## Abstract

New technique for cancellation of nonlinear cross-talk in polychromatic parametric sampling gate is described and quantified. The method relies on a newly derived look-up table method that achieves equalization and suppresses nonlinear response associated with parametric sampling operation. The new cancellation scheme is implemented in a framework of a specific parametric photonics assisted analog-to-digital conversion (ADC) copy-and-sample-all (CaSA) architecture. A 20 dB improvement in total harmonic distortion is demonstrated experimentally.

© 2013 OSA

## 1. Introduction

Nonlinear cross-talk in optical fibers generated by co-propagating frequency non-degenerate optical waves is a well-known impairment mechanism that leads to performance degradation in both analog and digital photonic applications. A distinct set of physical processes contributing to nonlinear cross-talk in silica fibers includes cross phase modulation (XPM), four-wave mixing (FWM) and stimulated Raman scattering (SRS). In digital wavelength division multiplexed (WDM) transmission systems, FWM-generated inter-channel cross-talk can be significantly reduced by decreasing the channel powers [1], by adopting non-equidistant channel spacing [2], or by tailoring a specific dispersion map of the transmission system [1]. In contrast, in photonic analog applications, particularly when fiber nonlinearity provides physical means for optical processing, cross-talk cannot be suppressed by such conventional means and is likely to set the ultimate performance limit.

Photonically-assisted processing has been identified as a path for overcoming analog to digital converter (ADC) limitations, specifically with regards to bandwidth constraints [3]. Fast optical gates can be constructed using short optical pulses enabling a broadband sampling operation; in such cases, photonics is used as the front-end (assist) while the electronics handles the backend digitization. The basic advantage of such architecture is its ability to parallelize the sampling operation and to lower the physical sampling rate seen by the electronics backplane. Similar schemes have been implemented for real-time monitoring with parallel gates [4], high frequency, sub-rate sampling at multiple wavelengths [5] and real-time digital signal processing by polychromatic sampling [6]. Among these, the polychromatic parametric sampling gate has been used in cases when simultaneous sampling of multiple high-bandwidth analog signals is required [7]. A physical device operates with ultrashort, high-peak power pump pulses and multiple analog signals at different frequencies; all waves are mixed in a dispersion engineered, highly nonlinear fiber (HNLF). In an ideal (cross-talk-free) case, the FWM interaction between each analog signal and the pump creates a gated signal at a new frequency that can be spectrally discriminated. In practice, input signals coexist in the temporal domain and necessarily lead to finite cross-talk (mixing) generated between participating optical carriers. Consequently, the polychromatic parametric gate performance is limited by the nonlinear cross-talk: pump-signal and signal-signal nonlinear interactions via the FWM and XPM processes resulting in non-negligible degradation in sampled signal accuracy.

Recognizing this basic limitation, this report investigates new means for cross-talk limitation in the backplane of the processor. Specifically, we demonstrate the increase in performance of the copy-and-sample-all (CaSA) polychromatic sampling gate recently used to achieve high resolution ADC [8]. Rather than relying on iterative, computationally intensive means for cross-talk suppression, we generate a single, multidimensional lookup table (LUT) and apply single-operation within the CaSA backplane. The use of equalization method based on lookup tables is a well-established methodology for ADC linearity improvement [9, 10], and was specifically selected to decouple the operational rate from the computational load expected with increased speeds. While we have previously demonstrated a simplified linearization of photonic ADC [11] operating at low frequencies using a multidimensional look-up table, the present report introduces a new, generalized cross-talk cancellation scheme applicable to CaSA photonic-assisted ADCs. The method was characterized experimentally and was capable of significant ADC resolution improvement: measured suppression of total harmonic distortion (THD) exceeded 20 dB for the first time.

