Open Access
Add to BibSonomy
Abstract
We propose what we believe to be a novel approach to enhance the dynamic range of a photonic analog-to-digital converter (PADC) without the need of additional custom-designed circuits or components. The method utilizes the unique characteristic of our previously reported multimode interference (MMI) coupler-based optical quantizer that exploits the periodicity of the optical phase to realize a modulo operation. Experiments were carried out to verify the effectiveness of the proposed method on our phase-shifted optical quantization ADC (PSOQ-ADC) chip. Experimental results show that our proposed method enhance the dynamic range from [$- {V_\pi }$, ${V_\pi }$] to [$- 2{V_\pi }$, $2{V_\pi }$] and has the potential to be further extended. Additionally, we successfully reconstructed radio frequency (RF) signals at a sampling rate of 30 Gs/s. Our work provides a promising solution for achieving a high dynamic range in on-chip PSOQ-ADC.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-performance analog-to-digital converters (ADCs) are essential in advanced information processing systems, such as radar and optical communication [1–3]. However, electronic ADCs (EADCs) face challenges in achieving higher performance due to aperture jitter and limited analog bandwidth of electronic components [4,5]. Photonic ADCs (PADCs) offer a promising solution, utilizing low-time-jitter optical sampling pulse trains generated by mode-locked lasers (MLLs) and ultra-wide analog bandwidth of electro-optic modulators [6–9]. According to the implementation principles, PADCs can be typically divided into three types: photonic-assisted ADC (PA-ADC) [10,11], photonic sampled and electronic quantized ADC (PS-ADC) [12,13], and photonic sampled and quantized ADC (PSQ-ADC) [14–21]. Among them, the PSQ-ADC realizes all-optical operation in sampling and quantization, which makes the best use of the advantages of optical components. Consequently, PSQ-ADCs have attracted significant attention in recent years.
PSQ-ADC architectures can be further classified into two categories by different quantization methods: utilizing intensity-to-wavelength optical quantization (IWOQ) [14,15] and phase-shifted optical quantization (PSOQ) [16–23]. However, the IWOQ-ADC requires relatively high optical input power and system stability, which makes it hard to achieve higher performance in monolithic integration. The PSOQ-ADC scheme, initially proposed by Taylor [16], offers promising advantages in terms of on-chip integration and required low input optical power. The major drawback of Taylor's system is that it requires the employed Mach-Zehnder modulators (MZMs) to have geometrically scaled half-wave voltages as resolution increases, which is impractical for improving resolution higher than 3-bit [15]. In contrast, exploiting multimode interference (MMI) couplers to accomplish PSOQ shows the simplest structures and on-chip integrability. As reported in [19,22], a 3-bit PSOQ-ADC based on MMI was successfully validated on silicon-on-insulator (SOI) and lithium niobate-on-insulator (LNOI) platform. Higher resolution (>=5-bit) can be achieved through structural optimization [21].
However, most reported PADCs mainly focus on increasing in resolution or RF bandwidth but ignore the dynamic range [19,24]. Dynamic range is an important parameter that cannot be neglected. For a radar sensor, the received echo power within range R from a radar target is scaled by ${R^{ - 4}}$ [25], meaning the target moving at different ranges exhibits wildly varying amplitudes. A limited dynamic range may result in the permanent loss of information due to amplitude saturation and clipping, which limits the ability of radar to simultaneously adapt both strong and weak received power. Therefore, achieving a high dynamic range in advanced receiver systems is an open issue [26].
Based on our previously reported PSOQ using MMI fabricated on the SOI platform [22], we find a unique characteristic: the periodicity of the optical phase can be exploited by the MMI-based optical quantizer to perform a modulo operation. That means when the amplitude of an input analog signal exceeds the dynamic range of the PADC based on the PSOQ, it preserves the information in a folded form rather than clipping it. As reported in [22], experiments were carried out as proof-of-concept to test the functionality of the PSOQ. However, it is challenging to meet the need of high-speed pure phase modulation based on the SOI platform because it is unavoidably accompanied by amplitude modulation. On the other hand, the LNOI platform is well-known for high-speed, pure phase modulation operation. In this paper, the unique characteristic of our previously proposed optical quantizer is fully utilized to enhance the dynamic range of a PADC. We then experimentally demonstrated the proposed method on our fabricated MMI-based PSOQ-ADC chip on the LNOI platform. Firstly, the quantization curves and the transfer function of the PSOQ-ADC were measured and analyzed. Experimental results show that the dynamic range is enhanced from [$- {V_\pi }$, ${V_\pi }$] to [$- 2{V_\pi }$, $2{V_\pi }$] and has the potential to be further extended. Additionally, we successfully reconstructed a sinewave and an arbitrary waveform at a sampling rate of 30 Gs/s. These results highlight the potential of our work as a solution for high-dynamic on-chip PADCs in advanced information processing systems.
