Abstract

Propagation-dominant instabilities and chaos were found under so-called good-cavity conditions in an axially pumped solid-state laser operated near the 1/3-degenerate cavity configuration that had not previously been studied numerically. By using the generalized Huygens integral together with rate equations, we obtained a V-shaped configuration that depends on a quasi-periodic threshold. We call the propagation dominant because the laser behaves as a conservative system governed by beam propagation. Although it had previously been predicted that chaos would be impossible under nearly degenerate conditions, we have recognized that the laser is transformed into chaos as a result of the interplay of beam propagation and gain dynamics as the cavity is tuned close to degeneracy.

© 2001 Optical Society of America

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References

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  1. L. A. Melnikov, S. A. Tatarkova, and G. N. Tatarkov, “Nonlinear dynamics of beam parameters and intensity in a unidirectional ring laser with a homogeneously broadened line,” J. Opt. Soc. Am. B 7, 1286–1292 (1990).
    [CrossRef]
  2. A. E. Siegman, Lasers (University Science, Mill Valley, California, 1986), p. 761.
  3. F. Hollinger and Chr. Jung, “Single-longitudinal-mode laser as a discrete dynamical system,” J. Opt. Soc. Am. B 2, 218–225 (1985).
    [CrossRef]
  4. R. Hauck, F. Hollinger, and H. Weber, “Chaotic and periodic emission of high power solid state lasers,” Opt. Commun. 47, 141–145 (1983).
    [CrossRef]
  5. F. Hollinger, Chr. Jung, and H. Weber, “Quasiperiodicity versus chaos in high power solid state lasers in multitransversal mode operation,” Opt. Commun. 75, 84–92 (1990).
    [CrossRef]
  6. L. A. Lugiato, F. Prati, L. M. Naducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A 37, 3847–3866 (1988).
    [CrossRef] [PubMed]
  7. L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and M. A. Pernigo, “Instabilities and spatial complexity in a laser,” J. Opt. Soc. Am. B 7, 1019–1033 (1990).
    [CrossRef]
  8. G. D’Alessandro and G. L. Oppo, “Gauss–Laguerre modes: a ‘sensible’ basis for laser dynamics,” Opt. Commun. 88, 130–136 (1992).
    [CrossRef]
  9. J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys. 20, 1183–1201 (1979).
    [CrossRef]
  10. M. D. Wei, W. F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
    [CrossRef]
  11. M. D. Wei and W. F. Hsieh, “Bifurcation of fundamental gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
    [CrossRef]
  12. M. D. Wei and W. F. Hsieh, “Cavity configuration dependent nonlinear dynamics in Kerr lens mode-locked lasers,” J. Opt. Soc. Am. B 17, 1335–1342 (2000).
    [CrossRef]
  13. H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid state laser with different fractionally degenerate resonator configurations,” J. Opt. Soc. Am. B 18, 7–12 (2001).
    [CrossRef]
  14. H. H. Wu, C. C. Sheu, T. W. Chen, M. D. Wei, and W. F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
    [CrossRef]
  15. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (Weinheim, New York, 1991), pp. 14, 26.
  16. Y. J. Cheng, P. L. Mussche, and A. E. Siegman, “Cavity decay rate and relaxation oscillation frequency in unconventional laser cavities,” IEEE J. Quantum Electron. 31, 391–398 (1995).
    [CrossRef]
  17. M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
    [CrossRef]
  18. M. L. Shih and P. W. Milonni, “Chaotic two-mode lasing,” Opt. Commun. 49, 155–160 (1984).
    [CrossRef]
  19. F. Hollinger, Chr. Jung, and H. Weber, “Simple mathematical model describing multitransversal solid-state lasers,” J. Opt. Soc. Am. B 7, 1013–1018 (1990).
    [CrossRef]

2001 (1)

2000 (1)

1999 (2)

M. D. Wei and W. F. Hsieh, “Bifurcation of fundamental gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
[CrossRef]

H. H. Wu, C. C. Sheu, T. W. Chen, M. D. Wei, and W. F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

1998 (2)

M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
[CrossRef]

M. D. Wei, W. F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

1995 (1)

Y. J. Cheng, P. L. Mussche, and A. E. Siegman, “Cavity decay rate and relaxation oscillation frequency in unconventional laser cavities,” IEEE J. Quantum Electron. 31, 391–398 (1995).
[CrossRef]

1992 (1)

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

1990 (4)

1988 (1)

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

1985 (1)

1984 (1)

M. L. Shih and P. W. Milonni, “Chaotic two-mode lasing,” Opt. Commun. 49, 155–160 (1984).
[CrossRef]

1983 (1)

R. Hauck, F. Hollinger, and H. Weber, “Chaotic and periodic emission of high power solid state lasers,” Opt. Commun. 47, 141–145 (1983).
[CrossRef]

1979 (1)

J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys. 20, 1183–1201 (1979).
[CrossRef]

Ackemann, T.

