Abstract

We describe a mode sorter for two-dimensional parity of transverse spatial states of light based on an out-of-plane Sagnac interferometer. Both Hermite-Gauss (HG) and Laguerre-Gauss (LG) modes can be guided into one of two output ports according to the two-dimensional parity of the mode in question. Our interferometer sorts HGnm input modes depending upon whether they have even or odd order n+m; it equivalently sorts LGlp modes depending upon whether they have an even or odd value of their orbital angular momentum l. It functions efficiently at the single-photon level, and therefore can be used to sort single-photon states. Due to the inherent phase stability of this type of interferometer as compared to those of the Mach-Zehnder type, it provides a promising tool for the manipulation and filtering of higher order transverse spatial modes for the purposes of quantum information processing. For example, several similar Sagnacs cascaded together may allow, for the first time, a stable measurement of the orbital angular momentum of a true single-photon state. Furthermore, as an alternative to well-known holographic techniques, one can use the Sagnac in conjunction with a multi-mode fiber as a spatial mode filter, which can be used to produce spatial-mode entangled Bell states and heralded single photons in arbitrary first-order (n+m = 1) spatial states, covering the entire Poincaré sphere of first-order transverse modes.

©2009 Optical Society of America

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  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 45, 8185–8189 (1992).
    [Crossref] [PubMed]
  2. 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]
  3. H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
    [Crossref]
  4. X. Xue, H. Wei, and A. G. Kirk, “Beam analysis by fractional Fourier transform,” Opt. Lett. 26, 1746–1748 (2001).
    [Crossref]
  5. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
    [Crossref]
  6. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and Analysis of Entangled Photonic Qubits in Spatial-Parity Space,” Phys. Rev. Lett. 99, 250502 (2007).
    [Crossref]
  7. S. J. van Enk, “Geometric phase, transformations of gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
    [Crossref]
  8. E. J. Galvez and C. D. Holmes, “Geometric phase of optical rotators,” J. Opt. Soc. Am. A 16, 1981–1985 (1999).
    [Crossref]
  9. B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
    [Crossref]
  10. L. L. Smith and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve inputoutput collinearity,” J. Opt. Soc. Am. A 13, 21022105 (1996).
    [Crossref]
  11. E. Mukamel, K. Banaszek, I. A. Walmsley, and C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003).
    [Crossref] [PubMed]
  12. A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449450 (1977).
    [Crossref]
  13. M. P. van Exter, P. S. K. Lee, S. Doesburg, and J. P. Woerdman, “Mode counting in high-dimensional orbital angular momentum entanglement,” Opt. Express 15, 6431–6438 (2007).
    [Crossref] [PubMed]
  14. A. Siegman, Lasers (University Science Books, Mill Valley CA, 1986), pp. 646–648.
  15. V. S. Liberman and B. Ya. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
    [Crossref] [PubMed]
  16. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).
  17. C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
    [Crossref] [PubMed]
  18. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
    [Crossref]
  19. S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A 71, 053812 (2005).
    [Crossref]
  20. N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
    [Crossref] [PubMed]
  21. D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of Orbital Angular Momentum from a Stressed Fiber-Optic Waveguide to a Light Beam,” Appl. Opt. 37, 469472 (1998)
    [Crossref]
  22. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, New York, 1991), p. 226, pp. 233–234.
  23. In order to interpret the final columns of Figs. 6(a) and 6(b), recall that the output beam has been rotated 90° with respect to the input beam, regardless of output port. This has no effect upon the rotationally symmetric HG00 mode, so the familiar interference fringes of a standard Mach-Zehnder interferometer are observed. However the 90° rotation does effect the first-order modes, so that the next two rows exhibit an interference pattern resulting form the superposition of the input HG mode and its 90° rotated counterpart. Note the characteristic “forking” of the vertical fringes in both of these plots, which shows the nontrivial phase structure of these modes as they interfere. For the HG45° mode, one would expect a similar “forked” pattern, but rotated by 90°. However, practical considerations required the presence of an extra mirror along the reference beam path in order to interfere the reference and output beams. Since an extra mirror reflection in the x-y plane transforms an HG45° mode into its 90° rotated counterpart, the presence of the extra mirror canceled the effect of the out-of-plane rotation in this case. In this case therefore the resulting interference pattern resembled that of the HG00 mode, which did not exhibit the phase structure of the mode. A similar issue occurs with the HG11 mode, which is identical to its 90° rotated counterpart up to an overall phase. Therefore, in order to more clearly demonstrate the desired phase structure of the HG45° and HG11 modes, we steered the output beam so that its propagation axis was transversely shifted with respect to the reference beam while still being (nearly) collinear with respect to it. For the case of the HG45° mode, the transverse shift was directed both down and to the right, while for the HG11 mode it was directed completely downwards. In this way, the interfering beams were only partially overlapping so that the resulting interference patterns, included in the fourth and fifth columns, clearly show the characteristic “forking” effect in their interference patterns.

