Cross-phase modulation based ultra-flat 90-line optical frequency comb generation

This paper proposed an approach to design an evenly spaced, 1.8 THz spectrally broad and 1.6 dB flat optical frequency comb (OFC) by exploiting the cross-phase modulation in highly nonlinear fiber. The OFC is realized by controlling the phase of the signals in two parallelly placed highly nonlinear fibers. The frequency and line spacing of the OFC can be tuned by simply varying the periodicity and central wavelength of input electrical and optical signal, respectively.


Introduction
The gradual increase in network traffic with 20-30% per year increases the demand for data transmission capacity (Lovkesh and Singh 2021;. To fulfil these demands, the future network needs to introduce optical wavelength division multiplexing (WDM) technology which uses the path routing technology where a particular wavelength behaves as an address of destination/output node (Singh and Singh 2017a, b). So, the future high-performance network needs to handle hundreds of wavelength channels at a time (Yamamoto et al. 2013;Love et al. 2017;Alberto et al. 2017). This can be done using multiple parallel optical links. Still, it is not favourable in terms of scalability, complexity and high energy consumption as the requirement of a large number of laser sources ("Single-source chip-based frequency comb enabling extreme parallel data transmission," 2018). Over worldwide, around 9% of all electricity is consumed for data transmission only (Lovkesh and Singh 2021;, which needs to be considered. These things can be overcome using an OFC as a multichannel source ("Single-source chip-based frequency comb enabling extreme parallel data transmission," 2018; Xiangjun 2018; Xin 2019). There are other applications of OFC also: optical coherence tomography (OCT), high precision frequency metrology, to stabilize the electric fields underneath the pulse envelope, etc. (Ursula keller 2003). Periodic optical ultra-short pulse generation using mode-locked laser (MLL) (Benkler et al. 2013;Kim et al. 2008), electrooptic modulators (Morohashi et al. 2010;Hmood et al. 2015;Liu et al. 2017;Li et al. 2014;Li and Lin 2020;Ozharar et al. 2008;Tsypkin et al. 2015;Jianjun 2011;Anoh Kelvin et al. 2014;Jianguo et al. 2013;Zheng 2013;Sakamoto et al. 2007;Dou et al. 2011;Zhang et al. 2012;Li and Xiao 2015a,b;Ullah 2018;Lv et al. 2019), Kerr nonlinearity (Sharma et al. 2021a,b) and micro-resonators (Pfeifle et al. 2014;Marin-Palomo et al. 2017) are some effective techniques to generate OFC.
MLL sources generate periodic pulse train of femto-second pulse width by controlling the cavity parameters (Benkler et al. 2013;Kim et al. 2008). The channel spacing of such OFC generators depends on the cavity length of the MLL, which makes the design untunable. The cavity dependency over environmental conditions makes the system unstable. The compression of optical signals can generate the tunable and stable OFC by controlling the RF signal parameters applied to the multiple parallelly or serially connected electro-optic modulators (Morohashi et al. 2010;Hmood et al. 2015;Liu et al. 2017;Li et al. 2014;Li and Lin 2020;Ozharar et al. 2008;Tsypkin et al. 2015;Jianjun 2011;Anoh Kelvin et al. 2014;Jianguo et al. 2013;Zheng 2013;Sakamoto et al. 2007;Dou et al. 2011;Zhang et al. 2012;Li and Xiao 2015a,b;Ullah 2018;Lv et al. 2019).  proposed a single polarization modulation based seven-line OFC and achieve the flatness of 1.47 dB. Yamamoto et al. (Yamamoto et al. 2013) proposed an 11-line frequency comb with 1.9 dB spectrum flatness by the proper adjustment in the phase of the sinusoidal signal and DC biased signal applied to the upper and lower branch of dual-drive Mach-Zehnder modulator (MZM). Similarly, Jassim et al. (Hmood et al. 2015) and Li et al. (Liu et al. 2017) has used the same approach based on the biasing of MZM and generate the frequency comb with 27 and 80 comb lines with 1 dB and 10 dB spectrum flatness, respectively. Bo et al. (Li et al. 2014) cascaded the multiple MZM modulator with a similar biasing approach and generated the frequency comb of 9, 45 and 47 frequency lines with 0.8 dB, 1 dB, and 1.9 dB. From the literature, many electro-optic modulators are required to increase the number of frequency lines which makes the system bulky, costly, and complex. The OFC generator can be realized by using micro-resonators (Pfeifle et al. 2014;Marin-Palomo et al. 2017). However, the design of micro-resonators is not fully understood to date, which makes the system complex. The comb expansion by using Kerr's nonlinearity is another approach to design OFC generators. Sharma et al. (Sharma et al. 2021b) exploit the FWM interaction to generate a 55-line, 1.375 THz broad and 3 dB flat OFC. However, a complex and bulkier system is required to realize such OFC.