## 2. Parametric CaSA photonic ADC and distortion correcting tables

The CaSA photonic-assisted ADC structure is constructed as shown in Fig. 1
. This preprocessor closely follows previously reported topology [8] and consists of three building blocks responsible for signal replication, sampling, and quantization. Analog optical input signal is replicated (multicast) via two-pump self-seeded parametric stage [12] to create multiple, frequency non-degenerate copies that are sent to a chromatic delay line inducing a precise copy-copy temporal offset. A single polychromatic parametric gate, operating at a sampling rate *R*, simultaneously samples all signal copies by a master optical pulse, providing a discrete measure of the original waveform. Subsequently, each sampled copy is filtered and quantized separately using an electronic (subrate) ADC. Electronic quantizer outputs are interleaved to reconstruct a discrete, digitized representation of the original input analog signal at a rate *N × R*; in practice, the CaSA preprocessor operates at *aggregate rate* of *N × R*, while physical sampling rate is only *R*.

In order to maintain the signal fidelity within the entire preprocessing chain, it is important that each section is operated in a near distortionless manner. Unfortunately, the two processing stages are essentially efficient mixers and have inherent FWM interaction that cannot be cancelled, even in principle. In the first, parametric multicasting stage, the newly generated signal copies experience quadratic nonlinearities induced by the pump depletion [13]. In the second, sampling stage, a sampled signal copy sees not only self-induced nonlinear distortion, but is also affected by a nonlinear cross-talk originating with copy-copy and copy-pump-copy processes [11]. While it is possible to minimize both nonlinear distortion types by specific design strategies controlling mixer interaction length and signal/pump powers [7, 12], the accumulated cross-talk level cannot be reduced *arbitrarily*. Consequently, physical means for distortion equalization in parametric preprocessor is generally insufficient for applications that require very high digitization precision or a high dynamic range requirement.

In practical terms, any parametric processor operates using very efficient FWM interaction; its very nature also dictates that FWM-induced distortions are always present in the system. In a well-known analogy, such processor can be compared to a conventional WDM fiber transmission link in which FWM-induced cross-talk also imposes a fundamental performance limit. However, in contrast to fiber transmission link in which the exact history of all WDM channels is not known a priori (since independent channels can be added and dropped unpredictably within the optical link), *the evolution of all channel copies in polychromatic parametric processor is strictly predefined*.

Consequently, all nonlinear distortions can be regarded as deterministic impairments, and can be, at least in principle, canceled (inverted) in practice. Intuitively, by mapping the preprocessor nonlinear transfer with sufficient accuracy, one should be able to accomplish the inversion in a general case, and not only in the CaSA architecture. The accuracy of the inversion operation is contingent on the absence of the noise in the system [14]. Consequently, a necessary condition that must be met before any inversion method can be successfully applied is that parametric processor operates in a low-noise regime [7].

Assuming that this condition is met, Fig. 2 describes the linearization technique for a memoryless, nonlinear single input/single output (SISO) ADC model. The ADC device is modeled as a cascade of a block possessing nonlinear transfer function and an ideal quantizer. In effect, the method rests on the assumption that any deviation from ideal linear response exhibited by the physical ADC output is a stationary scalar value that can be stored in one dimensional look-up table (LUT). As a result, the equalization, then, only requires adding the stored LUT value to the direct (physical) ADC output measure in real time, removing the computational burden from the backplane. In simple terms, the correction scheme maps the direct ADC output levels to a new set of levels, based on the distortion correction table. The latter becomes critically important as the operational rate is scaled: while a single-step vectorial addition can be accomplished in a trivial manner, this is not necessarily true for alternative methods relying on iterative computation approach [15].

In case when preprocessor has memory or possesses multiple inputs, dimensionality of the nonlinear transfer function is determined by its memory depth and the number of inputs. Subsequently, the dimensionality of distortion correction tables must strictly follow that of the transfer function. Indeed, the polychromatic parametric sampling gate is an example of a nonlinear multiple-inputs/multiple-output (MIMO) system where mutually delayed signal copies are identified as inputs while sampled idlers represent the system outputs. Consequently, each sampled idler is not only a function of its time/frequency content but also depends on all other signal copies due to the nonlinear cross-talk as well.

