Figure of merit for photonic differentiators

Reza Ashrafi; José Azaña

doi:10.1364/OE.20.002626

Figure of merit for photonic differentiators

Reza Ashrafi and José Azaña

Optics Express
Vol. 20,
Issue 3,
pp. 2626-2639
(2012)
•https://doi.org/10.1364/OE.20.002626

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Reza Ashrafi and José Azaña, "Figure of merit for photonic differentiators," Opt. Express 20, 2626-2639 (2012)

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Related Topics
Table of Contents Category
- Fourier Optics and Signal Processing
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber optics components (060.2340)
- Fiber Bragg gratings (060.3735)
- Ultrafast processing (070.7145)
- Analog optical signal processing (070.1170)
- All-optical devices (230.1150)
- Pulse shaping (320.5540)

About this Article
History
- Original Manuscript: October 21, 2011
- Revised Manuscript: January 5, 2012
- Manuscript Accepted: January 9, 2012
- Published: January 20, 2012

Accessible

Open Access

Abstract

We introduce a universal figure of merit to evaluate the processing speed (operation bandwidth) performance of arbitrary-order optical differentiators. In particular, we define the maximum-to-minimum bandwidth ratio (MMBR) as a main figure of merit of these devices, which essentially informs about the broadness of the acceptable input pulse bandwidth range. We derive and numerically confirm a general analytical expression for the MMBR of an arbitrary optical differentiator, showing that this can be expressed simply as a function of the differentiator’s amplitude resonance depth. The device MMBR can be improved by increasing the filter’s resonance depth, depending also on the differentiation order; in particular, the MMBR quickly deteriorates as the differentiator order is increased. In our analysis, photonic differentiators are considered in two main groups, namely (i) non-minimum phase and (ii) minimum phase optical filtering implementations. The derived analytical expression for the device MMBR is generalized for these two different solutions, and the validity of the obtained analytical estimates is verified through numerical simulations, including results for the cases of 1st-, 2nd-, and 3rd-order differentiators.

