Abstract

In exact analogy with their electronic counterparts, photonic temporal integrators are fundamental building blocks for constructing all-optical circuits for ultrafast information processing and computing. In this work, we introduce a simple and general approach for realizing all-optical arbitrary-order temporal integrators. We demonstrate that the Nth cumulative time integral of the complex field envelope of an input optical waveform can be obtained by simply propagating this waveform through a single uniform fiber/waveguide Bragg grating (BG) incorporating N π-phase shifts along its axial profile. We derive here the design specifications of photonic integrators based on multiple-phase-shifted BGs. We show that the phase shifts in the BG structure can be arbitrarily located along the grating length provided that each uniform grating section (sections separated by the phase shifts) is sufficiently long so that its associated peak reflectivity reaches nearly 100%. The resulting designs are demonstrated by numerical simulations assuming all-fiber implementations. Our simulations show that the proposed approach can provide optical operation bandwidths in the tensof-GHz regime using readily feasible photo-induced fiber BG structures.

© 2008 Optical Society of America

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References

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  1. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, "Photonics temporal integrator for all-optical computing," submitted (2008).
  2. N. Q. Ngo and L. N. Binh, "Optical realization of Newton-Cotes-Based Integrators for Dark Soliton Generation," J. Lightwave Technol. 24, 563-572 (2006).
    [CrossRef]
  3. N. Q. Ngo, "Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission," Opt. Lett. 32, 3020-3022 (2007).
    [CrossRef]
  4. J. Azaña, "Proposal of a uniform fiber Bragg grating as an ultrafast all-optical integrator," Opt. Lett. 33, 4-6 (2008).
    [CrossRef]
  5. N. Q. Ngo, "Optical integrator for optical dark-soliton detection and pulse shaping," Appl. Opt. 45, 6785-6791 (2006).
    [CrossRef] [PubMed]
  6. N. Q. Ngo and L. N. Binh, "A new approach for the design of optical square generator," Appl. Opt. 46, 35463560 (2007).
    [CrossRef]
  7. A. V. Oppenheim, A. S. Willsky, S. N. Nawab, and S. H. Nawab, Signals and Systems, 2nd ed., (Upper Saddle River, NJ: Prentice Hall 1996).
  8. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D.V. Plant, and J. Azaña, "Temporal differentiation of optical pulses using a phase-shifted fiber Bragg grating," Opt. Express 15, 371-381 (2007).
    [CrossRef] [PubMed]
  9. M. Kulishov and J. Azaña, "High-order all-optical temporal differentiation using multiple-phase-shifted fiber Bragg gratings," Opt. Express 15, 6152-6166 (2007).
    [CrossRef] [PubMed]
  10. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
    [CrossRef]
  11. T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]

2008 (1)

2007 (4)

2006 (2)

2002 (1)

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

1997 (1)

T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Azaña, J.

Berger, N. K.

Binh, L. N.

N. Q. Ngo and L. N. Binh, "A new approach for the design of optical square generator," Appl. Opt. 46, 35463560 (2007).
[CrossRef]

N. Q. Ngo and L. N. Binh, "Optical realization of Newton-Cotes-Based Integrators for Dark Soliton Generation," J. Lightwave Technol. 24, 563-572 (2006).
[CrossRef]

Erdogan, T.

T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Fischer, B.

Kulishov, M.

Levit, B.

Li, H.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Li, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Ngo, N. Q.

Plant, D.V.

Popelek, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Sheng, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Wang, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Wilcox, R. B.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Zweiback, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Appl. Opt. (2)

N. Q. Ngo, "Optical integrator for optical dark-soliton detection and pulse shaping," Appl. Opt. 45, 6785-6791 (2006).
[CrossRef] [PubMed]

N. Q. Ngo and L. N. Binh, "A new approach for the design of optical square generator," Appl. Opt. 46, 35463560 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (2)

Opt. Lett. (2)

Other (2)

A. V. Oppenheim, A. S. Willsky, S. N. Nawab, and S. H. Nawab, Signals and Systems, 2nd ed., (Upper Saddle River, NJ: Prentice Hall 1996).

R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, "Photonics temporal integrator for all-optical computing," submitted (2008).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of the proposed BG - based designs for first-order, second-order and Nth-order photonic temporal integrators. The vertical red lines indicate π phase shifts between the uniform grating sections.

Fig. 2.
Fig. 2.

Numerically simulated spectral amplitude responses of the tested BG-based designs for second-order (a) and third-order (b) integration compared with the spectral profiles of the ideal second and third-order photonic time integrators. The insets show the numerically simulated spectral phase responses of the tested BG-based designs for second-order (a) and third-order (b) integration.

Fig. 3.
Fig. 3.

Time-domain response of the MPS-BG (a) second-order and (b) third-order integrators compared with the ideal (a) second and (b) third time cumulative integrals of the considered input pulse waveforms. The input amplitude envelopes are defined as the (a) second and (b) third time derivatives of an ideal Gaussian pulse with a FWHM of 40 ps.

Equations (11)

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[ E A ( z 0 + L i ) E B ( z 0 + L i ) ] = T ( z 0 + L i ) [ E A ( z 0 ) E B ( z 0 ) ]
= [ T 11 ( L i ) T 12 ( z 0 , L i ) T 21 ( z 0 , L i ) T 22 ( L i ) ] [ E A ( z 0 ) E B ( z 0 ) ]
T 11 = T 22 * = [ cosh ( γ L i ) + j σ γ sinh ( γ L i ) ] exp [ j 2 π n eff L i λ B ] ,
T 12 = T 21 * = κ γ sinh ( γ L i ) exp [ j 2 π n eff L i λ B ] ,
Φ 11 = Φ 22 * = exp ( 2 ) ,
Φ 12 = Φ 21 = 0
H = 1 T 22 = H exp ( j ϕ τ )
T = T ( L 1 , L 2 ) Φ T ( 0 , L 1 )
H ( ω ) A 0 j n eff ck ( r 1 + r 2 ) ( ω ω 0 ) + n eff 2 c 2 k 2 r 1 r 2 ( ω ω 0 ) 2 r 1 r 2 exp ( ) 1
T = T ( L 1 + L 2 , L 3 ) Φ T ( L 1 , L 2 ) Φ T ( 0 , L 1 )
H ( ω ) B 0 3 ( ω ω 0 ) 2 + j n eff ck ( ω ω 0 ) 3

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