From the literature, the MLL based OFC generators show disadvantages in terms of tunability and stability. The electro-optic modulator based OFC generators have better tunability and stability but requires precise control over multiple parameters of multiple RF signals simultaneously, making the system complex. In addition to this, electro-optic modulator based OFC generators has less number frequency lines. The micro-resonator based OFC have complex structures. But XPM based OFC generators can realize a broad and tunable OFC. So, we realize a 90-line, 1.8 THz broad and 1.6 dB flat OFC in the proposed work. The structure of the paper is as follows. Section 2 covers the principle behind the generation of OFC. Section 3 covers the effects of various system parameters on the spectrum bandwidth and flatness of the OFC. The conclusion is made in Sect. 4 of the manuscript. The basic block diagram of the frequency comb generator is shown in Fig. 1. In the proposed design, the laser source provides a continuous wave (CW) optical signal at 1552 nm with 0 0 /0° elliptical/vertical polarization state and 5 MHZ FWHM followed by a power splitter. The CW signal in the upper branch is followed by a 90° polarization rotator and an optical attenuator. CW signal in the lower branch is followed by a Dual-Drive Mach-Zehnder modulator (DD-MZM), which is derived by periodic NRZ formatted electrical pulses. A binary source in the design gives the periodic signal to the NRZ-electrical pulse generator with the periodicity of 100 ps and 50% duty cycle. The timing diagram and spectrum diagram of the periodic optical signal at the DD-MZM output port is shown in Figs. 2a and 3a, respectively.
Further, the signal from DD-MZM is followed by a power splitter and propagates through the upper and lower branch of the USP generator. In the lower branch, periodic optical pulses propagate through an optical delay line (ODL) which induces the optical delay of 0.8 ps and modifies the phase of a periodic optical signal (Zhang et al. 2012) which is further followed by highly nonlinear fiber (HNLF). The CW signal is coming out from the polarization rotator, and the periodic optical signal co-propagates through the HNLF 1. The XPM and SPM interaction modulates the phase of the optical signals propagating through HNLF 1 and HNLF 2. The XPM interaction occurs in HNLF 1 only, and SPM interaction occurs in HNLF 1 and HNLF 2. The HNLF in both the branches has the same parameters to negate the effect of attenuation, SPM, dispersion, propagation delay, etc., in the design. Further, the HNLF 1 and HNLF 2 is followed by a power combiner followed by a y-axis polarization filter. An optical signal with 0.8 ps pulse duration and 50 ps periodicity is generated at the output port of the polarization filter. The output timing diagram and spectrum diagram are shown in Figs. 2b and 3b.
The proposed design gives the output period ultrashort pulse as the phase difference between periodic ultra-short pulses propagates in HNLF 1, and HNLF 2 become 180°, which is also given in Eq. (1).
Here in Eq. (1), θ u and θ l are the phase variation in the periodic optical signal HNLF 1 and HNLF 2. θ u, and θ l can be calculated using Eq. (2) and (3)  Here in Eq. (2), β is the phase propagation constant, L is the length of optical fiber, ϕ XPM is the phase variation due to cross-phase modulation, and ϕ SPM is the phase variation due to self-phase modulation. Similarly, in Eq. (3), β, L and ϕ SPM are the phase propagation constant, length of fiber and phase modulation due to SPM, respectively. The last term in Eq. (3) is the phase modulation due to the optical delay line, where c is the velocity of light, n is the refractive index of the medium, ∆T is the delay provided by ODL and λ o is the wavelength of an optical signal. The phase variation due to XPM in HNLF can be calculated using Eq. (4) (Agrawal 2008).
