Abstract

The development of ultrahigh-quality-factor ($\textit{Q}$) microresonators has been driving such technologies as cavity quantum electrodynamics (QED), high-precision sensing, optomechanics, and optical frequency comb generation. Here we report ultrahigh $Q$ crystalline microresonator fabrication with a $\textit{Q}$ exceeding ${10^8}$, for the first time to our knowledge, achieved solely by computer-controlled ultraprecision machining. Our machining fabrication method readily achieves the dispersion engineering and size control of manufactured devices via programmed machine motion, both of which were not possible with the conventional manual polishing method. We can achieve an ultrahigh ${{Q}}$ without the need for subsequent careful polishing that is generally required to ensure that surface integrity is maintained. We carefully addressed the cutting condition and crystal anisotropy to overcome the large surface roughness that has thus far been the primary cause of the low $Q$ in the machining process. Our result paves the way for a reliable fabrication with a view to various photonic applications utilizing ultrahigh-$\textit{Q}$ crystalline microresonators.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Ultrahigh-$\textit{Q}$ crystalline microresonators have been used as attractive platforms for studying nonlinear and quantum optics in the last few decades [15]. In particular, laser stabilization via self-injection locking and Kerr optical frequency comb generation are potential applications with the aim of realizing an optical-frequency synthesizer [6] and low-noise, compact photonic devices [7,8]. Injection locking to high-$\textit{Q}$ whispering-gallery mode (WGM) microresonators enables the laser linewidth to be reduced to less than hundreds of hertz [9]. Moreover, Kerr frequency comb generation [10] provides RF oscillators with high spectral purity [11,12] and an optical pulse train with a high repetition rate [13,14]. These applications rely on the high ${{Q}}$ of crystalline microresonators, typically up to ${10^9}$ and corresponding to a resonance linewidth of hundreds of kilohertz, which enhances the optical nonlinearity. The fundamental limit of the ${{Q}}$ factor in crystalline resonators is ${\sim}{10^{13}}$ [15] ($\!Q\; {\gt} {10^{11}}$ as observed in the experiment [16]), and this value surpasses that of resonators made with other materials (e.g., silica, silicon, etc.) [17]. In addition, they have a fully transparent window in the visible to mid-infrared wavelength region, which expands the available bandwidth as well as the telecom band [18].

Magnesium fluoride (${\rm{Mg}}{{\rm{F}}_{{2}}}$) and calcium fluoride (${\rm{Ca}}{{\rm{F}}_{{2}}}$) are crystalline materials that are commonly used for fabricating WGM microresonators thanks to their quality, commercial availability, and optical properties. We usually manufacture crystalline resonators by using diamond turning and a polishing process. They are accomplished either with a motion-controlled machine or manually [19]. A hard diamond tool enables us to fabricate WGM structures, but we have to employ subsequent manual polishing with diamond slurry to improve the ${{Q}}$ factor of the microresonator. Precision machining readily overcomes the geometrical limitation of the manual process; therefore, precise computer-controlled machining has achieved the predesigned mode structures needed when fabricating single-mode [19,20] and dispersion-engineered resonators to generate broadband microresonator frequency combs [2123]. However, a significant challenge remains because we need to employ additional hand polishing after the diamond turning process due to the relatively low ${{Q}}$ of ${10^6} \sim {10^7}$ at best that we obtain when using machining alone [19,21,22,24]. This value is usually not sufficient to generate the Kerr frequency comb, in which the threshold power is in proportion to ${\sim}V/\!{Q^2}$ [25]. The additional polishing improves the ${{Q}}$; however, subsequent polishing deforms the precisely fabricated structures despite the engineered dispersion realized by the programmed motion of the lathe [19].

In this paper, we describe an ultrahigh-$\textit{Q}$ crystalline microresonator fabrication technique that employs computer-controlled ultraprecision machining. The measured ${{Q}}$ of the ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystalline resonator reaches $1.4 \times {10^8}$, which is the highest value yet obtained without a subsequent polishing process. In addition, we achieved a comparably high ${{Q}}$ in ${\rm{Ca}}{{\rm{F}}_{{2}}}$ crystalline material. To obtain the ultrahigh ${{Q}}$, we addressed the single-crystal cutting condition by undertaking an orthogonal cutting experiment, which revealed the critical depth of cut for different cutting directions. Also, a precise cylindrical turning experiment revealed the relationship between crystal anisotropy and surface quality after machining and demonstrated the realization of nanometer-scale surface roughness with diamond turning alone. The results we obtained provide clear evidence that cutting parameters that have been optimized for fluoride crystals lead to a significant reduction in surface roughness.

An automated ultraprecision machining technique is compatible with dispersion engineering and a high ${{Q}}$ factor, which is often restricted by the trade-off relation encountered with conventional fabrication techniques. We confirmed that the dispersion of fabricated resonators agrees extremely well with the design, and this extends the potential of dispersion controllability in crystalline WGM microresonators. Furthermore, our approach, namely, the reliable production of high-$\textit{Q}$ crystalline microresonators, supports recent advances on the integration of crystalline microresonators with photonic waveguides towards a wide range of future applications [2629].

2. INVESTIGATION OF CRITICAL DEPTH OF CUT

A. Definition of Critical Depth of Cut

One important parameter that we need to know when fabricating a single crystalline resonator is the critical depth of cut. It is defined by the depth of cut at which the transition from ductile-mode to brittle-mode cutting is observed when machining single-crystal material [30]. In the ductile regime, a smooth crack-free surface can be maintained when generating a continuous ribbon chip, and this approach is considered more suitable for optical applications thanks to its ultrasmooth surface. On the other hand, the surface in the brittle regime is rougher and contains cracks, and hence it is generally inadequate for optical applications. ${\rm{Ca}}{{\rm{F}}_{{2}}}$ and ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystals are hard and brittle materials, and they have a crystal anisotropy, so they are challenging to cut. These features make it difficult to manufacture smooth optical elements with a designed shape such as spherical lenses and optical microresonators. In particular, high-$\textit{Q}$ microresonators require an ultrasmooth surface with a surface roughness of no more than a few nanometers. Thus, the critical depth of cut must be investigated before resonator fabrication if we are to cut the crystal in the ductile mode regime.

The cutting direction and plane of the crystal is expressed with the Miller index as shown in Fig. 1(a) (details of the Miller index and crystallographic structure can be found in Supplement 1). Figure 1(b) shows crystallographic images of ${\rm{Ca}}{{\rm{F}}_{{2}}}$ and ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystal. The difference in crystal structure influences the critical depth of cut and the cutting conditions.

 

Fig. 1. (a) Examples of the Miller index, where ($h k l$) and [$h k l$] indicate the corresponding plane and direction, respectively. (b) Crystallographic images of ${\rm{Ca}}{{\rm{F}}_{{2}}}$ and ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystal. The red arrows indicate examples of slip system. In contrast to the cubic symmetry system of ${\rm{Ca}}{{\rm{F}}_{{2}}}$, ${\rm{Mg}}{{\rm{F}}_{{2}}}$ is characterized by a more complex rutile structure.

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B. Orthogonal Cutting Experiment

To investigate the critical depth of cut, we performed an orthogonal cutting experiment on single-crystal ${\rm{Mg}}{{\rm{F}}_{{2}}}$. The experiment was carried out with an ultraprecision machining center (UVC-450C, TOSHIBA MACHINE), and a workpiece holder equipped with a dynamometer to detect the cutting force during the processing. As a workpiece, we used a prepolished single-crystal ${\rm{Mg}}{{\rm{F}}_{{2}}}$ substrate with a size of $38 \times 13\,\,{\rm{mm}}$ and a thickness of 1 mm, which was fixed on the workpiece holder with a vacuum chuck as shown in Fig. 2(a). Here, we tested two different crystal planes, (001) and (010), where we performed the cutting in every 30° rotational direction to investigate the critical depth of cut for different crystal orientations as shown in the left panel of Fig. 2(b). The direction of 0° was set at [$\bar1 00$] and [$00\bar 1$], respectively. The cutting tool was a single crystal diamond tool with a 0.2 mm nose radius, a ${-}{20^ \circ}$ rake angle, and a 10° clearance angle (details of the single-crystal diamond tool are provided in Supplement 1). The cutting slope $D/L$, which gives the cutting depth to cutting length ratio, and the feed rate, were set at 1/500 and 20 mm/min, respectively, with a numerical control (NC) program [the right panel of Fig. 2(b)].

