Abstract

We show how to design an optical device that can perform any linear function or coupling between inputs and outputs. This design method is progressive, requiring no global optimization. We also show how the device can configure itself progressively, avoiding design calculations and allowing the device to stabilize itself against drifts in component properties and to continually adjust itself to changing conditions. This self-configuration operates by training with the desired pairs of orthogonal input and output functions, using sets of detectors and local feedback loops to set individual optical elements within the device, with no global feedback or multiparameter optimization required. Simple mappings, such as spatial mode conversions and polarization control, can be implemented using standard planar integrated optics. In the spirit of a universal machine, we show that other linear operations, including frequency and time mappings, as well as nonreciprocal operation, are possible in principle, even if very challenging in practice, thus proving there is at least one constructive design for any conceivable linear optical component; such a universal device can also be self-configuring. This approach is general for linear waves, and could be applied to microwaves, acoustics, and quantum mechanical superpositions.

© 2013 Chinese Laser Press

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  39. M.-C. Wu, F.-C. Hsiao, and S.-Y. Tseng, “Adiabatic mode conversion in multimode waveguides using chirped computer-generated planar holograms,” IEEE Photon. Technol. Lett. 23, 807–809 (2011).
    [CrossRef]
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  41. S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  47. E. Palushani, H. C. Hansen Mulvad, M. Galili, H. Hu, L. K. Oxenløwe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical Fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18, 681–688 (2012).
    [CrossRef]

2013 (4)

V. Liu, D. A. B. Miller, and S. H. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proc. IEEE 101, 484–493 (2013).
[CrossRef]

P. P. Baveja, Y. Xiao, S. Arora, G. P. Agrawal, and D. N. Maywar, “All-optical semiconductor optical amplifier-based wavelength converters with sub-mW pumping,” IEEE Photon. Technol. Lett. 25, 78–80 (2013).
[CrossRef]

D. A. B. Miller, “How complicated must an optical component be?,” J. Opt. Soc. Am. A 30, 238–251 (2013).
[CrossRef]

D. A. B. Miller, “Self-aligning universal beam coupler,” Opt. Express 21, 6360–6370 (2013).
[CrossRef]

2012 (8)

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
[CrossRef]

E. Palushani, H. C. Hansen Mulvad, M. Galili, H. Hu, L. K. Oxenløwe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical Fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18, 681–688 (2012).
[CrossRef]

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. J. B. Yoo, “Demonstration of free space coherent optical communication using integrated silicon photonic orbital angular momentum devices,” Opt. Express 20, 9396–9402 (2012).
[CrossRef]

D. Dai, Y. Tang, and J. E. Bowers, “Mode conversion in tapered submicron silicon ridge optical waveguides,” Opt. Express 20, 13425–13439 (2012).
[CrossRef]

R. N. Mahalati, D. Askarov, J. P. Wilde, and J. M. Kahn, “Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling,” Opt. Express 20, 14321–14337 (2012).
[CrossRef]

P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express 20, 14705–14713 (2012).
[CrossRef]

D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express 20, 23985–23993 (2012).
[CrossRef]

2011 (5)

M.-C. Wu, F.-C. Hsiao, and S.-Y. Tseng, “Adiabatic mode conversion in multimode waveguides using chirped computer-generated planar holograms,” IEEE Photon. Technol. Lett. 23, 807–809 (2011).
[CrossRef]

M. P. J. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. J. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014 (2011).
[CrossRef]

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[CrossRef]

V. Liu, Y. Jiao, D. A. B. Miller, and S. Fan, “Design methodology for compact photonic-crystal-based wavelength division multiplexers,” Opt. Lett. 36, 591–593 (2011).
[CrossRef]

L. H. Gabrielli and M. Lipson, “Integrated Luneburg lens via ultra-strong index gradient on silicon,” Opt. Express 19, 20122–20127 (2011).
[CrossRef]

2010 (2)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010).
[CrossRef]

2009 (2)

2006 (4)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

C. K. Madsen, “Boundless-range optical phase modulator for high-speed frequency-shift heterodyne applications,” J. Lightwave Technol. 24, 2760–2767 (2006).
[CrossRef]

