An ultrafast all-optical temporal integrator is experimentally demonstrated. The demonstrated integrator is based on a very simple and practical solution only requiring the use of a widely available all-fiber passive component, namely a reflection uniform fiber Bragg grating (FBG). This design allows overcoming the severe speed (bandwidth) limitations of the previously demonstrated photonic integrator designs. We demonstrate temporal integration of a variety of ultrafast optical waveforms, including Gaussian, odd-symmetry Hermite Gaussian, and (odd-)symmetry double pulses, with temporal features as fast as ~6-ps, which is about one order of magnitude faster than in previous photonic integration demonstrations. The developed device is potentially interesting for a multitude of applications in all-optical computing and information processing, ultrahigh-speed optical communications, ultrafast pulse (de-)coding, shaping and metrology.
©2008 Optical Society of America
The implementation of all-optical circuits for computing, information processing, and networking could overcome the severe speed limitations of electronics-based systems [1–3]. The design and realization of photonics equivalents of fundamental devices that form basic building blocks in electronic circuits is a primary step towards the practical realization of all-optical circuits. Two very relevant examples of these fundamental devices are a photonic temporal differentiator [4–7] and a photonic temporal integrator [8–13]. These devices have been identified  as key elements to build up ultrafast all-optical circuits for real-time computing of complex engineering and scientific problems described by differential equations . Ultrafast photonic differentiators with operation bandwidths in the Terahertz range have been recently demonstrated [4–7]. An experimental photonic integrator has been reported only very recently . However, the demonstrated photonic integrator -based on an active resonant cavity design- was limited to operation bandwidths narrower than 100-GHz. An alternative photonic integrator design  capable of providing the performance that is required for future ultrafast applications (e.g. enabling operation bandwidths in the Terahertz regime) has been recently proposed. Here, we report the first experimental realization of this alternative photonic integrator design, demonstrating integration of arbitrary time-limited optical waveforms (pulses) with bandwidths about one order of magnitude higher than those achievable with previous photonics integrators .
A photonic integrator is a device capable of calculating the cumulative time integral of the complex electric field of an incoming arbitrary optical temporal waveform [8–13]. The photonic integrator demonstrated here is based on a recently proposed design  using a well-known and widely applied all-fiber component, namely a simple uniform fiber Bragg grating (FBG). A FBG is a fiber-optic component with a periodic perturbation of the refractive index that is permanently photo-induced along the length of a single-mode optical fiber . A uniform FBG operated in the linear regime behaves as a band-pass filter, reflecting only a narrow spectral band around its characteristic Bragg frequency. We have recently shown  that a weak-coupling uniform FBG operated in reflection provides the time integral of the electric field of an input arbitrary optical signal (i.e. temporal complex envelope) over a limited time window, which is fixed by the round-trip light propagation time along the total grating length. Currently, high-quality uniform FBGs with lengths above 10 cm, which corresponds to a round-trip propagation time (i.e. integration time windows) above 1 ns, can be routinely fabricated. More importantly, in contrast to the previous experimental integrator, which was based on resonant cavity filters [8–10, 13], the FBG-based integrator design does not exhibit a fundamental limitation in regards to the bandwidth of the optical signals to be processed. Hence, this approach is particularly well suited for application in ultrashort optical pulse generation and processing (ultra-broadband time-limited waveforms).
2. Operation principle
The FBG-based integrator is based on a so-called time-domain design method, where the functionality of an ideal integrator is emulated over a finite time window using a passive linear optical filter  (an ideal integrator would necessarily require the incorporation of an active medium in the device, which typically translates into a more sophisticated device implementation [8–10, 13]). Considering that the temporal impulse response of an ideal integrator (response of the device to an input ultrashort temporal impulse launched at the time t=0) is proportional to the unit step function u(t) (u(t)=0 for t<0; u(t)=1 for t≥0) [9, 13], it has been anticipated that a finite-time integrator can be implemented using a linear optical filter providing a constant temporal impulse response over a certain limited duration (operation time window of the integrator) . As mentioned above, this required impulse response can be practically achieved from a weak-coupling uniform FBG working in reflection , see Fig. 1(a). An FBG operates in the weak-coupling regime when κL≪π, where κ is the grating coupling coefficient and L is the grating length, which typically translates into a peak reflectivity lower than 10%. A weak-coupling uniform FBG provides a nearly constant temporal impulse response over a duration fixed by the round-trip propagation time Th through the grating length, i.e. Th≈2na ν L/c, where naν is the average refractive index of the fiber grating and c is the speed of light in vacuum . In this scheme, the optical signals to be processed must be spectrally centered at the Bragg frequency of the FBG, i.e. ω 0=cπ/(naνΛ), where ω 0 is the carrier optical frequency of the optical signals (both at the input and at the output of the integrator) and Λ is the grating period.