To help visualize the concept, Fig. 3 illustrates an example of the 2 × 2 MIMO nonlinear system characteristic and two corresponding distortion correction tables. For this specific example, the system transfer function for each output can be expressed as a surface in the three-dimensional space in which the two inputs serve as independent variables. In the linear and cross-talk-free case, the transfer function can be illustrated by a plain surface with no dependency on the cross-input (i.e. an affine surface parallel to the cross-input axis, as in Fig. 3). In the non-ideal case, any static deviation from the linear state, caused by either nonlinearity or cross-talk, can be stored in a two-dimensional LUT; these values can be added to the direct ADC’s outputs to revert the distortions. The distortion table equations for the discrete 2 × 2 MIMO system can be expressed as:

*in*and

*out*are the direct input and output of the system, $\overline{out}$ denotes the equalized output, and

*n*is the discrete sampling time instant. In Eq. (1),

*Q*represents the quantization function that rounds off the input amplitude values to the nearest digitized level, and

*cor*

_{1}and

*cor*

_{2}are the two dimensional matrices storing the correction values as their elements.

The distortion correction tables are generated by using calibration signals to extract the system nonlinear response with a specified degree of accuracy. In each sampling time instant, a pair of two direct ADC output levels jointly addresses a position in the correction table. The difference between the expected (linear) input and the system output gives an estimate of the correction value for that address. The final value for each address is determined as an ensemble average of multiple correction estimates. Several calibration signals have been suggested for error table population in the electrical ADCs. Among them, Gaussian white noise (GWN) signals have been recognized as the most effective for the ADC compensation [10]. However, with finite length GWN data record, the achievable accuracy in nonlinear response estimation cannot meet all requirements. Moreover, the estimation error increases as the record length decreases and compromise must be sought.

Recognizing this problem, the pseudo random multilevel signals (PRMS’s) were chosen as the training signals for our multi-dimensional ADC distortion compensation. In a PRMS, the input signal switches in a deterministic manner between the finite number of quantization levels. The spectra of the PRMS’s are similar to the white noise; therefore, they are efficient in exciting all processor frequency modes and have been widely deployed in nonlinear system identification [16]. Furthermore, the PRMS’s with *M* levels have been recognized as persistently exciting (PE) for the polynomial nonlinearity of order *M*-1; equivalently, a Volterra filter with nonlinearity up to polynomial degree *M*-1 can be uniquely identified using the PRMS’s with *M* levels.

## 3. Experimental results

In order to demonstrate the LUT MIMO distortion suppression in physical parametric preprocessor, the experimental setup for CaSA photonic ADC was constructed, as shown in Fig. 4
. The processor closely follows the functional topology shown in Fig. 1 and incorporates four blocks: parametric signal replicator, sampling gate (pump), sampling mixer, and detection (equalization) backend. The sampling gate relies on a cavity-less device [17] capable of variable sampling rate. The source employed a laser diode (LD_{1}) centered at 1557 nm, followed by a cascade of a phase modulator and an intensity modulator (MZM_{1}), driven by the 8 GHz and 2 GHz harmonic signals, respectively. The output of the modulator, after amplification (A_{2}) was transmitted via 1100 m of a single mode fiber (SMF_{1}) and 250 m of highly nonlinear fiber (HNLF_{1}). The HNLF was characterized by the zero dispersion wavelength (ZDW) of 1580 nm, a nonlinear coefficient of 18 W^{−1}km^{−1}, and a dispersion slope of 0.015 ps/km/nm^{2}. The output of HNLF_{1} was a 40 nm-wide frequency comb with 2-GHz line spacing. An 8 nm band of this comb was filtered with a band-pass filter centered at 1564 nm resulting in 2 ps pedestal-free optical pulses at a 2 GHz repetition rate. Figure 5(a)
and 5(b) show the spectral and temporal shape of the generated pulses respectively. In general, the analog bandwidth of the CaSA photonic ADC is determined by the temporal width of the optical sampling pulses (2 ps) and their timing jitter (100 fs), which satisfied the 50 GHz of analog bandwidth in our experiment.

