Abstract

We report experimental observations of the mode dynamics in a weakly multimode photorefractive ring oscillator with induced astigmatism. First a gallery of transverse, higher-order Gaussian modes in the system is presented. Subsequently the dynamics of beating modes and the interactions between modes belonging to different bases are examined. The latter show the transition of modes between the three equivalent mode families experimentally. Finally regions of cavity symmetry leading to different spatial modes and mode dynamics are identified.

© 2001 Optical Society of America

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  1. P. Pellat-Finet and J.-L. de la Tocnaye, “Optical generator of spheroidal wave functions using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
    [CrossRef]
  2. G. D’Alessandro, “Spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain,” Phys. Rev. A 46, 2791–2802 (1992).
    [CrossRef] [PubMed]
  3. D. Hennequin, L. Dambly, D. Dangoisse, and P. Glorieux, “Basic transverse dynamics of a photorefractive oscillator,” J. Opt. Soc. Am. B 11, 676–684 (1994).
    [CrossRef]
  4. J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
    [CrossRef] [PubMed]
  5. B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillations in unidirectional photorefractice ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
    [CrossRef] [PubMed]
  6. G. Balzer, C. Denz, O. Knaup, and T. Tschudi, “Circling vortices and pattern dynamics in a unidirectional photorefractive ring oscillator,” Chaos, Solitons Fractals 10, 725–730 (1999).
    [CrossRef]
  7. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
    [CrossRef] [PubMed]
  8. M. Vaupel, K. Staliunas, and C. O. Weiss, “Hydrodynamic phenomena in laser physics: Modes with flow and vortices behind an obstacle in an optical channel,” Phys. Rev. A 54, 880–892 (1996).
    [CrossRef] [PubMed]
  9. P. Yeh, “Two-wave mixing in nonlinear media,” Opt. Lett. 7, 484–486 (1989).
  10. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966).
    [CrossRef] [PubMed]
  11. J. T. Verdeyen, Laser Electronics (Prentice-Hall, Upper Saddle River, New Jersey, 1981).
  12. F. Encinas-Sanz, O. G. Calderón, R. Gutiérrez-Castrejón, and J. M. Guerra, “Measurement of the spatiotemporal dynamics of simple transverse patterns in a pulsed transversely excited atmospheric CO2 laser,” Phys. Rev. A 59, 4764–4772 (1999).
    [CrossRef]
  13. M. Ciofini, A. Labate, R. Meucci, and P.-Y. Wang, “Experimental evidence of selection and stabilization of spatial patterns in a CO2 laser by means of spatial perturbations,” Opt. Commun. 154, 307–312 (1998).
    [CrossRef]
  14. O. Svelto, ed. D. C. Hanna, translator and ed., Principles of Lasers 3rd ed. (Plenum Press, New York, 1989).
  15. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
    [CrossRef] [PubMed]
  16. M. Vaupel, C. O. Weiss, “Circling optical vortices,” Phys. Rev. A 51, 4078–4085 (1995).
    [CrossRef] [PubMed]

1999 (2)

G. Balzer, C. Denz, O. Knaup, and T. Tschudi, “Circling vortices and pattern dynamics in a unidirectional photorefractive ring oscillator,” Chaos, Solitons Fractals 10, 725–730 (1999).
[CrossRef]

F. Encinas-Sanz, O. G. Calderón, R. Gutiérrez-Castrejón, and J. M. Guerra, “Measurement of the spatiotemporal dynamics of simple transverse patterns in a pulsed transversely excited atmospheric CO2 laser,” Phys. Rev. A 59, 4764–4772 (1999).
[CrossRef]

1998 (1)

M. Ciofini, A. Labate, R. Meucci, and P.-Y. Wang, “Experimental evidence of selection and stabilization of spatial patterns in a CO2 laser by means of spatial perturbations,” Opt. Commun. 154, 307–312 (1998).
[CrossRef]

1996 (2)

