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Abstract
Backward stimulated Brillouin scattering (SBS) in optical microcavities has been widely used in nonlinear optics and microwave photonics. Compared with glass material microcavity, magnesium fluoride crystal microcavity has the advantages of small absorption coefficient, fewer defects and larger nonlinear coefficient, moreover, it usually has a narrow gain bandwidth of tens of megahertz. Here, we design a high precision machining system to produce ultra-smooth surface magnesium fluoride crystal disk cavities with a diameter of about 5 mm, Q value exceeding 108, FSR matching material Brillouin gain. By simply modulating the pump wavelength and coupling power, we observe SBS phenomena with a 13.47 GHz Brillouin frequency shift near 1.55µm and cascaded stimulated Brillouin lasers (SBL) of up to 12 orders. In addition, the Brillouin-Kerr optical frequency comb in this device is demonstrated, observing nearly 300 comb lines spanning across a spectral window of 250 nm. Our research provides a way to fabricate high-Q crystal microcavities and demonstrates the potential of these devices in applications such as microwave sources and nonlinear optics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, optical frequency comb [1,2], as a high-precision frequency standard and spectral analysis tool, has been widely used in the field of optics and photonics. Its principle is based on SBS in nonlinear optical processes [3], an optical effect with frequency conversion and bandwidth broadening capabilities. SBS can be described as the pump interacting with the Stokes field through sound waves [4], in which the pump photons annihilate and both Stokes photons and phonons are produced. Stokes photons are usually backscattered, with the Doppler frequency dropping relative to the sound frequency. In high-order stimulated Brillouin scattering, the light mode in the microcavity [5] interacts with the phonon mode, resulting in modulation of Brillouin frequency shift and photon number density [6–9]. By properly designing the geometrical parameters and material characteristics of the microcavity, the Brillouin optical frequency comb can be modulated to form a series of uniformly distributed optical frequency peaks [10,11]. The spacing between these peaks is determined by the geometry and refractive index of the microcavity, which can be adjusted to achieve modulation and optimization of the frequency comb [6]. In the past few years, SBS has been extensively studied in many optical fibers [12,13], microresonators [3,6,7,14], and chip-based waveguides [15,16]. Among them, whispering gallery mode (WGM) microresonators are more likely to generate low threshold cascaded SBS due to their high Q factor and low mode volume [3,6,7,14]. In crystalline materials, magnesium fluoride ($\textrm{Mg}{\textrm{F}_2}$) has a high refractive index and low absorption loss, which can support long optical path coupling and longitudinal mode spacing of a series of modes [17]. This makes the $\textrm{Mg}{\textrm{F}_2}$ microcavity have a higher quality factor ($Q$ value), which provides favorable conditions for realizing high-order Brillouin scattering. The Brillouin-Kerr effect is a phenomenon associated with higher-order Brillouin scattering and is used to describe photon number density modulation and frequency stability in Brillouin optical frequency combs [18–21]. By modulating photon number density, the Brillouin-Kerr effect can further increase the frequency stability and spectral purity of the optical frequency comb [22]. This is important for some applications such as frequency metrology, frequency synthesis and optical clock.
In this paper, the application of high-order stimulated Brillouin scattering and Brillouin-Kerr effect in microcavity of $\textrm{Mg}{\textrm{F}_2}$ crystal with ultra-high Q value will be emphasized. We discuss the design and optimization of microcavity for high-order Brillouin scattering, and discuss the effect of Brillouin-Kerr effect on the performance of optical comb. By further studying these nonlinear optical processes, we can further expand the application range and improve the performance of optical frequency combs.
2. Methods and theory
2.1 Principle of stimulated Brillouin scattering and optical frequency comb formation in microcavity
Stimulated Brillouin scattering (SBS) is an inelastic scattering process in which photons interact with phonons. SBS is the result of the interaction between mechanical wave and electromagnetic wave, and its intensity depends greatly on the intensity of the pump light. When a beam of pump light is coupled into a material medium, due to the thermal vibration of atoms, spontaneous optical inelastic scattering is formed, Stokes photons are generated, and the subsequently scattered Stokes light forms an interference effect with the pump light, resulting in non-uniform distribution of electric field, which leads to the electrostrictive effect of the medium and the formation of acoustic waves. Under the action of sound waves, the elastic optical effect of the medium is triggered to form a moving Bragg grating distribution. Due to the effect of Bragg diffraction, Stokes light gain is amplified, and finally stimulated Brillouin scattering is formed, and a certain frequency shift ${v_B}$ is generated [8,9].
