Abstract

It is generally assumed that a light beam with orbital angular momentum (OAM) per photon of lh̄, is transformed, when traversing a Dove prism, into a light beam with OAM per photon of -lh̄. In this paper, we show theoretically and experimentally that this OAM transformation rule does not apply for highly focused light beams. This result should be taken into account when designing classical and quantum algorithms that make use of Dove prims to manipulate the OAM of light.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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," Phys. Rev. A 458185 (1992).
    [CrossRef] [PubMed]
  2. G. Molina-Terriza, J. P. Torres anf L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002).
    [CrossRef] [PubMed]
  3. Graham Gibson, Johannes Courtial, Miles J. Padgett, Mikhail Vasnetsov, Valeriy Pasko, Stephen M. Barnett, and Sonja Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5448
    [CrossRef] [PubMed]
  4. R. J. Voogd, M. Singh, S. Pereira, A. van de Nes, and J. Braat, "The use of orbital angular momentum of light beams for super-high density optical data storage," OSA Annual Meeting, (Optical Society of America, 2004) paper FTuG14.
  5. Lluis Torner, Juan P. Torres, and Silvia Carrasco, "Digital spiral imaging," Opt. Express 13, 873 (2005).
    [CrossRef] [PubMed]
  6. A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentumstates of photons," Nature 412, 313 (2001).
    [CrossRef] [PubMed]
  7. A. Vaziri, G. Weihs and A. Zeilinger, "Experimental two-photon, three-dimensional entanglement for quantum communication," Phys. Rev. Lett. 89, 240401 (2002).
    [CrossRef] [PubMed]
  8. G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger," Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
    [CrossRef] [PubMed]
  9. JulioT. Barreiro, Nathan K. Langford, Nicholas A. Peters, and Paul G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005).
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics, Pergamon Press, 1993.
  11. A. N. de Oliveira, S. P. Walborn and C H Monken, "Implementing the Deutsch algorithm with polarization and transverse spatial modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
    [CrossRef]
  12. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef] [PubMed]
  13. R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006).
    [CrossRef] [PubMed]
  14. <jrn>. W. Chan, J.P. Torres, and J.H. Eberly, "Entanglement Migration of Biphotons in Spontaneous Parametric Downconversion," in CLEO/QELS 2006 Technical Digest (Optical Society of America, Long Beach, California, May 2006.</jrn>
  15. K. T. Kapale and J. P. Dowling, "Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams," Phys. Rev. Lett. 95173601 (2005).
    [CrossRef] [PubMed]
  16. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998).
    [CrossRef]
  17. M. J. Padgett and J. P. Lesso, "Dove prisms and polarized light," J. Mod. Opt. 46, 175-179 (1999)
  18. C. Cohen-Tannoudji, J. Dupont-Roc, and G Grynberg, "Atom-Photon Interactions : Basic Processes and Applications," (Wiley Science Paperback Series, 1992)
  19. <jrn>. J. Lekner, "Polarization of tightly focused beams," J. Opt. A: Pure Appl. Opt. bf 5, 6 (2003).</jrn>
    [CrossRef]
  20. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef] [PubMed]
  21. A. E. Siegman, Lasers, University Science Books, 1986.
  22. J. Visser and G. Nienhuis, "Orbital AngularMomentum of General AstigmaticModes," Phys. Rev. A 70, 013809 (2004).
    [CrossRef]
  23. I. S. Gradshteyn and I. M. Ryzhik, Tables of series, integrals and products, Academic Press, 1980. We make use of some useful properties of series of Bessel functions in chapters 8-9 about Special functions.

2006

R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

2005

K. T. Kapale and J. P. Dowling, "Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams," Phys. Rev. Lett. 95173601 (2005).
[CrossRef] [PubMed]

Lluis Torner, Juan P. Torres, and Silvia Carrasco, "Digital spiral imaging," Opt. Express 13, 873 (2005).
[CrossRef] [PubMed]

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger," Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

JulioT. Barreiro, Nathan K. Langford, Nicholas A. Peters, and Paul G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

A. N. de Oliveira, S. P. Walborn and C H Monken, "Implementing the Deutsch algorithm with polarization and transverse spatial modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

2004

2002

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

A. Vaziri, G. Weihs and A. Zeilinger, "Experimental two-photon, three-dimensional entanglement for quantum communication," Phys. Rev. Lett. 89, 240401 (2002).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres anf L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

2001

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentumstates of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

1999

M. J. Padgett and J. P. Lesso, "Dove prisms and polarized light," J. Mod. Opt. 46, 175-179 (1999)

1998

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

1992

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," Phys. Rev. A 458185 (1992).
[CrossRef] [PubMed]

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998).
[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," Phys. Rev. A 458185 (1992).
[CrossRef] [PubMed]

Barnett, S. M.