D. A. B. Miller, “On perfect cloaking,” Opt. Express 14, 12457–12466 (2006).
[CrossRef]

2005 (2)

2004 (2)

M. Gerken and D. A. B. Miller, “Multilayer thin-film stacks with steplike spatial beam shifting,” J. Lightwave Technol. 22, 612–618 (2004).
[CrossRef]

L. Hesselink, S. S. Orlov, and M. C. Bashaw, “Holographic data storage systems,” Proc. IEEE 92, 1231–1280 (2004).
[CrossRef]

2003 (1)

2000 (1)

1999 (1)

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5 μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

1998 (1)

S. Kawanishi, “Ultrahigh-speed optical time-division-multiplexed transmission technology based on optical signal processing,” IEEE J. Quantum Electron. 34, 2064–2079 (1998).
[CrossRef]

1994 (2)

F. Heismann, “Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems,” J. Lightwave Technol. 12, 690–699 (1994).
[CrossRef]

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

1992 (1)

1991 (1)

Y. Fujii, “High-isolation polarization-independent optical circulator coupled with single-mode fibers,” J. Lightwave Technol. 9, 456–460 (1991).
[CrossRef]

1982 (2)

R. C. Alferness, “Waveguide electrooptic modulators,” IEEE Trans. Microwave Theory 30, 1121–1137 (1982).
[CrossRef]

P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982).
[CrossRef]

1980 (1)

1978 (1)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

1962 (1)

F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Agrawal, G. P.

P. P. Baveja, Y. Xiao, S. Arora, G. P. Agrawal, and D. N. Maywar, “All-optical semiconductor optical amplifier-based wavelength converters with sub-mW pumping,” IEEE Photon. Technol. Lett. 25, 78–80 (2013).
[CrossRef]

Alferness, R. C.

R. C. Alferness, “Waveguide electrooptic modulators,” IEEE Trans. Microwave Theory 30, 1121–1137 (1982).
[CrossRef]

Arora, S.

P. P. Baveja, Y. Xiao, S. Arora, G. P. Agrawal, and D. N. Maywar, “All-optical semiconductor optical amplifier-based wavelength converters with sub-mW pumping,” IEEE Photon. Technol. Lett. 25, 78–80 (2013).
[CrossRef]

Askarov, D.

Baets, R.

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
[CrossRef]

F. Van Laere, W. Bogaerts, P. Dumon, G. Roelkens, D. Van Thourhout, and R. Baets, “Focusing polarization diversity grating couplers in silicon-on-insulator,” J. Lightwave Technol. 27, 612–618 (2009).
[CrossRef]

Baird, D.

F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Balram, K. C.

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[CrossRef]

Bashaw, M. C.

L. Hesselink, S. S. Orlov, and M. C. Bashaw, “Holographic data storage systems,” Proc. IEEE 92, 1231–1280 (2004).
[CrossRef]

Baveja, P. P.

P. P. Baveja, Y. Xiao, S. Arora, G. P. Agrawal, and D. N. Maywar, “All-optical semiconductor optical amplifier-based wavelength converters with sub-mW pumping,” IEEE Photon. Technol. Lett. 25, 78–80 (2013).
[CrossRef]

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

Bogaerts, W.

Bolle, C.

Bowers, J. E.

Boyd, R. W.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef]

Brener, I.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5 μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Brinkmeyer, E.

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
[CrossRef]

Brongersma, M. L.

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[CrossRef]

Buhl, L. L.

C. R. Doerr, N. K. Fontaine, M. Hirano, T. Sasaki, L. L. Buhl, and P. J. Winzer, “Silicon photonic integrated circuit for coupling to a ring-core multimode fiber for space-division multiplexing,” in European Conference on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.A.3.

Buhrer, F.

F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Burrows, E. C.

Cai, X.

Chaban, E. E.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5 μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

Chou, M. H.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5 μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Christman, S. B.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5 μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Clausen, A. T.

E. Palushani, H. C. Hansen Mulvad, M. Galili, H. Hu, L. K. Oxenløwe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical Fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18, 681–688 (2012).
[CrossRef]

Conwell, E. M.

F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Courtial, J.

M. P. J. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. J. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014 (2011).
[CrossRef]

Dai, D.

