Abstract

We designed a universal linear circuit by using microring resonators instead of conventional Mach-Zehnder interferometers and phase shifters. We illustrated that the footprint of the universal linear circuit can be drastically reduced (∼ 1/10). In addition, power consumption can also be reduced by using the sensitivity of the phase change in the vicinity of the resonant peak. Furthermore, as an important example of the application for optical communication, MIMO compensation operation is numerically demonstrated by our proposed universal linear circuit. The proposed design can be adapted to other experimentally reported devices, which will accelerate the integration of the universal linear circuit.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  13. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
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2019 (1)

T. Sato, T. Fujisawa, and K. Saitoh, “All-optical diode suppressing broadband backward transmission using single- and four-port photonic crystal cavities,” IEEE Photonics J. 11(1), 1–14 (2019).
[Crossref]

2018 (2)

2017 (2)

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Rep. 7(1), 15133–15133-10 (2017).
[Crossref]

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

2016 (1)

2015 (1)

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(6249), 711–716 (2015).
[Crossref]

2013 (1)

2011 (1)

2006 (1)

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

2002 (1)

1999 (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref]

Annoni, A.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Bannerman, R. H. S.

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 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(1), 58–61 (1994).
[Crossref]

Burgwal, R.

Carminati, M.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Carolan, 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(6249), 711–716 (2015).
[Crossref]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Clements, W. R.

Crespi, A.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Rep. 7(1), 15133–15133-10 (2017).
[Crossref]

Darmawan, S.

Fan, S.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Ferrari, G.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Flamini, F.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Rep. 7(1), 15133–15133-10 (2017).
[Crossref]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Frey, B. J.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

Fujisawa, T.

T. Sato, T. Fujisawa, and K. Saitoh, “All-optical diode suppressing broadband backward transmission using single- and four-port photonic crystal cavities,” IEEE Photonics J. 11(1), 1–14 (2019).
[Crossref]

Gates, J. C.

Ghosh, S.

Guglielmi, E.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Harrold, 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(6249), 711–716 (2015).
[Crossref]

Hashimoto, T.

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(6249), 711–716 (2015).
[Crossref]

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Humphreys, P. C.

Ikeda, K.

Itoh, M.

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(6249), 711–716 (2015).
[Crossref]

Joannopoulos, J. D.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Kawashima, H.

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Kolthammer, W. S.

Koshiba, M.

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Laing, A.

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(6249), 711–716 (2015).
[Crossref]

Leviton, D. B.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Madison, T. J.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Marshall, G. D.

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(6249), 711–716 (2015).
[Crossref]

Martín-López, E.

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(6249), 711–716 (2015).
[Crossref]

Matsuda, N.

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(6249), 711–716 (2015).
[Crossref]

Matthews, J. C. F.

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(6249), 711–716 (2015).
[Crossref]

Mei, T.

Melloni, A.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Mennea, P. L.

Metcalf, B. J.

Miller, D. A.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Miller, D. A. B.

Morichetti, F.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Nakano, Y.

O’Brien, J. L.

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(6249), 711–716 (2015).
[Crossref]

Oguma, M.

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(6249), 711–716 (2015).
[Crossref]

Osellame, R.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Rep. 7(1), 15133–15133-10 (2017).
[Crossref]

Reck, M.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref]

Renema, J. J.

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(6249), 711–716 (2015).
[Crossref]

Saitoh, K.

T. Sato, T. Fujisawa, and K. Saitoh, “All-optical diode suppressing broadband backward transmission using single- and four-port photonic crystal cavities,” IEEE Photonics J. 11(1), 1–14 (2019).
[Crossref]

Sampietro, M.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Sato, T.

T. Sato, T. Fujisawa, and K. Saitoh, “All-optical diode suppressing broadband backward transmission using single- and four-port photonic crystal cavities,” IEEE Photonics J. 11(1), 1–14 (2019).
[Crossref]

Sciarrino, F.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Rep. 7(1), 15133–15133-10 (2017).
[Crossref]

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(6249), 711–716 (2015).
[Crossref]

Silverstone, J. W.

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(6249), 711–716 (2015).
[Crossref]

Smith, D. H.

Smith, P. G. R.

Spagnolo, N.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Rep. 7(1), 15133–15133-10 (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(6249), 711–716 (2015).
[Crossref]

Suzuki, K.