Here in Eq. (4), n 2 is the nonlinear refractive index coefficient, P pump is pump signal power, A eff is the effective area, and α is the attenuation constant. In the proposed design, the CW signal having 90° polarization acts as a pump signal, and a periodic optical signal Here in Eq. (5), P pump is the CW signal co-propagating with the periodic signal in HNLF shown in Fig. (1). The periodic electric field intensity of the output signal with T periodicity of the proposed design can be calculated from Eq. (6) ).
Here in Eq. (6), E out is the output signal electric field intensity, E o is the electric field intensity of periodic optical pulse train, and ϕ odl is the phase change due to optical delay line. The power of output periodic ultrashort pulse can be calculated by using Eq. (7), which is given below (Failed 2019a): The output power (P out ) becomes zero for the time from ΔT to T as the condition given in Eq. (1) gets fulfilled and results in the periodic ultra-short pulse. From Eqs. (6) and (7), the periodicity (T) of the output periodic signal defines the frequency difference between two consecutive frequency lines of the frequency comb, and the duration of optical pulses (∆T) defines the flatness of generated frequency spectrum of output signal (Oppenheim and Willsky xxxx). The lower the duty cycle of periodic optical pulses, the higher the frequency comb's flatness. The various parameters of the components used in the proposed design are given in Table 1.

Results and discussion
The noise in the periodic ultra-short pulse directly affects the flatness of the frequency comb, which depends on the various aspects of design parameters like pump signal power, interaction length, effective area, and nonlinear refractive index coefficient. The attenuator controls pump power in the proposed design. So, the effect of attenuation due to attenuator in the design at a different effective area of HNLF with 25 dBm of input power on output optical signal to noise ratio is plotted in Fig. 4.
Optical signal to noise ratio (OSNR) plays a significant role in designing flat frequency comb design. It is defined as the ratio of output signal power and noise power. From the plot, it is analyzed that, with the increase in attenuation of attenuator at 10 μm 2 effective area, the output OSNR value also increases due to decrease in the XPM interaction and reaches to peak value as the overall phase difference between the probe signals become an odd multiple of 180°. Further increase in the attenuation value decreases the OSNR value. From the plot, it is also analyzed that, with the increase in effective area, due to the decrease in the extent of XPM interaction, the maximum OSNR value achieved at the lower value of attenuation. As the effective area becomes 20 μm 2 , the maximum OSNR value is not achieved even at the 0 dBm of attenuation value. To achieve maximum OSNR value, input pump power needs to increase. From this analysis, it can be concluded that the effective area of HNLF should be as small as possible for the high OSNR periodic pulse generation. OSNR in the periodic ultra-short pulse directly affects the flatness of the frequency comb. Whenever the phase difference between the probe signals is less than 180 0 , a partial destructive interference will occur. This partial destructive interference causes the generation of fixed optical levels for the common duration of the probe signals. This causes the decrease in the OSNR value and correspondingly degradation in the flatness of the spectrum. This OSNR increases as the phase difference between the probe signals shifts towards 180 0 . As the phase difference reaches 180 0 , complete destructive interference will occur between the probe signals, which causes the maximum OSNR value and flat OFC. From this, it is observed that higher the OSNR value, more will be the spectrum flatness which can also be observed by comparing the plots shown in Figs. 4 and 5. The spectral  diagrams of the proposed frequency comb for a fiber having effective area of 12.5 μm 2 at various attenuation levels of 0 dBm, 1.33 dBm, 2 dBm, and 3 dBm is shown in Fig. 5. From the spectrum diagrams, it is also observed that, whenever the probe signals phase difference is less than 180 0 , due to partial destructive interference, the power of the centre frequency of the optical signal around the centre frequency remain high causes the sag around the centre frequency of the OFC. From the spectrum diagrams, it is analyzed that, initially, with the decrease in attenuation value from 0 dBm to 1.33 dBm, the spectral flatness of frequency comb also increases and reaches the spectral flatness of 1.6 dB. Furthermore, the spectral flatness starts degrading with an increase in attenuation level beyond 1.33 dBm. The various parameters of HNLF also play a crucial role to achieve the maximum OSNR and spectral flatness of the frequency comb. The length and nonlinear refractive index of HNLF are the major ones. In Fig. 6, the effect of the length of HNLF and the effective nonlinear refractive index coefficient is analyzed. From the plot, it is concluded that with the increase in length, due to an increase in XPM interaction, initially OSNR of output periodic ultra-short pulses increases and reaches the peak value as the overall phase difference between the same polarized signals in upper and lower branches reaches to the odd multiple of 180°. Further increase in length results in a decrease in OSNR value. With the increase in HNLF length, the attenuation due to fiber and other nonlinear effects also increases significantly, decreasing the amplitude of output ultrashort pulse and decreasing the OSNR value.