 

Fig. 2. (a) Experimental setup for orthogonal cutting to investigate the critical depth of cut in ${\rm{Mg}}{{\rm{F}}_{{2}}}$ single crystal. (b) Schematics of a single-crystal substrate and the cutting direction in the orthogonal cutting experiment. The cutting was performed in every 30° rotation direction (left panel), and the diamond tool cut a ${{V}}$-shaped groove with a slope of $D/L$ (right panel). (c) Reconstructed image of machined surface using a scanning white-light interferometer. The critical depth of cut is given by the depth where the first brittle fracture appeared [black point in (c)]. (d) Cutting along the slip system (110) [001] and the inversion symmetry (110) [$00\bar 1$] promotes ductile mode cutting (left panel). Cleavage and subsequent brittle fractures are induced by the cutting force against cleavage plane (right panel). (e) Scanning electron micrographs showing machined surfaces of a (010) plane with [$00\bar 1$] direction (upper panel) and [$\bar 1 00$] direction (lower panel). The yellow shaded area and gray area correspond to the original uncut surfaces and machined surfaces, respectively. The difference between the machined surface conditions is attributed to the crystal anisotropy of the ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystal. (f), (g) The measured critical depth of cut versus cutting direction on a (001) and (010) plane, respectively. In comparison with the (001) plane, the (010) plane shows large variation in cutting direction due to crystal anisotropy. The results of 0° (green dot) and 270° (orange dot) correspond to the upper panel and lower panel in (e), respectively. These results confirmed that the measured critical depth of cut are consistent with images of the machined surface. To perform ductile mode cutting, the depth of cut must be kept below the critical depth of cut.

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The critical depth of cut, which is defined as the cutting depth at which the first brittle fracture appeared on the surface, was measured using a scanning white-light interferometer (New View TM6200, Zygo). Figure 2(c) shows an image of the machined surface, where the black points indicate fractures or cracks on the surface. It should be noted that ${\rm{Mg}}{{\rm{F}}_{{2}}}$ has a complex rutile structure with a different crystal plane configuration from ${\rm{Ca}}{{\rm{F}}_{{2}}}$ [see Fig. 1(b)]; hence, the two orthogonal planes are selected for the test to reveal the effect of crystal anisotropy. Figure 2(d) shows a schematic of the tensile stress model for single-crystal cutting, and this will be explained in more detail later.

Figure 2(e) presents scanning electron microscope (SEM) images showing the surface condition of the (010) plane after orthogonal machining (the yellow region is the original uncut surface). Even though the only difference is the cutting direction (i.e., [$00\bar 1$] and [$\bar 1 00$]), there is a significant impact on the surface quality of the machined region due to the crystal anisotropy. We observed large brittle fractures in the [$\bar 1 00$] direction, whereas overall the [$00\bar 1$] direction exhibited smooth surfaces. Figures 2(f) and 2(g) show the variation in the critical depth of cut as a function of cutting direction on each plane. On the (001) plane, the critical depth variation was approximately 120 nm, and the lower bound value was 86 nm in the 270° direction ([$0\bar 1 0$] direction). On the other hand, the variation with the (010) plane was more significant than that with the (001) plane, and the lower bound also decreased (i.e., worsened). These considerable differences in critical depth of cut are consistent with surface observations as shown in Fig. 2(e).

 

Fig. 3. Ultraprecise cylindrical turning and surface roughness measurement of ${\rm{Mg}}{{\rm{F}}_{{2}}}$ single crystal. (a) Experimental setup of an ultraprecision lathe for the cylindrical turning of a single crystal. (b) Micrograph showing a machined surface, where clear boundaries are observed between the rough turning and finish turning regions. The horizontal boundary in the rough turning region is evidence of the dependence of the cutting direction on the crystal anisotropy in ${\rm{Mg}}{{\rm{F}}_{{2}}}$ single crystal. (c), (d) Magnified views of the finish turning region in 135° and 180°, respectively. The machined surface of 135° is smoother than that of 180°, which agrees with the result of the surface roughness (RMS) measurement. (e) Schematic of surface roughness measurement. The yellow line and dot correspond to orientation flat [100] with an endface orientation of (001). The surface roughness at a total of 24 points was measured at 15° intervals. (f) Measured surface roughness (RMS) of the finish turning (red dots) and rough turning (blue dots) regions. A quarter symmetry is clearly observed in the finish turning condition due to crystal anisotropy. (g) Magnified plot on linear scale of finish turning in (f).

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We can understand the experimental results as follows. The difference in critical depth of cut could be considered to originate from the slip system and the cleavage plane since they are strongly related to the ductile-brittle mode transition. Cutting along the slip plane promotes ductile-mode cutting (i.e., plastic deformation), which contributes to the large critical depth of cut. On the other hand, the cutting force against cleavage induces crystal parting where brittle fractures are easily manifested. They are explained intuitively in Fig. 2(d).

The slip system and cleavage plane of single-crystal ${\rm{Mg}}{{\rm{F}}_{{2}}}$ are (110) [001] and (110), respectively (see also Fig. 1); therefore the influence of cutting on the (001) plane on the critical depth of cut is less susceptible to the cutting direction because the cutting on the (001) plane is always normal to both the slip system and the cleavage plane. With the (010) plane, however, we observed a large variation in the cutting direction, because the cutting periodically followed the same direction as the slip system (i.e., 0° and 180°) as shown in Fig. 2(g). In contrast, the smallest (i.e., worst) critical depth of cut was obtained for directions of 90° and 270°, corresponding to [100] and [$\overline 1 00$], respectively. They are in a configuration where the cutting force is applied in a direction almost perpendicular to the cleavage plane (110) and its inversion symmetry ($\overline 1 10$) as explained in Fig. 2(d) (right panel). As a result, a shallow depth of cut is needed to obtain a ductile mode for these directions. From the result of the orthogonal cutting experiment, we concluded that the depth of cut must be less than approximately 50 nm to maintain ductile mode cutting.

3. ULTRAPRECISION CYLINDRICAL TURNING

A. Procedure of Cylindrical Turning

Although the orthogonal cutting experiment provides information on the critical depth of cut for specific directions, the cutting direction continuously changes when cylindrical turning is performed to manufacture a crystalline cylinder workpiece. Therefore, the optimum turning parameters have to be investigated to achieve the smooth surface needed for a high-$\textit{Q}$ microresonator. An ${\rm{Mg}}{{\rm{F}}_{{2}}}$ cylinder workpiece was prepared with an end-face orientation of (001) because a $z$-cut ($c$-cut) resonator is used to avoid optical birefringence.

Cylindrical turning was performed using an ultraprecision aspheric surface machine (ULG-100 E, TOSHIBA MACHINE) as shown in Fig. 3(a). An ${\rm{Mg}}{{\rm{F}}_{{2}}}$ workpiece with a diameter of 6 mm was fixed to a brass jig and then mounted on a vacuum chuck. The ultraprecision turning was conducted in the following three steps. Rough turning was initially undertaken to form the desired diameter (here 3 mm). It should be noted that this initial rough turning was performed in the brittle regime. Next, pre-finish cutting was conducted to remove the large cracks that occurred in brittle mode cutting with a removal thickness of 8 µm. Finally, finish cutting in the ductile mode completed the ultraprecise turning under the following cutting conditions: $500\,\,{\rm{mi}}{{\rm{n}}^{- 1}}$ rotation speed, 0.1 mm/min feed rate, 50 nm depth of cut, and 2 µm removal thickness (the cutting conditions in each step are detailed in Supplement 1). We can see that cracks deeper than 10 µm that appeared during the rough turning could not be removed with pre-finish and finish cutting. Although the larger removal thickness results in lower production efficiency, it allows the removal of deep unwanted cracks.

The depth of cut at the finish turning step was set at 50 nm based on the result of the orthogonal cutting experiment. To achieve a smooth surface, other factors, such as the rotation speed, feed rate, and diamond tool, should be taken into account because these choices determine the effective cutting speed and cutting amount. In particular, previous studies have reported that the feed rate critically affects the quality of the machined surface as does the combination of the tool radius and depth of cut [3133]. These studies draw attention to the fact that a fast feed rate induces brittle mode cutting if the depth of cut is kept below the critical value. Thus, we chose a slow feed rate ($\le\! 1\,\,{\rm{mm/min}}$) when fabricating a smooth surface.

For the pre-finish and finish cutting, we used a single crystal diamond cutting tool with a 0.01 mm nose radius, a 0° rake angle, and a 10° clearance angle. In terms of the choice of the cutting tool, a smaller nose radius makes it possible to have a smaller contact area between tool and material during cylindrical turning, which helps to reduce any excess cutting force and leads to an improved surface quality. However, tools with a small nose radius are more fragile, which gives them a short lifespan; hence, in this work we use different tools for the rough turning and finish turning stages.

The machined surfaces were observed using an optical microscope (VHX-5000, Keyence) as shown in Figs. 3(b)3(d). Clear boundaries can be identified in the micrograph images between the rough turning and finish turning regions.

