Abstract

We introduce an all-optical arithmetic unit operating a weighted addition and subtraction between multiple phase-and-amplitude coded signals. The scheme corresponds to calculating the field dot-product of frequency channels with a static vector of coefficients. The system is reconfigurable and format transparent. It is based on Fourier-domain processing and multiple simultaneous four-wave mixing processes inside a single nonlinear element. We demonstrate the device with up to three channels at 40 Gb/s and evaluate its efficiency by measuring the bit-error-rate of a distortion compensation operation between two signals.

© 2014 Optical Society of America

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2013 (2)

2012 (3)

S. Singh, L. Lovkesh, “Ultrahigh speed optical signal processing logic based on an SOA-MZI,” IEEE J. Sel. Top. Quantum Electron. 18, 970–977 (2012).
[CrossRef]

L. Yan, A. Willner, X. Wu, A. Yi, “All-optical signal processing for ultrahigh speed optical systems and networks,” J. Lightwave Technol. 30, 3760–3770 (2012).
[CrossRef]

J. Kurumida, S. J. B. Yoo, “Nonlinear optical signal processing in optical packet switching systems,” IEEE J. Sel. Top. Quantum Electron. 18, 978–987 (2012).
[CrossRef]

2011 (4)

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

D. Schneider, “Trading at the speed of light,” IEEE Spectrum 48, 11–12 (2011).
[CrossRef]

P. Bower, I. Dedic, “High speed converters and DSP for 100G and beyond,” Opt. Fiber Technol. 17, 464–471 (2011).
[CrossRef]

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

2010 (3)

2009 (2)

J. Wang, Q. Sun, J. Sun, “All-optical 40 Gbit/s CSRZ-DPSK logic XOR gate and format conversion using four-wave mixing,” Opt. Express 17, 12555–12563 (2009).
[CrossRef] [PubMed]

D. E. Tamir, N. T. Shaked, P. J. Wilson, S. Dolev, “High-speed and low-power electro-optical DSP coprocessor,” J. Opt. Soc. Am. A 26, 11–20 (2009).
[CrossRef]

2006 (2)

G. Berrettini, A. Simi, A. Malacarne, A. Bogoni, L. Potí, “Ultrafast integrable and reconfigurable XNOR, AND, NOR, and NOT photonic logic gate,” IEEE Photonics Technol. Lett. 18, 917–919 (2006).
[CrossRef]

P. J. Winzer, R.-J. Essiambre, “Advanced modulation formats for high-capacity optical transport networks,” J. Lightwave Technol. 24, 4711–4728 (2006).
[CrossRef]

2002 (1)

K. Azadet, E. Haratsch, H. Kim, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid State Circuits 37, 317–327 (2002).
[CrossRef]

1994 (1)

J. Zhou, M. O’Mahony, S. Walker, “Analysis of optical crosstalk effects in multi-wavelength switched networks,” IEEE Photonics Technol. Lett. 6, 302–305 (1994).
[CrossRef]

1987 (1)

1985 (1)

A. Hill, D. Payne, “Linear crosstalk in wavelength-division-multiplexed optical-fiber transmission systems,” J. Lightwave Technol. 3, 643–651 (1985).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Athale, R. A.

R. A. Athale, “Optical matrix algebraic processors: a survey,” in 10th International Optical Computing Conference,S. Horvitz, ed. (International Society for Optics and Photonics, 1983), pp. 24–31.
[CrossRef]

Azadet, K.

K. Azadet, E. Haratsch, H. Kim, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid State Circuits 37, 317–327 (2002).
[CrossRef]

Becker, J.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Ben Ezra, S.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Berrettini, G.

G. Berrettini, A. Simi, A. Malacarne, A. Bogoni, L. Potí, “Ultrafast integrable and reconfigurable XNOR, AND, NOR, and NOT photonic logic gate,” IEEE Photonics Technol. Lett. 18, 917–919 (2006).
[CrossRef]

Bogoni, A.

