Optical pulse shaping capabilities of grating-assisted codirectional couplers

Microelectronics Journal 36 (2005) 289–293 www.elsevier.com/locate/mejo

Optical pulse shaping capabilities of grating-assisted codirectional couplers Jose´ Azan˜aa,*, Mykola Kulishovb Institut National de la Recherche Scientifique (INRS)—E´nergie, Mate´riaux et Te´le´communications, 800 de la Gauchetie`re Ouest, suite 6900, Montre´al, Que., Canada H5A 1K6 b Adtek Photomask Inc., 4950 Fisher Street, Montreal, Que., Canada H4T 1J6

a

Available online 17 March 2005

Abstract Ultrashort pulse propagation through grating-assisted codirectional couplers (GACCs) is numerically investigated. For this purpose, the linear temporal responses of uniform GACCs to ultrashort optical pulses are calculated and the influence of the different physical grating parameters (e.g. length and coupling strength) on these temporal responses is evaluated. It is observed that depending on the length and coupling strength of the uniform grating perturbation one can achieve very different temporal shapes at the output of the device, including square temporal waveforms as well as equalized multiple pulse sequences. These results open important new perspectives towards the implementation of practical, integrated optical pulse shapers operating in the picosecond and sub-picosecond regimes. q 2005 Elsevier Ltd. All rights reserved. Keywords: Integrated optics devices; Gratings; Optical pulse propagation; Optical pulse shaping

Recent research has demonstrated the potential of contradirectional couplers, e.g. short-period Bragg gratings (BGs), for diverse coherent pulse manipulation and control applications (see Refs. [1–4] and references cited therein). For instance, BGs operating in the linear regime have been used for optical pulse shaping and optical pulse repetition rate multiplication. Note that the BG approach is specially suited for synthesizing temporal features in the range of a few tens of picoseconds. The synthesis of faster temporal features would require a reduction of the spatial scale of the BG profiles to an extremely demanding or even unpractical level. GA codirectional couplers (GACCs) in their different implementation platforms, namely integrated planar waveguide couplers, long-period gratings (LPFGs) in single-mode fibers, or double-core fiber couplers, have been also widely investigated. Unlike a BG with its submicron periodicities, a GACC filter is specially suited for implementation in integrated platforms and for instance, it can be realized through an electro-optically (EO) induced grating where its parameters, such as the length and coupling coefficient, can * Corresponding author. Tel.: C1 514 875 1266; fax: C1 514 875 0344. E-mail addresses: [email protected] (J. Azan˜a), [email protected] (M. Kulishov).

0026-2692/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2005.02.015

be independently adjusted through electronic control of the interdigitated electrodes [5,6]. The GACC analysis (based on coupled-mode theory) has been generally limited to the spectral domain. This has similarly restricted the use of these devices to the conventional filtering—type applications (band-pass and band-stop filters, spectrum equalizers, etc.). Time-domain analysis of GACCs has received very little attention [7–9]. For instance, in [7] the idea of using weakcoupling GACCs as matched linear filters for encoding/ decoding optical pulses was proposed. More recently, the temporal properties of GACCs operating in the nonlinear regime for all-optical switching applications have been also investigated [8]. In more fundamental studies, optical path integration methods were developed for calculating the temporal impulse response of GACCs [9]. Our present work focuses on investigating the pulseshaping capabilities of GACCs. In particular, in this work we carry out an exhaustive theoretical (numerical) analysis of optical pulse propagation through GACCs within the linear regime of operation. We calculate the temporal response of uniform GACCs to ultrashort optical pulses and evaluate in detail the effects of varying the different grating physical parameters (coupling coefficient and length). In addition to the intrinsic physical interest of our results, they also show the strong potential of GACCs for optical pulse shaping applications.