2. Principle
Figure 1(a) depicts the schematic diagram of the MMI-based PSOQ-ADC chip. An optical sampling pulse train is generated from a sampling pulse source and then fed into the PSOQ-ADC chip. Inside the chip, the amplitude information of an input analog signal ${V_{in}}(t )$ is converted into the phase difference $\Delta \varphi (t )\; $ of the sampling pulse train through a phase modulator. $\Delta \varphi (t )\; $ is automatically wrapped in [$- \pi ,\pi $] in the optical quantizer due to its periodicity. After the optical quantizer, the output channels of the PSOQ-ADC chip are connected to an array of photodetectors (PDs) and comparators (Comps) to obtain the digital code. In the subsequent digital signal processing (DSP) unit, an unwrapping algorithm is employed to reconstruct the original phase information.
Fig. 1. (a) The schematic diagram of the MMI-based PSOQ-ADC. (b) The input analog signal is converted to the phase difference can be combined by (c) wrapped phase and (d) an integer multiple of $2\pi $. MMI: multimode interference. PSOQ-ADC: phase-shifted optical quantization analog-to-digital converter. PD: photodetectors. Comp: comparators. DSP: digital signal processing.
2.1 Modulo operation of the MMI-based PSOQ-ADC
After the sampling pulse train ${E_{in}}(t )$ enters the phase modulator, the phase difference $\Delta \varphi (t )$ generated by the phase modulator is proportional to the amplitude of ${V_{in}}(t )$ based on the Pockels effect, it can be equivalent to
where ${V_\pi }$ denotes the half-wave voltage of the phase modulator. After phase modulation, two inputs of the optical quantizer are expressed asAccording to the self-imaging theory, the output power transmission curves of K-output channels of the quantizer are represented as [20],
The power transmission curves have a fixed phase shift of 2π/K between the adjacent output channels, enabling phase-shifted quantization. For this paper, we adopt a 2 × 5 MMI structure, i.e., K = 5. In general, the unenhanced dynamic range of PSOQ-ADC is [-${V_\pi }$, ${V_\pi }$], corresponding to [-π, π] in phase domain.
In Eq. (3), when $\Delta \varphi $ swings large than π, it undergoes wrapping within [-π, π] due to its periodicity of 2π, as shown by the red line in Fig. 1(c). This phase-wrapping process is equivalent to performing a modulo operation on $\Delta \varphi $ with 2π as the modulus. Mathematically, we express the phase modulo operation as
And $\Delta \varphi $ can be rewritten in the form of a summation
where $\Delta {\varphi _\mathrm{{\cal M}}} = {\mathrm{{\cal M}}_\pi }({\Delta \varphi } )$ represents the phase residue after modulo operation, also known as the wrapped phase, ${\varphi _r}$ is an integer multiple of the modulus, given by ${\varphi _r} ={\pm} 2m\pi ,{\; \; }m \in \mathrm{\mathbb{Z}}$, as shown in Fig. 1(d). According to the above process, $\Delta \varphi $ is replaced by $\Delta {\varphi _\mathrm{{\cal M}}}$ in Eq. (3), which results in ${I_i}(t )$ presenting a modulo operation. Besides, it can be seen that the modulo operation is naturally achieved by the MMI-based optical quantizer without requiring additional electronic circuits or components.We can obtain an electrical signal after the optical-to-electrical (O/E) conversion performed by the PD arrays. Next, the detected electrical signals of the 5-output channels are injected into the electrical comparator arrays, and the comparator array judges the value as “0” or “1” based on an appropriate threshold. In scenarios where the dynamic range of $\Delta \varphi $ is within [-π, π], the quantization process results in 10 quantization levels [20], as indicated by the black box in Fig. 2(a-c). When the swing of $\Delta \varphi $ exceeds 2π, $\Delta \varphi $ is naturally wrapped within [-π, π] without clipping, as depicted by the gray curve in Fig. 2(c). Theoretically, if the wrapped phase $\Delta {\varphi _\mathrm{{\cal M}}}$ is unwrapping to its unfolded form, it becomes possible to perfectly reconstruct the phase difference beyond the range of [-π, π]. To illustrate this concept, here is an example of a dynamic range of [-2π, 2π]. The gray curve in Fig. 2(d) shows the expansion of the dynamic range, increasing the number of quantization codes from [0,9] to [0,19]. Thus, with each additional fold, the dynamic range expands proportionally. Next, we proceed to reconstruct the quantized wrapped signal.