B. Surface Roughness After Cylindrical Turning

The surface roughness after cylindrical turning was measured using a scanning white-light interferometer (New View TM6200, Zygo) at 15° intervals from the orientation flat [100] defined as 0° [Fig. 3(e)]. Figures 3(f) and 3(g) show the cylindrical surface roughness with an end-face orientation of (001) of an ${\rm{Mg}}{{\rm{F}}_{{2}}}$ workpiece. Unsurprisingly, the surface roughness after the rough cutting exceeded 200 nm for the entire cylindrical surface as shown with red dots in Fig. 3(f). The large roughness was caused by the brittle-regime cutting. In contrast, the smoothness improved significantly after the finish cutting, which was performed under the ductile cutting condition [blue dots in Fig. 3(f)]. The magnified plot on a linear scale is shown in Fig. 3(g). We confirmed that the turning condition for final cutting enabled us to achieve a smooth surface. Specifically, we obtained an RMS roughness of below 2 nm at 18 observation points. The result also revealed an interesting feature of 90° periodicity, namely, that specific observation points exhibited a slightly larger RMS roughness of 7.8 nm on average. Periodicity can also be seen in the micrograph shown in Figs. 3(b)3(d); for instance, 135° exhibits a smoother machined surface than that in the 180° direction [Figs. 3(c) and 3(d)]. This is evidence of the appearance of crystal anisotropy in ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystal as observed in the orthogonal cutting experiment.

This periodicity can also be understood from the slip system and cleavage configurations shown in Fig. 2(d). The relatively rough surfaces can be explained in terms of specific directions where the excess cutting force acts on the boundaries of cleavage planes. The 15° asymmetry is due to the rotation direction of the workpiece; the force on the cleavage plane exerts stress only in the clockwise direction (the opposite direction to workpiece rotation). There is nevertheless excellent surface integrity as long as the cutting remains on a cylindrical surface where the cutting force circumvents the crystal anisotropy. (Detailed discussions of the effect of crystal structure and the applicability to other crystal materials are presented in Supplement 1).

The cylindrical turning described here is a preprocess in microresonator fabrication. It should be noted that the measured roughness is the result of the implosion of a machined cylinder, not the dimensional resonator surface. However, the results we obtained allow us to predict the surface of the resonator under the employed cutting conditions.

C. Fabrication and Cleaning of Microresonators

Crystalline microresonators are fabricated using the same ultraprecision machine as that used in the cylindrical turning experiment (ULG-100 E, TOSHIBA MACHINE). The resonator diameter is determined after the finish cutting. Also, the resonator shape is carefully fabricated by feeding a diamond tool under the critical cutting depth. Here, the turning motion is fully and automatically controlled by the NC program. The manufacturing procedure is shown in Fig. 4, and the total fabrication time is about ten hours. We determined the cutting condition based on the cylindrical turning experiment and employed a finish turning condition at the resonator shaping step (also see Supplement 1).

 

Fig. 4. Fabrication flow of WGM microresonator when using ultraprecision turning. First, a rough turning determines the approximate diameter of resonator. Next, pre-finish and finish turning with ductile mode cutting are used to realize a cylindrical surface that is smooth and entirely crack free. Finally, fully programmed shaping steps are performed to fabricate the designed resonator structure.

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Once we had completed the fabrication, we cleaned the microresonator to remove lubricant and small chips attached to the surface. We emphasize that proper cleaning is essential for obtaining a high ${{Q}}$ as well as optimized cutting conditions. Since we used water-soluble oil as a machining lubricant during the ultraprecision turning, we first used acetone solution to clean the microresonator surface and remove the remaining lubricant. A lens cleaning tissue is usually used to wipe the resonator, but there is the possibility that it might scratch or damage the resonator surface, which could be a critical problem in terms of degrading the ${{Q}}$ factor. Alternatively, to avoid unwanted damage on the resonator, we can employ ultrasonic cleaning. (The cleaning method is detailed in Supplement 1). The use of an ultrasonic cleaner enables us to clean the surface without touching or rubbing it. It is also a great advantage for fully automated fabrication combined with ultraprecision turning.

4. ${{Q}}$ FACTOR AND DISPERSION OF FABRICATED MICRORESONATORS

The ${{Q}}$ factor and dispersion were measured in crystalline microresonators fabricated with the procedure described above. We fabricated an ${\rm{Mg}}{{\rm{F}}_{{2}}}$ WGM resonator and a ${\rm{Ca}}{{\rm{F}}_{{2}}}$ WGM resonator with the same curvature radii of 36 µm. The diameters were 508 µm for the ${\rm{Mg}}{{\rm{F}}_{{2}}}$ resonator and 512 µm for ${\rm{Ca}}{{\rm{F}}_{{2}}}$ resonator. Figures 5(a)5(c) show SEM images of the fabricated ${\rm{Mg}}{{\rm{F}}_{{2}}}$ microresonator. Although the two resonators were fabricated with the same motion program and cutting conditions, their diameters differ slightly due to the positioning misalignment of a workpiece that we mount manually. However, we can solve this problem by installing a real-time diameter measurement system or introducing zero adjustment of the workpiece offset position. Indeed, the additional measurement and turning of the workpiece enables us to achieve the practical precise control of the diameter at the submicrometer level.

 

Fig. 5. ${{Q}}$ factor and dispersion measurement of crystalline microresonators fabricated by ultraprecise turning. (a) SEM image of a fabricated ${\rm{Mg}}{{\rm{F}}_{{2}}}$ microresonator with a diameter of 508 µm and a curvature radius of 36 µm. (b), (c) Magnified views of the resonator. (d) Normalized transmission spectra of the fabricated ${\rm{Mg}}{{\rm{F}}_{{2}}}$ microresonator. The Lorentzian fitting (red line) yield loaded a ${{Q}}$ value of 139 million. (e) Normalized transmission spectra of the fabricated ${\rm{Ca}}{{\rm{F}}_{{2}}}$ microresonator. The fitting curves give a loaded ${{Q}}$ for the fundamental mode of 76.7 million. (f), (g) Measured dispersion ${D_{{\rm{int}}}}$ versus frequency. The red curve indicates the calculated dispersion of the fundamental TM mode, which agrees well with the experimental result.

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Figures 5(d) and 5(e) show the measured transmission spectra of the fabricated microresonators. We launched light from a frequency-tunable laser source, which was coupled into the resonator via a tapered optical silica fiber. A polarization controller was used to adjust the polarization before the light coupling. The transmitted light was monitored with a high-speed photodetector and oscilloscope, where we used a calibrated fiber Mach–Zehnder interferometer as the frequency reference. The full width at half-maximum (FWHM) linewidth of the ${\rm{Mg}}{{\rm{F}}_{{2}}}$ resonator was 1.40 MHz, which corresponds to a loaded $Q = 1.39 \times {10^8}$ at a wavelength of 1545 nm. Also, the ${\rm{Ca}}{{\rm{F}}_{{2}}}$ resonator had a linewidth of 2.53 MHz at 1546 nm, corresponding to $Q = 7.67 \times {10^7}$. We measured the ${{Q}}$ factor in different wavelengths regions and recorded comparably high-$\textit{Q}$ values for other resonant modes. The obtained ${{Q}}$, which exceeded 100 million, is the highest value recorded in a crystalline WGM microresonator fabricated solely by ultraprecision machining without a conventional polishing process. In other words, our approach has overcome the manufacturing limitation, namely, the need for skilled manual techniques throughout the fabrication process to obtain ultrahigh-$\textit{Q}$ crystalline microresonators.

Figures 5(f) and 5(g) show the measured integrated dispersion, defined as ${D_{{\rm{int}}}} = {\omega _\mu} - {\omega _0} - {D_1}\mu = {D_2}{\mu ^2}/2 + {D_3}{\mu ^3}/6 + \cdots$, where ${\omega _\mu}/2\pi$ is the resonance frequency of the µth mode ($\mu= 0$ designates the center mode), ${D_1}/2\pi$ is the equidistant free-spectral range (FSR), ${D_2}/2\pi$ is the second-order dispersion linked to group velocity dispersion, and ${D_3}/2\pi$ and higher-order terms correspond to higher-order dispersion. The microresonator dispersion measurement was performed assisted by a fiber laser comb and a wavelength meter [34]. The measured dispersion agrees well with the theoretical dispersion calculated with the finite element method (FEM) by using (COMSOL Multiphysics), and these results indicate that ultraprecision turning enables us to obtain the designed resonator shape.

Figure 6(a) and 6(b) show SEM images of fabricated triangular-shaped ${\rm{Ca}}{{\rm{F}}_{{2}}}$ microresonator whose triangular structure is designed with an apex angle of 120°. The measured dispersion is shown in Fig. 6(c), and the result also agrees with the calculated result, which exhibits strong normal dispersion compared with a spherical WGM. However, unfortunately, a typical ${{Q}}$ factor of the resonator is limited to 10 million. We suggest that the relatively low-${{Q}}$ (compared to spherical shape) is caused by the fragility of the edge of the apex or remaining specks of dust, thereby increasing scattering loss. The improvement of ${{Q}}$ factor in such a sophisticated resonator will be our specific challenge. Measured transmission and the fabrication flow of a WGM crystalline resonator with a triangular cross-sectional shape are provided in Supplement 1.