2007 (2)

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and Analysis of Entangled Photonic Qubits in Spatial-Parity Space,” Phys. Rev. Lett. 99, 250502 (2007).
[Crossref]

M. P. van Exter, P. S. K. Lee, S. Doesburg, and J. P. Woerdman, “Mode counting in high-dimensional orbital angular momentum entanglement,” Opt. Express 15, 6431–6438 (2007).
[Crossref] [PubMed]

2005 (2)

S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A 71, 053812 (2005).
[Crossref]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
[Crossref]

2004 (1)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

2003 (3)

E. Mukamel, K. Banaszek, I. A. Walmsley, and C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003).
[Crossref] [PubMed]

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

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[Crossref]

2002 (1)

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]

2001 (1)

1999 (1)

1998 (1)

1996 (1)

L. L. Smith and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve inputoutput collinearity,” J. Opt. Soc. Am. A 13, 21022105 (1996).
[Crossref]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
[Crossref]

1993 (1)

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

1992 (2)

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 45, 8185–8189 (1992).
[Crossref] [PubMed]

V. S. Liberman and B. Ya. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

1985 (1)

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref] [PubMed]

1977 (1)

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449450 (1977).
[Crossref]

Abouraddy, A. F.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and Analysis of Entangled Photonic Qubits in Spatial-Parity Space,” Phys. Rev. Lett. 99, 250502 (2007).
[Crossref]

Allen, L.

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 45, 8185–8189 (1992).
[Crossref] [PubMed]

Banaszek, K.

Barnett, S. M.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[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]

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[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 45, 8185–8189 (1992).
[Crossref] [PubMed]

Courtial, J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[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]

Dalton, R. B.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Doesburg, S.

Dorrer, C.

Enk, S. J. van

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

Exter, M. P. van

Franke-Arnold, S.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[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]

Galvez, E. J.

Gilchrist, A.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Harvey, M. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Holmes, C. D.

Hong, C. K.

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref] [PubMed]

Killett, B.

Kirk, A. G.

Koch, P. M.

L. L. Smith and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve inputoutput collinearity,” J. Opt. Soc. Am. A 13, 21022105 (1996).
[Crossref]

Kwiat, P. G.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
[Crossref]

Langford, N. K.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Leach, J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[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]

Lee, P. S. K.

Liberman, V. S.

V. S. Liberman and B. Ya. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Mandel, L.

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref] [PubMed]

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
[Crossref]

McGloin, D.

Monken, C. H.

S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A 71, 053812 (2005).
[Crossref]

Mukamel, E.

OBrien, J. L.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Okamoto, M.

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

Padgett, M. J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[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]

D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of Orbital Angular Momentum from a Stressed Fiber-Optic Waveguide to a Light Beam,” Appl. Opt. 37, 469472 (1998)
[Crossref]

Pádua, S.

S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A 71, 053812 (2005).
[Crossref]

Pryde, G. J.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Raymer, M. G.

Royer, A.

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449450 (1977).
[Crossref]

Saleh, B. E. A.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and Analysis of Entangled Photonic Qubits in Spatial-Parity Space,” Phys. Rev. Lett. 99, 250502 (2007).
[Crossref]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, New York, 1991), p. 226, pp. 233–234.

Sasada, H.

