We report the first realization of integrated, all-optical first- and higher-order photonic differentiators operating at terahertz (THz) processing speeds. This is accomplished in a Silicon-on-Insulator (SOI) CMOS-compatible platform using a simple integrated geometry based on (π-)phase-shifted Bragg gratings. Moreover, we achieve on-chip generation of sub-picosecond Hermite-Gaussian pulse waveforms, which are noteworthy for applications in next-generation optical telecommunications.
©2011 Optical Society of America
Research in ultra-high-bit-rate communications, ultra-fast computing and data processing has recently been oriented towards all-optical technologies to circumvent the speed (bandwidth) limitations imposed by electronics [1–3]. A particular emphasis has accordingly been dedicated to integrated optical systems allowing for complete on-chip operations, ideally in a format compatible with well-established electronic technologies [4–6]. However, several fundamental elements capable of performing basic signal processing operations in the optical domain have yet to be developed. In analogy with their electronic counterparts [7–9], arbitrary-order optical differentiators and integrators [6, 10] are key basic building blocks for the development of all-optical circuits devoted to a wide range of important functionalities.
An Nth-order (with N=1, 2, …) all-optical differentiator is a device capable of computing the Nth-time derivative of the complex-field envelope of an arbitrary incoming optical waveform . All-optical temporal differentiators are fundamental photonic signal-processing elements, enabling the implementation of a wide range of critical ultra-fast computing and information processing functionalities [10–19]. A relevant and well-known application of temporal differentiators is that of analog computing circuits dedicated to solving differential equations ; the possibility of implementing these circuits in the all-optical domain translates into potential processing speeds orders of magnitude higher than those enabled by electronics. For this specific application, not only first-order but also higher-order (with N≥2) differentiators are required. Besides their intrinsic interest for computing and data processing applications, all-optical differentiators are of great interest for optical pulse shaping [10, 13–15], and other applications in optical telecommunications [11, 13, 20], such as complex-field analysis and measurement of optical time waveforms , ultra-fast sensing and control , etc.
In the past decade, several different schemes for optical differentiation have been proposed, including the use of transversal photonic filter structures in silica-based circuits , bulk-optics interferometers , fiber grating devices [13–15], silicon micro-ring resonators , and semiconductor optical amplifiers . Unfortunately, most of these solutions suffer from at least one of the following critical limitations: they are either extremely restrictive in terms of processing bandwidth [14, 17], perform intensity-only processing , or rely on either bulk-optics  or all-fiber technology [13–15], thus preventing on-chip integration.
The devices demonstrated here are based on a novel design that combines the advantages of an integrated photonic platform with the linear spectral filtering capabilities offered by phase-shifted Bragg gratings [14, 19]. Through the use of this approach, we report the first realization of fully-integrated ultra-fast all-optical first- and second-order differentiators. In particular, the proposed devices are based on the use of a CMOS-compatible, deeply-etched SOI sidewall waveguide grating technology [21, 22], enabling the experimental realization of coupling coefficients more than an order of magnitude higher than in conventional all-fiber Bragg gratings (FBGs) . This remarkable coupling strength in the fabricated devices allowed us to achieve first- and second-order optical differentiation of sub-picosecond pulses on sub-millimeter length scales. We emphasize that this corresponds to a processing speed in the THz range, which significantly surpasses the capabilities of typical electronic systems (limited to speeds of a few GHz ), and of previously reported integrated-waveguide first-order photonic differentiators (processing speeds comparable to electronics, <10 GHz) , while also greatly exceeding the severe bandwidth limitations associated to phase-shifted FBG differentiators [14, 19]. Moreover, our results also represent the first demonstration of an integrated-waveguide (on-chip), sub-picosecond Hermite-Gaussian (H-G) waveform  generator, which is of fundamental interest for complex all-optical coding for network access applications [11, 25], for the analysis and shaping of ultra-short optical pulse waveforms , or for the generation of high-order dispersion-managed solitons (referred to as soliton molecules) .
