Spectral compression in a comb-like distributed fiber and its application in 7-bit all-optical quantization

Abstract. Subpicosecond pulse spectral compression without the limitation of the adiabatic soliton transform status in a comb-like distributed fiber (CDF) is proposed and numerically demonstrated. The principle of spectral compression in a single concatenation of single-mode fiber and high nonlinear fiber is theoretically analyzed and numerically proved. A three-stage CDF is designed and simulated following the principle. The simulation results show that an overall spectral compression ratio of 56.23 is obtained. The designed three-stage CDF is used in an all-optical analog-to-digital system to improve the quantization resolution to 7.1 bit.


Introduction
Since comb-like dispersion profiled fiber (CDPF) was first investigated in Ref. 1 for the soliton pulse train generation, a series of research has been implemented to analyze this cascaded structure. [2][3][4][5][6][7][8][9] CDPF is generally composed of a number of alternating lengths of high and low dispersion fibers [i.e., single mode fibers (SMFs) and dispersion shifting fibers (DSFs)] in which the dispersion effect and nonlinearity effect are spatially separated. 2 However, since the calculation and optimization for the dispersion, nonlinearity, and length of each fiber in CDPF are complex, a comb-like profiled fiber (CPF) was introduced to simplify the interconnected structure by using only two types of fibers, SMF and high nonlinearity fiber (HNLF). 3 CPF is widely used in the fields of pedestal-free adiabatic soliton pulse compression, high-repetition rate short optical soliton pulse train generation, wavelength-tunable femtosecond pulse source and, so on. [2][3][4][5][6][7][8][9] Recently, it has been found that a new application of CPF has the advantage of improving the resolution of an alloptical analog-to-digital conversion system. [10][11][12][13] In Ref. 10, the CPF composed of 19 segments of SMF and DSF is employed to realize a DSF to implement the adiabatic soliton spectral compression, giving a spectral compression ratio of up to 19.8 to 25.9. However, to the best of our knowledge, spectral compression without the limitation of adiabatic soliton status in a dispersion comb-like profiled fiber has never been studied and only a few works have demonstrated the wide wavelength-tunable abilities of spectral compression in CPF.
In this paper, the principle of spectral compression in a comb-like distributed fiber (CDF) based on a novel chirpcompensation method is analyzed by a strict theoretical derivation. The theoretical analysis shows that complete chirp compensation between the self-phase modulation effect (SPM) and group velocity dispersion (GVD) is the key to the design of the CDF. The three-stage CDF consisting of three concatenations of SMF and HNLF is carefully designed by solving the generalized nonlinear Schrödinger equation in the numerical simulation. The simulation results show that the spectral width is compressed from 11.8 to 0.21 nm, obtaining a spectral compression ratio up to 56.2. Using this technique, after shifting the soliton spectrum is sharply compressed and the all-optical quantization resolution of the all-optical analog-to-digital conversion (ADC) system is successfully improved towards 7.1-bit. The spectral compression in a single concatenation of SMF +HNLF is well understood as a process of chirp compensation. [14][15][16] The pulse is highly negatively chirped by the GVD effect by passing through the SMF (e.g., red line in Fig. 1). Then, chirp induced by the SPM in the HNLF has a nearly linear positive slope around the center of the pulse (e.g., blue line in Fig. 1), which can compensate the GVD-induced negative chirp. The central part of the spectrum is compressed since both the long and short wavelengths are shifted toward the central wavelength. This process is called the nonlinear chirp compensation in the paper. Moreover, the compensation extent of two different chirps around the pulse center determines the spectral compression quality. Figure 2(a) shows the spectral compression under the condition of three cases of chirp compensations: insufficient compensation, complete compensation, and overcompensation. The respective compensated chirps are presented in Fig. 3(b). It is obvious that the spectral compression is not finished when the compensation is insufficient, leading to a poor spectral compression ratio (SCR), which is defined as the ratio of the input to the output spectral width (FWHM).
A higher spectral compression ratio can be obtained under the condition of overcompensation, but the pedestal and the side-lobe components are drastically raised, significantly degrading the compressed spectrum quality. Hence, when the absolute value of two chirps is equal but the signs are opposite, the dispersion-induced chirp is totally cancelled by the SPM-induced chirp around the center wavelength, resulting in a high enough spectral compression ratio and suppressed side-lobe components. The conclusion can be drawn that an excellent trade-off between the spectral compression ratio and spectral compression quality has been made when the chirp is precisely compensated near the central wavelength.

