1. Introduction

A temporal integrator is a device that can calculate the integral of an input arbitrary time-domain waveform. Temporal integrators are fundamental basic blocks in many signal processing operations of interest, e.g., in computing, control, sensing and communication networks [1,2]. Temporal integrators in the electronic domain are limited to operation frequency bandwidths (processing speeds) typically smaller than a few GHz and they can only process real signals. In contrast, their photonic counterparts [2–11] can operate on the complex-field envelope of arbitrary optical signals with bandwidths easily above a few hundreds of GHz, well beyond the reach of any analog or digital electronic solution. Photonic temporal integrators have already been proposed for various interesting applications, including ultra-short pulse shaping [2–4], ultrafast computing of differential equations [2,5], all-optical memories [6] and photonic analog-to-digital (A/D) conversion [12]. Schemes for performing temporal integration of optical signals are based on either frequency or time-domain designs [2–11]. Solutions based on frequency domain designs emulate the target spectral response of an ideal integrator over a limited frequency bandwidth and they typically use active [3–6] or passive [7] optical resonant cavities (i.e. discrete-time filtering designs). As a main drawback, these methods exhibit a fundamental limitation on the operation frequency bandwidth (typically smaller than a few tens of GHz), which is inherently limited by the characteristic free-spectral-range of the resonant cavity. Ideally, active resonant cavities [3–6] can however provide accurate temporal integration over an unlimited time window.

Time-domain designs for photonic temporal integration [2,8–11] are based on the use of a passive optical linear filtering device approaching the temporal impulse response of an ideal integrator (unit-step function) over a finite time window. A particularly simple and practical solution is that based on a weak-coupling uniform fiber Bragg grating (FBG) operating in reflection [8,9]. This technique does not have a ‘fundamental’ limitation on its operation frequency bandwidth – in practice, this will be limited only by FBG technology constraints – and integration bandwidths up to a few hundreds of GHz have been experimentally demonstrated [9]. However, as a main critical drawback, similarly to any passive filtering solution for temporal integration, a uniform FBG integrator operates over a limited time window, which is fixed by the round-trip propagation time along the FBG length [8]. Besides the intrinsic time-limited performance of this integrator, it is challenging to fabricate low-reflectivity uniform FBGs with the desired performance (e.g. large operation frequency bandwidth and flat-top shape in the temporal impulse response) for grating lengths above a few cm (see section 2 for more discussions on this issue). To give a reference, a previously demonstrated 5-mm long uniform FBG integrator, capable of integrating optical signals with time-features as fast as ~6-ps, was limited to an operation time window of ~50 ps [9].

In this work, we propose and experimentally demonstrate a new ultrafast photonic temporal integrator design capable of providing simultaneously (i) a high processing bandwidth, as large as that of the FBG time-limited integrator solution [2,8–11], and (ii) a long operation time window, as long as that of a discrete-time photonic temporal integrator (resonant cavity design) [3–7] Our proposal is based on the concatenation in series of these two previous designs, namely a discrete-time photonic temporal integrator and a time-limited ultrafast passive integrator (e.g. uniform FBG). Generally speaking, a discrete-time photonic temporal integrator is a linear optical filtering element (e.g. coherent optical resonant cavity) capable of generating a periodic sequence of N amplitude and phase equalized time impulses in response to an input temporal impulse (N = 2, 3, 4 …). We demonstrate that if the time period of the resonant cavity is properly fixed, then this simple scheme enables increasing the operation time window of the original time-limited ultrafast optical integrator by N times without affecting the processing bandwidth of the original time-limited integrator (i.e. simultaneously offering the ultra-high processing bandwidth of the time-limited integrator). If an active optical resonant cavity, with N → ∞, were used, then an unlimited integration time window could be achieved. For proof-of-concept demonstrations, here we have experimentally extended the operation time window of a 5 mm FBG integrator, capable of accurate integration of optical signals with time features as fast as ~6 ps, by four times, from ~50 ps (original value) to ~200 ps, using a cascaded coherent two-arm interferometer operating as a 4-point discrete-time optical integrator.

