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Page 1: Ultrahigh-Q crystalline microresonator fabrication with ...

Ultrahigh-Q crystalline microresonator fabrication with only precision machining

Shun Fujii,1 Yuka Hayama,2 Kosuke Imamura,2 Hajime Kumazaki,1 Yasuhiro Kakinuma,2 and Takasumi Tanabe1, ∗

1Department of Electronics and Electrical Engineering,Faculty of Science and Technology, Keio University, Yokohama, 223-8522, Japan

2Department of System Design Engineering, Faculty of Science and Technology, Keio University, Yokohama, 223-8522, Japan(Dated: September 8, 2020)

The development of ultrahigh quality factor (Q) microresonators has been driving such technolo-gies as cavity quantum electrodynamics (QED), high-precision sensing, optomechanics, and opticalfrequency comb generation. Here we report ultrahigh-Q crystalline microresonator fabrication witha Q exceeding 108, for the first time, achieved solely by computer-controlled ultraprecision ma-chining. Our fabrication method readily achieved the dispersion engineering and size control offabricated devices via programmed machine motion. Moreover, in contrast to the conventionalpolishing method, our machining fabrication approach avoids the need for subsequent careful pol-ishing, which is generally required to ensure that surface integrity is maintained, and this enabledus to realize an ultrahigh-Q. We carefully addressed the cutting condition and crystal anisotropyto overcome the large surface roughness that has thus far been the primary cause of the low-Q inthe machining process. Our result paves the way for future mass-production with a view to variousphotonic applications utilizing ultrahigh-Q crystalline microresonators.

I. INTRODUCTION

Ultrahigh-Q crystalline microresonators have beenused as attractive platforms for studying nonlinear andquantum optics in the last few decades [1–5]. In par-ticular, laser stabilization via self-injection locking andKerr optical frequency comb generation are potential ap-plications with the aim of realizing an optical-frequencysynthesizer [6] and low-noise, compact photonic de-vices [7, 8]. Injection locking to high-Q whisperinggallery mode (WGM) microresonators enables the laserlinewidth to be reduced to less than hundreds of hertz [9].Moreover, Kerr frequency comb generation [10] providesRF oscillators with high spectral purity [11, 12] and anoptical pulse train with a high repetition rate [13, 14].These applications rely on the high-Q of crystalline mi-croresonators, typically up to 109 and corresponding toa resonance linewidth of hundreds of kilohertz, which en-hances the optical nonlinearity. The fundamental limitof the Q-factor in crystalline resonators is ∼1013 [15](Q > 1011 as observed in the experiment [16]), and thisvalue surpasses that of resonators made with other ma-terials (e.g., silica, silicon, etc) [17]. In addition, theyhave a fully transparent window in the visible to mid-infrared wavelength region, which expands the availablebandwidth as well as the telecom band [18].

Magnesium fluoride (MgF2) and calcium fluoride(CaF2) are crystalline materials that are commonly usedfor fabricating WGM microresonators thanks to theirquality, commercial availability, and optical properties.We usually manufacture crystalline resonators by usingdiamond turning and a polishing process. They are ac-complished either with a motion-controlled machine ormanually [19]. A hard diamond tool enables us to fab-

[email protected]

ricate WGM structures, but we have to employ subse-quent manual polishing with diamond slurry to improvethe Q-factor of the microresonator. Precision machiningreadily overcomes the geometrical limitation of the man-ual process; therefore, precise computer-controlled ma-chining has achieved the pre-designed mode structuresneeded when fabricating single-mode [19, 20] and dis-persion engineered resonators to generate broadband mi-croresonator frequency combs [21–23]. However, a sig-nificant challenge remains because we need to employadditional hand polishing after the diamond turning pro-cess due to the relatively low Q of 106∼107 at best thatwe obtain when using machining alone [19, 21, 22, 24].The additional polishing improves the Q; however, sub-sequent polishing deforms the precisely fabricated struc-tures despite the engineered dispersion realized by theprogrammed motion of the lathe [19].

In this article, we describe an ultrahigh-Q crys-talline microresonator fabrication technique that em-ploys computer-controlled ultraprecision machining. Themeasured Q of the MgF2 crystalline resonator reaches1.4×108, which is the highest value yet obtained withouta subsequent polishing process. In addition, we achieveda comparably high-Q in CaF2 crystalline material. Toobtain the ultrahigh-Q, we addressed the single-crystalcutting condition by undertaking an orthogonal cuttingexperiment, which revealed the critical depths of cutsfor different cutting directions. Also, a precise cylindri-cal turning experiment revealed the relationship betweencrystal anisotropy and surface quality after machiningand demonstrated the realization of nanometer-scale sur-face roughness with diamond turning alone. The resultswe obtained provide clear evidence that cutting param-eters that have been optimized for fluoride crystals leadto a significant reduction in surface roughness.

