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 HG_nm input modes depending upon whether they have even or odd order n+m; it equivalently sorts LG^l _p 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

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, 4337-4341 (1995). [CrossRef]
19. S. P. Walborn, S. P’adua, 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, 469-472 (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.

Appl. Opt.

• 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, 469-472 (1998) [CrossRef]

J. Opt. Soc. Am. A

• 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.

• 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

• 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]

Opt. Lett.

• 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]
• X. Xue, H. Wei, and A. G. Kirk, "Beam analysis by fractional Fourier transform," Opt. Lett. 26, 1746-1748 (2001). [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]

Phys. Rev. A

• 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. 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]
• A. Royer, "Wigner function as the expectation value of a parity operator," Phys. Rev. A 15, 449450 (1977). [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]
• 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. P’adua, 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.

• 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, 4337-4341 (1995). [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]
• 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]

Other

• 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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