Open Access
Add to BibSonomy
Abstract
We propose an on-chip all-optical multilevel amplitude regenerator scheme over a Mach Zehnder interferometer (MZI) configuration, enabling multiple amplitude-noise suppression on 16-QAM signals. Joint parameter optimization is carried out based on the general nonlinear model of the proposed scheme to significantly reduce the phase distortion caused by the nonlinear interferometer, which is the key to perform the phase preserving operation. The full function of the phase-preserving amplitude regeneration (PPAR) is verified by an experiment on an on-chip nonlinear waveguide with the length of the 2.31 cm. Furthermore, we perform thoughtful investigations on the oscillatory behavior achieved by the silicon MZI regenerator, enabling the full PPAR on 16-QAM signals through the optimized multiple power plateaus. A maximum 1.6 dB improvement of signal quality is achieved by the proposed on-chip amplitude regenerator at the input signal-to-noise ratio (SNR) of 25 dB. The impact from the two-photon absorption (TPA) effect as an positive role in the regenerator is also well discussed.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Modern optical communication system has been developing towards large capacity and high rate. The traditional on-off keying (OOK) signal and other low-order modulation format signals cannot meet the demand on high-speed transmission. By comparison, high-order modulation formats such as multi-level pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) become more and more popular in optical communication network although a higher optical signal-to-noise ratio (OSNR) is required [1]. All-optical regeneration technologies, in amplitude or/and phase domains, can be used to effectively improve the OSNR performance for the large-capacity and long-haul transmission [2]. All-optical regeneration can be performed through a variety of optical structures, such as nonlinear optical loop mirror (NOLM) [3–7], Mach Zehnder interferometer (MZI) [8–10], semiconductor optical amplifier (SOA) [11], and phase sensitive amplifier (PSA) [12,13]. These optical regenerators usually rely on bulky and expensive devices such as highly nonlinear fiber (HNLF) or SOA. Just like the development of optical switching chips [14], one began to pay attention to on-chip optical regenerators to reduce the volume and cost [14,15].
Nowadays, with the wide application of high-order modulation signals in coherent systems, the novel regenerators for quadrature phase shift keying (QPSK) or 16-QAM signal have a greater prospect more than those for OOK signals. However, most of the existed amplitude regeneration schemes for OOK or PAM-4 signals are unable to directly apply to the QPSK or QAM signals in that the phase of regenerated signals is several degraded through the amplitude-to-phase noise conversion. Therefore, it is necessary to preserve the signal’s phase in the process of the amplitude regeneration. At present, the mainstream method for phase-preserving amplitude regeneration (PPAR) is based on the NOLM structure with HNLF because of better stability more than the MZI-based counterpart [4]. On the contrary, it will be shown in the paper that, the structure of on-chip MZI regenerator has a promising advantage over the HNLF-based configurations in multi-level amplitude regeneration. Here, we focus on the on-chip MZI regenerators composed of silicon-on-insulator (SOI) waveguide and then analyze the PPAR performance, in terms of noise reduction ratio (NRR) and phase disturbation.
The rest of the paper is organized as follows. Section 2 introduces the nonlinear propagation equation for optical field in the silicon wire waveguide, and the expression of the power transfer function for the proposed MZI regenerator is also presented. In Section 3, the feasibility of PPAR regeneration based on the MZI structure is verified by the experiment on QPSK signals. Then, the structural parameters of on-chip MZI regenerator for triple-amplitude 16-QAM format are optimized in Section 4. Section 5 discusses the influence of two photon absorption (TPA) in silicon wire waveguides on the PPAR function of the on-chip MZI scheme. The final the conclusion is drawn in Section 6.
