1. INTRODUCTION

Temporal electromagnetic (EM) waveforms are essential to myriad important applications in science and technology [1,2]. In the optical domain, EM waveforms can be efficiently generated and processed over very short time scales, corresponding to frequency bandwidths in the THz range and above [1]. The ability to precisely manipulate and detect the temporal variations of the EM field on such fine time scales is essential to a wide range of fields, such as telecommunications and spectroscopy [1,3]. On the other hand, optical approaches are also increasingly employed for the generation, processing, and/or analysis of narrowband waveforms. Optical signals with bandwidths below a few GHz are crucial to many applications in biomedical imaging [4], radio-astronomy [5,6], remote sensing such as lidar [7], and the broad field of microwave photonics (MWP), concerned with the processing of radio-frequency (RF), microwave, or millimeter-wave signals using photonic solutions [2,8,9]. In applications such as vibration and acousto-optic sensing [10–12], the frequency extent of the signals of interests is often in the kHz range and even lower. Even though optics and photonics enable an efficient access to temporal waveforms across these different regimes, there is a prevalent challenge when it comes to the problem of noise mitigation. Whether dealing with broadband or narrowband waveforms, noise consisting of stochastic intensity and phase variations is unavoidably introduced at the generation, transmission, processing, and/or detection stages of the signals of interest. This noise can arise from many different sources, such as the amplified spontaneous emission (ASE) generated from optical amplifiers [13,14], or the incoherent fluorescence that may occur in a variety of pump–probe experiments [15]. To deal with this undesired component of a waveform, noise mitigation strategies are essential at one stage or another [16]. In many practical cases, noise reduction remains challenging, ultimately limiting the achievable performance or even preventing the realization of the desired task or operation.

The problem of noise mitigation is particularly difficult when dealing with low-energy or “weak” signals, a common situation in many science and technology applications. For example, in long-haul telecommunication systems, the considerable loss from long distance propagation through the optical fiber combined with the injected noise from multiple optical amplifiers [e.g., erbium-doped fiber amplifiers (EDFAs)] causes data signals to become weak and noisy, such that dedicated regeneration stages are required, which are costly and energy hungry [17]. In light-based biomedical imaging applications, the mere fact that a living tissue is imaged poses important limitations on the attainable light powers to avoid damaging the biological sample [18]. Similarly, in lidar applications, high optical powers can pose a threat to the eyes of a passersby [19]. There are many application examples such as the ones mentioned here that are ultimately limited by the ability to detect weak, noisy temporal waveforms.

The principal challenge to retrieve a weak signal that is additionally corrupted by noise lies in the fact that there exists no technique to simultaneously increase the energy of the waveform of interest while reducing its relative noise content. Increasing the amplitude of a signal under test (SUT) through active amplification unavoidably leads to deterioration of the relative noise content of the waveform [13]. Even though remarkable advancements have been reported to achieve noiseless amplification using parametric approaches, the noise figure of 0 dB remains an asymptotic value, and even under an ideal scenario, the noise already contained in the signal will also be amplified. On the other hand, the most widely established technique to denoise a temporal waveform, i.e., bandpass filtering (BPF), is inherently associated with an attenuation of the SUT. Moreover, in the optical domain, BPF is particularly difficult over narrow frequency bandwidths below a few GHz, and as such, this remains a highly active research topic [20–24]. A bandpass filter scheme also requires precise knowledge of the SUT specifications (e.g., bandwidth and central frequency), and it is especially susceptible to drifts in the SUT carrier frequency [20,25,26], an issue that is particularly problematic for narrowband waveforms. Additional signal frequency tracking and feedback mechanisms may need to be incorporated into the scheme to adjust the filter’s central frequency as per the signal specifications [27,28]. Thus, the recovery of a weak, noisy waveform is a long-standing problem [29] of importance across a wide range of applications, but it remains challenging under many practical conditions.

Recently, a new paradigm for noise mitigation in EM waveforms has emerged based on so-called passive amplification using concepts derived from the Talbot self-imaging effect [30,31]. High amplification, up to a factor of 27, has been achieved [30], and the potential of this method for denoising optical signals has been demonstrated and studied in recent work [31]. However, passive amplification based on Talbot effects has been strictly limited to periodic temporal waveforms, such as optical pulse trains. The effect of a passive amplifier in this case can be interpreted as a form of temporal averaging of the repetitive waveforms. Thus, as proposed to date, the method cannot be used on purely arbitrary (e.g., aperiodic) temporal waveforms, such as those found in most practical applications.

