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Abstract
We propose a novel format conversion scheme, which can implement multichannel format conversion from return-to-zero (RZ) to non-return-to-zero (NRZ) for both LP01 and LP11 simultaneously by designing a few-mode fiber Bragg grating (FM-FBG) with comb spectra. To achieve filtering for all channels of the two modes, the FM-FBG response spectra of LP11 is designed to shift with that of LP01 by the WDM-MDM channel spacing. This approach is realized by carefully selecting the specifications of the few-mode fiber (FMF) to fulfill the requirements of the effective refractive index difference between LP01 and LP11. Each single-channel outline of the FM-FBG response spectra is designed according to the algebraic difference between the RZ and NRZ spectra. Numerical results show that both LP01 and LP11 channels with 300-GHz-spaced RZ signals at 40 Gbit/s can be converted into NRZ signals simultaneously, and the converted NRZ signals have high Q-factor and their eye diagrams are clean and open.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The emergence of new generation communication applications have led to an ever increasing demand on the transmission capacity of optical networks [1]. Mode division multiplexing (MDM) based on few-mode fiber (FMF) has emerged as a promising new technology that because it can increase the transmission capacity of the network and it can readily be combined with the wavelength division multiplexing (WDM) and time division multiplexing (TDM) [2–4].
In recent years, many studies have focused on WDM-MDM systems [5–9]. Future WDM-MDM systems may employ different modulation formats depending on the network scale, application, and bit rate, similar to the existing WDM systems. In long-haul transmission networks, the return-to-zero (RZ) format is preferred to the non-return-zero (NRZ) format due to its robustness to the nonlinear effects [10,11]. The NRZ format is used in the low-speed access network due to its narrower spectral bandwidth and higher timing jitter tolerance [12]. Therefore, simultaneous multichannel RZ to NRZ format conversion for multiple spatial modes is an essential technology for the future WDM-MDM communication system to link and interface the high-speed and low-speed access networks. In single-mode communication systems, various solutions have been provided to solve the format conversion problem [13–20]. But as far as we know, no solution has been proposed to solve this problem in WDM-MDM systems.
In this paper, we analyze the spectral response characteristics of few-mode fiber Bragg gratings (FM-FBGs) between different spatial modes with a multi-channel response spectra, and propose a matching method named spectrum shift alignment, which can result in the reflection channels of $\rm {LP_{01}}$ and $\rm {LP_{11}}$ response spectra being aligned at target wavelengths. This is achieved by selecting the few-mode fiber specifications based on the requirement of effective refractive index difference between the two modes. Based on the proposed alignment method, we present a novel format conversion scheme for WDM-MDM systems that employ the wavelength-alignment (WA) transmission scheme, which can perform multichannel format conversion from RZ to NRZ for both $\rm {LP_{01}}$ and $\rm {LP_{11}}$ simultaneously using a single custom-designed FM-FBG. Numerical results show that both $\rm {LP_{01}}$ and $\rm {LP_{11}}$ channels with 300-GHz-spaced RZ signals at 40 Gbit/s can be converted into NRZ signals simultaneously. Moreover, the converted NRZ signals have a high Q-factor and their eye diagrams are clean and open.
2. Constructing the spectra of the FM-FBG
The problem of multi-channel random RZ-to-NRZ data conversion for $\rm {LP_{01}}$ and $\rm {LP_{11}}$ can be seen as constructing a FM-FBG as a filter to implement converting the input RZ data spectra of the two modes into the output NRZ data spectra.
To achieve format conversion for multi-channel input RZ signals, the reflectivity spectra of the FM-FBG should be designed as a comb shape. The outline of each reflection channel is determined according to the algebraic difference between the RZ and the NRZ spectral profiles [21].
In WDM-MDM systems using the WA transmission strategy, standard ITU-T WDM grids are still adopted in all spatial channels (the red pattern in Fig. 1). For such a multichannel reflectivity spectra, due to the effective refractive index difference between $\rm {LP_{01}}$ and $\rm {LP_{11}}$, the response spectra of the two modes are not the same (the difference between them can be seen as a shift). Due to the nature of the shift, when we align the reflection channel center of $\rm {LP_{01}}$ with the signal channel center, that of $\rm {LP_{11}}$ deviates (illustrated in Fig. 1(a)). The shift is proportional to the effective refractive index difference between the two modes. Also, the effective refractive difference between the two modes is related to the specification of the FMF. Therefore, we can control the shift by carefully selecting the FMF specification to fulfill the effective refractive index difference requirement. It should be noted the shift can be made equal to the channel spacing used in the WDM-MDM system so that all spatial channels can be filtered (shown in Fig. 1(b)). The procedure is described in detail below.
