauss–Laguerre modes: a ‘sensible’ basis for laser dynamics,” Opt. Commun. 88, 130–136 (1992).
[Crossref]

1991 (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

1990 (3)

1989 (3)

A. W. Yu, G. P. Agrawal, R. Roy, “Power spectra and spatial pattern dynamics of a ring laser,” J. Stat. Phys. 54, 1223–1241 (1989).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref] [PubMed]

1988 (1)

L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
[Crossref] [PubMed]

1981 (1)

N. N. Rasonov, V. E. Semenov, “The kinetics of the hysteresis change of the beam profile in nonlinear interferometers,” Opt. Commun. 38, 435–438 (1981).
[Crossref]

1979 (1)

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[Crossref]

1970 (1)

A. E. Siegmann, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2735 (1970); see also pages 570–577 of Ref. 12.
[Crossref]

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[Crossref]

Abraham, E.

Abraham, N. B.

W. Kaige, N. B. Abraham, L. A. Lugiato, “Leading role of optical phase instabilities in the formation of certain laser transverse patterns,” Phys. Rev. A 47, 1263–1273 (1993).
[Crossref]

N. B. Abraham, W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B 7, 951–961 (1990) and references therein.
[Crossref]

Adachihara, H.

Agrawal, G. P.

A. W. Yu, G. P. Agrawal, R. Roy, “Power spectra and spatial pattern dynamics of a ring laser,” J. Stat. Phys. 54, 1223–1241 (1989).
[Crossref]

Balle, S.

Bandy, D. K.

D. J. Jones, D. K. Bandy, “Hysteresis effects attributed to free-space diffraction in a passive optical system,” Opt. Commun. 92, 376–384 (1992).
[Crossref]

L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
[Crossref] [PubMed]

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

Bretenaker, F.

O. Emile, M. Brunel, F. Bretenaker, A. Le Floch, “Direct measurement of the transverse excess noise factor in a geometrically stable resonator,” Phys. Rev. A 57, 4889–4893 (1998).
[Crossref]

M. Brunel, G. Ropars, A. L. Floch, F. Bretenaker, “Transverse excess noise factor in geometrically stable resonators,” Phys. Rev. A 55, 4563–4567 (1997).
[Crossref]

Brunel, M.

O. Emile, M. Brunel, F. Bretenaker, A. Le Floch, “Direct measurement of the transverse excess noise factor in a geometrically stable resonator,” Phys. Rev. A 57, 4889–4893 (1998).
[Crossref]

M. Brunel, G. Ropars, A. L. Floch, F. Bretenaker, “Transverse excess noise factor in geometrically stable resonators,” Phys. Rev. A 55, 4563–4567 (1997).
[Crossref]

Burkhard, T.

T. Burkhard, M. O. Ziegler, I. Fischer, W. Elsaber, “Spatio-temporal dynamics of broad area semiconductor lasers and its characterization,” Chaos, Solitons and Fractals 10, 845–850 (1999).

Cheng, Yuh-Jen

Yuh-Jen Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[Crossref] [PubMed]

D’Alessandro, G.

G. D’Alessandro, G. L. Oppo, “Gauss–Laguerre modes: a ‘sensible’ basis for laser dynamics,” Opt. Commun. 88, 130–136 (1992).
[Crossref]

Danilova, A. P.

T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
[Crossref]

Danilova, T. N.

T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
[Crossref]

Dunlop, A. M.

A. M. Dunlop, W. J. Firth, E. M. Wright, “Master equation for spatio-temporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[Crossref]

Elsaber, W.

T. Burkhard, M. O. Ziegler, I. Fischer, W. Elsaber, “Spatio-temporal dynamics of broad area semiconductor lasers and its characterization,” Chaos, Solitons and Fractals 10, 845–850 (1999).

Emile, O.

O. Emile, M. Brunel, F. Bretenaker, A. Le Floch, “Direct measurement of the transverse excess noise factor in a geometrically stable resonator,” Phys. Rev. A 57, 4889–4893 (1998).
[Crossref]

Ershov, O. G.