In this communication, we present a Talbot-based approach for the passive amplification of arbitrary temporal waveforms through a lossless (energy-preserving) sampling process, which is shown to be ideally suited for the recovery of weak, noise-dominated signals. Specifically, we exploit the phenomenon of a temporal Talbot array illuminator (T-TAI) to transform an incoming waveform into a series of short temporal samples, outlining an amplified copy of the input signal of interest, as depicted in Fig. 1. Pioneered by the works of Lohmann, TAIs were first employed in the late 1980s to focus a uniform spatial optical wavefront into an array of bright light spots, with applications such as for power supply in optical parallel processors [32]. The concept was recently brought into the time domain, for CW-to-pulse conversion and invisibility cloaking [33–35], yet restricted to pulse generation from a uniform light wave (CW). A T-TAI can be practically implemented through discrete temporal phase modulation and dispersive delay. We demonstrate here that if properly designed, a T-TAI preserves the temporal envelope variations of an input arbitrary waveform, while enabling high amplification factors (${\gt}100$ demonstrated herein). Since the TAI operation relies on a coherent energy redistribution of the original signal, the deterministic (phase-coherent) SUT is sampled in a lossless fashion, while the noise is left essentially untouched, implementing a simultaneous passive amplification and noise mitigation of the waveform of interest. In recent work, we showed how a related concept could be used on the frequency spectrum representation of an input waveform, the so-called spectral TAI (S-TAI) [36]. This scheme enables in-band noise mitigation, and in particular, it is of specific interest for denoising the spectral representation of broadband time-limited waveforms, such as short optical pulses. In contrast, the T-TAI proposed here denoises the temporal representation of a waveform, such that it is suitable for application on arbitrary waveforms with no fundamental time restrictions (e.g., infinitely long signals). Furthermore, the scheme can be easily tailored for efficient signal denoising over a broad range of bandwidth specifications.

 

Fig. 1. Narrowband filtering concept. The denoising concept is implemented by a combination of temporal phase modulation and dispersive propagation according to the theory of the Talbot effect. Since the process relies on the coherence properties of the processed waveform, the weak coherent temporal waveform with peak intensity $I$ (blue, SUT) is reshaped into a periodic set of short samples outlining a copy of the input waveform amplified by an integer factor $q$, while the incoherent noise is left untouched. The discrete temporal phase modulation signal, shown in yellow, is implemented here through an electro-optic phase modulator (PM) driven by an electronic arbitrary waveform generator (AWG). The dispersive spectral phase filtering step, shown with a red dashed curve, here implemented by a linearly chirped fiber Bragg grating (LCFBG), realigns the energy into temporal bins of width ${t_s}$, separated by ${t_q = q{t_s}}$.

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We demonstrate the noise mitigation capabilities of the proposed T-TAI scheme through application on a wide range of different, practically relevant optical signals, from ultra-narrowband waveforms (with frequencies down to the kHz range) to high-speed telecommunication data signals (with bit rates up to the GHz range). The T-TAI method significantly outperforms a conventional BPF scheme, allowing for the recovery of noisy signals for which the BPF is simply unable to produce any notable reduction of the signal’s relative noise content. In a record experiment, passive amplification up to a measured factor exceeding 110 is demonstrated, enabling the sub-threshold detection of weak signals buried under a noise content more than 30 times stronger than the signal itself. In our T-TAI implementation (using electro-optic phase modulation), the effective operation passband of the T-TAI can be electronically reconfigured, such that it can be readily optimized for application on signals of different bandwidths. As another important advantage, the T-TAI does not require any a priori knowledge of the central frequency of the SUT, thus being able to provide the desired noise mitigation performance in a purely self-tracking fashion. Said other way, the TAI avoids the issues associated with signal frequency tracking by intrinsically selecting the coherent part of the incoming signal, regardless of its central frequency. In this report, we show how the self-tracking property of the T-TAI can be exploited for multi-wavelength signal processing tasks by simultaneously denoising four temporal waveforms (high-speed data streams) located around four different carrier wavelengths using a single T-TAI device.

2. CONCEPT AND OPERATION PRINCIPLE

The proposed denoising principle is based on a T-TAI involving a suitable temporal phase modulation of the SUT followed by spectral phase-only linear filtering (see Fig. 1). The T-TAI implements a lossless temporal sampling mechanism, such that it produces a sampled version of the input SUT in which the resulting short-pulse samples outline an amplified copy of the SUT’s temporal complex envelope. The concept can thus be understood as a simultaneous denoising and amplification process of the information contained within the SUT, which is readily accessible in the optical domain through the envelope of the output temporal pulses. The T-TAI is reminiscent of the mechanism of temporal compression of CW or quasi-CW light using a time lens (quadratic temporal phase modulation) followed by a dispersive line (quadratic spectral phase filtering) for periodic short-pulse generation [34,35,37–39]. Passive amplification of arbitrary temporal waveforms has been demonstrated using time–lens schemes, but the achieved gain factors (with values limited to well below 10) are largely insufficient for denoising applications [37,38]. In contrast, a TAI is based on discrete phase manipulations and exploits the autocorrelation properties of certain quadratic sequences stemming from number theory [40,41]. Through this process, the required phase manipulations along the time and frequency domains can be limited to a maximum excursion of ${2}\pi$, enabling the practical realization of significantly higher compression (i.e., amplification) factors. The TAI can then be compared to the well-known concept of a Fresnel lens, which also involves wrapping the quadratic temporal phase modulation (a conventional lens) in between zero and ${2}\pi$. However, the practical implementation of Fresnel lenses remains challenging as the requirements on modulation bandwidth are very stringent due to the large phase jumps that result from the wrapping of the phase [34]. For practical implementation of the TAI, one of the two phase manipulations can be made continuous rather than discrete while maintaining the overall focusing effect (see Supplement 1). Specifically, in our proposed scheme, the T-TAI is implemented using a discrete temporal phase modulation followed by a continuous quadratic phase filter, implemented through widely available group-velocity dispersion.

