## Abstract

Taylor’s scheme for photonic quantization and encoding based on Mach-Zehnder modulators (MZMs) requires that the employed MZMs have geometrically scaled half-wave voltages (*V _{π}*), which is impractical even with the state-of-art photonic fabrication techniques when the desired bit resolution is greater than 3 or 4 bits. The approaches based on the phase-shifting of modulation transfer functions eliminate the need of geometrically scaled

*V*, but they realize lower resolution than Taylor’s scheme as the realized resolution is log

_{π}_{2}(2

*N*), but not

*N*as in Taylor’s scheme, where

*N*is the number of optical channels (or MZMs). In this paper, we propose a novel photonic quantization and encoding scheme based on waveform folding using rectifier circuits, which aims to realize higher resolution with less MZMs (and less

*V*). In our design, a 4-bit quantization can be achieved using 2 MZMs with identical

_{π}*V*with the help of two rectifiers. A proof-of-concept experiment is implemented, which fully verifies the correctness of the approach. The scheme is modular extendable, i. e. an 8-bit quantization can be realized by using 4 MZMs (with 2 different

_{π}*V*), and 12-bit can be realized by using 6 MZMs (with 3 different

_{π}*V*). The impact of the rectifiers’ bandwidth on the system performance is also investigated. As less MZMs are employed and lower requirement on

_{π}*V*scaling, the proposed design provides a promising solution for high-performance photonic analog-to-digital conversion.

_{π}© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

High-speed analog-to-digital converters (ADCs) are demanded in many applications with wideband signal acquisition, such as high-speed optical and wireless communications, advanced radars, electronic monitor, real-time signal processing, measurement and modern instrumentations [1–3]. However, limited by the inherent aperture jitter of sampling clock, it is very difficult to achieve ADCs with ultrahigh sampling rate (i. e. higher than tens of GHz) without degrading conversion resolution [4]. Benefits from the high bandwidth of photonic components and the extremely low jitter of mode-locked lasers as well as the frequency comb lasers, photonic ADCs are widely regarded as promising candidates for wideband signal conversion [5]. Typical schemes of photonic ADCs include the technique of photonic sampling and electronic digitizing [6–8], the photonic time stretch for signal slowing down prior to electronic digitizing [9–11], the approaches based on optical nonlinearities for realizing spectral encoding and optical quantization [12,13], the photonic quantization and encoding schemes based on Mach-Zehnder interferometer(s) or modulator(s), etc [14,15].

In 1970s, Taylor firstly proposed a photonic quantization and encoding scheme using optical modulator array [14]. In his approach, a number of Mach-Zehnder modulators (MZMs) with geometrically scaled half-wave voltages (${V_\pi }$) are employed to realize quantization and encoding. In his scheme, the digital output is a Gray code, the number of achieved quantization levels is 2* ^{N}* and the resolution is the same as the number of employed MZMs, i.e.,

*N*. However, limited by the need for geometrically scaled ${V_\pi }$, it is hard to realize an ADC with the resolution higher than 3 or 4 bits even with the state-of-art photonic fabrication techniques. In order to overcome this limitation, the approaches based on cascaded MZMs or phase modulators (PMs) to realize equivalent ${V_\pi }$ scaling were proposed in [16,17]. In 2005, Stigwall et al. proposed a quantization scheme using a free-space Mach-Zehnder interferometer (MZI) with a PM in one arm [18]. They place photodetectors (PDs) at different positions in the interference pattern in order to realize the desired phase shifts of the modulation transfer functions. Unlike the waveform folding using modulation transfer functions with different ${V_\pi }$ in Taylor’s approach, Stigwall’s approach alleviates the difficulty relating to ${V_\pi }$ scaling based on the phase-shifting of modulation transfer functions. The concept of phase-shifting modulation curves rather than ${V_\pi }$ scaling was adopted by many other approaches, which include the approach using a PM and polarization interferometric phase shifters [19], that using properly biased MZMs with identical ${V_\pi }\; $[15], the differential encoding scheme with a single PM and delay line interferometers [20], that based on an unbalanced MZM [21], as well as the approach using an unbalanced MZM and filters [22]. The major drawback of the phase-shifting based quantization schemes lies in its relatively lower quantization levels, which is 2

