† Corresponding author. E-mail:

Project supported by the Fundamental Research Funds for the Central Universities, China (Grant No. FRF-TP-15-030A1) and China Postdoctoral Science Foundation (Grant No. 2015M580978).

An all-optical analog-to-digital converter (ADC) based on the nonlinear effect in a silicon waveguide is a promising candidate for overcoming the limitation of electronic devices and is suitable for photonic integration. In this paper, a lumped time-delay compensation scheme with 2-bit quantization resolution is proposed. A strip silicon waveguide is designed and used to compensate for the entire time-delays of the optical pulses after a soliton self-frequency shift (SSFS) module within a wavelength range of 1550 nm–1580 nm. A dispersion coefficient as high as –19800 ps/(km·nm) with ±0.5 ps/(km·nm) variation is predicted for the strip waveguide. The simulation results show that the maximum supportable sampling rate (MSSR) is 50.45 GSa/s with full width at half maximum (FWHM) variation less than 2.52 ps, along with the 2-bit effective-number-of-bit and Gray code output.

An analog-to-digital converter (ADC) is an important part of a communication system which needs to satisfy the requirements for the rapid development of ultra-wide-band applications such as an advanced radar system, high speed optical communication, etc.^{[1–3]} The electrical ADC whose sample rate and art resolution are several giga-samples per second (GSa/s) and 10 bit respectively cannot meet the requirements for ultra-wide-band applications because of its inherent electrical bottle-neck such as clock jitter and sampling aperture, etc.^{[1–4]} The all-optical ADC has attracted much attention because it can overcome the limitation of electronic devices, and meanwhile achieve appropriate quantization resolution and sampling rates. Optical quantization could be realized by soliton self-frequency shift (SSFS), higher-order soliton fission, cross-phase modulation effects, etc.^{[5–13]} Optical quantization utilizing the SSFS effect is a promising option among the above schemes because of its ultrafast response speed, which can realize the sampling rate transparency. The outstanding results in analog-to-digital conversion were given based on the SSFS effect.^{[14–20]} In the process of frequency shift, time-delays between optical pulses with different peak powers occur inherently, leading to the non-synchronization of the quantized optical pulses and coding error. The multiple optical time-delay lines (TDLs) are utilized to compensate for time-delay in the previous schemes whose disadvantages are complicated and expensive. Meanwhile, they cannot meet the demands of miniaturization and photonic integration. Silicon photonic technology is suitable for photonic integration, owing to its compatibility with the complementary metal–oxide–semiconductor (CMOS) process. The optical waves can be confined into the sub-micron region by the high refractive index of silicon by using the silicon-on-insulator (SOI) technology.^{[21]} Some SOI signal processing schemes have been proposed,^{[22–28]} but to the best of our knowledge, there is still no report on an all-optical ADC. The strip waveguide structure on the order of centimeters in length allows for a stronger negative dispersion than conventional fiber and photonic crystal fiber (PCF). However, the organic material requires a non-CMOS process and strict temperature limitation, which renders it unsuitable for mass-manufacturing.^{[29,30]}

In this paper, we propose a lumped time-delay compensation scheme with 2-bit quantization resolution to realize optical pulse synchronization for the first time by inserting a strip silicon waveguide instead of the multiple optical TDLs after the SSFS module. The time-delays between quantized pulses with different peak powers at the output of the silicon wavelength are compensated for. Results based on numerical simulation are discussed.

Figure *b*_{1} and *b*_{2} are

All-optical quantization is achieved by the SSFS effect, the central wavelength of the optical pulse presents a red shift. Meanwhile, the optical pulse has a time-delay at the output of the HNLF. The dynamic transmission of sampled optical pulses in HNLF can be described by the simplified generalized nonlinear Schrodinger equation (GNLSE). Solving the GNLSE with the moment method, the time-delay of the sampled pulse Δ*t* can be obtained as follows:^{[31,32]}

*Ω*is the frequency shift of the sampled pulse,

*β*

_{2}and

*L*are the second-order dispersion coefficient and the length of fiber, respectively. The dispersion parameter

*D*= −2

*πcβ*

_{2}/

*λ*

^{2},

*λ*

_{c}and

*λ*

_{0}are the central wavelengths of the quantized pulse and initial pulse, respectively. When the dispersion parameter

*D*and the fiber length

*L*are constants, the time-delay Δ

*t*is directly proportional to

*λ*

_{c}from Eq. (

*K*

_{1}and

*K*

_{2}, respectively. In order to compensate for the time-delay, the relation

*K*

_{2}= −

*K*

_{1}= (Δ

*T*

_{max}− Δ

*T*

_{min})/(

*λ*

_{max}−

*λ*

_{min}) needs satisfying, while

*D*

_{c}=

*K*

_{2}/

*L*

_{c}. Here,

*D*

_{c}and

*L*

_{c}are the dispersion parameter and the length of the lumped time-delay compensation device.

