Abstract

Working with finite numbers of modes to describe, generate, and detect optical fields can be both mathematically economical and physically useful. Such a modal basis can map directly to various applications in communications, sensing, and processing. But, we need a way to generate and analyze such fields, including measurement and control of both the relative amplitudes and phases of the modal components. Ideally such an analysis scheme would operate directly on the field, without needing a separate, mutually coherent reference beam. Here, we show first how to measure all those relative amplitudes and phases automatically and simultaneously. The method repurposes a self-configuring network of ${2} \times {2}$ blocks, such as integrated Mach–Zehnder interferometers, that can automatically align itself to the optical field by a sequence of simple one-parameter power minimizations when network elements, such as phase shifters, are adjusted. The optical field is then directly deduced from the resulting settings of those elements. We show how the entire network can be calibrated for such measurements, automatically and with just two light beams. Then, using the same calibration and running the mesh backwards, we can also controllably generate an arbitrary multimode field. Explicit algorithms and formulas are given for operating this system.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

2019 (6)

2018 (8)

D. Pérez, I. Gasulla, and J. Capmany, “Field-programmable photonic arrays,” Opt. Express 26, 27265–27278 (2018).
[Crossref]

P. L. Mennea, W. R. Clements, D. H. Smith, J. C. Gates, B. J. Metcalf, R. H. S. Bannerman, R. Burgwal, J. J. Renema, W. S. Kolthammer, I. A. Walmsley, and P. G. R. Smith, “Modular linear optical circuits,” Optica 5, 1087–1090 (2018).
[Crossref]

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

I. V. Dyakonov, I. A. Pogorelov, I. B. Bobrov, A. A. Kalinkin, S. S. Straupe, S. P. Kulik, P. V. Dyakonov, and S. A. Evlashin, “Reconfigurable photonics on a glass chip,” Phys. Rev. Appl. 10, 044048 (2018).
[Crossref]

P. J. Winzer, D. T. Neilson, and A. R. Chraplyvy, “Fiber-optic transmission and networking: the previous 20 and the next 20 years [Invited],” Opt. Express 26, 24190–24239 (2018).
[Crossref]

N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophotonic processors,” Optica 5, 1623–1631 (2018).
[Crossref]

D. Pérez, I. Gasulla, and J. Capmany, “Programmable multifunctional integrated nanophotonics,” Nanophotonics 7, 1351–1371 (2018).
[Crossref]

2017 (6)

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light–automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
[Crossref]

D. A. B. Miller, “Setting up meshes of interferometers–reversed local light interference method,” Opt. Express 25, 29233–29248 (2017).
[Crossref]

D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
[Crossref]

2016 (3)

2015 (4)

2014 (1)

2013 (4)

2010 (1)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Acín, A.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Aguiar, D.

Alsing, P. M.

Annoni, A.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light–automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
[Crossref]

Augusiak, R.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Bacco, D.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Baehr-Jones, T.

N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophotonic processors,” Optica 5, 1623–1631 (2018).
[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Bannerman, R. H. S.

Bell, B. A.

Birks, T. A.

Bobrov, I. B.

I. V. Dyakonov, I. A. Pogorelov, I. B. Bobrov, A. A. Kalinkin, S. S. Straupe, S. P. Kulik, P. V. Dyakonov, and S. A. Evlashin, “Reconfigurable photonics on a glass chip,” Phys. Rev. Appl. 10, 044048 (2018).
[Crossref]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Bogaerts, W.

Boller, K.-J.

Bonneau, D.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Bunandar, D.

N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophotonic processors,” Optica 5, 1623–1631 (2018).
[Crossref]

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Burgwal, R.

Cao, W.

D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
[Crossref]

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
[Crossref]

Capmany, J.

D. Pérez and J. Capmany, “Scalable analysis for arbitrary photonic integrated waveguide meshes,” Optica 6, 19–27 (2019).
[Crossref]

D. Pérez, I. Gasulla, and J. Capmany, “Field-programmable photonic arrays,” Opt. Express 26, 27265–27278 (2018).
[Crossref]

D. Pérez, I. Gasulla, and J. Capmany, “Programmable multifunctional integrated nanophotonics,” Nanophotonics 7, 1351–1371 (2018).
[Crossref]

D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
[Crossref]

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
[Crossref]

Carminati, M.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light–automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
[Crossref]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Carolan, J.