2. MZI structure composed of nonlinear silicon wire waveguides
2.1 Nonlinear phase shift induced by silicon wire waveguides
In silicon wire waveguides, the coupled-mode equation for guided optical wave involves in more physical effects besides those in HNLFs, such as TPA, free carrier absorption (FCA) and dispersion (FCD). The complex envelope E of the optical field is dependent on the position z and can be expressed as follows [16]:
Equation (1) can be numerically solved by the step-split Fourier method. For the silicon material, the dispersion coefficient and its slope are respectively 988.1ps/km/nm and 5.56ps/km/ nm2 at 1550 nm. The other parameters used in our calculation are listed as follows: ${\tau _c}$=0.5ns, ${\beta _{TPA}}$=5×10−12m/W, $\sigma $=1.45×10−21m2, $\mu $=7.5 [17]. According to the waveguide structure given in Ref. [18], i.e., the cross section of 470 nm (width) × 226 nm (height) and the length of 4 mm, we calculate the maximum phase shift ${\varphi _{max\; }}$ induced by self-phase modulation (SPM) related to the peak optical power of the Gaussian optical pulse. The calculated curve, along with two experimental points given in Ref. [18], is shown in Fig. 1. The agreement of the calculated results with the experimental data means the reliability of the calculation program for Eqs. (1) and (2). It is also shown from Fig. 1 that, with the increase of the peak power, the SPM-induced phase shift goes up to the maximum and then drops down since TPA loss is dependent on the power.
2.2 Phase and power transfer functions for MZI structures
Figure 2 shows the universal MZI structure, in which the upper and lower arms are based on the silicon wire waveguide or the optical phase shifter, respectively. The corresponding amplitude and phase transfer functions are represented by ${R_{1,2}}$ and ${\varphi _{1,2}}$, respectively. ${\rho _1}$ and ${\rho _2}$ is respectively the coupling efficiencies of the input and output couplers. The optical fields output from the upper and lower ports of the MZI structure, Eout and E2, can be expressed by
From Eq. (4), the corresponding output power can be derived from ${P_{out}} = {|{{E_{out}}} |^2}$ as follows:
From Eqs. (4) and (5), the phase disturbation and power transfer function (PTF) can be deduced from $\arg \left( {\frac{{{E_{out}}}}{{{E_{in\; }}}}} \right)$ and ${P_{out}}/{P_{in}}$, respectively. Here, we focus on the phase disturbation dependent on the coupling efficiencies ${\rho _1}$ and ${\rho _2}$, as shown in Fig. 3. From Fig. 3, the output coupling efficiency ${\rho _2}$ should be as large as possible in order to minimize the phase disturbation for phase preserving regeneration. In this paper, we choose ${\rho _2}$ = 0.99. On the other hand, the coupling efficiency ${\rho _1}$ affects the optical power input to the silicon waveguide, and then is limited by the damage threshold power.
3. Experimental verification of the PPAR function
In order to further validate the feasibility of the MZI-based regeneration scheme based on silicon wire waveguide, we build up the experimental system for QPSK signals as shown in Fig. 4. A continuous wave (CW) laser is launched into the IQ modulator for a QPSK signal, and an arbitrary waveform generator (AWG) is used to generate the data sequence. At the same time, as the reference source, the CW laser along with the regenerated signal is injected into the IQ demodulator for testing the regeneration performance through the real-time oscilloscope (OSC).
In our experiment, the degradation of QPSK signals is implemented by two erbium doped fiber amplifiers (EDFA1 and EDFA2), with the maximum output powers of 20 dBm and 37 dBm, respectively. By adjusting the output power of EDFA1 and fixing EDFA2, we are able to control the optical signal-to-noise ratios (OSNRs) of the degraded QPSK signals. The tunable attenuator is used to match the signal power with the optimal regenerative working point. Several polarization controllers (PCs) are useful for appropriately adjusting the polarization states of light. The MZI regenerator used here consists of two fiber couplers and a section of silicon waveguide with the length of 2.31 cm. The microscopic view of the silicon chip used in the experiment and the corresponding input and output ports for the silicon waveguide is depicted in Fig. 5. Detail structures, including the cross-section size of 606nm×220 nm, the total 21 sections with the 0.11 cm-long straight waveguide and their connection by the semicircle waveguide with a radius of 0.03 cm are also given in Fig. 5. The parameters of the silicon waveguide in the experiment are the same as those given in Ref. [19].
On the basis of the experimental setup, by monitoring the input and output powers of the regenerator with the 1550nm-CW light, we measure the normalized power transfer function curve relative to the output saturation power as shown in Fig. 6. According to the experimental parameters, the simulated curve is also plotted in the same figure. From Fig. 6, the experimental and simulated curves have the same saturation point or the maximum value. In the vicinity of the saturation point, the amplitude regeneration can be realized. This consistency shows the correctness of our computation program to a certain extent.