The FBG integrator demonstrated here was tested by processing different input pulse waveforms, including the following (see Fig. 2): (a) A transform-limited Gaussian optical pulse; (b) An odd-symmetry Hermite-Gaussian (OS-HG) pulse, obtained as the first-order derivative of the Gaussian pulse in (a); and (c) A symmetric double-pulse waveform, consisting of two temporally separated in-phase replicas of the same Gaussian optical pulse, and an odd-symmetry double-pulse waveform, consisting of two temporally separated out-of-phase replicas of the same Gaussian pulse. Fig. 2 shows the theoretical input waveforms and the expected outputs from an ideal uniform FBG-based photonic integrator. As discussed above, the cumulative time integral of each of the considered input waveforms can be obtained by simple linear reflection in the uniform FBG. This integral extends over a finite time window (illustrated in Fig. 2), as determined by the grating length. An important consideration in the design of an FBG-based integrator is that the optical signal at the integrator output is not necessarily zero outside the integration finite time window. For example, whilst the ideal integration of a signal containing a DC component should provide an infinite linear increase of output amplitude in time, the FBG passive method proposed here actually provides an output signal with an ending amplitude identical to its starting amplitude; this is achieved through the mentioned time-inverted integration process, perfectly compensating for the direct time-limited integration process. The odd-symmetric double pulse example shown in Fig. 2(c) clearly illustrates how the non-zero amplitude of the finite-time direct integration process is actually set back to the original idle amplitude through the subsequent time-inverted integration process. We emphasize that this feature is associated with the fact that a simple passive device is used to emulate a signal processing functionality (temporal integration) that intrinsically requires the use of an active (gain) medium. In particular, it can be easily shown that the complex envelope of the output waveform after the integration time window is actually given by the time-inverted cumulative integral of the time-inverted version of the input temporal complex envelope . If necessary, an additional temporal modulation mechanism may be used to extract the valid integrated signal (extending over an interval of duration Th) from the full temporal pattern. Possible practical implementations include electro-optic pulse pickers or nonlinear optical switchers. The associated processes, including the required time synchronization, may increase the complexity of the integration system
Fig. 2(a) shows that the well-known flat-top output temporal response of a weak-coupling grating to a broadband Gaussian-like input pulse can be easily interpreted on the basis of the newly described integration property of a uniform FBG. As illustrated in Fig. 2(b), the cumulative time integral of an OS-HG pulse (first-order derivative of a Gaussian pulse) is the input Gaussian pulse itself (integration is the complementary operation of differentiation); an out-of-phase replica of this pulse should be also obtained outside the temporal integration window. Concerning the double-pulse waveforms (Fig. 2(c)), the integrator will simply sum up the area under the two pulses in time when the two pulses are in phase, resulting in two consecutive steps respectively corresponding to the integration of the leading pulse and the subsequently added integration of the trailing pulse. In contrast, when the two input consecutive pulses are out of phase, the second optical pulse will compensate for the cumulative time integral of the first pulse, leading to a square-like (flat-top) output time profile with a duration fixed by the input inter-pulse delay. In any of these two cases, a timeinverted replica of the finite-time integration temporal pattern is again obtained after the integration window.
3. Experimental results and discussions
A weak-coupling 5-mm long uniform FBG sample was used for our experimental demonstrations. The grating physical length corresponds to an integration time window of ≈50 ps. The FBGs used in our experiments were written using a frequency-doubled argon-ion laser operating at 224 nm with a uniform phase mask having a period of 1071 nm. To increase the photosensitivity, the single mode fiber (SMF-28, Corning Inc.) used for the FBG fabrication was hydrogen-loaded for two weeks. To implement an FBG-based optical integrator, the coupling coefficient (defined as the optical power coupling between counter-propagating modes per unit of length) has to be sufficiently low to ensure that the FBG reflection temporal impulse response keeps constant over the duration corresponding to the total grating coupling region (total grating length). Operation in the weak-coupling regime was ensured by keeping the FBG peak reflectivity lower than 10%, which was achieved by control of the UV exposure time during the fabrication process.
The operation of this FBG as an ultrafast photonic integrator was demonstrated following the strategy defined in Fig. 1(b). We emphasize that the FBG needs to be operated in reflection, which requires the use of an additional device, e.g. optical circulator, as shown in Fig. 1(d), to retrieve the reflected signal. The inset of Fig. 1(d) shows the reflectivity (intensity of the reflection spectral response in log scale) of the tested FBG integrator with a 3-dB reflection spectrum bandwidth of 0.14 nm. In our experimental demonstrations, a passively mode-locked tunable fiber laser (Pritel Inc.) was used as the input pulse source, which generated nearly transform-limited Gaussian-like optical pulses, each with a Full-Width at Half-Maximum (FWHM) time duration of ≈6 ps, at a repetition rate of 16.7 MHz. The FWHM of the laser pulse spectrum centered at 1550.8nm was measured to be 0.47 nm. Notice that the FBG was mounted on a fiber holder and a weak axial strain was applied to the FBG with the aim of adjusting the resonance wavelength to coincide with the carrier wavelength of the input optical signal in each experiment.