 

Fig. 6. (a), (b) SEM images of a fabricated triangular-shaped ${\rm{Ca}}{{\rm{F}}_{{2}}}$ resonator with an apex of 120°. (c) Measured dispersion ${D_{{\rm{int}}}}$ versus frequency showing good agreement with the calculated dispersion for the fundamental mode (red curve). The perturbation around 201 THz is induced by mode coupling with a different transverse mode.

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In fact, we realized a Kerr frequency comb in the fabricated ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystalline microresonator by machining alone. The microresonator was designed to generate octave-wide parametric oscillation by engineering higher-order dispersion [35]. Hence, fully computer-controlled machining could be a great advantage as regards extending the potential of crystalline microresonators for optical frequency comb generation from the standpoint of dispersion engineering.

 

Fig. 7. ${{Q}}$-factor limitation caused by surface scattering loss in ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystalline microresonators derived from Eq. (2) ($\lambda = 1550\,\,{\rm{nm}}$ and $R = 250\,\,{\rm{\unicode{x00B5}{\rm m}}}$). The dashed contours show roughly estimated value from our measurement and previous studies [2,37,38].

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5. DISCUSSION

A. Outlook: Towards Further ${{Q}}$-Factor Improvement

The total (loaded) ${{Q}}$ factor is determined by folding the several loss contributions as

$$Q_{{\rm{tot}}}^{- 1} = Q_{{\rm{mat}}}^{- 1} + Q_{{\rm{surf}}}^{- 1} + Q_{{\rm{scatt}}}^{- 1} + Q_{{\rm{rad}}}^{- 1} + Q_{{\rm{ext}}}^{- 1},$$
where $Q_{{\rm{mat}}}^{- 1}$ is determined by material absorption, and $Q_{{\rm{surf}}}^{- 1}$ and $Q_{{\rm{scatt}}}^{- 1}$ are determined by surface absorption and scattering loss, respectively. In particular, scattering loss is mainly induced by surface roughness, dust particles, and subsurface damages. The radiation (tunneling) loss is given by $Q_{{\rm{rad}}}^{- 1}$, which is determined by the curvature radiation limit, and this loss is significant only with resonators having a diameter of several micrometers. $Q_{{\rm{ext}}}^{- 1}$ is given by the coupling rate between the resonator and the external waveguide (e.g.,  tapered fiber, prism). Since the total ${{Q}}$ factor can readily reach ${10^9}$ by polishing millimeter-scale fluoride crystalline resonators, the effect of $Q_{{\rm{mat}}}^{- 1}$, $Q_{{\rm{rad}}}^{- 1}$, and $Q_{{\rm{ext}}}^{- 1}$ should be negligible [36]. $Q_{{\rm{surf}}}^{- 1}$ is one possible reason for this, whereas single crystals such as ${\rm{Mg}}{{\rm{F}}_{{2}}}$ and ${\rm{Ca}}{{\rm{F}}_{{2}}}$ inhibit the diffusion of water into the crystal lattice, which makes $Q_{{\rm{surf}}}^{- 1}$ negligible in our case [2].

Then, we highlight $Q_{{\rm{scatt}}}^{- 1}$ as a fundamental limitation of ultraprecision machining. Since the surface roughness of the polished resonator reaches a subnanometer scale [2], it is reasonable to consider that the surface scattering limits ${{Q}}$ in diamond turning [Fig. 3(g) shows a surface roughness of a few nanometer scale]. The maximum ${{Q}}$ factor as regards surface roughness can be estimated as [2,37]

$${Q_{{\rm{scatt}}}} \approx \frac{{3{\lambda ^3}R}}{{8n{\pi ^2}{B^2}{\sigma ^2}}},$$
where $R$ is the resonator radius, $n$ is the refractive index, $B$ is the correlation length, and $\sigma$ is the surface roughness (RMS). The maximum ${{Q}}$ factor versus surface roughness and correlation length of ${\rm{Mg}}{{\rm{F}}_{{2}}}$ resonator is plotted in Fig. 7. Theoretically achievable values for ultraprecision machining correspond to ${{Q}}$ values of ${10^7} - {10^9}$, which are consistent with measured ${{Q}}$ factors. Although it is difficult to measure the correlation length directly due to instrumental limitations, we predict that the value is in the range of a few tens to hundreds of nanometers based on the ${{Q}}$ factor and surface roughness, which is also in a good agreement with the discussion in Ref. [38]. The plot indicates that the roughness of the machined surface could limit the present ${{Q}}$. A possible way to improve the surface roughness and correlation length is to optimize the cutting parameters, for example, by using a smaller depth of cut and a lower feed rate. Specifically, ideal conditions are believed to realize an ultrasmooth surface for the entire cylindrical position and consequently eliminate the effect of crystal anisotropy as seen in Fig. 3(g).

In addition, we should take the effect of subsurface damage into account since it causes the degeneration of the inner structure of the material. Subsurface damage occurs when machining a single crystal, and so it has been intensively studied in the field of micromachining and material science [39]. Such underlayer damage could degrade ${{Q}}$ in the same way as surface scattering; therefore, we investigated the surface and subsurface damage by using a SEM and a transmission electron microscope (TEM) in comparison with the results for polishing. As a result, we found that the damaged subsurface layers were around several tens of nanometers with single-crystal precise turning. (Details and results are presented in Supplement 1). It is generally known that the subsurface damage mechanism strongly depends on the crystal properties and cutting condition, and the efforts to reduce the subsurface damage are described elsewhere [40]. The reduction of underlayer damage could also help to improve the present ${{Q}}$.

6. CONCLUSION

In conclusion, we demonstrated the fabrication of an ultrahigh-$\textit{Q}$ crystalline microresonator by using ultraprecision turning alone. For the first time, we achieved a ${{Q}}$ value exceeding 100 million without polishing and thereby managed both an ultrahigh ${{Q}}$ and dispersion engineering simultaneously. We revealed the critical depth of cut needed to sustain ductile mode cutting, and this information contributes significantly to reducing surface roughness. Moreover, we proposed an optimal cutting condition for cylindrical turning for realizing an ultrasmooth surface throughout an entire cylindrical surface. This result provides the path towards the fabrication of a high-$\textit{Q}$ microresonator without the need for skilled manual work. Furthermore, we discussed the possibility of further improving the ${{Q}}$ factor from the standpoint of the theoretical limitation imposed by surface roughness. The described fabrication and cleaning procedure can be applied to various single-crystal materials and will raise the potential for realizing ultrahigh-Q crystalline microresonators with an all-precision-machining process.

Funding

Japan Society for the Promotion of Science (JP18J21797, JP18K19036); Strategic Information and Communications R&D Promotion Programme (191603001).

Acknowledgment

The authors thank Dr. Y. Mizumoto for fruitful discussion and also H. Amano, T. Takahashi, M. Fuchida, and K. Wada for technical support.

Disclosures

The authors declare no conflicts of interest.

 

See Supplement 1 for supporting content.

REFERENCES

1. I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006). [CrossRef]  

2. I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006). [CrossRef]  

3. G. Lin, A. Coillet, and Y. K. Chembo, “Nonlinear photonics with high-Q whispering-gallery-mode resonators,” Adv. Opt. Photon. 9, 828–890 (2017). [CrossRef]  

4. I. Breunig, “Three-wave mixing in whispering gallery resonators,” Laser Photon. Rev. 10, 569–587 (2016). [CrossRef]  

5. A. Kovach, D. Chen, J. He, H. Choi, A. H. Dogan, M. Ghasemkhani, H. Taheri, and A. M. Armani, “Emerging material systems for integrated optical kerr frequency combs,” Adv. Opt. Photon. 12, 135–222 (2020). [CrossRef]  

6. E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Sudmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11, 374 (2020). [CrossRef]  

7. W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015). [CrossRef]  

8. N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12, 694–698 (2018). [CrossRef]  

9. W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010). [CrossRef]  

10. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007). [CrossRef]  

11. W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015). [CrossRef]  

12. W. Weng, E. Lucas, G. Lihachev, V. E. Lobanov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Spectral purification of microwave signals with disciplined dissipative Kerr solitons,” Phys. Rev. Lett. 122, 013902 (2019). [CrossRef]  

13. T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014). [CrossRef]  

14. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. J. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015). [CrossRef]  

15. I. S. Grudinin, A. B. Matsko, and L. Maleki, “On the fundamental limits of Q factor of crystalline dielectric resonators,” Opt. Express 15, 3390–3395 (2007). [CrossRef]  

16. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007). [CrossRef]  

17. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804 (2004). [CrossRef]  

18. G. Lin and Y. K. Chembo, “On the dispersion management of fluorite whispering-gallery mode resonators for Kerr optical frequency comb generation in the telecom and mid-infrared range,” Opt. Express 23, 1594–1604 (2015). [CrossRef]  

19. A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006). [CrossRef]  

20. J. M. Winkler, I. S. Grudinin, and N. Yu, “On the properties of single-mode optical resonators,” Opt. Express 24, 13231–13243 (2016). [CrossRef]  