T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
[Crossref]

Fanning, C. G.

Yuh-Jen Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[Crossref] [PubMed]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 904–913.

Firth, W. J.

A. M. Dunlop, W. J. Firth, E. M. Wright, “Master equation for spatio-temporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[Crossref]

N. B. Abraham, W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B 7, 951–961 (1990) and references therein.
[Crossref]

Fischer, I.

T. Burkhard, M. O. Ziegler, I. Fischer, W. Elsaber, “Spatio-temporal dynamics of broad area semiconductor lasers and its characterization,” Chaos, Solitons and Fractals 10, 845–850 (1999).

Floch, A. L.

M. Brunel, G. Ropars, A. L. Floch, F. Bretenaker, “Transverse excess noise factor in geometrically stable resonators,” Phys. Rev. A 55, 4563–4567 (1997).
[Crossref]

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[Crossref]

Haderka, O.

Hess, O.

Hollinger, F.

Hurvits, T.

Imenkov, A. N.

T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
[Crossref]

Jones, D. J.

D. J. Jones, D. K. Bandy, “Hysteresis effects attributed to free-space diffraction in a passive optical system,” Opt. Commun. 92, 376–384 (1992).
[Crossref]

Joosten, K.

K. Joosten, G. Nienhuis, “Combined longitudinal and transverse noise enhancement in lasers,” Phys. Rev. A 58, 4937–4945 (1998).
[Crossref]

Å. M. Lindberg, M. A. van Eijkelenborg, K. Joosten, G. Nienhuis, J. P. Woerdman, “Observation of excess quantum noise in a geometrically stable laser,” Phys. Rev. A 57, 3036–3039 (1998).
[Crossref]

Jung, C.

Kaige, W.

W. Kaige, N. B. Abraham, L. A. Lugiato, “Leading role of optical phase instabilities in the formation of certain laser transverse patterns,” Phys. Rev. A 47, 1263–1273 (1993).
[Crossref]

Keselbrener, M.

Le Floch, A.

O. Emile, M. Brunel, F. Bretenaker, A. Le Floch, “Direct measurement of the transverse excess noise factor in a geometrically stable resonator,” Phys. Rev. A 57, 4889–4893 (1998).
[Crossref]

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[Crossref]

Lindberg, Å. M.

Å. M. Lindberg, M. A. van Eijkelenborg, K. Joosten, G. Nienhuis, J. P. Woerdman, “Observation of excess quantum noise in a geometrically stable laser,” Phys. Rev. A 57, 3036–3039 (1998).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[Crossref] [PubMed]

Lugiato, L. A.

W. Kaige, N. B. Abraham, L. A. Lugiato, “Leading role of optical phase instabilities in the formation of certain laser transverse patterns,” Phys. Rev. A 47, 1263–1273 (1993).
[Crossref]

L. A. Lugiato, “Spatio-temporal structures. Part I,” Phys. Rep. 219, 293–310 (1992).
[Crossref]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, M. A. Pernigo, “Instabilities and spatial complexity in a laser,” J. Opt. Soc. Am. B 7, 1019–1033 (1990).
[Crossref]

L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
[Crossref] [PubMed]

Martin-Regalado, J.

Miller, H. Y.

A. E. Siegmann, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2735 (1970); see also pages 570–577 of Ref. 12.
[Crossref]

Moloney, J. V.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 904–913.

Narducci, L. M.

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, M. A. Pernigo, “Instabilities and spatial complexity in a laser,” J. Opt. Soc. Am. B 7, 1019–1033 (1990).
[Crossref]

L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
[Crossref] [PubMed]

New, G. H. C.

M. A. Rippin, G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[Crossref]

Nienhuis, G.