Djordjevic, S. S.

Doerr, C. R.

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
[CrossRef]

C. R. Doerr, N. K. Fontaine, M. Hirano, T. Sasaki, L. L. Buhl, and P. J. Winzer, “Silicon photonic integrated circuit for coupling to a ring-core multimode fiber for space-division multiplexing,” in European Conference on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.A.3.

Dudley, A.

M. P. J. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. J. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014 (2011).
[CrossRef]

Dumon, P.

Eich, M.

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
[CrossRef]

Esmaeelpour, M.

Essiambre, R.-J.

Fan, S.

Fan, S. H.

V. Liu, D. A. B. Miller, and S. H. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proc. IEEE 101, 484–493 (2013).
[CrossRef]

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
[CrossRef]

Z. Wang and S. H. Fan, “Optical circulators in two-dimensional magneto-optical photonic crystals,” Opt. Lett. 30, 1989–1991 (2005).
[CrossRef]

Y. Jiao, S. H. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator,” Opt. Lett. 30, 141–143 (2005).
[CrossRef]

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M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5 μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
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Forbes, A.

M. P. J. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. J. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014 (2011).
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Freude, W.

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
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R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
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P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982).
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L. Hesselink, S. S. Orlov, and M. C. Bashaw, “Holographic data storage systems,” Proc. IEEE 92, 1231–1280 (2004).
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C. R. Doerr, N. K. Fontaine, M. Hirano, T. Sasaki, L. L. Buhl, and P. J. Winzer, “Silicon photonic integrated circuit for coupling to a ring-core multimode fiber for space-division multiplexing,” in European Conference on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.A.3.

Hsiao, F.-C.

M.-C. Wu, F.-C. Hsiao, and S.-Y. Tseng, “Adiabatic mode conversion in multimode waveguides using chirped computer-generated planar holograms,” IEEE Photon. Technol. Lett. 23, 807–809 (2011).
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E. Palushani, H. C. Hansen Mulvad, M. Galili, H. Hu, L. K. Oxenløwe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical Fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18, 681–688 (2012).
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E. Palushani, H. C. Hansen Mulvad, M. Galili, H. Hu, L. K. Oxenløwe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical Fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18, 681–688 (2012).
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Miller, D. A. B.

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T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
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Y. Jiao, S. H. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator,” Opt. Lett. 30, 141–143 (2005).
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Murnaghan, F. D.

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan, 1962).

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L. Hesselink, S. S. Orlov, and M. C. Bashaw, “Holographic data storage systems,” Proc. IEEE 92, 1231–1280 (2004).
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E. Palushani, H. C. Hansen Mulvad, M. Galili, H. Hu, L. K. Oxenløwe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical Fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18, 681–688 (2012).
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M. P. J. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. J. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014 (2011).
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E. Palushani, H. C. Hansen Mulvad, M. Galili, H. Hu, L. K. Oxenløwe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical Fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18, 681–688 (2012).
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S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
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C. R. Doerr, N. K. Fontaine, M. Hirano, T. Sasaki, L. L. Buhl, and P. J. Winzer, “Silicon photonic integrated circuit for coupling to a ring-core multimode fiber for space-division multiplexing,” in European Conference on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.A.3.

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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
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Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
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T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
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M.-C. Wu, F.-C. Hsiao, and S.-Y. Tseng, “Adiabatic mode conversion in multimode waveguides using chirped computer-generated planar holograms,” IEEE Photon. Technol. Lett. 23, 807–809 (2011).
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S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
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T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
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T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
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C. R. Doerr, N. K. Fontaine, M. Hirano, T. Sasaki, L. L. Buhl, and P. J. Winzer, “Silicon photonic integrated circuit for coupling to a ring-core multimode fiber for space-division multiplexing,” in European Conference on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.A.3.

Wu, M.-C.

M.-C. Wu, F.-C. Hsiao, and S.-Y. Tseng, “Adiabatic mode conversion in multimode waveguides using chirped computer-generated planar holograms,” IEEE Photon. Technol. Lett. 23, 807–809 (2011).
[CrossRef]

Xiao, Y.

