Microsphere-Based Optical Frequency Comb Generator for 200 GHz Spaced WDM Data Transmission System

: Optical frequency comb (OFC) generators based on whispering gallery mode (WGM) microresonators have a massive potential to ensure spectral and energy e ﬃ ciency in wavelength-division multiplexing (WDM) telecommunication systems. The use of silica microspheres for telecommunication applications has hardly been studied but could be promising. We propose, investigate, and optimize numerically a simple design of a silica microsphere-based OFC generator in the C-band with a free spectral range of 200 GHz and simulate its implementation to provide 4-channel 200 GHz spaced WDM data transmission system. We calculate microsphere characteristics such as WGM eigenfrequencies, dispersion, nonlinear Kerr coe ﬃ cient with allowance for thermo-optical e ﬀ ects, and simulate OFC generation in the regime of a stable dissipative Kerr soliton. We show that by employing generated OFC lines as optical carriers for WDM data transmission, it is possible to ensure error-free data transmission with a bit error rate (BER) of 4.5 × 10 − 30 , providing a total of 40 Gbit / s of transmission speed on four channels.


Introduction
Optical frequency combs (OFCs) generated in whispering gallery mode (WGM) microresonators, also called microcombs, are desirable for basic science and a considerable number of applications [1,2]. For example, microcombs have been used in spectroscopy [3,4], in radio-frequency photonics [5], and even in the search for exoplanets [6]. Significant activity is growing in the study of microresonator-based OFC applications in quantum optics [7]. Microcomb-based non-classical light sources are demanded in quantum information science for quantum communication [8][9][10], and can significantly advance the generation of entangled states for quantum computation [11,12].

Calculation of Microresonator Characteristics
FSR is defined as c/(2πRn eff ), where c is the speed of light, R is the radius of a microresonator, and n eff is the effective refractive index for an operating family of WGMs, so, it is essential to choose the appropriate size and take into account that not only the material but also the waveguide component gives a contribution to n eff . To find an optimal size of a silica microsphere for obtaining FSR = 200 GHz, we solve the characteristic equation for different radii [34]: n m (kR) 1/2 J l+1/2 (kR) where m = −1 for transverse magnetic (TM) and m = 1 for transverse electric (TE) modes; the prime denotes the derivative with respect to the argument in brackets; J l +1/2 is the Bessel function of the order of (l + 1/2) with the azimuthal index l; H l + 1/2 (1) is the Hankel function of the 1st kind of the order of (l + 1/2); k 0 = 2πν/c is the propagation constant in vacuum; ν is a frequency; k = n·k 0 and n is the refractive index of the silica glass given by the Sellmeier formula [35]: with constants C 1 = 0.6961663, C 2 = 0.4079426, C 3 = 0.8974794; λ 1 = 0.0684043 µm, λ 2 = 0.1162414 µm, λ 3 = 9.896161 µm [35], here λ m = 2πc/ ω m . To solve numerically Equation (1) and find the "cold" eigenfrequencies ν l , we use home-made software. We consider only a fundamental mode family corresponding to the first roots. The roots are localized by using approximation formulas for the eigenfrequencies ν l approx given, for example, in [34]: (l + 1/2) + 1.85576(l + 1/2) 1/3 − n m 1 These values ν l approx are used as the initiators of the algorithm for searching the roots of Equation (1) by the modified Powell method [36]. The iterative algorithm is implemented with allowance for silica glass dispersion given by Equation (2). After finding eigenfrequencies, we calculate the second-order dispersion [30]: where ∆ν l = ν l+1 − ν l−1 2 ; ∆(∆ν l ) = ν l+1 − 2ν l + ν l−1 .
FSR is calculated as the difference between two neighboring eigenfrequencies near 193.1 THz. The nonlinear Kerr coefficients γ are also estimated [33]: 2πR Expressions for fields corresponding to eigenmodes are very cumbersome, therefore they are not written here but can be found in [34].
We also estimate the contribution of thermo-optical effects on eigenfrequencies and FSR using the approach developed and applied previously for tellurite glass microspheres [37]. This approach is based on the numerical simulation of steady-state temperature distribution within the heat equation framework by the finite-element method with the COMSOL software [37]. In the model, the heat source is set near the microsphere surface in the equatorial area and originates due to partially dissipated pump energy. The volume of the heat source is equal to the effective WGM volume at the pump frequency. We use the boundary conditions "external natural convection" implemented in COMSOL. After finding the steady-state temperature field (an increase in temperature by ∆T), we estimate a shift of eigenfrequencies (WGM shift). This shift with temperature growth occurs due to both, the dependence of the refractive index on temperature 'dn/dT' and the increase in a radius by ∆R due to the thermal expansion (see Figure 1).