M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
[CrossRef]

Bandy, D. K.

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

Chen, T. W.

H. H. Wu, C. C. Sheu, T. W. Chen, M. D. Wei, and W. F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

Cheng, Y. J.

Y. J. Cheng, P. L. Mussche, and A. E. Siegman, “Cavity decay rate and relaxation oscillation frequency in unconventional laser cavities,” IEEE J. Quantum Electron. 31, 391–398 (1995).
[CrossRef]

D’Alessandro, G.

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

Gahl, A.

M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
[CrossRef]

Greene, J. M.

J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys. 20, 1183–1201 (1979).
[CrossRef]

Hauck, R.

R. Hauck, F. Hollinger, and H. Weber, “Chaotic and periodic emission of high power solid state lasers,” Opt. Commun. 47, 141–145 (1983).
[CrossRef]

Hoffer, L. M.

M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
[CrossRef]

Hollinger, F.

F. Hollinger, Chr. Jung, and H. Weber, “Simple mathematical model describing multitransversal solid-state lasers,” J. Opt. Soc. Am. B 7, 1013–1018 (1990).
[CrossRef]

F. Hollinger, Chr. Jung, and H. Weber, “Quasiperiodicity versus chaos in high power solid state lasers in multitransversal mode operation,” Opt. Commun. 75, 84–92 (1990).
[CrossRef]

F. Hollinger and Chr. Jung, “Single-longitudinal-mode laser as a discrete dynamical system,” J. Opt. Soc. Am. B 2, 218–225 (1985).
[CrossRef]

R. Hauck, F. Hollinger, and H. Weber, “Chaotic and periodic emission of high power solid state lasers,” Opt. Commun. 47, 141–145 (1983).
[CrossRef]

Hsieh, W. F.

H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid state laser with different fractionally degenerate resonator configurations,” J. Opt. Soc. Am. B 18, 7–12 (2001).
[CrossRef]

M. D. Wei and W. F. Hsieh, “Cavity configuration dependent nonlinear dynamics in Kerr lens mode-locked lasers,” J. Opt. Soc. Am. B 17, 1335–1342 (2000).
[CrossRef]

M. D. Wei and W. F. Hsieh, “Bifurcation of fundamental gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
[CrossRef]

H. H. Wu, C. C. Sheu, T. W. Chen, M. D. Wei, and W. F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

M. D. Wei, W. F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

Jung, Chr.

Lange, W.

M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
[CrossRef]

Lippi, G. L.

M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
[CrossRef]

Lugiato, L. A.

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

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

Melnikov, L. A.

Milonni, P. W.

M. L. Shih and P. W. Milonni, “Chaotic two-mode lasing,” Opt. Commun. 49, 155–160 (1984).
[CrossRef]

Moller, M.

M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
[CrossRef]

Mussche, P. L.

Y. J. Cheng, P. L. Mussche, and A. E. Siegman, “Cavity decay rate and relaxation oscillation frequency in unconventional laser cavities,” IEEE J. Quantum Electron. 31, 391–398 (1995).
[CrossRef]

Naducci, L. M.

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

Narducci, L. M.

Oppo, G. L.

G. D’Alessandro and 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, and M. A. Pernigo, “Instabilities and spatial complexity in a laser,” J. Opt. Soc. Am. B 7, 1019–1033 (1990).
[CrossRef]

Pernigo, M. A.

Prati, F.

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

Ru, P.

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

Sheu, C. C.

H. H. Wu, C. C. Sheu, T. W. Chen, M. D. Wei, and W. F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

Shih, M. L.

M. L. Shih and P. W. Milonni, “Chaotic two-mode lasing,” Opt. Commun. 49, 155–160 (1984).
[CrossRef]

Siegman, A. E.

Y. J. Cheng, P. L. Mussche, and A. E. Siegman, “Cavity decay rate and relaxation oscillation frequency in unconventional laser cavities,” IEEE J. Quantum Electron. 31, 391–398 (1995).
[CrossRef]

Sung, C. C.

M. D. Wei, W. F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

Tatarkov, G. N.

Tatarkova, S. A.

Tredicce, J. R.

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

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

Weber, H.

F. Hollinger, Chr. Jung, and H. Weber, “Quasiperiodicity versus chaos in high power solid state lasers in multitransversal mode operation,” Opt. Commun. 75, 84–92 (1990).
[CrossRef]

F. Hollinger, Chr. Jung, and H. Weber, “Simple mathematical model describing multitransversal solid-state lasers,” J. Opt. Soc. Am. B 7, 1013–1018 (1990).
[CrossRef]

R. Hauck, F. Hollinger, and H. Weber, “Chaotic and periodic emission of high power solid state lasers,” Opt. Commun. 47, 141–145 (1983).
[CrossRef]

Wei, M. D.