R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Beijersbergen, M. W.

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," Phys. Rev. A 458185 (1992).
[CrossRef] [PubMed]

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

de Oliveira, A. N.

A. N. de Oliveira, S. P. Walborn and C H Monken, "Implementing the Deutsch algorithm with polarization and transverse spatial modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

Dholakia, K.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

Dowling, J. P.

K. T. Kapale and J. P. Dowling, "Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams," Phys. Rev. Lett. 95173601 (2005).
[CrossRef] [PubMed]

Franke-Arnold, S.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Julio,

JulioT. Barreiro, Nathan K. Langford, Nicholas A. Peters, and Paul G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

Kapale, K. T.

K. T. Kapale and J. P. Dowling, "Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams," Phys. Rev. Lett. 95173601 (2005).
[CrossRef] [PubMed]

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Lesso, J. P.

M. J. Padgett and J. P. Lesso, "Dove prisms and polarized light," J. Mod. Opt. 46, 175-179 (1999)

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentumstates of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger," Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres anf L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Monken, C H

A. N. de Oliveira, S. P. Walborn and C H Monken, "Implementing the Deutsch algorithm with polarization and transverse spatial modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

Nienhuis, G.

J. Visser and G. Nienhuis, "Orbital AngularMomentum of General AstigmaticModes," Phys. Rev. A 70, 013809 (2004).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Padgett, M. J.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

M. J. Padgett and J. P. Lesso, "Dove prisms and polarized light," J. Mod. Opt. 46, 175-179 (1999)

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

Robertson, D. A.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998).
[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," Phys. Rev. A 458185 (1992).
[CrossRef] [PubMed]

Ursin, R.

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger," Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

Vaziri, A.

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger," Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

A. Vaziri, G. Weihs and A. Zeilinger, "Experimental two-photon, three-dimensional entanglement for quantum communication," Phys. Rev. Lett. 89, 240401 (2002).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentumstates of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

Visser, J.

J. Visser and G. Nienhuis, "Orbital AngularMomentum of General AstigmaticModes," Phys. Rev. A 70, 013809 (2004).
[CrossRef]

Walborn, S. P.

A. N. de Oliveira, S. P. Walborn and C H Monken, "Implementing the Deutsch algorithm with polarization and transverse spatial modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

Weihs, G.

A. Vaziri, G. Weihs and A. Zeilinger, "Experimental two-photon, three-dimensional entanglement for quantum communication," Phys. Rev. Lett. 89, 240401 (2002).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentumstates of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

Woerdman, J. P.

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," Phys. Rev. A 458185 (1992).
[CrossRef] [PubMed]

Zambrini, R.

R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

Zeilinger, A.

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger," Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

A. Vaziri, G. Weihs and A. Zeilinger, "Experimental two-photon, three-dimensional entanglement for quantum communication," Phys. Rev. Lett. 89, 240401 (2002).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentumstates of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

J. Mod. Opt.

M. J. Padgett and J. P. Lesso, "Dove prisms and polarized light," J. Mod. Opt. 46, 175-179 (1999)

J. Opt. B: Quantum Semiclass. Opt.

A. N. de Oliveira, S. P. Walborn and C H Monken, "Implementing the Deutsch algorithm with polarization and transverse spatial modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

Nature

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentumstates of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. A

J. Visser and G. Nienhuis, "Orbital AngularMomentum of General AstigmaticModes," Phys. Rev. A 70, 013809 (2004).
[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," Phys. Rev. A 458185 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

G. Molina-Terriza, J. P. Torres anf L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

A. Vaziri, G. Weihs and A. Zeilinger, "Experimental two-photon, three-dimensional entanglement for quantum communication," Phys. Rev. Lett. 89, 240401 (2002).
[CrossRef] [PubMed]

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger," Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

JulioT. Barreiro, Nathan K. Langford, Nicholas A. Peters, and Paul G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

K. T. Kapale and J. P. Dowling, "Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams," Phys. Rev. Lett. 95173601 (2005).
[CrossRef] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Other

A. E. Siegman, Lasers, University Science Books, 1986.

I. S. Gradshteyn and I. M. Ryzhik, Tables of series, integrals and products, Academic Press, 1980. We make use of some useful properties of series of Bessel functions in chapters 8-9 about Special functions.