Fig. 2. (a) Transmission curves of the MMI-based PSOQ-ADC when the dynamic range of phase difference exceeds [-π, π]. (b) Corresponding comparator results. (c) Wrapped quantized code. (d) Dynamic range expansion after using the unwrapping algorithm.
2.2 Phase unwrapping algorithm
As Fig. 2 shows, the quantized process can be defined as $Q(\cdot)$. The wrapped phase difference ${\varphi _\mathrm{{\cal M}}}$ after quantization process is expressed as
Since ${\varphi _r} ={\pm} 2m\pi ,{\; \; }m \in \mathrm{\mathbb{Z}}$, $Q({\Delta \varphi } )$ can be rewritten as,
Hence, when ${\varphi _r}$ is obtained, we can reconstruct $Q({\Delta \varphi } )$.
${\varphi _r}$ can be obtained from $Q({\Delta {\varphi_\mathrm{{\cal M}}}} )$ based on iterating the finite difference operation until ${\mathrm{\delta }^N}({\varphi _r}) = {\mathrm{{\cal M}}_\pi }({{\mathrm{\delta }^N}Q(\Delta {\varphi_\mathrm{{\cal M}}})} )$ is satisfied [27]. Besides, we define the first-order difference of a signal $s[k ]$ as ${\mathrm{\delta }^1}({\textrm{s}[k ]} )= s[{k + 1} ]- s[k ]$. The Nth-order difference operator, ${\mathrm{\delta }^N}({\textrm{s}[k ]} )$, can be computed iteratively as ${\mathrm{\delta }^N}({\textrm{s}[k ]} )= {\mathrm{\delta }^{N - 1}}({{\mathrm{\delta }^1}({\textrm{s}[k ]} )} )$ . The order of finite difference N is computed by
After getting ${\mathrm{\delta }^N}({\varphi _r})$, we need to compute the value of ${\varphi _r}$ from ${\mathrm{\delta }^N}({\varphi _r})$ through iteratively performing N-1 anti-difference operation. The anti-difference operator $\mathrm{{\cal S}}$ is defined as follows
In the first anti-difference iteration, we assume the initial value of ${\mathrm{\delta }^N}({\varphi _r}[0 ])$ as
By applying $\mathrm{{\cal S}}$ operation once, we obtain,
3. Experiments and results
3.1 Measurement of the transfer function of dynamic range of the PSOQ-ADC
We measured the transfer function to show the effectiveness of our proposed method in improving the dynamic range of the PSOQ-ADC. Figure 3 illustrates the experimental setup used to measure the transfer function of the MMI-based PSOQ-ADC chip on the LNOI platform. Firstly, a tunable semiconductor laser (TSL) was utilized as the incident light source to generate input light at 1534 nm and then connected to a polarization controller (PC). Next, the light was coupled into the PSOQ-ADC chip. A multi-channel programmable voltage source (PVS) was employed to control the applied voltage on the electrodes of the phase modulator. Therefore, the phase difference $\Delta \varphi $ changed over the applied voltage from PVS. We used a multi-port optical power meter (MPM) to measure the quantization curves from the PSOQ-ADC.
Fig. 3. Experimental setup for measuring the transfer function of the MMI-based PSOQ-ADC chip. PC: polarization controller. PVS: programmable voltage source. MPM: multi-port optical power meter. GPIB: general interface bus.