In order to achieve the laser effect with SBS gain in a microcavity, a double resonant structure is usually used, so a cavity resonance in both the pump and Brillouin gain regions is required [23]. As shown in Fig. 1(a), the figure above indicates the need for final control of the cavity geometry in a single echo wall mode (WGM), and the figure below shows that this strict condition is overcome by using a set of higher-order transverse WGM modes that differ from the fundamental free spectral range (FSR) frequency offset. In order to obtain higher order stimulated Brillouin scattering laser (SBL), a high precision machining device is used to control the geometry of the resonator so that the FSR of the resonator matches the Brillouin gain band [24]. Combined with the coupling mode equation established by SBS in the microcavity and the Brillouin gain ${g_B} = {c^2}{g_0}/{n_1}^2{V_{eff}}({1 + 4\Delta {\varOmega ^2}/{\Gamma ^2}} )$, the power threshold ${P_{th}}$ generated by SBL can be obtained [7], expressed as:
Fig. 1. Stimulated Brillouin scattering and Brillouin-Kerr optical frequency comb scheme in magnesium fluoride microresonators; (a) Brillouin WGM resonator scheme with double resonance structure, dotted line represents Brillouin gain band, solid Lorentz line and vertical line represent cavity resonance and excitation spectral line respectively; (b) FWM effect diagram, above is degenerate four-wave mixing, below is non-degenerate four-wave mixing; (c) Stimulated Brillouin scattering and Brillouin-Kerr optical comb diagram, input pump (gold), first order backward (red) and second order forward (dark orange) Stokes, backward equal frequency interval optical comb.
Microcavity-based optical frequency combs achieve spectral broadening primarily through the generation of cascaded four-wave mixing (FWM) [25,26] between lasers in a multi-wavelength cavity, which describes the interaction between photons, a process known in quantum mechanics as the creation of two new photons in the annihilation of two pumped photons. In a backward Brillouin-Kerr optical frequency comb, the pumped photons of FWM are supplied by Stokes photons generated by stimulated Brillouin scattering in the cavity, as shown in Fig. 1(b). Figure 1(c) shows the generation of SBL in the microcavity. The optical frequency generated by the first-order SBL and the backward Brillouin laser is combed back, and the second-order SBL is excited forward by the first-order SBL as a pump, so there is a phenomenon that the intensity of the forward-even order SBL is stronger, and the intensity of the back-odd-order SBL is stronger [23]. In order to better obtain high-order SBL and obtain microwave signals of hundreds of GHz, a high-precision machining device is used to control the geometry of the resonator so that the FSR of the resonator matches the Brillouin gain band.
2.2 Preparation of $Mg{F_2}$-WGMR disk
For certain pump wavelengths and microcavity materials, forward Brillouin scattering can occur in the phonon frequency range of ten MHz to several GHz, while the frequency shift of backward Brillouin is generally fixed. In this paper, the back-stimulated Brillouin scattering and Brillouin-Kerr optical frequency combs in the disk cavity of $\textrm{Mg}{\textrm{F}_2}$ crystal at 1550 nm are experimentally studied. In the $\textrm{Mg}{\textrm{F}_2}$ crystal material, the refractive index $n = 1.38$, the phonon rate ${V_a} = 7.75\textrm{km} \cdot {\textrm{s}^{ - 1}}$, the calculated ${v_B} = {\varOmega _B}/2\mathrm{\pi } = 13.8\textrm{GHz}$, corresponding to the wavelength interval of 0.1106 nm, because the longitudinal mode has a complex profile structure, The magnesium fluoride crystal cavity shows an important frequency change of 1 GHz, so its Brillouin frequency shift is 12.4-13.8 GHz in the 1550 nm pump band [17,27]. Therefore, in order to realize back Brillouin scattering by controlling the geometry of the resonator, the FSR in WGMs must be approximately equal to the Brillouin frequency shift ${v_B}$.
