Abstract

We generate optical vortex beams in a nanosecond optical parametric oscillator based on an image-rotating resonator. This efficient new method of vortex generation should be adaptable to pulsed or continuous lasers.

© 2003 Optical Society of America

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References

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  1. K. T. Gahagan and G. A. Swartzlander, Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16, 533–537 (1999).
    [CrossRef]
  2. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
    [CrossRef]
  3. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [CrossRef] [PubMed]
  4. A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optically written waveguide in an atomic vapor,” Phys. Rev. Lett. 82, 1438–1441 (1999).
    [CrossRef]
  5. G. A. Swartzlander, Jr. and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
    [CrossRef]
  6. C. T. Law, X. Zhang, and G. A. Swartzlander, Jr., “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25, 55–57 (2000).
    [CrossRef]
  7. P. DiTrapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
    [CrossRef]
  8. Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
    [CrossRef]
  9. E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 14159–64 (1997).
    [CrossRef]
  10. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
    [CrossRef]
  11. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phy. Rev. A 45, 8185–8189 (1992).
    [CrossRef]
  12. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  13. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  14. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83123–135 (1991).
    [CrossRef]
  15. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  16. A. V. Smith and D. J. Armstrong, “A nanosecond optical parametric oscillator with 90° image rotation: design and performance,” J. Opt. Soc. Am. B. 19, 1801–1814 (2002).
    [CrossRef]
  17. D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams using an image-rotating resonator,” to be published in Applied Optics.

2002 (1)

A. V. Smith and D. J. Armstrong, “A nanosecond optical parametric oscillator with 90° image rotation: design and performance,” J. Opt. Soc. Am. B. 19, 1801–1814 (2002).
[CrossRef]

2000 (3)

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

C. T. Law, X. Zhang, and G. A. Swartzlander, Jr., “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25, 55–57 (2000).
[CrossRef]

P. DiTrapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef]

1999 (2)

K. T. Gahagan and G. A. Swartzlander, Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16, 533–537 (1999).
[CrossRef]

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optically written waveguide in an atomic vapor,” Phys. Rev. Lett. 82, 1438–1441 (1999).
[CrossRef]

1997 (3)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[CrossRef]

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 14159–64 (1997).
[CrossRef]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (4)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phy. Rev. A 45, 8185–8189 (1992).
[CrossRef]

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr. and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef]

1991 (1)

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83123–135 (1991).
[CrossRef]

Abramochkin, E.

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 14159–64 (1997).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83123–135 (1991).
[CrossRef]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phy. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Armstrong, D. J.

A. V. Smith and D. J. Armstrong, “A nanosecond optical parametric oscillator with 90° image rotation: design and performance,” J. Opt. Soc. Am. B. 19, 1801–1814 (2002).
[CrossRef]

D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams using an image-rotating resonator,” to be published in Applied Optics.

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phy. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Chen, Z.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[CrossRef]

Chinaglia, W.

P. DiTrapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef]

Davidson, N.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

DiTrapani, P.

P. DiTrapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef]

Friese, M. E. J.

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optically written waveguide in an atomic vapor,” Phys. Rev. Lett. 82, 1438–1441 (1999).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Friesem, A. A.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Gahagan, K. T.

Hasman, E.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optically written waveguide in an atomic vapor,” Phys. Rev. Lett. 82, 1438–1441 (1999).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Law, C. T.

C. T. Law, X. Zhang, and G. A. Swartzlander, Jr., “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25, 55–57 (2000).
[CrossRef]

G. A. Swartzlander, Jr. and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef]

Losevsky, N.

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 14159–64 (1997).
[CrossRef]

Maker, P. D.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[CrossRef]

McDuff, R.

Minardi, S.

P. DiTrapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef]

Muller, R. E.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[CrossRef]

Oron, R.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Phillips, M. C.

D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams using an image-rotating resonator,” to be published in Applied Optics.

Piskarskas, A.

P. DiTrapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef]

Rubinsztein-Dunlop, H.

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optically written waveguide in an atomic vapor,” Phys. Rev. Lett. 82, 1438–1441 (1999).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Segev, M.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[CrossRef]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Smith, A. V.

A. V. Smith and D. J. Armstrong, “A nanosecond optical parametric oscillator with 90° image rotation: design and performance,” J. Opt. Soc. Am. B. 19, 1801–1814 (2002).
[CrossRef]

D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams using an image-rotating resonator,” to be published in Applied Optics.

Smith, C. P.

Soskin, M. S.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phy. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Swartzlander, Jr., G. A.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Truscott, A. G.

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optically written waveguide in an atomic vapor,” Phys. Rev. Lett. 82, 1438–1441 (1999).
[CrossRef]

Valiulis, G.

P. DiTrapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasnetsov, M. V.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Volostnikov, V.

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 14159–64 (1997).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83123–135 (1991).
[CrossRef]

White, A. G.

Wilson, D. W.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phy. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Zhang, X.

J. Mod. Opt. (1)

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

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

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

A. V. Smith and D. J. Armstrong, “A nanosecond optical parametric oscillator with 90° image rotation: design and performance,” J. Opt. Soc. Am. B. 19, 1801–1814 (2002).
[CrossRef]

Opt. Commun. (4)

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83123–135 (1991).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 14159–64 (1997).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Opt. Lett. (2)

Phy. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phy. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Phys. Rev. Lett. (6)

P. DiTrapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef]

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[CrossRef]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optically written waveguide in an atomic vapor,” Phys. Rev. Lett. 82, 1438–1441 (1999).
[CrossRef]

G. A. Swartzlander, Jr. and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef]

Other (1)

D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams using an image-rotating resonator,” to be published in Applied Optics.

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Figures (5)

Fig. 1.
Fig. 1.

Nonplanar ring OPO cavity with 90° image rotation on each cavity pass.

Fig. 2.
Fig. 2.

Far field signal fluence profile for on-axis seeding. The subscript ‖ denotes the direction parallel to birefringent walkoff in the KTP crystal, while ⊥ denotes the direction perpendicular to walkoff.

Fig. 3.
Fig. 3.

Formation of a vortex beam from an off-axis seed beam when the cavity is tuned to a four-times-around longitudinal mode. The circles represent the seed beam after 0, 1, 2, and 3 passes around the cavity. The arrows represent the relative phases in each portion of the beam.

Fig. 4.
Fig. 4.

Far field signal fluence profile for off-axis seeding with the cavity tuned to a four-times-around resonance.

Fig. 5.
Fig. 5.

Interference pattern of OPO beam and spherical wave showing spiral phase with charge 1.

Equations (1)

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M = ( r s 2 r p 1 e i ϕ 1 0 0 r s 1 r p 2 e i ϕ 2 )

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