M. D. Wei and W. F. Hsieh, “Cavity configuration dependent nonlinear dynamics in Kerr lens mode-locked lasers,” J. Opt. Soc. Am. B 17, 1335–1342 (2000).
[CrossRef]

M. D. Wei and W. F. Hsieh, “Bifurcation of fundamental gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
[CrossRef]

H. H. Wu, C. C. Sheu, T. W. Chen, M. D. Wei, and W. F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

M. D. Wei, W. F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

Wu, H. H.

H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid state laser with different fractionally degenerate resonator configurations,” J. Opt. Soc. Am. B 18, 7–12 (2001).
[CrossRef]

H. H. Wu, C. C. Sheu, T. W. Chen, M. D. Wei, and W. F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. J. Cheng, P. L. Mussche, and A. E. Siegman, “Cavity decay rate and relaxation oscillation frequency in unconventional laser cavities,” IEEE J. Quantum Electron. 31, 391–398 (1995).
[CrossRef]

J. Math. Phys. (1)

J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys. 20, 1183–1201 (1979).
[CrossRef]

J. Mod. Opt. (1)

M. Moller, L. M. Hoffer, G. L. Lippi, T. Ackemann, A. Gahl, and W. Lange, “Fabry–Perot and ring cavity configurations and transverse optical patterns,” J. Mod. Opt. 45, 1913–1926 (1998).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Commun. (7)

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

R. Hauck, F. Hollinger, and H. Weber, “Chaotic and periodic emission of high power solid state lasers,” Opt. Commun. 47, 141–145 (1983).
[CrossRef]

F. Hollinger, Chr. Jung, and H. Weber, “Quasiperiodicity versus chaos in high power solid state lasers in multitransversal mode operation,” Opt. Commun. 75, 84–92 (1990).
[CrossRef]

H. H. Wu, C. C. Sheu, T. W. Chen, M. D. Wei, and W. F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

M. L. Shih and P. W. Milonni, “Chaotic two-mode lasing,” Opt. Commun. 49, 155–160 (1984).
[CrossRef]

M. D. Wei, W. F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

M. D. Wei and W. F. Hsieh, “Bifurcation of fundamental gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
[CrossRef]

Phys. Rev. A (1)

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

Other (2)

A. E. Siegman, Lasers (University Science, Mill Valley, California, 1986), p. 761.

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (Weinheim, New York, 1991), pp. 14, 26.

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Figures (8)

Fig. 1
Fig. 1

Configuration of the laser system.

Fig. 2
Fig. 2

(a) Output power evolution and (b) beam profile of the period-2 steady state for g1g2=1/2 with ρ=0.95 above the instability threshold.

Fig. 3
Fig. 3

(a) Evolution of the output power of the quasi-periodic oscillation at g1g2=0.25466, ρ=0.95, Ψ=2.32, and Pp=210 mW. The inset is the magnification of six precession periods. (b) Phase space of (w, 1/R), as in Ref. 10, provides an explanation. The numbered filled circles stand for the number of iterations. (c) The corresponding spectrum of (a).

Fig. 4
Fig. 4

(a) Three-dimensional quasi-periodic bifurcation diagram in terms of Pp, Ψ, and g1g2 for ρ=0.95. (b) Dependence of the ratio of the instability threshold (P2) to the lasing threshold (P1) on parameter Ψ for g1g2=1/4 and ρ=0.95. (c) Dependence of P2/P1 on g1g2 for Ψ=1.3 with different values of ρ as indicated.

Fig. 5
Fig. 5

(a) Bifurcation diagram for higher pumping with ρ=0.95 and Ψ=2.78 near g1g2=1/4, (b) power evolution of a modulated quasi period, (c) modulated pulsing, (d) chaos.

Fig. 6
Fig. 6

(a) Precession oscillation and (b) power spectrum at g1g2=0.25466 with ρ=0.95, Ψ=2.78, and Pp=650 mW.

Fig. 7
Fig. 7

Frequency bifurcation plot using Pp as the parameter for (a) R=8.05 cm and (b) R=8.0075 cm.

Fig. 8
Fig. 8

Transverse beat frequency of the cold cavity and the precession frequency versus R for Pp fixed at 650 mW.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

Em+1-(r)=2πjBλ  exp(jk2L)Em+(r)exp[-(jπ/Bλ)×(Ar2+Dr2)]J0(2πrr/Bλ)rdr,
ABCD.
Em+1+(r)=ρEm+1-(r)exp(σΔNd)(r/a),
ΔNm+1=ΔNm+Rpm(N0-ΔNm)Δt-γΔNmΔt-(|Em|2/Es2)ΔNmΔt,
 RpmdV=Pp/hνp,

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