Further, the effect of length and n 2 of HNLF on the spectrum flatness is plotted in Fig. 7, which shows the spectrum diagram of the output signal at the various length of HNLF. From the spectrum diagrams, it is concluded that initially, with the increase in HNLF length, the spectrum flatness increases due to an increase in output OSNR value. Further increase in HNLF length results in a decrease in output spectrum flatness. Similarly, the duty cycle of periodic pulses plays a significant role in the flatness of the frequency comb (Ullah 2018). The power of the sideband of the periodic signal depends on the pulse width of the periodic signal and can be seen in Eqs. (8)-(10) (Sharma et al. 2021a). Equation (8) shows the Fourier series expansion of the periodic signal having the envelop shape of a gate signal. Here we consider the gate pulse for the simplification purpose.
Here in Eq. (8), n is an integer greater than 0 and o the frequency of the signal's envelope. In Eq. (9), a 0 and a n are the Fourier series coefficients and can be calculated from Eqs. (9) and (10) (Sharma et al. 2021a).
T p is the pulse duration of the periodic optical signal defined by the compression operation inside the active medium of SOA. T ′ is the periodic time of the optical signal at the power combiner's output port. Further, the effect of pulse duration on the spectrum bandwidth and flatness can be seen by comparing the spectrum diagrams shown in Fig. 8. The effect of a pulse width of periodic ultra-short pulse on the flatness of frequency comb can also be seen in Fig. 8. Figure 8 shows the variation in output frequency comb spectral flatness with respect to the variation in the duty cycle of a periodic ultra-short pulse at various frequency lines. This also holds a good agreement with Eq. (8).
The spectrum of the optical signal for different pulse duration is also shown in Fig. 9. From the spectrum diagrams, it can be seen that the spectrum bandwidth decreases with an increase in the pulse width of a periodic signal. It shows the optical spectrum of 0.5 ps, 1 ps, 1.5 ps, 2 ps, 2.5 ps, and 3 ps periodic optical pulses with 50 ps periodicity.
With the optimization in the various parameters of the proposed design, the periodic ultra-short pulse of 0.8 ps with the periodicity of 50 ps is realized, which results in the ultra-flat frequency comb of 90 comb lines with 20 GHz line spacing and 1.6 dB spectral flatness. The spectral diagram OFC is also shown in Fig. 10, which shows the improvement with respect to Singh and Singh, 2017b; Xin 2019; Morohashi et al.  Table 2.

Conclusion
In summary, we have proposed an approach to design an ultra-flat, tunable line spacing, 90-line and 1.6 dB flat OFC by exploiting the cross-phase modulation in highly nonlinear fiber. A frequency comb is obtained with the generation of periodic ultra-short pulses of 0.8 ps duration with 50 ps periodicity. The design analysis concludes that the line spacing in OFC directly depends on the periodicity of ultra-short pulses, which can be varied with the variation in the periodicity of input optical pulses. Further, the effect of the effective area of HNLF, attenuation due to attenuator in the design, nonlinear refractive index coefficient (n 2 ) and length of HNLF is analyzed and concluded that effective area and length of HNLF should be small, attenuation parameter depends on the XPM interaction requirement, and n 2 should be as high. Further, the effect of the duty cycle of periodic pulses on the flatness of frequency comb is also observed and concluded that the duty cycle of periodic pulses should be small for the flat optical spectrum. In future, the proposed design can be investigated for use as a multichannel source for high-performance data transmission.