Abstract

A spatial mode analyzer based on a Michelson interferometer with a bucket detector is experimentally implemented. The delay line in the interferometer is an optical implementation of the fractional Fourier transform (fFT) which enables the spatial mode analysis of a given input field in the Hermite-Gaussian (HG) mode basis. Modal weights for both 1D and 2D input fields are experimentally measured. Results for input fields comprising of multiple HG modes are also presented.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (1)

W. Farriss, T. Malhotra, A. N. Vamivakas, and J. R. Fienup, “Phase retrieval in generalized optical interferometry systems,” Optics Express 26, 2191–2202 (2018).
[Crossref] [PubMed]

2017 (1)

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

2016 (5)

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref] [PubMed]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photon. 10, 248–252 (2016).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
[Crossref]

D. Melati, A. Alippi, and A. Melloni, “Reconfigurable photonic integrated mode (de)multiplexer for SDM fiber transmission,” Opt. Express 24, 12625–12634 (2016).
[Crossref] [PubMed]

2015 (6)

B. Stern, X. Zhu, C. P. Chen, L. D. Tzuang, J. Cardenas, K. Bergman, and M. Lipson, “On-chip mode-division multiplexing switch,” Optica 2, 530–535 (2015).
[Crossref]

Z. Liu, O. P. Kocaoglu, T. L. Turner, and D. T. Miller, “Modal content of living human cone photoreceptors,” Biomed. Opt. Express 6, 3378–3404 (2015).
[Crossref] [PubMed]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
[Crossref]

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New Journal of Physics 17, 043024 (2015).
[Crossref]

2013 (4)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

S. Ackermann, B. Faatz, and V. Miltchev, “Modal analysis of a seeded free-electron laser,” Phys. Rev. ST Accel. Beams 16, 100702 (2013).
[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
[Crossref]

D. A. B. Miller, “Reconfigurable add-drop multiplexer for spatial modes,” Opt. Express 21, 20220–20229 (2013).
[Crossref] [PubMed]

2012 (3)

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, “Generalized optical interferometry for modal analysis in arbitrary degrees of freedom,” Opt. Lett. 37, 2889–2891 (2012).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

R. J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proc. IEEE 100, 1035–1055 (2012).
[Crossref]

2011 (2)

2009 (2)

2007 (2)

V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
[Crossref]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

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]

2003 (1)

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]

2001 (1)

2000 (1)

1998 (1)

1993 (2)

Abouraddy, A. F.

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, “Generalized optical interferometry for modal analysis in arbitrary degrees of freedom,” Opt. Lett. 37, 2889–2891 (2012).
[Crossref] [PubMed]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Ackermann, S.

S. Ackermann, B. Faatz, and V. Miltchev, “Modal analysis of a seeded free-electron laser,” Phys. Rev. ST Accel. Beams 16, 100702 (2013).
[Crossref]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Aiello, A.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New Journal of Physics 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

Alieva, T.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Programmable two-dimensional optical fractional fourier processor,” Opt. Express 17, 4976–4983 (2009).
[Crossref] [PubMed]

J. Rodrigo, T. Alieva, and M. J. Bastiaans, Phase-Space Rotators and their Applications in Optics (Wiley-VCH Verlag GmbH & Co. KGaA, 2011), pp. 251–271.

Alippi, A.

Alonso, M. A.

Arrizón, V.

Atia, G. K.

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

Banzer, P.

Bastiaans, M. J.

J. Rodrigo, T. Alieva, and M. J. Bastiaans, Phase-Space Rotators and their Applications in Optics (Wiley-VCH Verlag GmbH & Co. KGaA, 2011), pp. 251–271.

Baumgardner, L.

Berg-Johansen, S.

Bergman, K.

Borghi, R.

Boyd, R. W.

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” arXiv preprint arXiv:1701.05889 (2017).

Brüning, R.

Cai, X.-D.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Calvo, M. L.

Cardenas, J.

Carrada, R.

Chen, C. P.

Chen, M.-C.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Christodoulides, D. N.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Dudley, A.

Duparré, M.

