Abstract

To provide the underlying physical mechanism for formations of spatial- and polarization-entangled lasing patterns (namely, SPEPs), we performed experiments using a c-cut Nd:GdVO4 microchip laser with off-axis laser-diode pumping. This extends recent work on entangled lasing pattern generation from an isotropic laser, where such a pattern was explained only in terms of generalized coherent states (GCSs) formed by mathematical manipulation. Here, we show that polarization-resolved transverse patterns can be well explained by the transverse mode-locking of distinct orthogonal linearly polarized Ince-Gauss (IG) mode pairs rather than GCSs. Dynamic properties of SPEPs were experimentally examined in both free-running and modulated conditions to identify long-term correlations of IG mode pairs over time. The complete chaos synchronization among IG mode pairs subjected to external perturbation is also demonstrated.

© 2009 OSA

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  13. J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285(5425), 230–233 (1999).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  25. J.-Y. Ko, C.-C. Lin, K. Otsuka, Y. Miyasaka, K. Kamikariya, K. Nemoto, M. C. Ho, and I. M. Jiang, “Experimental Observations of Dual-Polarization Oscillations in Laser-Diode-Pumped Wide-Aperture Thin-Slice Nd:GdVO(4) Lasers,” Opt. Express 15(3), 945–954 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-945 .
    [CrossRef] [PubMed]

2009 (2)

2007 (6)

J.-Y. Ko, C.-C. Lin, K. Otsuka, Y. Miyasaka, K. Kamikariya, K. Nemoto, M. C. Ho, and I. M. Jiang, “Experimental Observations of Dual-Polarization Oscillations in Laser-Diode-Pumped Wide-Aperture Thin-Slice Nd:GdVO(4) Lasers,” Opt. Express 15(3), 945–954 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-945 .
[CrossRef] [PubMed]

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15(17), 10705–10717 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-15-17-10705 .
[CrossRef] [PubMed]

S.-C. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15(25), 16506–16519 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16506 .
[CrossRef] [PubMed]

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026614 (2007).
[CrossRef] [PubMed]

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

K. Otsuka, K. Nemoto, K. Kamikariya, Y. Miyasaka, J.-Y. Ko, and C.-C. Lin, “Chaos synchronization among orthogonally polarized emissions in a dual-polarization laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 ), 026204 (2007).
[CrossRef] [PubMed]

2006 (2)

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, and K. F. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96(3), 033901 (2006).
[CrossRef] [PubMed]

2004 (3)

R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 (2004).
[CrossRef]

M. A. Bandres and J. C. Gutirrez-Vega, “Ince-Gaussian beams,” Opt. Lett. 29(2), 144–146 (2004).
[CrossRef] [PubMed]

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 (2004).
[CrossRef]

2002 (2)

Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41(22), 4630–4637 (2002).
[CrossRef] [PubMed]

K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 (2002).
[CrossRef] [PubMed]

2001 (1)

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 (2001).
[CrossRef] [PubMed]

2000 (3)

K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 (2000).
[CrossRef] [PubMed]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

1999 (1)

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285(5425), 230–233 (1999).
[CrossRef] [PubMed]

1998 (1)

E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 (1998).
[CrossRef]

1996 (1)

E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53(6), 4435–4438 (1996).
[CrossRef] [PubMed]

1991 (1)

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991).
[CrossRef] [PubMed]

1979 (1)

K. Otsuka, “Effects of external perturbations on LiNdP4O12 lasers,” IEEE J. Quantum Electron. 15(7), 655–663 (1979).
[CrossRef]

Bandres, M. A.

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Brambilla, M.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991).
[CrossRef] [PubMed]

Breban, R.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 (2001).
[CrossRef] [PubMed]

Chen, Y. F.

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026614 (2007).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, and K. F. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96(3), 033901 (2006).
[CrossRef] [PubMed]

Chern, J.-L.

R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 (2004).
[CrossRef]

K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 (2000).
[CrossRef] [PubMed]

Chu, S.-C.

Dangoisse, D.

E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 (1998).
[CrossRef]

E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53(6), 4435–4438 (1996).
[CrossRef] [PubMed]

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

DeShazer, D. J.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 (2001).
[CrossRef] [PubMed]

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Furhapter, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Glorieux, P.

E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 (1998).
[CrossRef]

E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53(6), 4435–4438 (1996).
[CrossRef] [PubMed]

Gutiérrez-Vega, J. C.

Gutirrez-Vega, J. C.

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

Hell, S. W.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[CrossRef] [PubMed]

Hennequin, D.

E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53(6), 4435–4438 (1996).
[CrossRef] [PubMed]

Ho, M. C.

Huang, K. F.

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026614 (2007).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, and K. F. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96(3), 033901 (2006).
[CrossRef] [PubMed]

Hwong, S.-L.

