**120 GSa/s**equivalent sampling rate, high power of sampling copies, and low variance are obtained.

## 1.

## Introduction

Analog-to-digital convertor (ADC) as an important part of communication systems needs to satisfy requirements of the rapid development of ultrawide-band applications such as advanced Radar communication systems, high-speed optical communication, and so on.^{1}2.^{–}^{3} The electrical ADC whose sample rate is only several gage-samples per second ($\mathrm{GSa}/\mathrm{s}$) cannot meet the requirements of ultrawide-band applications because of its inherent electrical bottle-neck such as clock jitter and sampling aperture.^{1}2.3.^{–}^{4} Optical signal processing has attracted researchers’ interests because it can overcome the electrical limitations.^{5}6.7.^{–}^{8} Bres et al.^{9} demonstrated a $320\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gb}/\mathrm{s}$ optical demultiplexing sampling method that required a mode-locked laser with a very high repetition rate. In order to meet the requirement, a large number of copies are needed. Zhang et al.^{10} proposed a multicast parametric synchronous optical sampling scheme using only three copies based on a high nonlinear fiber (HNLF). However, a fiber length over 200 m was used to realize all-optical ADC, which is disadvantageous for photonics integration.

In this paper, we optimize a proposed multicast parametric synchronous sampling scheme utilizing a 2 cm high nonlinear spiral photonic crystal fiber (PCF) instead of a 200 m HNLF as the nonlinear medium based on the parametric process and a 1.8 km dispersion compensation fiber (DCF) instead of the 5 km single-mode fiber (SMF) as the time-stretched medium, respectively. The experimental results show the parametric process efficiency based on a high nonlinear spiral PCF and verify the feasibility of our optimized scheme.

## 2.

## Theory of Operation

The scheme is composed of signal multicast, time delay, and parametric sampling blocks. The original signal as the pump is copied utilizing a parametric process in a high nonlinear spiral ${\mathrm{PCF}}_{1}$. The idler gain is

where $\gamma $ is the nonlinear coefficient, $L$ is the fiber length, and ${E}_{p}(t)$ is the pump power. The idlers change with the original pump to realize signal multicast.In order to temporally overlap the pulse center of the pump and a nonreturn-to-zero (NRZ) code, these idlers are fed into an SMF to temporally delay each other by a factor

where $\mathrm{\Delta}\tau $ is the temporal delay and ${T}_{\mathrm{NRZ}}$ is the period of the NRZ signal. Hence, the temporal delays are $-\mathrm{\Delta}\tau $, 0, and $+\mathrm{\Delta}\tau $ between the pump and idler 1, idler 2, and idler 3, respectively.Time-stretched and linearly chirped pulses are used as the sampling pump signal based on high nonlinear spiral ${\mathrm{PCF}}_{2}$ in a parametric sampling block. The pump pulses are fed into a DCF to obtain linear chirp.^{11}^{,}^{12} The linear chirped pulses are delivered into a high nonlinear spiral ${\mathrm{PCF}}_{2}$ with idlers as ${A}_{i,n}(t)$ $(n=\mathrm{1,2},3)$. The phases of sampling copies given as ${A}_{\mathrm{sc},n}(t)$ $(n=\mathrm{1,2},3)$ are modulated by chirped pump pulses due to phase combination between pump and idlers in the high nonlinear spiral ${\mathrm{PCF}}_{2}$. The gain frequency variation of the sampling copies is

## (3)

$$\delta {\omega}_{\mathrm{sc}}(T)=-2\frac{\partial {\varphi}_{p}}{\partial T}=\frac{2\mathrm{sgn}({\beta}_{2})(z/{L}_{D})}{1+{(z/{L}_{D})}^{2}}\frac{T}{{T}_{0}^{2}},$$^{13}14.15.

^{–}

^{16}From Eq. (3), the chirp of the pump can be transferred to sampling copies and the chirp of sampling copies is twice that of the pump. Therefore, an arrayed waveguide grating (AWG) is utilized to extract different frequency components of the sampling copies. Finally, the sampling time points are sent into a data signal processing module to realize quantization and coding. In the sampling scheme, the equivalent sampling rate is

## (4)

$${R}_{s}=\frac{{|\delta {\omega}_{\mathrm{sc}}(T)|}_{\mathrm{max}}n{R}_{\mathrm{pump}}}{\mathrm{\Delta}f},$$## 3.