According to the formula $\mathrm{\Delta }{f_{\textrm{FSR}}} \approx \textrm{c}/2\mathrm{\pi }NR$, where N is the refractive index of the microcavity material, the radius of the disk cavity of $\textrm{Mg}{\textrm{F}_2}$ crystal should be about 2.5 mm. Figure 2(a) shows the main working area of high precision machining. Before processing, the commercial $\textrm{Mg}{\textrm{F}_2}$ crystal disk is fixed on one end of the metal aluminum rod, and the other end of the metal aluminum rod is fixed on the mouth of the gas bearing (as shown in the enlarged picture in the red box). The upper computer controls the speed of the air bearing and the position of the turning tool, and then the sample can be processed. The processing process is divided into three steps: rough grinding, rough polishing and fine polishing. After the rough grinding of diamond turning is completed, the following two steps are manually polished with diamond grinding paste and diamond suspension [24,28]. The whole process is observed in real time through the microscope. Figure 2(b) shows roughly and finely polished high definition surface images taken by a CCD camera. As you can see, the surface of the microcavity gradually becomes very smooth. Through the whole high-precision machining process, we obtained the $\textrm{Mg}{\textrm{F}_2}$ crystal disk cavity with diameter $D = 4.98\textrm{mm}$ and ultra-smooth surface, as shown in Fig. 2(c).
Fig. 2. High precision machining of crystal microcavity. (a) High precision machining device; (b) Surface images after rough and fine polishing; (c) 4.98 mm diameter $\textrm{Mg}{\textrm{F}_2}$ crystal disk cavity.
Figure 3(a) shows the transmission spectrum generated by the microcavity under CW laser pumping, where the black box mark is the selected spectral line for Lorentz fitting. The cavity ring-down method is used to excitation the microcavity at 1550 nm band to obtain its resonant spectral line, as shown in Fig. 3(b). The free spectral range of the microcavity is $\mathrm{\Delta }{\lambda _{\textrm{FSR}}} = 0.1052\textrm{nm}$(∼13.13 GHz), and the quality factor Q value reaches $4.71 \times {10^8}$.
Fig. 3. Transmission spectra of disk cavity of $\textrm{Mg}{\textrm{F}_2}$ crystal. (a) Transmission lines detected under CW laser pumping; (b) Lorentz fitted of the transmission spectral line marked in black box in (a).
3. Testing and results
3.1 Microcavity surface roughness detection
After the microcavity is prepared, the surface profile and surface roughness information can be detected by using white light interferometer without damaging the surface shape of the sample.
The surface information obtained from the disk cavity of $\textrm{Mg}{\textrm{F}_2}$ crystal in the preparation process is shown in Fig. 2(a) after coarse and fine polishing, and the corresponding Q value is shown in Fig. 3. It can be seen from the obtained three-dimensional image that there is a certain curvature on the side of the micro-cavity. The surface after coarse polishing is also marked as shown in Fig. 4(a), and its root-mean-square surface roughness ${\sigma _1} = 12.46\textrm{nm}$; while the surface after fine polishing is very smooth, as shown in Fig. 4(b), its root-mean-square surface roughness ${\sigma _2} = 0.34\textrm{nm}$. Using cavity ring-down method in 1550 nm band, the Q factors of the two states can reach $2.9 \times {10^7}$ and $4.71 \times {10^8}$, respectively.
Fig. 4. White light interferometer profilometer detection and corresponding Q value. (a) Surface profile and surface roughness information of the microcavity after coarse polishing and the Q value in this state. (b) Surface profile and surface roughness information of the polished cavity and the Q value in this state.
The experimental data of correlation between the Q factor of WGMRs and surface roughness that we collected are shown in the red dot mark in Fig. 5, and then these experiments are compared with the two theoretical estimates of the ${Q_{ss}}$ factor of surface scattering available in the literature. For a single crystal material with a given WGMR size, material and radiation effects can be ignored and we are led to consider that $Q \to {Q_{ss}}$. The first theory is the model derived by Vernooy et al. in [29], which can be expressed as:
Fig. 5. The Q factor measured by $\textrm{Mg}{\textrm{F}_2}$ WGMR as a function of surface roughness $\sigma $ and the roughness-limited ${Q_{ss}}$ estimated from the theoretical equation. The red dot marks the experimental data, the blue dashed line is the theoretical formula (2) of the correlation length $B = 50$, and the black solid line is the theoretical formula (3) of the correlation length $B = 200$ of the white light interferometer profiler.
Except for the spatial correlation length B in the formula, other parameters are the same as in Eq. (2). Due to the point diffusion function of the profiler objective, the correlation length should be larger than that obtained by atomic force microscope (AFM) [31]. As shown in Fig. 5, the Q factor estimated by the two theoretical formulas is consistent with the experimental data. For large-size crystal microcavity, the smoother the surface, the higher the Q value of quality factor. According to Eq. (1), the threshold power ${P_{th}}$ of stimulated Brillouin scattering in microcavity is inversely proportional to the square of Q value. Therefore, the preparation of ultra-smooth surface resonators provides a material basis for realizing low threshold SBL.