To focus an incoming waveform into bins of width ${t_s}$, periodically separated by ${t_q} = q{t_s}$, outlining a copy of the input temporal profile amplified by a factor $q$, the temporal phase must be modulated quadratically in $q$ discrete steps (bins), each of width ${t_s}$, according to

For realization of this sampling process, the SUT must be coherent in the sense that its temporal phase profile must be sufficiently smooth. Similar to any sampling process, the temporal separation ${t_q}$ between the resulting TAI pulses must satisfy the Nyquist criterion with respect to the fastest variations of the SUT to capture all relevant information [45]. Therefore, assuming that the shortest time features in the input signal are of the order of ${t_{\rm NY}}$, corresponding to a frequency extension (bandwidth) of the order of $\Delta \nu \sim1/{t_{\rm NY}}$, the TAI must be designed such that ${t_q} \le {t_{\rm NY}}$. Thus, the TAI’s frequency passband is approximately determined by $1/{t_q}$. Variations that occur on a faster time scale than this will not be properly captured by the TAI sampling process. Therefore, when one considers the case of stochastic noise, such as the ASE case studied here, the corresponding signal consists of random fluctuations in amplitude and in phase that vary much faster than the Nyquist period; as a result, the TAI-induced addition of these fluctuations will be essentially washed out such that the noise component remains nearly untouched by the TAI process (see Supplement 1 for further discussions on the noise mitigation process). As illustrated in Fig. S1, considering that a TAI is a linear process, the coherent signal of interest (SUT) will be passively amplified through the lossless sampling mechanism, whereas all additive stochastic noise along the input signal will not be processed by the TAI, maintaining its original level. This is the key observation at the core of the proposed denoising mechanism. As both the SUT and noise component undergo an identical insertion loss going through the TAI, the SUT envelope will be amplified over the background noise by the factor $q$. The value of this amplification factor as well as the sampling period (Nyquist bandwidth) can be tailored as per the design conditions defined above.

3. RESULTS

A TAI-based denoising scheme is implemented for application on optical waveforms. As illustrated in Fig. 1 and Fig. S1, the scheme consists of an electro-optic phase modulator (PM), driven by an RF arbitrary waveform generator (AWG), to impart the TAI temporal phase profile designed according to Eq. (1) on the input optical SUT, followed by a linearly chirped fiber Bragg grating (LCFBG) operated in reflection mode to implement the second-order dispersive propagation according to Eq. (2). Note that the T-TAI can be implemented through various other means (e.g., for a higher sampling speed), for example, through nonlinear temporal phase modulation [35] or through the use of discrete spectral phase filters [46]. Further discussions on design considerations of the T-TAI can be found in Supplement 1 through a relevant experimental demonstration on a chirped sinusoidal waveform (see Fig. S2), illustrating the Nyquist sampling frequency trade-offs of a T-TAI. Here, we focus on the key property of the T-TAI sampling mechanism pertaining to its denoising capability. We first demonstrate the potential of the T-TAI for high amplification and recovery of weak, noise-dominated signals in Section 3.A. We then follow with an in-depth quantitative analysis of the performance of the TAI method regarding its signal recovery and denoising features in Section 3.B. We finish this report with a demonstration of the self-tracking property of the T-TAI by simultaneously processing four data signals located at four different optical carriers, as one could encounter in a wavelength division multiplexing scheme, in Section 3.C.