K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 (2000).
[CrossRef] [PubMed]

Inavalli, V. V.

Jahn, R.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[CrossRef] [PubMed]

Jesacher, A.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Jiang, I. M.

Kamikariya, K.

Kawai, R.

R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 (2004).
[CrossRef]

K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 (2000).
[CrossRef] [PubMed]

Ko, J.-Y.

K. Otsuka, K. Nemoto, K. Kamikariya, Y. Miyasaka, J.-Y. Ko, and C.-C. Lin, “Chaos synchronization among orthogonally polarized emissions in a dual-polarization laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 ), 026204 (2007).
[CrossRef] [PubMed]

J.-Y. Ko, C.-C. Lin, K. Otsuka, Y. Miyasaka, K. Kamikariya, K. Nemoto, M. C. Ho, and I. M. Jiang, “Experimental Observations of Dual-Polarization Oscillations in Laser-Diode-Pumped Wide-Aperture Thin-Slice Nd:GdVO(4) Lasers,” Opt. Express 15(3), 945–954 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-945 .
[CrossRef] [PubMed]

R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 (2004).
[CrossRef]

K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 (2002).
[CrossRef] [PubMed]

K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 (2000).
[CrossRef] [PubMed]

Leger, J. R.

Lih, J.-S.

R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 (2004).
[CrossRef]

Lin, C.-C.

Louvergneaux, E.

E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 (1998).
[CrossRef]

E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53(6), 4435–4438 (1996).
[CrossRef] [PubMed]

Lu, T. H.

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026614 (2007).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, and K. F. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96(3), 033901 (2006).
[CrossRef] [PubMed]

Lugiato, L. A.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991).
[CrossRef] [PubMed]

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Miyasaka, Y.

Nemoto, K.

Novotny, L.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

Ohtomo, T.

Orenstein, M.

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285(5425), 230–233 (1999).
[CrossRef] [PubMed]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Otsuka, K.

K. Otsuka and S.-C. Chu, “Generation of vortex array beams from a thin-slice solid-state laser with shaped wide-aperture laser-diode pumping,” Opt. Lett. 34(1), 10–12 (2009).
[CrossRef] [PubMed]

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15(17), 10705–10717 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-15-17-10705 .
[CrossRef] [PubMed]

J.-Y. Ko, C.-C. Lin, K. Otsuka, Y. Miyasaka, K. Kamikariya, K. Nemoto, M. C. Ho, and I. M. Jiang, “Experimental Observations of Dual-Polarization Oscillations in Laser-Diode-Pumped Wide-Aperture Thin-Slice Nd:GdVO(4) Lasers,” Opt. Express 15(3), 945–954 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-945 .
[CrossRef] [PubMed]

K. Otsuka, K. Nemoto, K. Kamikariya, Y. Miyasaka, J.-Y. Ko, and C.-C. Lin, “Chaos synchronization among orthogonally polarized emissions in a dual-polarization laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 ), 026204 (2007).
[CrossRef] [PubMed]

S.-C. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15(25), 16506–16519 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16506 .
[CrossRef] [PubMed]

R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 (2004).
[CrossRef]

K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 (2002).
[CrossRef] [PubMed]

K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 (2000).
[CrossRef] [PubMed]

K. Otsuka, “Effects of external perturbations on LiNdP4O12 lasers,” IEEE J. Quantum Electron. 15(7), 655–663 (1979).
[CrossRef]

Ott, E.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 (2001).
[CrossRef] [PubMed]

Penna, V.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991).
[CrossRef] [PubMed]

Prati, F.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991).
[CrossRef] [PubMed]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Rizzoli, S. O.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[CrossRef] [PubMed]

Roy, R.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 (2001).
[CrossRef] [PubMed]

Scheuer, J.

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285(5425), 230–233 (1999).
[CrossRef] [PubMed]

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

Slekys, G.

E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 (1998).
[CrossRef]

Tamm, C.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991).
[CrossRef] [PubMed]

Viswanathan, N. K.

Weiss, C. O.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991).
[CrossRef] [PubMed]

Westphal, V.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[CrossRef] [PubMed]

Willig, K. I.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[CrossRef] [PubMed]

Yoshioka, A.

K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 (2002).
[CrossRef] [PubMed]

Zhan, Q.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Chaos (1)

K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 (2002).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

K. Otsuka, “Effects of external perturbations on LiNdP4O12 lasers,” IEEE J. Quantum Electron. 15(7), 655–663 (1979).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt. (1)

R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 (2004).
[CrossRef]

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

N. J. Phys. (1)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Nature (1)

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (3)

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991).
[CrossRef] [PubMed]

E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 (1998).
[CrossRef]

E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53(6), 4435–4438 (1996).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026614 (2007).
[CrossRef] [PubMed]

K. Otsuka, K. Nemoto, K. Kamikariya, Y. Miyasaka, J.-Y. Ko, and C.-C. Lin, “Chaos synchronization among orthogonally polarized emissions in a dual-polarization laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 ), 026204 (2007).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 (2001).
[CrossRef] [PubMed]

K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 (2000).
[CrossRef] [PubMed]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, and K. F. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96(3), 033901 (2006).
[CrossRef] [PubMed]

Science (1)

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285(5425), 230–233 (1999).
[CrossRef] [PubMed]

Other (1)

K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Springer, 2000) pp. 176–181.