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The results from our study are general and valid for any of the known configurations involving GA codirectional coupling. For the specific numerical analysis presented herein we assume a GACC with a short grating (14 mm) and a GACC with a long grating (60 mm). The gratings have a uniform period LZ40 mm. Specifically, the effective index of the ‘energy supplier’ mode was set to nsZ2 and that of the ‘energy receptor’ mode was set to nrZ1.96115. This ensures that the phase-matching condition between these two modes is satisfied at the wavelength of 1554 nm. The indices were fixed to the given values targeting integrated tunable GACC designs based on EO crystals (e.g. LiNbO3). The input pulse to the analyzed GACC structures was assumed to be a transform-limited Gaussian pulse centered at the resonance wavelength of the GACC filter (1554 nm), with a peak intensity 100 (arbitrary units) and a full-widthhalf-maximum (FWHM) duration depending on the grating under analysis (in all the cases, the input pulse bandwidth was fixed to be broader than the GACC spectral response in the ‘receptor’ mode, i.e. our analysis focuses on the socalled ultrashort pulse propagation regime). The temporal responses corresponding to the ‘supplier’ and ‘receptor’ modes, hs(t) and hr(t), were calculated by taking the inverse Fourier transform of the result of multiplying the input pulse spectrum Hin(n) by the GACC spectral transmission response corresponding to the ‘supsupplier’ mode, Hs(n), and to the ‘receptor’ mode Hr(n), respectively. We assume that only the ‘supplier’ mode is initially excited, i.e. that the input optical pulse is launched into the ‘supplier’ mode. In this case, it is well known from coupled-mode theory that for a uniform (unapodized and unchirped) grating, the associated spectral transmission responses are   s Hs ðnÞ Z cosðgLÞ C j sinðgLÞ expðjðbs K sÞLÞ (1) g

Hr ðnÞ Z j

k sinðgLÞ expðjðbr C sÞLÞ g

(2)

where k is the cross-coupling coefficient, L is the grating length, bs(n) and br(n) are the propagation constants corresponding to the ‘supplier’ and ‘receptor’ modes, respectively, s is the detuning factor, sðnÞZ ðbs ðnÞK br ðnÞÞ=2K p=L, L is the grating period, and gZ(s2Ck2)1/2. In our first set of simulations for the short grating (14 mm), shown in Fig. 1, we assumed an input pulse with a FWHM time-width of 450 fs. We show the GACC temporal responses (left column) and the associated spectra (right column) for different coupling strengths (the corresponding values for kL are giving within each figure). Specifically, each of the plots in the left column shows the average optical intensity (temporal waveform) at the output of the device for the ‘energy receptor’ mode (solid curves) and for

the ‘energy supplier’ mode (dotted curves). For comparison, the average optical intensity of the input pulse to the GACC is also shown in all the figures (dashed curves). Notice that in all the cases, the optical intensities are represented in normalized units, but one has to keep in mind that the output pulses have N times lower amplitude than the input pulse, where the respective values of N are given in the plots. The corresponding spectra are shown in the column at the right (again, solid curves are used for the ‘energy receptor’ mode, dotted curves for the ‘energy supplier’ mode and dashed curve for the input pulse). We observe that the temporal responses corresponding to the two interacting modes are very different and in fact, the most interesting pulse reshaping operations occur for the ‘energy receptor’ mode. An outstanding general property is that the temporal waveforms obtained for the ‘energy receptor’ mode are always symmetric, independently of the fixed physical grating parameters. In what follows we will focus our attention on the detailed analysis of the temporal waveforms in the ‘energy receptor’ mode. As evidenced by the results shown in Fig. 1, very different pulse shapes can be obtained depending on the coupling coefficient (peak modulation amplitude) of the uniform perturbation. Specifically, for the coupling coefficient k that provides kL Z0.28p, the original Gaussian pulse is re-shaped into a nearly square temporal waveform with a total duration of Dtz2 ps. It should be mentioned that square temporal waveforms are highly desired for a range of nonlinear optical switching and frequency conversion applications [8]. In our simulations, we observe some smoothing with respect to the ideal square pulse (e.g. rise/ fall times of finite duration), which is mainly associated with the limited input pulse bandwidth. Note that in this case a moderate coupling from the supplier mode to the receptor mode is induced by the grating, with a peak energy transmission (at the resonance wavelength) of z60%. For comparison, the generation of a square temporal waveform with a BG would require the use of a weak-coupling uniform BG (reflectivity !10%) [2]. More importantly, in order to generate a z2-ps square pulse, the BG should be around 200 mm long, thus requiring an extremely tough manufacturing precision. In general, our numerical results show that in addition to the differences in the temporal shapes, the optical pulses generated from a GACC exhibit a much shorter duration (generally in more than one order of magnitude) than those generated from an equivalent BG (with the same grating length). The discordance in the temporal scales that can be obtained with the BG and GACC approaches (for the same grating length) is mainly associated with the difference in their respective refractive index contrasts. The refractive index contrast is generally more than one order of magnitude larger for a BG (2neff) than for a GACC (n1Kn2). Note that the refractive index contrast determines the speed difference between the two coupled modes, which, in turn, fixes the temporal resolution associated with a given grating length.

