# Optical Frequency Combs Generated in Silica Microspheres in the Telecommunication C-, U-, and E-Bands

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Fabrication of Microspheres

^{7}on average. Within a couple of days, the Q-factors dropped on the average to a value of 4 × 10

^{7}, and then this value remained unchanged for a long time. Two months after the preparation of the samples, the Q-factors were about 2 × 10

^{7}. Note that the record large value is Q = 8 × 10

^{9}measured at 633 nm ~1 min after fabrication for a silica 750-μm microsphere [33]. However, typical Q-factors for silica microspheres at 1.55 μm used for OFC generation or Raman lasing are 10

^{7}–10

^{8}[28,29,34,35], which agrees with our results. The reasons limiting Q-factors are discussed in [33].

#### 2.2. Numerical

#### 2.2.1. Calculation of Dispersion and Nonlinear Coefficient

_{l}

_{+1/2}and H

_{l}

_{+1/2}

^{(1)}are the Bessel function and the Hankel function of the 1st kind of order l + 1/2, respectively; l is the azimuthal index; k

_{0}= 2∙π∙ν/c is the light propagation constant in vacuum; ν is the frequency; k = n(ν)·k

_{0}; n(ν) is the linear refractive index of the silica glass (calculated using the Sellmeier formula given in [39]). We selected the first roots of Equation (1) for the fundamental mode family. The roots were localized using a well-known approximation formula for the eigenfrequencies presented, for example, in [38]. These approximate eigenfrequencies were used as the initial values for finding the roots of Equation (1). The iterative algorithm was implemented with the dispersion of the silica glass taken into account [39]. After finding eigenfrequencies for the fundamental TE mode family, we calculated the 2nd-order dispersion [40]:

_{eff}and nonlinear Kerr coefficient γ were calculated

_{2}= 2.2·10

^{−20}m

^{2}/W is the nonlinear refractive index of the silica glass [39]. We found γ = 6.2 (W∙km)

^{−1}for ν ≈ 193 THz and neglected its frequency dependence in the simulations below.

#### 2.2.2. Calculation of Intracavity CW Power

_{R}is the round trip time; t and τ are slow and fast times, respectively; α is the loss coefficient including intrinsic and coupling losses; β

_{k}is the dispersion of the k-th order; θ is the coupling coefficient; and δ

_{0}is the frequency detuning of the pump field Ε

_{in}from the exact resonance.

_{in}|

^{2}(π∙d∙γ∙θ/α

^{3}); Y = |E|

^{2}(π∙d∙γ/α); and Δ = δ

_{0}/α). We found Y(Δ) from Equation (7), and in the case when this equation had three roots, we took the maximum value corresponding to the upper stable branch.

#### 2.2.3. Calculation of Gain Coefficients for the Degenerate FWM Processes under the Strong Intracavity CW Field Approximation

_{0}found from Equation (7) and two small-amplitude sidebands (|a

_{±}| << |E

_{0}|) symmetrically shifted by ±Ω from the pump frequency

_{0}+ πd∙D

_{e}(Ω) + 2πdγ|E

_{0}|

^{2}, D

_{o}(Ω), and D

_{e}(Ω) are odd and even dispersion, respectively

_{0}|

^{2}is the intracavity power.

_{±}) is observed if Re(λ

_{±}(Ω) > 0). For this case, the gain coefficient is Re(λ

_{±}(Ω)).

#### 2.2.4. Numerical Simulation Based on Lugiato–Lefever Equation with the Raman Nonlinearity Taken into Account

_{R}= 0.18 is the fraction of Raman contribution to the total nonlinear response; T

_{1}= 12.2 fs, and T

_{2}= 32 fs.

## 3. Results

#### 3.1. The 1st Experimental Series

_{in}|

^{2}= 50 mW, assumed α = θ/2, and set the finesse π/α = 5 × 10

^{4}. Since detuning was not known for sure, we considered various values presented in Figure 1d. We found that the experimental sidebands shown in Figure 1c correspond to the gain bands for reasonable values of detuning. In addition, the Raman scattering may also affect the low-frequency OFC, but here we did not consider it for qualitative understanding of the FWM process.

#### 3.2. Theoretical Study of FWM

_{p}= 10 mW and P

_{p}= 50 mW. We used the approach described in Section 2.2.2 for finding intracavity CW powers and the approach described in Section 2.2.3 for finding gain coefficients for a FWM process.

_{p}= 50 mW than for P

_{p}= 10 mW due to a higher intracavity power (see Figure 1a). We also analyzed large values of detuning Δ = 40, 60, 80 for P

_{p}= 50 mW (see also the corresponding marked points in Figure 2b). The calculated gain coefficients are given in Figure 2e (here, individual colormap scales are used for each value of detuning). The broadest gain-bands are obtained for the pump frequencies close to the zero dispersion frequency. Their spectral widths exceed 20 THz. The larger the detuning, the higher the gain coefficient.