## Results and Discussion

Based on the above theoretical analysis, we optimize the proposed multicast parametric synchronous sampling scheme utilizing a high nonlinear spiral PCF as the nonlinear processing medium. The optimized schematic diagram is exhibited in Fig. 1. First, the degenerated pump signal with 1550.4 nm center wavelength is phase modulated by 50, 150, and 300 MHz radio frequency harmonics to suppress stimulated Brillouin scattering. Next the pump signal is amplitude modulated by $10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gb}/\mathrm{s}$ NRZ bit sequence. Then the pump signal is amplified by erbium-doped fiber amplifier (${\mathrm{EDFA}}_{1}$) and filtered by an optical bandpass filter (${\mathrm{BPF}}_{1}$) with a 0.6 nm bandwidth to eliminate the amplified spontaneous emission noise, respectively. Subsequently, the amplified pump signal is coupled into a 1 cm high nonlinear spiral ${\mathrm{PCF}}_{1}$ with three continuous waves (CWs) whose peak powers are 1 dBm and whose center wavelengths are 1558.8, 1560.8, and 1562.8 nm for signal multicast. Three idlers are delivered into a 1.2 km SMF to realize a temporal delay with each other after optical ${\mathrm{BPF}}_{2}$ that removes the pump signal and three CWs at the output of ${\mathrm{PCF}}_{1}$. Ten gigahertz Gaussian optical pulses with a 1555 nm center wavelength and 10 ps pulse width are delivered into 1.8 km DCF instead of a 5 km SMF to stretch in time domain and gain linear chirp and are amplified by ${\mathrm{EDFA}}_{2}$. The amplified pulses as the sampling pulse train are coupled into ${\mathrm{PCF}}_{2}$ with the three idlers for parameter processing. Finally, the different frequencies are separated by an AWG, and a digital signal processor (DSP) realizes photoelectric conversion, quantization, and coding.

In our system, the 2 cm high nonlinear spiral PCF is utilized as a nonlinear medium instead of the 200 m HNLF. Its structure is shown in Fig. 2(a). The spiral PCF consists of an elliptical slot core with a low index silicon nanocrystals (Si-nc) rod surrounded by three rings of air holes in a spiral lattice. The PCF has six spiral arms, where each arm shapes a single spiral with $r$ radius and $\theta $ angular increment. The curves of the dispersion and the nonlinear coefficient of the fundamental mode versus the wavelength are shown in Fig. 2(b). Based on the spiral structure, the dispersion is only $-0.07\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}/(\mathrm{nm}\xb7\mathrm{km})$ and the dispersion slope is $-1.25\times {10}^{-3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}/({\mathrm{nm}}^{2}\xb7\mathrm{km})$ at the wavelength of 1550 nm. The nonlinear coefficients of the fundamental mode are as high as $224.36\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{W}}^{-1}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ and are $1.87\times {10}^{4}$ times as large as the one of the HNLF.^{17}18.^{–}^{19}

The spectra at the outputs of ${\mathrm{PCF}}_{1}$ and ${\mathrm{PCF}}_{2}$ are recorded as shown in Figs. 3(a) and 3(b) together with the eye diagrams of idler 1 and copy 1 (insets), respectively. The three idlers are generated at 1538.01, 1540.09, and 1541.98 nm with approximately the same peak power in Fig. 3(a), and further optical processing is conveniently based on the same peak power for each NRZ code. As shown in Fig. 3(b), three sampling copies are generated at 1568.22, 1570.20, and 1572.21 nm and the maximum peak power is $-5.235\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dBm}$ at 1570.20 nm. The simulation results are in very good agreement with the measurements. The spectrum widths of the three sampling copies are wider than the ones of the three idlers, which means the chirp of the pump transfers to the sampling copies. According to the above theoretical analysis, the chirp of the sampling copies is twice that of the pump. So the sampling copies can be filtered by an AWG with a 0.2 nm bandwidth and 12 wavelengths from 1567.91 to 1572.66 nm.

To ensure the same time interval for all of the samples and extract precisely different temporal sampling points, we chose 12 sampling channels as shown in Table 1. The peak powers of 12 sampling channels utilizing a 2 cm high nonlinear spiral PCF as a nonlinear medium after the photoelectric detector are recorded and compared with the ones utilizing a 200 m HNLF. There are peak power variations among samples whether PCF or HNLF is utilized as the nonlinear medium, because of the irregular amplitude of time-stretched sampling pulses and the gain ripples of sampled copies. As shown in Table 1, the peak powers using PCF as a nonlinear medium are higher than the ones using HNLF, where the maximum and minimum deviations are 15.84 and 31.01 dB, respectively. Because the spiral PCF has a very high nonlinear coefficient, the idlers with high peak power are easily obtained.

## Table 1

Sampling performance.

Sampling channel | Center wavelength (nm) | Peak power (dBm) (2 cm PCF) | Peak power (dBm) (200 m HNLF) |
---|---|---|---|

1 | 1567.91 | $-46.86$ | $-65.88$ |

2 | 1568.14 | $-48.06$ | $-63.90$ |

3 | 1568.35 | $-33.28$ | $-54.71$ |

4 | 1568.60 | $-23.80$ | $-53.98$ |

5 | 1569.86 | $-19.09$ | $-46.72$ |

6 | 1570.10 | $-30.32$ | $-56.87$ |

7 | 1570.37 | $-31.91$ | $-54.92$ |

8 | 1570.63 | $-19.99$ | $-46.33$ |

9 | 1571.86 | $-14.45$ | $-42.81$ |

10 | 1572.18 | $-29.69$ | $-56.77$ |

11 | 1572.38 | $-31.46$ | $-53.11$ |

12 | 1572.66 | $-25.41$ | $-56.42$ |

Note: PCF, photonic crystal fiber; HNLF, high nonlinear fiber.