Simulation of OFC Generation
The OFC generation in a single family of fundamental modes is modeled numerically in the framework of the Lugiato-Lefever equation [1,35]: where E(t, τ) is an electric field inside the resonator; T and t = NR·tR are the fast and slow times; NR is the number of a microresonator roundtrip; tR = 2πRneff/c is the roundtrip time; δp is the phase detuning of the continuous wave (CW) pump field Ep from the nearest resonance; θ is the coupling coefficient; βk = d k β/dω taken at the pump frequency ω0 (we set βk = 0 for k ≥ 4); β(ω) = neff·k0 is the propagation constant; γ0 is the nonlinear Kerr coefficient at ω0; α = (2π) 2 R/(Qλp) is the loss coefficient including intrinsic and coupling losses; and λp is a pump wavelength. The response function is approximated by where δ(t) is the delta function; fR = 0.18 is the fractional contribution of the delayed Raman response, and with constants τ1 = 12.2 fs and τ2 = 32 fs [35]. We use a home-made software based on the split-step Fourier method [35] to simulate OFC generation numerically in the frame of Equation (7). We previously simulated DKS generation with similar software without the Raman nonlinearity [38,39]. Although here, a more advanced version of the numerical code with allowance for the Raman nonlinearity is used.

Simulation of Silica Microsphere OFC Generator-Based 4-Channel 200 GHz Spaced IM/DD WDM-PON Transmission System
The purpose of our research model was to simulate the 4-channel 200 GHz spaced intensity modulation direct detection (IM/DD) WDM-PON data transmission system by the implementation of designed silica microsphere-based OFC generator as a portable light source. We evaluate the performance of such a communication system according to next-generation PON (NG-PON2) requirements for specified valid optical link distances of up to 60 km [40]. Therefore, we used "VPI Photonics Design Suite" software for the simulation of a 4-channel 10 Gbit/s per channel non-return-to-zero on-off keying (NRZ-OOK) modulated WDM-PON optical transmission system to show the implementation of an microsphere-based OFC generator providing a spectral and energy-efficient fiber optical telecommunication system solution.

Simulation of OFC Generation
The OFC generation in a single family of fundamental modes is modeled numerically in the framework of the Lugiato-Lefever equation [1,35]: where E(t, τ) is an electric field inside the resonator; T and t = N R ·t R are the fast and slow times; N R is the number of a microresonator roundtrip; t R = 2πRn eff /c is the roundtrip time; δ p is the phase detuning of the continuous wave (CW) pump field E p from the nearest resonance; θ is the coupling coefficient; β k = d k β/dω taken at the pump frequency ω 0 (we set β k = 0 for k ≥ 4); β(ω) = n eff ·k 0 is the propagation constant; γ 0 is the nonlinear Kerr coefficient at ω 0 ; α = (2π) 2 R/(Qλ p ) is the loss coefficient including intrinsic and coupling losses; and λ p is a pump wavelength. The response function is approximated by where δ(t) is the delta function; f R = 0.18 is the fractional contribution of the delayed Raman response, and with constants τ 1 = 12.2 fs and τ 2 = 32 fs [35]. We use a home-made software based on the split-step Fourier method [35] to simulate OFC generation numerically in the frame of Equation (7). We previously simulated DKS generation with similar software without the Raman nonlinearity [38,39]. Although here, a more advanced version of the numerical code with allowance for the Raman nonlinearity is used.

Simulation of Silica Microsphere OFC Generator-Based 4-Channel 200 GHz Spaced IM/DD WDM-PON Transmission System
The purpose of our research model was to simulate the 4-channel 200 GHz spaced intensity modulation direct detection (IM/DD) WDM-PON data transmission system by the implementation of designed silica microsphere-based OFC generator as a portable light source. We evaluate the performance of such a communication system according to next-generation PON (NG-PON2) requirements for specified valid optical link distances of up to 60 km [40]. Therefore, we used "VPI Photonics Design Suite" software for the simulation of a 4-channel 10 Gbit/s per channel non-return-to-zero on-off keying (NRZ-OOK) modulated WDM-PON optical transmission system to show the implementation of an microsphere-based OFC generator providing a spectral and energy-efficient fiber optical telecommunication system solution.
To achieve high precision of the measured results, it is crucial to provide the pseudo-random bit sequence (PRBS) with sufficient length, defined in VPI simulation scheme by using "VPI photonics design suite" simulation program modified built-in Wichman-Hill-Generator. Architecture and further simulation results of the above-described NRZ-OOK fiber optical communication system are presented in Section 3.3 of this article.