P. P. Baveja, Y. Xiao, S. Arora, G. P. Agrawal, and D. N. Maywar, “All-optical semiconductor optical amplifier-based wavelength converters with sub-mW pumping,” IEEE Photon. Technol. Lett. 25, 78–80 (2013).
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Yoo, S. J. B.

Yu, Z.

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
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Adv. Opt. Photon. (1)

Appl. Opt. (2)

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[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

E. Palushani, H. C. Hansen Mulvad, M. Galili, H. Hu, L. K. Oxenløwe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical Fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18, 681–688 (2012).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5 μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

P. P. Baveja, Y. Xiao, S. Arora, G. P. Agrawal, and D. N. Maywar, “All-optical semiconductor optical amplifier-based wavelength converters with sub-mW pumping,” IEEE Photon. Technol. Lett. 25, 78–80 (2013).
[CrossRef]

M.-C. Wu, F.-C. Hsiao, and S.-Y. Tseng, “Adiabatic mode conversion in multimode waveguides using chirped computer-generated planar holograms,” IEEE Photon. Technol. Lett. 23, 807–809 (2011).
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J. Lightwave Technol. (6)

J. Opt. Soc. Am. A (1)

Nano Lett. (1)

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[CrossRef]

Nat. Mater. (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

New J. Phys. (1)

M. P. J. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. J. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014 (2011).
[CrossRef]

Opt. Express (8)

Opt. Lett. (5)

Phys. Rep. (1)

P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

Proc. IEEE (2)

V. Liu, D. A. B. Miller, and S. H. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proc. IEEE 101, 484–493 (2013).
[CrossRef]

L. Hesselink, S. S. Orlov, and M. C. Bashaw, “Holographic data storage systems,” Proc. IEEE 92, 1231–1280 (2004).
[CrossRef]

Science (4)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef]

S. H. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘Nonreciprocal light propagation in a silicon photonic circuit’,” Science 335, 38 (2012).
[CrossRef]

Other (5)

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

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge, 2008).

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan, 1962).

C. R. Doerr, N. K. Fontaine, M. Hirano, T. Sasaki, L. L. Buhl, and P. J. Winzer, “Silicon photonic integrated circuit for coupling to a ring-core multimode fiber for space-division multiplexing,” in European Conference on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.A.3.

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Figures (10)

Fig. 1.
Fig. 1.

Schematic illustration of the self-configuring device. Diagonal gray rectangles are controllable reflectors. Vertical clear rectangles are controllable phase shifters. Dashed clear rectangles are optional phase shifters that may be present in the implementation, but are not necessary. Configurations for (a) one input and output beam pair, (b) two beam pairs, and (c) all four possible beam pairs.

Fig. 2.
Fig. 2.

Example planar layout of a device analogous to Fig. 1(c) with MZIs providing the variable reflectivities and the phase shifts. Not shown are devices, such as grating couplers, that would couple different segments of the input and output beams into and out of the waveguides WI1–WI4 and WO1–WO4, respectively. The self-aligning output coupler CO is reflected about a horizontal axis compared to Fig. 1(c) for compactness. Grayed arms of MZIs M14, M23, and M32 in both the input (CI) and the output (CO) self-aligning couplers are optional; these devices are operated only as phase shifters and could be replaced by simple phase shifters.

Fig. 3.
Fig. 3.

Polarization converter. (a) Plan view. (b) Perspective view. Light incident on the grating coupler in self-aligning coupler CI is split by its incident polarization into the two waveguides, and similarly light from the waveguides going into the grating coupler in self-aligning coupler CO appears on the two different polarizations on the output light beam. PI and PO are phase shifters; the similar but grayed boxes are optional dummy phase shifters. Optionally, a phase shifter and its dummy partner could instead be driven in push–pull to double the available relative phase shift. MZI and MZO are Mach–Zehnder interferometers, and DI and DO are detectors.

Fig. 4.
Fig. 4.

Red–blue interference device. A mixture of “red” and “blue” light at the input is split into its “red” and “blue” components by a dichroic beam splitter. Then the “red” component is converted to “blue” by a frequency shifter so both components are represented by “blue” light but in different waveguides. The device can be trained to look for any particular combination of “red” and “blue” and to output any particular combination of “red” and “blue” as a result.