N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophotonic processors,” Optica 5, 1623–1631 (2018).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

Carpenter, J.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre–Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schröder, “Complete spatiotemporal characterization and optical transfer matrix inversion of a 420 mode fiber,” Opt. Lett. 41, 5580–5583 (2016).
[Crossref]

Chen, C.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Chen, H.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre–Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Choutagunta, K.

Chraplyvy, A. R.

Clements, W. R.

Crudgington, L.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
[Crossref]

D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
[Crossref]

Denolle, B.

Ding, Y.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Dyakonov, I. V.

I. V. Dyakonov, I. A. Pogorelov, I. B. Bobrov, A. A. Kalinkin, S. S. Straupe, S. P. Kulik, P. V. Dyakonov, and S. A. Evlashin, “Reconfigurable photonics on a glass chip,” Phys. Rev. Appl. 10, 044048 (2018).
[Crossref]

Dyakonov, P. V.

I. V. Dyakonov, I. A. Pogorelov, I. B. Bobrov, A. A. Kalinkin, S. S. Straupe, S. P. Kulik, P. V. Dyakonov, and S. A. Evlashin, “Reconfigurable photonics on a glass chip,” Phys. Rev. Appl. 10, 044048 (2018).
[Crossref]

Eckstein, A.

Eggleton, B. J.

Englund, D.

N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophotonic processors,” Optica 5, 1623–1631 (2018).
[Crossref]

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

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D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
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N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
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D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
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N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre–Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
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N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
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X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
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X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
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D. A. B. Miller, “Waves, modes, communications, and optics: a tutorial,” Adv. Opt. Photon. 11, 679–825 (2019).
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D. A. B. Miller, “Reconfigurable add-drop multiplexer for spatial modes,” Opt. Express 21, 20220–20229 (2013).
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S. Pai, I. A. D. Williamson, T. W. Hughes, M. Minkov, O. Solgaard, S. Fan, and D. A. B. Miller, “Parallel programming of an arbitrary feedforward photonic network,” IEEE J. Sel. Topics Quantum Electron. (Early Access) (2020), 10.1109/JSTQE.2020.2997849.

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M. Milanizadeh, D. Aguiar, A. Melloni, and F. Morichetti, “Canceling thermal cross-talk effects in photonic integrated circuits,” J. Lightwave Technol. 37, 1325–1332 (2019).
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A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light–automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
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J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
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J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
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J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach–Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
[Crossref]

Pai, S.

S. Pai, I. A. D. Williamson, T. W. Hughes, M. Minkov, O. Solgaard, S. Fan, and D. A. B. Miller, “Parallel programming of an arbitrary feedforward photonic network,” IEEE J. Sel. Topics Quantum Electron. (Early Access) (2020), 10.1109/JSTQE.2020.2997849.

Perez, D.

D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
[Crossref]

Pérez, D.

D. Pérez and J. Capmany, “Scalable analysis for arbitrary photonic integrated waveguide meshes,” Optica 6, 19–27 (2019).
[Crossref]

D. Pérez, I. Gasulla, and J. Capmany, “Field-programmable photonic arrays,” Opt. Express 26, 27265–27278 (2018).
[Crossref]

D. Pérez, I. Gasulla, and J. Capmany, “Programmable multifunctional integrated nanophotonics,” Nanophotonics 7, 1351–1371 (2018).
[Crossref]

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
[Crossref]

Pinkse, P. W. H.

Pogorelov, I. A.

I. V. Dyakonov, I. A. Pogorelov, I. B. Bobrov, A. A. Kalinkin, S. S. Straupe, S. P. Kulik, P. V. Dyakonov, and S. A. Evlashin, “Reconfigurable photonics on a glass chip,” Phys. Rev. Appl. 10, 044048 (2018).
[Crossref]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Prabhu, M.

N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophotonic processors,” Optica 5, 1623–1631 (2018).
[Crossref]

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Qiang, X.

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach–Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
[Crossref]

Ralph, T. C.

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

Renema, J. J.

Ribeiro, A.

Roberts, I.

Roeloffzen, C. G. H.

Rottwitt, K.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Ruocco, A.

Russell, N. J.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

Ryf, R.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre–Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Salavrakos, A.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Sampietro, M.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light–automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
[Crossref]

Sanadgol Nezami, M.

