Abstract

We demonstrate continuously tunable optical delays as large as 1.1 μs range for 10 Gb/s NRZ optical signals based on four-wave mixing (FWM) process in silicon waveguide. The large delay range is made possible by a novel wavelength-optimized optical phase conjugation scheme, which allows for tunable dispersion compensation to minimize the residual group-velocity dispersion (GVD) for the entire tuning range.

© 2009 Optical Society of America

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References

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  1. M. K. Dhodhi, S. Tariq, and K. A. Saleh, "Bottlenecks in next generation DWDM-based optical networks," Comput. Commun. 24, 1726-1733 (2001).
    [CrossRef]
  2. S. J. B. Yoo, "Optical packet and burst switching technologies for the future photonic internet," J. Lightwave Technol. 24, 4468-4492 (2006).
    [CrossRef]
  3. E. Choi, J. H. Na, Y. Ryu, G. Mudhana, and B. H. Lee, "All-fiber variable optical delay line for applications in optical coherence tomography: feasibility study for a novel delay line," Opt. Express 13, 1334-1345 (2005).
    [CrossRef] [PubMed]
  4. J. L. Corral, J. Marti, J. M. Fuster, and R. I. Laming, "True time-delay scheme for feeding optically controlled phased-array antennas using chirped-fiber gratings," Photon. Technol. Lett. 9, 1529-1531 (1997).
    [CrossRef]
  5. G. N. Pearson, K. D. Ridley, and D. V. Willetts, "Chirp-pulse-compression three dimensional lidar imager with fiber optics," Appl. Opt. 44, 257-265 (2005).
    [CrossRef] [PubMed]
  6. Y. Han and B. Jalali, "Photonic time-stretched analog-to-digital converter: Fundamental concepts and practical considerations," J. Lightwave Technol. 21, 3085-3103 (2003).
    [CrossRef]
  7. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, 2002).
  8. R. W. Boyd and D. J. Gauthier, "‘Slow’ and ‘fast’ light," Progress in Optics 43, edited by E. Wolf (Elsevier, Amsterdam, 2002), Chap. 6, p. 497-530.
  9. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Applications of slow light in telecommunications," Opt. Photon. News 17, 18-22 (2006).
    [CrossRef]
  10. M. Burzio, P. Cinato, R. Finotti, P. Gambini, M. Puleo, E. Vezzoni, and L. Zucchelli, "Optical cell synchronization in an ATM optical switch," in Proc. ECOC ’94, Florence, Italy, 2, 581-584 (1994).
  11. L. Zucchelli, M. Burzio, and P. Gambini, "New solutions for optical packet delineation and synchronization in optical packet switched networks," in Proc. ECOC ’96, Oslo, Norway, 3, 301-304 (1996).
  12. K. Shimizu, G. Kalogerakis, K. Wong, M. Marhic, and L. Kazovsky, "Timing jitter and amplitude noise reduction by a chirped pulsed-pump fiber OPA," in Proc. OFC ’03, Anaheim, USA, 1, 197-198 (2003).
  13. J. van Howe and C. Xu, "Ultrafast optical delay line using soliton propagation between a time-prism pair," Opt. Express 13, 1138-1143 (2005).
    [CrossRef] [PubMed]
  14. Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
    [CrossRef]
  15. L. C. Christen, O. F. Yilmaz, S. R. Nuccio, X. Wu, I. Fazal, A. E. Willner, C. Langrock, and M. M. Fejer, "Tunable 105 ns optical delay for 80 Gb/s RZ-DQPSK, 40 Gb/s RZ-DPSK, and 40 Gb/s RZ-OOK signals using wavelength conversion and chromatic dispersion," Opt. Lett. 34, 542-544 (2009).
    [CrossRef] [PubMed]
  16. J. E. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. E. Willner, and A. L. Gaeta, "All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion," Opt. Express 13, 7872-7877 (2005).
    [CrossRef] [PubMed]
  17. Y. Okawachi, M. A. Foster, X. Chen, A. C. Turner-Foster, R. Salem, M. Lipson, C. Xu, and A. L. Gaeta, "Large tunable delays using parametric mixing and phase conjugation in Si nanowaveguides," Opt. Express 16, 10349-10357 (2008).
    [CrossRef] [PubMed]
  18. E. Myslivets, N. Alic, J. R. Windmiller, R. M. Jopson, and S. Radic, "400 ns continuously tunable delay of 10 Gbps intensity modulated optical signal," Photon. Technol. Lett. 21, 251-253 (2009).
    [CrossRef]
  19. M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
    [CrossRef]
  20. S. Namiki, "Wide-band and -range tunable dispersion compensation through parametric wavelength conversion and dispersion optical fibers," J. Lightwave Technol. 26, 28-35 (2008).
    [CrossRef]
  21. A. Yariv, D. Fekete, and D. M. Pepper, "Compensation for channel dispersion by nonlinear optical phase conjugation," Opt. Lett. 4, 52-54 (1979).
    [CrossRef] [PubMed]
  22. S. Ayotte, S. Xu, H. Rong, O. Cohen, and M. J. Paniccia, "Dispersion compensation by optical phase conjugation in silicon waveguide," Electron. Lett. 43, 1037-1039 (2007).
    [CrossRef]
  23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, Boston, 1989).
  24. B. G. Lee, A. Biberman, M. A. Foster, A. C. Turner, M. Lipson, A. L. Gaeta, and K. Bergman, "Bit-error-rate characterization of Silicon four-wave-mixing wavelength converters at 10 and 40 Gb/s," CLEO 2008, paper CPDB4.