Eberly, J. H.

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref] [PubMed]

Erhard, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photon. 10, 248–252 (2016).
[Crossref]

Essiambre, R. J.

R. J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proc. IEEE 100, 1035–1055 (2012).
[Crossref]

Faatz, B.

S. Ackermann, B. Faatz, and V. Miltchev, “Modal analysis of a seeded free-electron laser,” Phys. Rev. ST Accel. Beams 16, 100702 (2013).
[Crossref]

Farriss, W.

W. Farriss, T. Malhotra, A. N. Vamivakas, and J. R. Fienup, “Phase retrieval in generalized optical interferometry systems,” Optics Express 26, 2191–2202 (2018).
[Crossref] [PubMed]

W. Farriss, J. R. Fienup, T. Malhotra, and A. N. Vamivakas, “Single-Pixel Phase Retrieval in Generalized Interferometry,” in Imaging and Applied Optics, OSA Technical Digest (online), paper CTu1B.2 (2017).

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Fickler, R.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photon. 10, 248–252 (2016).
[Crossref]

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” arXiv preprint arXiv:1701.05889 (2017).

Fienup, J. R.

W. Farriss, T. Malhotra, A. N. Vamivakas, and J. R. Fienup, “Phase retrieval in generalized optical interferometry systems,” Optics Express 26, 2191–2202 (2018).
[Crossref] [PubMed]

W. Farriss, J. R. Fienup, T. Malhotra, and A. N. Vamivakas, “Single-Pixel Phase Retrieval in Generalized Interferometry,” in Imaging and Applied Optics, OSA Technical Digest (online), paper CTu1B.2 (2017).

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
[Crossref]

Flamm, D.

Forbes, A.

Giacobino, E.

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New Journal of Physics 17, 043024 (2015).
[Crossref]

Ginoya, M.

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” arXiv preprint arXiv:1701.05889 (2017).

Giuseppe, G. Di

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

González, L. A.

Gori, F.

Grafe, M.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Guattari, G.

Heilmann, R.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Huber, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photon. 10, 248–252 (2016).
[Crossref]

Jahromi, A. K.

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

Kagalwala, K. H.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Kahn, J. M.

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
[Crossref]

Kaiser, T.

Keil, R.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Kirk, A. G.

Kocaoglu, O. P.

Kondakci, H.

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

Krenn, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photon. 10, 248–252 (2016).
[Crossref]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Larson, W. D.

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

Leary, C.

Lebugle, M.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Leuchs, G.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New Journal of Physics 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

Li, G.

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
[Crossref]

Li, L.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Li, X.

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
[Crossref]

Lipson, M.

Liu, N.-L.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Liu, Z.

Lohmann, A. W.

Lu, C.-Y.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Malhotra, T.

W. Farriss, T. Malhotra, A. N. Vamivakas, and J. R. Fienup, “Phase retrieval in generalized optical interferometry systems,” Optics Express 26, 2191–2202 (2018).
[Crossref] [PubMed]

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref] [PubMed]

W. Farriss, J. R. Fienup, T. Malhotra, and A. N. Vamivakas, “Single-Pixel Phase Retrieval in Generalized Interferometry,” in Imaging and Applied Optics, OSA Technical Digest (online), paper CTu1B.2 (2017).

Malik, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photon. 10, 248–252 (2016).
[Crossref]

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]

Mardani, D.

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

Marquardt, C.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New Journal of Physics 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

Marrucci, L.

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]

Martin, L.

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

McLaren, M.

Melati, D.

Melloni, A.

Mendlovic, D.

Miller, D. A. B.

Miller, D. T.

Miltchev, V.

S. Ackermann, B. Faatz, and V. Miltchev, “Modal analysis of a seeded free-electron laser,” Phys. Rev. ST Accel. Beams 16, 100702 (2013).
[Crossref]

Moya-Cessa, H.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
[Crossref]

Nolte, S.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

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]

Ornigotti, M.

Ozaktas, H. M.

Pan, J.-W.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

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]

Perez-Leija, A.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Qian, X.-F.