F. Shokraneh, S. Geoffroy-Gagnon, M. Sanadgol Nezami, and O. Liboiron-Ladouceur, “A single layer neural network implemented by a 4 × 4 MZI-based optical processor,” IEEE Photon. J. 11, 4501612 (2019).
[Crossref]

Santagati, R.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach–Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
[Crossref]

Schröder, J.

Shadbolt, P. J.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

Shen, Y.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Shokraneh, F.

F. Shokraneh, S. Geoffroy-Gagnon, M. Sanadgol Nezami, and O. Liboiron-Ladouceur, “A single layer neural network implemented by a 4 × 4 MZI-based optical processor,” IEEE Photon. J. 11, 4501612 (2019).
[Crossref]

Silverstone, J. W.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

Skirlo, S.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Skrzypczyk, P.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Smith, A. M.

Smith, D. H.

Smith, P. G. R.

Solgaard, O.

S. Pai, I. A. D. Williamson, T. W. Hughes, M. Minkov, O. Solgaard, S. Fan, and D. A. B. Miller, “Parallel programming of an arbitrary feedforward photonic network,” IEEE J. Sel. Topics Quantum Electron. (Early Access) (2020), 10.1109/JSTQE.2020.2997849.

Soljacic, M.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Sparrow, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

Steinbrecher, G. R.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Straupe, S. S.

I. V. Dyakonov, I. A. Pogorelov, I. B. Bobrov, A. A. Kalinkin, S. S. Straupe, S. P. Kulik, P. V. Dyakonov, and S. A. Evlashin, “Reconfigurable photonics on a glass chip,” Phys. Rev. Appl. 10, 044048 (2018).
[Crossref]

Sun, X.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Taballione, C.

Thompson, M. G.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach–Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

Thomson, D. J.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
[Crossref]

D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
[Crossref]

Thomson, R. R.

Tison, C. C.

Treps, N.

Tura, J.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Vanacker, L.

Visscher, I.

Walmsley, I. A.

Wang, J.

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach–Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
[Crossref]

Wang, J. B.

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

Wilkes, C. M.

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach–Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
[Crossref]

Williamson, I. A. D.

S. Pai, I. A. D. Williamson, T. W. Hughes, M. Minkov, O. Solgaard, S. Fan, and D. A. B. Miller, “Parallel programming of an arbitrary feedforward photonic network,” IEEE J. Sel. Topics Quantum Electron. (Early Access) (2020), 10.1109/JSTQE.2020.2997849.

Winzer, P. J.

Wolterink, T. A. W.

Wong, F. N. C.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Yerolatsitis, S.

Zhao, S.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Zhou, X.

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach–Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
[Crossref]

Zhuang, L.

Adv. Opt. Photon. (2)

IEEE Photon. J. (1)

F. Shokraneh, S. Geoffroy-Gagnon, M. Sanadgol Nezami, and O. Liboiron-Ladouceur, “A single layer neural network implemented by a 4 × 4 MZI-based optical processor,” IEEE Photon. J. 11, 4501612 (2019).
[Crossref]

J. Lightwave Technol. (3)

Laser Photon. Rev. (1)

D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
[Crossref]

Light Sci. Appl. (1)

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light–automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
[Crossref]

Nanophotonics (1)

D. Pérez, I. Gasulla, and J. Capmany, “Programmable multifunctional integrated nanophotonics,” Nanophotonics 7, 1351–1371 (2018).
[Crossref]

Nat. Commun. (2)

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre–Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
[Crossref]

Nat. Photonics (3)

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
[Crossref]

Opt. Express (7)

Opt. Lett. (2)

Optica (6)

Photon. Res. (1)

Phys. Rev. Appl. (1)

I. V. Dyakonov, I. A. Pogorelov, I. B. Bobrov, A. A. Kalinkin, S. S. Straupe, S. P. Kulik, P. V. Dyakonov, and S. A. Evlashin, “Reconfigurable photonics on a glass chip,” Phys. Rev. Appl. 10, 044048 (2018).
[Crossref]

Phys. Rev. Lett. (1)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Science (2)

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

Other (2)

S. Pai, I. A. D. Williamson, T. W. Hughes, M. Minkov, O. Solgaard, S. Fan, and D. A. B. Miller, “Parallel programming of an arbitrary feedforward photonic network,” IEEE J. Sel. Topics Quantum Electron. (Early Access) (2020), 10.1109/JSTQE.2020.2997849.