3.2 Cascaded stimulated Brillouin scattering
The experimental setup of SBL is shown in Fig. 6. Toptica laser is used to output a wideband tunable continuous laser as the pump light, which is then input into an erbium-doped fiber amplifier through a polarization controller (PC) for synchronous modulation and amplification. After power amplification, the pump light passes through an optical circulator (OC) and is coupled to the high Q value $\textrm{Mg}{\textrm{F}_2}$ crystal disk cavity through the fiber taper. The generated forward signal passes through a variable optical attenuator (VOA) to reduce the optical power of the input detection instrument, and then through a 1 × 2 fiber coupler (FC) to beam the attenuated forward signal according to the spectral ratio of 50:50, one beam directly into the optical spectrum analyzer 2 (OSA2, AE8600), and the other beam is received by a photodetector (PD) and converted into an electrical signal input to an oscilloscope (OSC, Wavesurfer 104Xs-A) whose time scale is expressed in wavelength to detect the transmission spectrum. The backward signal is extracted by the same fiber taper and collected by a circulator, recorded by the optical spectrum analyzer 1 (OSA1, AE8600).
Fig. 6. Experimental apparatus for stimulated Brillouin scattering. TOPTICA, single-frequency tunable pump laser; PC, polarization controller; EDFA, erbium-doped fiber amplifier; OC, optical circulator; VOA, variable optical attenuator; FC, 50:50 fiber coupler; PD, photodetector; OSA, optical spectrum analyzer; OSC, oscilloscope.
By controlling the pump light power coupled into the microcavity, the SBL can produce both single-stage and cascaded phenomena. As shown in Fig. 7(a), when the pump wavelength is 1549.25 nm and the coupled pump power ${P_{th}} = 1.8\textrm{mW}$, the micro-cavity generates a single-order backward SBL at 1549.358 nm, and the relative pump wavelength offset is 0.108 nm(∼13.47 GHz), which matches the theoretical value of 12.4-13.8 GHz. The threshold value of the microcavity can be derived from Eq. (1), where $\eta = 1$, ${n_1} = 1.38$, ${V_{eff}} = 7.01 \times {10^6}\mathrm{\mu }{\textrm{m}^3}$, ${g_0} = 3.38 \times {10^{ - 10}}$, ${\lambda _p} = 1549.25\textrm{nm}$, ${\lambda _B} = 1549.358\textrm{nm}$, ${Q_p} = {Q_B} = 1 \times {10^8}$, the theoretical value of the microcavity is 16.2$\mathrm{\mu W}$, which is much smaller than the experimental Brillouin min value of 1.8 mW, which may be caused by more coupling loss and incomplete overlap of the Brillouin gain spectrum and the resonant spectrum of the microcavity. Gradually increase the pumping power. The single-order SBL power increases almost linearly. The threshold graph shown in Fig. 7(b) is obtained by using the single-order SBL power as a function fitting data set of coupled pump power. After increasing the coupling pump power to 8.5 mW, the power of the single-stage SBL began to transform to the higher-stage, and gradually produced cascade SBL signals. In this experiment, while the pump power is increased above 1.3W, a cascade SBL of up to 12 orders is finally observed near 1550.3 nm, and the laser interval of each order is about 0.11 nm(∼13.72 GHz), as shown in Fig. 7(c). The blue signal is a forward signal, and only a second-order SBL signal is significant, because cascade SBL requires that pump waves and all Stokes waves resonate in the cavity and the forward Brillouin scattering phenomenon is generally weak [32]. In addition, because of the high power of the forward pumping wave and only part of the pump power involved in the nonlinear process, the forward spectral line almost covers the signal of the odd-order SBL Rayleigh scattering back. Red is the backward SBL signal, and the intensity of the low-order Brillouin laser is stronger than that of the pump light scattered back by Rayleigh and the high-order Brillouin laser converted from the low-order. However, due to the overlap between the Brillouin gain spectrum and the resonance mode of the micro-cavity, there are also cases where the intensity of the backward third-order SBL is relatively weak [8,14]. A third-order anti-Stokes SBL also appeared in the backward signal. Figure 7(d)-(g) show a backward cascade SBL process from order 3 to 12 by increasing the coupled pump power near 1550.3 nm.
Fig. 7. Experimental results of stimulated Brillouin scattering in a microcavity. (a) Generation of first-order SBL. (b) The relationship between the output power of the first-order SBL experiment and the coupled pump power, with a threshold of 1.8 mW. (c) Second-order forward SBL and 12th-order backward cascade SBL excited near 1550.3 nm. (d)-(g) The process of increasing the coupled pump power to excite order 3 to 12 backward cascaded SBL.