C. Cohen-Tannoudji, J. Dupont-Roc, and G Grynberg, "Atom-Photon Interactions : Basic Processes and Applications," (Wiley Science Paperback Series, 1992)

<jrn>. J. Lekner, "Polarization of tightly focused beams," J. Opt. A: Pure Appl. Opt. bf 5, 6 (2003).</jrn>
[CrossRef]

<jrn>. W. Chan, J.P. Torres, and J.H. Eberly, "Entanglement Migration of Biphotons in Spontaneous Parametric Downconversion," in CLEO/QELS 2006 Technical Digest (Optical Society of America, Long Beach, California, May 2006.</jrn>

M. Born and E. Wolf, Principles of Optics, Pergamon Press, 1993.

R. J. Voogd, M. Singh, S. Pereira, A. van de Nes, and J. Braat, "The use of orbital angular momentum of light beams for super-high density optical data storage," OSA Annual Meeting, (Optical Society of America, 2004) paper FTuG14.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Geometrical configuration of a Dove prism. (a) Lateral view (yz-plane) and (b) Top view (xz-plane). Solid and dashed lines represent the typical path of two optical rays.

Fig. 2.
Fig. 2.

Location of the center of the light beam at the output plane. (a) The angle in the x-plane (ix ) is changed. (b) The angle in the y-plane (iy ) is changed. Dots: experimental results. Solid line: theoretical results.

Fig. 3.
Fig. 3.

Ellipticity of the output beam at the output plane, after traversing the Dove prism. Filled circles: Experimental results with the Dove prism. Triangles: experimental results when the Dove prism is removed. The solid and dashed lines are the theoretical results, as explained in the text. The dashed line corresponds to the theoretical value of the ellipticity (e=1) when the Dove prism is removed. Inset: Filled circles: x-axis; Empty circles: y-axis. Input beam waist: w 0≃50µm.

Fig. 4.
Fig. 4.

Spatial light intensity measured at the output plane, with the Dove prism removed (a) and (c), and with the Dove prism, (b) and (d). (a) and (b): w 0=560µm, (c) and (d) w 0=50µm. All dimensions are in µm.

Fig. 5.
Fig. 5.

OAMdecomposition of the output beam. (a) Input beam width w 0=20µm, winding number l=0; (b) w 0=100µm, l=0; (c) w 0=20µm, l=1; (d) w 0=100µm, l=1.

Fig. 6.
Fig. 6.

Weight of the central mode of the output beam. Solid line: weight of the mode m=0, for an input gaussian beam (l=0). Dashed line: weight of the m=-1 mode, for an input l=1 vortex beam.

Equations (23)

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

x 2 = x 1 + [ L n + h 0 tan α ( 1 1 n ) ] i x
o x = i x
y 2 = ( h 0 y 1 ) h 0 ( η n + 1 tan α ) i y
o y = i y
h 0 = L { tan [ α + sin 1 ( cos α n ) ] + 1 tan α } 1
η = h 0 sin α L [ 1 ( cos α n ) 2 ] 1 2 cos 2 { α + sin 1 ( cos α n ) }
z ̅ x = z x + [ L n h 0 tan α ( 1 1 n ) ]
z ̅ y = z y + h 0 [ η n 1 tan α ]
A out ( ρ , φ ) = N ( x w ̅ x + i y w ̅ y ) l exp ( x 2 w ̅ x 2 y 2 w ̅ y 2 ) exp ( i kx 2 2 R ̅ x + i ky 2 2 R ̅ y ) exp ( il φ )
A out ( ρ , φ ) = 1 ( 2 π ) 1 2 m a m ( ρ ) exp ( im φ )
C m = ( 1 2 l 2 l ! w ̅ x w ̅ y ) ρ 2 l + 1 d ρ exp [ ρ 2 ( 1 w ̅ x 2 + 1 w ̅ y 2 ) ]
× k = 0 l ( l k ) i k ( 1 w ̅ x 1 w ̅ y ) k ( 1 w ̅ x + 1 w ̅ y ) l k J ( l + m ) 2 k ( S ) 2
s = k ρ 2 4 ( 1 R ̅ x 1 R ̅ y ) + i ρ 2 2 ( 1 w ̅ x 2 1 w ̅ y 2 )
l { C m }
x 2 = x 1 + L tan ( i x ) + tan ( α ) tan ( α + i y ) tan ( i x ) 1 + tan ( α ) tan ( α + i y ) ,
y 2 = L ( tan ( α ) tan ( i y ) 1 + tan ( α ) tan ( α + i y ) ) y 1 ,
o x = i x , o y = i y .
x 2 = x 1 + [ L n + h 0 tan α ( 1 1 n ) ] i x
o x = i x
( y 2 h 0 2 ) = ( y 1 h 0 2 ) h 0 ( η n + 1 tan α ) i y
o y = i y
( x 2 o x ) = ( A x B x C x D x ) ( x 1 i x ) ,
q ̅ i = A i + B i q i C i + D i q i ,

Metrics