Fig. 6. Normalized power transfer curves of the MZI regenerator obtained by experiment and simulation
In what follows, we demonstrate the regeneration performance for QPSK signals by means of the constellation diagram and the noise reduction ratio (NRR), which can be derived from the real-time oscilloscope after the IQ demodulator. The input signal-to-noise ratios (SNRs) of the QPSK signals are adjusted by controlling the optical power output from EDFA1. Correspondingly, the input error vector magnitude (EVMin) changes as well. For amplitude regeneration, the signal power input into the MZI regenerator should keep at the above-mentioned saturation point. Here, we define the NRR parameter to measure the regeneration capability, that is, $NRR = 20\textrm{lg}(EV{M_{in}}/EV{M_{out}})$ [20,21], where EVMout is the EVM after regeneration. Figure 7 shows the experimental NRR dependent on the input SNR. The corresponding simulation curve is plotted in Fig. 7, in which the receiver noise is neglected. It leads to a better regeneration performance obtained in the theoretical calculation compared to the experiment. From Fig. 7, the experiment and simulation have a similar trend. When SNRin=17.3 dB, the experimental and simulated NRR is 2.4 dB and 2.5 dB respectively, and the corresponding constellation diagrams of the QPSK signals before (blue) and after (red) regeneration are also plotted in the inset. It is shown that, the amplitude regeneration with phase preservation can be well effectively implemented by the MZI structure with silicon wire waveguide.
4. Optimal design of the on-chip MZI regenerator
4.1 Chip architecture
The on-chip regenerator designed here is a modified version of the experimented MZI structure. Two fiber couplers used in the experiment are substituted with the integrated counterparts, each of which consists of two waveguides with the spacing of 1.744µm. When the coupling lengths are 14.738µm and 96.24µm, we can obtain ${\rho _1} = 5\%$ and ${\rho _2} = 99\%$, respectively. The upper arm of on-chip MZI structure is still a section of silicon wire waveguide and the lower arm is thermally adjustable to compensate for the fabrication process error. The cross-section of silicon waveguide is a key factor for the optimal design of on-chip MZI regenerator and needs to keep a balance between nonlinearity and confinement loss. Figure 8(a) illustrates the optimized cross section of the rib waveguide. The top and bottom widths are 800 nm and 900 nm, and the corresponding etched depths are 70 nm and 150 nm, respectively. The nonlinear coefficient reaches 150/W/m, 14000 times higher than that of the ordinary HNLF. The optimized single-mode waveguide has a smaller linear loss of about 0.3 dB/cm at 1550 nm, which helps to increase the number of regenerative levels.
Fig. 8. The optimization of silicon wire waveguide: (a) Waveguide cross-section; (b) Length dependency of power; (c) Layout of rid waveguide.
We utilize the same analysis method as used in Fig. 6 to simulate the influence of waveguide length on the input power at the first working point, as shown in Fig. 8(b). According to Eq. (5), the input power of the first working point can be approximated as ${P_1}\frac{{3\pi }}{{2\gamma ({1 - {\rho_1}} )L}}$. From Fig. 8(b), the optimal waveguide length could take 30 cm from the viewpoint of cost and loss. Too long silicon waveguide will increase the overall (linear and nonlinear) loss as shown in Fig. 8(b). The layout of the 30cm-long silicon wire waveguide is designed in Fig. 8(c). The bending radius and spacing of the wire waveguide are 30µm and 4µm, respectively. The 30cm-long waveguide occupies the total area of 1092×1040µm. Thus, the regeneration chip can be fabricated on a 5mm×5 mm active wafer and coupled to the tail fibers through optical grating for easy test. The grating coupling loss at each port is about 5 ∼ 6 dB.