An automated ultra-precision machining technique iscompatible with dispersion engineering and a high-Q fac-tor, which is often restricted by the trade-off relation en-

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countered with conventional fabrication techniques. Weconfirmed that the dispersion of fabricated resonatorsagrees extremely well with the design, and this extendsthe potential of dispersion controllability in crystallineWGM microresonators. Furthermore, our approach,namely the reliable production of high-Q crystalline mi-croresonators, supports recent advances on the integra-tion of crystalline microresonators with photonic waveg-uides towards a wide range of future applications [25, 26].

II. INVESTIGATION OF CRITICAL DEPTH OFCUT

A. Definition of critical depth of cut

One important parameter that we need to know whenfabricating a single crystal is the critical depth of cut.It is defined by the depth of cut at which the transi-tion from ductile-mode to brittle-mode cutting is ob-served when machining single-crystal material [27]. Inthe ductile regime, a smooth crack-free surface can bemaintained when generating a continuous ribbon chip,and this approach is considered more suitable for opticalapplications thanks to its ultra-smooth surface. On theother hand, the surface in the brittle regime is rougherand contains cracks, hence it is generally inadequate foroptical applications. CaF2 and MgF2 crystals are hardand brittle materials, and have a crystal anisotropy, sothey are challenging to cut. These features make it dif-ficult to manufacture smooth optical elements with adesigned shape such as spherical lenses and optical mi-croresonators. In particular, high-Q microresonators re-quire an ultra-smooth surface with a surface roughness ofno more than a few nanometers. Thus, the critical depthof cut must be investigated before resonator fabricationif we are to cut the crystal in the ductile mode regime.

The cutting direction and plane of the crystal is ex-pressed with the Miller index, as shown in Fig. 1(a) (De-tails of the Miller index and crystallographic structurecan be found in Supplement 1). Figure 1(b) shows crys-tallographic images of CaF2 and MgF2 crystal. The dif-ference in crystal structure influences the critical depthof cut and the cutting conditions.

B. Orthogonal cutting experiment

To investigate the critical depth of cut, we performedan orthogonal cutting experiment on single-crystal MgF2.The experiment was carried out with an ultra-precisionmachining center (UVC-450C, TOSHIBA MACHINE),and a workpiece holder equipped with a dynamometerto detect the cutting force during the processing. As aworkpiece, we used a pre-polished single-crystal MgF2

substrate with a size of 38×13 mm and a thickness of1 mm, which was fixed on the workpiece holder with avacuum chuck as shown in Fig. 2(a). The cutting tool

was a single crystal diamond tool with a 0.2 mm nose ra-dius, a −20◦ rake angle, and a 10◦ clearance angle (De-tails of the single-crystal diamond tool are provided inSupplement 1). The cutting slope D/L, which gives thecutting depth to cutting length ratio, and the feed rate,were set at 1/500 and 20 mm/min, respectively, with anumerical control (NC) program [Fig. 2(b)]. The criticaldepth of cut, which is defined as the cutting depth atwhich 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 imageof the machined surface, where the black points indicatefractures or cracks on the surface.

Here, we tested two different crystal planes, (001) and(010), where we performed the cutting in every 30◦ rota-tional direction to investigate the critical depth of cut fordifferent crystal orientations. The direction of 0◦ was setat [100] and [001], respectively. It should be noted thatMgF2 has a complex rutile structure with a different crys-tal plane configuration from CaF2 [see Fig. 1(b)]; hencethe two orthogonal planes are selected for the test to re-veal the effect of crystal anisotropy. Figure 2(d) showsa schematic of the tensile stress model for single-crystalcutting, 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 theoriginal uncut surface). Even though the only differenceis the cutting direction (i.e., [001] and [100]), there is asignificant impact on the surface quality of the machinedregion due to the crystal anisotropy. We observed largebrittle fractures in the [100] direction, whereas overall the[001] direction exhibited smooth surfaces. Figures 2(f)and 2(g) show the variation in the critical depth of cutas a function of cutting direction on each plane. Onthe (001) plane, the critical depth variation was approxi-mately 120 nm, and the lower bound value was 86 nm inthe 270◦ direction ([010] direction). On the other hand,the variation with the (010) plane was more significantthan that with the (001) plane, and the lower bound alsodecreased (i.e., worsened). These considerable differencesin critical depth of cut are consistent with surface obser-vations, as shown in Fig. 2(e).