Microresonator Characteristics
The simulated spatial distribution of the electric field of the fundamental TE mode at the frequency of about 193.1 THz is shown in Figure 2. For the fundamental TM mode, the spatial distribution of the absolute value of the field is very similar and not depicted here.
Photonics 2020, 7, x FOR PEER REVIEW 5 of 16 To achieve high precision of the measured results, it is crucial to provide the pseudo-random bit sequence (PRBS) with sufficient length, defined in VPI simulation scheme by using "VPI photonics design suite" simulation program modified built-in Wichman-Hill-Generator. Architecture and further simulation results of the above-described NRZ-OOK fiber optical communication system are presented in Section 3.3 of this article.

Microresonator Characteristics
The simulated spatial distribution of the electric field of the fundamental TE mode at the frequency of about 193.1 THz is shown in Figure 2. For the fundamental TM mode, the spatial distribution of the absolute value of the field is very similar and not depicted here. We solve Equation (1) for different microsphere diameters (d = 2R) to find the optimal one giving FSR of 200 GHz. FSR as a function of the microsphere diameter is plotted in Figure 3 for TE and TM families. These curves for TE and TM modes differ by ~10 MHz but almost coincide for the chosen scale in Figure 3.  We solve Equation (1) for different microsphere diameters (d = 2R) to find the optimal one giving FSR of 200 GHz. FSR as a function of the microsphere diameter is plotted in Figure 3 for TE and TM families. These curves for TE and TM modes differ by~10 MHz but almost coincide for the chosen scale in Figure 3. To achieve high precision of the measured results, it is crucial to provide the pseudo-random bit sequence (PRBS) with sufficient length, defined in VPI simulation scheme by using "VPI photonics design suite" simulation program modified built-in Wichman-Hill-Generator. Architecture and further simulation results of the above-described NRZ-OOK fiber optical communication system are presented in Section 3.3 of this article.