2. Spectral response and operational bandwidth of an Nth-order differentiator
Using a transfer matrix method combined with coupled-mode theory , it can be analytically demonstrated  that the spectral transfer function required for the realization of an Nth-order all-optical temporal differentiator  can be provided by the reflection of a uniform multiple-π-phase-shifted Bragg grating, i.e. characterized by an unvarying grating period Λ and a constant coupling coefficient κ (coupling strength per unit length) with N discrete π-phase-shifts along its length. Notice that a π-phase-shift refers to a half-period retardation in the grating period. The desired spectral response of these devices is only achieved within their reflection notch over a limited spectral range around the Bragg resonance frequency, f B. We recall that the spectral filtering response of an Nth-order differentiator is proportional to r~(f−f 0)N, where f is the optical frequency variable and f 0 is the central frequency of the signal to be processed (to coincide with the Bragg resonance f 0=f B). Specific conditions for a particular design are obtained by expanding the complex-field reflectivity of the relevant Bragg grating structure in a Taylor series (around the Bragg resonance) to the required power of the frequency f. It can be shown  that an Nth-order derivative requires (N+1) uniform Bragg grating sections, symmetrically distributed and each separated by a π-phase-shift, as illustrated in Fig. 1 . In particular, for a first-order differentiator (with its reflection depending linearly on the base-band frequency, f−f 0) the Bragg grating segments should have identical lengths (L 1=L −1=L) separated by a single π-phase shift, whereas second-order temporal differentiation (with the spectral response described by a parabolic function of the base-band frequency, f−f 0) can be obtained using a three-segment-grating with the lengths satisfying the condition: L −2=L 2=0.5L 1=L. The total lengths of the devices LT are thus equal to 2L and 4L, respectively.
For the device geometry considered above it is possible to derive analytical formula for the spectral response of our differentiators. The complex reflectivity coefficient r as a function of the optical frequency f, for each of the described phase-shifted Bragg gratings (first- and second-order differentiators) can be approximated by the following functions:Fig. 1.
By expanding these functions using a Taylor series around the resonant wavelength, λB, one obtains:
The operational bandwidth of the devices (defined as the maximal spectral bandwidth along the resonance notch over which the Bragg grating provides the desired filtering response) can be directly enlarged by either increasing the coupling coefficient κ or decreasing the length of the gratings L. However, simulations revealed that the functionality of the devices is optimal (defined by the requirement that reflectivity changes from 0 to 100% within the resonance notch and that the ratio of the operational bandwidth to the FWHM of the reflectivity notch is maximized) when 2κL≈π. This poses limitations on the physical parameters of the devices and consequently, on their overall operational bandwidth (processing speed). For SOI based gratings with κ=1150/cm, the above condition leads to operational bandwidths of ~1nm and ~2nm for first- and second order- differentiators respectively, as shown in Fig. 2 (curves denoted by the subscripts 'A').
Nonetheless, we determined numerically that the optimal condition can be somewhat relaxed in order to increase the operational bandwidth while still achieving the desired differentiation functionality. By compromising on the energetic efficiency (i.e. by accepting that less energy is reflected from the device), it is possible to reduce this requirement such that the intensity reflection amplitude changes from 0 to 70% only (out-of-notch reflection), instead of reaching the 100% peak reflectivity of a fully optimized device. Differentiation can thus be realised using the condition 2κL≈0.5π for first-order differentiators, and 2κL≈0.3π for second-order differentiators, leading to a reduction of the required device length for the same coupling coefficient and the associated wider operational bandwidth. As illustrated in Fig. 2, for a coupling coefficient of 1150/cm an operational bandwidth (marked as Δf2B in Fig. 2) of ~0.7THz (Δλ~5.6nm at 1550nm) for the first- and ~1.9THz (~15.2nm) for the second-order differentiators can be obtained. A further increase of the operational bandwidth, i.e. achieved by reducing the device length and consequently the reflectivity, results in excessive distortions (e.g. attenuation and asymmetry) in the output temporal waveforms.
