1. Introduction

Physical processing along the Fourier-domain frequency representation plays an important role in numerous research fields such as spectroscopy [1], LiDAR [2], telecommunications [3,4], quantum technologies [5–8], and optical computing [9]. Within the realm of spectroscopic technologies, the spectral peaks in Fourier transform spectroscopy are employed for spectral characterization, providing valuable insights into the identification of biomolecules [10,11]. In quantum technologies, the spectral degree of freedom holds particular interest as it allows for information encoding and coherent interfaces [6–8]. Optical frequency combs (OFCs), encompassing broadband spectra, also demonstrate remarkable capabilities in chaotic LiDAR applications [12,13]. Despite the success of photonics in accessing the Fourier domain spectrum across these diverse fields, a persistent challenge arises in the presence of noise. Noise characterized by stochastic intensity and phase variations inevitably arises during photonic generation, transmission, processing, and detection processes [4,14,15]. Consequently, the stochastic noise superimposed on the optical spectrum leads to degraded optical signal-to-noise ratio (OSNR). This degradation significantly impacts performance and compromises the accuracy of detections and measurements. This challenge becomes particularly pronounced when dealing with broadband spectral waveforms with weak power [16].

To increase the power of desired signals, optical amplifiers are commonly employed through active gain processes. Erbium-doped fiber amplifiers (EDFAs) have emerged as a reliable and straightforward platform for compensating transmission losses and amplifying communication signals to desired levels in long-haul optical communication systems. Optical parametric amplifiers, leveraging the third-order nonlinearity, offer another appealing alternative due to their notable properties such as high gain and broad operating bandwidth [17]. However, these amplification techniques amplify both the signal and pre-existing noise simultaneously and introduce additional noise into the system, thereby resulting in a deterioration of the OSNR. To address this challenge, phase-sensitive optical parametric amplification has been developed to enable noiseless amplification [18]. Nonetheless, the requirement for phase matching between the pump and signal restricts the amplification capabilities for unknown arbitrary incoming signals under test (SUT). Apart from directly amplifying the desired signals, noise mitigation involving the use of filtering schemes to reject noisy portions of the spectra is another approach to improve the signal quality. Bandpass filters (BPFs) and comb filters are widely employed for denoising the SUT [19]. However, suppressing in-band noise during the filtering process remains a key challenge. Alternative analog signal processing strategies have been proposed to mitigate in-band noise [20]. Nevertheless, these approaches typically require prior knowledge of the SUT, making it relatively difficult to denoise unknown waveforms obscured by noise.

Recently, Talbot amplifiers have attracted significant attention from both theoretical and practical standpoints due to their property of noiseless amplification [21–24]. This innovative approach leverages a periodic energy redistribution process to convert coherent waveforms into a series of high-power peaks, effectively amplifying the envelope of the input waveform. Importantly, the incoherent noise background remains unaffected, resulting in noiseless amplification. Talbot amplifiers enable passive sampling of the signal under test (SUT) while providing an accompanying power gain. Conventional Talbot amplifiers manipulate temporal phases of signal waveforms by using an electro-optic (EO) phase modulator driven by an arbitrary waveform generator. Additionally, a dispersive medium serves as a spectral phase filter to fulfill the spectral phase modulation requirement [22]. However, the use of a fixed dispersive medium in the system imposes a fixed spectral phase profile, thus limiting the flexibility of the amplification gain factor for a given input SUT. Moreover, the adopted electrical devices and equipment impose a fundamental constraint on the operation bandwidth and achievable amplification of the Talbot amplifier. In dealing with Fourier domain spectra characterized by initial frequency tone spacing of a few gigahertz, the amplification factor is typically limited to a factor of four [25].