R. Loudon, Quantum Theory of Light, 3rd ed. (Oxford, 2000), pp. 88–91.

Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Concept and architectures of a self-configuring mesh layer with optical inputs. (a) Example optical input, here with eight waveguides driven by the outputs from an example square array of grating couplers illuminated by some input light field. (b) Binary tree mesh, which we can self-configure to give all the resulting input power in the one output waveguide. Parameters $\Delta \theta$ and $\Delta \phi$ control the “split ratio” and one other phase shift in each ${2} \times {2}$ block ${K}$, respectively. Each such block can be implemented as an MZI. Elements D are detectors at the “drop ports” of these blocks or MZIs. (c) Alternative, diagonal line mesh architecture. The successive “columns” of MZIs are shown for both architectures.
Fig. 2.
Fig. 2. Algorithm outlines (a) for automatic analysis of a multimode input field, (b) generating a desired “backwards” multimode output field.
Fig. 3.
Fig. 3. (a) Schematic of a general waveguide MZI with up to four phase shifters (colored rectangles) on waveguides (gray lines), with two 50:50 beamsplitters BS1 and BS2, shown as directional couplers. (b) Corresponding conceptual BS showing “Top,” “Left,” “Bottom,” and “Right” surfaces.
Fig. 4.
Fig. 4. Outline of the calibration process.

Equations (18)

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θ a v = θ P + θ W 2
Δ θ = θ P θ W
ϕ a v = ϕ T + ϕ L 2
Δ ϕ = ϕ T ϕ L
ϕ T o t = ϕ a v + θ a v + π 2
[ a R a B ] = M ( ϕ T o t , Δ θ , Δ ϕ ) [ a T a L ] .
M ( ϕ T o t , Δ θ , Δ ϕ ) = exp ( i ϕ T o t ) M s ( Δ θ ) M ϕ ( Δ ϕ ) ,
M ϕ ( Δ ϕ ) = [ exp ( i Δ ϕ 2 ) 0 0 exp ( i Δ ϕ 2 ) ] .
M s ( Δ θ ) = [ sin ( Δ θ 2 ) cos ( Δ θ 2 ) cos ( Δ θ 2 ) sin ( Δ θ 2 ) ] .
B ( ϕ T o t , Δ θ , Δ ϕ ) = M T ( ϕ T o t , Δ θ , Δ ϕ ) = exp ( i ϕ T o t ) M ϕ T ( Δ ϕ ) M s T ( Δ θ ) ,
[ b T b L ] = B ( ϕ T o t , Δ θ , Δ ϕ ) [ b R b B ] ,
| c [ c 1 c 2 c N ] T = ( [ d 1 d 2 d N ] T ) ( | d ) ,
B = [ α μ β ν ] ,
| d [ d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 ] = [ μ 11 μ 21 α 31 ν 11 μ 21 α 31 α 12 v 21 α 31 β 12 ν 21 α 31 μ 13 α 22 β 31 ν 13 α 22 β 31 α 14 β 22 β 31 β 14 β 22 β 31 ] .
| c [ c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 ] = [ ( μ 11 μ 21 α 31 ) ( ν 11 μ 21 α 31 ) ( α 12 v 21 α 31 ) ( β 12 ν 21 α 31 ) ( μ 13 α 22 β 31 ) ( ν 13 α 22 β 31 ) ( α 14 β 22 β 31 ) ( β 14 β 22 β 31 ) ] ( | d ) ,
| c = [ ( α 7 ) ( α 6 β 7 ) ( α 5 β 6 β 7 ) ( α 4 β 5 β 6 β 7 ) ( α 3 β 4 β 5 β 6 β 7 ) ( α 2 β 3 β 4 β 5 β 6 β 7 ) ( α 1 β 2 β 3 β 4 β 5 β 6 β 7 ) ( β 1 β 2 β 3 β 4 β 5 β 6 β 7 ) ] .
ϕ av = 0 i f Δ ϕ = 0 a n d θ a v = 0 i f Δ θ = 0.
Δ θ = θ P , Δ ϕ = ϕ T , ϕ T o t = ϕ a v + θ a v + π 2 = ϕ T 2 + θ P 2 + π 2 ,