*N*for a system with

*N*channels but not 2

*as in Taylor’s scheme. Accordingly, its resolution is $\textrm{lo}{\textrm{g}_2}({2N} )$, but not*

^{N}*N*bits. Hence, how to improve the bit resolution is a major issue for phase-shifting based ADC schemes. The possible solutions include the scheme using symmetrical number system encoding [23,24], the approach using electrical circuits for linear combination [25], as well as the method with cascaded quantization modules [26]. However, these solutions improve the resolution at the cost of a large number of comparators, complex logic circuits or specially designed MZMs with cascaded optical couplers. Therefore, it is still an open question to find a relatively simple design of phase-shifting based quantization scheme with improved resolution.

In this paper, we propose a novel design of photonic quantization and encoding scheme based on waveform folding using rectifier circuits, which aims to realize higher resolution with less MZMs as well as less ${V_\pi }$ scaling. The rectifiers are employed to implement waveform folding, as that realized by MZMs with geometrically scaled ${V_\pi }$ in Taylor’s scheme. In our design, a 4-bit quantization can be achieved using 2 MZMs with identical ${V_\pi }$ with the help of two rectifiers. We implement a proof-of-concept experiment with a 4-bit A/D conversion to verify the correctness and feasibility of the approach. We further discuss the extendibility of the approach in order to achieve higher resolution and present a modular extendable scheme. It is shown that an 8-bit quantization can be realized by using 4 MZMs (with 2 different ${V_\pi }$), and 12-bit can be realized by 6 MZMs (with 3 different ${V_\pi }$). We also present a comparison of the required number of MZMs and minimum normalized ${V_\pi }$ between the proposed scheme and Taylor’s scheme as well as a typical phase-shifting based quantization scheme in [15]. In addition, we investigate the impact of the bandwidth of rectifiers on the system performance in terms of ENOB.

## 2. Design of a 4-bit quantization and encoding module

The schematic diagram of the proposed 4-bit photonic quantization and encoding module is shown in Fig. 1. The system consists of a pulsed laser source, two MZMs with identical ${V_\pi }$, two rectifiers, two PDs and four comparators. A pulse train generated by the pulsed laser is sent to two parallel MZMs to sample the input analog signal to be digitized, which is applied to the MZMs via the radio frequency (RF) ports. The two MZMs should be properly biased. The modulated optical signal from MZM 1 is detected by PD 1 to perform the optical to electrical (O/E) conversion and then sent to Comp. 1 to generate the MSB. The output optical signal from MZM 2 is converted into an electrical signal by PD 2 and then split into two equal channels (Channel 2 and 3). Channel 2 is directly linked to Comp. 2 to generate the MSB-1, and Channel 3 is connected with Rect. 1 to obtain desired signal rectification. One output from Rect. 1 is delivered to Comp. 3 to achieve the MSB-2; the other output is sent to Rect. 2, which drives Comp. 4 to generate the LSB.

Note that the waveforms in Channel 3 and 4 (corresponding to MSB-2 and LSB) are reshaped by the rectifiers, which realize the demanded waveform folding and period scaling. By properly setting the threshold values of comparators, a digital Gray code can be obtained. The optical intensity from the *j*-th MZM (*j *= 1, 2) can be expressed as

The output of Rect. 1 is then sent to Rect. 2 to further reshape the waveform. Rect. 2 also contains a subtractor and a full-wave rectifier. Different from Rect. 1 that folding the waveform at half, Rect. 2 folds the input waveform at the value of $\sqrt 2 {I_{3max}}/2$ (denoted by dotted line in Fig. 2), where ${I_{3max}}$ means the maximum value of ${I_3}$. Therefore, the function of Rect. 2 can be given by