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

G. Berrettini, A. Simi, A. Malacarne, A. Bogoni, L. Potí, “Ultrafast integrable and reconfigurable XNOR, AND, NOR, and NOT photonic logic gate,” IEEE Photonics Technol. Lett. 18, 917–919 (2006).
[CrossRef]

Bonk, R.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Bower, P.

P. Bower, I. Dedic, “High speed converters and DSP for 100G and beyond,” Opt. Fiber Technol. 17, 464–471 (2011).
[CrossRef]

Caulfield, H. J.

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261–263 (2010).
[CrossRef]

Chiou, A. E. T.

Dedic, I.

P. Bower, I. Dedic, “High speed converters and DSP for 100G and beyond,” Opt. Fiber Technol. 17, 464–471 (2011).
[CrossRef]

Dolev, S.

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261–263 (2010).
[CrossRef]

D. E. Tamir, N. T. Shaked, P. J. Wilson, S. Dolev, “High-speed and low-power electro-optical DSP coprocessor,” J. Opt. Soc. Am. A 26, 11–20 (2009).
[CrossRef]

Dreschmann, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Du, L. B.

Eggleton, B. J.

Ellermeyer, T.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Essiambre, R.-J.

Freiberger, M.

M. Freiberger, D. Templeton, E. Mercado, “Low latency optical services,” in National Fiber Optic Engineers Conference(2012), paper NTu2E.1.
[CrossRef]

Freude, W.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Frey, F.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Frisken, S.

Haratsch, E.

K. Azadet, E. Haratsch, H. Kim, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid State Circuits 37, 317–327 (2002).
[CrossRef]

Hill, A.

A. Hill, D. Payne, “Linear crosstalk in wavelength-division-multiplexed optical-fiber transmission systems,” J. Lightwave Technol. 3, 643–651 (1985).
[CrossRef]

Hillerkuss, D.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Hoh, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Huber, G.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Huebner, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Ishikawa, H.

H. Ishikawa, Ultrafast All-Optical Signal Processing Devices (John Wiley, 2008).
[CrossRef]

Jordan, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Kim, H.

K. Azadet, E. Haratsch, H. Kim, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid State Circuits 37, 317–327 (2002).
[CrossRef]

Kleinow, P.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Koenig, S.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Koos, C.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Kurumida, J.

J. Kurumida, S. J. B. Yoo, “Nonlinear optical signal processing in optical packet switching systems,” IEEE J. Sel. Top. Quantum Electron. 18, 978–987 (2012).
[CrossRef]

Leuthold, J.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Li, J.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Li, M.

Lovkesh, L.

S. Singh, L. Lovkesh, “Ultrahigh speed optical signal processing logic based on an SOA-MZI,” IEEE J. Sel. Top. Quantum Electron. 18, 970–977 (2012).
[CrossRef]

Lowery, A. J.

Ludwig, A.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Lutz, J.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Malacarne, A.

G. Berrettini, A. Simi, A. Malacarne, A. Bogoni, L. Potí, “Ultrafast integrable and reconfigurable XNOR, AND, NOR, and NOT photonic logic gate,” IEEE Photonics Technol. Lett. 18, 917–919 (2006).
[CrossRef]

Marculescu, A.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Mercado, E.

M. Freiberger, D. Templeton, E. Mercado, “Low latency optical services,” in National Fiber Optic Engineers Conference(2012), paper NTu2E.1.
[CrossRef]

Meyer, J.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Moeller, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Narkiss, N.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Nebendahl, B.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Nuccio, S. R.

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

O’Mahony, M.

J. Zhou, M. O’Mahony, S. Walker, “Analysis of optical crosstalk effects in multi-wavelength switched networks,” IEEE Photonics Technol. Lett. 6, 302–305 (1994).
[CrossRef]

Oehler, A.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Paquot, Y.