3.3 Generation of Brillouin-Kerr optical frequency comb
Continue to increase the pump power to 1.5W, and adjust the pump wavelength in the long wavelength direction. When the pump wavelength is 1549.72 nm, because the pump light is only used to produce Brillouin laser, and only the mode family corresponding to the Brillouin mode meets the anomalous dispersion condition, the single-order SBL and the optical frequency comb generated by its four-wave mixing can be observed at the same time.
As shown in Fig. 8(a), the spacing of single-order SBL is about 0.11 nm (∼13.72 GHz), and the spacing of each optical frequency comb is equal to about 0.54 nm, which is equivalent to 5 integer multiples of FSR. In order to observe a dense optical frequency comb with a wider spectral range, the observation spectral range is widened to 250 nm, and then the pump wavelength is continued to be modulated in the direction of longer wavelengths while the pump power remains unchanged. When the pumping wavelength was 1549.75 nm, a large number of optical combs were observed in the spectral range of 1450nm-1700nm, as shown in Fig. 8(b). Then, when the pump wavelength was stepped by 0.3 nm in the direction of long wavelength, it was found that the optical frequency combs became denser due to cascaded four-wave mixing but then became sparse due to incomplete mode matching. The process is shown in Fig. 8(b)-(e), and Fig. 8(f) is the Brillouin-Kerr optical frequency combs amplification near the 1550 nm pump. Due to the light amplification of the Raman scattering effect, the power of the optical combs near 1630 nm are increased, as shown in Fig. 8(g). At a pump power of 1.5W, the precise cavity FSR can separate the OFC lines generated by Raman near the amplification region, and we can observe the evident pump-Laman-FWM interaction in Fig. 8(g) which is the magnified view of amplification region in Fig. 8(e). It is due to the FWM interaction between the OFC lines in the Raman wavelength region near 1630 nm and that of the pump. When the power satisfies the threshold power of both the OFC lines of Kerr and the OFC lines of Raman, and at the same time the excited resonant modes are both in the MI gain range and the Raman gain bandwidth, we are able to observe a number of other combinations in the vicinity of the pump-Raman-FWM combinations, which may be generated by the FWM interactions between the Kerr optical combs and the Raman optical combs [33]. It indicates that there is competition between various nonlinear effects in the microcavity during the experiment. At its densest, nearly 300 comb lines of the 250 nm spectral window can be observed.
Fig. 8. Backward first-order SBL triggers optical frequency combs. (a) The first order SBL near 1549.72 nm induces optical frequency combs in the spectral range of 9 nm, and the comb tooth spacing is 5 integer multiple FSR. (b)-(e) Modulation of pump light in the long wavelength direction near 1549.75 nm to obtain the optical frequency comb in the spectral range of 1450-1700nm from sparse to dense and then to sparse. (f) The magnification spectrum near the 1550 nm pump in (d). (g) The magnification spectrum near 1630 nm in (e).
4. Conclusions and outlook
In summary, we design a high precision machining device to fabricate a magnesium fluoride crystal disk cavity with a diameter of 4.98 mm and a super-smooth surface. The FSR is equivalent to the Brillouin frequency shift of the material, and the maximum Q value can reach $4.71 \times {10^8}$. A nonlinear experimental device for stimulated Brillouin scattering based on a microcavity is constructed. The experimental results show that a backward cascaded SBL of up to 12 orders is generated in the microcavity, and the cascaded SBL beat frequency can generate microwave signals of hundreds of GHz. In addition, we observe the generation and change of the optical frequency comb induced by the first order backward SBL with wavelength modulation in the spectral range of 1450-1700nm at 1.5W pump power, and observe nearly 300 comb lines in the spectral window of 250 nm. We hope that the high Q value magnesium fluoride crystal microcavity prepared by high precision machining can be used as an optical component in microwave photonics and nonlinear optics. In the future work, we will further explore the packaging model of the crystal microcavity coupled with the fiber cone and study the Kerr soliton optical frequency comb phenomenon stimulated by Kerr self-phase modulation in Brillouin laser.
Funding
National Natural Science Foundation of China (62101230); Natural Science Foundation of Jiangxi Province (20224BAB202006, 20232ACB212008, 20232BAB212016, 20232BCJ23096); Chinese Aeronautical Establishment (2018ZC56006); Jiangxi Provincial Department of Education Science and Technology Project (GJJ200915).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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