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Abstract
High-quality-factor optical microresonators have become an appealing object for numerous applications. However, the mid-infrared band experiences a lack of applicable materials for nonlinear photonics. Crystalline germanium demonstrates attractive material properties such as high nonlinear refractive index, large transparency window including the mid-IR band, particularly long wave multiphonon absorption limit. Nevertheless, the reported optical losses in germanium microresonators might not allow the potential of the Ge-based devices to be revealed. In this study, we report the fabrication of germanium microresonators with radii of 1.35 and 1.5 mm, exhibiting exceptional quality factors (Q-factors) exceeding 20 million, approaching the absorption-limited values at a wavelength of 2.68 µm. These Q-factors are a hundred times higher than previously reported, to the best of our knowledge. We measured the two-photon absorption coefficient combined with free-carrier absorption leveraging the high-Q of the resonators (obtained βTPA = (0.71 ± 0.12) · 10−8 m/W at 2.68 µm). This research underscores the potential of whispering gallery mode microresonators as valuable tools for measuring absorption coefficients at different wavelengths, providing a comprehensive analysis of various loss mechanisms. Furthermore, the exceptional Q-factors observed in germanium microresonators open intriguing opportunities for the advancement of germanium-based photonics within the mid-infrared spectral band.
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1. Introduction
Whispering gallery mode (WGM) optical microresonators have become indispensable tools for both fundamental and applied applications [1–3]. Their unique combination of small mode volumes and high-quality factors enables the development of compact, high-performance devices across various fields. Theses fields include the stabilization of lasers using self-injection locking [4–7], chemical and biological sensing [8,9], nonlinear optical devices [10–13], and spectroscopy [14–16].
In recent years, significant advancements have been made in the creation of high-Q WGM microresonators tailored for the mid-infrared spectral range (mid-IR), utilizing various crystalline (fluorides [17–19], silicon [20] and germanium [21]), and amorphous [22–24] materials. Among semiconductor materials used for WGM microresonators - silicon and germanium have similar properties. Silicon is the predominant semiconductor material and holds paramount significance in microelectronics and photonics [20,25,26]. Germanium offers distinct advantages, namely, it has a wider optical transparency window from 1.7 to 15 µm (for silicon from 1.2 to 8 µm), uniquely large refractive index $n$ = 4.06 at 2.5 µm (3.4 at 5 µm for silicon), and a tenfold higher nonlinear optical coefficient ($n_2$ = $6\cdot 10^{-17}$ m$^2$/W) [27–29]. Besides, germanium platform is CMOS compatible and thus can be used for effective, low-cost and high-performance IR integrated photonics [30–33]. This study focuses on the properties of semiconductor WGM microresonators made from crystalline germanium for the mid-IR band. Germanium-based WGM microresonators due to their optical properties offer highly promising capabilities for different nonlinear optical processes.
The theoretical investigation of the frequency combs generation in germanium shows the possibility of achieving it beyond 3.5 µm [34,35]. This prospect opens up new possibilities for germanium application in various fields, including quantum optics, communications, and spectroscopy. To date, crystalline [21], amorphous [36] and on-chip [31,37] microresonators made of germanium have been demonstrated. The highest reported quality factor in the crystalline Ge WGM microresonator was $2.5\cdot 10^5$ at 8 µm [21]. In the range of 2–3 µm, germanium suffers from a strong two-photon absorption (TPA) due to the small value of the band gap (0.67 eV) [27], which goes along with a free-carrier absorption (FCA) and a free-carrier dispersion (FCD). As far as the TPA and FCA have the same physical origin and contributions to the optical losses are difficult to separate, below we use the therm "TPA" for the sum of these two. These effects appear simultaneously and lead to additional optical losses in the range of 1.7–3 µm. It worth noting, that the reported TPA coefficients for Ge have discrepancies as in [38] and [39] the values differ by two orders of magnitude. That motivates us to consider the TPA coefficient experimental measurements.