To measure the quantization curves of the 5-output channels, we gradually increased the applied voltage of the PVS from -5.45 V to 5.45 V and recorded the 5-output optical power of the PSOQ-ADC chip. Figure 4(a) represents the normalized quantization curves of the 5-output channels of the PSOQ-ADC. We used half of the normalized power, i.e., 0.5, as the decision threshold for the comparator array. The crossing points between the normalized power quantization curves and the threshold line, indicated by blue stars, determine the starting point of each quantization zone. Then, we derived the transfer function from the measured quantization curves. As Fig. 4(b) shows, the red dashed line and the black line in the plot represent the ideal transfer function and the measured wrapped transfer function, respectively. It can be observed from the obtained transfer function that the quantization levels are periodically limited to [0,9] due to the modulo operation occurring in the optical quantizer. In the phase domain, the modulus is 2π, corresponding to 10 in the digitized output of the PSOQ-ADC. Therefore, the modulus was set as 10 to unfold the unwrapped transfer function. The quantization levels can be extended to [0,19], which achieves a doubling of the dynamic range, as the blue line shows in Fig. 4(c).
Fig. 4. Experimental results of measuring the transfer function of the MMI-based PSOQ-ADC chip. (a) Normalized quantization curves of the 5-output channels of the PSOQ-ADC, with phase error remaining unchanged before and after modulo operation. (b) The measured wrapped transfer function. (c)The measured transfer function after unwrapping. Ch: channel.
As shown in Fig. 4(b) and (c), the measured transfer curves, represented by black and blue lines, deviate from the ideal transfer curves represented by red dashed lines. The deviations are caused by nonlinear errors generated from the quantizer design and fabrication process. Although this work does not currently consider the nonlinear errors of the quantizer, in practical applications, they can introduce harmonic components that degrade the signal-to-noise ratio (SNR). Therefore, further research is essential to explore DSP algorithms for mitigating the nonlinear errors to improve the overall system performance.
In addition, while Fig. 4 only illustrates experimental results of a doubling of the dynamic range, when the amplitude of an input signal is several times the unenhanced dynamic range of the PSOQ-ADC, the modulo operation remains effective, preserving the signal in a folded form. Therefore, the folded signal can be reconstructed by the unwrapping algorithm, thereby further expanding the dynamic range.
3.2 Analog to digital conversion experiments
Analog to digital conversion experiments were carried out to further confirm the feasibility of the proposed algorithm on PADC. The experimental setup is shown in Fig. 5. A light generated from a continuous wave (CW) source was modulated using a commercial Mach-Zehnder modulator (MZM). The MZM was biased at 0.64V using a DC source, and the input microwave signal denoted as RF1, had a frequency of 15 GHz. This configuration resulted in generating a 30 GS/s optical sampling pulse train. After being connected to an erbium-doped fiber amplifier 1 (EDFA1) and a polarization controller (PC), the optical sampling pulse train passed into the PSOQ-ADC chip. We employed either a microwave source (MWS) or an arbitrary waveform generator (AWG) to generate the analog signals to be sampled. Next, the 5-output channels optical signals from the PSOQ-ADC were amplified using EDFA2 and filtered using a band pass filter (BPF). Finally, the output waveforms were recorded by an oscilloscope (OSC), and the data was exported to a computer for offline processing. RF1, Sig1, Sig2, and OSC were all kept synchronized.
Fig. 5. The experimental setup of analog to digital conversion. CW: continuous wave. MZM: Mach-Zehnder modulator. RF: radio frequency signal. EDFA: erbium-doped fiber amplifier. ASI: analog signal input. MWS: microwave source. LNA: low-noise amplifier. AWG: arbitrary waveform generator. Sig: signal. BPF: bandpass filter. OSC: oscilloscope.
Firstly, we used a sinusoidal signal (Sig1) as the analog signal input (ASI). The Sig1 was generated using a MWS and with the frequency of 1 GHz. Based on the measurement results from the previous section, the half-wave voltage ${V_\pi }$ of the phase modulator is 2.73 V. To ensure modulo operation occurs in the phase modulator, we set the amplitude of Sig1 as 22 dBm. In Fig. 6(a), the blue lines depict the normalized recorded waveforms of the 5-output channels of our PSOQ-ADC, while the red dashed lines represent the envelope of the output signals. After extracting envelope information from Fig. 6(a), we generated Fig. 6(b). Next, we simulated the threshold decision process in the computer to obtain the quantized wrapped samples.
Fig. 6. The results of the analog to digital conversion experiment for the 1GHz sinewave signal. (a) Normalized recorded waveforms of the 5-output channels of PSOQ-ADC. (b) Extracted envelope information from the 5-output normalized recorded waveforms. (c)Wrapped 1GHz sinewave samples. (d) Unwrapped 1GHz sinewave samples.