M. Vaupel, K. Staliunas, and C. O. Weiss, “Hydrodynamic phenomena in laser physics: Modes with flow and vortices behind an obstacle in an optical channel,” Phys. Rev. A 54, 880–892 (1996).
[CrossRef] [PubMed]

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

1995 (2)

B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillations in unidirectional photorefractice ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
[CrossRef] [PubMed]

M. Vaupel, C. O. Weiss, “Circling optical vortices,” Phys. Rev. A 51, 4078–4085 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

G. D’Alessandro, “Spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain,” Phys. Rev. A 46, 2791–2802 (1992).
[CrossRef] [PubMed]

1991 (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

1990 (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

1989 (1)

P. Yeh, “Two-wave mixing in nonlinear media,” Opt. Lett. 7, 484–486 (1989).

1985 (1)

P. Pellat-Finet and J.-L. de la Tocnaye, “Optical generator of spheroidal wave functions using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
[CrossRef]

1966 (1)

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Balzer, G.

G. Balzer, C. Denz, O. Knaup, and T. Tschudi, “Circling vortices and pattern dynamics in a unidirectional photorefractive ring oscillator,” Chaos, Solitons Fractals 10, 725–730 (1999).
[CrossRef]

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Calderón, O. G.

F. Encinas-Sanz, O. G. Calderón, R. Gutiérrez-Castrejón, and J. M. Guerra, “Measurement of the spatiotemporal dynamics of simple transverse patterns in a pulsed transversely excited atmospheric CO2 laser,” Phys. Rev. A 59, 4764–4772 (1999).
[CrossRef]

Ciofini, M.

M. Ciofini, A. Labate, R. Meucci, and P.-Y. Wang, “Experimental evidence of selection and stabilization of spatial patterns in a CO2 laser by means of spatial perturbations,” Opt. Commun. 154, 307–312 (1998).
[CrossRef]

D’Alessandro, G.

G. D’Alessandro, “Spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain,” Phys. Rev. A 46, 2791–2802 (1992).
[CrossRef] [PubMed]

Dambly, L.

Dangoisse, D.

de la Tocnaye, J.-L.

P. Pellat-Finet and J.-L. de la Tocnaye, “Optical generator of spheroidal wave functions using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
[CrossRef]

Denz, C.

G. Balzer, C. Denz, O. Knaup, and T. Tschudi, “Circling vortices and pattern dynamics in a unidirectional photorefractive ring oscillator,” Chaos, Solitons Fractals 10, 725–730 (1999).
[CrossRef]

Encinas-Sanz, F.

F. Encinas-Sanz, O. G. Calderón, R. Gutiérrez-Castrejón, and J. M. Guerra, “Measurement of the spatiotemporal dynamics of simple transverse patterns in a pulsed transversely excited atmospheric CO2 laser,” Phys. Rev. A 59, 4764–4772 (1999).
[CrossRef]

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Glorieux, P.

Guerra, J. M.

F. Encinas-Sanz, O. G. Calderón, R. Gutiérrez-Castrejón, and J. M. Guerra, “Measurement of the spatiotemporal dynamics of simple transverse patterns in a pulsed transversely excited atmospheric CO2 laser,” Phys. Rev. A 59, 4764–4772 (1999).
[CrossRef]

Gutiérrez-Castrejón, R.

F. Encinas-Sanz, O. G. Calderón, R. Gutiérrez-Castrejón, and J. M. Guerra, “Measurement of the spatiotemporal dynamics of simple transverse patterns in a pulsed transversely excited atmospheric CO2 laser,” Phys. Rev. A 59, 4764–4772 (1999).
[CrossRef]

Hennequin, D.

Jost, B. M.

B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillations in unidirectional photorefractice ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
[CrossRef] [PubMed]

Knaup, O.

G. Balzer, C. Denz, O. Knaup, and T. Tschudi, “Circling vortices and pattern dynamics in a unidirectional photorefractive ring oscillator,” Chaos, Solitons Fractals 10, 725–730 (1999).
[CrossRef]

Kogelnik, H.

Labate, A.

