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.

©2012 Optical Society of America

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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).

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]

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).

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]

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

2009 (6)

2008 (6)

2007 (2)

2006 (4)

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]

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

2005 (2)

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

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

2003 (1)

2001 (1)

2000 (2)

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]

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]

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)

1983 (1)

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]

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. 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]

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. 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]

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]

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]

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]

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]

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.

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.

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]

da Silva, H. J. A.

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]

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]

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.

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]

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.

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).

Huang, D.

Janner, D.

Jannson, T.

Jeppesen, P.

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]

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.

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).

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]

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.

Mulvad, H. C. H.

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]

Muriel, M. A.

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]

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]

Ngo, N. Q.

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]

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Figures (10)

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

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Fig. 2
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
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
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
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
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
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
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
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
Fig. 10 MMBRWDOB 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)

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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 dmin 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 MMBRWDOB for the assumed NMP-based NOODs in Fig. 5.

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

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

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Equations (9)

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

H(ω)∝ ( −jω ) Ν
Φ(ω)=HT{ Ln[ | H(ω) | ] }
C c = ∫ −∞ +∞ P out (t) P ideal (t)dt ( ∫ −∞ +∞ P out 2 (t)dt )( ∫ −∞ +∞ P ideal 2 (t)dt ) ×100%
MMBR=  B W max B W min
MMB R WDOB =  DOB B W min
| H(f) |= H min + H max − H min ( DOB /2 ) N | f | N
H 0 = H min + ( B W min /2 DOB /2 ) N ( H max − H min )
H 0 H min =1+ ( B W min DOB ) N ( H max H min −1 )
MMB R WDOB =  d−1 d min −1 N

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