Our study presents a comprehensive investigation of high-Q crystalline germanium WGM microresonator. The measured quality factor exceeds 20 million at 2.68 µm which is close to the material absorption limit according to the optical loss coefficient given by the manufacturer. This is the first demonstration of high-Q microresonators made of germanium with an improvement in quality factor of more than hundred times compared to demonstrated previously, to the best of our knowledge. The possibility of using various coupling elements made of a material with a high refractive index is shown using the examples of germanium prism and hemisphere, which can be effectively used to measure the quality factor, nonlinear absorption and as a prospective laser stabilization via self-injection locking at mid-IR. We consider WGM microresonators as a powerful and sensitive tool for measuring optical losses including nonlinear losses such as two-photon absorption. By measuring the intrinsic quality factor (with careful analysis of other contributions to losses, such as the radiation loss, nonlinear absorption, scattering, etc.), we can distinguish the nonlinear absorption coefficient. Measuring the broadening of the resonance curve width at different powers for the same microresonator mode, we define nonlinear absorption coefficient of Ge at 2.68 µm caused by TPA. The measured coefficient is $\beta _{\text {TPA}} = (0.71 \pm 0.12) \cdot 10^{-8}$ m/W at 2.68 µm, which matches the value presented in Ref. [27] within the error margin. This approach allows for a comprehensive assessment of material losses in WGM microresonators made of different materials. We also demonstrate the self-injection locking of a semiconductor laser to the germanium microresonator at 2.68 µm which can be used not only for laser frequency stabilization but also potentially for Kerr frequency combs generation [7,40,41]. It worth noting, that optical losses tend to decrease at longer wavelengths as Rayleigh scattering reduces and in this way the obtained quality factor make the crystalline germanium microresonator essential and powerful element for the photonic application and opens the way to improving germanium-based planar technology for mid-IR. The level of optical losses in germanium together with high nonlinear refractive index makes this material almost the best choice for frequency combs generation in mid-IR, specifically in a fingerprint region around 5 µm. In particular experimentally demonstrated Q-factors for wavelengths longer than 4 µm tend to decrease with record values of $1.4 \cdot 10^{8}$ in BaF$_2$ and $1.6 \cdot 10^{8}$ in SrF$_2$ [17,19]. The increase in optical losses in fluorides is related to multiphonon absorption at long wavelengths, which is not the issue for germanium in this band [27]. The high nonlinearity and Q-factor highlight the germanium as one of the most promising materials for microresonators in the mid-IR band along with silicon and As$_2$S$_3$. The investigation considering frequency comb generation in crystalline germanium resonators and comparison with some other materials may become an actual topic for future research.
2. Q-factor contribution
The total quality factor of a WGM resonator is defined by different loss mechanisms and can be estimated as
The quality factor $Q$ as a function of losses can be expressed as following [42]:
where $n$ denotes the refractive index, $\alpha$ represents the loss per unit length, and $\lambda$ is wavelength. Taking the material parameters for the germanium used in our research ($\alpha _{mat}$ = 0.005 cm$^{-1}$ at 2.7 µm according to manufacturer specifications) we estimated $Q_{mat}$ as $\sim 1.8\cdot 10^7$ for the low in-coupled power.Germanium microresonators are susceptible to nonlinear losses namely multiphoton absorption and free-carrier absorption since the material is a semiconductor. We may divide losses contribution in following way:
where $\alpha _{\text {int}}$ is the internal loss in the microresonator including bulk absorption and scattering in the resonator volume and surface, $\alpha _{\text {coupl}}$ is the coupling losses, $\alpha _{\text {TPA}}$ is the loss coefficient due to TPA. The two-photon absorption can be observed with the increase of the input power: $\alpha _{\text {TPA}} = \beta _{\text {TPA}} \cdot I$, where $\beta _{\text {TPA}}$ is the nonlinear TPA coefficient, $I$ is the electromagnetic field intensity, which can be expressed as $I = \frac {P_{\text {in}} 2\pi n a}{V_{\text {eff}}}$, where $P_{\text {in}}$ is the pump power in-coupled in the microresonator, $V_{\text {eff}} \simeq 15.12a^2\sqrt {ab}m^{-7/6}$ is the effective mode volume [42,45], $m \approx 2\pi an / \lambda$ is the azimuthal index of the mode, $a$ and $b$ are the semiaxes of the spheroid. For the wavelength above 4 µm, instead of TPA, the three-photon absorption appears. It is replaced by the four-photon absorption above 6 µm [39].In experiment, we measure the full width at half maximum (FWHM) $\omega _{\text {FWHM}}$ of the resonance curve. Using Eq. (2) and (3) one may obtain:
3. Fabrication
Germanium disk microresonators are fabricated on a lathe machine (DAC ALM Lathe) using the single-point diamond turning method. This method starts with blank material produced by Crystran Ltd, utilizing the Czochralski technique [44]. The fabrication process for the germanium microresonator unfolds as follows: the germanium blank workpiece is first cut and then glued onto a holder, which is subsequently placed on the rotational spindle of the lathe machine. The turning process is automated, and the program allows for setting almost any desired shape, size, and side profile of the microresonator. Using this method, we fabricated two microresonators with parameters a = 1.35 and 1.5 mm, b = 1 and 1.5 mm, respectively. The photos in Fig. 1(a) displays the SEM images of a germanium microresonator after fabrication.