Frequency non-degenerate signal copies were generated in a self-seeded parametric multicaster: the original signal (LD_{2}) centered at 1550 nm, followed by the MZM_{2} and driven by the signal under test was mixed with two 1 W pumps, LD_{3} and LD_{4} centered at 1535 nm and 1557 nm, in the 60 m gradient strained HNLF_{2} (ZDW of 1552 nm, dispersion slope of 0.03 ps/km/nm^{2}, and nonlinear coefficient of 12 W^{−1}km^{−1}) to generate two equalized signal copies at 1529 nm and 1542 nm respectively.

Figure 5(c) shows the optical spectrum after the HNLF_{2}. Two signal copies were de-multiplexed, amplified (A_{6} and A_{7}), delayed, and multiplexed to produce the signal copies mutually delayed by 250ps.

Amplified (A_{3}) pulse source and delayed signal copies were combined and launched into a 10 m-long HNLF_{3} characterized with a ZDW of 1554 nm, a dispersion slope of 0.025 ps/km/nm^{2}, and a nonlinear coefficient of 22 W^{−1}km^{−1}. Figure 5(d) shows the optical spectrum of the polychromatic parametric sampling after the HNLF_{3}. A severe cross-talk is readily identified by the appearance of higher order mixing products at 1578 nm. The sampled idlers were 2 ps-wide and carried mutually-delayed measures of the original waveform. The sampled pulses were dispersively reshaped using 200 m of SMF and amplified (A_{8} and A_{9}), prior to detection by a 2 GHz receiver. Receiver outputs were passed to a high-resolution (12-bit, 8.5 ENOB) electronic ADC operating at 2 GS/s. Finally, the output of the ADC was passed to the real-time equalization to measure the performance of a new cross-talk cancellation technique.

Figure 6(a)
shows the calibration sequence used for constructing the LUT during the processor training period. The sequence was specifically chosen to be a 3rd order pseudo random 32-level sequence (PR32S-3) at 4 GBaud/s generated from a 10-bit resolution digital-to-analog converter (DAC). The all zero state was added the PR32S-3 to make it De Bruijn sequences. The 3rd order of the generated De Bruijn sequence guarantees the occurrence of all possible level combinations in the adjacent symbol pairs of the training waveform. The primitive polynomial for generating the pseudo random sequence was chosen to be *x*^{3} + *x*^{2} + 6 over the Galois field of *GF*(32).

Figure 6(b) shows the RF spectrum of the calibration signal. At 2 GS/s, the calibration sequence indeed exhibits desired white-noise-like characteristic, as required for exciting the full ADC frequency band during the calibration phase. The cross blue and diamond red markers in Fig. 6(a) represent the optical parametric sampling points corresponding to the delayed signal copies.

Ensemble averaging over 128 realizations of the training sequence has been performed to reduce the system noise in the look-up table formation period. A 7th order, two-dimensional polynomial surface has been fitted to the collected 32 × 32 distortion correction matrices to fill the empty ADC levels between the 32 levels of the training signal. The calibration time required for 128 periods of the PR32S-3 sequence with a total number of 128 × (32^{3}-1) symbols was about 10 seconds, mainly dominated by the data transfer (I/O) latency. Figure 7
shows the distortion correction values for both idlers measured during the experiment. This representation was chosen as it is particularly instructive in case when two idlers are processed simultaneously: in this case a three-dimensional visualization is still possible – an additional idler will increase the dimensionality of the map and prevent simple plotting shown in Fig. 7.