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References

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Publication

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, Special Issue on “Optical signal processing,” J. Lightwave Technol. 24(7), 2484–2486 (2006).
[Crossref]
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[Crossref]
N. Q. Ngo and L. N. Binh, “Optical realization of Newton-Cotes-based integrators for dark soliton generation,” J. Lightwave Technol. 24(1), 563–572 (2006).
[Crossref]
R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008).
[Crossref] [PubMed]
N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1-3), 115–129 (2004).
[Crossref]
R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14(22), 10699–10707 (2006).
[Crossref] [PubMed]
L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. Larochelle, A. Carballar, and J. Azaña, “Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators,” Opt. Lett. 34(12), 1792–1794 (2009).
[Crossref] [PubMed]
M. Kulishov, D. Krcmarík, and R. Slavík, “Design of terahertz-bandwidth arbitrary-order temporal differentiators based on long-period fiber gratings,” Opt. Lett. 32(20), 2978–2980 (2007).
[Crossref] [PubMed]
R. Slavík, Y. Park, M. Kulishov, and J. Azaña, “Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings,” Opt. Lett. 34(20), 3116–3118 (2009).
[Crossref] [PubMed]
F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008).
[Crossref] [PubMed]
M. Li, D. Janner, J. Yao, and V. Pruneri, “Arbitrary-order all-fiber temporal differentiator based on a fiber Bragg grating: design and experimental demonstration,” Opt. Express 17(22), 19798–19807 (2009).
[Crossref] [PubMed]
D. Gatti, T. T. Fernandez, S. Longhi, and P. Laporta, “Temporal differentiators based on highly-structured fibre Bragg gratings,” Electron. Lett. 46(13), 943–945 (2010).
[Crossref]
M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. 33(21), 2458–2460 (2008).
[Crossref] [PubMed]
M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30(20), 2700–2702 (2005).
[Crossref] [PubMed]
R. Ashrafi, M. H. Asghari, and J. Azaña, “Ultrafast optical arbitrary-order differentiators based on apodized long period gratings,” IEEE Photon. J. 3(3), 353–364 (2011).
[Crossref]
M. H. Asghari and J. Azaña, “All-optical Hilbert transformer based on a single phase-shifted fiber Bragg grating: design and analysis,” Opt. Lett. 34(3), 334–336 (2009).
[Crossref] [PubMed]
M. Li and J. Yao, “Experimental demonstration of a wideband photonic temporal Hilbert transformer based on a single fiber Bragg grating,” J. Lightwave Technol. 22, 1559–1561 (2010).
T. Jannson, “Real-time Fourier transformation in dispersive optical fibers,” Opt. Lett. 8(4), 232–234 (1983).
[Crossref] [PubMed]
J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36(5), 517–526 (2000).
[Crossref]
F. Li, Y. Park, and J. Azaña, “Linear characterization of optical pulses with durations ranging from the picosecond to the nanosecond regime using ultrafast photonic differentiation,” J. Lightwave Technol. 27(21), 4623–4633 (2009).
[Crossref]
H. J. A. da Silva and J. J. O’Reilly, “Optical pulse modeling with Hermite-Gaussian functions,” Opt. Lett. 14(10), 526–528 (1989).
[Crossref] [PubMed]
M. H. Asghari and J. Azaña, “Proposal and analysis of a reconfigurable pulse shaping technique based on multi-arm optical differentiators,” Opt. Commun. 281(18), 4581–4588 (2008).
[Crossref]
M. Stratmann, T. Pagel, and F. Mitschke, “Experimental observation of temporal soliton molecules,” Phys. Rev. Lett. 95(14), 143902 (2005).
[Crossref] [PubMed]
L. K. Oxenløwe, R. Slavík, M. Galili, H. C. H. Mulvad, A. T. Clausen, Y. Park, J. Azaña, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]
L. K. Oxenløwe, M. Galili, H. Hu, H. Ji, E. Palushani, J. L. Areal, J. Xu, H. C. H. Mulvad, A. T. Clausen, and P. Jeppesen, “Serial optical communications and ultra-fast optical signal processing of Tbit/s data signals,” IEEE Topical Meeting on Microwave Photonics (MWP2010), Montreal Quebec, Canada, 361–364, 5–9 Oct. 2010.
N. Q. Ngo, L. N. Binh, and X. Dai, “Optical dark-soliton generators and detectors,” Opt. Commun. 132(3-4), 389–402 (1996).
[Crossref]
P. Velanas, A. Bogris, A. Argyris, and D. Syvridis, “High-speed all-optical first- and second-order differentiators based on cross-phase modulation in fibers,” J. Lightwave Technol. 26(18), 3269–3276 (2008).
[Crossref]
Z. Li and C. Wu, “All-optical differentiator and high-speed pulse generation based on cross-polarization modulation in a semiconductor optical amplifier,” Opt. Lett. 34(6), 830–832 (2009).
[Crossref] [PubMed]
Y. Park, M. H. Asghari, R. Helsten, and J. Azaña, “Implementation of broadband microwave arbitrary-order time differential operators using a reconfigurable incoherent photonic processor,” IEEE Photon. J. 2, 1040–1050 (2010).
J. Zhou, S. Fu, S. Aditya, P. P. Shum, C. Lin, V. Wong, and D. Lim, “Photonic temporal differentiator based on polarization modulation in a LiNbO3 phase modulator,” IEEE International Topical Meeting on Microwave Photonics MWP '09, 1–3, 2009.
A. V. Okishev, “Optical differentiation and multimillijoule approximately 150 ps pulse generation in a regenerative amplifier with a temperature-tuned intracavity volume Bragg grating,” Appl. Opt. 49(8), 1331–1334 (2010).
[Crossref] [PubMed]
J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, “All-optical differentiator based on cross-gain modulation in semiconductor optical amplifier,” Opt. Lett. 32(20), 3029–3031 (2007).
[Crossref] [PubMed]
X. Li, J. Dong, Y. Yu, and X. Zhang, “A tunable microwave photonic filter based on an all-optical differentiator,” IEEE Photon. Technol. Lett. 23(5), 308–310 (2011).
[Crossref]
J. Niu, K. Xu, X. Sun, Q. Lv, J. Dai, J. Wu, and J. Lin, “Instantaneous microwave frequency measurement using a photonic differentiator and an opto-electric hybrid implementation,” Asia-Pacific Microwave Photonics Conference 2010 (APMP2010), Hong Kong, China, 26–28 April 2010.
A. V. Oppenheim, A. S. Willsky, S. N. Nawab, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1996).
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[Crossref]
A. Papoulis, The Fourier Integral and Its Applications (New York McGraw-Hill, 1962).
J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave Technol. 21(1), 254–263 (2003).
[Crossref]
K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9(11), 1481–1492 (1991).
[Crossref]
G.-W. Chern and L. A. Wang, “Analysis and design of almost-periodic vertical-grating-assisted codirectional coupler filters with nonuniform duty ratios,” Appl. Opt. 39(25), 4629–4637 (2000).
[Crossref] [PubMed]
T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]
R. Slavík, “Extremely deep long-period fiber grating made with CO2 laser,” IEEE Photon. Technol. Lett. 18(16), 1705–1707 (2006).
[Crossref]