K. Joosten, G. Nienhuis, “Combined longitudinal and transverse noise enhancement in lasers,” Phys. Rev. A 58, 4937–4945 (1998).
[Crossref]

Å. M. Lindberg, M. A. van Eijkelenborg, K. Joosten, G. Nienhuis, J. P. Woerdman, “Observation of excess quantum noise in a geometrically stable laser,” Phys. Rev. A 57, 3036–3039 (1998).
[Crossref]

Oppo, G. L.

G. D’Alessandro, G. L. Oppo, “Gauss–Laguerre modes: a ‘sensible’ basis for laser dynamics,” Opt. Commun. 88, 130–136 (1992).
[Crossref]

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, M. A. Pernigo, “Instabilities and spatial complexity in a laser,” J. Opt. Soc. Am. B 7, 1019–1033 (1990).
[Crossref]

Penns, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

Pernigo, M. A.

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

Pratt, F.

L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
[Crossref] [PubMed]

Rasonov, N. N.

N. N. Rasonov, V. E. Semenov, “The kinetics of the hysteresis change of the beam profile in nonlinear interferometers,” Opt. Commun. 38, 435–438 (1981).
[Crossref]

Rippin, M. A.

M. A. Rippin, G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[Crossref]

Ropars, G.

M. Brunel, G. Ropars, A. L. Floch, F. Bretenaker, “Transverse excess noise factor in geometrically stable resonators,” Phys. Rev. A 55, 4563–4567 (1997).
[Crossref]

Roy, R.

A. W. Yu, G. P. Agrawal, R. Roy, “Power spectra and spatial pattern dynamics of a ring laser,” J. Stat. Phys. 54, 1223–1241 (1989).
[Crossref]

Ru, P.

H. Adachihara, O. Hess, E. Abraham, P. Ru, J. V. Moloney, “Spatiotemporal chaos in broad-area semiconductor lasers,” J. Opt. Soc. Am. B 10, 658–665 (1993).
[Crossref]

L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
[Crossref] [PubMed]

Ruschin, S.

San Miguel, M.

Semenov, V. E.

N. N. Rasonov, V. E. Semenov, “The kinetics of the hysteresis change of the beam profile in nonlinear interferometers,” Opt. Commun. 38, 435–438 (1981).
[Crossref]

Sherstnev, V. V.

T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
[Crossref]

Siegman, A. E.

Yuh-Jen Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[Crossref] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref] [PubMed]

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[Crossref]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 626–652.

Siegmann, A. E.

A. E. Siegmann, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2735 (1970); see also pages 570–577 of Ref. 12.
[Crossref]

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

Thijssen, M. S.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[Crossref] [PubMed]

Tredicce, J. R.

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, M. A. Pernigo, “Instabilities and spatial complexity in a laser,” J. Opt. Soc. Am. B 7, 1019–1033 (1990).
[Crossref]

L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
[Crossref] [PubMed]

van Eijkelenborg, M. A.

M. A. van Eijkelenborg, M. P. van Exter, J. P. Woerdman, “Threshold characteristics and intensity fluctuations of lasers with excess quantum noise,” Phys. Rev. A 57, 571–579 (1998).
[Crossref]

Å. M. Lindberg, M. A. van Eijkelenborg, K. Joosten, G. Nienhuis, J. P. Woerdman, “Observation of excess quantum noise in a geometrically stable laser,” Phys. Rev. A 57, 3036–3039 (1998).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[Crossref] [PubMed]

van Exter, M. P.

M. A. van Eijkelenborg, M. P. van Exter, J. P. Woerdman, “Threshold characteristics and intensity fluctuations of lasers with excess quantum noise,” Phys. Rev. A 57, 571–579 (1998).
[Crossref]

Weber, H.

Weiss, C. O.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

Woerdman, J. P.

M. A. van Eijkelenborg, M. P. van Exter, J. P. Woerdman, “Threshold characteristics and intensity fluctuations of lasers with excess quantum noise,” Phys. Rev. A 57, 571–579 (1998).
[Crossref]

Å. M. Lindberg, M. A. van Eijkelenborg, K. Joosten, G. Nienhuis, J. P. Woerdman, “Observation of excess quantum noise in a geometrically stable laser,” Phys. Rev. A 57, 3036–3039 (1998).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[Crossref] [PubMed]

Wright, E. M.

