Abstract

We report the first experimental demonstration of pulsed pump wavelength exchange in a highly nonlinear dispersion-shifted fiber for alloptical time de-multiplexing of 80 Gb/s return-to-zero (RZ) signals. Orthogonal pulsed pump and continuous wave (cw) pump are used in the fiber-based wavelength exchange for ultra-fast power switching. Error-free operation was achieved for the proposed all-optical 1:8 de-multiplexer over all time de-multiplexed 10 Gb/s RZ signals with ≤ 2.1 dB power penalty at 10-9 bit-error rate. The experimental results show that a wide-band phasematching is feasible for fiber-based nonlinear parametric processes.

© 2008 Optical Society of America

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  1. Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
    [CrossRef]
  2. S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
    [CrossRef]
  3. K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, "Wavelength exchange: a novel function for optical networks," Inf. Sci. 149, 161-169 (2003).
    [CrossRef]
  4. K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments," IEEE J. Sel. Top. Quantum Electron. 8, 560-568 (2002).
    [CrossRef]
  5. G. Kalogerakis, M. Marhic, and L. Kazovsky, "Low Noise Figure Efficient Wavelength Exchange in an Optical Fibre," in Proc. of ECOC, Paper Th1.3.1 (2006).
  6. J. L. Blows and S. E. French, "Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump," Opt. Lett. 27, 491-493 (2002).
    [CrossRef]
  7. R. W. L. Fung, H. K. Y. Cheung, and K. K. Y. Wong, "Widely Tunable Wavelength Exchange in Anomalous-Dispersion Regime," IEEE Photon. Technol. Lett. 19, 1846-1848 (2007).
    [CrossRef]
  8. K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Polarization-insensitive wavelength exchange in highly-nonlinear dispersion shifted fiber," in Proc. of OFC, Paper ThY3 (2002).
  9. R. W. L. Fung, H. K. Y. Cheung, B. P. P. Kuo, and K. K. Y. Wong, "Wavelength Exchange with Enhanced Extinction Ratio in Highly Nonlinear Dispersion-Shifted Fiber," in Proc. of CLEO, Paper JTuA39 (2007).
  10. H. C. H. Mulvad, L. K. Oxenlowe, A. T. Clausen, M. Galili, L. Gruner-Nielsen, and P. Jeppesen, "Error-Free 320 Gb/s Simultaneous Add-Drop Multiplexing," in Proc. of OFC, Paper OTuI5 (2007).
  11. J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
    [CrossRef]
  12. C. H. Kwok, C. W. Chow, H. K. Tsang, C. Lin, and A. Bjarklev, "Nonlinear polarization rotation in a dispersion-flattened photonic-crystal fiber for ultrawideband (>100 nm) all-optical wavelength conversion of 10 Gbit/s nonreturn-to-zero signals," Opt. Lett. 31, 1782-1784 (2006).
    [CrossRef] [PubMed]

2007

R. W. L. Fung, H. K. Y. Cheung, and K. K. Y. Wong, "Widely Tunable Wavelength Exchange in Anomalous-Dispersion Regime," IEEE Photon. Technol. Lett. 19, 1846-1848 (2007).
[CrossRef]

2006

C. H. Kwok, C. W. Chow, H. K. Tsang, C. Lin, and A. Bjarklev, "Nonlinear polarization rotation in a dispersion-flattened photonic-crystal fiber for ultrawideband (>100 nm) all-optical wavelength conversion of 10 Gbit/s nonreturn-to-zero signals," Opt. Lett. 31, 1782-1784 (2006).
[CrossRef] [PubMed]

2005

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

2004

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

2003

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, "Wavelength exchange: a novel function for optical networks," Inf. Sci. 149, 161-169 (2003).
[CrossRef]

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

2002

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments," IEEE J. Sel. Top. Quantum Electron. 8, 560-568 (2002).
[CrossRef]

J. L. Blows and S. E. French, "Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump," Opt. Lett. 27, 491-493 (2002).
[CrossRef]

Agrawal, G. P.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

Berg, K. S.

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

Bjarklev, A.

C. H. Kwok, C. W. Chow, H. K. Tsang, C. Lin, and A. Bjarklev, "Nonlinear polarization rotation in a dispersion-flattened photonic-crystal fiber for ultrawideband (>100 nm) all-optical wavelength conversion of 10 Gbit/s nonreturn-to-zero signals," Opt. Lett. 31, 1782-1784 (2006).
[CrossRef] [PubMed]

Blows, J. L.