21. I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express 20, 6604–6609 (2012). [CrossRef]  

22. I. S. Grudinin and N. Yu, “Dispersion engineering of crystalline resonators via microstructuring,” Optica 2, 221–224 (2015). [CrossRef]  

23. Y. Nakagawa, Y. Mizumoto, T. Kato, T. Kobatake, H. Itobe, Y. Kakinuma, and T. Tanabe, “Dispersion tailoring of a crystalline whispering gallery mode microcavity for a wide-spanning optical Kerr frequency comb,” J. Opt. Soc. Am. B 33, 1913–1920 (2016). [CrossRef]  

24. N. G. Pavlov, G. Lihachev, S. Koptyaev, E. Lucas, M. Karpov, N. M. Kondratiev, I. A. Bilenko, T. J. Kippenberg, and M. L. Gorodetsky, “Soliton dual frequency combs in crystalline microresonators,” Opt. Lett. 42, 514–517 (2017). [CrossRef]  

25. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004). [CrossRef]  

26. M. Soltani, V. Ilchenko, A. Matsko, A. Savchenkov, J. Schlafer, C. Ryan, and L. Maleki, “Ultrahigh Q whispering gallery mode electro-optic resonators on a silicon photonic chip,” Opt. Lett. 41, 4375–4378 (2016). [CrossRef]  

27. M. Anderson, N. G. Pavlov, J. D. Jost, G. Lihachev, J. Liu, T. Morais, M. Zervas, M. L. Gorodetsky, and T. J. Kippenberg, “Highly efficient coupling of crystalline microresonators to integrated photonic waveguides,” Opt. Lett. 43, 2106–2109 (2018). [CrossRef]  

28. G. Liu, V. S. Ilchenko, T. Su, Y.-C. Ling, S. Feng, K. Shang, Y. Zhang, W. Liang, A. A. Savchenkov, A. B. Matsko, L. Maleki, and S. J. B. Yoo, “Low-loss prism-waveguide optical coupling for ultrahigh-Q low-index monolithic resonators,” Optica 5, 219–226 (2018). [CrossRef]  

29. Y. Zhuang, H. Kumazaki, S. Fujii, R. Imamura, N. A. B. Daud, R. Ishida, H. Chen, and T. Tanabe, “Coupling of a whispering gallery mode to a silicon chip with photonic crystal,” Opt. Lett. 44, 5731–5734 (2019). [CrossRef]  

30. P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73, 949–957 (1990). [CrossRef]  

31. W. Blackley and R. Scattergood, “Ductile-regime machining model for diamond turning of brittle materials,” Precis. Eng. 13, 95–103 (1991). [CrossRef]  

32. K. Liu, D. Zuo, X. P. Li, and M. Rahman, “Nanometric ductile cutting characteristics of silicon wafer using single crystal diamond tools,” J. Vac. Sci. Technol. B 27, 1361–1366 (2009). [CrossRef]  

33. S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan, and Y. Kakinuma, “Experimental study of crystal anisotropy based on ultra-precision cylindrical turning of single-crystal calcium fluoride,” Precis. Eng. 40, 172–181 (2015). [CrossRef]  

34. S. Fujii and T. Tanabe, “Dispersion engineering and measurement of whispering gallery mode microresonator for Kerr frequency comb generation,” Nanophotonics (2020), (early posting). [CrossRef]  

35. S. Fujii, S. Tanaka, M. Fuchida, H. Amano, Y. Hayama, R. Suzuki, Y. Kakinuma, and T. Tanabe, “Octave-wide phase-matched four-wave mixing in dispersion-engineered crystalline microresonators,” Opt. Lett. 44, 3146–3149 (2019). [CrossRef]  

36. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996). [CrossRef]  

37. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000). [CrossRef]  

38. G. Lin, R. Henriet, A. Coillet, M. Jacquot, L. Furfaro, G. Cibiel, L. Larger, and Y. K. Chembo, “Dependence of quality factor on surface roughness in crystalline whispering-gallery mode resonators,” Opt. Lett. 43, 495–498 (2018). [CrossRef]  

39. J. Yan, T. Asami, H. Harada, and T. Kuriyagawa, “Fundamental investigation of subsurface damage in single crystalline silicon caused by diamond machining,” Precis. Eng. 33, 378–386 (2009). [CrossRef]  

40. Y. Mizumoto, H. Amano, H. Kangawa, K. Harano, H. Sumiya, and Y. Kakinuma, “On the improvement of subsurface quality of single crystal machined by boron-doped nano-polycrystalline diamond tools,” Precis. Eng. 52, 73–83 (2018). [CrossRef]  

References

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  1. I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
    [Crossref]
  2. I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006).
    [Crossref]
  3. G. Lin, A. Coillet, and Y. K. Chembo, “Nonlinear photonics with high-Q whispering-gallery-mode resonators,” Adv. Opt. Photon. 9, 828–890 (2017).
    [Crossref]
  4. I. Breunig, “Three-wave mixing in whispering gallery resonators,” Laser Photon. Rev. 10, 569–587 (2016).
    [Crossref]
  5. A. Kovach, D. Chen, J. He, H. Choi, A. H. Dogan, M. Ghasemkhani, H. Taheri, and A. M. Armani, “Emerging material systems for integrated optical kerr frequency combs,” Adv. Opt. Photon. 12, 135–222 (2020).
    [Crossref]
  6. E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Sudmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11, 374 (2020).
    [Crossref]
  7. W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
    [Crossref]
  8. N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12, 694–698 (2018).
    [Crossref]
  9. W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
    [Crossref]
  10. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
    [Crossref]
  11. W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
    [Crossref]
  12. W. Weng, E. Lucas, G. Lihachev, V. E. Lobanov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Spectral purification of microwave signals with disciplined dissipative Kerr solitons,” Phys. Rev. Lett. 122, 013902 (2019).
    [Crossref]
  13. T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
    [Crossref]
  14. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. J. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
    [Crossref]
  15. I. S. Grudinin, A. B. Matsko, and L. Maleki, “On the fundamental limits of Q factor of crystalline dielectric resonators,” Opt. Express 15, 3390–3395 (2007).
    [Crossref]
  16. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
    [Crossref]
  17. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804 (2004).
    [Crossref]
  18. G. Lin and Y. K. Chembo, “On the dispersion management of fluorite whispering-gallery mode resonators for Kerr optical frequency comb generation in the telecom and mid-infrared range,” Opt. Express 23, 1594–1604 (2015).
    [Crossref]
  19. A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
    [Crossref]
  20. J. M. Winkler, I. S. Grudinin, and N. Yu, “On the properties of single-mode optical resonators,” Opt. Express 24, 13231–13243 (2016).
    [Crossref]
  21. I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express 20, 6604–6609 (2012).
    [Crossref]
  22. I. S. Grudinin and N. Yu, “Dispersion engineering of crystalline resonators via microstructuring,” Optica 2, 221–224 (2015).
    [Crossref]
  23. Y. Nakagawa, Y. Mizumoto, T. Kato, T. Kobatake, H. Itobe, Y. Kakinuma, and T. Tanabe, “Dispersion tailoring of a crystalline whispering gallery mode microcavity for a wide-spanning optical Kerr frequency comb,” J. Opt. Soc. Am. B 33, 1913–1920 (2016).
    [Crossref]
  24. N. G. Pavlov, G. Lihachev, S. Koptyaev, E. Lucas, M. Karpov, N. M. Kondratiev, I. A. Bilenko, T. J. Kippenberg, and M. L. Gorodetsky, “Soliton dual frequency combs in crystalline microresonators,” Opt. Lett. 42, 514–517 (2017).
    [Crossref]
  25. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
    [Crossref]
  26. M. Soltani, V. Ilchenko, A. Matsko, A. Savchenkov, J. Schlafer, C. Ryan, and L. Maleki, “Ultrahigh Q whispering gallery mode electro-optic resonators on a silicon photonic chip,” Opt. Lett. 41, 4375–4378 (2016).
    [Crossref]
  27. M. Anderson, N. G. Pavlov, J. D. Jost, G. Lihachev, J. Liu, T. Morais, M. Zervas, M. L. Gorodetsky, and T. J. Kippenberg, “Highly efficient coupling of crystalline microresonators to integrated photonic waveguides,” Opt. Lett. 43, 2106–2109 (2018).
    [Crossref]
  28. G. Liu, V. S. Ilchenko, T. Su, Y.-C. Ling, S. Feng, K. Shang, Y. Zhang, W. Liang, A. A. Savchenkov, A. B. Matsko, L. Maleki, and S. J. B. Yoo, “Low-loss prism-waveguide optical coupling for ultrahigh-Q low-index monolithic resonators,” Optica 5, 219–226 (2018).
    [Crossref]
  29. Y. Zhuang, H. Kumazaki, S. Fujii, R. Imamura, N. A. B. Daud, R. Ishida, H. Chen, and T. Tanabe, “Coupling of a whispering gallery mode to a silicon chip with photonic crystal,” Opt. Lett. 44, 5731–5734 (2019).
    [Crossref]
  30. P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73, 949–957 (1990).
    [Crossref]
  31. W. Blackley and R. Scattergood, “Ductile-regime machining model for diamond turning of brittle materials,” Precis. Eng. 13, 95–103 (1991).
    [Crossref]
  32. K. Liu, D. Zuo, X. P. Li, and M. Rahman, “Nanometric ductile cutting characteristics of silicon wafer using single crystal diamond tools,” J. Vac. Sci. Technol. B 27, 1361–1366 (2009).
    [Crossref]
  33. S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan, and Y. Kakinuma, “Experimental study of crystal anisotropy based on ultra-precision cylindrical turning of single-crystal calcium fluoride,” Precis. Eng. 40, 172–181 (2015).
    [Crossref]
  34. S. Fujii and T. Tanabe, “Dispersion engineering and measurement of whispering gallery mode microresonator for Kerr frequency comb generation,” Nanophotonics (2020), (early posting).
    [Crossref]
  35. S. Fujii, S. Tanaka, M. Fuchida, H. Amano, Y. Hayama, R. Suzuki, Y. Kakinuma, and T. Tanabe, “Octave-wide phase-matched four-wave mixing in dispersion-engineered crystalline microresonators,” Opt. Lett. 44, 3146–3149 (2019).
    [Crossref]
  36. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [Crossref]
  37. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
    [Crossref]
  38. G. Lin, R. Henriet, A. Coillet, M. Jacquot, L. Furfaro, G. Cibiel, L. Larger, and Y. K. Chembo, “Dependence of quality factor on surface roughness in crystalline whispering-gallery mode resonators,” Opt. Lett. 43, 495–498 (2018).
    [Crossref]
  39. J. Yan, T. Asami, H. Harada, and T. Kuriyagawa, “Fundamental investigation of subsurface damage in single crystalline silicon caused by diamond machining,” Precis. Eng. 33, 378–386 (2009).
    [Crossref]
  40. Y. Mizumoto, H. Amano, H. Kangawa, K. Harano, H. Sumiya, and Y. Kakinuma, “On the improvement of subsurface quality of single crystal machined by boron-doped nano-polycrystalline diamond tools,” Precis. Eng. 52, 73–83 (2018).
    [Crossref]