A. M. Dunlop, W. J. Firth, E. M. Wright, “Master equation for spatio-temporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[Crossref]

Yakovlev, Yu. P.

T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
[Crossref]

Yu, A. W.

A. W. Yu, G. P. Agrawal, R. Roy, “Power spectra and spatial pattern dynamics of a ring laser,” J. Stat. Phys. 54, 1223–1241 (1989).
[Crossref]

Ziegler, M. O.

T. Burkhard, M. O. Ziegler, I. Fischer, W. Elsaber, “Spatio-temporal dynamics of broad area semiconductor lasers and its characterization,” Chaos, Solitons and Fractals 10, 845–850 (1999).

Appl. Opt. (1)

A. E. Siegmann, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2735 (1970); see also pages 570–577 of Ref. 12.
[Crossref]

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[Crossref]

Chaos, Solitons and Fractals (1)

T. Burkhard, M. O. Ziegler, I. Fischer, W. Elsaber, “Spatio-temporal dynamics of broad area semiconductor lasers and its characterization,” Chaos, Solitons and Fractals 10, 845–850 (1999).

J. Mod. Opt. (1)

M. A. Rippin, G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (4)

J. Stat. Phys. (1)

A. W. Yu, G. P. Agrawal, R. Roy, “Power spectra and spatial pattern dynamics of a ring laser,” J. Stat. Phys. 54, 1223–1241 (1989).
[Crossref]

Opt. Commun. (5)

N. N. Rasonov, V. E. Semenov, “The kinetics of the hysteresis change of the beam profile in nonlinear interferometers,” Opt. Commun. 38, 435–438 (1981).
[Crossref]

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[Crossref]

A. M. Dunlop, W. J. Firth, E. M. Wright, “Master equation for spatio-temporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[Crossref]

D. J. Jones, D. K. Bandy, “Hysteresis effects attributed to free-space diffraction in a passive optical system,” Opt. Commun. 92, 376–384 (1992).
[Crossref]

G. D’Alessandro, G. L. Oppo, “Gauss–Laguerre modes: a ‘sensible’ basis for laser dynamics,” Opt. Commun. 88, 130–136 (1992).
[Crossref]

Opt. Lett. (1)

Phys. Rep. (1)

L. A. Lugiato, “Spatio-temporal structures. Part I,” Phys. Rep. 219, 293–310 (1992).
[Crossref]

Phys. Rev. A (10)

W. Kaige, N. B. Abraham, L. A. Lugiato, “Leading role of optical phase instabilities in the formation of certain laser transverse patterns,” Phys. Rev. A 47, 1263–1273 (1993).
[Crossref]

L. A. Lugiato, F. Pratt, L. M. Narducci, P. Ru, J. R. Tredicce, D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
[Crossref] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref] [PubMed]

M. Brunel, G. Ropars, A. L. Floch, F. Bretenaker, “Transverse excess noise factor in geometrically stable resonators,” Phys. Rev. A 55, 4563–4567 (1997).
[Crossref]

K. Joosten, G. Nienhuis, “Combined longitudinal and transverse noise enhancement in lasers,” Phys. Rev. A 58, 4937–4945 (1998).
[Crossref]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penns, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. II. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[Crossref] [PubMed]

M. A. van Eijkelenborg, M. P. van Exter, J. P. Woerdman, “Threshold characteristics and intensity fluctuations of lasers with excess quantum noise,” Phys. Rev. A 57, 571–579 (1998).
[Crossref]

Å. M. Lindberg, M. A. van Eijkelenborg, K. Joosten, G. Nienhuis, J. P. Woerdman, “Observation of excess quantum noise in a geometrically stable laser,” Phys. Rev. A 57, 3036–3039 (1998).
[Crossref]