We can understand the experimental results as follows.The difference in critical depth of cut could be consid-ered to originate from the slip system and the cleavageplane since they are strongly related to the ductile-brittlemode transition. Cutting along the slip plane promotesductile-mode cutting (i.e., plastic deformation), whichcontributes to the large critical depth of cut. On theother hand, the cutting force against cleavage inducescrystal parting where brittle fractures are easily mani-fested. They are explained intuitively in Fig. 2(d).

The slip system and cleavage plane of single-crystalMgF2 are (110)[001] and (110), respectively; thereforethe influence of cutting on the (001) plane on the criticaldepth of cut is less susceptible to the cutting directionbecause the cutting on the (001) plane is always normal

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to both the slip system and the cleavage plane. Withthe (010) plane, however, we observed a large variationin the cutting direction, because the cutting periodicallyfollowed the same direction as the slip system (i.e., 0◦

and 180◦) as shown in Fig. 2(g). In contrast, the small-est (i.e., worst) critical depth of cut was obtained fordirections of 90◦ and 270◦. They are in a configurationwhere the cutting force is applied in a direction almostperpendicular to the cleavage plane (110), as explained inFig. 2(d) (right panel). As a result, a shallow depth of cutis 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 thanapproximately 50 nm to maintain ductile mode cutting.

III. ULTRA-PRECISION CYLINDRICALTURNING

A. Procedure of cylindrical turning

Although the orthogonal cutting experiment providesinformation on the critical depth of cut for specific direc-tions, the cutting direction continuously changes whencylindrical turning is performed to manufacture a crys-talline cylinder workpiece. Therefore, the optimum turn-ing parameters have to be investigated to achieve thesmooth surface needed for a high-Q microresonator. AMgF2 cylinder workpiece was prepared with an end-faceorientation of (001) because a z-cut (c-cut) resonator isused to avoid optical birefringence.

Cylindrical turning was performed using an ultra-precision aspheric surface machine (ULG-100E,TOSHIBA MACHINE), as shown in Fig. 3(a). AnMgF2 workpiece with a diameter of 6 mm was fixed toa brass jig, and then mounted on a vacuum chuck. Theultra-precision turning was conducted in the followingthree steps. Rough turning was initially undertakento form the desired diameter (here 3 mm). It shouldbe noted that this initial rough turning was performedin the brittle regime. Next, pre-finish cutting wasconducted to remove the large cracks that occurred inbrittle mode cutting with a removal thickness of 8 µm.Finally, finish cutting in the ductile mode completedthe ultra-precise turning under the following cuttingconditions: 500 min−1 rotation speed, 0.1 mm/min feedrate, 50 nm depth of cut, and 2 µm removal thickness(The cutting conditions in each step are detailed inSupplement 1). We can see that cracks deeper than10 µm that appeared during the rough turning could notbe removed with pre-finish and finish cutting. Althoughthe larger removal thickness results in lower productionefficiency, it allows the removal of deep unwanted cracks.

The depth of cut at the finish turning step was setat 50 nm based on the result of the orthogonal cuttingexperiment. 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 deter-

mine the effective cutting speed and cutting amount. Inparticular, previous studies have reported that the feedrate critically affects the quality of the machined surfaceas does the combination of the tool radius and depth ofcut [28–30]. These studies draw attention to the factthat a fast feed rate induces brittle mode cutting if thedepth of cut is kept below the critical value. Thus, wechose a slow feed rate (≤ 1 mm/min) when fabricating asmooth surface.

For the pre-finish and finish cutting, we used a singlecrystal diamond cutting tool with a 0.01 mm nose radius,a 0◦ rake angle, and a 10◦ clearance angle. In terms ofthe choice of the cutting tool, a smaller nose radius makesit possible to have a smaller contact area between tooland material during cylindrical turning, which helps toreduce any excess cutting force and leads to an improvedsurface quality. However, tools with a small nose radiusare more fragile, which gives them a short lifespan; hencein this work we use different tools for the rough turningand finish turning stages.

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

B. Surface roughness after cylindrical turning

The surface roughness after cylindrical turning wasmeasured using a scanning white-light interferometer(New View TM6200, Zygo) at 15◦ intervals from the ori-entation flat [100] defined as 0◦ [Fig. 3(e)]. Figure 3(f)and 3(g) show the cylindrical surface roughness with anend-face orientation of (001) of a MgF2 workpiece. Un-surprisingly, the surface roughness after the rough cut-ting exceeded 200 nm for the entire cylindrical surface,as shown with red dots in Fig. 3(f). The large rough-ness was caused by the brittle-regime cutting. In con-trast, the smoothness improved significantly after thefinish cutting, which was performed under the ductilecutting condition [blue dots in Fig. 3(f)]. The magnifiedplot on a linear scale is shown in Fig. 3(g). We confirmedthat the turning condition for final cutting enabled us toachieve a smooth surface. Specifically, we obtained anRMS roughness of below 2 nm at 18 observation points.The result also revealed an interesting feature of 90◦ peri-odicity, namely that specific observation points exhibiteda slightly larger RMS roughness of 7.8 nm on average.Periodicity can also be seen in the micrograph shown inFig. 3(b)-3(d); for instance, 135◦ exhibits a smoother ma-chined surface than that in the 180◦ direction [Fig. 3(c)and 3(d)]. This is evidence of the appearance of crystalanisotropy in MgF2 crystal, as observed in the orthogonalcutting experiment.