Fig. 5.
Fig. 5.

Example general apparatus for performing arbitrary linear mappings from input fields with spatial, polarization and frequency content to corresponding output fields, illustrated here for four spatial modes and three different frequency components. Each of the resulting 4×2×3=24 orthogonal channels can be separately modulated using the modulators in the middle column, corresponding to the elements of Ddiag.

Fig. 6.
Fig. 6.

Mode transformer for the operator U for M=4 with the reflectivities and phase shifts labeled for each beam-splitter block. The diagonal mirror has 100% reflectivity.

Fig. 7.
Fig. 7.

Beam splitter with definitions of field reflection and transmission factors and nominal labels of the beam-splitter ports as top, bottom, left, and right.

Fig. 8.
Fig. 8.

Symmetric Mach–Zehnder waveguide modulator configuration with 50% (“3 dB”) splitters notionally implemented here with coupled waveguides and two arms each with a phase-shifting element. The gray rectangles represent the phase-shifting control elements (e.g., electrodes). The labeling of the ports corresponds with the notation used in Fig. 7.

Fig. 9.
Fig. 9.

Use of optical circulators with forward and backward modes. (a) Schematic of a three-port optical circulator. The dashed lines show the effective paths of waves in different directions between the three ports. (b) Universal four-port “two-way”, potentially nonreciprocal device, with input and output beams in each of two paths at both the left and right of the device. The central U, Ddiag, and V units form a general spatial mode converter as in Figs. 1, 2, and 5.

Fig. 10.
Fig. 10.

Illustration of an idealized time-delay unit. The switches rotates through positions 1, 2, and 3, with a dwell time of Δt at each position, taking a total time of 3Δt to cycle through all three positions before returning to position 1. (a) Switch used at input side. (b) Switch used at output side.

Equations (31)

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|ϕO=D|ϕI.
D=msDm|ϕDOmϕDIm|,
D=VDdiagU.
D=[v11v21v31v41v12v22v32v42][sD100sD2][u11*u21*u31*u41*u12*u22*u32*u42*],
|ϕDI1=[u11u21u31u41]|ϕDI2=[u12u22u32u42]|ϕDO1=[v11v21v31v41]|ϕDO2=[v12v22v32v42].
ND=2MC(MI+MOMC),
red+blueblueredbluered.
t(TB)=|t(TB)|exp(iθ(TB)),
|t(TB)|2=1|r(TR)|2=|t(LR)|2=1|r(LB)|2,
θ(TR)+θ(LB)θ(TB)θ(LR)=±π
|ϕDIm=n=1Mamn|ϕ1n,
r11(TR)exp(iθ11)=a11*.
t11(LR)r12(TR)exp[i(θ11+θ12)]=a12*,
r12(TR)exp(iθ12)=a12*exp(iθ11)/t11(LR).
r1n(TR)exp(iθ1n)=a1n*exp(ip=1n1θ1p)/q=1n1t1q(LR),
|ϕ(u)=j=1Mu+1aj(u)|ϕuj,
|ϕ(2)=C(1)|ϕ(1),
C(u)=[tu1(TB)c12(u)c13(u)c1(Mu)(u)c1(Mu+1)(u)0tu2(TB)c23(u)0tu3(TB)000tu(Mu)(TB)c(Mu)(Mu+1)(u)],
css(u)=tus(TB).
csj(u)=ruj(TR)rus(LB)[p=s+1j1tup(LR)]exp[ip=s+1jθup].
|ϕDI2(2)j=1M1a2j(2)|ϕ2j=C(1)|ϕDI2
|ϕDI3(3)j=1M2a3j(3)|ϕ3j=C(2)C(1)|ϕDI3.
|ϕDOm=n=1Mbmn|β1n.
θrefl=θTC=θRD=θFL=θGB.
θtrans=θTD=θLC=θFB=θGR.
t(TB)=t(LR)=texp(iθS)exp(iθav),
t=cos(Δθ/2),
θS=θex+θtrans+θrefl.
θtrans=θrefl±π/2.
r(TR)=r(LB)=rexp(iθS)exp(iθav),
r=sin(Δθ/2).

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