Parmigiani, F.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Payne, D.

A. Hill, D. Payne, “Linear crosstalk in wavelength-division-multiplexed optical-fiber transmission systems,” J. Lightwave Technol. 3, 643–651 (1985).
[CrossRef]

Pelusi, M. D.

Peng, J.

Petropoulos, P.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Potí, L.

G. Berrettini, A. Simi, A. Malacarne, A. Bogoni, L. Potí, “Ultrafast integrable and reconfigurable XNOR, AND, NOR, and NOT photonic logic gate,” IEEE Photonics Technol. Lett. 18, 917–919 (2006).
[CrossRef]

Resan, B.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Roeger, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Roelens, M. A. F.

Schellinger, T.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Schmogrow, R.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Schneider, D.

D. Schneider, “Trading at the speed of light,” IEEE Spectrum 48, 11–12 (2011).
[CrossRef]

Schröder, J.

Shaked, N. T.

D. E. Tamir, N. T. Shaked, P. J. Wilson, S. Dolev, “High-speed and low-power electro-optical DSP coprocessor,” J. Opt. Soc. Am. A 26, 11–20 (2009).
[CrossRef]

Shan, O.

Shen, Z.

Simi, A.

G. Berrettini, A. Simi, A. Malacarne, A. Bogoni, L. Potí, “Ultrafast integrable and reconfigurable XNOR, AND, NOR, and NOT photonic logic gate,” IEEE Photonics Technol. Lett. 18, 917–919 (2006).
[CrossRef]

Singh, S.

S. Singh, L. Lovkesh, “Ultrahigh speed optical signal processing logic based on an SOA-MZI,” IEEE J. Sel. Top. Quantum Electron. 18, 970–977 (2012).
[CrossRef]

Sun, J.

Sun, Q.

Tamir, D. E.

D. E. Tamir, N. T. Shaked, P. J. Wilson, S. Dolev, “High-speed and low-power electro-optical DSP coprocessor,” J. Opt. Soc. Am. A 26, 11–20 (2009).
[CrossRef]

Templeton, D.

M. Freiberger, D. Templeton, E. Mercado, “Low latency optical services,” in National Fiber Optic Engineers Conference(2012), paper NTu2E.1.
[CrossRef]

Vallaitis, T.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Walker, S.

J. Zhou, M. O’Mahony, S. Walker, “Analysis of optical crosstalk effects in multi-wavelength switched networks,” IEEE Photonics Technol. Lett. 6, 302–305 (1994).
[CrossRef]

Wang, J.

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

J. Wang, Q. Sun, J. Sun, “All-optical 40 Gbit/s CSRZ-DPSK logic XOR gate and format conversion using four-wave mixing,” Opt. Express 17, 12555–12563 (2009).
[CrossRef] [PubMed]

Wang, X.

Weingarten, K.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Willner, A.

L. Yan, A. Willner, X. Wu, A. Yi, “All-optical signal processing for ultrahigh speed optical systems and networks,” J. Lightwave Technol. 30, 3760–3770 (2012).
[CrossRef]

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

Wilson, P. J.

D. E. Tamir, N. T. Shaked, P. J. Wilson, S. Dolev, “High-speed and low-power electro-optical DSP coprocessor,” J. Opt. Soc. Am. A 26, 11–20 (2009).
[CrossRef]

Winter, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

Winzer, P. J.

Wu, X.

L. Yan, A. Willner, X. Wu, A. Yi, “All-optical signal processing for ultrahigh speed optical systems and networks,” J. Lightwave Technol. 30, 3760–3770 (2012).
[CrossRef]

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

Yan, L.

Yeh, P.

Yi, A.

Yilmaz, O.

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

Yoo, S. J. B.

J. Kurumida, S. J. B. Yoo, “Nonlinear optical signal processing in optical packet switching systems,” IEEE J. Sel. Top. Quantum Electron. 18, 978–987 (2012).
[CrossRef]

Zhang, L.