Fig. 1. a. SEM images of the germanium microresonator after the turning process; b. Schematic illustration of the experimental setup for measuring the Q-factor with different coupling elements and microresonators; c. Germanium prism used as a coupling element; d. Germanium hemisphere used as a coupler.
The polishing process is designed to reduce the surface roughness of the microresonator and eliminate surface scattering. For this purpose, we utilized diamond slurries with various grain sizes and a homemade rotary machine. The grain sizes of the slurries were progressively decreased from 4 µm to 30 nm. Between each change in grain size, the surface was thoroughly wiped with methanol, and frequent surface inspections were conducted under a microscope after each adjustment in the slurry’s grain size. This meticulous polishing process enabled us to significantly reduce surface roughness and achieve Q-factors exceeding one billion in fluorides [46], thereby rendering surface scattering a minor contributor to total optical losses.
4. Experimental setup of Q-measurements
The experimental setup for Q-factor measurements for germanium microresonators is presented in Fig. 1(b). A tunable 2.68 µm continuous wave DFB laser without isolator (Nanoplus) is used to excite WGM. For diode laser changing the operating current modifies the output power and generation frequency. To obtain the same output frequency at different pump powers the temperature of the laser is adjusted. The light is coupled into the microresonator through the coupling element. The transmitted light is collected at the photodetector (PD27 by IoffeLed). The laser frequency is calibrated using a Fabry-Pérot germanium interferometer with a free spectral range (FSR) of 725 MHz. Three-coordinate translation stage with a piezoelectric controller (PZT) is used to control the alignment and precise positioning of the microresonator. The excitation of the WGMs is provided with two types of coupling elements: prism and hemisphere (Fig. 1(c),(d), and in order to satisfy the phase-match condition, we chose a coupler made of the same material as the microresonator [20,42,47]. Crystalline microresonators are overmoded and various WGMs families can be excited.
The calculation of the quality factor in our research was provided by measuring the FWHM of the resonance curve. By varying the distance between the resonator and the coupling element, the loading of the microresonator is changed. The maximum transmission dip of the resonant curve corresponds to the critical coupling regime, when the intrinsic and external coupling losses of the microresonator are equal. Thus, to obtain information about the intrinsic quality factor, it is necessary to provide measurements in the critical coupling regime [47]. We determine critical coupling in experiment by measuring several transmission traces with close to maximum transmission dip. Then, the case with maximum mode contrast which decreases with both receding and approaching of the microresonator to the coupler is considered as critically coupled.
4.1 Prism as a coupling element
The transmission spectra for different coupling rates are measured for each of the microresonators. With a germanium prism as a coupling element, the maximum obtained in-coupled power is several percent. A large contrast in the refractive indices between air and germanium prism, and large multiple reflections from rectangular walls of the prism decrease pump power throughout an optical path. The in-coupled power in this case was up to 0.1 mW. Low in-coupled power provides perfect conditions for observations of the WGMs in linear regime, avoiding nonlinear effects and self-injection locking of the laser to a microresonator.
