Abstract

The design and analysis of an Nth-order optical integrator using the digital filter technique is presented. The optical integrator is synthesized using planar-waveguide technology. It is shown that a first-order optical integrator can be used as an optical dark-soliton detector by converting an optical dark-soliton pulse into an optical bell-shaped pulse for ease of detection. The optical integrators can generate an optical step function, staircase function, and paraboliclike functions from input optical Gaussian pulses. The optical integrators may be potentially used as basic building blocks of all-optical signal processing systems because the time integrals of signals may sometimes be required for further use or analysis. Furthermore, an optical integrator may be used for the shaping of optical pulses or in an optical feedback control system.

© 2006 Optical Society of America

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    [CrossRef]
  3. K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, "Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit," Electron. Lett. 40, 537-538 (2004).
    [CrossRef]
  4. S. Osawa, N. Wada, K. Kitayama, and W. Chujo, "Arbitrarily shaped optical pulse train synthesis using weight/phase-programmable 32-tapped delay line waveguide filter," Electron. Lett. 37, 1356-1357 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  20. M. Nakazawa and K. Suzuki, "10 Gbit/s pseudorandom dark soliton data transmission over 1200 km," Electron. Lett. 31, 1076-1077 (1995).
    [CrossRef]
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    [CrossRef]
  22. M. Nakazawa and K. Suzuki, "Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit-shifting with a Mach-Zehnder interferometer," Electron. Lett. 31, 1084-1085 (1995).
    [CrossRef]
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2006

A. M. Weiner, J. P. Heritage, and R. N. Thurston, "Synthesis of phase-coherent, picosecond optical square pulses," Opt. Lett. 11, 153-155 (2006).
[CrossRef]

S. Mookherjea, "Using gain to tune the dispersion relation of coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 715-717 (2006).
[CrossRef]

2005

2004

K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, "Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit," Electron. Lett. 40, 537-538 (2004).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

2002

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

2001

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, "Arbitrarily shaped optical pulse train synthesis using weight/phase-programmable 32-tapped delay line waveguide filter," Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

P. Petropoulos, M. Isben, A. D. Ellis, and D. J. Richardson, "Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating," J. Lightwave Technol. 19, 746-752 (2001).
[CrossRef]

2000

S. Longhi, M. Marano, P. Laporta, and V. Pruneri, "Multiplication and reshaping of high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings," IEEE Photon. Technol. Lett. 12, 1498-1500 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

1999

K. Okamoto, T. Kominato, H. Yamada, and T. Goh, "Fabrication of frequency spectrum synthesiser consisting of arrayed-waveguide grating pair and thermo-optic amplitude and phase controllers," Electron. Lett. 35, 733-734 (1999).
[CrossRef]

1997

Ph. Emplit, M. Haelterman, R. Kashyap, and M. De Lathouwer, "Fiber Bragg grating for optical dark soliton generation," IEEE Photon. Technol. Lett. 9, 1122-1124 (1997).
[CrossRef]

1996

N. Q. Ngo and L. N. Binh, "Optical dark-soliton generators and detectors," Opt. Commun. 132, 389-402 (1996).
[CrossRef]

1995

M. Nakazawa and K. Suzuki, "Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit-shifting with a Mach-Zehnder interferometer," Electron. Lett. 31, 1084-1085 (1995).
[CrossRef]

M. Nakazawa and K. Suzuki, "10 Gbit/s pseudorandom dark soliton data transmission over 1200 km," Electron. Lett. 31, 1076-1077 (1995).
[CrossRef]

1994

Y. S. Kivshar, M. Haelterman, Ph. Emplit, and J. P. Hamaide, "Gordon-Haus effect on dark solitons," Opt. Lett. 19, 19-21 (1994).
[CrossRef] [PubMed]

N. Q. Ngo and L. N. Binh, "Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators," J. Lightwave Technol. 12, 827-841 (1994).
[CrossRef]

S. Suzuki, M. Yanagisawa, Y. Hibino, and K. Oda, "High-density integrated planar lightwave circuits using SiO2-GeO2 waveguides with a high-refractive index difference," J. Lightwave Technol. 12, 790-796 (1994).
[CrossRef]

1989

Absil, P. P.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Baker, C. C.