To achieve ultra-broadband operation bandwidth, M. Fernandez et al. have successfully demonstrated the use of nonlinear optical processes including cross-phase modulation (XPM) and parametric four-wave mixing (FWM) process to implement ultrafast temporal Talbot array illuminators [26–28]. These temporal Talbot amplifiers have proved effective in denoising high-speed temporal signals. Apart from temporal denoising assisted with nonlinear phase modulation, we have proposed and experimentally verified parametric-assisted spectral Talbot effect for purifying noisy optical frequency comb [29,30]. However, the center wavelength of the SUT is unavoidably altered. Recently, a simple demonstration of the all-optical XPM based spectral Talbot effect has been reported in [31]. The approach offers the capability to flexibly recover and amplify broadband spectral waveforms over a wide range of gain factors.

In this study, we further elaborate our work in the tunable XPM based spectral Talbot amplifier. This all-optical spectral Talbot amplifier is designed for the recovery of broadband spectral waveforms with large initial frequency tone spacing, offering adjustable gain factors ranging from 3 to 10. Our approach leverages ultrafast coherent energy redistribution in the Fourier frequency domain, enabling the transformation of a SUT with a free spectral range (FSR) of 17.5 GHz into a newly defined spectrum with a user-defined wider frequency spacing. Importantly, this process ideally preserves the entire energy of the SUT, resulting in intensified and reshaped spectral peaks. Concurrently, non-coherent stochastic noise remains largely unaffected throughout this operation. Consequently, this leads to a substantial enhancement in the OSNR of the SUT. In our experimental setup, we achieve OSNR enhancements of 4.7 dB, 6.0 dB, 9.5 dB, and 10.0 dB, corresponding to gain factors of 3, 5, 9, and 10, respectively. Notably, these enhancements are achieved through a programmable approach that eliminates the need for any modification to the existing components of the setup. Additionally, the proposed spectral Talbot amplifier can provide more discretion on the wavelength allocation of recovered spectral waveform, resulting in multiple copies of the purified spectra. This feature enables the retrieval of desired outputs with a new degree of flexibility supported by our all-optical spectral recovery system.

2. Operating principle and conceptual design

The conceptual scheme of our approach and its distinction with respect to a commonly adopted spectral control system are presented in Fig. 1. Figure 1(a) illustrates a method for spectral processing and compression widely used in the spectral-temporal control of pulses [6–8]. This spectral bandwidth compression consists of two main steps, including phase manipulation of both spectral and temporal envelopes. First, a broadband optical waveform is chirped through a spectral phase filter where a spectral phase mask is applied to the input spectrum. This chirping process separates distinct spectral components in the time domain, leading to temporal waveform broadening. Subsequently, a quadratic temporal phase is imposed on the signal. This operation results in the transformation of a broadband optical pulse into a narrower bandwidth configuration.

 

Fig. 1. Conceptual schemes of (a) spectral bandwidth control of light through linear temporal and spectral phase manipulation. (b) Spectral Talbot amplifier through linear spectral phase manipulation and nonlinear XPM based temporal phase modulation, with an amplification factor of p. (c) Spectral Talbot amplifier through linear spectral phase manipulation and cascaded nonlinear XPM based temporal phase modulation, with an amplification factor of p/q.

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Different from the aforementioned technique, the concept of our proposed scheme for spectral sampling is shown in Fig. 1(b) and (c). In Fig. 1(b), we initiate our process with spectral phase filtering using either a programmable optical filter (POF) or a dispersive medium. This spectral filtering process results in a phase profile that is equal to, under mod 2π, the one introduced by a dispersion governed by [32]:

Here, ${\ddot{\varPhi }}$ is the total linear group velocity dispersion, f is the width of spectral sampling peaks [23] and p is an integer factor. The spectral phase filtering leads to multiple copies of the original temporal waveform, with the period of the temporal traces become 1/(pf). Each replica of the original temporal input carries a certain phase step, and the kth phase step can be expressed as:

Notably, when the temporal waveforms exhibit an ultrashort temporal interval (small value of 1/(pf)), the waveform-to-waveform phase variation can be extremely rapid, exceeding hundreds of GHz. Traditional EO phase modulation is inadequate for compensating such ultra-fast phase variation due to the limited bandwidth of EO system. To overcome this limitation and enable ultra-fast temporal phase modulation, we employ an XPM based time lens (XPM-TL) to apply an ultrafast temporal phase pattern by coupling the SUT to a suitable optical pump. Subsequently, in the following step, an XPM-TL compensates the phase pattern carried by the temporal waveform of the pump, described as:

The all-optical XPM based spectral Talbot amplifier provides a lossless spectral sampling mechanism, enhancing the OSNR in the output spectrum. Figure 2 illustrates the operating principle of the spectral Talbot amplifier for recovering the signal from noisy spectra. A noisy spectral waveform, represented by red lines, serves as the SUT, while an in-phase temporal counterpart is used as an example. The SUT passes through a spectral phase filter to provide group velocity dispersion satisfying Eq. (1). Consequently, replicas of the temporal waveform are generated, each carrying a different phase. With the assistance of a well-designed pump, XPM is performed along a highly nonlinear fiber (HNLF) to compensate for the phase variation of the temporal waveforms. The pump modulates the phase of the SUT according to ${\varphi _{XPM}}(t )= 2\gamma L{P_{pump}}(t )$, where $\gamma $ and L are the nonlinear coefficient and effective length of the HNLF, respectively. Therefore, the temporal profile of the pump corresponds to the phase function described in Eq. (4) and Eq. (5). Pulse shaping via a POF allows for the generation of various target intensity profiles of the pump [26,36]. Phase compensation through the XPM process results in coherent sum of the powers in consecutive frequency tones of the SUT, while stochastic noise remains unchanged. Consequently, the energy of the SUT is redistributed, generating OSNR-enhanced sampling peaks while preserving the spectral envelope.

 

Fig. 2. Schematic of the all-optical XPM based spectral Talbot amplifier and its working principle.

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A key advantage of our proposed system lies in its tunability and flexibility in recovering arbitrary noisy spectral waveforms. In the initial step, we can introduce various spectral phase patterns using a POF. Once the spectral phase pattern is defined in this first step, we can calculate the required temporal power profile of the pump from Eq. (4) and Eq. (5). These pump traces can be conveniently generated using pulse shapers. After the XPM process, we can successfully compensate for the temporal phase profile of the SUT. Consequently, by changing the spectral phase patterns in the first step and flexibly generating corresponding pump traces via POFs, our method offers tunability in the amplification or gain factor, allowing us to control the spectral spacing of the output spectrum as needed. Another crucial feature of the all-optical spectral Talbot amplifier is its remarkable operational bandwidth of over 1 THz. This wide bandwidth capability allows us to effectively denoise SUTs with an initial frequency spacing of over 10 GHz. Specifically, the pump generated through pulse shaping can have ultra-short temporal bins, facilitating ultrafast temporal phase compensation. As a result, the spectrum is denoised and reshaped, leading to an enhanced OSNR and expanded frequency tone spacing up to hundreds of GHz. This capability holds potential promise for increasing the carrier-to-noise ratio of wide-spacing soliton combs [37,38].