The resonance curves, showcasing the narrowest obtained FWHM for TE and TM laser polarizations in the critical coupling regime for the 1.5 mm radius microresonator, are shown in Fig. 2(a-d) respectively. FWHM of the resonant curves for up and down frequency scanning are equal which indicates the absence of nonlinearity and thus resonance curves can be approximated with the Lorentzian fit (the solid lines in Fig. 2). The loaded quality factor was measured as $Q$ = $(1.1\pm 0.1)\cdot 10^7$ for TE and $Q$ = $(1.2\pm 0.1)\cdot 10^7$ for TM polarization which leads to the intrinsic $Q_0$ = $(2.2\pm 0.2)\cdot 10^7$ for TE and $Q_0$ = $(2.4\pm 0.3)\cdot 10^7$ for TM polarization accordingly and thus we reached the limit of Q defined by material absorption value provided by manufacturer.
Fig. 2. Transmission spectra of the germanium microresonator in critically coupling regime for forward (a,c) and backward (b,d) frequency scans for TE and TM laser polarizations with germanium prism as a coupling element. The blue solid lines correspond to the Lorentzian approximation.
4.2 Hemisphere as a coupling element
Using a hemisphere as a coupling element (Fig. 1(d)) allows us to increase the in-coupled power ratio that provides possibility to study the impact of nonlinear losses on the microresonator quality factor. The geometric shape of the hemisphere directs incident rays to its center without refraction, thus simplifying the alignment process to some extent. The mode matching between the microresonator and pump is very dependent on the alignment relative to the center of the hemisphere. By precise shift around the center of the hemisphere, we managed to increase it up to $\sim$ 40${\% }$. The in-coupled power with the germanium hemisphere coupler increased up to 1.4 mW, and the reflected wave excited by the Rayleigh scattering going to the laser also was increased. This led to the self-injection locking (SIL) of a generation frequency to a microresonator mode [7,48].
This effect can be effectively used for suppression laser phase noise and narrowing laser line, as reported in Refs. [7,48]. Additionally, it holds potential for enabling frequency comb generation in germanium [34,40,41] for wavelength longer than 3.5 µm. A characteristic feature of the SIL regime is the dependence of the resonance curve profile on the phase of the backscattered wave (locking phase) [49]. Reference [6] describes a technique of microresonator quality factor determination in the SIL regime by measuring the widths of resonance curves for forward and backward frequency scans for different gaps between the microresonator and the coupling element. In this case, measurements should be carried out for the same value of the locking phase. By adjusting the distance between the laser and microresonator using a PZT we can alter the phase of the backscattering wave to obtain different shapes of the resonance curve, as depicted in Fig. 3 and indicated by different colors at constant laser parameters. The optimal phase (phi = 0) corresponds to the green line. In our experiment, the locking phase changed rapidly, resulting in the resonance curve shape in the SIL regime quickly transitioning from curve shape I to III in Fig. 3 under constant laser parameters. The lack of phase stability may be attributed to thermal effects, which are nearly unavoidable in real-life high-quality-factor microresonator platforms and can lead to frequency drift [28]. Significant disturbance at the edges of the transmission curves in Fig. 3 are presumably related to thermal responsivity of germanium.
Fig. 3. Transmission spectrum for different phases (I-III) of the backward wave for forward frequency scan.
We observed thermal nonlinearity at high in-coupled power ($\sim$1.4 mW). It results in a broadening of the resonance curve during the forward frequency scanning and a narrowing during the backward frequency scanning which cannot be equalized by backscattering phase adjusting. The width of the resonances in Fig. 4 are 47 and 27 MHz for the forward and backward frequency scans consequently. Pronounced thermo-optical oscillations are observed [50].
Fig. 4. Transmission spectrum demonstrating the influence of thermal nonlinearity along with thermal fluctuations in the transmission spectrum during forward and backward frequency scan. The width of the resonance curve is 47 MHz for forward frequency scan and 27 MHz for backward frequency scan.
To measure the quality factor, we carefully select the resonances (to avoid SIL and the influence of thermal nonlinearity) by the form of the curve and the equality of FWHM values. Since highest measured Q-factors (Fig. 2) are almost the same for both polarizations the following Q-factor measurements were carried out without changing polarization - only for TE modes. Transmission spectra with resonances matching all the criteria are found and fitted by the Lorentzian approximation shown for up and down frequency scanning in Fig. 5. The obtained transmission spectrum give the same value of the FWHM and intrinsic quality factor of $Q_0$ = $(2.2\pm 0.2)\cdot 10^7$ within the error margin that was obtained previously with the prism coupler, but with higher value of in-coupled power. The obtained values of Q-factor is hundred times higher than previously reported [21] to the best of our knowledge. For the microresonator with radius 1.35 mm, we obtained the same values of the quality factor. Thus, the use of a hemisphere has an advantage over a prism, in ease of alignment and in higher value of in-coupled power which allow to observe different nonlinear effects in germanium microresonator.