2011 (2)

R. Ashrafi, M. H. Asghari, and J. Azaña, “Ultrafast optical arbitrary-order differentiators based on apodized long period gratings,” IEEE Photon. J. 3(3), 353–364 (2011).
[Crossref]

X. Li, J. Dong, Y. Yu, and X. Zhang, “A tunable microwave photonic filter based on an all-optical differentiator,” IEEE Photon. Technol. Lett. 23(5), 308–310 (2011).
[Crossref]

2010 (5)

Y. Park, M. H. Asghari, R. Helsten, and J. Azaña, “Implementation of broadband microwave arbitrary-order time differential operators using a reconfigurable incoherent photonic processor,” IEEE Photon. J. 2, 1040–1050 (2010).

M. Li and J. Yao, “Experimental demonstration of a wideband photonic temporal Hilbert transformer based on a single fiber Bragg grating,” J. Lightwave Technol. 22, 1559–1561 (2010).

J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. J. 2(3), 359–386 (2010).
[Crossref]

D. Gatti, T. T. Fernandez, S. Longhi, and P. Laporta, “Temporal differentiators based on highly-structured fibre Bragg gratings,” Electron. Lett. 46(13), 943–945 (2010).
[Crossref]

A. V. Okishev, “Optical differentiation and multimillijoule approximately 150 ps pulse generation in a regenerative amplifier with a temperature-tuned intracavity volume Bragg grating,” Appl. Opt. 49(8), 1331–1334 (2010).
[Crossref] [PubMed]

2009 (6)

M. H. Asghari and J. Azaña, “All-optical Hilbert transformer based on a single phase-shifted fiber Bragg grating: design and analysis,” Opt. Lett. 34(3), 334–336 (2009).
[Crossref] [PubMed]

Z. Li and C. Wu, “All-optical differentiator and high-speed pulse generation based on cross-polarization modulation in a semiconductor optical amplifier,” Opt. Lett. 34(6), 830–832 (2009).
[Crossref] [PubMed]

L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. Larochelle, A. Carballar, and J. Azaña, “Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators,” Opt. Lett. 34(12), 1792–1794 (2009).
[Crossref] [PubMed]

R. Slavík, Y. Park, M. Kulishov, and J. Azaña, “Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings,” Opt. Lett. 34(20), 3116–3118 (2009).
[Crossref] [PubMed]

M. Li, D. Janner, J. Yao, and V. Pruneri, “Arbitrary-order all-fiber temporal differentiator based on a fiber Bragg grating: design and experimental demonstration,” Opt. Express 17(22), 19798–19807 (2009).
[Crossref] [PubMed]

F. Li, Y. Park, and J. Azaña, “Linear characterization of optical pulses with durations ranging from the picosecond to the nanosecond regime using ultrafast photonic differentiation,” J. Lightwave Technol. 27(21), 4623–4633 (2009).
[Crossref]

2008 (6)

F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008).
[Crossref] [PubMed]

M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. 33(21), 2458–2460 (2008).
[Crossref] [PubMed]

R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008).
[Crossref] [PubMed]