2. Time-limited passive photonic temporal integrators

Briefly, the temporal impulse response (complex envelope) of an ideal integrator, h(t), i.e. response to the temporal impulse δ(t) (Dirac-delta), is proportional to the unit step function u(t) [1]

 

Fig. 1 Effect of the limited spatial grating profile resolution on the reflection field spectrum (a) and temporal impulse response (b) of a 5 mm long weak-coupling FBG integrator. The ideal responses of the FBG integrator are plotted with red-solid lines and the results for the cases of grating spatial resolutions of 0.1 mm, 0.5 mm and 1 mm are plotted with blue-dotted, green-dashed and brown-dash-dotted lines, respectively. The plots are in normalized units (n.u.). The vertical axis in (a) is in logarithmic scale.

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A photonic temporal integrator based on a weak-coupling uniform FBGs does not have a ‘fundamental’ limitation concerning the input optical signal frequency bandwidth; however, in practice, the FBG integrator’s operation bandwidth will be limited by imperfections in the grating spatial profile induced during the fabrication process. The temporal impulse response of the fabricated FBG deviates from the ideal square-like profile due to the limited spatial resolution and amplitude and period fluctuations in the refractive index modulation profile of the grating [9,13]. As a result, the resulting reflection field spectrum of the actual FBG approaches the target sinc function only over a limited bandwidth around the grating resonance (Bragg) frequency. Thus, the processing bandwidth of the FBG integrator is ultimately determined by the frequency bandwidth over which the grating reflection spectrum approaches the ideal sinc function (notice that this bandwidth is typically larger than the frequency extend of the main lobe in the FBG reflectivity).

As discussed, there is a spatial resolution limitation for imprinting the refractive index modulation of FBGs using current grating fabrication technology. This causes the reflection spectrum of the grating to be filtered by a Gaussian-like profile, which effectively decreases the operation bandwidth of the grating. This effect is simulated here for the case of a 5 mm long weak-coupling FBG integrator (providing an operation time window of T≈50 ps). The effect of limited spatial resolution in the refractive index modulation of the FBG is modeled here by convolving the target grating’s apodization profile (i.e. uniform in this case) by a Gaussian waveform with full width at half maximum (FWHM) equal to the spatial resolution of the grating fabrication process. The reflection amplitude spectrum and temporal impulse response of the simulated grating are plotted in Fig. 1 (a) and (b), respectively. In this figure the results for the cases of 0.1 mm, 0.5 mm and 1 mm spatial resolutions for the grating’s apodization profile are plotted with blue-dotted, green-dashed and brown-dash-dotted lines, respectively. The effect of spectral filtering the reflection spectrum, caused by the limited resolution of the fabrication process, leads to the observed deviation in the reflection side-lobes from the ideal sinc spectrum. These simulations reveal that the poorer the spatial resolution in the grating fabrication process, the narrower the operation bandwidth of the FBG photonic integrator (related inversely to the rising-time of the impulse response) will be.

In [13], the authors have studied the effect of small period or amplitude fluctuations imprinted on the grating during the fabrication process over the spectral response of the filter. The background noise induced by amplitude and period fluctuations in the grating’s refractive index modulation also affects the processing bandwidth of the FBG integrator, as defined above (see for instance, Figs. 6 and 7 in [13]). This processing bandwidth limitation is more critical for longer gratings. To be more concrete, in Fig. 3(a) the spectral response of an (ideal) FBG integrator is plotted with red-dotted line compared to the spectral response of a 4-times longer FBG integrator, plotted with green-solid line. From this figure one can easily infer that if the two fabricated FBGs were to be affected by the same amount of background noise in their reflection spectra, as induced by practical amplitude and period fluctuations in the fabricated grating profile, the processing bandwidth of the integrator (bandwidth support over which the reflection spectrum do not significantly deviate from the ideal sinc function) will be more limited for the longer FBG. This is related with the fact that a longer FBG reaches a lower reflectivity over a narrower bandwidth. Moreover, to ensure operation in the weak-coupling regime, a lower refractive index modulation is necessary as the grating length is increased [14], making longer gratings more vulnerable to the presence of amplitude and period noise in the grating spatial profiles. Hence, even though weak-coupling FBGs can be fabricated with lengths of a few centimeters (corresponding to operation time windows of a few hundreds of picoseconds), the resulting integrator processing bandwidth will be significantly reduced as the grating length is increased. To give a reference, we have measured the processing bandwidth of a 1-cm long weak-coupling FBG integrator presently available in our laboratories (fabricated at the University of Ottawa). This device provides a processing bandwidth of ~100 GHz, corresponding to a background noise level of ~−20 dB.