2009 (2)

2008 (2)

2007 (3)

S. Ayotte, S. Xu, H. Rong, O. Cohen, and M. J. Paniccia, "Dispersion compensation by optical phase conjugation in silicon waveguide," Electron. Lett. 43, 1037-1039 (2007).
[CrossRef]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
[CrossRef]

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

2006 (2)

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Applications of slow light in telecommunications," Opt. Photon. News 17, 18-22 (2006).
[CrossRef]

S. J. B. Yoo, "Optical packet and burst switching technologies for the future photonic internet," J. Lightwave Technol. 24, 4468-4492 (2006).
[CrossRef]

2005 (4)

2003 (1)

2001 (1)

M. K. Dhodhi, S. Tariq, and K. A. Saleh, "Bottlenecks in next generation DWDM-based optical networks," Comput. Commun. 24, 1726-1733 (2001).
[CrossRef]

1997 (1)

J. L. Corral, J. Marti, J. M. Fuster, and R. I. Laming, "True time-delay scheme for feeding optically controlled phased-array antennas using chirped-fiber gratings," Photon. Technol. Lett. 9, 1529-1531 (1997).
[CrossRef]

1979 (1)

Alic, N.

E. Myslivets, N. Alic, J. R. Windmiller, R. M. Jopson, and S. Radic, "400 ns continuously tunable delay of 10 Gbps intensity modulated optical signal," Photon. Technol. Lett. 21, 251-253 (2009).
[CrossRef]

Ayotte, S.

S. Ayotte, S. Xu, H. Rong, O. Cohen, and M. J. Paniccia, "Dispersion compensation by optical phase conjugation in silicon waveguide," Electron. Lett. 43, 1037-1039 (2007).
[CrossRef]

Boyd, R. W.

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Applications of slow light in telecommunications," Opt. Photon. News 17, 18-22 (2006).
[CrossRef]

Chen, X.

Choi, E.

Christen, L. C.

Cohen, O.

S. Ayotte, S. Xu, H. Rong, O. Cohen, and M. J. Paniccia, "Dispersion compensation by optical phase conjugation in silicon waveguide," Electron. Lett. 43, 1037-1039 (2007).
[CrossRef]

Corral, J. L.