P. Velanas, A. Bogris, A. Argyris, and D. Syvridis, “High-speed all-optical first- and second-order differentiators based on cross-phase modulation in fibers,” J. Lightwave Technol. 26(18), 3269–3276 (2008).
[Crossref]

M. H. Asghari and J. Azaña, “Proposal and analysis of a reconfigurable pulse shaping technique based on multi-arm optical differentiators,” Opt. Commun. 281(18), 4581–4588 (2008).
[Crossref]

L. K. Oxenløwe, R. Slavík, M. Galili, H. C. H. Mulvad, A. T. Clausen, Y. Park, J. Azaña, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

2007 (2)

M. Kulishov, D. Krcmarík, and R. Slavík, “Design of terahertz-bandwidth arbitrary-order temporal differentiators based on long-period fiber gratings,” Opt. Lett. 32(20), 2978–2980 (2007).
[Crossref] [PubMed]

J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, “All-optical differentiator based on cross-gain modulation in semiconductor optical amplifier,” Opt. Lett. 32(20), 3029–3031 (2007).
[Crossref] [PubMed]

2006 (4)

N. Q. Ngo and L. N. Binh, “Optical realization of Newton-Cotes-based integrators for dark soliton generation,” J. Lightwave Technol. 24(1), 563–572 (2006).
[Crossref]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14(22), 10699–10707 (2006).
[Crossref] [PubMed]

R. Slavík, “Extremely deep long-period fiber grating made with CO2 laser,” IEEE Photon. Technol. Lett. 18(16), 1705–1707 (2006).
[Crossref]

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, Special Issue on “Optical signal processing,” J. Lightwave Technol. 24(7), 2484–2486 (2006).
[Crossref]

2005 (2)

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental observation of temporal soliton molecules,” Phys. Rev. Lett. 95(14), 143902 (2005).
[Crossref] [PubMed]

M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30(20), 2700–2702 (2005).
[Crossref] [PubMed]

2004 (1)

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1-3), 115–129 (2004).
[Crossref]

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36(5), 517–526 (2000).
[Crossref]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

1996 (1)

N. Q. Ngo, L. N. Binh, and X. Dai, “Optical dark-soliton generators and detectors,” Opt. Commun. 132(3-4), 389–402 (1996).
[Crossref]

1991 (1)

K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9(11), 1481–1492 (1991).
[Crossref]

1989 (1)

H. J. A. da Silva and J. J. O’Reilly, “Optical pulse modeling with Hermite-Gaussian functions,” Opt. Lett. 14(10), 526–528 (1989).
[Crossref] [PubMed]

1983 (1)

T. Jannson, “Real-time Fourier transformation in dispersive optical fibers,” Opt. Lett. 8(4), 232–234 (1983).
[Crossref] [PubMed]

Ahn, T.-J.

Argyris, A.

Asghari, M. H.

R. Ashrafi, M. H. Asghari, and J. Azaña, “Ultrafast optical arbitrary-order differentiators based on apodized long period gratings,” IEEE Photon. J. 3(3), 353–364 (2011).
[Crossref]

M. H. Asghari and J. Azaña, “All-optical Hilbert transformer based on a single phase-shifted fiber Bragg grating: design and analysis,” Opt. Lett. 34(3), 334–336 (2009).
[Crossref] [PubMed]

M. H. Asghari and J. Azaña, “Proposal and analysis of a reconfigurable pulse shaping technique based on multi-arm optical differentiators,” Opt. Commun. 281(18), 4581–4588 (2008).
[Crossref]

Ashrafi, R.

R. Ashrafi, M. H. Asghari, and J. Azaña, “Ultrafast optical arbitrary-order differentiators based on apodized long period gratings,” IEEE Photon. J. 3(3), 353–364 (2011).
[Crossref]

Ayotte, N.

Azaa, J.

J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. J. 2(3), 359–386 (2010).
[Crossref]

Azaña, J.