A. Noisy Weak Signal Recovery below the Detection Threshold

We first demonstrate the main feature of the TAI, namely, its simultaneous amplification and denoising capabilities, which allows us to perform the difficult task of extracting weak, noisy signals, such as those often found in many of the abovementioned fields [17–19]. For this purpose, the T-TAI is designed to achieve an ultrahigh amplification factor, namely, $q = {120}$ with ${t_s} = {26}\;{\rm ps}$ (corresponding to a sampling period ${t_q} = {3.12}\;{\rm ns}$), implementing an effective ultra-narrow passband of ${\sim}{320}\;{\rm MHz}$. As further outlined in Fig. S3 and Supplement 1, the T-TAI is implemented with a 40-GHz PM driven by a 120-GSa/s AWG and an LCFBG as a dispersive line offering $\ddot \phi$$= {12{,}930}\;{{\rm ps}^2}$, equivalent to about 595 km of single mode fiber (SMF28). For the example reported here, the optical SUT was generated by electro-optic intensity modulation of a 1550-nm CW laser. These measurements are captured using a 100-GHz photodiode connected to a 70-GHz bandwidth electrical sampling oscilloscope. The experimental amplification factor is measured as 110.3, corresponding to the ratio of the value at the TAI sampling peaks to that of the signal with the PM turned off, both measured at the output of the T-TAI system [Fig. 2(a3)]. To the best of our knowledge, this is the largest amplification factor demonstrated to date using a Talbot-based amplification system. Moreover, by using state-of-the art, yet commercially available, components, we believe that even higher amplification factors could be achieved (see Fig. S4). After including the insertion loss of the T-TAI processor (measured as 4.7 dB), the net gain from input to output is found to be 36.8. The pulse widths of the output samples are measured to be 25 ps, very close to the expected theoretical value of 26 ps. Additional detailed plots of the TAI sampling pulses and the RF signal employed for phase modulation are shown in Fig. S5, while the spectral amplitude and phase responses of the LCFBGs used in the reported experiments are shown in Fig. S6. We note that the output TAI peaks near the edges of the recovered waveform do not follow exactly the shape of the input square waveform. This can be attributed to both the fact that the Nyquist criterion is not satisfied for fast transitions at the leading and trailing edges of the input waveform, and because the SUT duration is not an exact integer multiple of the modulation signal period (see Section 2 of Supplement 1 for further discussions). Nonetheless, this result demonstrates that the amplification effect can still be observed for a SUT of arbitrary duration, allowing for single-shot and real-time processing of an incoming waveform of unknown parameters, within the Nyquist criterion.

 

Fig. 2. Amplification of a weak waveform. A temporal TAI providing a high net amplification factor (${\sim}36.8$) is utilized, and its performance is compared with that of a BPF. Their effects are compared by measuring the signal at various points, namely, (1) at the source, corresponding to the input of the noise mitigation schemes (gray trace), (2) after the BPF (red trace), (3) after the T-TAI apparatus, with the PM off (gray trace) and on (green trace), and (4) after the T-TAI apparatus preceded by the BPF, with the PM off (gray trace) and on (blue trace). (a1) A 10.5-ns isolated pulse with an average power of $-7.5\;{\rm dBm}$ is generated at the input (SUT). (a2) Since the waveform contains little noise, the effect of the BPF results in an insertion loss of only about 3.5 dB. (a3) On the other hand, the TAI redistributes the input waveform into a series of peaks, implementing an experimental amplification by a factor of 110.3 (before insertion loss). An example of a measured output sampling pulse is shown in the inset. Note that all waveforms are normalized to the peak value of the waveform measured at the output of the TAI with the PM off [gray trace in (a3), not visible here due to the linear vertical scaling]. (b) 7.7 dBm of noise is injected in the waveform, lowering the SNR to $-15.2\;{\rm dB}$, as measured by an optical power meter (see detailed definition in Section 4 of Supplement 1). The input signal is effectively buried under noise, and its features cannot be extracted. (c) The power of the input waveform is attentuated to $-21.6\;{\rm dBm}$ such that it is below the threshold of the detector (given as $-20\;{\rm dBm}$ from the manufacturer), and the SNR is maintained at the same level as in (b). Here, the TAI scheme allows to clearly recover and identify the underlying waveform, whereas the BPF approach requires optimal post-processing, i.e., (d) bandpass digital filtering with a very narrow (370 MHz) filter, to recover the input signal.

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We demonstrate the noise mitigation capabilities of the T-TAI by injecting ASE noise with a power level more than 30 times higher than that of the SUT. The noise is spectrally confined to a 10-dB bandwidth of 80 GHz and centered at the SUT’s wavelength (see Fig. S7). The SUT is now buried under so much noise that no features can be resolved [Fig. 2(b1)]. We attempt to denoise this waveform employing a 5-GHz bandwidth fiber Bragg grating (FBG)-based BPF (centered at the SUT’s wavelength; see Fig. S7 for the filter’s transfer function), which is representative of state-of-the-art performance for a narrowband optical BPF. The resulting waveform remains significantly noisy. On the other hand, the obtained results after the TAI setup produces a sampled version of the input waveform that is clearly more visible than the one obtained at the BPF output. This is due to both the narrower effective filtering bandwidth of the T-TAI (${\sim}320\;{\rm MHz}$) and simultaneous signal amplification and noise mitigation implemented by the TAI. Yet, in this case, the best outcome is achieved when the two denoising mechanisms are used in tandem, with the BPF operating first [subplots in the last row, Fig. 2(b4), labeled ${\rm BPF} + {\rm TAI}$]. This combines both the BPF ability to remove a large portion of the out-of-band noise, as well the high amplification and noise mitigation capabilities of the TAI. An in-depth quantitative study of the noise mitigation performance of these different approaches is provided in Section 3.B.

