Abstract

We studied both numerically and experimentally the transverse modes near degenerate resonator configurations such as g1g2=1/4, 1/2, 3/4 in a simple plano–concave Nd:YVO4 laser with a small pump size. We found that there are stationary modes that show an additional beam waist besides the well-known waist on the flat mirror end near g1g2=1/4, 3/4. When the specific modes of the three degeneracies propagate through a lens, they are capable of exhibiting multiple beam waists. The multibeam-waist mode possesses a small waist size without a ring structure in the near field on the flat mirror end; nevertheless, it has a far-field pattern with many concentric rings. The numerical results show good agreement with the experimental observations. Furthermore, by simultaneously considering the wavelike and the raylike character of the multibeam-waist mode, we found that it can be represented as a superposition of N consecutive round-trip electric fields of period-N solution in the degenerate empty cavity, where N=2 for g1g2=1/2 and N=3 for g1g2=1/4, 3/4.

© 2003 Optical Society of America

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References

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  1. V. Couderc, O. Guy, A. Barthelemy, C. Froehly, and F. Louradour, “Self-optimized resonator for optical pumping of solid-state lasers,” Opt. Lett. 19, 1134–1136 (1994).
    [CrossRef] [PubMed]
  2. J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
    [CrossRef]
  3. N. J. van Druten, S. S. R. Oemrawsingh, Y. Lien, C. Serrat, M. P. van Exter, and J. P. Woerdman, “Observation of transverse modes in a microchip laser with combined gain and index guiding,” J. Opt. Soc. Am. B 18, 1793–1803 (2001).
    [CrossRef]
  4. 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]
  5. 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]
  6. S. A. Collins, “Lens-system diffraction-integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1174 (1970).
    [CrossRef]
  7. 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]
  8. A. G. Fox and T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. 2, 774–783 (1966).
    [CrossRef]
  9. 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]
  10. P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state laser,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
    [CrossRef]
  11. 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]
  12. I. A. Ramsay and J. J. Degnan, “A ray analysis of optical resonators formed by two spherical mirrors,” Appl. Opt. 9, 385–398 (1970).
    [CrossRef] [PubMed]

2001

1999

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

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

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]

1994

1991

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state laser,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

1970

1966

A. G. Fox and T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. 2, 774–783 (1966).
[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]

Barthelemy, A.

Brussard, M.

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state laser,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

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]

Collins, S. A.

Couderc, V.

Degnan, J. J.

Dingjan, J.

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[CrossRef]

Fox, A. G.

A. G. Fox and T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. 2, 774–783 (1966).
[CrossRef]

Froehly, C.

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]

Guy, O.

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]

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]

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]

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]

Laporta, P.

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state laser,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

Li, T.

A. G. Fox and T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. 2, 774–783 (1966).
[CrossRef]

Lien, Y.

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]

Louradour, F.

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]

Oemrawsingh, S. S. R.

Ramsay, I. A.

Serrat, C.

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]

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]

van Druten, N. J.

van Exter, M. P.

Wei, M. D.

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]

Woerdman, J. P.

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]

Appl. Opt.

IEEE J. Quantum Electron.

A. G. Fox and T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. 2, 774–783 (1966).
[CrossRef]

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]

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state laser,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

J. Mod. Opt.

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.

J. Opt. Soc. Am. B

Opt. Commun.

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]

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[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]

Opt. Lett.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup: BS, beam splitter; A, attenuator; other abbreviations defined in text.

Fig. 2
Fig. 2

(a) Numerical intensity profiles on the flat mirror end for L=6 cm (solid curve) and L=6.15 cm (dashed curve) together with the corresponding experimental photographs of beam-waist patterns in the near field. The graininess in the photographs is due to reflection from the screen. (b) Far-field intensity profiles that corresponding to the profiles in (a). Solid curve, degeneracy. (c) Experimental far-field mode pattern as a knife set in the cavity.

Fig. 3
Fig. 3

Propagating behavior for L=6 cm in Fig. 2(a): (a) through intracavity space and (b) through a transform lens. Note the correspondence between photographs in (a) and (b).

Fig. 4
Fig. 4

(a) Ray diagram for illuminating three sources of three beam waists for g1g2=1/4. Each round-trip ray emanating from the flat mirror toward the output coupler goes through a focus behind the convergent lens. (b) Periodic orbits of the q parameter for the empty cavity. The two concentric circles mean that there are infinite sets of period-N solutions. (c) Gaussian-beam evolution in the empty cavity.

Fig. 5
Fig. 5

Normalized intensity profiles of the period-3 superposition mode (open circles) and the self-consistent mode from the mode calculation (filled circles). Inset, their central parts with a linear scale on the vertical axis.

Fig. 6
Fig. 6

Ray picture similar to Fig. 4(a) for g1g2=3/4 together with three photographs taken before, at, and after the additional beam waist before the beam has gone through the transform lens. The elliptic central parts are due to astigmatism of the attenuator.

Fig. 7
Fig. 7

Lasing threshold versus wp for L=6, 6.15 cm.

Equations (5)

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

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