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Fig. 1. Results corresponding to a GACC with a 14-mm long uniform grating: temporal waveforms (left column) and spectra (right column) of the output pulses in the ‘receptor’ mode (solid curves), output pulses in the ‘supplier’ mode (dotted curves) and input pulses to the GACC (dashed curves) for different values of the grating strength (kL).

Following with our analysis of the results in Fig. 1, other interesting temporal shapes can be obtained by simply varying the grating coupling strength. For instance, sequences of multiple optical pulses can be obtained from the uniform GACC (in the receptor mode) by simply increasing the coupling coefficient. In fact, if this coupling coefficient is properly fixed one can synthesize sequences of nearly identical optical pulses. For instance, for kL Z0.785p, the temporal response of the GACC is composed by two consecutive, identical optical pulses

which are well separated by z1.75 ps. It is important to note that each one of the generated individual optical pulses is practically an undistorted replica (in temporal shape and duration) of the optical pulse launched at the input of the GACC device. As another example of this outstanding behaviour of a uniform GACC, for kLZ0.945p, the input optical pulse is re-shaped into a periodic pulse sequence comprising three nearly identical optical pulses (temporal period z1 ps). Again, the generated individual pulses are practically undistorted replicas of the input

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optical pulse. We emphasize that the generation of customized optical pulse trains is important for several applications (e.g. all-optical pulse repetition rate multiplication) and in fact, this topic has been the subject of recent intensive research [2–4]. In fact, pulse repetition rate multiplication where the individual pulses retain the original features (e.g. temporal duration) is not a trivial task. For instance, if BGs are used for this purpose, very complex grating profiles (e.g. multiple grating structures, such as concatenated, sampled or superimposed BGs [2–4]) are required. In our second set of simulations, we assume a GACC with a longer uniform grating of length LZ6 cm and an input

pulse with a FWHM time-width of 1.5 ps. Fig. 2 shows the results corresponding to this second set of simulations, using the same definitions as for Fig. 1. As expected, for sufficiently low coupling coefficients, nearly square temporal waveforms can be synthesized (in the ‘energy receptor’ mode). As an example, for kLZ0.2p, a flat-top pulse with a total temporal duration of Dtz10 ps is generated. The change in duration with respect to the optical pulses generated in our previous example is obviously associated with the increase in grating length. As shown in Fig. 2, sequences of equalized multiple optical pulses can be also obtained from the long uniform GACC (in the receptor mode) by simply increasing the coupling coefficient. For instance, for kLZ0.775p,

Fig. 2. Results corresponding to a GACC with a 60-mm long uniform grating. Same definitions as for Fig. 1.

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the temporal response of the GACC is composed by two consecutive, identical optical pulses which are well separated by z6 ps. As another example, for kLZ0.96p, the input optical pulse is re-shaped into a periodic pulse sequence comprising three nearly identical optical pulses (temporal period z3.75 ps). The observed temporal evolution in the ‘supplier’ and ‘receptor’ modes is general for any arbitrary uniform GACC, and the whole cycle (e.g. square-double pulse-triple pulse) is completed along each grating section of length L so that kLZp. This is true as long as it is ensured that the device is operated within the so-called ultrashort pulse propagation regime (where the bandwidth of the input pulse is broader than the GACC transmission bandwidth). In general, the fact that one can achieve optical pulse shapes with customized picosecond/subpicosecond features by simply propagating an ultrashort optical pulse through a simple and practical uniform GACC, where the features of the generated temporal structure (temporal shapes, durations, etc.) can be engineered by suitably fixing the physical parameters of the grating structure (grating length and coupling coefficient), would definitely confirm the strong potential of GACCs for optical pulse shaping/processing applications and should stimulate future research in this direction.

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Acknowledgements This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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