J. L. Corral, J. Marti, J. M. Fuster, and R. I. Laming, "True time-delay scheme for feeding optically controlled phased-array antennas using chirped-fiber gratings," Photon. Technol. Lett. 9, 1529-1531 (1997).
[CrossRef]

Dhodhi, M. K.

M. K. Dhodhi, S. Tariq, and K. A. Saleh, "Bottlenecks in next generation DWDM-based optical networks," Comput. Commun. 24, 1726-1733 (2001).
[CrossRef]

Fazal, I.

Fejer, M. M.

L. C. Christen, O. F. Yilmaz, S. R. Nuccio, X. Wu, I. Fazal, A. E. Willner, C. Langrock, and M. M. Fejer, "Tunable 105 ns optical delay for 80 Gb/s RZ-DQPSK, 40 Gb/s RZ-DPSK, and 40 Gb/s RZ-OOK signals using wavelength conversion and chromatic dispersion," Opt. Lett. 34, 542-544 (2009).
[CrossRef] [PubMed]

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

Fekete, D.

Foster, M. A.

Y. Okawachi, M. A. Foster, X. Chen, A. C. Turner-Foster, R. Salem, M. Lipson, C. Xu, and A. L. Gaeta, "Large tunable delays using parametric mixing and phase conjugation in Si nanowaveguides," Opt. Express 16, 10349-10357 (2008).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
[CrossRef]

Fuster, J. M.

J. L. Corral, J. Marti, J. M. Fuster, and R. I. Laming, "True time-delay scheme for feeding optically controlled phased-array antennas using chirped-fiber gratings," Photon. Technol. Lett. 9, 1529-1531 (1997).
[CrossRef]

Gaeta, A. E.

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

Gaeta, A. L.

Gauthier, D. J.

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Applications of slow light in telecommunications," Opt. Photon. News 17, 18-22 (2006).
[CrossRef]

Han, Y.

Jalali, B.

Jopson, R. M.

E. Myslivets, N. Alic, J. R. Windmiller, R. M. Jopson, and S. Radic, "400 ns continuously tunable delay of 10 Gbps intensity modulated optical signal," Photon. Technol. Lett. 21, 251-253 (2009).
[CrossRef]

Laming, R. I.

J. L. Corral, J. Marti, J. M. Fuster, and R. I. Laming, "True time-delay scheme for feeding optically controlled phased-array antennas using chirped-fiber gratings," Photon. Technol. Lett. 9, 1529-1531 (1997).
[CrossRef]

Langrock, C.

L. C. Christen, O. F. Yilmaz, S. R. Nuccio, X. Wu, I. Fazal, A. E. Willner, C. Langrock, and M. M. Fejer, "Tunable 105 ns optical delay for 80 Gb/s RZ-DQPSK, 40 Gb/s RZ-DPSK, and 40 Gb/s RZ-OOK signals using wavelength conversion and chromatic dispersion," Opt. Lett. 34, 542-544 (2009).
[CrossRef] [PubMed]

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

Lee, B. H.

Lipson, M.

Y. Okawachi, M. A. Foster, X. Chen, A. C. Turner-Foster, R. Salem, M. Lipson, C. Xu, and A. L. Gaeta, "Large tunable delays using parametric mixing and phase conjugation in Si nanowaveguides," Opt. Express 16, 10349-10357 (2008).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
[CrossRef]

Marti, J.

J. L. Corral, J. Marti, J. M. Fuster, and R. I. Laming, "True time-delay scheme for feeding optically controlled phased-array antennas using chirped-fiber gratings," Photon. Technol. Lett. 9, 1529-1531 (1997).
[CrossRef]

Mudhana, G.

Myslivets, E.

E. Myslivets, N. Alic, J. R. Windmiller, R. M. Jopson, and S. Radic, "400 ns continuously tunable delay of 10 Gbps intensity modulated optical signal," Photon. Technol. Lett. 21, 251-253 (2009).
[CrossRef]

Na, J. H.

Namiki, S.