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

Zhou, J.

J. Zhou, M. O’Mahony, S. Walker, “Analysis of optical crosstalk effects in multi-wavelength switched networks,” IEEE Photonics Technol. Lett. 6, 302–305 (1994).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (3)

S. Singh, L. Lovkesh, “Ultrahigh speed optical signal processing logic based on an SOA-MZI,” IEEE J. Sel. Top. Quantum Electron. 18, 970–977 (2012).
[CrossRef]

J. Kurumida, S. J. B. Yoo, “Nonlinear optical signal processing in optical packet switching systems,” IEEE J. Sel. Top. Quantum Electron. 18, 978–987 (2012).
[CrossRef]

A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17, 320–332 (2011).
[CrossRef]

IEEE J. Solid State Circuits (1)

K. Azadet, E. Haratsch, H. Kim, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid State Circuits 37, 317–327 (2002).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

G. Berrettini, A. Simi, A. Malacarne, A. Bogoni, L. Potí, “Ultrafast integrable and reconfigurable XNOR, AND, NOR, and NOT photonic logic gate,” IEEE Photonics Technol. Lett. 18, 917–919 (2006).
[CrossRef]

J. Zhou, M. O’Mahony, S. Walker, “Analysis of optical crosstalk effects in multi-wavelength switched networks,” IEEE Photonics Technol. Lett. 6, 302–305 (1994).
[CrossRef]

IEEE Spectrum (1)

D. Schneider, “Trading at the speed of light,” IEEE Spectrum 48, 11–12 (2011).
[CrossRef]

J. Lightwave Technol. (3)

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

D. E. Tamir, N. T. Shaked, P. J. Wilson, S. Dolev, “High-speed and low-power electro-optical DSP coprocessor,” J. Opt. Soc. Am. A 26, 11–20 (2009).
[CrossRef]

Nat. Photonics (2)

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5, 364–371 (2011).
[CrossRef]

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261–263 (2010).
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Opt. Express (4)

Opt. Fiber Technol. (1)

P. Bower, I. Dedic, “High speed converters and DSP for 100G and beyond,” Opt. Fiber Technol. 17, 464–471 (2011).
[CrossRef]

Opt. Lett. (1)

Other (4)

H. Ishikawa, Ultrafast All-Optical Signal Processing Devices (John Wiley, 2008).
[CrossRef]

R. A. Athale, “Optical matrix algebraic processors: a survey,” in 10th International Optical Computing Conference,S. Horvitz, ed. (International Society for Optics and Photonics, 1983), pp. 24–31.
[CrossRef]

M. Freiberger, D. Templeton, E. Mercado, “Low latency optical services,” in National Fiber Optic Engineers Conference(2012), paper NTu2E.1.
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G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

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Figures (8)

Fig. 1
Fig. 1

Principle of the all-optical dot-product. Relative phase and magnitude coefficients are applied to three WDM signals before propagating through a Kerr nonlinear medium for generation of idlers at a same frequency. Signals with opposite phases (represented with opposite arrows), lead to idlers canceling each other.

Fig. 2
Fig. 2

Experimental diagram. A comb source is split into a pump pulse train and a data signal encoded using a Mach-Zehnder modulator. Spectral carving and selective delaying of the data signal creates three distinct bit streams. After following different path, pumps and signals are recombined and controlled in phase and intensity by a Fourier domain programmable optical processor (FD-POP) and generate an idler in a highly nonlinear fibre.

Fig. 3
Fig. 3

Optical spectrum after the HNLF. (dashed trace) Signal 1 and Signal 2 are equal, ϕS1 = ϕS2 + π; (dotted trace) Signal 2 and Signal 3 are equal, ϕS2 = ϕS3 + π; (solid trace) dot-product with three signals involved. (Right) Spectrum of the corresponding idlers superimposed and filtering setup.