Fig. 1. (a) Demonstration of the response spectra of the shift between ${\rm LP_{01}}$ and ${\rm LP_{11}}$. (b) Principle of spectrum shift alignment
2.1 Alignment of the reflection channels of ${LP_{\textit {01}}}$ and ${LP_{\textit {11}}}$
The uniform FM-FBG as an optical filter would select those wavelengths that meet the phase conditions to resonate [22,23]. Mode coupling in FM-FBGs is controllable. Our strategy imposes certain restrictions on the specification of the FMF, where the effective refractive index difference between $\rm {LP_{01}}$ and $\rm {LP_{11}}$ will typically be greater than ${10^{ - 3}}$, satisfying the weak coupling condition [24]. Furthermore, we can select two excited modes with a small field overlap and carefully choose the UV intensity and UV exposure time in fabricating the FM-FBG to suppress mode coupling [25,26]. Thus, the coupling between $\rm {LP_{01}}$ and $\rm {LP_{11}}$ has been neglected in this paper. The reflection spectra of $\rm {LP_{01}}$ and $\rm {LP_{11}}$ can be calculated separately, and their resonant wavelengths is given by
where ${\lambda _{01}}$ and ${\lambda _{11}}$ denote the resonant wavelength of $\rm {LP_{01}}$ and $\rm {LP_{11}}$,respectively. ${n_{\rm eff,01}}$, ${n_{\rm eff,11}}$ represent the mode effective refractive indices of $\rm {LP_{01}}$ and $\rm {LP_{11}}$, respectively, and $\Lambda$ is the period of the uniform FM-FBG.Due to the effective refractive index difference between modes, when uniform FM-FBGs are excited by different modes with equal power, the difference between their response spectra is mainly at the resonance wavelength, which can be seen as a shift. The multichannel response spectra of our designed FM-FBG can be regarded as constructed many segments each of which is a uniform FM-FBG having a different period. Therefore, the response spectra difference between the two modes can also be cosidered as a shift. Generally, the variation range of the period is small enough to be neglected when we calculate the shift, hence the period can be set as a constant and used as the Bragg period of the central wavelength of full $\rm {LP_{01}}$ reflection spectra in the calculation. From Eq. (1) and Eq. (2), the shift can be calculated as
where $\Delta {\lambda _{\rm shift}}$ is the shift between the spectra of $\rm {LP_{01}}$ and $\rm {LP_{11}}$, ${\Lambda _{\rm c}}$ denote the Bragg period of the central wavelength of full $\rm {LP_{01}}$ spectra.It is obvious that the FM-FBG response spectra of $\rm {LP_{01}}$ and $\rm {LP_{11}}$ can not entirely overlap to achieve filtering for all spatial channels. To achieve format conversion for all signal channels of $\rm {LP_{01}}$ and $\rm {LP_{11}}$, we propose a matching method named spectrum shift alignment, which aligns the response spectra channels of two spatial modes at all central wavelength of carriers by WDM-MDM channel spacing (illustrated in Fig. 1(b), i.e., the first reflection channel of $\rm {LP_{01}}$ is aligned with the second one of $\rm {LP_{11}}$; the second reflection channel of $\rm {LP_{01}}$ is aligned with the third one of $\rm {LP_{11}}$, etc.).
To implement spectrum shift alignment, two conditions need to be met. Firstly, designing one more channel for the response spectra of FM-FBG than carriers used in the WDM-MDM system. Secondly, the shift should be set equal to the channel spacing used in the WDM-MDM system. Mathematically, these conditions are as follows:
According to Eq. (3), the shift is determined by the effective refractive index difference between $\rm {LP_{01}}$ and $\rm {LP_{11}}$. However, the difference depends on the specification of the FMF used for transmission. Thus, we should carefully choose the specification of the FMF by Eq. (3), Eq. (5) and the eigenvalue equation of optical fibers to meet the effective refractive index difference requirement of spectrum shift alignment.
2.2 Design of the multi-channel spectra for ${LP_{\textit {01}}}$
To implement the conversion of the input RZ data spectra into the output NRZ data spectra, we must carefully design the single-channel outline for each channel of the FM-FBG response spectra, which significantly affects the Q-factor of the converted NRZ signal. Based on the considerations we discussed above, the algebraic difference between the RZ and the NRZ spectra should be calculated before ascertaining the appropriate transfer function of the FM-FBG single-channel response spectra. One of the methods to evaluate the algebraic difference between the RZ and the NRZ is first to construct isolated RZ pulses and isolated NRZ pulses and then calculate the algebraic difference between their spectra (shown in Fig. 2) [21].
