Abstract

We propose and demonstrate experimentally a single lens design for an astigmatic mode converter that transforms the transverse mode of paraxial optical beams. As an application, we implement a controlled-not gate based on a Michelson interferometer in which the photon polarization is the control bit and the first order transverse mode is the target. As a further application, we also build a transverse mode parity sorter which can be useful for quantum information processing as a measurement device for the transverse mode qubit.

© 2010 Optical Society of America

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References

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  1. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum,” Nature 412, 313–316 (2001).
    [Crossref] [PubMed]
  2. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature 3, 305–310 (2007).
  3. C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
    [Crossref]
  4. C. E. R. Souza, P. Milman, J. A. O. Huguenin, and A. Z. Khoury, “Topological Phase for Spin-Orbit Transformations on a Laser Beam,” Phys. Rev. Lett. 99, 160401 (2007).
    [Crossref] [PubMed]
  5. 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 momenntum,” Opt. Commun. 96, 123–132 (1993).
    [Crossref]
  6. V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis, maZh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990)].
  7. J. A. O. Huguenin, B. Coutinho dos Santos, P. A. M. dos Santos, and A. Z. Khoury, “Topological defects in moire fringes with spiral zone plates,” J. Opt. Soc. Am. A 20, 1883–1889 (2003).
    [Crossref]
  8. S. J. van Enk, “Geometric phase, transformations of gaussian light beams and angular momentum transfer,” Opt. Comm. 102, 59–64 (1993).
    [Crossref]
  9. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24 No. 7 430–432 (1999).
    [Crossref]
  10. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).
  11. L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96, 163905 (2006).
    [Crossref] [PubMed]
  12. E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
    [Crossref] [PubMed]
  13. E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745 (2009).
    [Crossref]
  14. H. Sasada and M. Okamoto, “Transverse-mode beam spliter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
    [Crossref]
  15. C.C. Leary, L.A. Baumgardner, and M. G. Raymer, “Stable mode sorting by two-dimensional parity of photonic transverse spatial states,” Opt. Express 17, 2435 (2009).
    [Crossref] [PubMed]
  16. 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 Semiclassic. Opt. 7, 288 (2005).
    [Crossref]
  17. M. Fiorentino and F. N. C. Wong, “Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic,” Phys. Rev. Lett. 93070502 (2004).
    [Crossref] [PubMed]

2009 (3)

2008 (1)

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[Crossref]

2007 (2)

C. E. R. Souza, P. Milman, J. A. O. Huguenin, and A. Z. Khoury, “Topological Phase for Spin-Orbit Transformations on a Laser Beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature 3, 305–310 (2007).

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

2005 (1)

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 Semiclassic. Opt. 7, 288 (2005).
[Crossref]

2004 (1)

M. Fiorentino and F. N. C. Wong, “Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic,” Phys. Rev. Lett. 93070502 (2004).
[Crossref] [PubMed]

2003 (2)

H. Sasada and M. Okamoto, “Transverse-mode beam spliter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[Crossref]

J. A. O. Huguenin, B. Coutinho dos Santos, P. A. M. dos Santos, and A. Z. Khoury, “Topological defects in moire fringes with spiral zone plates,” J. Opt. Soc. Am. A 20, 1883–1889 (2003).
[Crossref]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

1999 (1)

1993 (2)

S. J. van Enk, “Geometric phase, transformations of gaussian light beams and angular momentum transfer,” Opt. Comm. 102, 59–64 (1993).
[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 momenntum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis, maZh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990)].

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 momenntum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Aolita, L.

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[Crossref]

Baumgardner, L.A.

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis, maZh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990)].

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 momenntum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Borges, C. V. S.

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).

Courtial, J.

Coutinho dos Santos, B.

De Martini, F.

E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745 (2009).
[Crossref]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

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 Semiclassic. Opt. 7, 288 (2005).
[Crossref]

dos Santos, P. A. M.

Fiorentino, M.

M. Fiorentino and F. N. C. Wong, “Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic,” Phys. Rev. Lett. 93070502 (2004).
[Crossref] [PubMed]

Huguenin, J. A. O.

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[Crossref]

C. E. R. Souza, P. Milman, J. A. O. Huguenin, and A. Z. Khoury, “Topological Phase for Spin-Orbit Transformations on a Laser Beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

J. A. O. Huguenin, B. Coutinho dos Santos, P. A. M. dos Santos, and A. Z. Khoury, “Topological defects in moire fringes with spiral zone plates,” J. Opt. Soc. Am. A 20, 1883–1889 (2003).
[Crossref]

Karimi, E.