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

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
[Crossref]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
[Crossref]

Siegman, A.

A. Siegman, Lasers (University Science Books, Mill Valley CA, 1986), pp. 646–648.

Simpson, N. B.

Smith, B. J.

Smith, L. L.

L. L. Smith and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve inputoutput collinearity,” J. Opt. Soc. Am. A 13, 21022105 (1996).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

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 45, 8185–8189 (1992).
[Crossref] [PubMed]

Teich, M. C.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and Analysis of Entangled Photonic Qubits in Spatial-Parity Space,” Phys. Rev. Lett. 99, 250502 (2007).
[Crossref]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, New York, 1991), p. 226, pp. 233–234.

Walborn, S. P.

S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A 71, 053812 (2005).
[Crossref]

Walmsley, I. A.

Wei, H.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[Crossref]

X. Xue, H. Wei, and A. G. Kirk, “Beam analysis by fractional Fourier transform,” Opt. Lett. 26, 1746–1748 (2001).
[Crossref]

Weinfurter, H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
[Crossref]

White, A. G.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Woerdman, J. P.

M. P. van Exter, P. S. K. Lee, S. Doesburg, and J. P. Woerdman, “Mode counting in high-dimensional orbital angular momentum entanglement,” Opt. Express 15, 6431–6438 (2007).
[Crossref] [PubMed]

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 45, 8185–8189 (1992).
[Crossref] [PubMed]

Xue, X.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[Crossref]

X. Xue, H. Wei, and A. G. Kirk, “Beam analysis by fractional Fourier transform,” Opt. Lett. 26, 1746–1748 (2001).
[Crossref]

Ya. Zeldovich, B.

V. S. Liberman and B. Ya. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

Yaoc, E.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[Crossref]

Yarnall, T.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and Analysis of Entangled Photonic Qubits in Spatial-Parity Space,” Phys. Rev. Lett. 99, 250502 (2007).
[Crossref]

Zeilinger, A.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

E. J. Galvez and C. D. Holmes, “Geometric phase of optical rotators,” J. Opt. Soc. Am. A 16, 1981–1985 (1999).
[Crossref]

L. L. Smith and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve inputoutput collinearity,” J. Opt. Soc. Am. A 13, 21022105 (1996).
[Crossref]

Opt. Commun. (2)

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, “Simplified measurement of the orbital angular momentum of single photons ,” Opt. Commun. 223, 117–122 (2003).
[Crossref]

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

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (6)

V. S. Liberman and B. Ya. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[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 45, 8185–8189 (1992).
[Crossref] [PubMed]

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449450 (1977).
[Crossref]

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref] [PubMed]

S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A 71, 053812 (2005).
[Crossref]

Phys. Rev. Lett. (4)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. OBrien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 43374341 (1995).
[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]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and Analysis of Entangled Photonic Qubits in Spatial-Parity Space,” Phys. Rev. Lett. 99, 250502 (2007).
[Crossref]

Other (4)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

A. Siegman, Lasers (University Science Books, Mill Valley CA, 1986), pp. 646–648.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, New York, 1991), p. 226, pp. 233–234.

In order to interpret the final columns of Figs. 6(a) and 6(b), recall that the output beam has been rotated 90° with respect to the input beam, regardless of output port. This has no effect upon the rotationally symmetric HG00 mode, so the familiar interference fringes of a standard Mach-Zehnder interferometer are observed. However the 90° rotation does effect the first-order modes, so that the next two rows exhibit an interference pattern resulting form the superposition of the input HG mode and its 90° rotated counterpart. Note the characteristic “forking” of the vertical fringes in both of these plots, which shows the nontrivial phase structure of these modes as they interfere. For the HG45° mode, one would expect a similar “forked” pattern, but rotated by 90°. However, practical considerations required the presence of an extra mirror along the reference beam path in order to interfere the reference and output beams. Since an extra mirror reflection in the x-y plane transforms an HG45° mode into its 90° rotated counterpart, the presence of the extra mirror canceled the effect of the out-of-plane rotation in this case. In this case therefore the resulting interference pattern resembled that of the HG00 mode, which did not exhibit the phase structure of the mode. A similar issue occurs with the HG11 mode, which is identical to its 90° rotated counterpart up to an overall phase. Therefore, in order to more clearly demonstrate the desired phase structure of the HG45° and HG11 modes, we steered the output beam so that its propagation axis was transversely shifted with respect to the reference beam while still being (nearly) collinear with respect to it. For the case of the HG45° mode, the transverse shift was directed both down and to the right, while for the HG11 mode it was directed completely downwards. In this way, the interfering beams were only partially overlapping so that the resulting interference patterns, included in the fourth and fifth columns, clearly show the characteristic “forking” effect in their interference patterns.