The detailed quantization and encoding process of the 4-bit module is shown in Fig. 3. Figures 3(a)–3(c) show the modulation transfer functions of four channels, where the horizontal axis denotes the phase shift induced by the input analog signal and the vertical axis indicates the normalized signal intensity prior to comparators. We set the threshold value of both the comparators for MSB and MSB-1 as half of the full scale, which is denoted by the horizontal solid line in Fig. 3(a). In order to partition the full range of input voltage into 4 equal parts, we set the threshold value for MSB-2 as $1/\sqrt 2 $ of the maximum, which is denoted by the horizontal dashed line in Fig. 3(b). For LSB, the threshold value is set as *M* times of the maximum so as to divide the voltage range (0∼${V_\pi }$) into 8 equal parts, where *M* is expressed as

From Figs. 3(b) and 3(c), we can see that the modulation transfer function curves are folded horizontally with factors of 1/2 and 1/4 with the help of the rectifiers and proper threshold settings, which are equivalent to the folding functions induced by the ${V_\pi }$ scaling down of MZMs in Taylor’s scheme.

At the comparators, the signal intensity greater than the preset threshold is digitized as “1”, otherwise as “0”. As shown in Fig. 3(d), the output digital code is a Gray code, which significantly reduces the possible readout error since any two consecutive output codes differ in only one bit.

Note that the above scheme with four channels and two MZMs achieves 16 quantization levels and therefore a resolution of 4 bits. In addition, two employed MZMs have identical ${V_\pi }$. As a comparison, to achieve the same resolution in Taylor’s scheme, four MZMs with geometrically scaled ${V_\pi }$ are necessary. Meanwhile, the quantization schemes based on the phase-shifting technique should have 8 independent channels to achieve a resolution of 4 bits. Therefore, the proposed 4-bit quantization and encoding module dramatically reduces the system complexity and is therefore much more feasible for practical applications and future system integration in a chip. Another advantage of the given approach is that it can be easily extended into higher resolution, e.g. 8 bits or 12 bits, in a modular way, which will be discussed in the next section.

## 3. Results and discussions

A proof-of-concept experiment demonstration of the above 4-bit quantization and encoding module is carried out. In the experiment, a continuous-wave laser diode (Yokogawa AQ 2201) with a wavelength of 1550 nm and an output power of 5 dBm is employed as the light source. A 4-GHz RF signal with a power of 18 dBm provided by a signal generator (Agilent E8254A) is used to drive a MZM (20-GHz, JDS-Uniphase). A PD (45-GHz, New Focus 1104) is used to perform the O/E conversion. The output temporal waveform is observed and recorded by a digital sampling oscilloscope (Agilent DCA 86100C). The captured waveforms are processed in an off-line program in computer, which includes the functions of the rectifiers and comparators.

In the experiment, the waveform data of MSB and MSB-1 channels are obtained with the oscilloscope by setting the bias voltage of the MZM at ${\textrm{V}_\pi }/8\; $ and $5{\textrm{V}_\pi }/8$, respectively. The waveform data of MSB-2 and LSB are obtained by processing the recorded data with emulating the functions of two rectifiers. The recorded and emulated 4 waveforms are shown in Fig. 4(a). Based on the waveform data, we sample the waveforms to get discrete waveform data at a fixed time interval. We set the threshold values of four channels as $1/2$, $1/2$, $\sqrt 2 /2$, *M* times of the maximum (according to Eq. (2)), respectively, as described in the previous section. The quantized signal (normalized) is shown in Fig. 4(b). For comparison, the fitted sinusoidal signal is also given. Figure 4(c) shows the errors between the quantized and the fitted signals. The errors are introduced by the quantization noise and the noises on signal, which include the random noises (thermal, shot noise) and nonlinearities in system. Moreover, the phase bias-drift of MZMs and the ambiguity of comparators also induce errors between the quantized and fitted signals.

Based on the errors between the quantized and the fitted signals, the digital signal-to-noise ratio (dSNR) is estimated to be around 23.8 dB, which corresponds to an effective number of bits (ENOB) of 3.67, according to the formula ENOB*=(*dSNR-1.76)/6.02. Since the resolution is 4 bits, the ENOB deviation from the ideal case is 0.33 bit. We believe major sources of the ENOB degradation include the system noises (e. g. thermal noise and shot noise), the bias voltage setting errors and the threshold setting errors.