Furthermore, the three-dimensional map provides an insight to distortion origins created in the parametric preprocessor. In the self-induced nonlinear distortion direction (indicated in Fig. 7) it is possible to clearly distinguish the sinusoidal excitation (input signal) generated by the Mach-Zehnder modulator (MZM). However, the waveform trace in this (self-induced) direction is not an ideal sine since the distortions include nonlinear contributions from the multicaster and the sampling gate. In the case when parametric stages are physically designed to minimize the nonlinear distortions (such in this experiment), the MZM-induced nonlinearity (distortion) will dominate. In contrast, the cross-talk between the participating waves (idlers) in the parametric processor is dominated by the mixing in the parametric gate. In particular, distinct cross-talk mechanisms within the parametric gate distort the idlers: The idler** _{1}** (Fig. 7(a)) is mainly affected by the cross-gain modulation (XGM), whereas the idler

**depletes the pump, which, in turn, leads to the parametric gain reduction. As a consequence, the correction table aims to increases the idler**

_{2}**level when the idler**

_{1}**is at a high level.**

_{2}In contrast, the idler** _{2}** is affected by a mixing product that consisting of the idler

**copy, which is a result of a FWM among pump, idlers and the newly generated wave at 1578 nm. Accordingly, the correction table decreases the idler**

_{1}**level when the idler**

_{2}**level is elevated. While this qualitative description is possible in the case of three-dimensional map shown in Fig. 7, its equivalent still exists as the number of idlers is increased, but in a multidimensional space not amenable to a simple graphical representation shown here. However, even in such a case, it is very important to note that**

_{1}*a single integer addition is the only operation required for each digitized sample in the introduced equalization scheme*.

The FFT outputs of the ADCs for the two idlers before and after applying the new equalization scheme are shown in Fig. 8
. The input analog signals under test were 201 MHz, 7.532 GHz, and 11.081 GHz harmonic waves and the signal modulator (MZM_{2}) was biased at the quadrature point to guarantee the absence of even harmonics in the signal spectrum entering the parametric multicaster. The quadratic distortion imposed by the parametric multicasting and sampling is clearly illustrated in Fig. 8, which exhibits a strong second harmonic distortion across the spectrum. As stated earlier, this particular idler distortion is generated as a combination of self-induced penalty and cross-talk from other channels. The total harmonic distortion for different test signals was calculated and depicted in the corresponding sub-plot in Fig. 8. For the Nyquist rate sampling case, 201 MHz signal, a 20 dB improvement in total harmonic distortion (THD) was achieved for both idlers. More importantly, this corresponds to as much as 3.3 ENOB enhancement in total ADC resolution. The total THD improvement was lower in cases of sub-Nyquist rate sampling, 7.351 GHZ and 11.081 GHz, and was attributed to the frequency dependent nonlinearity of the RF amplifier driving the signal modulator (MZM_{2}): the RF amplifier was not capable of achromatic performance and provided different response in calibration (low frequencies) and measurement regime (high frequencies). The performance results are summarized in Table 1
.

Considering that the inherent MZM and parametric nonlinearities are typically characterized by saturation-like nonlinearities, the inverse operation necessarily increases the input noise variance. In particular, for the case of the driving condition at 75% of the MZM_{2} *Vπ* and signal powers ensuring operation away from the mixer deep saturation, the noise enhancement was characterized and determined to be less than 3 dB. Finally, the two corrected idler samples were interleaved achieving the overall 4 GS/s of the analog signal in the CaSA photonics ADC structure. The overall (joint) performance of the two interleaved channels is shown in Fig. 9
. The overall performance improvement after linearization is also summarized in Table 1.

## 4. Conclusion

We have reported a new technique for nonlinear cross-talk cancellation in polychromatic parametric preprocessors. The new method relies on a multidimensional mapping of the input signals and construction of device-specific distortion correction tables (DCTs). The DCTs were generated by extracting the input/output nonlinear transfer function of the sampling gate by means of a calibration (training) sequence. The new method was implemented for a specific case of parametric CaSA photonics-assisted ADC using sub-Nyquist rate sampled channels. The performance of the technique in suppressing the cross-talk, as well as removing other nonlinearities (such as MZM) has been experimentally validated. Specifically, we have observed a maximal 20 dB reduction in total harmonic distortion, corresponding to a 3.3 ENOB enhancement in ADC resolution. The described approach removes the basic obstacle to high resolution, high speed photonic ADC realization and enables significant increase in tolerances of the parametric pre-processor design and construction in a wide area of photonic applications.

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