Fig. 4
Fig. 4

Sampling scope traces showing the modulus squared of the dot-product output signal. The blue traces are simulations of the expected intensity levels.

Fig. 5
Fig. 5

Two WDM signals with controlled amount of distortion are generated. The dot-product device is configured to revert the distortion so as to restore the initial binary signal.

Fig. 6
Fig. 6

Attenuation and phase profiles implemented inside the FD-POP to distort channel 1 (left) and equivalent delay line interferometer (right).

Fig. 7
Fig. 7

Optical spectrum after the HNLF. (black) Signal 1 and Signal 2 are equal, ϕS1 = ϕS2 ; (blue) Signal 1 and Signal 2 are equal, ϕS1 = ϕS2 + π; (green) Signal 1 and Signal 2 are different and affected by a power transfer of 30%, ϕS1 = ϕS2 + π.

Fig. 8
Fig. 8

Bit-error rate testing of the compensator. Back-to-back (B2B) and compensated curves are measured respectively at the wavelength of the signal and of the converted idler.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

E I tot = a 1 E S 1 + a 2 E S 2 + a 3 E S 3
= | a 1 | e i ϕ 1 E S 1 + | a 2 | e i ϕ 2 E S 2 + | a 3 | e i ϕ 3 E S 3
E S 1 ( t ) = E S 1 e i 2 π ν S 1 t
E S 2 ( t ) = E S 2 e i 2 π ( ν S 1 + Δ ν ) t
E S 3 ( t ) = E S 3 e i 2 π ( ν S 1 + 2 Δ ν ) t
E P 1 ( t ) = E P 1 e i 2 π ν P 1 t
E P 2 ( t ) = E P 2 e i 2 π ( ν P 1 + Δ ν 2 ) t
E P 3 ( t ) = E P 3 e i 2 π ( ν P 1 + Δ ν ) t
k I 1 = k S 1 + 2 k P 1
k I 2 = k S 2 + 2 k P 2
k I 3 = k S 3 + 2 k P 3
ϕ I 1 = ϕ S 1 + 2 ϕ P 1 = ϕ S 1 + 2 ϕ P
ϕ I 2 = ϕ S 2 + 2 ϕ P 2 = ϕ S 2 + 2 ϕ P
ϕ I 2 = ϕ S 3 + 2 ϕ P 3 = ϕ S 3 + 2 ϕ P
A I 1 | a 1 | A S 1 A P 1 2
A I 2 | a 2 | A S 2 A P 2 2
A I 3 | a 3 | A S 3 A P 3 2
E I tot = A I 1 e i ϕ I 1 + A I 2 e i ϕ I 2 + A I 3 e i ϕ I 3
( | a 1 | A S 1 e i ( ϕ S 1 + ϕ 1 ) + | a 2 | A S 2 e i ( ϕ S 2 + ϕ 2 ) + | a 3 | A S 3 e i ( ϕ S 3 + ϕ 3 ) ) × e i . 2 ϕ P absolute phase offset
= a 1 E S 1 + a 2 E S 2 + a 3 E S 3
X = ( a b c d )
X 1 = α ( d b c a )
E tot = a E S 1 + b E S 2 = A S 1 e i ϕ S 1 ( a e i k Δ L 2 + b e + i k Δ L 2 )
= A S 1 e i ϕ S 1 ( ( a + b ) cos ( k Δ L 2 ) + i ( b a ) sin ( k Δ L 2 ) )
1 = | E tot | 2 | A S 1 | 2 = ( a + b 2 ) cos 2 ( k Δ L 2 ) + ( b a ) 2 sin 2 ( k Δ L 2 )
Φ 1 = angle ( E tot ) = a tan ( ( b a ) sin ( k Δ L 2 ) ( a + b ) cos ( k Δ L 2 ) )
X 0.2 = ( 0.8 0.2 0.2 0.8 )
X 0.3 = ( 0.7 0.3 0.3 0.7 )

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