Based on the proposed 4-bit quantization and encoding module, we can construct ADC systems with higher bit resolution in a modular way. A system with a resolution of 4*n* (*n*$> 1$) is designed as shown in Fig. 5, which contains *n* modules. The modules within the system are the same as that in Fig. 1 and two MZMs in each module have the identical ${\textrm{V}_\pi }$. Each module corresponds to 4 bits of ADC output. Half-wave voltages of MZMs in different modules are configured in geometrically scaling-down from the first module to the last module, i.e.${V^0}$, ${V^0}/2$, ${V^0}/4$,······. Therefore, by a system with 2 modules, we can realize a resolution of 8 bits, which needs 4 MZMs with 2 different ${\textrm{V}_\pi }$. Similarly, a 12-bit system needs 6 MZMs with 3 different ${\textrm{V}_\pi }$. We can see that the requirements on the number of MZMs and the ${\textrm{V}_\pi }$ scaling are much lower than that of Taylor’s scheme, under the same resolution. For an n-bit system, the required number of MZMs in proposed scheme is reduced to half of that in Taylor’s scheme and the needed ${\textrm{V}_\pi }$ is $1/{2^{3\textrm{n}}}$ of that in Taylor’s scheme. Furthermore, compared with the phase-shifting based schemes, which also avoids the ${\textrm{V}_\pi }\; $ scaling, the proposed approach largely improves the bit resolution and simplifies the system configuration, i.e. 4n MZMs are used in proposed scheme while ${2^{4\textrm{n} - 1}}$ MZMs are required in work in [15] to realize an n-bit quantization system. A detailed comparison among this work, Taylor’s approach and a typical phase-shifting based approach in [15], including the number of employed MZMs and the minimum normalized ${\textrm{V}_\pi }$, is presented in Table 1, where we assume the ${\textrm{V}_\pi }$ for MSB as 1 in all approaches.

As Taylor's scheme, the waveform folding and period scaling in rectifiers would lead to the frequency multiplication of the input signal. The bandwidth of rectifiers has major influence on waveform distortion and therefore the performance degradation. Rectifiers with a bandwidth higher than 35 GHz has been reported [27]. Here we investigate the impact of the bandwidth of rectifiers on the system performance in terms of ENOB in a 4-bit module by simulation. It is assumed a 4 GHz signal as an input signal. In the simulation, the bandwidth of rectifiers in MSB-2 and LSB channels varies from 0 GHz to 30 GHz. Figure 6 gives the dependence of ENOB on the bandwidth of rectifiers. It is observed that ENOB increases in a step-wise fashion with the increase of the rectifiers’ bandwidth. ENOB reaches 3.67 bits when the bandwidth of rectifiers is 21 GHz or higher. The worst ENOB is around 2 bits, corresponding to the bandwidth of 0, which means the rectifiers output 0.

## 4. Conclusions

In this paper, a photonic ADC approach based on waveform shaping using rectifier circuits was presented and demonstrated. A 4-bit quantization and encoding module with two MZMs (identical ${\textrm{V}_\pi }$) and two rectifiers was designed to improve bit resolution with less requirement on ${\textrm{V}_\pi }$ scaling. The principle of quantization and encoding has been described and a proof-of-concept experiment was presented to verify the scheme. We also present a modular extension scheme for realizing A/D conversion with higher bit resolution. It is shown that the proposed approach can greatly improve the bit resolution by using fewer MZMs and avoiding ultralow ${V_\pi }$. Furthermore, the proposed ADC approach overcomes the limitation of low bit resolution relating to the phase-shifting based ADC schemes. In addition, the ideal bandwidth of the used rectifiers for a 4 bit quantization system is given.

## Funding

Zhejiang Provincial Natural Science Foundation (LQ18F050002, LZ20F010003); National Natural Science Foundation of China (61901148, 61975048).

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