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Figures (13)

Fig. 1
Fig. 1

(a) Experimental setup. LD: laser diode, AP: anamorphic prism pair, P: polarizer, BS: beam splitter, SFP: scanning Fabry-Perot interferometer (free spectral range: 2 GHz; resolution: 6.6 MHz), DO: digital oscilloscope. IR viewer: PbS phototube with a TV monitor. (b) Typical far-field pattern of the SPEPs and the polarization-resolved structural change. Optical spectra for IG mode pairs are also shown. Polarization directions α (i.e., polarizer angles) and tilt directions ϕ of the IG mode pairs are indicated by arrows and dashed lines, respectively. Pump power P = 526 mW.

Fig. 2
Fig. 2

Experimental SPEPs formed from even- and odd-parity IGp,p modes, oscillation spectrum, and the associated polarization-resolved patterns. Pump power: (a) 534 mW, (b) 509 mW, (c) 498 mW. α1 ⊥ α2. Polarization directions of patterns in the right column: α1 + 45° for upper patterns; α2 + 45° for lower patterns.

Fig. 3
Fig. 3

Example analytical patterns of IG modes. (a) IGe 6,6 , (b) IGe 7,3 , (c) IGo 6,4 .

Fig. 4
Fig. 4

Theoretically reconstructed patterns corresponding to Fig. 2. Polarization directions α and tilt angles ϕ were set identical to those in Fig. 2.

Fig. 5
Fig. 5

(a) Experimental SPEPs formed from the IGe 4,4 and IGo 4,2 modes, oscillation spectrum, and associated polarization-resolved patterns. P = 530 mW. (b) Theoretically reconstructed patterns.

Fig. 6
Fig. 6

Typical example of single-frequency SPEPs and polarization-resolved patterns observed in the larger c-cut Nd:GdVO4 laser cavity. (a) Experimental result. Pump power, P = 91 mW. α1 = 55°, ϕ1 = 70°, α2 = 65°, ϕ2 = - 20°. (b) Theoretically reconstructed patterns.

Fig. 7
Fig. 7

Effect of phase difference ΔΨ on polarization-resolved transverse structures corresponding to Fig. 5(a). (a) Δψ = 0. (b) ± π/2.

Fig. 8
Fig. 8

Experimental setup for measuring temporal correlations among IG mode pairs of odd- and even IG8,8 shown in the inset. PD: photodiode (bandwidth, DC−125 MHz); DO: digital oscilloscope (bandwidth, DC−200 MHz).

Fig. 9
Fig. 9

Intensity fluctuations of odd- and even-party IG8.8 modes, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots under the free-running condition. P = 478 mW.

Fig. 10
Fig. 10

Synchronized chaotic oscillations with SPEP (i.e., total) beam feedback: Intensity fluctuations of odd- and even-party IG8.8 modes, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots when the self-mixing modulation was applied at fD = f1 = 200 kHz. P = 478 mW.

Fig. 11
Fig. 11

(a) Total beam (SPEP) feedback. (α: polarization direction of an observed pattern), (b) IGo 8,8 mode feedback, (c) IGe 8,8 mode feedback. fD = f1 = 200 kHz. P = 478 mW.

Fig. 12
Fig. 12

Failure of transverse mode locking: Polarization-resolved modal intensity fluctuations, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots of noise-driven random relaxation-oscillation fluctuations when the pump power was increased to P = 526 mW.

Fig. 13
Fig. 13

Unsynchronized chaotic relaxation oscillations with SPEP (total) beam feedback: polarization-resolved modal intensity fluctuations, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots when the self-mixing modulation was applied at fD = f1 = 480 kHz. P = 526 mW.

Equations (2)

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I G e p , m ( r , ε ) = C [ w 0 / w ( z ) ] C p m ( i ξ , ε ) C p m ( η , ε ) exp [ r 2 / w 2 ]                        × exp i [ k z + { k r 2 / 2 R ( z ) } ( p + 1 ) ψ z ( z ) ] ,
I G o p , m ( r , ε ) = S [ w 0 / w ( z ) ] S p m ( i ξ , ε ) S p m ( η , ε ) exp [ r 2 / w 2 ]                        × exp i [ k z + { k r 2 / 2 R ( z ) } ( p + 1 ) ψ z ( z ) ] ,

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