Nuccio, S. R.

Okawachi, Y.

Paniccia, M. J.

S. Ayotte, S. Xu, H. Rong, O. Cohen, and M. J. Paniccia, "Dispersion compensation by optical phase conjugation in silicon waveguide," Electron. Lett. 43, 1037-1039 (2007).
[CrossRef]

Pearson, G. N.

Pepper, D. M.

Radic, S.

E. Myslivets, N. Alic, J. R. Windmiller, R. M. Jopson, and S. Radic, "400 ns continuously tunable delay of 10 Gbps intensity modulated optical signal," Photon. Technol. Lett. 21, 251-253 (2009).
[CrossRef]

Ridley, K. D.

Rong, H.

S. Ayotte, S. Xu, H. Rong, O. Cohen, and M. J. Paniccia, "Dispersion compensation by optical phase conjugation in silicon waveguide," Electron. Lett. 43, 1037-1039 (2007).
[CrossRef]

Roussev, R.

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

Ryu, Y.

Saleh, K. A.

M. K. Dhodhi, S. Tariq, and K. A. Saleh, "Bottlenecks in next generation DWDM-based optical networks," Comput. Commun. 24, 1726-1733 (2001).
[CrossRef]

Salem, R.

Y. Okawachi, M. A. Foster, X. Chen, A. C. Turner-Foster, R. Salem, M. Lipson, C. Xu, and A. L. Gaeta, "Large tunable delays using parametric mixing and phase conjugation in Si nanowaveguides," Opt. Express 16, 10349-10357 (2008).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
[CrossRef]

Sharping, J. E.

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

J. E. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. E. Willner, and A. L. Gaeta, "All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion," Opt. Express 13, 7872-7877 (2005).
[CrossRef] [PubMed]

Tariq, S.

M. K. Dhodhi, S. Tariq, and K. A. Saleh, "Bottlenecks in next generation DWDM-based optical networks," Comput. Commun. 24, 1726-1733 (2001).
[CrossRef]

Turner, A. C.

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
[CrossRef]

Turner-Foster, A. C.

van Howe, J.

Wang, Y.

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

J. E. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. E. Willner, and A. L. Gaeta, "All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion," Opt. Express 13, 7872-7877 (2005).
[CrossRef] [PubMed]

Willetts, D. V.

Willner, A. E.

Windmiller, J. R.

E. Myslivets, N. Alic, J. R. Windmiller, R. M. Jopson, and S. Radic, "400 ns continuously tunable delay of 10 Gbps intensity modulated optical signal," Photon. Technol. Lett. 21, 251-253 (2009).
[CrossRef]

Wu, X.

Xu, C.

Xu, S.

S. Ayotte, S. Xu, H. Rong, O. Cohen, and M. J. Paniccia, "Dispersion compensation by optical phase conjugation in silicon waveguide," Electron. Lett. 43, 1037-1039 (2007).
[CrossRef]

Yan, L.

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

Yariv, A.

Yilmaz, O. F.

Yoo, S. J. B.

Yu, C.

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

Appl. Opt. (1)

Comput. Commun. (1)

M. K. Dhodhi, S. Tariq, and K. A. Saleh, "Bottlenecks in next generation DWDM-based optical networks," Comput. Commun. 24, 1726-1733 (2001).
[CrossRef]

Electron. Lett. (1)

S. Ayotte, S. Xu, H. Rong, O. Cohen, and M. J. Paniccia, "Dispersion compensation by optical phase conjugation in silicon waveguide," Electron. Lett. 43, 1037-1039 (2007).
[CrossRef]

J. Lightwave Technol. (3)

Opt. Express (5)

Opt. Lett. (2)

Opt. Photon. News (1)

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Applications of slow light in telecommunications," Opt. Photon. News 17, 18-22 (2006).
[CrossRef]

Photon. Technol. Lett. (3)