Microresonator Characteristics
The simulated spatial distribution of the electric field of the fundamental TE mode at the frequency of about 193.1 THz is shown in Figure 2. For the fundamental TM mode, the spatial distribution of the absolute value of the field is very similar and not depicted here. We solve Equation (1) for different microsphere diameters (d = 2R) to find the optimal one giving FSR of 200 GHz. FSR as a function of the microsphere diameter is plotted in Figure 3 for TE and TM families. These curves for TE and TM modes differ by ~10 MHz but almost coincide for the chosen scale in Figure 3.  Furthermore, using Expressions (4) and (5), we calculate dispersion for the optimal microsphere diameter of~328.5 µm and for comparison for d = 322 µm and for d = 335 µm for which FSR differs by ±2% from 200 THz and is 204 and 196 THz, respectively. The frequency-dependent dispersions for TE and TM modes for these diameters are plotted in Figure 4a in the 188-198 THz range. It is seen that curves differ slightly. Figure 4b demonstrates the dispersion of the TE family at a wider frequency range for the diameter of 328.5 µm. The dispersion is anomalous at 193.1 THz and the zero-dispersion frequency is about 209 THz. The nonlinear Kerr coefficient is about 3.5 (W·km) −1 at 193.1 THz, and its frequency dependence can be neglected in the C-band.
Photonics 2020, 7, x FOR PEER REVIEW 6 of 16 Furthermore, using Expressions (4) and (5), we calculate dispersion for the optimal microsphere diameter of ~328.5 μm and for comparison for d = 322 μm and for d = 335 μm for which FSR differs by ±2% from 200 THz and is 204 and 196 THz, respectively. The frequency-dependent dispersions for TE and TM modes for these diameters are plotted in Figure 4a in the 188-198 THz range. It is seen that curves differ slightly. Figure 4b demonstrates the dispersion of the TE family at a wider frequency range for the diameter of 328.5 μm. The dispersion is anomalous at 193.1 THz and the zero-dispersion frequency is about 209 THz. The nonlinear Kerr coefficient is about 3.5 (W·km) −1 at 193.1 THz, and its frequency dependence can be neglected in the C-band. Next, we estimate the influence of thermal effects on shifts of WGM frequencies and FSR for the optimal diameter. Figure 5a,b demonstrate the calculated shift of eigenfrequencies and FSRs as functions of the temperature increase, respectively. For Figure 5, we assume uniform temperature increase distribution. WGM shifts are almost the same for TE and TM mode families, but FSRs differ by ~10 MHz. FSR difference occurs due to initial "cold" FSR non-equality between TM and TE modes. To understand when such temperature increases can be achieved, we simulate temperature fields at different powers of the heat source. Temperature distributions for the heat powers of 1, 3, and 5 mW are presented in Figure 6a,b, in Figure 6c,d, and in Figure 6e,f, respectively. Figure 6a,c,e show the complete simulated geometry, including the microsphere itself and the fiber stem, but Figure 6b,d,f show an enlarged microsphere. The temperature inside the microsphere is distributed fairly uniformly. For greater clarity, Figure 6g also shows the dependences of the temperature averaged over the sphere, the temperature averaged over the WGM volume, and the maximum Next, we estimate the influence of thermal effects on shifts of WGM frequencies and FSR for the optimal diameter. Figure 5a,b demonstrate the calculated shift of eigenfrequencies and FSRs as functions of the temperature increase, respectively. For Figure 5, we assume uniform temperature increase distribution. WGM shifts are almost the same for TE and TM mode families, but FSRs differ by~10 MHz. FSR difference occurs due to initial "cold" FSR non-equality between TM and TE modes.
Photonics 2020, 7, x FOR PEER REVIEW 6 of 16 Furthermore, using Expressions (4) and (5), we calculate dispersion for the optimal microsphere diameter of ~328.5 μm and for comparison for d = 322 μm and for d = 335 μm for which FSR differs by ±2% from 200 THz and is 204 and 196 THz, respectively. The frequency-dependent dispersions for TE and TM modes for these diameters are plotted in Figure 4a in the 188-198 THz range. It is seen that curves differ slightly. Figure 4b demonstrates the dispersion of the TE family at a wider frequency range for the diameter of 328.5 μm. The dispersion is anomalous at 193.1 THz and the zero-dispersion frequency is about 209 THz. The nonlinear Kerr coefficient is about 3.5 (W·km) −1 at 193.1 THz, and its frequency dependence can be neglected in the C-band. Next, we estimate the influence of thermal effects on shifts of WGM frequencies and FSR for the optimal diameter. Figure 5a,b demonstrate the calculated shift of eigenfrequencies and FSRs as functions of the temperature increase, respectively. For Figure 5, we assume uniform temperature increase distribution. WGM shifts are almost the same for TE and TM mode families, but FSRs differ by ~10 MHz. FSR difference occurs due to initial "cold" FSR non-equality between TM and TE modes. To understand when such temperature increases can be achieved, we simulate temperature fields at different powers of the heat source. Temperature distributions for the heat powers of 1, 3, and 5 mW are presented in Figure 6a,b, in Figure 6c,d, and in Figure 6e,f, respectively. Figure 6a,c,e show the complete simulated geometry, including the microsphere itself and the fiber stem, but Figure 6b,d,f show an enlarged microsphere. The temperature inside the microsphere is distributed fairly uniformly. For greater clarity, Figure 6g also shows the dependences of the temperature averaged over the sphere, the temperature averaged over the WGM volume, and the maximum To understand when such temperature increases can be achieved, we simulate temperature fields at different powers of the heat source. Temperature distributions for the heat powers of 1, 3, and 5 mW are presented in Figure 6a,b, in Figure 6c,d, and in Figure 6e,f, respectively. Figure 6a,c,e show the complete simulated geometry, including the microsphere itself and the fiber stem, but Figure 6b,d,f show an enlarged microsphere. The temperature inside the microsphere is distributed fairly uniformly. For greater clarity, Figure 6g also shows the dependences of the temperature averaged over the sphere, the temperature averaged over the WGM volume, and the maximum temperature on the heat power. Despite the fact that the full nonlinear heat equation was solved, the dependencies of these temperatures on heat power appear to be almost linear for the considered heat power range.
Photonics 2020, 7, x FOR PEER REVIEW 7 of 16 temperature on the heat power. Despite the fact that the full nonlinear heat equation was solved, the dependencies of these temperatures on heat power appear to be almost linear for the considered heat power range. After finding the temperature distributions and dependencies of temperature increases on the heat power, we plot the FSR versus the power of the heat source (see Figure 7). We also verified that the change in dispersion curves with increasing temperature is negligible for the considered heat powers of a few mW.  After finding the temperature distributions and dependencies of temperature increases on the heat power, we plot the FSR versus the power of the heat source (see Figure 7). We also verified that the change in dispersion curves with increasing temperature is negligible for the considered heat powers of a few mW.
Photonics 2020, 7, x FOR PEER REVIEW 7 of 16 temperature on the heat power. Despite the fact that the full nonlinear heat equation was solved, the dependencies of these temperatures on heat power appear to be almost linear for the considered heat power range. After finding the temperature distributions and dependencies of temperature increases on the heat power, we plot the FSR versus the power of the heat source (see Figure 7). We also verified that the change in dispersion curves with increasing temperature is negligible for the considered heat powers of a few mW.