After the quantization process, firstly, we calculated the order of finite difference to be 1 using Eq. (9). Then, a modulo operation was performed on the wrapped samples with a modulus of 10. The result of the modulo operation was added to the wrapped samples to obtain the unwrapped samples. Figure 6(c) displays the wrapped samples, while Fig. 6(d) shows the unwrapped samples obtained after applying the phase unwrapping algorithm. The red dots in Fig. 6(d) represent the recovered signal, and the gray dashed line represents the ideal input microwave signal. Discrepancies between the two can be attributed to inherent quantization noise, optical sampling pulse time jitter, EDFA noise, and quantizer phase errors.
Secondly, considering the practical case, we conducted an experiment that chose the analog signal (Sig2) with a more complex waveform and varying amplitudes. To simulate a practical high-dynamic-range analog signal, firstly, we generated an arbitrary signal with a bandwidth of 1GHz by an AWG, and then the generated signal was amplified by a low-noise amplifier (LNA). The spectrum of Sig2 is shown in Fig. 7(a). With our proposed method, the clipped information is preserved in a folded form in Fig. 7(b), and Fig. 7(c) demonstrates the successful reconstruction of Sig2, indicating that the proposed PSOQ-ADC method effectively digitizes signals with large amplitude variations without saturation. The results in Fig. 7 highlight the feasibility of our proposed PSOQ-ADC method for digitizing signals with wide dynamic ranges.
Fig. 7. The results of the analog to digital conversion experiment for an arbitrary waveform. (a) The spectrum of the arbitrary waveform is captured from a spectrum analyzer. (b) Wrapped and (c) Unwrapped samples using our proposed method.
4. Conclusions
We propose a novel method for enhancing the dynamic range of PADC. The method utilizes a modulo operation automatically occurring in the MMI-based optical quantizer, allowing for an enhancement of the dynamic range without the need of extra electronic circuits or components. We establish a mathematical model to describe the principle of dynamic range enhancement and implement a phase unwrapping algorithm adapted to quantized phase signals. Experimental validation is based on our fabricated MMI-based PSOQ-ADC chip on LNOI platform. Experimental results show an improvement in dynamic range from [$- {V_\pi }$, ${V_\pi }$] to [$- 2{V_\pi }$, $2{V_\pi }$] and have the potential to be further extended. Additionally, we successfully digitized and reconstructed a sinewave and an arbitrary waveform at a 30 Gs/s sampling rate. Furthermore, the method can also be applied to other PSOQ-ADC structures that support phase modulo operation, offering a promising solution for high-performance PADCs.
Funding
National Natural Science Foundation of China (61935003, 62021005, 62275029).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. Awny, R. Nagulapalli, M. Kroh, J. Hoffmann, P. Runge, D. Micusik, G. Fischer, A. C. Ulusoy, M. Ko, and D. Kissinger, “A linear differential transimpedance amplifier for 100-gb/s integrated coherent optical fiber receivers,” IEEE Trans. Microw. Theory Technol. 66(2), 973–986 (2018). [CrossRef]
2. P. J. Winzer, D. T. Neilson, and A. R. Chraplyvy, “Fiber-optic transmission and networking: the previous 20 and the next 20 years,” Opt. Express 26(18), 24190–24239 (2018). [CrossRef]
3. P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014). [CrossRef]
4. B. Murmann, “Adc performance survey 1997-2022,” http://web.stanford.edu/murmann/adcsurvey.html.
5. R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Select. Areas Commun. 17(4), 539–550 (1999). [CrossRef]
6. A. Khilo, S. J. Spector, M. E. Grein, et al., “Photonic adc: overcoming the bottleneck of electronic jitter,” Opt. Express 20(4), 4454–4469 (2012). [CrossRef]
7. M. Hasegawa, T. Satoh, T. Nagashima, M. Mendez, and T. Konishi, “Below 100-fs timing jitter seamless operations in 10-gsample/s 3-bit photonic analog-to-digital conversion,” IEEE Photonics J. 7(3), 1–7 (2015). [CrossRef]
8. X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-ghz large-signal modulation bandwidth in hybrid silicon photonics mach zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019). [CrossRef]
9. J. Meng, M. Miscuglio, J. K. George, A. Babakhani, and V. J. Sorger, “Electronic bottleneck suppression in next-generation networks with integrated photonic digital-to-analog converters,” Adv. Photonics Res. 2(2), 2000033 (2021). [CrossRef]
10. F. Laghezza, F. Scotti, P. Ghelfi, A. Bogoni, and S. Pinna, “Jitter-limited photonic analog-to-digital converter with 7 effective bits for wideband radar applications,” in proceeding of Radar Conference (IEEE, 2013), pp. 1–5.