 

Fig. 6 Experimentally obtained photonic integration (b-d) of a ~100 ps square-like input pulse (a) using the proposed scheme (c-d) compared to the integration using the original integrator (b). Results for the case of a 2-point discrete-time optical integrator (b) and a 4-point discrete-time optical integrator (c) are shown. For comparison, the numerical integral of the input waveform in (a) is also represented (dashed curves).

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Fig. 7 Experimentally obtained photonic integration of a complex-field optical signal using the proposed scheme (red-solid line). Input is an optical waveform consists of two π-phase shifted ultrafast Gaussian pulses each with FWHM duration of ~6 ps (blue-dotted line). For comparison, the numerical integral of the input waveform in is also represented (green-dashed line).

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Fig. 3 Spectral responses of the proposed configuration (green-solid lines) from concatenation of a time-limited integrator (red-dotted lines) and a discrete-time optical integrator (blue-dashed lines): (a) 4-point (N = 4) discrete-time integrator; (b) ideal active optical resonant cavity (N → ∞). The vertical axes are in logarithmic scale. A value of T = 50 ps was fixed for these representations.

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3. Operation principle of the proposed solution for extending the operation time-window of the time-limited integrators

To increase the operation time window of the time-limited FBG integrators, we propose here concatenating in series a discrete-time photonic temporal integrator with the FBG-based integrator, the former having a free-spectral-range identical to the inverse of the FBG integration time window (T), see Fig. 2 . Mathematically, in the temporal domain, the impulse response of the proposed configuration, h(t), can be derived as the convolution of the temporal impulse response of the discrete-time optical integrator, $h_D\left(t\right)\propto {\displaystyle {\sum }_{n=0}^{N-1}\delta \left(t-nT\right)}$ (N = 2, 3, 4 …), with that of the original time-limited integrator, as defined by Eq. (2):

 

Fig. 2 Conceptual diagram of the proposed temporal integrator design.

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The corresponding reflected output from the concatenated uniform FBG (temporal operation window = T) for each impulse in the generated train, is a squared temporal function extending over the FBG operation time window, T (the FBG impulse response is plotted with a dashed red line). In this way, a squared temporal impulse response with a duration N times longer than that of the original FBG integrator is obtained at the device’s output. In the case of using an active optical resonant cavity as the discrete-time optical integrator, in which N→∞, an unlimited operation time window could be achieved using this configuration. Considering the case of using a passive discrete-time optical integrator, the operation time window of the proposed device is still limited (N × T), thus an additional time-modulation mechanism may be used to eliminate the unwanted signals outside the extended operation time window (N × T), e.g. using EOM, as described above.

To find the spectral response of the proposed configuration, H(f), the reflection field spectrum of the time-limited FBG integrator, H_T(f) ∝ sinc(fT)exp(-jπfT) [10], is multiplied by the spectral field response of an N-point discrete-time optical integrator with a free-spectral-range equal to the inverse of the operation time window of the time-limited integrator, $H_D\left(f\right)\propto {\displaystyle {\sum }_{n=0}^{N-1}{e}^{-j2\pi nfT}}$. The ideal spectral amplitude responses of the time-limited and discrete-time (for the case of N = 4) integrators, with the definitions given above, are plotted in Fig. 3(a) with red-dotted and blue-dashed lines, respectively. The spectral response of a system which cascades these two integrators in series, obtained by numerical multiplication of the two individual spectral responses, is plotted in Fig. 3(a) with a green-solid line. In general, the combined spectral response can be analytically derived as follows

In the case of using an ideal active optical resonant cavity, with N → ∞, as the discrete optical integrator, the spectral field response of the discrete-time integrator can be expressed as H_C(f) = 1/(1-e^-j2πfT) [7] (plotted in Fig. 3(b) with a blue-dashed line) and the frequency response resulting from concatenation of this cavity with the original time-limited integrator is