OFC Generation in DKS Regime
Next, we simulate OFC generation in the DKS regime in the framework of Equation (7). We consider TE modes (for TM modes, the results will be almost the same). It is known that stable DKS can exist in an anomalous dispersion region only for specific conditions for a pump power and normalized detuning ∆ (∆ = δ p / α) [33,[41][42][43]. If the solution in the form of DKS exists in this case, one particular value of DKS peak power corresponds to each admissible detuning [41,43]. For a pump power less than a threshold for this detuning, DKS cannot exist, but CW is a solution. We set pump power slightly higher than this threshold and study properties of generated DKS for different values of ∆. We set pump frequency at 193.1 THz assuming that the nearest resonant WGM can be shifted to 193.1 THz due to the thermal effects caused by partially dissipated pump power and/or external heating of a microsphere. Figure 8a shows the DKS spectrum calculated for ∆ = 50. This spectrum is asymmetric with respect to the pump frequency, which is explained by the influence of the Raman nonlinearity and agrees with the results presented in [43,44]. Figure 8b demonstrates the spectral envelopes of stable DKSes simulated for different ∆. The larger the normalized detuning, the broader the spectrum is. For example, for ∆ = 10, the spectral width at the level of -30 dB is 3.8 THz, but for ∆ = 70, the spectral width is 8.8 THz. Next, we count a quantity of spectral lines (harmonics) in OFC spectra with intensity higher than -30 dB (see Figure 8c). The quantity of lines satisfying this condition increases from 19 for ∆ = 10 up to 44 for ∆ = 70. For ∆ > 70, DKS is unstable and we do not consider intracavity nonlinear dynamics for this case. Note that due to higher-order dispersion, the range of DKS stability is slightly wider than with allowance for only the second-order dispersion presented in [43]. We also find DKS duration (full width at half maximum, FWHM) in the time domain as a function of ∆ (see Figure 8d). For larger ∆ (when the spectrum is wider), the duration is shorter according to the Fourier-transform limitation (379 fs for ∆ = 10 and 169 fs for ∆ = 70).

OFC Generation in DKS Regime
Next, we simulate OFC generation in the DKS regime in the framework of Equation (7). We consider TE modes (for TM modes, the results will be almost the same). It is known that stable DKS can exist in an anomalous dispersion region only for specific conditions for a pump power and normalized detuning Δ (Δ = δp/α) [33,[41][42][43]. If the solution in the form of DKS exists in this case, one particular value of DKS peak power corresponds to each admissible detuning [41,43]. For a pump power less than a threshold for this detuning, DKS cannot exist, but CW is a solution. We set pump power slightly higher than this threshold and study properties of generated DKS for different values of Δ. We set pump frequency at 193.1 THz assuming that the nearest resonant WGM can be shifted to 193.1 THz due to the thermal effects caused by partially dissipated pump power and/or external heating of a microsphere. Figure 8a shows the DKS spectrum calculated for Δ = 50. This spectrum is asymmetric with respect to the pump frequency, which is explained by the influence of the Raman nonlinearity and agrees with the results presented in [43,44]. Figure 8b demonstrates the spectral envelopes of stable DKSes simulated for different Δ. The larger the normalized detuning, the broader the spectrum is. For example, for Δ = 10, the spectral width at the level of -30 dB is 3.8 THz, but for Δ = 70, the spectral width is 8.8 THz. Next, we count a quantity of spectral lines (harmonics) in OFC spectra with intensity higher than -30 dB (see Figure 8c). The quantity of lines satisfying this condition increases from 19 for Δ = 10 up to 44 for Δ = 70. For Δ > 70, DKS is unstable and we do not consider intracavity nonlinear dynamics for this case. Note that due to higher-order dispersion, the range of DKS stability is slightly wider than with allowance for only the second-order dispersion presented in [43]. We also find DKS duration (full width at half maximum, FWHM) in the time domain as a function of Δ (see Figure 8d). For larger Δ (when the spectrum is wider), the duration is shorter according to the Fourier-transform limitation (379 fs for Δ = 10 and 169 fs for Δ = 70).