Fig. 5. Transmission spectra of the germanium microresonator in critical coupling regime for forward (a) and backward (b) frequency scans with germanium hemisphere as a coupling element. The blue curve corresponds to the Lorentzian approximation.
4.3 Optical nonlinear absorption
To reveal directly the effect of TPA, one needs to measure the dependence of FWHM on in-coupled power. We excite the same WGM for different pump powers. Since the maximum optical power of the used laser diode is limited (up to 10 mW) we use full range of operating currents from 80 mA to 190 mA to obtain maximum power difference. On the other hand, diode laser frequency depends on operating current. We gradually change the diode laser temperature along changing operating current to keep the same operating frequency. Moreover, we have to suppress backscattering wave to avoid SIL. The germanium optical window is added between collimating and focusing lenses, see Fig. 1(a), at the angle of 45 °C to the laser beam. So that, we obtain the same WGM at 80 mA, 28 °C, 35 µW in-coupled power and 190 mA, 22 °C, 138 µW in-coupled power with Lorentzian profile resonance unperturbed by SIL.
We measure the quality factor at critically coupling regime. FWHM is $10.1 \pm 2.6$ MHz which corresponded to intrinsic quality factor of $(2.2 \pm 0.4)\cdot 10^7$ at lower pump power. FWHM at higher pump power is wider - $19.5 \pm 2.2$ MHz, which corresponded to intrinsic quality factor of $(1.2 \pm 0.1)\cdot 10^7$. This technique made it possible to measure and compare resonance width for the one observing mode with a power difference of $\sim$ 4 times - from $\sim$ 35 to 138 µW in-coupled optical power. At maximum power FWHM gets wider comparing to the FWHM at minimum power and thus the quality factor decreases. Figure 6 represents transmission spectra for minimum power (gray dots) and maximum power (red dots). The resonance curves are approximated with the Lorentzian profile (the gray and red solid lines).
Fig. 6. Transmission spectrum of the germanium microresonator in critically coupled regime for minimum (gray line) and maximum (red line) power for one mode. Dashed curves correspond to the experimental data, solid curves correspond to the Lorentzian approximation
Based on the experimental results, we obtained the TPA coefficient using Eq. (5) as $\beta _{\text {TPA}} = (0.71 \pm 0.12) \cdot 10^{-8}$ m/W. This result is close to the experimental results of [38] within the error margin and slightly higher than in theoretical model [27] based on different experimentally obtained results for several spectral bands. However it is two orders of magnitude higher than in [39] that might be connected with experimental method used in that research which minimizes the contribution of FCA.
5. Conclusion
In summary, we successfully fabricate germanium microresonators with 1.35 and 1.5 mm radii. These microresonators exhibit a high-quality factor for WGMs which leads to the intrinsic $Q_0$ = $(2.2\pm 0.2)\cdot 10^7$ for TE polarization and $Q_0$ = $(2.4\pm 0.3)\cdot 10^7$ for TM polarization, which close to the absorption-limited Q-factor at a wavelength of 2.68 µm.
We reveal the impact of TPA directly by broadening of the resonance curve at different in-coupled power. The obtained coefficient for TPA and associated optical losses is $\beta _{\text {TPA}} = (0.71 \pm 0.12) \cdot 10^{-8}$ m/W that agrees with the results obtained in [38]. Additionally, we have highlighted potential applications of these fabricated germanium microresonators in laser diode frequency stabilization and in the linewidth narrowing of lasers operating at 2.68 $\mu$m.
Our study validates the utility of WGM microresonators as beneficial tools for measuring absorption. The observed high Q-factor in the germanium microresonators demonstrates the potential of germanium-based infrared photonics.
Funding
Russian Science Foundation (20-12-00344).
Acknowledgments
We thank O. Benderov for assistance with equipment and fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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