Theoretic Derivation of Comb-Like Distributed Fiber Based on Nonlinear Chirp Compensation Model
CDF is composed of several concatenations of SMF and HNLF such as the three-stage CDF shown in Fig. 3. Each concatenation of SMF and HNLF should satisfy the condition that the GVD-induced chirp in SMF is completely compensated by the SPM-induced chirp in HNLF, so that the central part of the pulse is chirp-free and can be prechirped in the next SMF. In general, the process of spectral compression in CDF is a repetition of the complete chirp compensation around the central wavelength. As the other parameters are decided when the fibers are chosen, the lengths of SMF and HNLF are the only factor which could determine the extent of chirp compensation. Therefore, how to calculate the fibers' length is the key to the design of the whole system based on the chirp compensation model. In order to prove the theoretical analysis, a strict derivation to calculate the fibers' length is provided next. A chirp-free Gaussian pulse, with a central wavelength of 1550 nm, a duration of T 0 (FWHM), and a peak power of P 0 (P 0 ¼ E 2 0 ), is used as an the input of the system. The SMF has a dispersion coefficient of β 2 while the nonlinear coefficient of HNLF is a constant of λ. The lengths of the first concatenation of SMF and HNLF (named SMF1 and HNLF1) are S 1 and H 1 , respectively. The input pulse can be expressed in the time and frequency domains. 17

<
: Only considering the group velocity dispersion effect in the SMF, the distribution in the frequency domain E 1 ðωÞ after pulse transmission in SMF1 is Using the Fourier transform pair of we can obtain the respective distribution in the time domain E 1 ðTÞ.
Simplifying Eq. (3): From Eq. (4), we can gain the linear phase shift ϕ GVD−S 1 in the SMF1: Then, the chirp induced by GVD (δω GVD−S 1 ) in the SMF1 can be calculated by taking the derivative with respect to ϕ GVD−S 1 .
After the transmission in the SMF1, the pulse E 1 keeps transmitting in the HNLF1. Similarly, only considering the self-phase shifting effect in the HNLF1, the nonlinear phase shifting in the HNLF1 can be calculated as follows: 17 The chirp induced by SPM (δω SPM−H 1 ) in the SMF1 can be calculated by taking the derivative with respect to ϕ SPM−H 1 .
In order to simplify Eq. (8), the nonlinear part of exp½−T 2 ∕T 2 0 þ ðβ 2 2 S 2 1 ∕T 2 0 Þ can be neglected as the constant of 1. The reason is described as follows: First, the magnitude order of a typical β 2 is 10 −27 ðs 2 ∕mÞ, otherwise a typical pulse duration has a number scale of 10 −15 ðsÞ. Therefore, the part of β 2 2 S 2 1 ∕T 2 0 is approximate to the value of zero. Besides, the chirp compensation would happen around the central wavelength of the pulse, e.g., −0.3T 0 < T < 0.3T. Assuming that T ¼ 0.3T 0 , the value of expð−T 2 ∕T 2 0 Þ is equal to 0.9139, which is close to 1. Therefore, we can make an approximation that the nonlinear part of exp½−T 2 ∕T 2 0 þ ðβ 2 2 S 2 1 ∕T 2 0 Þ can be neglected as the constant of 1. Then, The key of the chirp compensation model is the precisely complete chirp compensation between the GVD-induced and SPM-induced chirps around the central wavelength of the pulse. Therefore, − From Eq. (11), we can gain the relationship between the lengths of the SMF1 and HNLF1 as follows: It can be concluded from Eq. (11) that the length of HNLF1 is only determined by the length of SMF1 when the fibers are selected, which just simplifies the design of the CDF.
Similarly, the length of the next concatenation of SMF and HNLF ðS 2 ; H 2 ; S 3 ; H 3 ; : : : Þ can be precisely calculated following the derivation above and is shown as follows: Equations (12) to (14) reveal the relationship between the length of HNLF, SMF, and the parameters of two types of fibers based on the complete nonlinear chirp compensation in the CDF. The result of derivation is used in the CDF design in the next section.