Architecture and Simulation of 4-Channel 200 GHz Spaced IM/DD WDM-PON Transmission System
The simulation setup of a 4-channel 200 GHz spaced WDM-PON transmission system is depicted in Figure 9.

Architecture and Simulation of 4-Channel 200 GHz Spaced IM/DD WDM-PON Transmission System
The simulation setup of a 4-channel 200 GHz spaced WDM-PON transmission system is depicted in Figure 9. The output of an amplified spontaneous emission (ASE) optical light source with high output power of up to 23 dBm and spectrum power density of −6 dBm/nm within the 1528-1630 nm band is connected to the input of the user-defined optical band-pass filter (OBPF) where the above-simulated silica microsphere-based OFC output spectrum is implemented. Afterward, the output of comb spectral lines is filtered out by an optical band-pass filter with 760 GHz 3-dB bandwidth to obtain four optical carrier signals. The spectral lines from OFC-WGMR are filtered and de-multiplexed utilizing arrayed-waveguide-grating (AWG) de-multiplexer (de-MUX), which corresponds to wavelength-routed WDM-PON (WR-WDM-PON) architecture. The 3-dB bandwidth of each AWG channel for 200 GHz channel spacing was set to 87.3 GHz. The spectrum of the optical signal at the output of the user-defined OBPF with implemented microsphere-based OFC, the obtained band-pass filtering for four optical carriers, and the output of AWG de-multiplexer after back-to-back (B2B) transmission are shown in Figure 10a-c, respectively. Note that Figure 10a contains OFC lines simulated in Section 3.2 and presented in Figure 8a. The Lorentzian line shape is assumed. The output of an amplified spontaneous emission (ASE) optical light source with high output power of up to 23 dBm and spectrum power density of −6 dBm/nm within the 1528-1630 nm band is connected to the input of the user-defined optical band-pass filter (OBPF) where the above-simulated silica microsphere-based OFC output spectrum is implemented. Afterward, the output of comb spectral lines is filtered out by an optical band-pass filter with 760 GHz 3-dB bandwidth to obtain four optical carrier signals. The spectral lines from OFC-WGMR are filtered and de-multiplexed utilizing arrayed-waveguide-grating (AWG) de-multiplexer (de-MUX), which corresponds to wavelength-routed WDM-PON (WR-WDM-PON) architecture. The 3-dB bandwidth of each AWG channel for 200 GHz channel spacing was set to 87.3 GHz. The spectrum of the optical signal at the output of the user-defined OBPF with implemented microsphere-based OFC, the obtained band-pass filtering for four optical carriers, and the output of AWG de-multiplexer after back-to-back (B2B) transmission are shown in Figure 10a-c, respectively. Note that Figure 10a contains OFC lines simulated in Section 3.2 and presented in Figure 8a. The Lorentzian line shape is assumed.
Carriers (comb spectral lines) separated with AWG de-MUX are fed to the optical input of the Mach-Zehnder modulators (MZMs). The electrical data signals are provided by PRBS from PRBS generator through NRZ driver, which encodes the logical data by using the non-return-to-zero (NRZ) technique generating electrical NRZ signals with a bit rate of 10 Gbit/s. Each MZM, having a 3-dB bandwidth of 12 GHz and 20 dB extinction ratio, is driven by NRZ signal S 1 (t) [45].
Optical signals from each transmitter's (Tx) MZM are coupled together using an AWG multiplexer (MUX). The combined, modulated optical signals are transmitted over 20 up to 60 km ITU-T G.652 single-mode fiber (SMF) span, with 0.02 dB/km attenuation and 16 ps/nm/km dispersion coefficients at 1550 nm reference wavelength. According to NG-PON2 (ITU-T G.989.2) recommendation, the specified valid optical link distances are up to 40 km, but longer lengths are supported and reach network distance up to 60 km. Therefore, we extend the optical link section to maximal PON transmission distance of 60 km.
The receiver (Rx) consists of a PIN photodiode with 3-dB bandwidth of 12 GHz, sensitivity of −18 dBm for BER of 10 −12 , and responsivity of 0.65 A/W [46]. Afterward, the received, modulated signal is filtered by an electrical low-pass filter (LPF) with 7.5 GHz 3-dB electrical bandwidth. The electrical signal analyzer is used to measure the received signal, e.g., showing bit pattern and BER. Carriers (comb spectral lines) separated with AWG de-MUX are fed to the optical input of the Mach-Zehnder modulators (MZMs). The electrical data signals are provided by PRBS from PRBS generator through NRZ driver, which encodes the logical data by using the non-return-to-zero (NRZ) technique generating electrical NRZ signals with a bit rate of 10 Gbit/s. Each MZM, having a 3-dB bandwidth of 12 GHz and 20 dB extinction ratio, is driven by NRZ signal S1(t) [45].