Subsequently, these electrical signals are sent into the signal processing block for equalization. The distortion of sampling can be eliminated from the shape of the linearly chirped pump and the gain ripples. In the DSP, the equalization function is

## (5)

$$S=({e}_{1},{e}_{2}\cdots {e}_{12})\left(\begin{array}{ccc}{s}_{1}& & 0\\ & \ddots & \\ 0& & {s}_{12}\end{array}\right),$$Changing the length of the PCF will induce different gain ripples and this will induce sampling distortion. The powers of the idler 1 and copies as a function of the length of ${\mathrm{PCF}}_{1}$ and ${\mathrm{PCF}}_{2}$ are recorded in Figs. 5(a) and 5(b), respectively. The power of the idler 1 and three sampling copies increases with the lengths of ${\mathrm{PCF}}_{1}$ and ${\mathrm{PCF}}_{2}$, respectively. The simulation results are in very good agreement with the experimental data. The powers of the three idlers are almost the same at the output of ${\mathrm{PCF}}_{1}$, but there are power variations among the three sampling copies at the output of ${\mathrm{PCF}}_{2}$ due to the irregular shape of the sampling pulses. With the increase in the length of ${\mathrm{PCF}}_{1}$ and ${\mathrm{PCF}}_{2}$, the power increasing amount of idlers and copies decreases monotonically and approaches zero, respectively. Consequently, it is inadvisable that high power for idlers or copies be obtained by increasing the fiber length.

The sampling distortion comes not only from the fiber length, but also the wavelength space between each CW. Here, we define the variance as

## (6)

$$\text{variance}=\frac{1}{3}{\left[\sum _{i=1}^{3}{({\text{power}}_{i}-\text{average})}^{2}\right]}^{1/2},$$^{20}When the wavelength space between each CW is 3 nm, the maximal variance is 0.037 dB at the output of ${\mathrm{PCF}}_{1}$. When the length of ${\mathrm{PCF}}_{1}$ is 1 cm, the variance gets large with the increase in the wavelength space as shown in Fig. 6(b). Due to the gain ripples of the idlers and the irregular shape of the sampling pulse, the variance at the output of ${\mathrm{PCF}}_{2}$ is larger than the one at the output of ${\mathrm{PCF}}_{1}$. When the wavelength space between each CW is 3 nm, the maximal variance is 2.159 dB at the output of ${\mathrm{PCF}}_{2}$. However, the minimal variance at the output of ${\mathrm{HNLF}}_{1}$ is 0.6 dB in the previous scheme when the wavelength space is 3 nm, leading to a large sampling distortion.

Changing the power of the pump or sampling pulse will also induce different gain ripples and this will also induce sampling distortion. The power of idlers and sampling copies versus the input power of pump and sampling pulse diagrams with different nonlinear coefficients is simulated, as shown in Figs. 7(a) and 7(b), respectively. The power of idlers or sampling copies monotonously increases and slopes gently with the increase in the input power of the pump or sampling pulse in any case of nonlinear coefficient $\gamma $. In Fig. 7(a), the curves are fitted with the same nonlinear coefficient $\gamma $, and the power of idlers is higher by 4.51 dB than the one utilizing HNLF as a nonlinear medium. In Fig. 7(b), the curves have a deviation with the same nonlinear coefficient $\gamma $ due to the gain ripples of idlers and the irregular shape of the sampling pulse; the power of the sampled copies is 12.44 dB higher than the one utilizing HNLF as a nonlinear medium.

## 4.

## Conclusion

In summary, we optimize a proposed multicast parametric synchronous sampling scheme utilizing 2 cm high nonlinear spiral PCF instead of 200 m HNLF as nonlinear medium based on parametric process and 1.8 km DCF instead of 5 km SMF as time-stretched medium, respectively. By using the high nonlinear spiral PCF, the high power of sampling copies and low variance are obtained to avoid sampling distortion. In order to realize real-time sampling and reduce the requirements of the electronic devices, the chirped sampling pulses are utilized and a $120\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GSa}/\mathrm{s}$ equivalent sampling rate is obtained. Compared with the pre-existing schemes, the size of the parametric processor can be much smaller than those previously proposed based on HNLF. Therefore, our optimized scheme is more in line with the concept of integration and miniaturization.

## Acknowledgments

This work is jointly supported by Beijing Natural Science Foundation under Grant No. 4122049, Beijing Higher Education Young Elite Teacher (No. YETP0381), and the Fundamental Research Funds for the Central Universities (FRF-JX-12-002, FRF-TP-12-083A, and FRF-TP-14-049A2).

## References

## Biography

**Sha Li** received her PhD in optical engineering at Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 2014. She is currently a postdoctoral researcher in School of Computer and Communication Engineering, University of Science and Technology Beijing (USTB), Beijing, China. Her research interests include optical signal processing, all-optical analog-to-digital conversion, and nonlinear fiber optics.

**Jianping Wang** is now the vice president in School of Computer and Communication Engineering, University of Science and Technology Beijing (USTB), Beijing, China. She has published more than 50 papers, of which more than 20 articles were in SCI. Her current research interests include optical signal processing, visible light communication, and deep space communication.