To test the T-TAI on weak, noise-dominated waveforms, the noisy signal in Fig. 2(b) is attenuated by 14.1 dB, such that the SUT is below the detection threshold of the photo-detection system, making it undetectable even without the presence of noise (see Fig. S6). The obtained results are shown in Figs. 2(c) and 2(d). At this point, the SUT is barely discernable unless the TAI is utilized [see results in Fig. 2(c)]. Otherwise, the SUT could be recovered only after an optimal digital filtering of the photo-detected signal [see results in Fig. 2(d)]. The digital filter employed for the input and BPF waveforms has a full width at half maximum (FWHM) of 370 MHz (185 MHz at baseband). This optimal digital filter, enabling severe denoising of the input waveform with minimum alteration of its original square-like shape, was found following a systematic, extensive optimization procedure, assuming a precise a priori knowledge of the input SUT. For the sake of comparison, the T-TAI waveform is also digitally filtered, but with a much larger FWHM of 79 GHz to keep the TAI peaks nearly undistorted, as shown in Fig. S8. An important observation is that the T-TAI returns a waveform with an amplitude more than 30 times larger than the recovered input waveform (normalized peak intensities of 3.6 versus 0.11). Evidently, the visibility of the signal is not improved by the 5-GHz bandwidth BPF, due to the low power of the signal. Additionally, it is important to note that in this case, the use of an active amplifier, such as an EDFA, would also fail to improve the detectability of the signal since the SNR would be further degraded. On the other hand, the TAI processor offers the unique capability to amplify the coherent waveform (the SUT) while decreasing the relative amount of noise, enabling the detection and effective recovery of the signal of interest directly in the wave domain, even without the help of any digital post-processing.

Finally, we note that the T-TAI method requires the use of a detector with a much higher bandwidth (at least $ q $ times higher) than the one needed for a direct measurement of the waveform of interest. This implies that a correspondingly higher amount of noise would be introduced by the detection scheme [47]. However, in our reported work, for demonstration and evaluation of the T-TAI concept, we consider the case of signals affected by an amount of noise that is orders of magnitude higher than the additional noise that may be expected from the use of a larger bandwidth detector. In particular, we estimate that the combined contribution of thermal and quantum shot noise from the photo-detectors used in all reported experiments here is limited to the nW range (versus the µW to mW power levels of the injected noise). Whereas we believe that the proposed T-TAI approach may offer and interesting solution for ultra-low noise detection of weak waveforms, a detailed study of this potential application for the T-TAI concept remains outside the scope of the present communication.

B. Noise Mitigation Performance of the T-TAI

For demonstration and evaluation of the narrowband filtering capabilities of the TAI, we target the recovery of ultra-narrowband signals corrupted by noise and compare the obtained performance against an optical BPF with a bandwidth of ${\sim}5\;{\rm GHz}$. As discussed above, optical BPF for noise reduction remains challenging over narrow bandwidths. Current practical optical filtering technologies (such as the FBG filter used herein) can be hardly designed for an operation bandwidth just below a few GHz. Solutions have been proposed that can overcome this limitation but using more complex, costly, and/or exotic techniques [20–24]; still, to the best of our knowledge, the narrower optical BPF bandwidth demonstrated to date is well within the MHz range [24–26]. A key drawback of a conventional BPF is that this requires very precise knowledge of the central frequency of the SUT for proper alignment between the filter and signal. However, in practice, the SUT frequency may vary over time, such that precise monitoring and control of the SUT spectrum may be required [27,28]. Practical monitoring of the SUT carrier frequency is not always possible, and it is particularly challenging in the case of a narrowband signal. Another approach to deal with the SUT’s varying central frequency or bandwidth is to actively adjust the parameters of the BPF. However, most BPFs are based on mechanical or thermal control that makes them unsuited for dynamic reconfigurability. Alternatively, coherent detection techniques may be employed to achieve narrowband noise mitigation of optical signals. However, these approaches still face the difficulties described above on signal tracking and carrier drift, and they ultimately recover the SUT in the RF domain at an intermediate frequency or at baseband rather than in the optical analog domain. When considering the signal after detection, narrowband processing techniques have been widely studied and applied using baseband RF signal processing, and numerous sophisticated digital signal processing (DSP) techniques [48], such as matched filters [49,50]. However, many of the applications mentioned above require the information to be available in the optical domain, rendering these approaches ill suited. Thus, for the purpose of this communication, we underline the capabilities of the T-TAI to denoise signals directly in the optical domain, a problem that presents the important challenges mentioned above. In any case, we note that baseband and/or digital signal processing could still be used after detection of the waveform processed by the T-TAI to improve further the performance of the noise mitigation scheme. On the other hand, as demonstrated here, the effective bandpass of the T-TAI can be reconfigured electronically so that it can quickly adapt to the SUT’s varying bandwidth, while the SUT’s central frequency is inherently tracked by the TAI mechanism, as demonstrated in Section 3.C.