Fig. 2. The spectra of the isolated RZ pulse with 67% duty cycle (yellow dashed dotted curve), the isolated NRZ pulse (bule dashed curve), for interval between pulses ${T_{\rm {p}}} = 25$ $\rm {ps}$, the wavelength of the carrier ${\lambda _{\rm c}} = 1550.12$ $\rm {nm}$ and the roll off factor ${\beta } = 0.2$. The red dotted curve is their algebraic difference (the ideal transfer function of the filter) and the cyan solid curve is the well-designed single-channel target spectra.
As a single-channel response spectra of the FM-FBG, it ought to have a limited passband. Additionally, it should be borne in mind that most of the NRZ signal power is concentrated in the range of $\left | {\lambda {\rm {\ }\hbox{-}{\rm \ }}{\lambda _{\rm c,NRZ}}} \right |{\rm {\ <\ }}\lambda _{\rm c,NRZ}^2{(c \cdot {T_{\rm p})}^{ - 1}}$. [14] Taking into account these considerations, after setting a uniform attenuation for out-of-band components, we can construct a band-pass filter (cyan curve in Fig. 2) as routinely done in the reported schemes [13,15,27,28]. Therefore, the well-designed single-channel simplified transfer function of FM-FBGs filter is given by
According to Eq. (4), after linearly superimposing the single-channel spectral response calculated above and setting appropriate characteristic length of the group delay, we can get the comb response spectra of the FM-FBG. The multi-channel response spectra of $\rm {LP_{01}}$ can therefore be written as
3. FM-FBG design
To demonstrate the principle of the operation of the proposed FM-FBG, we have chosen two 300-GHz-spaced channels (1547.72 and 1550.12 nm) for the WMD-MDM system to simulate. The carriers have been chosen with large channel spacing to reduce crosstalk between the adjacent channels. According to the discussion in Section. 2, three channels (1547.72, 1550.12 and 1552.52 nm) are selected as FM-FBGs designed channels for $\rm {LP_{01}}$. As for the FMF specification, the refractive indices and core diameter of the FMF selected for this exampple are $n_0$ = 1.4681, $n_1$ = 1.4620 and $a$ = 13.5714 $\mu m$, and the effective refractive indices of $\rm {LP_{01}}$ and $\rm {LP_{11}}$ are 1.4665 and 1.4642, respectively. The specifications chosen for the FMF are not unique and the specifications correspond to a type of two-mode fiber that only allows for stable transmission of LP01 and LP11 modes within the fiber.
Taking Eq. (7) as the object function, we can synthesize the structure of FM-FBG by the well-known discrete layer-peeling method [29–31], and use the transfer matrix method to simulate the reflection spectra of $\rm {LP_{01}}$ and $\rm {LP_{11}}$ respectively [32]. In our calculations, we set [${d_j}$] = [0.021, 0.056, 0.091]. The FM-FBG is divided uniformly into 7985 segments and the total length of it is 12 cm.
In Fig. 3(a), the local chirp results in a stair-like graph whose variation span is about $8 \times {10^{ - 9}}$ m. As is shown in Fig. 3(b), the maximum refractive index modulation is about $2.7 \times {10^{ - 4}}$ (i.e. much smaller than ${10^{ - 3}}$) making the FM-FBG practically realizable.
As is shown in Fig. 4(a), the simulated $\rm {LP_{01}}$ response spectra of the synthesized FBG (blue dotted curve) is in excellent agreement with the target multichannel spectra (red curve) obtained from Eq. (7). We plot the response spectra of $\rm {LP_{01}}$ and $\rm {LP_{11}}$ in Fig. 4(b) to demonstrate the concept of spectrum shift alignment proposed above. It is clear that the response spectra of $\rm {LP_{01}}$ and $\rm {LP_{11}}$ are mostly overlapped at all carriers (1547.72 and 1550.12 nm) used in the WDM-MDM system.
Fig. 4. (a) Simulated reflection spectra (dotted blue curve), target reflection spectra (solid red curve), and simulated group delay (dotted green curve). (b) Simulated reflection spectra of $\rm {LP_{01}}$ (dotted blue curve), simulated reflection spectra of $\rm {LP_{11}}$ (solid red curve).