4. Experimental demonstrations and discussions

The experimental setup used for proof-of-concept demonstrations is shown in Fig. 4 . A weak-coupling 5-mm long uniform FBG operated in reflection was used as the original time-limited integrator with an FBG bandwidth of Δλ ≈8 nm around the FBG central wavelength of λ₀≈1550.5 nm. The coherence time of this FBG integrator is τ_c ≈λ₀ ²/cΔλ = 1 ps where c is the speed of light in the vacuum. The grating physical length fixed a maximum integration time window of T ~50 ps. This same grating was previously demonstrated for temporal integration of ultrafast optical signals with time features as fast as ~6 ps, corresponding to a processing bandwidth of a few hundreds of GHz [9]. In a first experiment, we examined the ultrashort impulse response of a photonic integrator implemented according to the proposed idea, providing up to a 4-fold increasing in the device’s operation time window. A passively mode-locked wavelength-tunable fiber laser (Pritel Inc.) was used as the input pulse source, which generated nearly transform-limited Gaussian-like optical pulses, each with an FWHM time duration of ~6 ps, at a repetition rate of 16.9 MHz. The ultra-short laser pulses were spectrally centered at the FBG resonance wavelength (~1550.5 nm). As shown Fig. 4, a passive 4-point discrete-time optical integrator was implemented using two cascaded interferometers (I₁ and I₂ in the figure) with relative delays Δτ₁ ~Δτ₂/2 ~T ~50 ps. The relative time delays of the two stage interferometric setup were adjusted accurately using high-resolution actuators (with 30 nanometer spatial resolution) (Newport Inc.) to generate four in-phase replicas of the input ultra-short impulse with a repetition period equal to T. The time period of the input optical carrier at a wavelength of ~1550.5 nm is ~5.2 fs. Since this period is much shorter than the coherence time of the FBG integrator (~1 ps), the copies of the output signal are coherently superimposed. This shows that the proposed integrator is working as a coherent integrator.

 

Fig. 4 Experimental setup for the proof-of-concept demonstration of the proposed photonic integration design.

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Notice that if used separately, this concatenated interferometer setup could be employed as a temporal integrator only over signals with a frequency bandwidth notably smaller than the device free-spectral-range of ~20 GHz (i.e. with time features longer than ~50 ps). The temporal output intensity was then detected with a 40 GHz photo-detector connected to a high-speed sampling oscilloscope (measurements regarding the ultrafast time response of the original FBG integrator (being capable of processing time features as short as ~6 ps) were reported in [9] using Fourier transform spectral interferometry (FTSI) [15]). The measured ultrashort impulse responses corresponding to the original FBG integrator (i.e. excludinginterferometers), an integrator with twice the original operation time window (i.e. using a 2-point discrete-time integrator, based on a single interferometer, I₁) and an integrator with four times the original operation time window (using the 4-point discrete-time integrator based on two concatenated interferometers, I₁ and I₂) are plotted with solid-curves in Figs. 5(a) to (c) , respectively. For comparison, the numerically calculated integral of the input temporal impulse waveform is also plotted in each figure with a dashed-curve (shifted from origin for better comparison). As expected, the operation time window of the original FBG integrator was increased up to four times using a passive 4-point discrete-time optical integrator (two cascaded interferometers). Using our experimental setup, we have demonstrated accurate temporal integration of complex-field optical signals with time-features as fast as ~6 ps, over time durations as long as ~200 ps. This translates into a time-bandwidth product (TBP) of ~33.33 which is ~4 times higher than that of the original time-limited integrator.

 

Fig. 5 Experimentally measured (solid curves) ultra-short temporal pulse response of the proposed photonic integrator (b-c) compared to that of the original integrator (a). Results for the case of a 2-point passive discrete-time optical integrator (b) and a 4-point passive discrete-time optical integrator (c) are shown. For comparison, the ultra-short temporal response of an ideal device is also represented (dashed curves). Input is a Gaussian pulse with FWHM ~6 ps, spectrally centered at the FBG resonance frequency.

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In another experiment, we successfully integrated a ~100 ps square-like input optical pulse, i.e. with time duration twice longer than the original integration time window. This temporal waveform was generated by reflecting the input ultra-short laser pulse from a weak- coupling 1-cm long uniform FBG followed by amplification using an Erbium Doped Fiber Amplifier (EDFA). The measured input optical waveform to be integrated (square-like pulse) is plotted in Fig. 6(a). The temporal responses to this input waveform from the original integrator, the integrator with twice the original operation time window, and the integrator with four times the original operation time window are plotted in Figs. 6(b) to (d), respectively. The numerically calculated cumulative integral of the input square-like pulse is also shown with a dash-dotted-brown curve in each figure. The results in Fig. 6 confirm the precision provided by the proposed temporal integration scheme as well as the anticipated 4-fold increasing over the original FBG’s integration time window.