The narrowband optical signals tested in our experiments have a sinusoidal temporal envelope with frequencies ranging from 100 kHz to 250 MHz, generated through intensity modulation of CW light with RF sinusoids. Three different carrier optical signal-to-noise ratios (OSNRs) are tested, namely, 70.1 dB, 30.1 dB, and 25.4 dB, through the addition of ASE noise filtered to a 3-dB bandwidth of 26 GHz near the carrier frequency [see experimental setup in Fig. S3(b)]. The OSNR values are determined by extrapolating the noise near the carrier frequency, as measured by an optical spectrum analyzer (OSA) with a resolution bandwidth of 5 MHz (0.04 pm) (see Fig. S9 and the detailed definition in Section 4 of Supplement 1). All results presented in the remainder of this paper were captured using a 50-GHz photodetector connected to a 28-GHz real-time oscilloscope. Two different TAI configurations were used for this analysis. In both cases, the temporal phase modulation signals were generated by a 24-GSa/s AWG. The first one was designed for $q = 12$ and a sampling pulse width ${t_s} = 82.5\;{\rm ps}$, leading to a corresponding sampling rate of $1/{t_q} = 1.01\;{\rm GHz}$ and a measured mean amplification of 9.1 (see Supplement 1, Section 4). The second configuration was implemented with $q = 20$ and ${t_s} = 64.1\;{\rm ps}$, leading to a sampling rate of $1/{t_q} = 780\;{\rm MHz}$ and a measured mean amplification of 14.2. Both TAI configurations employ an LCFBG with a dispersion coefficient $| {\ddot \phi} | = 12{,}930\;{{\rm ps}^2}$, such that the TAI specifications can be easily switched from one to another by simply programing the corresponding temporal phase modulation in the AWG. Four different noise mitigation configurations are tested: using the BPF alone, TAI with $q = 12$, TAI with $q = 20$, and BPF followed by TAI with $q = 12$. As an example, the results corresponding to the denoising of a 10-MHz tone with OSNR of 30.1 dB are shown in Fig. 3(a), and further example traces are shown in Fig. S10 of Supplement 1. A mere visual inspection of the measured waveforms provides clear evidence that the T-TAI process denoises all the tested sinusoidal signals more effectively than the BPF approach. To assess the noise mitigation performance obtained for each of the studied cases, the ${{\rm SNR}_{\mu ,\sigma}}$ and squared Pearson correlation coefficient (SPCC) were calculated for each evaluated frequency and OSNR configuration, results shown in Figs. 3(b) and 3(c). Further details on these metrics are included in Supplement 1.

 

Fig. 3. Ultra-narrowband filtering from noise. Sinusoidal waveforms on an optical carrier are denoised to compare the performance of a standard narrow optical BPF versus the TAI and a combination of the BPF followed by the TAI. (a) Example temporal traces for a sinusoidal waveform with a frequency of 10 MHz and OSNR of 30.1 dB. (b) Measured ${{\rm SNR}_{\mu ,\sigma}}$ and (c) SPCC as a function of the input sinusoidal frequency for OSNRs of (1) 72.2 dB, (2) 30.1 dB, and (3) 25.4 dB.

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The ${{\rm SNR}_{\mu ,\sigma}}$ is defined here as ${{\rm SNR}_{\mu ,\sigma}} = {10}{\rm log}_{10}[({\mu _1} - {\mu _{0})}/({\sigma _1} + {\sigma _{0})}]$, where $ \mu $ and $ \sigma $ correspond to the mean and standard deviations of the measured sinusoidal profile around the crest and through (maximum and minimum points of the sinusoids), as indicated by subscripts 1 and 0, respectively [51] (see detailed definition in Section 4 of Supplement 1). The obtained results confirm that the TAI mechanism reduces the relative amount of noise along the input signal, significantly outperforming the BPF. The noise mitigation capability of the TAI, evaluated as the relative increase in the ${{\rm SNR}_{\mu ,\sigma}}$, is also more notable as the input OSNR is decreased. Due to the inherent insertion loss of the BPF (measured to be about 3.5 dB in the example reported here), the BPF is evidently at a disadvantage in the case of high OSNR waveforms because in this case, the BPF simply leads to a decrease in the detected signal’s intensity with almost no variation of the original noise content. In sharp contrast, the TAI approach induces an additional passive amplification of the signal, which is achieved in all cases. The obtained results also show that it is more advantageous to employ a TAI with a higher amplification factor, as indicated by the traces corresponding to $q = 20$ (versus the traces for $q = 12$). Specifically, for a given sample pulse width ${t_s}$, a higher amplification factor leads to both a narrower effective filtering bandwidth and a higher relative intensity increase in the output waveform. This further lowers the noise content relative to this waveform, which in turn translates into a higher degree of noise mitigation. We also note that the best performance is achieved when the BPF and TAI are used in tandem, leveraging the out-of-band noise reduction from both these methods, as well as the passive amplification offered by the T-TAI. This is most beneficial with moderate amounts of noise, since only a small amount of noise is then contained within the narrow 5-GHz bandwidth. On the other hand, when the amount of injected noise is high, a significant amount of noise is left within the BPF’s bandwidth after filtering, such that the added value of the BPF is less significant.