4. Simulation results and discussion
In order to evaluate the capability of the designed FM-FBG, we simulated the operation of the synthesized FBG to perform multichannel RZ-to-NRZ format conversion for $\rm {LP_{01}}$ and $\rm {LP_{11}}$ in the case of a pseudo-random binary sequence (PRBS) of length ${2^{31}} - 1$ bits. Figure 5 shows the input RZ signal spectra for the duty cycles of 67% of $\rm {LP_{01}}$ and $\rm {LP_{11}}$ and the output NRZ signal spectra of them, respectively. It should be noted that the waveform used in our simulations are unipolar electric field waveforms.
Fig. 5. (a) Spectra of the input RZ signal and (b) spectra of the output NRZ signal for $\rm {LP_{01}}$ mode. (c) Spectra of the input RZ signal and (d) spectra the output NRZ signal for $\rm {LP_{11}}$ mode.
As is shown in Fig. 5(b) and Fig. 5(d), the spectra are reshaped at in-band wavelengths of all carriers, and the amplitude of out-of-band frequency is mostly suppressed. Inevitably, some frequency components will not be suppressed (circled in the red dotted curve in Fig. 5(b) and Fig. 5(d)), because we designed the FM-FBG response spectra with one more channel than transmitted carriers. However, if we can choose the appropriate channel spacing between carriers according to the duty cycle of waveform we used or set the channel spacing large enough so that the unfiltered frequencies fall within the partial lobe of the spectra, the impact brought by this situation can be greatly mitigated or be made negligible. The converted NRZ signal waveforms and their eye diagrams are shown in Fig. 6. It is evident that the quality of the converted NRZ waveform has not been greatly affected by the unfiltered frequency components when we set appropriate channel spacing.
Fig. 6. Input RZ waveforms, the spectra, and the eye diagrams for the output NRZ signals (from top to bottom). From left to right, the columns correspond to channels 1-4. The first and the second column correspond to the NRZ carrier wavelengths of 1547.72 and 1550.12 nm of $\rm {LP_{01}}$, respectively. The third and the fourth column correspond to that of $\rm {LP_{11}}$, respectively.
The second row shows the output NRZ waveform. In contrast to the DI-based and SOA-based scheme previously reported for single-mode systems [15,19,33], the FM-FBG-based scheme produces a converted waveform with smaller fluctuations in peak power between a single "1" bit and several consecutive "1" bits, while without obvious pattern-dependent effects.
In the third row, the clean and open eye diagrams show that the input four-channel RZ signals are converted to NRZ signals simultaneously with a high Q-factor ($> 25{\rm {dB}}$) thanks to the well-designed spectral response and the appropriately selected channel spacing. Furthermore, the Q-factor of different channels does not change significantly with the wavelength of the carrier or the spatial mode used. All Q-factors in our simulations are calculated with 160 points sampled in a period ranging from 0.34T to 0.66T. Compared to the Mach-Zehnder interferometric-based schemes reported previously for single-mode systems [20], the FM-FBG-based scheme output waveform exhibits no obvious bottom residual ripples in the eye diagram of the output NRZ signal. On the basis of these results, it can be concluded that the proposed FM-FBG filter gives rise to a high Q-factor RZ-to-NRZ format conversion for both $\rm {LP_{01}}$ and $\rm {LP_{11}}$.
5. Conclusion
We have proposed a single FM-FBG-based scheme to realize multichannel format conversion from RZ to NRZ for both $\rm {LP_{01}}$ and $\rm {LP_{11}}$ modes. The comb spectra of the FM-FBG is constructed based on the algebraic difference between the optical spectra of RZ and NRZ signals. To implement format conversion for all spatial mode channels, the response spectra of $\rm {LP_{11}}$ and $\rm {LP_{01}}$ are shifted by the WDM-MDM channel spacing. This shift is achieved by carefully selecting the specifications of the FMF to meet the effective refractive index difference requirements between $\rm {LP_{01}}$ and $\rm {LP_{11}}$. Simulation results show that two-channel two-mode 300-GHz-spaced RZ signals at 40 Gbit/s can be converted into NRZ signals simultaneously. The converted NRZ signals have high Q-factor, and their eye diagrams are also clean and open. The FBG-based scheme has a simpler system structure and higher robustness in achieving RZ-to-NRZ format conversion for multiple modes and multiple channels using only a single FM-FBG. In addition, the passive nature of FBG and the absence of active components in our proposed scheme make it free from ASE noise issues.
Funding
Bureau of Education of Guangzhou Municipality (202234641); Department of Education of Guangdong Province (2020ZDZX3072, 2022GXJK308, 2022ZDJS096, 2022ZDJS098, SXK20220201035); Department of Education of Guangdong Province (2021KTSCX098).
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.
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