3. Experimental results

3.1 Experimental setup

Figure 3 depicts the experimental setups for both the generation of the broadband comb source (BCS) and the tunable all-optical XPM based spectral Talbot amplifier in a proof-of-concept demonstration. In Fig. 3(a), a 1553 nm continuous wave (CW) light source is transformed into an optical pulse train at a repetition rate of 17.5 GHz. We achieve this using a fiber-based cavity-less configuration, which includes an intensity modulator, a phase modulator, and a dispersion compensating fiber (DCF). Specifically, we employ a sinusoidal wave at 17.5 GHz frequency from a signal generator, splitting it into two branches. One branch directly drives the intensity modulator, while the other is used to drive the phase modulator. Subsequently, the generated pulses are amplified by a high-power erbium-doped fiber amplifier (EDFA) to reach an average power of 26 dBm. These pulses are then directed into a dispersion-decreasing fiber (DDF). Self-phase modulation (SPM) of these pulses effectively broadens the spectrum, resulting in a 17.5-GHz spacing optical comb covering over 35 nm bandwidth, as shown in Fig. 3(b). In the following, as illustrated in Fig. 3(c), the BCS is split into two branches. In the upper branch, we use a POF with employed transfer functions to shape the super-flat region of the BCS, generating the desired pump centered at 1540 nm. Determination of the requisite transfer function stems from the target temporal waveform, characterized by the desired intensity profile expressed as ${e^{j \cdot sin({f \cdot t} )}} \cdot \mathop \sum \nolimits_{l ={-} \infty }^{l ={+} \infty } P\left( {t - l \cdot \frac{1}{f}} \right)$, where f = 17.5 GHz. Upon application of the computed complex transfer function via the POF, the target pump exhibiting the desired temporal intensity profile is realized. This pump is subsequently directed to a tunable optical delay line (TODL), followed by a high-power EDFA and a polarization controller (PC). In the lower branch, we extract the SUT centered at ∼1559 nm, using another POF. To intentionally degrade the OSNR of the SUT, we introduce additional amplified spontaneous emission (ASE) noise to the SUT. The noisy SUT then passes through a single-mode fiber (SMF) providing a dispersion of 45 ps/nm. After polarization alignment with PCs, the noisy SUT and the designed pump are combined and sent into a 200-m HNLF with a nonlinear coefficient γ = 30 W^-1 ·km^-1. An optical spectrum analyzer (OSA) with a spectral resolution of 0.02 nm is used to measure the spectral waveforms after the processing.

 

Fig. 3. Experimental setups for broadband comb source (BCS) generation and tunable all-optical XPM based spectral Talbot amplifier. (a) Fiber-based cavity-less system for generating a BCS; (b) measured broadband comb source; (c) tunable all-optical spectral Talbot amplifier, POF: programmable optical filter, TODL: tunable optical delay line, EDFA: erbium-doped optical fiber amplifier, PC: polarization controller, SMF: single-mode fiber, HNLF: highly nonlinear fiber, OSA: optical spectrum analyzer.

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3.2 Tunable all-optical spectral recovery

We demonstrate tunable all-optical spectral Talbot amplification by controlling the two POFs in the upper and lower branches shown in Fig. 3(c), providing different noiseless amplification factors in a fully programmable fashion. In the lower branch, the POF can provide additional chromatic dispersion ranging from -20 ps/nm to 20 ps/nm with a minimum step of 0.1 ps/nm for precise adjustment on the total group velocity dispersion (GVD). It can facilitate the realization of wide-range amplification factors for the broadband spectral signals. Conversely, in the upper branch, the pump profile is flexibly designed through line-by-line shaping using the POF. We first design the tunable all-optical spectral Talbot amplifier for different amplification factors. Initially, we design the operation for amplification factors including 3, 5, 9, and 10. Figure 4 illustrates the temporal traces of the pump for each amplification case measured by a 500-GHz bandwidth optical sampling oscilloscope (PicoSolve PSO101B). The measured temporal traces, shown in blue, are compared with the ideal pump profiles, depicted in brown. The temporal waveforms of the pump for m = 3 (p = 9 and q = 3), m = 5 (p = 10 and q = 2), m = 9 (p = 9 and q = 1), and m = 10 (p = 10 and q = 1) are derived from Eq. (5), while the pump power for each amplification case is 23.1 dBm (m = 3), 23.6 dBm (m = 5), 23 dBm (m = 9), and 24.5 dBm (m = 10), respectively. Notably, each time-bin exhibits an extremely short temporal duration of 6.35 ps (for m = 3 and 9) and 5.71 ps (for m = 5 and 10), respectively. It is observed that the synthesized pump pattern shows a high degree of similarity with the ideal target pump profile.

 

Fig. 4. Measured temporal traces of the pump (shown in blue) and comparison against the ideal pump profile (shown in brown) for different amplification factors. (a) m = 3; (b) m = 5; (c) m = 9; (d) m = 10.