M. Ciofini, A. Labate, R. Meucci, and P.-Y. Wang, “Experimental evidence of selection and stabilization of spatial patterns in a CO2 laser by means of spatial perturbations,” Opt. Commun. 154, 307–312 (1998).
[CrossRef]

Li, T.

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Malos, J.

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

Meucci, R.

M. Ciofini, A. Labate, R. Meucci, and P.-Y. Wang, “Experimental evidence of selection and stabilization of spatial patterns in a CO2 laser by means of spatial perturbations,” Opt. Commun. 154, 307–312 (1998).
[CrossRef]

Pellat-Finet, P.

P. Pellat-Finet and J.-L. de la Tocnaye, “Optical generator of spheroidal wave functions using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
[CrossRef]

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Saleh, B. E. A.

B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillations in unidirectional photorefractice ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
[CrossRef] [PubMed]

Staliunas, K.

M. Vaupel, K. Staliunas, and C. O. Weiss, “Hydrodynamic phenomena in laser physics: Modes with flow and vortices behind an obstacle in an optical channel,” Phys. Rev. A 54, 880–892 (1996).
[CrossRef] [PubMed]

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Tschudi, T.

G. Balzer, C. Denz, O. Knaup, and T. Tschudi, “Circling vortices and pattern dynamics in a unidirectional photorefractive ring oscillator,” Chaos, Solitons Fractals 10, 725–730 (1999).
[CrossRef]

Vaupel, M.

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

M. Vaupel, K. Staliunas, and C. O. Weiss, “Hydrodynamic phenomena in laser physics: Modes with flow and vortices behind an obstacle in an optical channel,” Phys. Rev. A 54, 880–892 (1996).
[CrossRef] [PubMed]

M. Vaupel, C. O. Weiss, “Circling optical vortices,” Phys. Rev. A 51, 4078–4085 (1995).
[CrossRef] [PubMed]

Wang, P.-Y.

M. Ciofini, A. Labate, R. Meucci, and P.-Y. Wang, “Experimental evidence of selection and stabilization of spatial patterns in a CO2 laser by means of spatial perturbations,” Opt. Commun. 154, 307–312 (1998).
[CrossRef]

Weiss, C. O.

M. Vaupel, K. Staliunas, and C. O. Weiss, “Hydrodynamic phenomena in laser physics: Modes with flow and vortices behind an obstacle in an optical channel,” Phys. Rev. A 54, 880–892 (1996).
[CrossRef] [PubMed]

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

M. Vaupel, C. O. Weiss, “Circling optical vortices,” Phys. Rev. A 51, 4078–4085 (1995).
[CrossRef] [PubMed]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Yeh, P.

P. Yeh, “Two-wave mixing in nonlinear media,” Opt. Lett. 7, 484–486 (1989).

Appl. Opt. (1)

Chaos, Solitons Fractals (1)

G. Balzer, C. Denz, O. Knaup, and T. Tschudi, “Circling vortices and pattern dynamics in a unidirectional photorefractive ring oscillator,” Chaos, Solitons Fractals 10, 725–730 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

P. Pellat-Finet and J.-L. de la Tocnaye, “Optical generator of spheroidal wave functions using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
[CrossRef]

M. Ciofini, A. Labate, R. Meucci, and P.-Y. Wang, “Experimental evidence of selection and stabilization of spatial patterns in a CO2 laser by means of spatial perturbations,” Opt. Commun. 154, 307–312 (1998).
[CrossRef]

Opt. Lett. (1)

P. Yeh, “Two-wave mixing in nonlinear media,” Opt. Lett. 7, 484–486 (1989).

Phys. Rev. A (7)

M. Vaupel, K. Staliunas, and C. O. Weiss, “Hydrodynamic phenomena in laser physics: Modes with flow and vortices behind an obstacle in an optical channel,” Phys. Rev. A 54, 880–892 (1996).
[CrossRef] [PubMed]