E. Myslivets, N. Alic, J. R. Windmiller, R. M. Jopson, and S. Radic, "400 ns continuously tunable delay of 10 Gbps intensity modulated optical signal," Photon. Technol. Lett. 21, 251-253 (2009).
[CrossRef]

Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping, and A. E. Gaeta, "44-ns continuously tunable dispersionless optical delay element using a PPLN waveguide with two-pump configuration, DCF, and a dispersion compensator," Photon. Technol. Lett. 19, 861-863 (2007).
[CrossRef]

J. L. Corral, J. Marti, J. M. Fuster, and R. I. Laming, "True time-delay scheme for feeding optically controlled phased-array antennas using chirped-fiber gratings," Photon. Technol. Lett. 9, 1529-1531 (1997).
[CrossRef]

Other (7)

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, 2002).

R. W. Boyd and D. J. Gauthier, "‘Slow’ and ‘fast’ light," Progress in Optics 43, edited by E. Wolf (Elsevier, Amsterdam, 2002), Chap. 6, p. 497-530.

M. Burzio, P. Cinato, R. Finotti, P. Gambini, M. Puleo, E. Vezzoni, and L. Zucchelli, "Optical cell synchronization in an ATM optical switch," in Proc. ECOC ’94, Florence, Italy, 2, 581-584 (1994).

L. Zucchelli, M. Burzio, and P. Gambini, "New solutions for optical packet delineation and synchronization in optical packet switched networks," in Proc. ECOC ’96, Oslo, Norway, 3, 301-304 (1996).

K. Shimizu, G. Kalogerakis, K. Wong, M. Marhic, and L. Kazovsky, "Timing jitter and amplitude noise reduction by a chirped pulsed-pump fiber OPA," in Proc. OFC ’03, Anaheim, USA, 1, 197-198 (2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, Boston, 1989).

B. G. Lee, A. Biberman, M. A. Foster, A. C. Turner, M. Lipson, A. L. Gaeta, and K. Bergman, "Bit-error-rate characterization of Silicon four-wave-mixing wavelength converters at 10 and 40 Gb/s," CLEO 2008, paper CPDB4.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Delay generator using two wavelength conversions via FWM and two dispersion links. (b) The principle of zero residual dispersion based on the wavelength-optimized optical phase conjugation.

Fig. 2.
Fig. 2.

Experimental setup for the delay scheme based on wavelength-optimized optical phase conjugation. C: coupler; TBF: tunable bandpass filter; SSMF: standard single mode fiber, PC: polarization controller. NF: notch filter. Each of the four Raman amplifiers consists of four semiconductor LDs. Polarization multiplexing is used to combine the pump diodes, and the pump power for each amplifier is 23 dBm. The spectrum after each waveguide is inset. The resolution in the measurement is 0.1 nm.

Fig. 3.
Fig. 3.

Measured delay as a function of λ1. Inset: the measured delay for λ1 = 1535.5 nm (top) and 1572 nm (bottom).

Fig. 4.
Fig. 4.

Measured eye diagrams and BER curves of back-to-back (B2B) signals and delayed signals when λ 1 is at different wavelengths.

Equations (6)

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

χ 1 ( λ 1 ) = χ 2 ( λ 2 ) .
τ ( λ 1 ) = λ 10 λ 1 χ 1 ( λ ) + λ 20 λ 2 χ 2 ( λ ) ,
φ ( Δ ω ) = Δ ω 2 2 [ β 2,2 ( ω 2 ) L 2 β 2,1 ( ω 1 ) L 1 ] + Δ ω 3 6 [ β 3,2 ( ω 2 ) L 2 + β 2,1 ( ω 1 ) L 1 ] ,
+ Δ ω 4 24 [ β 4,2 ( ω 2 ) L 2 β 4,1 ( ω 1 ) L 1 ] +
φ max ( Δ ω ) = Δ ω 2 4 β 3 L ̅ Δ ω C ,
φ ' max ( Δ ω ) = Δ ω 3 6 β 4 L ̅ Δ ω C ,

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