R. Ashrafi, M. H. Asghari, and J. Azaña, “Ultrafast optical arbitrary-order differentiators based on apodized long period gratings,” IEEE Photon. J. 3(3), 353–364 (2011).
[Crossref]

M. H. Asghari and J. Azaña, “All-optical Hilbert transformer based on a single phase-shifted fiber Bragg grating: design and analysis,” Opt. Lett. 34(3), 334–336 (2009).
[Crossref] [PubMed]

M. H. Asghari and J. Azaña, “Proposal and analysis of a reconfigurable pulse shaping technique based on multi-arm optical differentiators,” Opt. Commun. 281(18), 4581–4588 (2008).
[Crossref]

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, Special Issue on “Optical signal processing,” J. Lightwave Technol. 24(7), 2484–2486 (2006).
[Crossref]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14(22), 10699–10707 (2006).
[Crossref] [PubMed]

M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30(20), 2700–2702 (2005).
[Crossref] [PubMed]

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36(5), 517–526 (2000).
[Crossref]

Binh, L. N.

N. Q. Ngo and L. N. Binh, “Optical realization of Newton-Cotes-based integrators for dark soliton generation,” J. Lightwave Technol. 24(1), 563–572 (2006).
[Crossref]

N. Q. Ngo, L. N. Binh, and X. Dai, “Optical dark-soliton generators and detectors,” Opt. Commun. 132(3-4), 389–402 (1996).
[Crossref]

Bogris, A.

Boudreau, S.

Brenne, J. K.

J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave Technol. 21(1), 254–263 (2003).
[Crossref]

Carballar, A.

Chern, G.-W.

Cincontti, G.

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, Special Issue on “Optical signal processing,” J. Lightwave Technol. 24(7), 2484–2486 (2006).
[Crossref]

Clausen, A. T.

da Silva, H. J. A.

H. J. A. da Silva and J. J. O’Reilly, “Optical pulse modeling with Hermite-Gaussian functions,” Opt. Lett. 14(10), 526–528 (1989).
[Crossref] [PubMed]

Dai, X.

N. Q. Ngo, L. N. Binh, and X. Dai, “Optical dark-soliton generators and detectors,” Opt. Commun. 132(3-4), 389–402 (1996).
[Crossref]

Dong, J.

X. Li, J. Dong, Y. Yu, and X. Zhang, “A tunable microwave photonic filter based on an all-optical differentiator,” IEEE Photon. Technol. Lett. 23(5), 308–310 (2011).
[Crossref]

Doucet, S.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Fernandez, T. T.

D. Gatti, T. T. Fernandez, S. Longhi, and P. Laporta, “Temporal differentiators based on highly-structured fibre Bragg gratings,” Electron. Lett. 46(13), 943–945 (2010).
[Crossref]

Galili, M.

Gatti, D.

D. Gatti, T. T. Fernandez, S. Longhi, and P. Laporta, “Temporal differentiators based on highly-structured fibre Bragg gratings,” Electron. Lett. 46(13), 943–945 (2010).
[Crossref]

Helsten, R.

Huang, D.

Janner, D.

Jannson, T.

T. Jannson, “Real-time Fourier transformation in dispersive optical fibers,” Opt. Lett. 8(4), 232–234 (1983).
[Crossref] [PubMed]

Jeppesen, P.

Kam, C. H.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1-3), 115–129 (2004).
[Crossref]

Krcmarík, D.

Kulishov, M.

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14(22), 10699–10707 (2006).
[Crossref] [PubMed]

M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30(20), 2700–2702 (2005).
[Crossref] [PubMed]

Laporta, P.

D. Gatti, T. T. Fernandez, S. Longhi, and P. Laporta, “Temporal differentiators based on highly-structured fibre Bragg gratings,” Electron. Lett. 46(13), 943–945 (2010).
[Crossref]

Larochelle, S.

Li, F.

Li, M.

M. Li and J. Yao, “Experimental demonstration of a wideband photonic temporal Hilbert transformer based on a single fiber Bragg grating,” J. Lightwave Technol. 22, 1559–1561 (2010).

Li, X.

X. Li, J. Dong, Y. Yu, and X. Zhang, “A tunable microwave photonic filter based on an all-optical differentiator,” IEEE Photon. Technol. Lett. 23(5), 308–310 (2011).
[Crossref]

Li, Z.