3. Experimental results
We experimentally demonstrated temporal differentiation by using π-phase-shifted waveguide Bragg gratings fabricated via deeply etching the ridge waveguide sidewalls (see Fig. 3c ), allowing for an unprecedented control of the (extremely high, as previously stated) coupling coefficient [21,22]. The waveguides were fabricated on a standard SOITEC SOI wafer, with a 220-nm-thick silicon (Si) core layer and a 1-μm-thick silica (SiO2) lower cladding on a silicon substrate; see Fig. 3b. Electron-beam lithography (into a negative tone hydrogen silsesquioxane mask), and subsequently SF6 reactive ion etching were used to form the single mode ridge waveguide (w=500nm) with a periodic perturbation along the propagation direction [21,22]. A uniform ~500-nm-thick coating layer of spin-on glass was deposited over the entire sample (top layer shown in Fig. 3b). The designed grating structures possess a periodic relief along the waveguide sidewalls (with a period of Λ=316nm, corresponding to a Bragg wavelength of ~1550nm) with specifically located phase shifts (Fig. 3a). A sinusoidal shaped grating perturbation was implemented in order to minimize scattering losses, while maintaining the required high coupling coefficient. In the fabricated devices, a phase shift was defined lithographically as a non-perturbed waveguide section of length Λ/2. The grating coupling strength could be controlled directly by adjusting the recess depth, d, as presented in Fig. 3d.
A wavelength-tunable TE-polarized CW laser was first used to obtain the spectral intensity transfer function of the devices. Light was coupled into the waveguides via a standard end-fire rig setup by way of a 50x microscope objective lens. Using a fixed input power of ~7mW, a Newport power meter (model 1830C) was employed to record the output power as a function of the wavelength (with a wavelength resolution of 10pm).
The spectral transfer functions of the grating devices were measured for the first- and second-order differentiators and are presented in Figs. 4a and 4b, together with their simulated counterparts (red lines). The observed high frequency fringes are a result of Fabry-Pérot (F-P) oscillations induced by the waveguide facets. The resonance notches exhibit bandwidths of ~4nm and ~3nm, for the first- and second-order differentiators, respectively.
By analyzing the F-P fringes both outside of the reflection band and in the reflectivity notch, it is possible to estimate the transmission losses of the full system (composed of an input waveguide, a grating section and an output waveguide) . We obtained overall loss values of around 2-3dB/cm, as typically reported for propagation losses in similar SOI waveguides . This implies that any additional loss arising from the grating section (with a typical length of tens of microns) is very minor. Moreover, since the chip length is 4.5mm, the full on-chip losses are approximately equal to 1dB. The coupling losses are estimated to be around 10dB per facet, which can be reduced to ~2dB when polymer inverse taper spot size convertors  are used.
The experimental verification of the differentiation functionality of the integrated devices was performed by processing optical pulses from a passively mode-locked fiber laser (PriTel, Inc.) operating at a repetition rate of 16.9MHz. Autocorrelation measurements were used to verify the nearly transform-limited assumption of the pulses under differentiation. These traces are shown in Fig. 5a and 5d (black lines). By experimentally recording the power spectra of the input pulses, and numerically taking an inverse Fourier transform of their square root (no chirp assumption), we were able to obtain the time domain profile of the pulses and simulate the autocorrelation. These traces are also shown in Fig. 5a and 5d (in red), and agree well with the autocorrelation obtained directly (i.e., experimentally).
