Abstract

We experimentally observe the formation of high-order transverse patterns in a microchip laser with a high degree of frequency degeneracy. With a doughnut pump profile the spontaneous transverse patterns are well localized in the Lissajous trajectories. The observed transverse patterns are reconstructed with a high degree of accuracy by use of the coherent state of quantum theory. The accurate reconstruction suggests that laser resonators can be designed to obtain a more thorough understanding of the quantum–classical connection.

© 2003 Optical Society of America

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References

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  1. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  2. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  3. Y. F. Chen and Y. P. Lan, Appl. Phys. B 72, 167 (2001).
    [CrossRef]
  4. Y. F. Chen and Y. P. Lan, Phys. Rev. A 66, 053812 (2002).
    [CrossRef]
  5. I. A. Ramsay and J. J. Degnan, Appl. Opt. 9, 385 (1970).
    [CrossRef] [PubMed]
  6. Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 046215 (2002).
    [CrossRef]
  7. Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 066210 (2002).
    [CrossRef]
  8. K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
    [CrossRef]
  9. D. Dragoman and M. Dragoman, Prog. Quantum Electron. 23, 131 (1999).
    [CrossRef]
  10. V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, Phys. Rev. Lett. 88, 014102 (2002).
    [CrossRef]

2002 (5)

Y. F. Chen and Y. P. Lan, Phys. Rev. A 66, 053812 (2002).
[CrossRef]

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 046215 (2002).
[CrossRef]

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 066210 (2002).
[CrossRef]

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, Phys. Rev. Lett. 88, 014102 (2002).
[CrossRef]

2001 (1)

Y. F. Chen and Y. P. Lan, Appl. Phys. B 72, 167 (2001).
[CrossRef]

1999 (1)

D. Dragoman and M. Dragoman, Prog. Quantum Electron. 23, 131 (1999).
[CrossRef]

1986 (1)

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

1984 (1)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

1970 (1)

Chen, Y. F.

Y. F. Chen and Y. P. Lan, Phys. Rev. A 66, 053812 (2002).
[CrossRef]

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 046215 (2002).
[CrossRef]

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 066210 (2002).
[CrossRef]

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef]

Y. F. Chen and Y. P. Lan, Appl. Phys. B 72, 167 (2001).
[CrossRef]

Degnan, J. J.

Doya, V.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, Phys. Rev. Lett. 88, 014102 (2002).
[CrossRef]

Dragoman, D.

D. Dragoman and M. Dragoman, Prog. Quantum Electron. 23, 131 (1999).
[CrossRef]

Dragoman, M.

D. Dragoman and M. Dragoman, Prog. Quantum Electron. 23, 131 (1999).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Huang, K. F.

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 046215 (2002).
[CrossRef]

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef]

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 066210 (2002).
[CrossRef]

Lai, H. C.

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef]

Lan, Y. P.

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef]

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 066210 (2002).
[CrossRef]

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 046215 (2002).
[CrossRef]

Y. F. Chen and Y. P. Lan, Phys. Rev. A 66, 053812 (2002).
[CrossRef]

Y. F. Chen and Y. P. Lan, Appl. Phys. B 72, 167 (2001).
[CrossRef]

Legrand, O.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, Phys. Rev. Lett. 88, 014102 (2002).
[CrossRef]

Miniatura, C.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, Phys. Rev. Lett. 88, 014102 (2002).
[CrossRef]

Mortessagne, F.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, Phys. Rev. Lett. 88, 014102 (2002).
[CrossRef]

Ramsay, I. A.

Siegman, A. E.

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

Appl. Opt. (1)

Appl. Phys. B (1)

Y. F. Chen and Y. P. Lan, Appl. Phys. B 72, 167 (2001).
[CrossRef]

Phys. Rev. A (1)

Y. F. Chen and Y. P. Lan, Phys. Rev. A 66, 053812 (2002).
[CrossRef]

Phys. Rev. E (2)

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 046215 (2002).
[CrossRef]

Y. F. Chen, K. F. Huang, and Y. P. Lan, Phys. Rev. E 66, 066210 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, Phys. Rev. Lett. 88, 014102 (2002).
[CrossRef]

Prog. Quantum Electron. (1)

D. Dragoman and M. Dragoman, Prog. Quantum Electron. 23, 131 (1999).
[CrossRef]

Other (2)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

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

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Figures (3)

Fig. 1
Fig. 1

Schematic of a fiber-coupled diode end-pumped laser; a typical pump profile of a fiber-coupled laser diode away from the focal plane.

Fig. 2
Fig. 2

Experimental results for the typical transverse patterns observed in a cavity length near ΔνL/ΔνT=3.

Fig. 3
Fig. 3

Numerically reconstructed patterns for the results shown in Fig. 2, calculated with Eq. (5). (a) Ψ20,54,1x,y;ϖo2, (b) Ψ16,55,2x,y;ϖo2, (c) Ψ10,36,3x,y;ϖo2, (d) Ξ10,37,4x,y;ϖo2, (e) Ψ8,38,5x,y;ϖo2, (f) Ψ8,39,6x,y;ϖo2.

Equations (5)

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

HGm,nx,y;ϖo=12m+n-1πm!n!1/21ϖoHm2xϖo×Hn2yϖoexp-x2+y2ϖo2,
νl,m,n=lΔνL+m+n+1ΔνT,
ΔνT=ΔνL1πcos-11-dR,
LG0,±Nx,y;ϖo=2-N/2K=0NNK1/2×±iKHGK,N-Kx,y;ϖo.
ΨN,Mp,qx,y;ϖo=K=JN-JNK-1/2K=JN-JNK1/2×HGpK,qN-Kx,y;ϖo,

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