Banerjee, S.

Binh, L. N.

N. Q. Ngo and L. N. Binh, "Optical dark-soliton generators and detectors," Opt. Commun. 132, 389-402 (1996).
[CrossRef]

N. Q. Ngo and L. N. Binh, "Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators," J. Lightwave Technol. 12, 827-841 (1994).
[CrossRef]

Bourkoff, E.

Calhoun, C.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Chujo, W.

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, "Arbitrarily shaped optical pulse train synthesis using weight/phase-programmable 32-tapped delay line waveguide filter," Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

De Lathouwer, M.

Ph. Emplit, M. Haelterman, R. Kashyap, and M. De Lathouwer, "Fiber Bragg grating for optical dark soliton generation," IEEE Photon. Technol. Lett. 9, 1122-1124 (1997).
[CrossRef]

Dwight, H. B.

H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, Collier-Macmillan Canada, 1961).

Ellis, A. D.

Emplit, Ph.

Ph. Emplit, M. Haelterman, R. Kashyap, and M. De Lathouwer, "Fiber Bragg grating for optical dark soliton generation," IEEE Photon. Technol. Lett. 9, 1122-1124 (1997).
[CrossRef]

Y. S. Kivshar, M. Haelterman, Ph. Emplit, and J. P. Hamaide, "Gordon-Haus effect on dark solitons," Opt. Lett. 19, 19-21 (1994).
[CrossRef] [PubMed]

Franklin, G. F.

G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. (Addison-Wesley, 1990).

Gill, D.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Goh, T.

K. Okamoto, T. Kominato, H. Yamada, and T. Goh, "Fabrication of frequency spectrum synthesiser consisting of arrayed-waveguide grating pair and thermo-optic amplitude and phase controllers," Electron. Lett. 35, 733-734 (1999).
[CrossRef]

Grover, R.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Haelterman, M.

Ph. Emplit, M. Haelterman, R. Kashyap, and M. De Lathouwer, "Fiber Bragg grating for optical dark soliton generation," IEEE Photon. Technol. Lett. 9, 1122-1124 (1997).
[CrossRef]

Y. S. Kivshar, M. Haelterman, Ph. Emplit, and J. P. Hamaide, "Gordon-Haus effect on dark solitons," Opt. Lett. 19, 19-21 (1994).
[CrossRef] [PubMed]

Hamaide, J. P.

Heritage, J. P.

Hibino, Y.

S. Suzuki, M. Yanagisawa, Y. Hibino, and K. Oda, "High-density integrated planar lightwave circuits using SiO2-GeO2 waveguides with a high-refractive index difference," J. Lightwave Technol. 12, 790-796 (1994).
[CrossRef]

Ho, P.-T.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Hryniewicz, J. V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Ibrahim, T. A.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Isben, M.

Johnson, F. G.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Kashyap, R.

Ph. Emplit, M. Haelterman, R. Kashyap, and M. De Lathouwer, "Fiber Bragg grating for optical dark soliton generation," IEEE Photon. Technol. Lett. 9, 1122-1124 (1997).
[CrossRef]

King, O.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Kitayama, K.

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, "Arbitrarily shaped optical pulse train synthesis using weight/phase-programmable 32-tapped delay line waveguide filter," Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

Kivshar, Y. S.

Klotzkin, D.

Kominato, T.

K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, "Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit," Electron. Lett. 40, 537-538 (2004).
[CrossRef]

K. Okamoto, T. Kominato, H. Yamada, and T. Goh, "Fabrication of frequency spectrum synthesiser consisting of arrayed-waveguide grating pair and thermo-optic amplitude and phase controllers," Electron. Lett. 35, 733-734 (1999).
[CrossRef]

Laporta, P.

S. Longhi, M. Marano, P. Laporta, and V. Pruneri, "Multiplication and reshaping of high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings," IEEE Photon. Technol. Lett. 12, 1498-1500 (2000).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Longhi, S.