E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745 (2009).
[Crossref]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Khoury, A. Z.

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[Crossref]

C. E. R. Souza, P. Milman, J. A. O. Huguenin, and A. Z. Khoury, “Topological Phase for Spin-Orbit Transformations on a Laser Beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

J. A. O. Huguenin, B. Coutinho dos Santos, P. A. M. dos Santos, and A. Z. Khoury, “Topological defects in moire fringes with spiral zone plates,” J. Opt. Soc. Am. A 20, 1883–1889 (2003).
[Crossref]

Leary, C.C.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Marrucci, L.

E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745 (2009).
[Crossref]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Milman, P.

C. E. R. Souza, P. Milman, J. A. O. Huguenin, and A. Z. Khoury, “Topological Phase for Spin-Orbit Transformations on a Laser Beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature 3, 305–310 (2007).

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 Semiclassic. Opt. 7, 288 (2005).
[Crossref]

Nagali, E.

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745 (2009).
[Crossref]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).

Okamoto, M.

H. Sasada and M. Okamoto, “Transverse-mode beam spliter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[Crossref]

Padgett, M. J.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Piccirillo, B.

E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745 (2009).
[Crossref]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Raymer, M. G.

Santamato, E.

E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745 (2009).
[Crossref]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Sasada, H.

H. Sasada and M. Okamoto, “Transverse-mode beam spliter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[Crossref]

Sciarrino, F.

E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745 (2009).
[Crossref]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Soskin, M. S.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis, maZh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990)].

Souza, C. E. R.

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[Crossref]

C. E. R. Souza, P. Milman, J. A. O. Huguenin, and A. Z. Khoury, “Topological Phase for Spin-Orbit Transformations on a Laser Beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature 3, 305–310 (2007).

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature 3, 305–310 (2007).

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 momenntum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

van Enk, S. J.

S. J. van Enk, “Geometric phase, transformations of gaussian light beams and angular momentum transfer,” Opt. Comm. 102, 59–64 (1993).
[Crossref]

Vasnetsov, M. V.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis, maZh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990)].

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Walborn, S. P.

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[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 Semiclassic. Opt. 7, 288 (2005).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

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 momenntum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Wong, F. N. C.

M. Fiorentino and F. N. C. Wong, “Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic,” Phys. Rev. Lett. 93070502 (2004).
[Crossref] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

J. Opt. B: Quantum Semiclassic. Opt. (1)

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 Semiclassic. Opt. 7, 288 (2005).
[Crossref]

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

Nature (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature 3, 305–310 (2007).

Opt. Comm. (1)

S. J. van Enk, “Geometric phase, transformations of gaussian light beams and angular momentum transfer,” Opt. Comm. 102, 59–64 (1993).
[Crossref]

Opt. Commun. (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 momenntum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (2)

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[Crossref]

H. Sasada and M. Okamoto, “Transverse-mode beam spliter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[Crossref]

Phys. Rev. Lett. (4)

M. Fiorentino and F. N. C. Wong, “Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic,” Phys. Rev. Lett. 93070502 (2004).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

C. E. R. Souza, P. Milman, J. A. O. Huguenin, and A. Z. Khoury, “Topological Phase for Spin-Orbit Transformations on a Laser Beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

Pis, maZh. Eksp. Teor. Fiz. (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis, maZh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990)].

Other (1)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).

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Figures (5)

Fig. 1.
Fig. 1.

Setup of the single lens mode converter. (a) Scheme for the π/2 mode converter; (b) Images of the input modes; (c) Images of the output modes obtained with the single-lens π/2 converter.

Fig. 2.
Fig. 2.

Setup for the Michelson CNOT gate using the polarization and first order transverse modes

Fig. 3.
Fig. 3.

(a) Schematic representation of the action of the CNOT gate; (b) Experimental images in the input and output of the CNOT gate.

Fig. 4.
Fig. 4.

Setup of the transverse mode sorter with the single-lens mode converter.

Fig. 5.
Fig. 5.

Experimental results: (a) Image 1: input beam prepared in a vertical HG mode. Images 2-4: output of the TMS for different voltages and the SLMC oriented at θ = 0°; (b) Image 1: input beam prepared in a vertical HG mode. Images 2-4: output of the TMS for different voltages and the SLMC oriented at θ = 45°. (c) Image 1: input beam prepared in an LG mode. Images 2-4: output of the TMS for different voltages and the SLMC oriented at θ = 45°.

Equations (2)

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ϕ 1 = 1 2 ( H h + V ν ) ,
ϕ 2 = 1 2 ( H ν + V h ) .

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