Liu, D.

Liu, F.

Longhi, S.

D. Gatti, T. T. Fernandez, S. Longhi, and P. Laporta, “Temporal differentiators based on highly-structured fibre Bragg gratings,” Electron. Lett. 46(13), 943–945 (2010).
[Crossref]

Madsen, C. K.

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, Special Issue on “Optical signal processing,” J. Lightwave Technol. 24(7), 2484–2486 (2006).
[Crossref]

Mitschke, F.

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental observation of temporal soliton molecules,” Phys. Rev. Lett. 95(14), 143902 (2005).
[Crossref] [PubMed]

Morandotti, R.

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Fig. 1 A schematic of an Nth-order optical differentiator (the curves correspond to N = 2).

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Fig. 2 A schematic of the spectral amplitude response of an NOOD, defining the main specifications of a general NOOD filter.

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Fig. 3 Some different possible spectral amplitude responses of a 3rd-order differentiator. As it can be seen in this figure the spectral amplitude outside the differentiation frequency band (DOB) can be different depending on the specific optical filter device technology.

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Fig. 4 Cross-correlation coefficient versus input Gaussian pulse bandwidth for each of the spectral amplitude profiles shown in Fig. 3. The differentiators are considered to be NMP-based with an ideal spectral phase profile. The depth of H(f) for all the cases (H₁-H₅) is considered to be d = 70dB.

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Fig. 5 The spectral amplitude profiles in dB scale with three different depths of d = 50dB, d = 70dB and d = infiniy for the considered NMP-based 3rd-order differentiator. The differentiator has an amplitude spectral profile with a shape similar to that of H₅ in Fig. 3 and an ideal linear spectral phase profile, including an exact π-phase-jump at the NOOD’s central frequency.

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Fig. 6 Cross correlation coefficient versus input Gaussian pulse bandwidth for the considered NMP-based 3rd-order differentiator with three different depths of d = 50dB, 70dB and infinity.

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Fig. 7 The spectral amplitude response of 1st-, 2nd- and 3rd-order differentiators in linear and dB scales. The depth of H(f) for all the cases (1st-, 2nd- and 3rd-order differentiators) is considered to be d = 50dB.

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Fig. 8 Cross-correlation coefficient versus input Gaussian pulse bandwidth for the considered 1st-, 2nd- and 3rd-order differentiators, shown in Fig. 7, all having the resonance depth, d = 50dB.

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Fig. 9 Cross-correlation coefficient versus input Gaussian pulse bandwidth for MP-based and NMP-based differentiators, both having an identical spectral amplitude response profile and the same resonance depth of d = 50dB.

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Fig. 10 MMBR_WDOB curves versus resonance depth (d) for the NOODs based on (a) non-minimum phase and (b) minimum-phase filtering systems. The solid curves in (a) and (b) plot the obtained theoretical expression, Eq. (9), and the circle points are obtained from numerical simulations.

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Tables (5)

Table 1 Previously demonstrated optical differentiators with different device operation bandwidths (DOBs) based on various photonic devices.

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Table 2 Estimation of d_min of NOODs (N = 1,2,3) based on MP and NMP systems considering a Gaussian input pulse. For the odd-order NMP-based NOODs, an ideal π-phase-jump for the spectral phase profile in the NOOD’s central frequency (i.e. ω = 0) has been considered.

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Table 3 Evaluation of MMBR_WDOB for the assumed NMP-based NOODs in Fig. 5.

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Table 4 Evaluation of MMBR_WDOB for the assumed NMP-based NOODs in Fig. 7.

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Table 5 Evaluation of MMBR_WDOB for the assumed 1st-order NMP-based and MP-based differentiators.