To further confirm this analysis, the SPCC was also determined for each case, results shown in Fig. 3(c). This coefficient is widely recognized as a relevant criterion to derive an optimal noise-reduction filtering process [52]. In contrast to the ${{\rm SNR}_{\mu ,\sigma}}$, the SPCC gives a measure of the fidelity of the signal recovery process with regards to the temporal shape of the signal, irrespective of the waveform intensity value. In these measurements, the SPCC is calculated between the measured output signal (after the BPF or TAI process) and a numerically filtered version of the high OSNR version of the SUT, shown with the black traces in Fig. 3(a). The obtained SPCC varies from zero (for entirely dissimilar temporal shapes) to one (for identical shapes). As expected, the lower performance of the BPF is not as pronounced in the high OSNR case as for the evaluation using the ${{\rm SNR}_{\mu ,\sigma}}$ (note the vertical axis range), since the insertion loss does not impact the SPCC as much. Nonetheless, the TAI-based approaches remain more advantageous in all cases, with the tandem approach consistently offering the best outcome. Note that both the ${{\rm SNR}_{\mu ,\sigma}}$ and SPCC measurements show a deterioration of the performance of the TAI-based approaches for the highest frequency sine wave, 250 MHz. This degradation is expected because the output sampling rate of the T-TAI approaches here the threshold of the Nyquist criterion, such that the signal itself is starting to be averaged out.

To test the noise filtering capabilities of the TAI on high-speed (broadband) signals, we carried out a similar analysis by recovering noisy telecommunication data signals, specifically, non-return-to-zero on–off keying (NRZ-OOK) pseudorandom binary sequence (PRBS) data signals, with a bit length ${2^7} {-} 1$. In this case, only the TAI configuration with $q = 20$ and a sampling rate $1/{t_q} = 780\;{\rm MHz}$ was employed. The study was carried out for the same OSNR levels mentioned above, for data rates of 19.5 Mb/s and 195 Mb/s, such that the TAI produces 40 peaks per bit and four peaks per bit, respectively. The results corresponding to the signal with an OSNR of 30.1 and a bit rate of 195 Mb/s are shown as an example in Fig. 4(a), and the other traces are shown in Fig. S11. Again, a simple visual inspection of the measured temporal waveforms shows that the TAI implements the predicted denoising and enhancement of the input signal in all evaluated cases, beyond the performance of the BPF. For each case, we estimate the bit error rate (BER), as obtained from the $Q$-factor [53], and the SPCC, results in Figs. 4(b) and 4(c). These results further confirm the significant performance advantage of the TAI over the use of a BPF and that the use of a BPF combined with the TAI is mostly advantageous in the low OSNR cases.

 

Fig. 4. Denoising and enhancement of noisy telecommunication data (NRZ-OOK) signals. (a1) A weak data signal with a rate of 195 Mb/s and OSNR of 30.1 is recovered using (a2) an optical BPF, (a3) TAI with ${q = 20}$, and (a4) BPF followed by a TAI with $q = 20$. Extracted performance metrics for noise mitigation of telecommunication data signal, including (b) SPCC and (b) bit error rate (BER).

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It is worth noting that although here we demonstrate the performance of the TAI on intensity-modulated signals, a TAI preserves the full complex information of the sampled signal, in both magnitude and phase [36]. Thus, the proposed method could also be used on data signals with more advanced (e.g., complex) modulation formats, such as phase shift keying (PSK), pulse-amplitude modulation (PAM), or quadrature amplitude modulation (QAM). This potential is demonstrated through the numerical simulations presented in Fig. S12 of Supplement 1 on a 16-QAM optical data signal.

C. Self-Tracking Capabilities of the T-TAI

Finally, to showcase the unique self-tracking capabilities of the TAI, we process a signal composed of four data streams located at four different central frequencies (wavelengths), namely, 1559.4 nm, 1557.8 nm, 1556.2 nm, and 1554.5 nm, corresponding to the dense wavelength division multiplexing (DWDM) telecommunication channels H22, H24, H26, and H28, respectively. Simultaneous denoising and enhancement of the four wavelength-multiplexed waveforms is achieved using a single TAI unit, results shown in Fig. 5. The TAI employed in these measurements was designed for $q = 12$ and ${t_s} = 48.9\;{\rm ps}$, yielding an output sampling rate of $1/{t_q}\sim1.7\;{\rm GHz}$, using an LCFBG with $| {\ddot \phi} | = 5,076\;{{\rm ps}^2}$. The SUTs are NRZ-OOK data signals similar to those used in Section 3.B but with a PRBS length of ${2^{10}}{-}1$. Two different data rates are tested, namely, 425 Mb/s and 1.7 Gb/s. Furthermore, three different noise configurations are tested for each data rate, corresponding to mean OSNRs of 44.0 dB, 27.0 dB, and 15.1 dB. These OSNR values are determined in a similar fashion as for the experiments shown in Section 3.B, but with a resolution bandwidth of 10 pm (${\sim}1.25\;{\rm GHz}$). The entire set of results is shown in Fig. S11, while Fig. 5 shows the results corresponding to the case with a low OSNR of 15.1 dB and a data rate of 1.7 Gb/s. In this case, the input waveforms are so noisy that it is not possible to extract any meaningful information from the original bit stream; strikingly, the T-TAI signal is able to effectively mitigate the noise contained in each waveform, all done in a simultaneous fashion, without any spectral stabilization or alignment procedures, enabling access to the encoded information across the four DWDM channels. The measured amplification factors across the four wavelength channels range from ${\sim}9$ to ${\sim}10.5$ for the 425 Mb/s data rate and from ${\sim}8.75$ to ${\sim}10.75$ for the 1.7 Gb/s data rate; see details in the plot shown in Fig. S13.