There are many different ways to realize time-delay compensation as long as the *T*_{0} is the initial pulse width and *β*_{2} is the GVD coefficient. The self-phase modulation has nonlinear length *L*_{NL} = (*γP*_{0})^{−1}, where *γ* is the nonlinear parameter and *P*_{0} is the initial peak power. The dispersion plays a major role, while the nonlinearity is relatively weak, requiring *L* ≪ *L*_{NL} and *L* ≈ *L*_{D}, for waveguide length *L*.

Fortunately, the tight mode confinement in an SOI waveguide helps introduce significant waveguide dispersion and thus allows one to obtain a dramatically large normal group-velocity dispersion. In other words, a large and flattened negative dispersion parameter is obtained by designing the SOI waveguide appropriately.^{[33–38]} Figure ^{2}/m and a dispersion of –19800 ps/(km·nm) at 1550 nm. Meanwhile, the dispersion variation is ±0.5 ps/(km·nm) from 1400 nm to 1600 nm. This value is 1000 times larger than that of standard silica fiber (*β*_{2} < 0.02 ps^{2}/m). With such a high dispersion, the length of the strip silicon waveguide is shorter than that of the conventional fiber or PCF.

The lumped time-delay compensation scheme is shown in Fig.

*A*is the slowly varying envelope,

*β*

_{m}is the

*m*-th dispersion coefficient,

*γ*

_{0}and

*γ*

_{TPA}are the real and imaginary part of the complex nonlinear coefficient,

*ω*

_{0}is the center angular frequency of the optical pulse,

*α*

_{l}is the linear loss of the silicon waveguide,

*α*

_{FCA}and

*n*

_{FCD}are the coefficients of FCA and FCD, which are given by

*λ*

_{0}and

*λ*

_{ref}being the input center wavelength and the reference wavelength,

*N*

_{c}being the free-carrier density and given by

*β*

_{TPA}is the TPA parameter,

*h*is the Plank constant,

*f*

_{0}is the center frequency,

*A*

_{eff}is the effective mode field area of the silicon waveguide, and

*τ*is the effective carrier lifetime.

The nonlinear response function *R*(*t*) in Eq. (*R*(*t*) = (1 − *f*_{R}) *δ* (*t*) + *f*_{R}*h*_{R} (*t*), with *f*_{R} being the fractional contribution of the nuclei to the total nonlinear polarization. The Raman response function *h*_{R} (*t*) can be deduced from the Raman response spectrum. The Fourier transform *H*_{R} (*Ω*) of *h*_{R} (*t*) is given by

*Ω*

_{R}and

*Γ*

_{R}are the Raman shift and the bandwidth of the Raman gain spectrum at room temperature.

Solving the GNLSE with the split-step-Fourier method,^{[31]} the dynamic behaviors of the pulses in the time and frequency domain in the quantization process is numerically analyzed. In Fig. *Q*_{1}, *Q*_{2}, and *Q*_{3} of the sampled pulses are 9.58 W, 12.80 W, and 13.99 W, respectively. *Q*_{0} with a peak power of 0 is of no use in the coding module, so it is neglected in our simulation. After transmitting in a span of HNLF (*γ*_{HNLF} = 16 W^{−1}·km^{−1}, *D*_{HNLF} = +3 ps/(nm·km), *α*_{HNLF} = 0.9 dB/km) for SSFS, the central wavelengths of the sampled pulses are shifted to specific positions. The spectral profiles of the optical pulses *Q*_{1}–*Q*_{3} after the HNLF are shown in Fig.