Tanemura, T.

Tang, R.

Tanizawa, K.

Thompson, M. G.

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(6249), 711–716 (2015).
[Crossref]

Tobing, L. Y. M.

Tsuji, Y.

Viggianiello, N.

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Rep. 7(1), 15133–15133-10 (2017).
[Crossref]

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Walmsley, I. A.

Walsmley, I. A.

Zeilinger, A.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref]

Zhang, D. H.

Zhang, Y.

IEEE J. Quantum Electron. (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

IEEE Photonics J. (1)

T. Sato, T. Fujisawa, and K. Saitoh, “All-optical diode suppressing broadband backward transmission using single- and four-port photonic crystal cavities,” IEEE Photonics J. 11(1), 1–14 (2019).
[Crossref]

J. Lightwave Technol. (2)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20(3), 463–468 (2002).
[Crossref]

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

Light: Sci. Appl. (1)

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—Automatically undoing strong mixing between modes,” Light: Sci. Appl. 6(12), e17110 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optica (2)

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(1), 58–61 (1994).
[Crossref]

Proc. SPIE (1)

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

Sci. Rep. (1)

F. Flamini, N. Spagnolo, N. Viggianiello, A. Crespi, R. Osellame, and F. Sciarrino, “Benchmarking integrated linear-optical architectures for quantum information processing,” Sci. Rep. 7(1), 15133–15133-10 (2017).
[Crossref]

Science (1)

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(6249), 711–716 (2015).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Schematic diagram of the proposed universal linear circuit using ring resonators. This example illustrates the 3 × 3 unitary operation (b1, b2, b3)T = T(a1, a2, a3) T in the planar lightwave circuit.
Fig. 2.
Fig. 2. Comparison of the (a) phase shift and (b) transmission that occur by changing the refractive index in three types of waveguides: a wire waveguide, a wire waveguide connected to a single two-port ring resonator, and cascaded two-port ring resonators. The illustrations on the right represent these three waveguides. The phase is defined as tan−1(b/a), where a and b are the incoming and outgoing amplitudes, respectively.
Fig. 3.
Fig. 3. Transmission characteristics of the four-port double ring resonator. (a) Transmission spectra for through ports (red lines) and drop ports (blue lines) and (b) the transmission for a wavelength of 1.5496 µm as a function of refractive index change Δn. The illustration on the right represents the four-port double ring resonator. The gap between the rings is g2 = 360 nm; the other parameters are identical to the single two-port ring resonator in Fig. 2. The transmissions for the thorough and drop ports are defined as |b/a|2 and |c/a|2, respectively, where a, b, and c are the incoming amplitude from the input port, and the outgoing amplitudes to the thorough port and drop port, respectively.
Fig. 4.
Fig. 4. Illustration of the MIMO compensation using the proposed universal linear circuit. It is assumed that the signals are linearly mixed by the unitary transfer matrix T.
Fig. 5.
Fig. 5. Field distribution (logarithmic plot) of the universal linear circuit for MIMO compensation by a 3 × 3 unitary operation. Mixed three lightwaves (Figs. 5(a), (b) and (c)) are input from the left side and are successfully divided among three right ports. The values in the left- and right- side labels denote the input and output amplitudes, respectively.

Equations (5)

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U = A ( φ ) B ( θ ) ,
A ( φ ) = ( cos φ j sin φ j sin φ cos φ ) ,
B ( θ ) = ( 1 0 0 e j θ ) ,
T 23 ( 3 ) ( φ , θ ) = ( 1 0 0 0 cos φ j sin φ e j θ 0 j sin φ cos φ e j θ ) .
T ( 3 ) = D ( 3 ) ( θ 1 , θ 2 , θ 3 ) T 12 ( 3 ) ( φ 12 , θ 12 ) T 13 ( 3 ) ( φ 13 , θ 13 ) T 23 ( 3 ) ( φ 23 , θ 23 ) = D ( 3 ) ( θ 1 + π , θ 2 + π , θ 3 + π 2 ) T 12 ( 3 ) ( φ 12 , θ 12 ) S 12 ( 3 ) T 23 ( 3 ) ( φ 13 , θ 13 π 2 ) S 12 ( 3 ) T 23 ( 3 ) ( φ 23 , θ 23 ) ,

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