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Figures (10)

Fig. 1.
Fig. 1.

Previously realized 1-D and 2-D parity sorters of the Mach-Zehnder type, each with a tiltable phase-shifting glass plate (GP) in one arm. (a) The 1-D parity sorter is distinguished by having an extra mirror in one arm. Upon entering this interferometer, an LG 01 mode is sorted into its constituent HG 10 and HG 01 components due to complete constructive and destructive interference at the output ports. (b) In contrast, the 2-D parity sorter has a Dove prism in each arm. One of the Dove prisms is rotated 90° with respect to the other, which causes a 180° relative rotation of the two interfering beams in the transverse plane. In this case, both HG 10 and HG 01 modes exit the same port, so that an incident LG 01 mode is not decomposed.

Fig. 2.
Fig. 2.

(a) A tabulation of the action of the the 1-D sorter upon the HG modes of order n+m ≤ 3. The plus signs designate modal lobes which are in phase with each other and 180° out of phase with the unmarked lobes. Modes with an even value for n exit port A, while modes with odd n-value exit port B (see Fig. 1(a)). (b) A tabulation of the action of the the 2-D sorter upon the HG modes of order n+m ≤ 3. The plus signs designate the relative phases of the lobes as in Fig. 2(a). Modes with an even value for n+m exit port A, while modes with odd n+m-value exit port B (see Fig. 1(b)).

Fig. 3.
Fig. 3.

The 2-D parity sorting Sagnac interferometer. Mirrors M1–M3 cause the beam path to travel out of the x-z plane, tracing out two congruent sides of an isosceles triangle in the x-y plane, with congruent base angles θ and apex angle β = π - 2θ. Incident HG 10 and HG 01 modes exit the same port of this interferometer, as do LG 01 modes, as is the case for the sorter in Fig. 1(b). See text for further discussion.

Fig. 4.
Fig. 4.

(a) The unit sphere construction corresponding to the beam path in Fig. 3 as discussed in the text. The beam path angles θ and β from Fig. 3 define the helicity vectors ĥ1 - ĥ4, which in turn define the points P1–P4 on the spherical surface as shown. The dashed lines denote the triangular closed loop that results from successively connecting these points with geodesic curves. The Fig. is drawn for the special case where θ = 45°, so that β = 90°. In this case, the angles between the distinct helicity vectors are all 90°, so that the area enclosed by the loop is exactly one eighth of the area of the entire sphere, which corresponds to a transverse rotation of Ω = π 2 rads, or 90°. (b) Plot of the beam rotation angle Ω vs. the parameter θ. (c) Plot of the relative rotation angle Ψ (modulo π) of the two counter-propagating beams after passing through the Sagnac interferometer vs. θ, emphasizing the two special cases where Ψ = 180° and Ψ = 90°. These two cases correspond to the first and second cascaded interferometer stages for the OAM sorting scheme in [2].

Fig. 5.
Fig. 5.

The Experimental Apparatus. The propagating and counter-propagating beams interfere at a 50:50 beam splitter (BS1), while a second 50:50 beam splitter (BS2) separates the backward-propagating output mode (port A) from the forward-propagating input mode. An external cavity helium neon laser (HeNe) acts as the source, with thin crossed wires (W) and an iris (I) inserted into the cavity in order to select the higher order HG modes. A third beam splitter (BS3) picks off part of the source beam to use a a reference beam (Ref) for the interferometry experiments discussed in the main text. Two Berek polarization compensators (BC) are placed inside the interferometer to correct for the Fresnel polarization changes due to reflections from the out-of-plane dielectric mirrors. The output fields at both port A and port B were measured with CCD cameras.

