Abstract

We present techniques for designing pulses for linear slow-light delay systems which are optimal in the sense that they maximize the signal-to-noise ratio (SNR) and signal-to-noise-plus-interference ratio (SNIR) of the detected pulse energy. Given a communication model in which input pulses are created in a finite temporal window and output pulse energy in measured in a temporally-offset output window, the SNIR-optimal pulses achieve typical improvements of 10 dB compared to traditional pulse shapes for a given output window offset. Alternatively, for fixed SNR or SNIR, window offset (detection delay) can be increased by 0.3 times the window width. This approach also invites a communication-based model for delay and signal fidelity.

©2008 Optical Society of America

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References

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  1. R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” in Progress in Optics, Vol. 43, E. Wolf, ed. (Elsevier, Amsterdam, 2002), pp. 497–530.
  2. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
    [Crossref] [PubMed]
  3. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
    [Crossref] [PubMed]
  4. Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14, 12022–12027 (2006) http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12022.
    [Crossref] [PubMed]
  5. B. Zhang, L. Yan, I. Fazal, L. Zhang, A. E. Willner, Z. Zhu, and D. J. Gauthier, “Slow light on Gbit/s differential-phase-shift-keying signals,” Opt. Express 15, 1878–1883 (2007) http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-4-1878.
    [Crossref] [PubMed]
  6. B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
    [Crossref]
  7. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
    [Crossref]
  8. M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995–10002 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-9995.
    [Crossref] [PubMed]

2007 (1)

2006 (2)

2005 (4)

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[Crossref] [PubMed]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[Crossref]

M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995–10002 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-9995.
[Crossref] [PubMed]

A. L.,

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Boyd, R. W.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[Crossref]

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” in Progress in Optics, Vol. 43, E. Wolf, ed. (Elsevier, Amsterdam, 2002), pp. 497–530.

Dawes, A. M. C.

Fazal, I.

Gaeta,

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Gaeta, A. L.

Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14, 12022–12027 (2006) http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12022.
[Crossref] [PubMed]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[Crossref]

Gauthier, D. J.

B. Zhang, L. Yan, I. Fazal, L. Zhang, A. E. Willner, Z. Zhu, and D. J. Gauthier, “Slow light on Gbit/s differential-phase-shift-keying signals,” Opt. Express 15, 1878–1883 (2007) http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-4-1878.
[Crossref] [PubMed]

M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995–10002 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-9995.
[Crossref] [PubMed]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[Crossref]

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” in Progress in Optics, Vol. 43, E. Wolf, ed. (Elsevier, Amsterdam, 2002), pp. 497–530.

Hamann, H. F.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[Crossref] [PubMed]

Macke, B.

B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
[Crossref]

McNab, S. J.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[Crossref] [PubMed]

Neifeld, M. A.

O’Boyle, M.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[Crossref] [PubMed]

Okawachi, Y.

Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14, 12022–12027 (2006) http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12022.
[Crossref] [PubMed]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Schweinsberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Ségard, B.

B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
[Crossref]

Sharping, J. E.

Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14, 12022–12027 (2006) http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12022.
[Crossref] [PubMed]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Stenner, M. D.

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[Crossref] [PubMed]

Willner, A. E.

Xu, C.

Yan, L.

Zhang, B.

Zhang, L.

Zhu, Z.

Nature (1)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[Crossref] [PubMed]

Opt. Express (3)

Phys. Rev. A (2)

B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
[Crossref]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[Crossref]

Phys. Rev. Lett. (1)

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L., and Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Other (1)

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” in Progress in Optics, Vol. 43, E. Wolf, ed. (Elsevier, Amsterdam, 2002), pp. 497–530.

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Figures (7)

Fig. 1.
Fig. 1. (a) The detection model considered herein. A pulse is generated within some temporal window (the input window) and detected within a corresponding output window. Only energy falling within the output window contributes to the identification of the bit. (b) The discrete linear representation of pulse propagation. The submatrix H s describes the pulse in the output window in terms of the input pulse. The matrices H i1 and H i2 similarly describe the interference, the energy that falls outside the output window.
Fig. 2.
Fig. 2. a) Input pulse intensity for several candidate pulse shapes: SNIR-optimal (solid), SNR-optimal (dashed), square (dot-dashed), and impulse (dotted). The light vertical lines bound the input window. b) Output pulses, normalized to unit height. The light vertical lines bound the output window. c) Power spectra for SNIR- and SNR-optimal pulses and the square magnitude of the medium transfer function (solid line with squares).
Fig. 3.
Fig. 3. Eye diagrams for the pulses from Fig. 2. The shaded region in each eye diagram indicates the “open” region.
Fig. 4.
Fig. 4. a) SNR of SNIR-optimal (solid), SNR-optimal (dashed), square (dot-dashed), and impulse (dotted) pulses vs window offset for an example gain/window-size/noise. b) SNIR of the same pulses. c) Centroid (with dots) and peak (without) delay vs window offset.
Fig. 5.
Fig. 5. Input pulses (a), output pulses (b), and power spectra (c) of SNIR-optimal (solid) and SNR-optimal (dashed) pulses for several different window widths. From top to bottom, the window widths are {14,10,6,2}×(2π/γ). All other parameters are the same as for Fig. 2. The boxes indicate the input and output windows.
Fig. 6.
Fig. 6. SNR (a) and SNIR (b) vs window offset for different values of gain. The plots show both SNIR-optimal (solid) and SNR-optimal (dashed) pulses. The symbols correspond to gain with gL taking on the values 10 (plus), 20 (dot), 30 (asterisk). All other values are identical to those used in Fig. 2.
Fig. 7.
Fig. 7. Input pulses (a), output pulses (b), and power spectra (c) for SNIR-optimal (solid) and SNR-optimal (dashed) pulses for different values of noise. There is only one SNR-optimal pulse because it does not depend on noise. With decreasing noise, both input and output SNIR-optimal pulses shift to the right. The corresponding power spectra get wider. The SNR values are 0, 10, 30, 70, and 90 dB, as defined in the text. The line with dots is the SNIR-optimal pulse used in Fig. 2.

Equations (4)

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g = Hf ,
SNIR H s f 2 H i f 2 + σ 2 = f H s H s f f H i H i f + σ 2 f f , where H i = [ H i 1 H i 2 ] ,
0 = [ d d f ( f H s H s f ) ] [ f H i H i f + σ 2 f f ] [ f H s H s f ] [ d d f ( f H i H i f + σ 2 f f ) ]
= [ H s H s SNIR ( H i H i + σ 2 I ) ] f .

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