 

Fig. 5. Self-tracking denoising and enhancement of DWDM signals. (a) As an example, temporal traces of the 1.7-Gb/s data signal with an input OSNR of 15.1 dB are shown for the four DWDM channels. (b) The data streams (SUTs, green trace, with the additional injected noise in gray) are combined in a single optical fiber using a multiplexing DWDM module [see detailed experimental schematic in Fig. S3(c)], as shown through the measured spectra displayed in the plot here. (c) All waveforms are simultaneously processed by the T-TAI unit, leading to a broadening of the spectrum of each of the wavelength-multiplexed SUT (green trace). The gray dashed trace shows the output of the DWDM, with part of the noise mitigated by the inherent BPF process implemented by the demultiplexing DWDM stage (3-dB bandwidth of ${\sim}200\;{\rm GHz}$). (d) The temporal traces of the demultiplexed waveforms are measured individually. The amplification and noise mitigation are clearly visible by comparing the waveforms with the phase modulator on (blue trace) and with the phase modulator off (red trace). Note that the spectral broadening caused by the TAI must be such that the spectrum of each of the multiplexed waveforms remains within the bandwidth assigned to each channel to avoid any distortion caused by the demultiplexing stage.

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4. CONCLUSION

We have proposed a lossless temporal sampling technique based on the Talbot effect that is ideally suited for recovering weak arbitrary signals corrupted by noise. It offers a way to simultaneously amplify the temporal envelope of a signal while decreasing its relative noise content, a performance that is beyond the potential of active amplifiers (which necessarily degrade the SNR) and BPFs (which necessarily attenuate the signal of interest). Furthermore, the T-TAI comprises a fairly simple scheme, involving a combination of temporal phase modulation and dispersive delay, and it offers a great versatility in the design of its performance specifications, including the amplification factor and sampling rate (or the equivalent effective filtering bandwidth); specifically, using the scheme proposed here, involving electro-optic phase modulation, the filtering passband can be electronically reconfigured to operate on vastly different signal bandwidths, from the kHz to the GHz regime. As another important advantage, the T-TAI approach avoids the need to have precise knowledge of the central frequency of the SUT by inherently selecting the coherent part of the incoming signal. In contrast to typical active amplifiers, there is little interest in concatenating multiple T-TAI processors, since these subsequent processors could be reduced to a single module with optimized amplification and sampling rate specifications.

We note that the use of nonlinear techniques for temporal phase modulation, such as cross-phase modulation (XPM), would allow to process signals with much higher bandwidths, e.g., enabling the enhancement of data signals with tens of Gbaud and well beyond. There is no restriction on the signal shape, and a TAI process has recently been shown to preserve both the amplitude and phase information of the SUT [36]. In a practical telecommunication setting, we predict that the TAI could benefit from the dispersion that normally occurs along the optical fiber telecommunication link. This dispersion is typically considered undesirable since it distorts the data signal, requiring the use of dispersion compensating strategies. On the other hand, a transmitter in a fiber link could modulate the phase of the signal before sending it, as per the relevant Talbot design equations, such that the signal would get passively amplified as it reaches the detection stage. Furthermore, we note that the T-TAI can be interpreted, and thus utilized, as an efficient sampling method for photonic analog-to-digital conversion (ADC). Generally, in this scheme, one may expect the vertical resolution to increase with the amplification factor. The TAI ADC design would be similar to the scheme proposed and demonstrated by Ru et al. [39], but in which the continuous quadratic temporal lens modulation is implemented using a discrete Talbot phase profile, which would potentially enable realization of high-rate sampling with amplification factors significantly higher than those achievable with the conventional lens approach. As another future line of work, we believe that an all-optical mechanism might be implemented on the output T-TAI waveform to recover an amplified version of the original continuous waveform (e.g., through envelope amplification) rather than the sampled version directly produced by the TAI process. This approach could potentially offer a different, interesting strategy for low-noise amplification of optical waveforms.

We conclude by underlining the fact that the TAI scheme involves the use of very general wave manipulations, available in nearly all wave systems, offering a universal framework for the recovery of weak, noise-dominated waveforms. The proposed concept could thus be utilized on a large variety of physical supports, such as across the entire EM spectrum as well as for plasmonics, acoustics, quantum wavefunctions, etc. Hence, we predict that this concept will prove useful across a wide range of science and technology fields.