Fig. 6.
Fig. 6.

Observed (a) and predicted (b) output intensity profiles and interference patterns for a given input field.

Fig. 7.
Fig. 7.

A spatial filter for zero and first-order HG modes. (a) A three-mode optical fiber (3MF) with input and output coupling lenses (CL). The fiber, which has a parabolic refractive index profile, acts as a mode filter upon an arbitrary transverse input beam ∣Ψ〉 = Σ n,m=0 cnm HGnm 〉 such that upon coupling into and out of the fiber the resulting beam state is of the form ∣ψ〉 = c 00HG 00〉 + c 10HG 10〉 + ∣HG 01〉. (b) Inserting a 2-D Parity Sorter (S) after the fiber separates the zero and first-order modes into ports A and B, respectively.

Fig. 8.
Fig. 8.

Possible applications of a 2-D parity sorter to quantum information processing. (a) Proposed scheme to produce Bell states entangled in first-order transverse spatial modes. (b) Proposed scheme to produce heralded single photons in arbitrary first-order transverse spatial states.

Fig. 9.
Fig. 9.

A device that imparts a relative phase shift between counter-propagating fields, consisting of a tiltable birefringent waveplate (WP) with identical lengths of Faraday glass (FG) on each side, distinguished only by the direction of the applied magnetic fields (B) permeating them. (a) A vertically polarized right-propagating photon becomes aligned with the fast axis of a birefringent waveplate (axis is denoted by the bold line), which induces a phase shift. (b) A vertically polarized left-propagating photon becomes aligned with the slow axis of the waveplate, thereby experiencing an unequal phase shift when compared to the forward-propagating beam.

Fig. 10.
Fig. 10.

A Faraday isolator, consisting of a piece of Faraday glass (FG) with a polarizing beam splitter on each side (PBS1 and PBS2). (a) An input photon (propagating left to right) polarized at positive 45° angle with respect to the vertical is completely transmitted and becomes vertically polarized. (b) A back-propagating (from right to left) output photon from port A of the Sagnac with vertical polarization is completely transmitted through PBS2, but is subsequently completely deflected from PBS1.

Equations (25)

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cos Ω 2 = 1 + cos α + cos β + cos γ 4 cos α 2 cos β 2 cos γ 2
cos Ω 2 = sin θ
ψ 0 = E 0 n , m = 0 c nm H G nm
H G nm = A nm 1 w 0 H n ( 2 x w 0 ) H m ( 2 y w 0 ) e ( x 2 + y 2 ) w 0 2
L G q , m = ( ρ a ) m L q m ( V ρ 2 a 2 ) e V 2 ρ 2 a 2 e i m ϕ
ψ F = c 00 H G 00 + c 10 H G 10 + c 01 H G 01
ψ 2 ph = ϕ nm ( H V + V H )
ϕ nm = j , k , s , t = 0 c jkst nm H G jk H G st
ϕ 00 = c 0 H G 00 H G 00 + c 1 ( H G 10 H G 10 + H G 01 H G 01 )
+ c 2 ( H G 00 H G 02 + H G 02 H G 00 + H G 00 H G 20 + H G 20 H G 00 ) +
ψ B = 1 2 ( H G 10 H G 10 + H G 01 H G 01 ) ( H V + V H )
ϕ 45 ° ϕ 10 + ϕ 01 = c 1 ( H G 10 H G 00 + H G 00 H G 10
+ H G 01 H G 00 + H G 00 H G 01 ) +
ψ H G 00 A H G 45 ° B ( H A V B + V A H B )
ψ B = L G 0 + 1 ( H V + V H )
E even A = T ( tr + rt ) E even in = 2 T tr E even in
E even B = T ( t 2 r 2 ) E even in
E odd A = T ( tr rt ) E odd in = 0
E odd B = T ( t 2 + r 2 ) E odd in ,
I even A T I even in
I even B ε 2 T I even in
I odd A = 0
I odd B T I odd in ,
E S 1 I dark I bright 1 ε 2
E T I bright I in T .

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