11. Y. Mandalawi, J. Meier, K. Singh, M. I. Hosni, S. De, and T. Schneider, “Analysis of bandwidth reduction and resolution improvement for photonics-assisted adc,” J. Lightwave Technol. (2023).
12. H. Gevorgyan, K. Al Qubaisi, M. S. Dahlem, and A. Khilo, “Silicon photonic time-wavelength pulse interleaver for photonic analog-to-digital converters,” Opt. Express 24(12), 13489–13499 (2016). [CrossRef]
13. G. Yang, W. Zou, L. Yu, K. Wu, and J. Chen, “Compensation of multi-channel mismatches in high-speed high-resolution photonic analog-to-digital converter,” Opt. Express 24(21), 24061–24074 (2016). [CrossRef]
14. T. Nagashima, M. Hasegawa, and T. Konishi, “40 gsample/s all-optical analog to digital conversion with resolution degradation prevention,” IEEE Photonics Technol. Lett. 29(1), 74–77 (2017). [CrossRef]
15. F. Mehdizadeh, M. Soroosh, H. Alipour-Banaei, and E. Farshidi, “All optical 2-bit analog to digital converter using photonic crystal based cavities,” Opt. Quantum Electron. 49(1), 38 (2017). [CrossRef]
16. H. Taylor, M. Taylor, and P. Bauer, “Electro-optic analog-to-digital conversion using channel waveguide modulators,” Appl. Phys. Lett. 32(9), 559–561 (1978). [CrossRef]
17. J. Stigwall and S. Galt, “Demonstration and analysis of a 40-gigasample/s interferometric analog-to-digital converter,” J. Lightwave Technol. 24(3), 1247–1256 (2006). [CrossRef]
18. H. Chi, Z. Li, X.-M. Zhang, S. Zheng, X. Jin, and J. Yao, “Proposal for photonic quantization with differential encoding using a phase modulator and delay-line interferometers,” Opt. Lett. 36(9), 1629–1631 (2011). [CrossRef]
19. D. Tu, X. Huang, H. Yu, Y. Yin, Z. Yu, Z. Wei, and Z. Li, “Photonic sampled and quantized analog-to-digital converters on thin-film lithium niobate platform,” Opt. Express 31(2), 1931–1942 (2023). [CrossRef]
20. Y. Tian, J. Qiu, Z. Huang, Y. Qiao, Z. Dong, and J. Wu, “On-chip integratable all-optical quantizer using cascaded step-size mmi,” Opt. Express 26(3), 2453–2461 (2018). [CrossRef]
21. Y. Tian, Z. Kang, J. He, Z. Zheng, J. Qiu, J. Wu, and X. Zhang, “Cascaded all-optical quantization employing step-size 259 mmi and shape-optimized power splitter,” Opt. Laser Technol. 158, 108820 (2023). [CrossRef]
22. C. Liu, J. Qiu, Y. Tian, R. Tao, Y. Liu, Y. He, B. Zhang, Y. Li, and J. Wu, “Experimental demonstration of an optical quantizer with enob of 3.31 bit by using a cascaded step-size mmi,” Opt. Express 29(2), 2555–2563 (2021). [CrossRef]
23. S. Yang, H. He, H. Chi, and R. Zeng, “Photonic quantization and encoding scheme with improved bit resolution based on waveform folding,” Opt. Express 27(24), 35565–35573 (2019). [CrossRef]
24. Y. Liu, J. Qiu, C. Liu, Y. He, R. Tao, and J. Wu, “An optical analog-to-digital converter with enhanced enob based on mmi-based phase-shift quantization,” Photonics 8(2), 52 (2021). [CrossRef]
25. M. A. Richards and J. Scheer, Principles of Modern Radar (SciTech Publishing, 2012).
26. J. D. McKinney, “Photonics illuminates the future of radar,” Nature 507(7492), 310–312 (2014). [CrossRef]
27. A. Bhandari, F. Krahmer, and R. Raskar, “On unlimited sampling and reconstruction,” IEEE Trans. Signal Process. 69, 3827–3839 (2021). [CrossRef]