Design and Simulation of the Comb-Like
Distributed Fiber To prove the analytical derivation above, a three-stage CDF is designed which is strictly calculated following the derivation. The calculation results of the length of each SMF and HNLF are shown in Fig. 4. The numerical simulation is simultaneously carried out for certification. The pulse propagation in the CDF is described by the nonlinear Schrödinger equation: where β 2 and β 3 are second-and third-order dispersions, respectively, α is the loss coefficient, and γ is the nonlinear coefficient. Equation (15) is solved by using the split-step Fourier method. 17 Figure 4 shows the design of a threestage CDF composed of SMF and HNLF, in which each fiber length is labeled. The parameters of SMF and HNLF are listed in Table 1. In the simulation, a chirp-free Gaussian pulse, with a central wavelength of 1550 nm, a duration of 300 fs (FWHM), and a peak power of 6.7 W, is used as the input of the CDF.
The simulation results are shown in Figs. 5-7. In the first concatenation, the input spectrum is compressed from 11.8 to 1.5 nm, obtaining a SCR of 7.87. The linear chirp induced by GVD in SMF1 is completely compensated near the pulse center (−1 ps < T < 1 ps). The second concatenation acquires a second stage SCR of 2.79 and a two-level SCR of 21.85, compressing the output spectrum of HNLF1 from 1.5 to 0.54 nm, and the precise compensated chirp is distributed in the area of −2.5 ps < T < 2.5 ps. A further SCR of 2.57 is gained in the last cascading, leading to a total SCR of 56.2 for the three-stage CDF. The simulation results have proved the feasibility of subpicosecond pulse spectral compression in a carefully designed CDF. An SCR of 56.2, which is the highest to our knowledge, is obtained. Following the principle of complete chirp compensation, more stages of CDF (>3) can be designed to realize a higher spectral compression ratio.