Optical signals from each transmitter's (Tx) MZM are coupled together using an AWG multiplexer (MUX). The combined, modulated optical signals are transmitted over 20 up to 60 km ITU-T G.652 single-mode fiber (SMF) span, with 0.02 dB/km attenuation and 16 ps/nm/km dispersion coefficients at 1550 nm reference wavelength. According to NG-PON2 (ITU-T G.989.2) recommendation, the specified valid optical link distances are up to 40 km, but longer lengths are supported and reach network distance up to 60 km. Therefore, we extend the optical link section to maximal PON transmission distance of 60 km.
The receiver (Rx) consists of a PIN photodiode with 3-dB bandwidth of 12 GHz, sensitivity of −18 dBm for BER of 10 −12 , and responsivity of 0.65 A/W [46]. Afterward, the received, modulated signal is filtered by an electrical low-pass filter (LPF) with 7.5 GHz 3-dB electrical bandwidth. The electrical signal analyzer is used to measure the received signal, e.g., showing bit pattern and BER.
The performance indicators as BER and eye diagrams of the received signal verify the feasibility of the designed transmission. The obtained BER results of each optical channel with respect to optical network link section length of up to 60 km over SMF for NRZ modulated 4-channel IM/DD WDM-PON transmission system with 200 GHz spacing is shown in Figure 11. We observed that the worst-performing channel in terms of BER performance was the 1st channel (193.1 THz). In the best scenario, the highest system performance was observed for the 4th optical channel (193.7 THz), where BER of the received signal after transmission over 60 km SMF fiber link was 4.5 × 10 −30 . The drop in the BER performance is mainly affected by the power and noise floor variation between comb lines and phase noise. In such a case, the comb source for the data transmission system must ensure minimal optical carrier-to-noise power ratio (OCNR) that a comb line must have to be useful for data transmission.
We have demonstrated 4-channel 200 GHz spaced IM/DD WDM-PON transmission system with operating data rates of 10 Gbit/s per channel over different SMF fiber link section lengths up to 60 km, please see Figure 12. As shown in Figure 12a-c, for the 1st optical channel with worst BER performance in B2B configuration, as well after 20 and 40 km, the signal quality is very good, the eye We observed that the worst-performing channel in terms of BER performance was the 1st channel (193.1 THz). In the best scenario, the highest system performance was observed for the 4th optical channel (193.7 THz), where BER of the received signal after transmission over 60 km SMF fiber link was 4.5 × 10 −30 . The drop in the BER performance is mainly affected by the power and noise floor variation between comb lines and phase noise. In such a case, the comb source for the data transmission system must ensure minimal optical carrier-to-noise power ratio (OCNR) that a comb line must have to be useful for data transmission.
We have demonstrated 4-channel 200 GHz spaced IM/DD WDM-PON transmission system with operating data rates of 10 Gbit/s per channel over different SMF fiber link section lengths up to 60 km, please see Figure 12. As shown in Figure 12a-c, for the 1st optical channel with worst BER performance in B2B configuration, as well after 20 and 40 km, the signal quality is very good, the eye is open, and error-free transmission can be provided. After 60 km transmission, the BER of the received signal was 9.1 × 10 −4 , please see Figure 12d. Therefore, our investigated 200 GHz spaced OFC-WGMR light source-based IM/DD WDM-PON transmission system is fully capable of providing 10 Gbit/s of NRZ-OOK modulated signal transmission according to NG-PON2 recommendation specified valid optical link distances of 40 km. That means it is technically challenging to ensure such transmission stability for longer distances by the use of OFC-WGMR as an optical light source for telecommunication applications. More comprehensive future research on the limits of the OFCs parameters for implementing the PON transmission systems segment of such a long transmission distance (60 km) is desirable.