2020 (2)

A. Kovach, D. Chen, J. He, H. Choi, A. H. Dogan, M. Ghasemkhani, H. Taheri, and A. M. Armani, “Emerging material systems for integrated optical kerr frequency combs,” Adv. Opt. Photon. 12, 135–222 (2020).
[Crossref]

E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Sudmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11, 374 (2020).
[Crossref]

2019 (3)

2018 (5)

2017 (2)

2016 (4)

2015 (6)

W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
[Crossref]

G. Lin and Y. K. Chembo, “On the dispersion management of fluorite whispering-gallery mode resonators for Kerr optical frequency comb generation in the telecom and mid-infrared range,” Opt. Express 23, 1594–1604 (2015).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. J. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
[Crossref]

W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

I. S. Grudinin and N. Yu, “Dispersion engineering of crystalline resonators via microstructuring,” Optica 2, 221–224 (2015).
[Crossref]

S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan, and Y. Kakinuma, “Experimental study of crystal anisotropy based on ultra-precision cylindrical turning of single-crystal calcium fluoride,” Precis. Eng. 40, 172–181 (2015).
[Crossref]

2014 (1)

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

2012 (1)

2010 (1)

2009 (2)

K. Liu, D. Zuo, X. P. Li, and M. Rahman, “Nanometric ductile cutting characteristics of silicon wafer using single crystal diamond tools,” J. Vac. Sci. Technol. B 27, 1361–1366 (2009).
[Crossref]

J. Yan, T. Asami, H. Harada, and T. Kuriyagawa, “Fundamental investigation of subsurface damage in single crystalline silicon caused by diamond machining,” Precis. Eng. 33, 378–386 (2009).
[Crossref]

2007 (3)

2006 (3)

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
[Crossref]

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
[Crossref]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006).
[Crossref]

2004 (2)

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804 (2004).
[Crossref]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

2000 (1)

1996 (1)

1991 (1)

W. Blackley and R. Scattergood, “Ductile-regime machining model for diamond turning of brittle materials,” Precis. Eng. 13, 95–103 (1991).
[Crossref]

1990 (1)

P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73, 949–957 (1990).
[Crossref]

Amano, H.

S. Fujii, S. Tanaka, M. Fuchida, H. Amano, Y. Hayama, R. Suzuki, Y. Kakinuma, and T. Tanabe, “Octave-wide phase-matched four-wave mixing in dispersion-engineered crystalline microresonators,” Opt. Lett. 44, 3146–3149 (2019).
[Crossref]

Y. Mizumoto, H. Amano, H. Kangawa, K. Harano, H. Sumiya, and Y. Kakinuma, “On the improvement of subsurface quality of single crystal machined by boron-doped nano-polycrystalline diamond tools,” Precis. Eng. 52, 73–83 (2018).
[Crossref]

Anderson, M.

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[Crossref]

Armani, A. M.

Asami, T.

J. Yan, T. Asami, H. Harada, and T. Kuriyagawa, “Fundamental investigation of subsurface damage in single crystalline silicon caused by diamond machining,” Precis. Eng. 33, 378–386 (2009).
[Crossref]

Azami, S.

S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan, and Y. Kakinuma, “Experimental study of crystal anisotropy based on ultra-precision cylindrical turning of single-crystal calcium fluoride,” Precis. Eng. 40, 172–181 (2015).
[Crossref]

Baumgartel, L.

Bilenko, I. A.

Blackley, W.

W. Blackley and R. Scattergood, “Ductile-regime machining model for diamond turning of brittle materials,” Precis. Eng. 13, 95–103 (1991).
[Crossref]

Blake, P. N.

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Eliyahu, D.

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W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
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W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
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W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
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W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
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A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
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A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
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I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
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I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006).
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A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804 (2004).
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M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
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M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
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Imamura, R.

Ishida, R.

Itobe, H.

Jacquot, M.

Jost, J.

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
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Kakinuma, Y.

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Y. Mizumoto, H. Amano, H. Kangawa, K. Harano, H. Sumiya, and Y. Kakinuma, “On the improvement of subsurface quality of single crystal machined by boron-doped nano-polycrystalline diamond tools,” Precis. Eng. 52, 73–83 (2018).
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Y. Nakagawa, Y. Mizumoto, T. Kato, T. Kobatake, H. Itobe, Y. Kakinuma, and T. Tanabe, “Dispersion tailoring of a crystalline whispering gallery mode microcavity for a wide-spanning optical Kerr frequency comb,” J. Opt. Soc. Am. B 33, 1913–1920 (2016).
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S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan, and Y. Kakinuma, “Experimental study of crystal anisotropy based on ultra-precision cylindrical turning of single-crystal calcium fluoride,” Precis. Eng. 40, 172–181 (2015).
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Y. Mizumoto, H. Amano, H. Kangawa, K. Harano, H. Sumiya, and Y. Kakinuma, “On the improvement of subsurface quality of single crystal machined by boron-doped nano-polycrystalline diamond tools,” Precis. Eng. 52, 73–83 (2018).
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Kato, T.

Kippenberg, T. J.

E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Sudmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11, 374 (2020).
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W. Weng, E. Lucas, G. Lihachev, V. E. Lobanov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Spectral purification of microwave signals with disciplined dissipative Kerr solitons,” Phys. Rev. Lett. 122, 013902 (2019).
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T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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Kondratiev, N. M.

Koptyaev, S.

N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12, 694–698 (2018).
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N. G. Pavlov, G. Lihachev, S. Koptyaev, E. Lucas, M. Karpov, N. M. Kondratiev, I. A. Bilenko, T. J. Kippenberg, and M. L. Gorodetsky, “Soliton dual frequency combs in crystalline microresonators,” Opt. Lett. 42, 514–517 (2017).
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Kudo, H.

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Kuriyagawa, T.

J. Yan, T. Asami, H. Harada, and T. Kuriyagawa, “Fundamental investigation of subsurface damage in single crystalline silicon caused by diamond machining,” Precis. Eng. 33, 378–386 (2009).
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Larger, L.

Li, X. P.

K. Liu, D. Zuo, X. P. Li, and M. Rahman, “Nanometric ductile cutting characteristics of silicon wafer using single crystal diamond tools,” J. Vac. Sci. Technol. B 27, 1361–1366 (2009).
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G. Liu, V. S. Ilchenko, T. Su, Y.-C. Ling, S. Feng, K. Shang, Y. Zhang, W. Liang, A. A. Savchenkov, A. B. Matsko, L. Maleki, and S. J. B. Yoo, “Low-loss prism-waveguide optical coupling for ultrahigh-Q low-index monolithic resonators,” Optica 5, 219–226 (2018).
[Crossref]

W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
[Crossref]

W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
[Crossref]

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Lihachev, G. V.