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref] [PubMed]

Raymer, M.

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Richardson, D. J.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
[Crossref]

Rodrigo, J.

J. Rodrigo, T. Alieva, and M. J. Bastiaans, Phase-Space Rotators and their Applications in Optics (Wiley-VCH Verlag GmbH & Co. KGaA, 2011), pp. 251–271.

Rodrigo, J. A.

Ruiz, U.

Saleh, B. E. A.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, “Generalized optical interferometry for modal analysis in arbitrary degrees of freedom,” Opt. Lett. 37, 2889–2891 (2012).
[Crossref] [PubMed]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Santarsiero, M.

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]

Schmidt, O. A.

Schröter, S.

Schulze, C.

Shabahang, S.

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

Stern, B.

Stiller, B.

Su, Z.-E.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Szameit, A.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Teich, M. C.

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Tichy, M.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Tkach, R. W.

R. J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proc. IEEE 100, 1035–1055 (2012).
[Crossref]

Töppel, F.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New Journal of Physics 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Turner, T. L.

Tzuang, L. D.

Vamivakas, A. N.

W. Farriss, T. Malhotra, A. N. Vamivakas, and J. R. Fienup, “Phase retrieval in generalized optical interferometry systems,” Optics Express 26, 2191–2202 (2018).
[Crossref] [PubMed]

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref] [PubMed]

W. Farriss, J. R. Fienup, T. Malhotra, and A. N. Vamivakas, “Single-Pixel Phase Retrieval in Generalized Interferometry,” in Imaging and Applied Optics, OSA Technical Digest (online), paper CTu1B.2 (2017).

Voelz, D. G.

D. G. Voelz, Computational Fourier optics: a MATLAB tutorial (SPIE, 2011).

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Wang, X.-L.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Wei, H.

Weihs, G.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Weimann, S.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Wu, D.

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Xue, X.

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Yarnall, T.

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Yarnall, T. M.

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Zeilinger, A.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photon. 10, 248–252 (2016).
[Crossref]

Zhao, N.

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
[Crossref]

Zhu, X.

Adv. Opt. Photon. (2)

Appl. Opt. (1)

Biomed. Opt. Express (1)

J. Opt. Soc. Am. A (3)

Nat. Comm. (1)

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Grafe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional fourier transforms,” Nat. Comm. 7, 11027 (2016).
[Crossref]

Nat. Photon. (5)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photon. 10, 248–252 (2016).
[Crossref]

Nature (1)

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

New Journal of Physics (1)

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New Journal of Physics 17, 043024 (2015).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Optica (2)

Optics Express (1)

W. Farriss, T. Malhotra, A. N. Vamivakas, and J. R. Fienup, “Phase retrieval in generalized optical interferometry systems,” Optics Express 26, 2191–2202 (2018).
[Crossref] [PubMed]

Phys. Rev. A (2)

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. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Phys. Rev. Lett. (2)

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]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref] [PubMed]

Phys. Rev. ST Accel. Beams (1)

S. Ackermann, B. Faatz, and V. Miltchev, “Modal analysis of a seeded free-electron laser,” Phys. Rev. ST Accel. Beams 16, 100702 (2013).
[Crossref]

Proc. IEEE (1)

R. J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proc. IEEE 100, 1035–1055 (2012).
[Crossref]

Sci. Rep. (1)

L. Martin, D. Mardani, H. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral hilbert-space analyzers,” Sci. Rep. 744995 (2017).
[Crossref] [PubMed]

Other (5)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” arXiv preprint arXiv:1701.05889 (2017).

J. Rodrigo, T. Alieva, and M. J. Bastiaans, Phase-Space Rotators and their Applications in Optics (Wiley-VCH Verlag GmbH & Co. KGaA, 2011), pp. 251–271.

D. G. Voelz, Computational Fourier optics: a MATLAB tutorial (SPIE, 2011).

W. Farriss, J. R. Fienup, T. Malhotra, and A. N. Vamivakas, “Single-Pixel Phase Retrieval in Generalized Interferometry,” in Imaging and Applied Optics, OSA Technical Digest (online), paper CTu1B.2 (2017).