J. L. Blows and S. E. French, "Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump," Opt. Lett. 27, 491-493 (2002).
[CrossRef]

Centanni, J. C.

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

Cheung, H. K. Y.

R. W. L. Fung, H. K. Y. Cheung, and K. K. Y. Wong, "Widely Tunable Wavelength Exchange in Anomalous-Dispersion Regime," IEEE Photon. Technol. Lett. 19, 1846-1848 (2007).
[CrossRef]

Chinlon Lin, H. K.

C. H. Kwok, C. W. Chow, H. K. Tsang, C. Lin, and A. Bjarklev, "Nonlinear polarization rotation in a dispersion-flattened photonic-crystal fiber for ultrawideband (>100 nm) all-optical wavelength conversion of 10 Gbit/s nonreturn-to-zero signals," Opt. Lett. 31, 1782-1784 (2006).
[CrossRef] [PubMed]

Chow, C. W.

C. H. Kwok, C. W. Chow, H. K. Tsang, C. Lin, and A. Bjarklev, "Nonlinear polarization rotation in a dispersion-flattened photonic-crystal fiber for ultrawideband (>100 nm) all-optical wavelength conversion of 10 Gbit/s nonreturn-to-zero signals," Opt. Lett. 31, 1782-1784 (2006).
[CrossRef] [PubMed]

Chraplyvy, A. R.

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

Clausen, A. T.

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

Ford, J.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

French, S. E.

J. L. Blows and S. E. French, "Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump," Opt. Lett. 27, 491-493 (2002).
[CrossRef]

Fung, R. W. L.

R. W. L. Fung, H. K. Y. Cheung, and K. K. Y. Wong, "Widely Tunable Wavelength Exchange in Anomalous-Dispersion Regime," IEEE Photon. Technol. Lett. 19, 1846-1848 (2007).
[CrossRef]

Gnauck, A. H.

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

Ibsen, M.

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

Jeppesen, P.

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

Jiang, R.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

Jopson, R.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

Jopson, R. M.

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

Kazovsky, L. G.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, "Wavelength exchange: a novel function for optical networks," Inf. Sci. 149, 161-169 (2003).
[CrossRef]

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments," IEEE J. Sel. Top. Quantum Electron. 8, 560-568 (2002).
[CrossRef]

Kwok, C. H.

C. H. Kwok, C. W. Chow, H. K. Tsang, C. Lin, and A. Bjarklev, "Nonlinear polarization rotation in a dispersion-flattened photonic-crystal fiber for ultrawideband (>100 nm) all-optical wavelength conversion of 10 Gbit/s nonreturn-to-zero signals," Opt. Lett. 31, 1782-1784 (2006).
[CrossRef] [PubMed]

Lee, J. H.

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

Lin, Q.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

Marhic, M. E.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, "Wavelength exchange: a novel function for optical networks," Inf. Sci. 149, 161-169 (2003).
[CrossRef]

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments," IEEE J. Sel. Top. Quantum Electron. 8, 560-568 (2002).
[CrossRef]

Marki, C. F.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

McKinstrie, C. J.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

Oxenlwe, L. K.

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

Radic, S.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

Richardson, D. J.

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

Tsang, H. K.

C. H. Kwok, C. W. Chow, H. K. Tsang, C. Lin, and A. Bjarklev, "Nonlinear polarization rotation in a dispersion-flattened photonic-crystal fiber for ultrawideband (>100 nm) all-optical wavelength conversion of 10 Gbit/s nonreturn-to-zero signals," Opt. Lett. 31, 1782-1784 (2006).
[CrossRef] [PubMed]

Uesaka, K.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, "Wavelength exchange: a novel function for optical networks," Inf. Sci. 149, 161-169 (2003).
[CrossRef]

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments," IEEE J. Sel. Top. Quantum Electron. 8, 560-568 (2002).
[CrossRef]

Wong, K. K. Y.

R. W. L. Fung, H. K. Y. Cheung, and K. K. Y. Wong, "Widely Tunable Wavelength Exchange in Anomalous-Dispersion Regime," IEEE Photon. Technol. Lett. 19, 1846-1848 (2007).
[CrossRef]

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, "Wavelength exchange: a novel function for optical networks," Inf. Sci. 149, 161-169 (2003).
[CrossRef]

Wong, K. K.-Y.