Application in the 7-Bit All-Optical ADC System
The soliton self-frequency shift effect is usually employed as a critical quantization principle in the all-optical analog-todigital conversion system, which has been carefully studied in Refs. 18-21. The amount of frequency shift increases in proportion to the peak power of the input pulse to realize the power-to-wavelength conversion. The quantization resolution, which is a key factor for the whole system, is defined as 20 Fig. 4 Design of three-stage comb-like distributed fibers composed of SMF and HNLF. Table 1 Second-order dispersion β 2 , third-order dispersion β 3 , the nonlinear coefficient γ, and the loss coefficient α of the singlemode fiber (SMF) and high nonlinearity fiber (HNLF).
where λ shift and λ FWHM are the amount of the center wavelength shift and the spectral width of the pulse after Raman self-frequency shift, respectively. For a certain value of the input pulse's peak power, the amount of the center wavelength shift (λ shift ) is a constant. Therefore, compressing the spectral width of the pulse after raman soliton frequency shifting (RSFS) is one feasible approach to increase the resolution of an all-optical ADC system. In this section, we propose and simulate a 7-bit all-optical analog-todigital scheme based on Raman self-frequency shift and the wide-band wavelength-tunable spectral compression in CDF as shown in Fig. 8. In the simulation, a chirp-free     Tables 1 and 2, respectively. Figure 9 presents the simulated results of 128 output compressed spectra after propagating in the three-stage CDF. The maximum and minimum spectral compression ratios are 50.6 and 35.8, which correspond to the minimum and maximum spectral widths of 0.17 and 0.23 nm after compression. The system resolution is determined by the maximum spectral width. Therefore, the 0.23-nm width is used in the Eq. (16) calculation and the calculated resolution of the all-optical quantization system is 7.1, which is the highest right now to the best of our knowledge.
It is easy to find from Fig. 9 that the spectral compression ratio is not a constant but keeps changing. The discussion should be made to explain why and how the spectral compression ratio changes in the wide-band wavelength-tunable spectral compression of CDF, and how it may affect the spectral compression quality.
As described above, the essential part of spectral compression in CDF is the balanced chirp compensation in each concatenation of SMF and HNLF. For the input pulse with different central wavelengths but with unchanged peak powers (i.e., Fig. 8), the balance would retain its stability and the GVD or SPM effect cannot break the tie, which leads to maintaining the spectral compression ratio as a constant. However, the pulses after RSFS not only have different central wavelengths but also have different peak powers. Different peak powers have little influence on the GVD effect, but sharply affect the SPM process, Fig. 8 Schematic diagram of all-optical ADC system based on RSFS and spectral compression in CDF. Table 2 Second-order dispersion β 2 , third-order dispersion β 3 , the nonlinear coefficient γ, and the loss coefficient α of the singlemode fiber (SMF) and high nonlinearity fiber (HNLF). Optical Engineering 126106-6 December 2014 • Vol. 53 (12) which may totally break the balance of chirp compensation.
Here, we just assume that the chirp compensation is balanced for a pulse with a lower peak power, because as the peak power grows, the chirp induced by SPM would rapidly increase and make the chirp overcompensated and the spectral compression quality would be degraded [e.g., green line shown in the Figs. 2(a) and 2(b)]. Conversely, the balanced chirp compensation for a pulse with a higher peak power would result in insufficient chirp compensation and would decrease the spectral compression ratio for the pulses with lower a peak power [e.g., blue line shown in the Figs. 2(a) and 2(b)]. Therefore, it can be concluded that balanced chirp compensation for every spectrum in a wide-band wavelength-tunable range cannot be realized in the wide-band wavelength-tunable spectral compression of CDF. What we call in this paper the "position" of balanced chirp compensation (i.e., the value of the pulse's peak power when the chirp compensation is balanced) is the key to determining the spectral compression ratio and the spectral compression quality.
Taking the simulation shown in Fig. 9 as an example, we resimulate the spectral compression in a three-stage CDF 128 times under the condition of three different balanced chirpcompensation positions: the lowest peak power (25 W), the highest peak power (31 W), and the middle point (28 W). The simulation setups are same as those in Fig. 9 above.
The simulated results are shown in Fig. 10 Figure 10(d) exhibits the spectral compression ratios' change as a function of the input peak power.
The results just certify the analysis above. A higher spectral compression ratio can be obtained from 41 to 53.75 when the balanced chirp-compensation position is 25 W, leading to a maximum quantization resolution of 7.25-bit compared with the other two situations. However, simultaneously, spectral pedestal side-lobe components rapidly arise (−9.6 dB when 31 W) which may directly degrade the spectral compression quality and even disable the resolution improvement of the all-optical analog-to-digital conversion system. 22 On the other hand, although successfully avoiding the generation of spectral pedestal side-lobe components, the spectral compression ratio is too low to improve the quantization resolution when the balanced chirp-compensation position is 31 W. Therefore, as we can see, the middle position of the peak powers, which is 28 W here, just provides a trade-off between the spectral compression ratio and quality. A high enough quantization resolution and sufficient low spectral pedestal can be gained at the same time. This feature is vital for the design of CDF applied in the all-optical analog-to-digital conversion system.
As shown in Fig. 11, which gathers three-stage compressed spectra from Figs. 5-7, it is clearer that the sidemode suppression ratio increases with the spectral compression. A qualitative discussion and explanation are given in this paper. The principle of spectral compression in the CDF is the complete chirp compensation around the pulse's center which had been presented above. In the first concatenation of SMF + HNLF, the spectrum is simultaneously compressed and the spectral side-lobe component is inevitably generated at the leading and trailing edges of the spectrum. The reason for the spectral side-lobe component generation is carefully analyzed in authors' previous papers. 22,23 Then, in the second stage, the central part of the spectrum continues to be compressed. However, it cannot be ignored that spectral side-lobe components at the leading and trailing edges are compressed at the same time. The spectral compression ratio is proportional to the peak power in some sense. Therefore, the central part of the spectrum with a higher peak power (corresponding to the central pulse) gains a bigger spectral compression ratio than that of the spectral side-lobe component, leading to an increase of the side-mode suppression ratio with the compression of the spectral width. This feature of spectral side-lobe component suppression in CDF is especially beneficial to the all-optical quantization resolution improvement proposed in this paper.

Conclusions
In summary, a three-stage CDF using a novel chirp-compensation method is proposed and numerically demonstrated. A strict mathematical derivation based on the complete nonlinear chirp-compensation principle is provided to calculate the fibers' lengths in CDF. Then, we design a three-stage CDF to verify low-pedestal wide-band wavelength-tunable spectral compression. The results show that a spectral compression ratio of 56.2 has been gained. The CDF's application in the 7-bit all-optical ADC system based on the soliton self-frequency shift is also introduced and simulated. The results show that the quantization resolution after spectral compression in CDF has been improved to 7.1-bit. Finally, we discuss how to select the balanced chirp-compensation position can make a compromise between the spectral compression ratio and spectral compression quality in the CDF and draw the conclusion that the middle points of the input peak powers are the optimum choice.