Femtosecond Optical Waveform Generation Based on Space-to-Time Mapping in Long Period Gratings

Reza Ashrafi, Ming Li, 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 and simple all-fiber approach for femtosecond optical arbitrary waveform generation based on superluminal space-to-time mapping in long period gratings is proposed and numerically demonstrated.

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

I. INTRODUCTION

Techniques for the precise synthesis and control of the temporal shape of ultra-short optical pulses with durations down to the femtosecond regime have become increasingly important for a wide range of applications in such diverse fields as ultrahigh-bit-rate (~Tb/s) optical communications [1,2], and nonlinear optics [3]. To give a few examples, femtosecond saw-tooth optical pulses are highly desired for nonlinear optical switching [1] as well as for a range of wavelength conversion applications [2]. Femtosecond parabolic pulse shapes are also of great interest, e.g. to achieve ultra-flat self-phase modulation (SPM) – induced spectral broadening in supercontinuum generation experiments, extinction ratio enhancement of optical signals, retiming operations, etc. [3].

Ultrashort optical arbitrary waveform generation (OAWG) has been implemented mainly based on two linear pulse shaping approaches, namely spectral shaping [2] and temporal coherence synthesization [4,5]. Aiming at the implementation of OAWG in optical fibers, linear spectral shaping based on fiber Bragg gratings (FBGs) have been extensively used [2]. In this approach, a single-unit filter is required for implementation of the target pulse re-shaping operation. This should be contrasted with approaches based on temporal coherence synthesization [4,5], which require the use of a sufficiently large number of suitably delayed [4] or arbitrary-order differentiated [5] copies of the original input pulse, thus generally leading to a more complex pulse shaping device. As a general rule, in optical grating-based filters, to achieve a faster temporal signal, a smaller spatial feature (i.e. a higher spatial resolution) is required in the grating coupling profile [6]. Previous investigations have revealed that under certain conditions (i.e. first-order Born approximation), the output time-domain optical field complex envelope variation follows the spatial variation of the coupling coefficient (apodization profile) in counter-directional coupling structures, i.e. FBGs [6]. This phenomenon, referred to as space-to-time mapping, provides a very straightforward mechanism to synthesize optical waveforms with prescribed complex temporal shapes.

However, in FBGs, the ratio (v) between the resolution of the mentioned variations in space (Δz) and time (Δt) is necessarily lower than the propagation speed of light in vacuum (c), i.e. v=Δz/Δt<c [6]. Considering a typically achievable sub-mm resolution for fiber grating apodization profiles, FBG pulse shapers are limited to temporal resolutions of at least several picoseconds.

In this work, we extend the Born approximation to the case of co-directional coupling filters, e.g. long period fiber gratings (LPGs), see Fig. 1(a). Similarly to the case of FBGs [6], we show here that for a sufficiently weak coupling strength, the grating complex apodization profile can be directly mapped into the LPG filter’s temporal impulse response, with a suitable space-to-time scaling. In contrast to the FBG case, by employing co-directional coupling filters, the space-to-time mapping speed (v=Δz/Δt) can be much higher than the propagation speed of light in vacuum. This superluminal space-to-time mapping speed in LPGs enables the synthesis of waveforms with temporal features orders of magnitude faster than those achievable by FBGs (assuming the same spatial resolution in the grating apodization profile). Based on this finding, we numerically demonstrate here the capability of the newly introduced OAWG approach for generation of two relevant ultra-fast optical arbitrary waveforms (i.e. saw-tooth and parabolic pulses) down to the femtosecond regime using feasible LPG designs, i.e. with mm resolutions.

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Fig. 1. (a) Proposed all-fiber approach for femtosecond OAWG based on superlminal space-to-time mapping in LPGs, where k(z) represents the complex coupling-coefficient variation along the grating length. (b) An illustration of a previously demonstrated fiber-optic approach [7] to transfer the cross-coupling signal in the fiber cladding-mode to the fiber core-mode by concatenating (1) a core-mode blocker and (2) a short, strong uniform LPG.

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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MM2 (Contributed Oral)11:00 AM – 11:15 AM

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II. OPERATION PRINCIPLE

A schematic of the proposed general architecture for an all-fiber LPG-based OAWG is shown in Fig. 1(a). The LPG for implementation of the OAWG is properly apodized along its length and operates in the cross-coupling mode (e.g. single-mode fiber LPG working in the core-to-cladding operation mode). The complex apodization profile (k(z)) must be a spatial-domain scaled version of the target output temporal pulse waveform (h(t)), with a space-to-time scaling law defined by h(t) [k(z)]z=t.c/∆N , where ∆N is the difference between the effective refractive indices of the two coupled modes in LPG. Notice the anticipated ‘superluminal speed’ is implied by this law. Fig. 1(b) shows a schematic of a previously demonstrated all-fiber approach for practical implementation of the cross-coupling operation mode in LPGs [7].

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Fig. 2. (a,b) The apodized-LPG designs for femtosecond optical saw-tooth and parabolic waveform generation respectively. (c,d) The corresponding simulation results for spectral power responses of the designed LPGs. (e,f) The corresponding simulation results for temporal responses (complex envelopes) of the designed LPGs to an ultrashort (~100fs) Gaussian pulse.

III. LPG DESIGNS AND SIMULATION RESULTS

Standard single-mode fiber (Corning SMF-28) has been considered as the optical waveguide platform. Also we have considered the same LPG design parameters as the well experimentally characterized LPG made on SMF-28 in [8]. 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 [8]. The following wavelength dependence has been assumed for the effective refractive indices of the two coupled modes [8]: 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.

The LPG apodization profiles along the grating structure for implementation of the sub-picosecond optical saw-tooth and parabolic waveforms generation are shown in Fig. 2(a,b) respectively. The corresponding simulation results for power spectral responses of the designed LPGs using coupled-mode theory combined with a transfer-matrix method are presented in Fig. 2(c,d) respectively. The corresponding temporal responses of the designed LPGs to an ultrashort (~100fs) input Gaussian pulse are shown in Fig. 2(e,f) respectively. Our simulation results reveal that the designed LPGs implement very nearly the desired pulse re-shaping operations even when the weak-coupling strength condition is clearly not satisfied, see the results corresponding to the design cases with the highest maximum coupling coefficient in Fig. 2, leading to a nearly 100% cross-coupling peak. The reported results illustrate the capability of the proposed approach for synthesizing ultrafast waveforms well into the femtosecond regime using readily feasible grating profiles (mm resolutions).

IV. CONCLUSION

We have proposed and numerically demonstrated a novel all-fiber approach for femtosecond-regime OAWG based on superluminal space-to-time mapping in LPGs. The proposed approach provides a simple and feasible method to design ultrafast pulse shapers based on integrated-waveguide or all-fiber technologies with temporal resolutions orders of magnitude faster than those achievable by their FBG filter counterparts.

REFERENCES

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[2] F. Parmigiani, M. Ibsen, T.T. Ng, L. Provost, P. Petropoulos, and D.J. Richardson, “An efficient wavelength converter exploiting a grating-based saw-tooth pulse shaper,” IEEE Photon. Technol. Lett., vol. 20, pp. 1461–1463, 2008.

[3] C. Finot, J.M. Dudley, B. Kibler, D.J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron., vol. 45, pp. 1482–1489, 2009.

[4] Y. Park, M.H. Asghari, T.J. Ahn, and J. Azaña, “Transform-limited picosecond pulse shaping based on temporal coherence synthesization,” Opt. Express, vol. 15, pp. 9584–9599, 2007.

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

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

[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] 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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