O. Emile, M. Brunel, F. Bretenaker, A. Le Floch, “Direct measurement of the transverse excess noise factor in a geometrically stable resonator,” Phys. Rev. A 57, 4889–4893 (1998).
[Crossref]

Phys. Rev. Lett. (2)

Yuh-Jen Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[Crossref] [PubMed]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[Crossref] [PubMed]

Semiconductors (1)

T. N. Danilova, A. P. Danilova, O. G. Ershov, A. N. Imenkov, V. V. Sherstnev, Yu. P. Yakovlev, “Spatial distribution of the radiation in the far zone of InAsSb/InAsAbP mesastrip lasers as a function of current,” Semiconductors 32, 339–342 (1998).
[Crossref]

Other (3)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 626–652.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), pp. 887–890.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 904–913.

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Figures (4)

Fig. 1
Fig. 1

Loss and matrix order required for 1% error as functions of mode number; g=0.8, h=3.88.

Fig. 2
Fig. 2

Dependence on α of loss and matrix order required; error limited to 1%, h=2.0, 4.0, and 6.0.

Fig. 3
Fig. 3

Dependence on mode number of loss, phase shift, photon number distortion, and field distortion; h=3.5, g=0.999.

Fig. 4
Fig. 4

Dependence on mode number of loss, phase shift, and photon number distortion; h=3.5, g=0.8.

Equations (39)

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f2(r2)=-iko2π|B| -dr1 f1(r1)Ω(r1)exp(ikoσ),
σ=12(r1B-1Ar1+r2DB-1r2-2r1B-1r2),
λψ(r2)=-iko2π|B| -dr1 ψ(r1)Ω(r1)exp(ikoσ).
K(r1, r2)=-ikoΩ(r1)exp(ikoσ)2π|B|
-dr Ω(r)expiko2(rB-1Ar-rDB-1r)
×ψi(r)ψj(r)=δij.
ABT=BAT,BTD=DTB
expikor2(D-D)B-1r22,
ψ(r)=ψ(r)expikor(D-D)B-1r2
B-1A+DB-1=B-1A+DB-1=S,
λψ(r2)=-iko2π|B| -dr1 ψ(r1)Ω(r1)exp(ikoσ),
σ=12(r1B-1Ar1+r2DB-1r2-2r1B-1r2).
λ=-λ*,
ψ=ψ*.
koσ=Axxu12+Ayyv12+Dxxu22+Dyyv22-2(u1u2+u1u2),
ui=xiko/Bxx,vi=yiko/Byyfori=1, 2,
Axx=Ayy=A,
Bxx=Byy=B,
Dxx=Dyy=D.
λψ(aou2)=-iξ0h2du1 J0(2ξu1u2)×exp[iξ(Au1+Du2)]ψ(aou1),
ξ=2[4-(D+A)2]1/2.
f=0hr dr|ϕ-ψ|20hr dr|ϕ|21/2,
p=0hr dr(|ϕ|2-|ψ|2)20hr dr|ϕ|41/2.
A=D>0.
gi=1-Λ2Ri,i=1, 2,
12<g1=g2=g<1.0.
g=1+A21/2.
λψ(aoxi)=j=1NMijψ(aoxj),
Mij=-iξwjJ0(2ξxixj)exp[iξA(xi+xj)]
0exp(-z)f(z)dz=j=1Nwjf(zj),
Mij=-iξwj exp(zj)J0(2ξzizj)exp[iξA(zi+zj)],
ξ=ξh2/zM
ψ(r)=i=1Nciϕi(r)
λci=j=1NMijcj,
Mij=0duϕi*(aou)0h2du1K(u, u1)ϕj(aou1).
Mlj=exp[-iθ(2l+1)]0h2duϕj(aou)ϕl(aou).
percenterror=100 0hr dr|ψ800-ψn|20hr dr|ψ800|2.
α=hξ=h(2g1-g2)-1/2,
matrixorderrequired=20α-30.

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