4.2 Phase preserving amplitude regeneration for 16-QAM signals
Firstly, we simulate the power and phase transfer curves of the on-chip MZI regenerator before analyzing the PPAR performance for 16-QAM signals. According to the equations given in Section 2 and the optimized parameters of the on-chip MZI regenerator, we can calculate the normalized PTF and phase disturbation for the input signals, as shown in Fig. 9(a) and (b), in which the TPA effect in the silicon wire waveguide is taken into account. From Fig. 9(a), the optimized PTF has three flat plateaus around the working points (WPs), useful for regeneration of three-amplitude signals. From Eq. (5) the power period between the WPs is $P = \frac{{2\pi {\alpha _l}}}{{\gamma ({1 - {\rho_1}} )({1 - {e^{ - {\alpha_l}L}}} )}}$. The first working point has a maximum phase disturbation of 6.8° caused by the SPM effect, as shown in Fig. 9(b). For comparison, Fig. 9 also gives the transfer curves before optimization, corresponding to the experimental parameters used in Section 3. It is shown that the optimization can bring about some advantages over the case before optimization in the number of flat steps, the input power and phase disturbation at the first WP.
Then, we carry out the PPAR investigation on 16-QAM signals. The phase-preserving amplitude regenerators can be used for quadrature QPSK and QAM signals. QAM formats can be classificated as standard rectangular QAM, circular QAM, probability shaping QAM and geometric shaping QAM. In this paper, we focus on a geometric shaping 16-QAM, with iterative polar quantization 16-QAM (IPQ-16-QAM) as an example [8], a promising format in soft decision and reducing the bit error rate [22]. The amplitudes of the constellation points are aligned with the three working points shown in Fig. 9(a). The 16-QAM signals input to the on-chip MZI regenerator are degraded by ASE noise, as shown in Fig. 10(a). Figure 10(b) shows the output constellation diagram of regenerated 16-QAM when SNRin=24 dB. By comparing Fig. 10(a) with (b), it is known that the 16-QAM signal is regenerated in amplitude, with negligible phase degradation. For different input SNRs, the PPAR performance, in terms of NRR and average phase disturbation caused by nonlinear effect and noise, is shown in Fig. 11(a). It is found that the NRR increases with the input SNR, and the average phase disturbance is 3.8°at the input SNR of 18.6 dB, close to the error-free threshold [5,19]. Figure 11(b) also gives the curve of EVM reduction, defined as ΔEVM = EVMin - EVMout [20]. When SNRin=25 dB, ΔEVM has the maximum value of 0.7%.
For the QAM signals, the power of each constellation point should match with the corresponding working point for multi-level amplitude regeneration. The number of regenerative levels depends on the flatness of the PTF steps. For the PTF curve with multiple flat steps, just like that of the on-chip MZI regenerator, the constellation point distribution has no influence on the regeneration performance. Thus, this on-chip MZI regenerator can as well support other QAM formats.
5. Discussion
5.1 Optimization of TPA
The optical nonlinearity is the fundamental basis of the different nonlinear responses to the signal and the noise [23]. For the signals, especially the multi-level signals, it requires a linear mapping on each amplitude level when passing through the regenerator. However, the noise on amplitude levels is sufficiently suppressed due to the power plateau observed around the WPs. In the proposed silicon MZI regenerator, the TPA effect as a nonlinear process plays a positive role to the power transfer function, which could support the more flat response on each WP. Based on the structural parameters given in Section 4, we simulate the PTF curve in the absence of the TPA effect, as shown in Fig. 12. For the case without TPA, only one ideal WP occurs on the plat plateau of the PTF curve, just as in HNLFs [8]. On the contrary, the existence of a proper TPA loss is helpful for supporting multiple WPs suitable for multi-amplitude regeneration. Therefore, the TPA effect improves the noise-suppression efficiency in our scheme.
Following on the results in Fig. 12, we can obtain a more desirable power transfer function for the amplitude regeneration by optimizing the TPA coefficient. The TPA effect in the silicon waveguide is sensitive to the carrier concentration [24], which is characterized by the TPA coefficient ${\beta _{TPA}}$. Therefore, one can adjust the TPA impact through applying the voltages to the active waveguide [25]. Figure 13 gives the simulation results on PTF curves over different TPA coefficient. The optimal TPA coefficient corresponding to the flat response on each working point is 5×10−12m/W for the multi-level amplitude regeneration.