N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12, 694–698 (2018).
[Crossref]

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Ling, Y.-C.

Liu, G.

Liu, J.

Liu, K.

K. Liu, D. Zuo, X. P. Li, and M. Rahman, “Nanometric ductile cutting characteristics of silicon wafer using single crystal diamond tools,” J. Vac. Sci. Technol. B 27, 1361–1366 (2009).
[Crossref]

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W. Weng, E. Lucas, G. Lihachev, V. E. Lobanov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Spectral purification of microwave signals with disciplined dissipative Kerr solitons,” Phys. Rev. Lett. 122, 013902 (2019).
[Crossref]

Lucas, E.

E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Sudmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11, 374 (2020).
[Crossref]

W. Weng, E. Lucas, G. Lihachev, V. E. Lobanov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Spectral purification of microwave signals with disciplined dissipative Kerr solitons,” Phys. Rev. Lett. 122, 013902 (2019).
[Crossref]

N. G. Pavlov, G. Lihachev, S. Koptyaev, E. Lucas, M. Karpov, N. M. Kondratiev, I. A. Bilenko, T. J. Kippenberg, and M. L. Gorodetsky, “Soliton dual frequency combs in crystalline microresonators,” Opt. Lett. 42, 514–517 (2017).
[Crossref]

Maleki, L.

G. Liu, V. S. Ilchenko, T. Su, Y.-C. Ling, S. Feng, K. Shang, Y. Zhang, W. Liang, A. A. Savchenkov, A. B. Matsko, L. Maleki, and S. J. B. Yoo, “Low-loss prism-waveguide optical coupling for ultrahigh-Q low-index monolithic resonators,” Optica 5, 219–226 (2018).
[Crossref]

M. Soltani, V. Ilchenko, A. Matsko, A. Savchenkov, J. Schlafer, C. Ryan, and L. Maleki, “Ultrahigh Q whispering gallery mode electro-optic resonators on a silicon photonic chip,” Opt. Lett. 41, 4375–4378 (2016).
[Crossref]

W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
[Crossref]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
[Crossref]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
[Crossref]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “On the fundamental limits of Q factor of crystalline dielectric resonators,” Opt. Express 15, 3390–3395 (2007).
[Crossref]

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
[Crossref]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006).
[Crossref]

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
[Crossref]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804 (2004).
[Crossref]

Matsko, A.

Matsko, A. B.

G. Liu, V. S. Ilchenko, T. Su, Y.-C. Ling, S. Feng, K. Shang, Y. Zhang, W. Liang, A. A. Savchenkov, A. B. Matsko, L. Maleki, and S. J. B. Yoo, “Low-loss prism-waveguide optical coupling for ultrahigh-Q low-index monolithic resonators,” Optica 5, 219–226 (2018).
[Crossref]

W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
[Crossref]

W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
[Crossref]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “On the fundamental limits of Q factor of crystalline dielectric resonators,” Opt. Express 15, 3390–3395 (2007).
[Crossref]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
[Crossref]

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
[Crossref]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006).
[Crossref]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804 (2004).
[Crossref]

Mizumoto, Y.

Y. Mizumoto, H. Amano, H. Kangawa, K. Harano, H. Sumiya, and Y. Kakinuma, “On the improvement of subsurface quality of single crystal machined by boron-doped nano-polycrystalline diamond tools,” Precis. Eng. 52, 73–83 (2018).
[Crossref]

Y. Nakagawa, Y. Mizumoto, T. Kato, T. Kobatake, H. Itobe, Y. Kakinuma, and T. Tanabe, “Dispersion tailoring of a crystalline whispering gallery mode microcavity for a wide-spanning optical Kerr frequency comb,” J. Opt. Soc. Am. B 33, 1913–1920 (2016).
[Crossref]

S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan, and Y. Kakinuma, “Experimental study of crystal anisotropy based on ultra-precision cylindrical turning of single-crystal calcium fluoride,” Precis. Eng. 40, 172–181 (2015).
[Crossref]

Mohageg, M.

Morais, T.

Nakagawa, Y.

Pavlov, N. G.

Polonsky, S. V.

N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12, 694–698 (2018).
[Crossref]

Pryamikov, A. D.

Rahman, M.

K. Liu, D. Zuo, X. P. Li, and M. Rahman, “Nanometric ductile cutting characteristics of silicon wafer using single crystal diamond tools,” J. Vac. Sci. Technol. B 27, 1361–1366 (2009).
[Crossref]

Ryabko, M. V.

N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12, 694–698 (2018).
[Crossref]

Ryan, C.

Savchenkov, A.

Savchenkov, A. A.

G. Liu, V. S. Ilchenko, T. Su, Y.-C. Ling, S. Feng, K. Shang, Y. Zhang, W. Liang, A. A. Savchenkov, A. B. Matsko, L. Maleki, and S. J. B. Yoo, “Low-loss prism-waveguide optical coupling for ultrahigh-Q low-index monolithic resonators,” Optica 5, 219–226 (2018).
[Crossref]

W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
[Crossref]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
[Crossref]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
[Crossref]

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
[Crossref]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006).
[Crossref]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804 (2004).
[Crossref]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[Crossref]

Scattergood, R.

W. Blackley and R. Scattergood, “Ductile-regime machining model for diamond turning of brittle materials,” Precis. Eng. 13, 95–103 (1991).
[Crossref]

Scattergood, R. O.

P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73, 949–957 (1990).
[Crossref]

Schilt, S.

E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Sudmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11, 374 (2020).
[Crossref]

Schlafer, J.

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[Crossref]

Seidel, D.

W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
[Crossref]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
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Shang, K.

Soltani, M.

Spillane, S. M.

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[Crossref]

Strekalov, D.

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
[Crossref]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006).
[Crossref]

Su, T.

Sudmeyer, T.

E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Sudmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11, 374 (2020).
[Crossref]

Suh, M.-G.

Sumiya, H.

Y. Mizumoto, H. Amano, H. Kangawa, K. Harano, H. Sumiya, and Y. Kakinuma, “On the improvement of subsurface quality of single crystal machined by boron-doped nano-polycrystalline diamond tools,” Precis. Eng. 52, 73–83 (2018).
[Crossref]

Suzuki, R.

Taheri, H.

Tanabe, T.

Tanaka, S.

Vahala, K. J.

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. J. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
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T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

Voloshin, A. S.

N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12, 694–698 (2018).
[Crossref]

Wang, C.

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Weng, W.

W. Weng, E. Lucas, G. Lihachev, V. E. Lobanov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Spectral purification of microwave signals with disciplined dissipative Kerr solitons,” Phys. Rev. Lett. 122, 013902 (2019).
[Crossref]

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[Crossref]

Winkler, J. M.

Yan, J.

S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan, and Y. Kakinuma, “Experimental study of crystal anisotropy based on ultra-precision cylindrical turning of single-crystal calcium fluoride,” Precis. Eng. 40, 172–181 (2015).
[Crossref]

J. Yan, T. Asami, H. Harada, and T. Kuriyagawa, “Fundamental investigation of subsurface damage in single crystalline silicon caused by diamond machining,” Precis. Eng. 33, 378–386 (2009).
[Crossref]

Yang, K. Y.

Yang, Q.-F.

Yi, X.

Yoo, S. J. B.

Yu, N.

Zervas, M.

Zhang, Y.

Zhuang, Y.

Zuo, D.

K. Liu, D. Zuo, X. P. Li, and M. Rahman, “Nanometric ductile cutting characteristics of silicon wafer using single crystal diamond tools,” J. Vac. Sci. Technol. B 27, 1361–1366 (2009).
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Adv. Opt. Photon. (2)

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[Crossref]

J. Opt. Soc. Am. B (2)

J. Vac. Sci. Technol. B (1)

K. Liu, D. Zuo, X. P. Li, and M. Rahman, “Nanometric ductile cutting characteristics of silicon wafer using single crystal diamond tools,” J. Vac. Sci. Technol. B 27, 1361–1366 (2009).
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Laser Photon. Rev. (1)

I. Breunig, “Three-wave mixing in whispering gallery resonators,” Laser Photon. Rev. 10, 569–587 (2016).
[Crossref]

Nat. Commun. (3)

E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Sudmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11, 374 (2020).
[Crossref]

W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Ultralow noise miniature external cavity semiconductor laser,” Nat. Commun. 6, 7371 (2015).
[Crossref]

W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

Nat. Photonics (2)

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12, 694–698 (2018).
[Crossref]

Nature (1)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[Crossref]

Opt. Commun. (1)

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33–38 (2006).
[Crossref]

Opt. Express (5)

Opt. Lett. (9)

N. G. Pavlov, G. Lihachev, S. Koptyaev, E. Lucas, M. Karpov, N. M. Kondratiev, I. A. Bilenko, T. J. Kippenberg, and M. L. Gorodetsky, “Soliton dual frequency combs in crystalline microresonators,” Opt. Lett. 42, 514–517 (2017).
[Crossref]

