Arbitrary-Order Photonic Differentiators Based on Phase-Shifted Long-Period Gratings

Reza Ashrafi, and José Azaña Institut National de la Recherche Scientifique – Energie, Matériaux et Télécommunications (INRS-EMT)

800 de la Gauchetière Ouest, Suite 6900, Montréal, Québec H5A 1K6, Canada ashrafi@emt.inrs.ca

Abstract—A novel design of THz-bandwidth all-optical arbitrary-order differentiators based on phase-shifted long period fiber/waveguide gratings is proposed and numerically demonstrated. This solution offers a dramatically increased tolerance against practical variations in the grating parameters (e.g. coupling strength).

Keywords-all-optical devices; all-optical signal processing; pulse shaping; ultrafast processing; fiber optics components, gratings, differentiation.

I. INTRODUCTION All-optical circuits for computing, information processing,

and networking could overcome the severe speed limitations currently imposed by electronics-based systems [1]. Photonic temporal differentiators are of fundamental interest as basic building blocks in ultrahigh-speed all-optical analog/digital signal processing and computing circuits [1]. We refer to an Nth-order optical differentiator as a device that provides the Nth-time derivative of the temporal complex envelope of an arbitrary input optical signal. These devices have been already employed for various important applications, including methods for the measurement and characterization of optical signals and devices [2], ultra-short optical pulse shaping [1], and generation and processing of ultrahigh bitrate (~Tbit/s) serial optical communication signals [3].

It has been demonstrated that a uniform long period fiber grating (LPG) operating in full-coupling condition implements a first-order time differentiator over processing bandwidths easily of a few THz [4]. Aiming at the implementation of higher order THz-bandwidth temporal differentiators, it has also been demonstrated that a uniform-period LPG incorporating N-1 π-phase shifts working in the core-to-core operation mode can serve as an Nth-order temporal differentiator [5]. A critical drawback of these approaches [4,5] is that they are extremely sensitive to small variations in the grating parameters, particularly variations in the coupling strength, which must be fixed to satisfy very specific conditions. To overcome this problem, a 1st-order differentiation design based on a phase-shifted LPG working in the cross-coupling (e.g. core-to-cladding) operation mode has been proposed [6]. In this work, we generalize and optimize this previous approach, showing that an Nth-order optical differentiator can be realized by incorporating N suitably located phase-shifts in a uniform-period LPG working in the cross-coupling operation mode. Our numerical studies prove that this design is nearly insensitive to variations in the grating coupling strength over extremely large coupling ranges, thus

dramatically relaxing the specifications for practical fabrication of these devices.

II. OPERATION PRINCIPLE An Nth-order optical differentiator can be implemented

using a linear optical filter with a spectral transfer function H(ω)=(-jω)N, where ω is the baseband frequency defined by ω=ωopt-ω0, with ωopt being the optical frequency variable and ω0 being the carrier angular frequency of the input optical signal to be processed. A schematic of the proposed general architecture for an all-fiber arbitrary-order time differentiator is shown in Fig. 1(a). The LPG for implementation of the Nth-order optical differentiator has N π-phase shifts along its length and operates in the cross-coupling mode (e.g. single-mode fiber LPG working in the core-to-cladding operation mode). Notice that the LPG’s cross-coupling operation mode can be practically implemented based on either a fiber-optic approach [7] or integrated-waveguide technology [8]. Fig. 1(b) shows a schematic of a previously demonstrated all-fiber approach for implementation of the cross-coupling operation mode in LPGs [7], i.e. to ensure that both the input and output signals are carried by the fiber core mode. Our proposed simplified approach for implementation of the target transfer function (i.e. H(ω)) is based on synthesizing the corresponding time-domain impulse response of the Nth-order (i.e. N=2,3,4,…) optical differentiator. This time-domain impulse response is linearly mapped along the grating complex apodization profile by using the first-order Born approximation (so-called space-to-time mapping) [9].

proposed phase‐shifted LPG

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Fig. 1. (a) Proposed phase-shifted LPG-based Nth-order all-optical ultrafast differentiator (output curve corresponds to N=2). (b) An illustration of a previously demonstrated fiber-optic approach [7] to transfer the cross-coupling signal from the fiber cladding-mode into the fiber core-mode by concatenating (1) a core-mode blocker and (2) a short, strong uniform LPG.

III. LPG DESIGNS AND SIMULATION RESULTS Standard single-mode fiber (Corning SMF-28) has been

considered as the optical waveguide platform. We have considered the same LPG design parameters as the experimentally characterized LPG made on SMF-28 in Ref.

This research was supported in part by the Natural Sciences andEngineering Research Council of Canada (NSERC), le Fonds Québécois de laRecherche sur la Nature et les Technologies (FQRNT), and Institut Nationalde la Recherche Scientifique (INRS).