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Figure 5 depicts the results of spectral denoising and reshaping of the SUT with a Gaussian-like profile. First, the original BCS is carved to obtain a spectral waveform with a Gaussian profile and an initial frequency spacing of 17.5 GHz for experimental validation. The bandwidth of such an original SUT exceeds 1 THz, as illustrated in Fig. 5(a). Subsequently, the ASE noise is loaded to create a noisy SUT, as depicted in Fig. 5(b). The noisy SUT is then processed by our spectral Talbot amplifier. Significant improvement of the OSNR has been achieved. In Fig. 5(c), when m = 3, the noisy SUT is reshaped into an output spectrum (shown in red) with a 52.5 GHz spacing and 22 spectral sampling peaks, where each sampling peak has an enhanced power. The black dashed curve refers to the spectral envelope of the original SUT. It is observed that the reshaped and purified output perfectly reconstructs the original information of the envelope. For m = 5, the output purified spectrum features a frequency spacing of 87.5 GHz, while each spectral peak has an OSNR improvement of ∼7 dB. As to m = 9, illustrated in Fig. 5(e), the OSNR of the SUT sees a remarkable OSNR improvement of ∼9.5 dB, while the frequency spacing of the output spectrum increases to 157.5 GHz. At m = 10, the original noisy SUT undergoes further reshaping, resulting in a more purified spectrum with a frequency spacing of 175 GHz and enhanced OSNR of 10 dB. We observe a slight mismatch between the output spectral sampling peaks and the black dashed envelope. This variation can be attributed to minor deviation in the measured pump waveforms from the ideal traces.

 

Fig. 5. Experimental results on tunable spectral recovery of a SUT with Gaussian-like profile. (a) Clean SUT; (b) Noisy SUT loaded with ASE noise; (c) recovered spectrum with m = 3; (d) recovered spectrum with m = 5; (e) recovered spectrum with m = 9; (f) recovered spectrum with m = 10.

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By increasing the amount of noise injected into the SUT which inherently possesses a lower power, we reach a point where the weak SUT is entirely submerged in noise, rendering any information retrieval at the receiver nearly impossible. In such scenarios, the output spectral waveform can be effectively recovered from the noise background and accurately measured, essentially presenting a sampled representation of the original waveform. This capability is realized through the application of the tunable all-optical spectral Talbot amplifier. Figure 6(a) shows a new SUT with a peak power of -55.5 dBm and a spectral bandwidth of 1.2 THz carved from the BCS. Subsequently, a substantial amount of ASE noise is introduced into the input SUT to the extent that it is completely submerged in the noise background, as illustrated in Fig. 6(b). Adopting an amplification factor m = 9, the all-optical spectral Talbot amplifier facilitates an impressive improvement of ∼10 dB in the selected spectral peaks. Referring to the eight spectral sampling peaks successfully extracted from the background noise, we observe that the entire spectral envelope is preserved, accurately representing a sampled version of the original waveform.

 

Fig. 6. Recovery of a low-power SUT buried under noise. (a) Clean low-power SUT; (b) undetectable SUT buried under ASE noise; (c) recovered SUT with well-preserved spectral envelope.

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3.3 Reconfigurable reconstruction of noisy spectrum