Y. Zhuang, H. Kumazaki, S. Fujii, R. Imamura, N. A. B. Daud, R. Ishida, H. Chen, and T. Tanabe, “Coupling of a whispering gallery mode to a silicon chip with photonic crystal,” Opt. Lett. 44, 5731–5734 (2019).
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M. Soltani, V. Ilchenko, A. Matsko, A. Savchenkov, J. Schlafer, C. Ryan, and L. Maleki, “Ultrahigh Q whispering gallery mode electro-optic resonators on a silicon photonic chip,” Opt. Lett. 41, 4375–4378 (2016).
[Crossref]

M. Anderson, N. G. Pavlov, J. D. Jost, G. Lihachev, J. Liu, T. Morais, M. Zervas, M. L. Gorodetsky, and T. J. Kippenberg, “Highly efficient coupling of crystalline microresonators to integrated photonic waveguides,” Opt. Lett. 43, 2106–2109 (2018).
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G. Lin, R. Henriet, A. Coillet, M. Jacquot, L. Furfaro, G. Cibiel, L. Larger, and Y. K. Chembo, “Dependence of quality factor on surface roughness in crystalline whispering-gallery mode resonators,” Opt. Lett. 43, 495–498 (2018).
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A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
[Crossref]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
[Crossref]

S. Fujii, S. Tanaka, M. Fuchida, H. Amano, Y. Hayama, R. Suzuki, Y. Kakinuma, and T. Tanabe, “Octave-wide phase-matched four-wave mixing in dispersion-engineered crystalline microresonators,” Opt. Lett. 44, 3146–3149 (2019).
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Optica (3)

Phys. Rev. A (2)

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804 (2004).
[Crossref]

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
[Crossref]

Phys. Rev. Lett. (2)

W. Weng, E. Lucas, G. Lihachev, V. E. Lobanov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Spectral purification of microwave signals with disciplined dissipative Kerr solitons,” Phys. Rev. Lett. 122, 013902 (2019).
[Crossref]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

Precis. Eng. (4)

J. Yan, T. Asami, H. Harada, and T. Kuriyagawa, “Fundamental investigation of subsurface damage in single crystalline silicon caused by diamond machining,” Precis. Eng. 33, 378–386 (2009).
[Crossref]

Y. Mizumoto, H. Amano, H. Kangawa, K. Harano, H. Sumiya, and Y. Kakinuma, “On the improvement of subsurface quality of single crystal machined by boron-doped nano-polycrystalline diamond tools,” Precis. Eng. 52, 73–83 (2018).
[Crossref]

S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan, and Y. Kakinuma, “Experimental study of crystal anisotropy based on ultra-precision cylindrical turning of single-crystal calcium fluoride,” Precis. Eng. 40, 172–181 (2015).
[Crossref]

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[Crossref]

Other (1)

S. Fujii and T. Tanabe, “Dispersion engineering and measurement of whispering gallery mode microresonator for Kerr frequency comb generation,” Nanophotonics (2020), (early posting).
[Crossref]

Supplementary Material (1)

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Figures (7)

Fig. 1.
Fig. 1. (a) Examples of the Miller index, where ($h k l$) and [$h k l$] indicate the corresponding plane and direction, respectively. (b) Crystallographic images of ${\rm{Ca}}{{\rm{F}}_{{2}}}$ and ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystal. The red arrows indicate examples of slip system. In contrast to the cubic symmetry system of ${\rm{Ca}}{{\rm{F}}_{{2}}}$, ${\rm{Mg}}{{\rm{F}}_{{2}}}$ is characterized by a more complex rutile structure.
Fig. 2.
Fig. 2. (a) Experimental setup for orthogonal cutting to investigate the critical depth of cut in ${\rm{Mg}}{{\rm{F}}_{{2}}}$ single crystal. (b) Schematics of a single-crystal substrate and the cutting direction in the orthogonal cutting experiment. The cutting was performed in every 30° rotation direction (left panel), and the diamond tool cut a ${{V}}$-shaped groove with a slope of $D/L$ (right panel). (c) Reconstructed image of machined surface using a scanning white-light interferometer. The critical depth of cut is given by the depth where the first brittle fracture appeared [black point in (c)]. (d) Cutting along the slip system (110) [001] and the inversion symmetry (110) [$00\bar 1$] promotes ductile mode cutting (left panel). Cleavage and subsequent brittle fractures are induced by the cutting force against cleavage plane (right panel). (e) Scanning electron micrographs showing machined surfaces of a (010) plane with [$00\bar 1$] direction (upper panel) and [$\bar 1 00$] direction (lower panel). The yellow shaded area and gray area correspond to the original uncut surfaces and machined surfaces, respectively. The difference between the machined surface conditions is attributed to the crystal anisotropy of the ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystal. (f), (g) The measured critical depth of cut versus cutting direction on a (001) and (010) plane, respectively. In comparison with the (001) plane, the (010) plane shows large variation in cutting direction due to crystal anisotropy. The results of 0° (green dot) and 270° (orange dot) correspond to the upper panel and lower panel in (e), respectively. These results confirmed that the measured critical depth of cut are consistent with images of the machined surface. To perform ductile mode cutting, the depth of cut must be kept below the critical depth of cut.
Fig. 3.
Fig. 3. Ultraprecise cylindrical turning and surface roughness measurement of ${\rm{Mg}}{{\rm{F}}_{{2}}}$ single crystal. (a) Experimental setup of an ultraprecision lathe for the cylindrical turning of a single crystal. (b) Micrograph showing a machined surface, where clear boundaries are observed between the rough turning and finish turning regions. The horizontal boundary in the rough turning region is evidence of the dependence of the cutting direction on the crystal anisotropy in ${\rm{Mg}}{{\rm{F}}_{{2}}}$ single crystal. (c), (d) Magnified views of the finish turning region in 135° and 180°, respectively. The machined surface of 135° is smoother than that of 180°, which agrees with the result of the surface roughness (RMS) measurement. (e) Schematic of surface roughness measurement. The yellow line and dot correspond to orientation flat [100] with an endface orientation of (001). The surface roughness at a total of 24 points was measured at 15° intervals. (f) Measured surface roughness (RMS) of the finish turning (red dots) and rough turning (blue dots) regions. A quarter symmetry is clearly observed in the finish turning condition due to crystal anisotropy. (g) Magnified plot on linear scale of finish turning in (f).
Fig. 4.
Fig. 4. Fabrication flow of WGM microresonator when using ultraprecision turning. First, a rough turning determines the approximate diameter of resonator. Next, pre-finish and finish turning with ductile mode cutting are used to realize a cylindrical surface that is smooth and entirely crack free. Finally, fully programmed shaping steps are performed to fabricate the designed resonator structure.
Fig. 5.
Fig. 5. ${{Q}}$ factor and dispersion measurement of crystalline microresonators fabricated by ultraprecise turning. (a) SEM image of a fabricated ${\rm{Mg}}{{\rm{F}}_{{2}}}$ microresonator with a diameter of 508 µm and a curvature radius of 36 µm. (b), (c) Magnified views of the resonator. (d) Normalized transmission spectra of the fabricated ${\rm{Mg}}{{\rm{F}}_{{2}}}$ microresonator. The Lorentzian fitting (red line) yield loaded a ${{Q}}$ value of 139 million. (e) Normalized transmission spectra of the fabricated ${\rm{Ca}}{{\rm{F}}_{{2}}}$ microresonator. The fitting curves give a loaded ${{Q}}$ for the fundamental mode of 76.7 million. (f), (g) Measured dispersion ${D_{{\rm{int}}}}$ versus frequency. The red curve indicates the calculated dispersion of the fundamental TM mode, which agrees well with the experimental result.
Fig. 6.
Fig. 6. (a), (b) SEM images of a fabricated triangular-shaped ${\rm{Ca}}{{\rm{F}}_{{2}}}$ resonator with an apex of 120°. (c) Measured dispersion ${D_{{\rm{int}}}}$ versus frequency showing good agreement with the calculated dispersion for the fundamental mode (red curve). The perturbation around 201 THz is induced by mode coupling with a different transverse mode.
Fig. 7.
Fig. 7. ${{Q}}$-factor limitation caused by surface scattering loss in ${\rm{Mg}}{{\rm{F}}_{{2}}}$ crystalline microresonators derived from Eq. (2) ($\lambda = 1550\,\,{\rm{nm}}$ and $R = 250\,\,{\rm{\unicode{x00B5}{\rm m}}}$). The dashed contours show roughly estimated value from our measurement and previous studies [2,37,38].

Equations (2)

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Q t o t 1 = Q m a t 1 + Q s u r f 1 + Q s c a t t 1 + Q r a d 1 + Q e x t 1 ,
Q s c a t t 3 λ 3 R 8 n π 2 B 2 σ 2 ,

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