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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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

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

[Crossref]
[PubMed]

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

[Crossref]

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]

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]

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]

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. 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]

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]

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]

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]

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]

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

[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]

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]

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]

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]

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

[Crossref]
[PubMed]

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]

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]

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]

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]

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]

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

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

[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]

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]

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]

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]

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]

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]

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]

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

[Crossref]

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.

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]

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]

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]

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

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]

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

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]

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.

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]

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. 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]

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]

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]

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]

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]

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. 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]

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]

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]

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]

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

[Crossref]

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]

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]

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]

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]

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.