S. Longhi, M. Marano, P. Laporta, and V. Pruneri, "Multiplication and reshaping of high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings," IEEE Photon. Technol. Lett. 12, 1498-1500 (2000).
[CrossRef]

Marano, M.

S. Longhi, M. Marano, P. Laporta, and V. Pruneri, "Multiplication and reshaping of high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings," IEEE Photon. Technol. Lett. 12, 1498-1500 (2000).
[CrossRef]

Mookherjea, S.

S. Mookherjea, "Using gain to tune the dispersion relation of coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 715-717 (2006).
[CrossRef]

Nakazawa, M.

M. Nakazawa and K. Suzuki, "10 Gbit/s pseudorandom dark soliton data transmission over 1200 km," Electron. Lett. 31, 1076-1077 (1995).
[CrossRef]

M. Nakazawa and K. Suzuki, "Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit-shifting with a Mach-Zehnder interferometer," Electron. Lett. 31, 1084-1085 (1995).
[CrossRef]

Ngo, N. Q.

N. Q. Ngo and L. N. Binh, "Optical dark-soliton generators and detectors," Opt. Commun. 132, 389-402 (1996).
[CrossRef]

N. Q. Ngo and L. N. Binh, "Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators," J. Lightwave Technol. 12, 827-841 (1994).
[CrossRef]

Oda, K.

S. Suzuki, M. Yanagisawa, Y. Hibino, and K. Oda, "High-density integrated planar lightwave circuits using SiO2-GeO2 waveguides with a high-refractive index difference," J. Lightwave Technol. 12, 790-796 (1994).
[CrossRef]

Okamoto, K.

K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, "Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit," Electron. Lett. 40, 537-538 (2004).
[CrossRef]

K. Okamoto, T. Kominato, H. Yamada, and T. Goh, "Fabrication of frequency spectrum synthesiser consisting of arrayed-waveguide grating pair and thermo-optic amplitude and phase controllers," Electron. Lett. 35, 733-734 (1999).
[CrossRef]

Osawa, S.

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, "Arbitrarily shaped optical pulse train synthesis using weight/phase-programmable 32-tapped delay line waveguide filter," Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

Petropoulos, P.

Powell, J. D.

G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. (Addison-Wesley, 1990).

Pruneri, V.

S. Longhi, M. Marano, P. Laporta, and V. Pruneri, "Multiplication and reshaping of high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings," IEEE Photon. Technol. Lett. 12, 1498-1500 (2000).
[CrossRef]

Richardson, D. J.

Seiferth, F.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Shibata, T.

K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, "Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit," Electron. Lett. 40, 537-538 (2004).
[CrossRef]

Steckl, A. J.

Suzuki, K.

M. Nakazawa and K. Suzuki, "Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit-shifting with a Mach-Zehnder interferometer," Electron. Lett. 31, 1084-1085 (1995).
[CrossRef]

M. Nakazawa and K. Suzuki, "10 Gbit/s pseudorandom dark soliton data transmission over 1200 km," Electron. Lett. 31, 1076-1077 (1995).
[CrossRef]

Suzuki, S.

S. Suzuki, M. Yanagisawa, Y. Hibino, and K. Oda, "High-density integrated planar lightwave circuits using SiO2-GeO2 waveguides with a high-refractive index difference," J. Lightwave Technol. 12, 790-796 (1994).
[CrossRef]

Takahashi, H.

K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, "Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit," Electron. Lett. 40, 537-538 (2004).
[CrossRef]

Takiguchi, K.

K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, "Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit," Electron. Lett. 40, 537-538 (2004).
[CrossRef]

Thurston, R. N.

Trakalo, M.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Van, V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Wada, N.

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, "Arbitrarily shaped optical pulse train synthesis using weight/phase-programmable 32-tapped delay line waveguide filter," Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

Weiner, A. M.

Wilson, R. A.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Workman, M. L.

G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. (Addison-Wesley, 1990).

Yamada, H.