As mentioned above, the Gaussian-like pulse directly generated from the fiber laser was re-shaped into various different waveforms using an optical pulse shaper based on a single two-arm (Michelson) bulk-optics interferometer [7, 17], see Fig. 1(c). These properly re-shaped waveforms were used as the input optical signals in the tested FBG integrator. A schematic is shown in Fig. 1 (c). The input and the output beams were collimated and focused, respectively, by collimating lenses. A non-polarizing cube beam splitter was used to split and combine the beams. A fiber-optic polarization controller (not shown in the diagram) was used in front of the interferometer to properly select the polarization axis of the beam splitter as it was slightly polarization dependent. For generating an OS-HG waveform from the input Gaussian-like optical pulse, a temporal differentiation process was implemented ; for this purpose, the inter-arm relative delay in the Michelson interferometer was fixed to achieve a destructive spectral interference at the center wavelength of the input Gaussian-like pulse. In this case, the spectral transfer function of the interferometer approximates the spectral response of an ideal differentiator over a bandwidth of about 30% of the spectral period between destructive interferences (i.e. 30% of the interferometer’s free-spectral-range, FSR) ; thus, the relative delay in the interferometer was adjusted to ensure that the resulting differentiation band coincided with the full spectral bandwidth of the input pulse. Concerning the double-pulse generation, the interferometer’s FSR was adjusted to be narrower than the input pulse spectral bandwidth so that the pulse replicas in time were separated. (We remind the reader that the interferometer’s FSR is inversely proportional to the relative time delay between the interferometer arms). In-phase pulse replicas were achieved by tuning the interferometer’s constructive interference to be centered at the pulse carrier wavelength whereas out-of-phase replicas were obtained by tuning the interferometer’s destructive interference to be centered at the pulse carrier wavelength.
For full (amplitude and phase) time-domain characterization of the ultrafast pulse waveforms, we used a Fourier-transform spectral interferometry (FTSI) method [18, 19] where the input optical pulse itself was used as the reference pulse. Briefly, a Mach-Zehnder interferometer was formed by two fiber couplers; the input pulse shaper (followed by the integrator for output pulse diagnosis) was located in one of the inteferometer’s arms together with two polarization controllers for polarization control of the light incident to the (shaper + integrator) and to the interferometer. The spectral pattern resulting from interference between the input (reference) optical pulse and the signal at the shaper (+ integrator) output was captured at the interferometer output using an optical spectrum analyzer (OSA); this interference pattern was used for the full (amplitude and phase) reconstruction of the waveform generated at the differentiator output. For this purpose, the well-known FTSI algorithm was applied .
Figure 3 (a–d) show the measured temporal complex envelopes of the input and output optical signals in some of the performed experiments. The results in Fig. 3(a) are for the integration experiment of an OS-HG pulse. This pulse was obtained by first-order time differentiation of the input Gaussian-like pulse (laser pulse) using the Michelson interferometer; for this purpose, the interferometer was configured to provide a relative delay between its two arms shorter than the input pulse duration and with a relative π-phase shift between them . The generated OS-HG waveform, Fig. 3(a)-i, consisted of two π-phase shifted consecutive temporal lobes, each with an individual FWHM time-width of ≈7.5 ps, temporally separated by ≈21.8 ps. As expected, the reflected waveform from the FBG, Fig. 3(a)-o, over the integration window (marked with a gray box) was a nearly transform-limited (constant-phase) Gaussian-like pulse, almost identical to the original laser pulse, in good agreement with the theoretical predictions in Fig. 2(b)
In fact, the experimentally measured temporal pulse profile agrees very well with the numerical time integral of the measured input OS-HG pulse field (shown with red hollow circles). The results shown in Fig. 3(b) and 3(c) correspond to integration of two odd-symmetry double-pulse waveforms, each consisting of two π phase-shifted replicas of the 6-ps (FWHM) input laser pulse, with different time separations (28.3 ps and 35.9 ps). Obviously, the inter-pulse time separation (determined by the relative time delay between the interferometer arms) could be easily tuned with our free-space interferometer setup. The reconstructed temporal waveforms at the FBG integrator output are shown in Fig. 3(b-o) and Fig. 3(c-o), respectively, showing again an excellent agreement both with the numerical integrations of the corresponding measured input pulse waveforms (over the indicated integration time window) and with the theoretical predictions in Fig. 2(c). As expected, flat-top optical pulses were generated over the integration time window, each with a different FWHM time duration, as fixed by the time separation between the input odd-symmetry twin pulses. Finally, we also tested the integration of a symmetric double-pulse structure, consisting of two in-phase replicas of the 6-ps (FWHM) input laser pulse temporally separated by ≈30.5 ps, results shown in Fig. 3(d). The obtained temporal pulse profile is in excellent agreement both with the finite-time numerical integration of the measured input waveform and with the theoretically predicted temporal profile in Fig. 2(c). Notice that the field integral of the two identical in-phase pulses is twice the field integral of the single pulse and as a result, in terms of the measured average optical intensity (i.e. square of the field waveform integration), the second step should be three times higher than the first one. The reported results clearly confirm that the developed FBG-based integrator operates on the complex envelope (amplitude and phase) of the incoming optical waveforms. Notice also that the time-inverted replicas of the integrated waveforms in the output signals should all have the same intensity levels as the integrated waveforms. The observed relatively lower intensity levels in these replicas may be fundamentally associated with artifacts introduced by the used FTSI pulse waveform measurement method.