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[10]. The grating period is Λ=430μm, which corresponds to coupling of the fiber core mode into the LP06 cladding mode at a central wavelength of 1550nm [10]. The following wavelength dependence has been assumed for the effective refractive indices of the two coupled modes [10]: n0,1(λ)=1.4884-0.031547λ+0.012023λ2 for the core mode and n0,6(λ)=1.4806-0.025396λ+0.009802λ2 for the cladding mode, where 1.2<λ<1.7 is the wavelength variable in μm.

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Fig. 2. (1-3) The phase-shifted LPG designs (local coupling coefficient) for 2nd-, 3rd- and 4th-order ultrafast optical differentiators, respectively. (4-6) The corresponding simulation results for spectral amplitude and phase responses of the designed phase-shifted LPGs. (7-9) The corresponding simulation results for temporal responses (complex envelopes) of the designed LPGs to an ultrashort Gaussian pulse (250fs-FWHM).

The coupling coefficient vs. length (complex grating apodization profile, including the required π-phase-shifts along the grating structure), for each of the target LPG designs, i.e. for implementation of 2nd-, 3rd-, and 4th-order 6THz-bandwidth optical differentiators, are shown in Fig. 2(1-3, respectively). The corresponding amplitude and phase spectral responses of the designed LPGs, numerically simulated using coupled-mode theory combined with a transfer-matrix method, are presented in Fig. 1(4-6, respectively). The corresponding temporal responses of the designed LPGs to an ultrashort (250fs-FWHM) input Gaussian pulse are shown in Fig. 2(7-9, respectively). The designed LPGs provide very nearly the required spectral response and ideal temporal responses. The performance of the designed differentiator devices is estimated by changing the LPG’s peak coupling coefficient (k) and evaluating the level of similarity (cross-correlation coefficient, Cc) between the ideal output and the actual LPG’s output). Fig. 3 shows this performance evaluation for our approach in this work (a,c,e) and for the previously demonstrated LPG-based approach for arbitrary-order differentiation in Ref. [5] (b,d,f),

for the cases of 2nd-, 3rd- and 4th-order differentiation, respectively. As anticipated, the newly proposed designs are nearly insensitive to variations in the LPG’s coupling strength over extremely large coupling ranges; this compares very favorably with the very narrow tolerance to deviations in the coupling-strength value observed for previous LPG designs.

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Fig. 3. Simulation results of Cc (solid curves) as a function of the grating coupling coefficient for the proposed LPG-based optical differentiator designs in our work (a,c,e) and in a previously demonstrated LPG-based approach (b,d,f), for the cases of 2nd-, 3rd-, and 4th-order differentiation, respectively.

IV. CONCLUSION

We have proposed and numerically demonstrated a novel design approach for THz-bandwidth arbitrary-order optical differentiators based on phase-shifted LPGs. The proposed designs offer a dramatically increased tolerance to deviations in the gratings’ coupling strength, thus overcoming the narrow tolerance to deviations in the grating parameters suffered by previous LPG-based designs. Our simulations also show that this design can provide processing bandwidths of up to several THz using readily feasible LPG specifications.

REFERENCES

[1] J. Azaña, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. J., vol. 2, pp. 359–386, 2010.

[2] R.T. Watts, K. Shi, L.P. Barry, “Time-resolved chirp measurement for 100GBaud test systems using an ideal frequency discriminator,” Opt. Commun., vol. 285, pp. 2039–2043, 2012.

[3] E. Palushani, H. Hu, L.K. Oxenløwe, R. Slavík, M. Galili, H.C.H. Mulvad, A.T. Clausen and P. Jeppesen, “640 Gb/s timing tolerant demultiplexing using a cascaded long-period fiber grating pulse shaper,” European Conference on Optical Communication (ECOC), paper 4.3.3, 2009.

[4] R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express, vol. 14, pp. 10699–10707, 2006.

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

[6] J. Azaña, and M. Kulishov, “All-fibre ultrafast optical differentiator based on π-phase-shifted long-period grating,” Electron. Lett., vol. 41, pp. 1368-1369 , 2005.

[7] R. Slavík, M. Kulishov, Y. Park, and J. Azaña, “Long-period-fiber-grating-based filter configuration enabling arbitrary linear filtering characteristics,” Opt. Lett., vol. 34, pp. 1045–1047, 2009.

[8] J. Jiang, C. L. Callender, J. P. Noad, and J. Ding, “Hybrid silica/polymer long period gratings for wavelength filtering and power distribution,” Appl. Opt., vol. 48, pp. 4866–4873, 2009.

[9] H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J., vol. 55, pp. 109–126, 1976.

[10] M. Smietana, W. J. Bock., P. Mikulic, and J. Chen, “Increasing sensitivity of arc-induced long-period gratings—pushing the fabrication technique toward its limits,” Meas. Sci. Technol., vol. 22, p. 015201, 2011.

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