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Figures (4)

Fig. 1
Fig. 1 Schematic of a Michelson interferometer designed for spatial mode analysis in the Hermite-Gaussian (HG) basis. The delay line in the interferometer is replaced with a fractional Fourier transform (fFT delay line) of parameters αx, αy. An exampleinput field which comprises of three HG modes is shown (intensity of this field is shown as ‘Input’). The intensities of the constituent HG modes in the input, and their weights, are shown adjacent to the input in the schematic (see text for details). The phase delays imparted to each of these HG modes by the fFT, which depend on the index (m, n) of the mode and the delay parameter αx, αy, are shown above the fFT delay line. The corresponding numerically simulated interferogram, recorded by the bucket detector, is shown as ‘Interferograms’. Finally, the modal weights revealed by Fourier analysis are shown next to the interferogram. Note that the fFT delay line in the interferometer is implemented using a set of three generalized lenses. 1(a) shows the schematic of this fFT set-up in transmission for simplicity.
Fig. 2
Fig. 2 Schematic of the optical set-up: the gray box shows the Michelson interferometer. The fFT is implemented using SLMs in the upper arm. The stabilization channel sends a second optical beam cross-polarized with respect to the signal input in order to generate an error signal which is used to path stabilize the interferometer. Experimentally generated test modes are also shown.
Fig. 3
Fig. 3 Results for 1D mode analysis: First column shows the intensities of the various input distributions as recorded on the CCD (located in place of the bucket detector): 3(a) HG1,0(x), 3(d) HG2,0(x), (g) HG10,0(x) and 3(j) 1D slit, rect(x/a), with width a= 2.3 mm. Numerically modeled interferograms are shown in red circles in upper panels and experimentally generated interferograms are shown in blue circles in lower panels of 3(b), 3(e), 3(h) and 3(k) for each of the inputs respectively. The red lines in the experimentally generated interferograms in 3(b), 3(e) and 3(h) show the fits from which the modal weights are extracted. Extracted modal weights are shown in 3(c), 3(f), 3(i) and 3(l) (red bars show theoretical values of |cth|2 for slit input). The error in the measured modal weights for the slit input are shown in blue bars in lower panel of 3(l).
Fig. 4
Fig. 4 Results for 2D mode analysis: First column shows the intensities of the various input distributions: 4(a) HG2,2(x, y), 4(d) HG4,1(x) and 4(i) superposition state L G 0 , 1 w ( x , y ) = H G 1 , 0 ( x , y ) + i H G 1 , 0 ( x , y ), with w=1.6 mm. Second and third columns show the numerically modeled interferograms and experimentally generated interferograms in 4(b), 4(e) and 4(h) for each of the inputs respectively. Modal weights extracted from data are shown in 4(d), 4(f) and 4(i).

Equations (6)

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P ( α x , α y ) = m , n | c m n | 2 + m , n | c m n | 2 cos ( m α x + n α y ) .
F α x , α y ( x o , y o ) = exp [ i ( α x + α y ) / 2 ] i sin α x sin α y × f ( x , y ) exp { i π [ x 2 + x o 2 ] cot α x 2 x x o csc α x ] } × exp { i π [ ( y 2 + y o 2 ) cot α y 2 y y o csc α y ] } d x d y .
p x , y ( L 1 , L 3 ) = [ 1 cot ( α x , y / 2 ) 2 ] / z p x , y ( L 2 ) = 2 [ 1 sin ( α x , y ) ] / z α x , y = q x , y × π / 2 and q x , y [ 0 , 4 ] .
t A ( x , y ) = exp { i ϕ ( x , y ) } ; ϕ ( x , y ) = ( π p x ( L ) x 2 λ ) + ( π p y ( L ) y 2 λ ) .
H ( f x , f y ) = exp ( i 2 π λ z ) exp [ i π λ z ( f x 2 + f y 2 ) ]
U out ( x , y ) = F π / 2 [ F π / 2 [ U 1 ( x , y ) ] H ( f x , f y ) ] .

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