In the last experiment we have proved the capability of this integrator to process complex-field optical signals. In this case the input was generated by temporally cascading two ultrafast Gaussian-like pulses, each with an FWHM duration of ~6 ps. These two Gaussian signals were relatively delayed in the time-domain by ~124.7 ps and also adjusted to be π-phase shifted with respect to each other using a one-stage bulk-optics interferometer and a high-resolution actuator. The measured intensity of the input optical signal using a fast photo-detector (12 ps rising time) followed by a 40-GHz sampling oscilloscope is plotted in Fig. 7 with a blue-dotted line. The measured output optical signal from the proposed experimental configuration (with a 4-fold increase in the operation time window over the original FBG integrator) is showed in this same figure with a red-solid line. Numerical integration of the input bi-polar optical signal is also shown with a green-dashed line for comparison. This later experiment clearly confirms the sensitivity of the newly proposed photonic integrator design to the input phase variation, as expected for a complex-field temporal integrator, over the extended operation time window. Considering that the two consecutive optical pulses are π-phase shifted with respect to each other, the time-domain integral of the trailing pulse waveform compensates for the cumulative integral of the leading pulse waveform, leading to the observed squared-like signal generation with a duration fixed by the relative delay between the two pulses (relative time delay between 2 and 3 times longer than the operation window of the original integrator). As discussed in section 2, the ripples in the temporal response presented in Fig. 7 are due to birefringence and amplitude and phase fluctuations of the imprinted FBG integrator.

To finalize, we notice again that we target the extension of the operation time-window of a coherent optical temporal integration process. As a result, the four replicas of the input signal generated in the cascaded interferometers must be all in phase while also having the exact relative delay fixed by the operation time window of the original integrator. High-resolution actuators in our bulk-optics interferometric setup were necessary to achieve these stringent specifications. We believe that the proposed concept would greatly benefit from implementation in a robust and stable integrated-waveguide format.

Other than the characteristics of the device by itself, the energetic efficiency (EE) of photonic temporal integrators (defined as the ratio between the output signal and input signal energies) is also dependent on the shape, bandwidth and temporal duration of the input optical signals. Detailed information about the EE of the time-limited FBG integrator used in the experiments presented in this work is discussed in [9] and [10]. As for any linear passive filtering process, the EE of a passive device is decreased as the filter’s pass-band spectral width is decreased since a larger portion of the input signal energy is then filtered out. In the scheme proposed in this paper (i.e. a time-limited integrator concatenated in series with an N-point discrete-time optical integrator), the temporal impulse response of the new integrator is N times longer than that of the original FBG integrator and as a result, the corresponding spectral response exhibits a pass-band frequency width that is N times narrower than that of the original FBG integrator. Thus, the EE of an integrator composed by a time-limited integrator concatenated with an N-point discrete-time optical integrator is N times lower than that of the time-limited integrator. In other words, in this new scheme, the longer operation time window for the integrator is achieved at the expense of a reduced power efficiency of the device. This trade-off could be overcome by use of an active resonant cavity for implementing the discrete-time optical integrator.

5. Conclusions

In conclusion, a novel design for temporal integration of ultrafast complex-field optical waveforms with an extended operation time window has been proposed and experimentally demonstrated. This concept is based on cascading in series a discrete-time optical integrator (e.g. coherent optical resonant cavity) and an FBG-based time-limited ultrafast photonic integrator. This new technique combines the advantages of the two cascaded integrators, offering the processing bandwidth of the FBG time-limited integrator and the operation time window of the discrete-time photonic integrator. Proof-of-concept experiments of this proposed idea have been reported, demonstrating up to a 4-fold improvement in the operation time window of an FBG ultra-fast photonic integrator. The proposed scheme would potentially enable implementing complex-field photonic integrators offering operation bandwidths well in the THz range over time windows ideally extending to infinity by use of active optical resonant cavities. This scheme would be particularly well suited for practical implementation in an integrated waveguide platform.