5.2 Comparison with HNLF based MZI regenerator
In terms of multi-level amplitude regeneration, we make a comparison between the HNLF-based regenerator and the silicon structure to identify the importance of the proposed scheme. We listed five key aspects to a multi-stage amplitude regenerator, including “nonlinear waveguide length”, “power transfer function”, “phase preserving”, “regenerated signal” and “structural feature” to give a thoughtful comparison between the typical HNLF-based MZI regenerators with the proposed silicon scheme, see in Table 1. In general, the silicon MZI regenerator proposed in the paper has the smallest size thanks to the high Kerr nonlinearity in the silicon ship compared to the several hundred or even kilometer-long HNLF. Consequently, it dramatically reduces the cost of the nonlinear device, with the budget ratio only of 1:50 or less compared to the HNLF cases [5]. Moreover, the amplitude-regeneration operation with the phase preserving function could also been performed in the proposed scheme through the experimental test, to the best of our knowledge, which is the first experiment on such multiple functions over a chip-level regenerator. Although the multi-level operation only on 16-QAM signals has been discussed based on the proposed scheme, the more regenerative-level could be performed following the similar optimization process in the paper. Therefore, the proposed on-chip multi-level amplitude regenerator offers an integrated and multiple-functions solution.
Table 1. Comparison of the on-chip MZI regenerator with the HNLF-based counterparts
It should be noticed there are some variations of the structures to achieve the multi-level amplitude regeneration. Based on our experiences on NOLM structures [4,5,21], the on-chip NOLM regenerator is an alternative approach to perform the amplitude noise suppression. Because it is no need to synchronize the two signals from the clockwise and counter-clockwise arms in the NOLM structure, it may be more piratical in the real implementation. Moreover, to perform the uniform regenerative-behaviour on each level, a polarization-orthogonal continuous-light is necessary to the MZI or NOLM regenerator chip.
In a word, the on-chip MZI structure composed of silicon wire waveguide is a desirable nonlinear platform, useful for building up phase-persevering amplitude regenerators for higher-order modulation signals in the future.
6. Conclusions
In this paper, we propose the on-chip MZI scheme for PPAR function of high-order modulation formats. The PPAR performance is simulated by utilizing the propagation equation of optical pulse in silicon wire waveguide, and investigating the power and phase transfer functions of the resulting MZI-based regenerator. The correction of the theoretical model and the feasibility of the proposed scheme are verified by the experiments on QPSK signals. Furthermore, we optimize the cross-section of silicon wire waveguide and design the regeneration chip for the 16-QAM signals with three amplitudes. Our simulation shows that, the EVM reduction of the 16-QAM signals can be up to the maximum of 0.7% at the input SNR of 25 dB. The positive influence of the TPA effect in silicon waveguide on the on-chip MZI regenerator is also discussed.
Funding
National Natural Science Foundation of China (61671108, 61975027, 62001086); Sichuan Science and Technology Program (2021YFG0143); National Key Research and Development Program of China (2019YFB2203103).
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. Carena, V. Curri, P. Poggiolini, G. Bosco, and F. Forghieri, “Maximum reach versus transmission capacity for Terabit super channels based on 27.75-GBaud PM-QPSK, PM-8QAM, or PM-16-QAM,” IEEE Photonics Technol. Lett. 22(11), 829–831 (2010). [CrossRef]
2. P. V. Mamyshev, “All-optical data regeneration based on self-phase modulation effect,” in European Conference and Exhibition on Optical Communication (1998).
3. T. Roethlingshoefer, T. Richter, C. Schubert, G. Onishchukov, B. Schmauss, and G. Leuchs, “All-optical phase-preserving multilevel amplitude regeneration,” Opt. Express 22(22), 27077–27085 (2014). [CrossRef]
4. F. Wen, C. P. Tsekrekos, Y. Geng, X. Zhou, B. Wu, K. Qiu, S. K. Turitsyn, and S. Sygletos, “All-optical multilevel amplitude regeneration in a single nonlinear optical loop mirror,” Opt. Express 26(10), 12698–12706 (2018). [CrossRef]
5. B. Guo, F. Wen, B. Wu, F. Sun, and K. Qiu, “All-optical multilevel amplitude regeneration based on polarization-orthogonal continuous-wave-light-assisted nonlinear-optical loop mirror (PC-NOLM) subsystem,” IEEE Access 7(1), 149666–149671 (2019). [CrossRef]
6. M. Sorokina, “Design of multilevel amplitude regenerative system,” Opt. Lett. 39(8), 2499–2502 (2014). [CrossRef]
7. F. Wen, B. Wu, K. Qiu, and S. Sygletos, “Conjugate nonlinear-optical loop mirror (Conj-NOLM)-based phase-preserving multilevel amplitude regenerator,” Opt. Express 27(14), 19940–19949 (2019). [CrossRef]
8. X. Kong, B. Wu, X. Zhou, Q. Wan, S. Jiang, F. Wen, and K. Qiu, “Design of all-optical multi-level regenerators based on Mach–Zehnder interferometer,” Opt. Commun. 380, 377–381 (2016). [CrossRef]
9. B. Guo, B. Wu, F. Wen, and K. Qiu, “All-optical phase-preserving amplitude regeneration of high-order modulated signals based on Mach-Zehnder interferometers,” in 12th International Photonics and OptoElectronics Meetings, paper JW4A.11 (2019).