#### 3.3. The 2nd Experimental Series

#### 3.4. The 3rd Experimental Series and Its Interpretation

_{p}= 197.7 THz and attained only two OFCs: the first one was near the pump frequency, and the second was the Raman-assisted OFC (Figure 4b). In this case, the pump frequency was in the normal dispersion range, while the soliton-like Raman-assisted OFC was in the anomalous dispersion range.

^{9}modes. The numerically modeled stable spectrum and spectral phase of the Raman-assisted OFC are plotted in Figure 4c. Here, we set Δ = 6 and X = 25 to attain a sufficiently good agreement between the experimental and the simulated spectra (compare Figure 4b,c). The measured spectral intensity at the pump frequency f

_{p}is higher than the calculated spectral intensity at f

_{p}because in the experiment the transmitted unconverted pump is also measured. The slight difference in the shape of the experimental and measured spectra is explained by the fact that we did not exactly know several experimental parameters (for example, detuning and coupling coefficient). The simulated intensity distribution in the time domain is shown in Figure 4d. Next, we filtered out the Raman-assisted OFC in the spectral domain (using the super-Gaussian filter for the complex spectral envelope: exp(-(f

_{p}+ Δf

_{R})

^{6}/δf

^{6}), Δf

_{R}= 15 THz, δf = 7 THz) and found its field distribution in the time domain. The temporal intensity and phase of the Raman soliton are plotted in Figure 4e. We indeed obtained the sech

^{2}-shape soliton demonstrated in Figure 4e. This soliton has almost flat spectral and temporal phases. Its duration is 180 fs (full width at half maximum, FWHM) and TBP = 0.315 (time-bandwidth product, TBP). Note that for the sech

^{2}-shape Fourier transform limited pulses, TBP is also 0.315.

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Simplified experimental scheme for the 1st series. (

**b**) Numerically calculated dispersion for a silica microsphere with a diameter of 165 μm. (

**c**) Experimental spectrum demonstrating three OFCs: near the pump and in two frequency-separated sidebands. (

**d**) Numerically calculated gain coefficient under the strong pump approximation for a pump power of 50 mW. The dashed vertical line through (

**b**–

**d**) indicates zero dispersion.

**Figure 2.**(

**a**,

**b**) Intracavity CW power versus normalized detuning calculated for pump powers of 10 and 50 mW ((

**a**) is on a magnified scale). Numerically calculated gain coefficients under the strong pump approximation as functions of pump frequency and sideband frequency for pump powers P

_{p}= 10 mW (

**c**) and P

_{p}= 50 mW (

**d**,

**e**). For (

**c**,

**d**) the same colormap scale is used. For (

**e**) colormap scales are individual for each subplot.

**Figure 3.**(

**a**) Simplified experimental scheme for the second series. (

**b**–

**d**) The experimental spectra demonstrating frequency-tunable Raman-assisted soliton-like OFCs in the U-band as well as OFCs near the pump frequency in the C-band (and partially in the L-band in (

**b**)) and OFCs due to four-wave mixing in the E-band (and partially in the S-band in (

**b**)). The plateau in (

**b**–

**d**) between ~192 and 195 THz is due to using a bandpass filter before the tapered fiber.

**Figure 4.**(

**a**) A simplified experimental scheme for the 3rd series. (

**b**) The experimental spectrum demonstrating Raman-assisted soliton-like OFC and (

**c**) the corresponding numerically simulated spectrum (vertical light blue lines, left axis), the spectral envelope of the Raman-assisted soliton (solid line, left axis), and the spectral phase of the Raman-assisted soliton (dash-dotted line, right axis). (

**d**) Numerically simulated intensity distribution in the time domain. (

**e**) Numerically simulated intensity distribution of the filtered out Raman-assisted soliton-like OFC in the time domain (solid line, left axis) and its phase (dash-dotted line, right axis). Broadband low intensity background in (

**b**) is due to spontaneous emission from the pump laser.

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**MDPI and ACS Style**

Anashkina, E.A.; Marisova, M.P.; Salgals, T.; Alnis, J.; Lyashuk, I.; Leuchs, G.; Spolitis, S.; Bobrovs, V.; Andrianov, A.V.
Optical Frequency Combs Generated in Silica Microspheres in the Telecommunication C-, U-, and E-Bands. *Photonics* **2021**, *8*, 345.
https://doi.org/10.3390/photonics8090345

**AMA Style**

Anashkina EA, Marisova MP, Salgals T, Alnis J, Lyashuk I, Leuchs G, Spolitis S, Bobrovs V, Andrianov AV.
Optical Frequency Combs Generated in Silica Microspheres in the Telecommunication C-, U-, and E-Bands. *Photonics*. 2021; 8(9):345.
https://doi.org/10.3390/photonics8090345

**Chicago/Turabian Style**

Anashkina, Elena A., Maria P. Marisova, Toms Salgals, Janis Alnis, Ilya Lyashuk, Gerd Leuchs, Sandis Spolitis, Vjaceslavs Bobrovs, and Alexey V. Andrianov.
2021. "Optical Frequency Combs Generated in Silica Microspheres in the Telecommunication C-, U-, and E-Bands" *Photonics* 8, no. 9: 345.
https://doi.org/10.3390/photonics8090345