The tunable all-optical spectral Talbot amplifier exhibits the ability to flexibly generate reconstructed outputs from a noisy input SUT. The flexibility is achieved through simple adjustment of the temporal delay of the pump trace via the TODL. In Fig. 7, we present the experimental results of this capability. Initially, we generate a trapezoidal SUT with a 17.5-GHz spacing, a 1.1-THz bandwidth, and a relatively low peak power of approximately -40 dBm, as depicted in Fig. 7(a). This SUT is then deliberately loaded with a strong ASE noise, leading to severe distortion and almost complete submergence under the background noise. Using the all-optical spectral Talbot amplifier with m = 9, the noisy trapezoidal SUT is effectively purified and recovered from noise with an OSNR improvement of ∼10 dB, as shown in Fig. 7(c). The black dashed curve indicates the original trapezoidal spectral envelope. It is observed that the purified spectral peaks perfectly reconstruct the original information of the SUT. Subsequently, we produce other outputs of the purified spectrum by tuning the TODL for several temporal bins, expressed as n/(mf), where n is an integer ranging from 0 to m-1. This temporal delay affects the pump trace and results in different compensated phase profiles of the SUT. Figure 7(d) illustrates two outputs of the purified spectrum. The newly generated output spectrum using n = 4, shown in red, also preserves the improved OSNR and the original trapezoidal envelope, while each spectral peak is shifted by 70 GHz compared to the reference output shown in blue. Furthermore, we present a comprehensive overview of all the recovered outputs of the purified waveform by continuously tuning the TODL for several temporal bins. Figure 7(e) summarizes these cases under different relative delay times between the pump and the SUT. Importantly, each selectable case maintains a consistent spectral envelope.

 

Fig. 7. Selective recovery of noisy trapezoidal spectral waveform. (a) Clean spectral waveform with a trapezoidal shape; (b) noisy and distorted trapezoidal waveform; (c) purified spectral waveform with preserved trapezoidal envelope; (d) two recovery cases with a 70-GHz difference in the frequency tones; (e) superposition of 9 output spectra obtained under different temporal delays of the pump.

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4. Numerical analysis and discussion

The accuracy and variation of the pump power plays a crucial role in the performance of the all-optical spectral Talbot amplifier, as it relies on ultrafast phase modulation achieved by XPM. To evaluate the impact of the pump power deviation on spectral amplification, we conduct a numerical investigation on deterioration of the pump power profiles. It is believed that the deterioration mainly stems from practical accuracy and resolution of the POF. We here introduce a relative deviation, denoted as $\delta $, to the power of the pump trace, which causes fluctuation of the imparted phase pattern from the pump to the SUT during the XPM process. Therefore, the deviation of the imparted phase pattern through the XPM process can be expressed as [39]:

The effect of the pump deterioration under the case of amplification factor m = 10 is exhibited in Fig. 8. The phase patterns are calculated based on Eq. (4). Figure 8(a) shows the ideal pump trace where the deviation $\delta $ is set to zero (σ = 0 and μ = 0), while Fig. 8(b) illustrates the input and output spectra. The input spectrum exhibits a frequency spacing of 17.5 GHz and a suppression ratio of ∼29 dB, as illustrated by the blue curve. With an ideal XPM process, the input spectrum experiences perfect amplification, leading to an output spectrum (red curve) with increased peak power while maintaining the background noise floor, resulting in an OSNR improvement of 10 dB. When we introduce a deviation with σ = 0.05 and μ = 0, representing random fluctuations around a mean of 0 in the phase deviation δ, the temporal pump trace is disturbed as shown in Fig. 8(b)(i). Remarkably, Fig. 8(b)(ii) reveals an output spectrum that remains nearly unchanged with almost the same purified spectral peaks compared to the result obtained for σ = 0 in Fig. 8(b)(i). Under this case, the deviation of the pump will not significantly affect the spectral denoising process. As σ keeps increasing to 0.10 while μ is still 0, residual spectral energy is found in the output spectrum in Fig. 8(c)(ii). The residual energy indicates incomplete energy redistribution in the all-optical spectral Talbot amplifier, attributed to the imperfect phase compensation caused by a relatively large pump deviation. Nevertheless, the powers of the denoised sampling peaks decrease within a relatively small range (<1 dB), thanks to the substantial suppression of residual spectral energy. By contrast, if μ becomes 0.1 while σ = 0.1, there is a bias that is included in the deviation δ. Under this case, the power of residual energy become larger as shown in Fig. 8(d)(ii), with a suppression ratio of ∼20 dB. This shows that the all-optical spectral Talbot amplification process is primarily sensitive to the mean deviation value μ of the pump trace. This factor is identified as the primary contributor to residual spectral energy observed in our experimental demonstrations. Nonetheless, the spectral denoising peaks of the output spectrum still possess a high power, which is just 0.5 dB below that obtained with δ=0 shown in Fig. 8(a)(i), highlighting the robustness of our denoising system against pump deterioration.