F. Encinas-Sanz, O. G. Calderón, R. Gutiérrez-Castrejón, and J. M. Guerra, “Measurement of the spatiotemporal dynamics of simple transverse patterns in a pulsed transversely excited atmospheric CO2 laser,” Phys. Rev. A 59, 4764–4772 (1999).
[CrossRef]

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillations in unidirectional photorefractice ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
[CrossRef] [PubMed]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singulary crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

M. Vaupel, C. O. Weiss, “Circling optical vortices,” Phys. Rev. A 51, 4078–4085 (1995).
[CrossRef] [PubMed]

G. D’Alessandro, “Spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain,” Phys. Rev. A 46, 2791–2802 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Other (2)

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Upper Saddle River, New Jersey, 1981).

O. Svelto, ed. D. C. Hanna, translator and ed., Principles of Lasers 3rd ed. (Plenum Press, New York, 1989).

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Figures (12)

Fig. 1
Fig. 1

Experimental configuration of the unidirectional PRO with induced asymmetry. FI, Faraday rotator; BSs, are beam splitters, Ms, mirrors; Ls, lenses, I, iris aperture, PZT, is a piezo-driven mirror; CCDs, CCD cameras; C optical c axis of the BaTiO3 crystal. The arrows indicate the beam polarization and translation of PZT.

Fig. 2
Fig. 2

Transverse mode arrangement of the PRO. νl and νT are the longitudinal and transverse mode spacing, Δνq is the width of a transverse mode, q is the mode family number, and ΔνP is the narrow linewidth of the two-beam coupling gain.

Fig. 3
Fig. 3

Experimental intensity patterns of transverse Hermite modes. Mode families up to the order of q=4 are shown. The modes in the dashed boxes are theoretical plots.

Fig. 4
Fig. 4

Experimental intensity patterns of transverse Laguerre modes. Mode families up to the order of q=4 are shown. The modes in the dashed boxes are theoretical plots.

Fig. 5
Fig. 5

Experimental intensity patterns of transverse doughnut modes. (a) First-order doughnut mode, (b) phase dislocation in the center of the mode, (c)–(e) second-, third-, and fourth-order doughnut mode, respectively.

Fig. 6
Fig. 6

Two examples of periodic mode beatings in the second-order mode family. I shows the sequence A021H021A104HA021. The first half of sequence II is almost equal to I, but ends at A021, the second half of II shows an alternative rout from A021 to A021.

Fig. 7
Fig. 7

Mode transition in the first-order mode family. The first-order doughnut mode B01 is obtained as a superposition of the Hermite mode H10 and H01 or the Laguerre modes A011 and A012.

Fig. 8
Fig. 8

Example of a mode transition in the second-order mode family, from the H02 to the H20 Hermite mode. The second-order doughnut mode B02 is obtained as a superposition of the modes H02, H20, and H11. The H11 mode can be identified in the picture just before the B02 mode.

Fig. 9
Fig. 9

Another example of a mode transition in the second-order mode family, from the H20 to the H02 mode. The Laguerre modes A10 and A021 are obtained as superpositions of H20 and H02 with a phase shift of 0 and π, respectively.

Fig. 10
Fig. 10

Example of a mode transition in the third-order mode family, passing from the H21 Hermite mode to the A111 Laguerre mode.

Fig. 11
Fig. 11

Another example of a mode transition in the third-order mode family, passing from the H30 to the H12 Hermite mode. The Laguerre modes A111 and A031 are obtained as superpositions of the H30 and H12 modes with appropriate phase shifts.

Fig. 12
Fig. 12

Example of interfamily mode interactions. Superposition of the fundamental mode, A00 and the third- and fourth-order doughnut mode, B031 and B042 are shown, resulting in circling optical vortices. The interferograms show the phase dislocations.

Equations (3)

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

νt=12π arccos1-L2 f cL=114 MHz,
Δνm=ν11-(Req)1/2π(Req)1/4=42 MHz.
νosc=νpΔνp+νmΔνm1Δνp+1Δνm>νmΔνp=νp+νm ΔνpΔνm.

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