The time-domain complex-field response of the pulse reflected from the device was indirectly analyzed from power spectrum measurements using a Fourier Transform Spectral Interferometry (FTSI) technique  applied in reflection, in which the input pulse itself was used as the reference signal. This simple method has been extensively used in the recent past for time-domain characterization of linear optical pulse shaping and processing devices [10, 11, 13]. In our case, free-space beam splitters were employed to interfere the input signal with the beam reflected from the device, and the interference pattern was recorded on an optical spectrum analyzer (OSA). Complete (amplitude and phase) temporal information about the differentiated signals was found via suitable numerical post-processing (allowing for the separation of the actual grating response from the F-P oscillations induced by the waveguide facets). Further details can be found in  and references therein.
The results for the first- (Figs. 5b-c) and second-order (Figs. 5e-f) differentiation were obtained by processing pulses with intensity FWHM of around 4nm and 3nm, respectively, in order to match the operational bandwidths of the devices, presented in Figs. 4a and 4b. For comparison, analytically calculated temporal derivatives of ideal 700-fs and 1250-fs Gaussian input pulses are also shown (dashed lines) in Figs. 5b and 5e, respectively. Overall, the measured results are in good agreement with their simulated counterparts, although slight asymmetries and incomplete π-phase-shifts in the recovered differentiated temporal signals can be observed. This is most likely due either to a degree of asymmetry in the input pulse temporal profile (please note that ideal Gaussian pulses were used in the theoretical calculations) or because of inexact π-phase-shifts in the final fabricated gratings. More generally, distortions in the temporal profile of the differentiated signal may also arise from excessive deviations from the 2κL≈π condition, from the use of a pulse bandwidth exceeding that of the device operational bandwidth, and/or from using a pulse with a central wavelength differing from the Bragg wavelength. Nevertheless, most of these circumstances can be eliminated by careful control of the fabrication and experimental conditions.
It is also worth to underline that the results reported above also represent the first generation of sub-picosecond H-G pulse waveforms from an integrated device. These waveforms are highly desired for advanced coding and soliton transmission experiments in fiber-optics communication links [11, 24, 26]. As shown in the reported experiments, temporally orthogonal second- and third-order H-G wave-packets (Figs. 5b-c and Figs. 5e-f, respectively) can be obtained by first- and second-order differentiation of an input Gaussian pulse, respectively (the Gaussian itself is the first-order H-G waveform) [11, 24, 30]. (Sub-) picosecond H-G optical waveforms were previously generated using bulk-optics [11, 26] or all-fiber platforms , and hence, our experiments represent the first generation of these important optical pulse shapes using a compact and efficient integrated-waveguide device. By using a higher-order design (i.e. with 4 or more Bragg grating segments separated by π-phase-shifts, see Fig. 1), higher-order mutually orthogonal H-G waveforms could be directly generated on chip.
In conclusion, we have reported the first experimental demonstration of first- and higher-order ultra-fast all-optical temporal differentiation obtained in a CMOS-compatible integrated photonic platform (SOI waveguide Bragg gratings). A high coupling coefficient in the fabricated Bragg gratings, achieved via deep-etching of the ridge SOI waveguide structures, has enabled the realization of very wide operational bandwidths (allowing for processing speeds in the terahertz range). These devices are capable of performing temporal differentiation at least two orders of magnitude faster than today's state-of-the-art electronic-based systems. Optical differentiation and the sub-picosecond H-G waveforms generated in the reported experiments are extremely promising for ultra-fast all-optical computing and information processing, ultra-high-bit-rate optical communications, sensing, advanced optical coding and optical pulse shaping. The presented technique based on π-phase-shifted gratings can easily be extended and implemented for temporal differentiation of arbitrarily higher order, as well as for partial-order (i.e. fractional) differentiation . Moreover, devices with a similar geometry but operating in transmission can be potentially used as all-optical temporal integrators .
The authors would like to acknowledge support from: NSERC (Natural Sciences and Engineering Research Council of Canada), in particular the NSERC Strategic GRANT program, FQRNT (Fonds Québécois de la Recherche sur la Nature et les Technologies), and CIPI (Canadian Institute for Photonic Innovations). K. Rutkowska acknowledges a Marie Curie Outgoing International Fellowship (MOIF-CT-2006-039600).
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