 

Fig. 8. Numerical investigation of the effect of pump power deviation. (a)(i)-(d)(i): temporal trace of the pump under different deviation parameters; (a)(ii)-(d)(ii): input and output optical spectra. (a) σ = 0 and μ = 0; (b) σ = 0.05 and μ = 0; (c) σ = 0.10 and μ = 0; (d) σ = 0.10 and μ = 0.10.

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Notably, in most denoising scenarios, the spectral sampling peaks play a critical role since they reconstruct the original information of the noisy SUT. The presence of residual spectral energy indicates an incomplete energy redistribution process, potentially leading to a lower OSNR improvement. To further investigate the effect, we set the standard deviation parameter to σ = 0, implying that the relative deviation δ in Eq. (6) is solely defined by the mean parameter μ. Figure 9 assesses the achieved gain of the OSNR by varying the imparted phase error due to the mean parameter μ under different denoising cases. The gain gradually decreases as δ increases from 1% to 20% across all cases. Under the case of phase error = 20%, the gain factor only decreases by about 0.6 dB for m = 10 and m = 9. In the similar scenario, the gain factor reduces by approximately 0.8 dB for m = 5 and m = 3. This underscores the robustness of the OSNR gain against deviations in the imparted phase pattern through the XPM process.

 

Fig. 9. Achieved gain of the all-optical spectral Talbot amplifier against the phase error.

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Our proposed all-optical Talbot amplifier is not limited to integer amplification factor, but can support any desired rational gain factor m (m = p/q). Specifically, based on Eq. (5), p and q could be set to be coprime with each other, resulting in a final rational gain factor equal to p/q. Moreover, while the SUTs investigated in this study comprise discrete spectral waveforms with an initial spacing of 17.5 GHz, it is noteworthy that the XPM-based Talbot amplifier can be effectively applied to arbitrary spectral waveforms, including transient events carrying random phase information. In such cases, real-time optical Fourier transform techniques will be required in the final detection stage for a single-shot measurement, relieving the need of synchronization between the SUTs and the XPM pump [23]. In addition, we anticipate the capability of achieving larger amplification factors (m > 10), which can be realized by adopting a shorter SMF in our setup. Furthermore, it is worth noting that a compact well-designed nonlinear Kerr medium can be employed in our system, such as one implemented with silicon nitride waveguides, which enable high nonlinearity and low propagation loss [40]. Moreover, an on-chip dispersive medium can be employed to replace the bulky SMF [41,42]. Therefore, we anticipate that miniaturization would promise advantages of our system in terms of size, weight, latency, and power consumption, all of which contribute significantly for noise mitigation in numerous potential research fields ranging from telecommunications to sensing.

5. Conclusion

In summary, we propose and experimentally demonstrate a tunable all-optical spectral Talbot amplifier capable of recovering arbitrary noisy spectral waveforms with THz bandwidth. Our simple all-fiber system exhibits the ability to flexibly amplify the SUT and improve its OSNR across a wide range of amplification factors from 3 to 10. Leveraging the ultrafast phase modulation of XPM, our system effectively purifies THz-bandwidth spectral waveforms with wide initial spectral spacing. By programming the required spectral phase patterns and generating desired pump traces via programmable optical filters, we have demonstrated various OSNR gain ranging from 4.7 dB to 10 dB in a fully programmable fashion. Moreover, the frequency tones of purified spectra can be tuned over a wide range of tens of GHz, which has been experimentally realized by simply tuning an optical delay line. Our reconfigurable all-optical spectral Talbot amplifier can become a fundamental building block for frequency information processing and denoising that offers tremendous benefits to quantum computing and communications, LiDAR, spectroscopy, telecommunications, and microwave photonics.