Discussion and Conclusions
We proposed and numerically investigated a simple design of silica microsphere-based OFC generator in the optical C-band producing stable DKS and simulated its implementation in the WDM data transmission system. The optimal microsphere diameter of 328.5 μm provides an FSR of 200 GHz, according to ITU-T G. 694.1 recommendation. We considered the TE and TM families of fundamental WGMs and found that the FSR values practically coincide. With a slight deviation of the diameter from the optimal value, FSR changes linearly. For example, if a diameter changes by ±2%, FSR changes by ±2% too (for diameters of 322 μm and 335 μm, FSRs are 204 and 196 THz, respectively). We also investigated the influence of thermo-optical effects on WGM shift and FSR. The WGM shift and FSR are almost linear functions of temperature. When the temperature rises by 60 C, the WGM eigenfrequencies decrease by 100 GHz, but FSR decreases by 0.1 GHz. We simulated steady-state temperature distribution originated from partial pump power dissipation. Although the heat source was located in a small volume equal to the effective mode volume, the temperature distribution was fairly uniform over the microsphere.
The dispersion as a function of frequency and the nonlinear Kerr coefficient were calculated. In the C-band, the dispersion is anomalous (of the order of −10 ps 2 /km). Both material and waveguide

Discussion and Conclusions
We proposed and numerically investigated a simple design of silica microsphere-based OFC generator in the optical C-band producing stable DKS and simulated its implementation in the WDM data transmission system. The optimal microsphere diameter of 328.5 µm provides an FSR of 200 GHz, according to ITU-T G. 694.1 recommendation. We considered the TE and TM families of fundamental WGMs and found that the FSR values practically coincide. With a slight deviation of the diameter from the optimal value, FSR changes linearly. For example, if a diameter changes by ±2%, FSR changes by ±2% too (for diameters of 322 µm and 335 µm, FSRs are 204 and 196 THz, respectively). We also investigated the influence of thermo-optical effects on WGM shift and FSR. The WGM shift and FSR are almost linear functions of temperature. When the temperature rises by 60 C, the WGM eigenfrequencies decrease by 100 GHz, but FSR decreases by 0.1 GHz. We simulated steady-state temperature distribution originated from partial pump power dissipation. Although the heat source was located in a small volume equal to the effective mode volume, the temperature distribution was fairly uniform over the microsphere.
The dispersion as a function of frequency and the nonlinear Kerr coefficient were calculated. In the C-band, the dispersion is anomalous (of the order of −10 ps 2 /km). Both material and waveguide contributions are important here. The zero-dispersion frequency is about 209 THz. Dispersion curves for TE and TM fundamental modes differ slightly. The nonlinear Kerr coefficient is about 3.5 (W·km) −1 . It was also verified that the change in dispersion with temperature increasing was negligible for the considered heat powers of a few mW.
We simulated OFC generation in the DKS regime for the range of normalized detuning ∆ where stable DKS can exist. When ∆ changes from 10 to 70, the spectral width at the level of −30 dB grows from 3.8 THz to 8.8 THz, and the DKS duration in the time domain shortened from 379 fs to 169 fs (FWHM). DKS spectra are asymmetric with respect to the pump frequency at 193.1 THz, which is explained by the influence of the Raman nonlinearity and agrees with the results presented in [43]. We assumed that OFC with FSR of 200 GHz could be anchored to 193.1 THz using temperature control. Narrow-band CW pump can be coupled to a microresonator using a fiber taper and generated OFC can be extracted from a microresonator with the same taper [28]. Note that a reliable way to generate DKS and influence of thermal effects on the dynamics of DKS formation is considered in [47]. The experimental features of the DKS generation in a silica microsphere are reported in [27]. For attaining OFCs with lower FSRs, bottle microresonators can be used as predicted in [48,49].
We have also simulated a 4-channel 200 GHz spaced IM/DD WDM-PON transmission system with operating data rates of 10 Gbit/s per channel over different SMF fiber link lengths up to 60 km. The variation in the BER performance of the OFC-WGMR portable light source is observed and is mainly affected by the power and noise floor variation between carriers. Therefore, the proposed 200 GHz spaced OFC-WCOMB light source-based IM/DD WDM-PON transmission system is fully capable of providing 10 Gbit/s of NRZ modulated signal transmission to NG-PON2 recommendation specified valid optical link distances up to 40 km.
Note that the considered system is attractive to service providers for substituting existing central office (CO) architecture. The major drawback of the current CO architecture is that individual lasers' array is used to sustain WDM data channels; however, here we use an OFC generator, employing only one single laser. An OFC generator based on silica microspheres is a cost-effective solution, making the possibility to add more end-users to one PON. The latter is especially interesting for those countries where service providers still use copper infrastructure, considering that service providers will need to install fewer PONs to connect a more significant number of end-users. So, our solution can potentially lower expenses to upgrade existing fiber infrastructure and change infrastructure in favor of optical fiber systems.

Conflicts of Interest:
The authors declare no conflict of interest.