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments," IEEE J. Sel. Top. Quantum Electron. 8, 560-568 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments," IEEE J. Sel. Top. Quantum Electron. 8, 560-568 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device, " IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

R. W. L. Fung, H. K. Y. Cheung, and K. K. Y. Wong, "Widely Tunable Wavelength Exchange in Anomalous-Dispersion Regime," IEEE Photon. Technol. Lett. 19, 1846-1848 (2007).
[CrossRef]

Inf. Sci.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, "Wavelength exchange: a novel function for optical networks," Inf. Sci. 149, 161-169 (2003).
[CrossRef]

J. Lightwave Technol.

J. H. Lee, L. K. Oxenlwe, M. Ibsen, K. S. Berg, A. T. Clausen, D. J. Richardson, and P. Jeppesen, "All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology," J. Lightwave Technol. 21, 2518-2523 (2003).
[CrossRef]

Opt. Lett.

C. H. Kwok, C. W. Chow, H. K. Tsang, C. Lin, and A. Bjarklev, "Nonlinear polarization rotation in a dispersion-flattened photonic-crystal fiber for ultrawideband (>100 nm) all-optical wavelength conversion of 10 Gbit/s nonreturn-to-zero signals," Opt. Lett. 31, 1782-1784 (2006).
[CrossRef] [PubMed]

J. L. Blows and S. E. French, "Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump," Opt. Lett. 27, 491-493 (2002).
[CrossRef]

Other

G. Kalogerakis, M. Marhic, and L. Kazovsky, "Low Noise Figure Efficient Wavelength Exchange in an Optical Fibre," in Proc. of ECOC, Paper Th1.3.1 (2006).

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, "Polarization-insensitive wavelength exchange in highly-nonlinear dispersion shifted fiber," in Proc. of OFC, Paper ThY3 (2002).

R. W. L. Fung, H. K. Y. Cheung, B. P. P. Kuo, and K. K. Y. Wong, "Wavelength Exchange with Enhanced Extinction Ratio in Highly Nonlinear Dispersion-Shifted Fiber," in Proc. of CLEO, Paper JTuA39 (2007).

H. C. H. Mulvad, L. K. Oxenlowe, A. T. Clausen, M. Galili, L. Gruner-Nielsen, and P. Jeppesen, "Error-Free 320 Gb/s Simultaneous Add-Drop Multiplexing," in Proc. of OFC, Paper OTuI5 (2007).

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Figures (7)

Fig. 1.
Fig. 1.

Wavelength allocation and SOP of the pumps and the signals in an orthogonal-polarized pulsed pump wavelength exchange.

Fig. 2.
Fig. 2.

Schematic illustration of (upper) de-multiplexing, and (lower) simultaneous add-drop of OTDM signal in a pulsed pump wavelength exchange. Inset shows the wavelength allocation and the fiber dispersion curve.

Fig. 3.
Fig. 3.

Experimental setup of the pulsed pump wavelength exchange for de-multiplexing of OTDM signals.

Fig. 4.
Fig. 4.

Measured eye diagrams of (a) the original 10 Gb/s RZ signal and (b) the bit-rate multiplied 80 Gb/s signals. (c) Waveform of the wavelength converted pulsed pump at 1548 nm. Time base: (a) and (b) 20 ps/div; (c) 10 ps/div.

Fig. 5.
Fig. 5.

Measured eye diagrams of (a) the 10 Gb/s de-multiplexed signals and (b) the 70 Gb/s surviving signals at different time delays of the pulsed pump. Time base: 20 ps/div.

Fig. 6.
Fig. 6.

Optical spectrum of the 80 Gb/s OTDM signal, the wavelength-converted pulsed pump, the output of the HNLF, and the filtered and amplified de-multiplexed 10 Gb/s signal.

Fig. 7.
Fig. 7.

BER plots of the back-to-back and the 8 de-multiplexed 10 Gb/s signals.

Equations (3)

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

1 λ p 1 + 1 λ s 1 = 1 λ p 2 + 1 λ s 2 = 2 λ 0
Δ β = β p 1 + β s 1 β p 2 β s 2
max { P p 2 } = P p 1 ¯ = π 2 L ( 3 2 ) 1 γ

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