Figure *P*_{LSB} and the full-scale of the power range *P*_{total} are 1.6 W and 6.4 W, respectively. Meanwhile, a peak DNL error of 0.2 LSB and a maximum INL error of 0.2 are obtained.

The initial sampled pulses are recorded in Fig. *Q*_{1}, *Q*_{2}, and *Q*_{3}”, the correct codes should be “10 01 11”. However, as shown in Fig. *Q*_{1}, *Q*_{2}, and *Q*_{0}”, which are inconsistent with the input ones. So the time-delay compensation is quite necessary for ensuring the coding correctness.

Here, a strip silicon waveguide is utilized to realize lumped time-delay compensation. As shown in Fig. *K*_{1} = 1.752 ps/nm in the scheme. Thus, the slope of the time-delay compensation curve is *K*_{2} = −1.752 ps/nm. The strip silicon waveguide can be fabricated, starting from SOI wafers patterned with deep-UV lithography, which is a mature technique at present. In Fig.

_{broadened}is the FWHM of the broadened pulses, respectively. From Eq. (

_{broadened}are both directly proportional to the reciprocal of the sampling rate. Considering the cases with the GVD fluctuation, the effect of

*β*

_{2}on the FWHM of the compensated pulses is shown in Fig.

*β*

_{2}, the FWHM values of the three compensated pulses present fluctuations. However, the variations are relatively slight, which are all less than 2.52 ps. For example, the FWHM of

*Q*

_{2}is 16.44 ps at the GVD value of 24.44 ps

^{2}/m, and it is 18.96 ps at the GVD value of 25.24 ps

^{2}/m; their difference is only 2.52 ps. In Fig.

*β*

_{2},

*Q*

_{1},

*Q*

_{2}, and

*Q*

_{3}have almost the same time error, and the maximum is 30 ps at the GVD value of 26.24 ps

^{2}/m. Considering both the pulse broadening and GVD fluctuation issues, the curve of MSSR versus GVD is shown in Fig.

^{2}/m and the maximum central wavelength is 1572 nm in our scheme, and the sampling rate should be no more than 50.45 GSa/s. With the increase or decrease of GVD relative to 25.24 ps

^{2}/m, the MSSR decreases due to the large dispersion of the strip silicon waveguide.

For the cases where the values of slope *K*_{2} are different, while the dispersion of the silicon waveguide remains constant and the length of the silicon waveguide is changed, the diagram of FWHM, time error and MSSR versus slope *K*_{2} are shown in Figs. *N* < 1.4) is met by a VOA before the SSFS module; the effect from the nonlinearity of the silicon waveguide can be neglected.

The quantization resolution *N* is defined as

*λ*

_{shift}and Δ

*λ*

_{FWHM}are the magnitude of the center wavelength shift and the spectral width of the optical pulse at the output of HNLF. It is obvious that to obtain a large resolution

*N*, a larger

*λ*

_{shift}and a smaller Δ

*λ*

_{FWHM}should be achieved. In the present paper, the strip silicon waveguide is utilized to realize time-delay compensation after SSFS instead of DCF or PCF, as well as to meet the demands of miniaturization and photonic integration, which is the development trend of all-optical ADC. Owing to the inherent limitations of length, TPA, FCA, and FCD, it is difficult for the proposed all-optical ADC to realize more than 2-bit quantization. However, it is possible to combine the proposed all-optical ADC with the spectral compression scheme to obtain a smaller Δ

*λ*

_{FWHM}and a higher quantization resolution

*N*after the SSFS module as shown in Fig.

In this paper, we propose a lumped time-delay compensation scheme based on the silicon waveguide for simplifying the structure and ensuring the coding correctness of the all-optical ADC by SSFS effect. The corresponding TDLs could be replaced by a short silicon waveguide to realize time-delay compensation and integration. The simulation results show that the strip silicon waveguide can compensate for the time-delay accurately with an FWHM variation of less than 2.52 ps. The proposed scheme can support an MSSR of 50.45 GSa/s with no coding error. When a spectral compression module can be added by utilizing a span of HNLF after the SSFS module, a potential 3-bit resolution can be realized since the total wavelength range is 30 nm. The lumped time-delay compensation scheme utilizing the large dispersion strip silicon waveguide instead of the multiple TDLs and fiber reduces the structure complexity in practical applications, which conforms to the trend of integration and miniaturization of all-optical ADC.

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