K. Okamoto, T. Kominato, H. Yamada, and T. Goh, "Fabrication of frequency spectrum synthesiser consisting of arrayed-waveguide grating pair and thermo-optic amplitude and phase controllers," Electron. Lett. 35, 733-734 (1999).
[CrossRef]

Yanagisawa, M.

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Figures (6)

Fig. 1
Fig. 1

Schematic of the proposed first-order optical rectangular integrator.

Fig. 2
Fig. 2

(a) Magnitude response of the ideal first-order optical integrator (dashed curve) and magnitude response of the proposed first-order optical rectangular integrator (solid curve). (b) Phase response of the first-order optical rectangular integrator. (c) Impulse response of the first-order optical rectangular integrator. The normalized frequency corresponds to ω T / ( 2 π ) and the normalized time corresponds to the number of sampling periods, T, of the integrator.

Fig. 3
Fig. 3

(a) Intensity profile of a fundamental dark-soliton pulse where σ = 50 T . (b) Using (a) as an input pulse, a bell-shaped optical pulse is generated at the output of the proposed optical rectangular integrator. (c) Using (a) as an input pulse, a bell-shaped optical pulse [which is almost the same as (b)] is generated at the output of an ideal optical integrator. (d) Intensity profiles of (b) (proposed integrator, solid curve) and (c) (ideal integrator, dashed curve), which are almost the same and overlap with each other.

Fig. 4
Fig. 4

Absolute values of the relative processing errors of the first-order integrator for three different input pulse widths of σ = 20 T , σ = 50 T , and σ = 100 T .

Fig. 5
Fig. 5

(Color online) (a) Input-modulated Gaussian pulse (with a pulse width of 2 σ = 100 T ) into the Nth-order optical rectangular integrator. (b) Using (a) as an input signal, an optical step function is generated at the output of the first-order ( N = 1 ) optical rectangular integrator. (c) Using (a) as an input signal, an optical paraboliclike function is generated at the output of the second-order ( N = 2 ) optical rectangular integrator. (d) Using (a) as an input signal, an optical paraboliclike function is also generated at the output of the third-order ( N = 3 ) optical rectangular integrator. The optical paraboliclike function shown in (d) has a higher degree than that shown in (c).

Fig. 6
Fig. 6

(Color online) (a) Input-modulated super-Gaussian pulse train into the first- and second-order optical rectangular integrators, where A k = [ 1010111000 ] , FWHM = 100 T , T b = 200 T , and m = 2 . (b) Using (a) as an input signal, an optical staircase function is generated at the output of the first-order optical rectangular integrator. (c) Using (a) as an input signal, an optical paraboliclike function is generated at the output of the second-order optical rectangular integrator.

Equations (13)

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y ( t ) | t = n T = 0 nT x ( t ) d t .
H I ( ω ) = { 1 j ω T , 0 ω T / ( 2 π ) 1 / 2 , 1 j ( 2 π ω T ) , 1 / 2 < ω T / ( 2 π ) 1 ,
H ^ 1 ( z ) = T 1 z 1 ,
H ^ N ( z ) = [ H ^ ( z ) ] N = [ T 1 z 1 ] N , N 1 ,
H 1 ( z ) = G d 2 1 [ G ( 1 d ) 2   exp ( - 2 α L ) ] 1 / 2 z 1 .
H N ( z ) = [ H 1 ( z ) ] N = { G d 2 1 [ G ( 1 d ) 2   exp ( - 2 α L ) ] 1 / 2 z 1 } N ,
1 = [ G ( 1 d ) 2   exp ( - 2 α L ) ] 1 / 2 .
G = 1 ( 1 d ) 2   exp ( - 2 α L ) .
tan h ( t / σ ) d t = σ ln [ cosh ( t / σ ) ] max { σ   ln [ cosh ( t / σ ) ] } ,
x ( t ) = tanh ( t / σ ) .
    y ( t ) = x ( t ) h 1 ( t ) σ ln [ cosh ( t / σ ) ] max { σ ln [ cosh ( t / σ ) ] } ,
P ( t ) = k A k x ( t k T b ) ,
x ( t ) = exp [ - ln   2 2 ( 2 t FWHM ) 2 m ] ,

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