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Equations (9)

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H (ω) \propto {(- j ω)}^{Ν}

Φ (ω) = HT {Ln [| H (ω) |]}

C_{c} = \frac{\int_{- \infty}^{+ \infty} P_{o u t} (t) P_{i d e a l} (t) d t}{\sqrt{(\int_{- \infty}^{+ \infty} P_{o u t}^{2} (t) d t) (\int_{- \infty}^{+ \infty} P_{i d e a l}^{2} (t) d t)}} \times 100 %

M M B R = \frac{B W_{\max}}{B W_{\min}}

M M B R_{WDOB} = \frac{D O B}{B W_{\min}}

| H (f) | = H_{\min} + \frac{H_{\max} - H_{\min}}{{(D O B / 2)}^{N}} {| f |}^{N}

H_{0} = H_{\min} + {(\frac{B W_{\min} / 2}{D O B / 2})}^{N} (H_{\max} - H_{\min})

\frac{H_{0}}{H_{\min}} = 1 + {(\frac{B W_{\min}}{D O B})}^{N} (\frac{H_{\max}}{H_{\min}} - 1)

M M B R_{WDOB} = \sqrt[N]{\frac{d - 1}{d_{\min} - 1}}

Photonic devices for optical differentiation	DOB
Silicon microring resonator [11], Apodized FBGs working in reflection [12,13]	10s of GHz
Apodized chirped FBG working in transmission [8]	100s of GHz
Uniform long period gratings [7,10]	A few THz
Apodized long period gratings [16]	> 10 THz

NOODs	NMP-based	MP-Based
1st-order	d_min = 21.31dB	d_min ≥40.70dB d_min≈40.70 + 0.25tanh[(d-42)/4)] dB
2nd-order	d_min = 34.68dB	d_min ≥54.67dB d_min≈54.671 + 1.041tanh[(d-55)/10)] dB
3rd-order	d_min = 38.57dB	d_min ≥81.45dB d_min≈83.535 + 3.535tanh[(d-89.5)/11)] dB

NOOD	MMBR_WDOB from analytical expression: Eq. (9)	MMBR_WDOB from simulation: Fig. 6
3rd-order, d = 50dB	$\sqrt[3]{\frac{10^{50 / 20} - 1}{10^{38.57 / 20} - 1}} = 1.56$	$\frac{D O B}{B W_{\min}} = \frac{7.093 THz}{4 .561THZ} = 1.56$
3rd-order, d = 70dB	$\sqrt[3]{\frac{10^{70 / 20} - 1}{10^{38.57 / 20} - 1}} = 3.35$	$\frac{D O B}{B W_{\min}} = \frac{7.093 THz}{2 .134THZ} = 3.35$
3rd-order, d = inf.	$\sqrt[3]{\frac{10^{\infty / 20} - 1}{10^{38.57 / 20} - 1}} = \infty$	$\frac{D O B}{B W_{\min}} = \frac{7.093 THz}{0} = \infty$

NOOD	MMBR_WDOB from analytical expression: Eq. (9)	MMBR_WDOB from simulation: Fig. 8
1st-order, d = 50dB	$\frac{10^{50 / 20} - 1}{10^{21.31 / 20} - 1} = 29.661$	$\frac{D O B}{B W_{\min}} = \frac{7.093 THz}{0 .239THz} = 29.661$
2nd-order, d = 50dB	$\sqrt[]{\frac{10^{50 / 20} - 1}{10^{34.68 / 20} - 1}} = 2.43$	$\frac{D O B}{B W_{\min}} = \frac{7.093 THz}{2 .92THz} = 2.43$
3rd-order, d = 50dB	$\sqrt[3]{\frac{10^{50 / 20} - 1}{10^{38.57 / 20} - 1}} = 1.56$	$\frac{D O B}{B W_{\min}} = \frac{7.093 THz}{4 .56THz} = 1.56$

NOOD	MMBR_WDOB from analytical expression: Eq. (9)	MMBR_WDOB from simulation: Fig. 9
1st-order, d = 50dB, MP-based	$\frac{10^{50 / 20} - 1}{10^{40.941 / 20} - 1} = 2.854$	$\frac{D O B}{B W_{\min}} = \frac{7.093 THz}{2 .485THz} = 2.854$
1st-order, d = 50dB, NMP-based	$\frac{10^{50 / 20} - 1}{10^{21.31 / 20} - 1} = 29.661$	$\frac{D O B}{B W_{\min}} = \frac{7.093 THz}{0 .239THz} = 29.661$

Abstract

References

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