10. X. Jiang, B. Wu, B. Guo, F. Wen, and K. Qiu, “A novel Mach–Zehnder-interferometer regenerator for any level number of optical PAM signals using cross phase modulation of orthogonally polarized light,” Opt. Commun. 458, 124840 (2020). [CrossRef]
11. K. R. H. Bottrill, R. Kakarla, F. Parmigiani, D. Venkitesh, and P. Petropoulos, “Phase Regeneration of QPSK Signal in SOA using Single-stage, Wavelength Converting PSA,” IEEE Photonics Technol. Lett. 28(2), 205–208 (2016). [CrossRef]
12. R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,’,” Nat. Photonics 4(10), 690–695 (2010). [CrossRef]
13. Z. Xing, H. Wang, and Y. Ji, “QPSK Signal Regeneration Based on Vector Phase Sensitive Amplification With Low Pump Powers,” IEEE Access 7(1), 63936–63943 (2019). [CrossRef]
14. W. Shi, Y. Tian, and A. Gervais, “Scaling Capacity of Fiber-Optic Transmission Systems via Silicon Photonics,” Nanophotonics 1(16), 4629–4663 (2020). [CrossRef]
15. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “All-optical regeneration on a silicon chip,” Opt. Express 15(12), 7802–7809 (2007). [CrossRef]
16. Y. Long, A. Wang, L. Zhou, and J. Wang, “All-optical wavelength conversion and signal regeneration of PAM-4 signal using a silicon waveguide,” Opt. Express 24(7), 7158–7167 (2016). [CrossRef]
17. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007). [CrossRef]
18. E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood, “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14(12), 5524–5534 (2006). [CrossRef]
19. Y. Geng, B. Wu, F. Wen, H. Zhou, X. Zhou, and K. Qiu, “Clock-pump four-wave-mixing-based multichannel all-optical regeneration in silicon waveguide,” Opt. Eng. 56(11), 1 (2017). [CrossRef]
20. B. Guo, B. Wu, F. Wen, and K. Qiu, “Cascaded nonlinear-optical loop mirror-based all-optical PAM regenerator,” Appl. Sci. 10(1), 206 (2019). [CrossRef]
21. S. Zhang, B. Wu, F. Sun, and F. Wen, “Optimization method for universal NOLM-based phase-preserving amplitude regenerator with application to Att-NALM scheme,” Opt. Commun. 475, 126188 (2020). [CrossRef]
22. H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Iterative Polar Quantization Based Modulation to Achieve Channel Capacity in Ultra-High-Speed Optical Communication Systems,” IEEE Photonics J. 2(4), 593–599 (2010). [CrossRef]
23. Q. Lin, O. Painter, and G. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15(25), 16604–16644 (2007). [CrossRef]
24. A. Smolyaninov, M. Yang, L. Pang, and Y. Fainman, “Plasmonic enhanced two-photon absorption in silicon photodetectors for optical correlators in the near-infrared,” Opt. Lett. 41(19), 4445–4448 (2016). [CrossRef]
25. M. Sorokina, S. Sygletos, A. Ellis, and S. Turitsyn, “Regenerative Fourier transformation for dual-quadrature regeneration of multilevel rectangular QAM,” Opt. Lett. 40(13), 3117–3120 (2015). [CrossRef]
26. F. Wan, B. Wu, F. Wen, and K. Qiu, “In-Band OSNR Measurement Method for All-Optical Regenerators in Optical Domain,” Appl. Sci. 9(24), 5438 (2019). [CrossRef]