The energetic efficiencies of the reported integration examples, i.e. integrations of an OS-HG, an odd-symmetry double-pulse and a symmetric double-pulse, (defined by the ratio of the output time-average power to the input time-average power) were measured to be 0.3%, 0.4%, and 1.1%, respectively. This low energy efficiency is due to the fact that the integration signal is reflected from the weak-coupling FBG, while the non-integrated signal transmits through the FBG, (i.e. this integrator utilizes only a small portion of the input signal power). Moreover, the energetic efficiency of the integration process is degraded when processing an optical signal with a broader spectral bandwidth , which practically limits the maximum signal bandwidth that can be processed with a FBG of a given length.
Temporal integration is the signal processing counterpart of temporal differentiation. Photonics differentiators have already proved very useful for a wide range of applications in ultrafast optical pulse processing and coding [6, 7, 20, 21], shaping [22, 23] and metrology [24, 25]. Applications in a similar range of fields can be anticipated for a photonic integrator [9–13]. For instance, the proof-of-concept experiments presented above suggest various interesting specific applications for the realized photonic integrator, namely reconfigurable flat-top pulse generation and phase-encoded pulse pattern recognition. The generation of ultrashort flat-top temporal intensity profiles is highly desired for a range of non-linear optical switching and frequency conversion applications [23, 26]. In these applications, the flat-top pulse time width needs to be optimized according to the features (e.g bit rate) of the transmission link. While several FBG-based designs have been demonstrated for flat-top pulse shaping [23, 26], none of these previous designs allows for flat-top pulse width tuning. A very simple mechanism for reconfigurable flat-top pulse generation based on the integration property of a uniform FBG has been demonstrated above, see results in Fig. 3(b) and (c). A similar approach enabling reconfigurable flat-top pulse generation has been also proposed in Ref. . In particular, a flat-top optical pulse can be generated by simple reflection of an odd-symmetry double-pulse waveform in a uniform FBG; the flat-top width can be easily reconfigured by simply tuning the time separation between the input twin pulses. Concerning the second mentioned application, in view of the behavior described in Fig. 2(c) and the corresponding experimental demonstrations in Fig. 3(b)–(d), it is obvious that the integrator response (output intensity profile) to a sequence of consecutive input pulses depend on the discrete phase shifts among these pulses, thus suggesting a direct and simple mechanism for recovering the phase keys encrypted in a sequence of intensity bits.
We emphasize again that the importance of an ultrafast optical integrator goes far beyond the above discussed applications. In direct analogy with the electronic-domain developments [27, 28], a photonic temporal integrator is the key element for implementing all-optical computing systems aimed to real-time solving of scientific and engineering problems that can be described by differential equations . The possibility of realizing these computations all-optically translates into a potential for speeds several orders of magnitude higher than with conventional electronic systems. The photonic integrator design demonstrated here offers the performance (e.g. in terms of processing speeds/bandwidths) that is required for future ultrafast all-optical computers.
We have reported the experimental demonstration of an ultrafast photonic (all-fiber) integrator capable of overcoming the speed (bandwidth) limitations of the previously demonstrated photonic integrator design, which was based on an active resonant cavity filter . In particular, we have demonstrated temporal integration of a variety of optical waveforms with temporal features as fast as ≈6-ps, which is about one order of magnitude faster than in previous photonic integration demonstrations . Moreover, the demonstrated design is based on a very simple and practical solution only requiring the use of a widely available all-fiber passive component, namely a reflection uniform FBG. The demonstrated functionality should prove very useful for a wide range of applications in all-optical computing and information processing, ultrahigh-speed optical communications, ultrafast pulse (de-)coding, shaping and metrology.
This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through its Strategic Grants Program.
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