This periodicity can also be understood from the slipsystem and cleavage configurations shown in Fig. 2(d).The relatively rough surfaces can be explained in terms

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of specific directions where the excess cutting force actson the boundaries of cleavage planes. The 15◦ asymme-try is due to the rotation direction of the workpiece; theforce on the cleavage plane exerts stress only in the clock-wise direction (the opposite direction to workpiece rota-tion). There is nevertheless excellent surface integrity aslong as the cutting remains on a cylindrical surface wherethe cutting force circumvents the crystal anisotropy. (Adetailed discussion of the effect of crystal structure oncutting condition is presented in Supplement 1.)

The cylindrical turning described here is a pre-processin microresonator fabrication. It should be noted thatthe measured roughness is the result of the implosionof a machined cylinder, not the dimensional resonatorsurface. However, the results we obtained allow us topredict the surface of the resonator under the employedcutting conditions.

C. Fabrication and cleaning of microresonators

Crystalline microresonators are fabricated using thesame ultra-precision machine as that used in the cylin-drical turning experiment (ULG-100E, TOSHIBA MA-CHINE). The resonator diameter is determined after thefinish cutting. It should be noted that the diameter canbe precisely controlled by measuring the diameter andundertaking additional turning prior to microresonatorfabrication. The resonator shape is carefully fabricatedby feeding a diamond tool under the critical cuttingdepth. Here, the turning motion is fully and automati-cally controlled by the NC program. The manufacturingprocedure is shown in Fig. 4, and the total fabricationtime is about ten hours. We determined the cuttingcondition based on the cylindrical turning experimentand employed a finish turning condition at the resonatorshaping step (Also see Supplement 1).

Once we had completed the fabrication, we cleaned themicroresonator to remove lubricant and small chips at-tached to the surface. We emphasize that proper cleaningis essential for obtaining a high-Q as well as optimizedcutting conditions. Since we used water-soluble oil asa machining lubricant during the ultra-precision turning,we first used acetone solution to clean the microresonatorsurface and remove the remaining lubricant. A lens clean-ing tissue is usually used to wipe the resonator, but thereis the possibility that it might scratch or damage theresonator surface, which could be a critical problem interms of degrading the Q-factor. Alternatively, to avoidunwanted damage on the resonator, we can employ ul-trasonic cleaning. (The cleaning method is detailed inSupplement 1.) The use of an ultrasonic cleaner enablesus to clean the surface without touching or rubbing it. Itis also a great advantage for fully automated fabricationcombined with ultra-precision turning.

IV. Q-FACTOR AND DISPERSION OFFABRICATED MICRORESONATORS

The Q-factor and dispersion were measured in crys-talline microresonators fabricated with the procedure de-scribed above. We fabricated an MgF2 WGM resonatorand a CaF2 WGM resonator with the same curvatureradii of 36 µm. The diameters were 508 µm for theMgF2 resonator and 512 µm for CaF2 resonator. Fig-ures 5(a)-5(c) show SEM images of the fabricated MgF2

microresonator. Although the two resonators were fabri-cated with the same motion program and cutting condi-tions, their diameters differ slightly as a result of differ-ences in the positioning accuracy in the cylindrical turn-ing process and the original diameter of the cylindricalworkpiece. As described in the previous section, the ad-ditional measurement of the workpiece dimension enablesus to achieve the practical precise control of the diameterat the sub-micrometer level.

Figures 5(d) and 5(e) show the measured transmissionspectra of the fabricated microresonators. We launchedlight from a frequency tunable laser source, which wascoupled into the resonator via a tapered optical silicafiber. A polarization controller was used to adjust thepolarization before the light coupling. The transmit-ted light was monitored with a high-speed photodetec-tor and oscilloscope, where we used a calibrated fiberMach-Zehnder interferometer as the frequency reference.The full-width at half-maximum (FWHM) linewidth ofthe MgF2 resonator was 1.40 MHz, which corresponds toa loaded Q = 1.39 × 108 at a wavelength of 1545 nm.Also, the CaF2 resonator had a linewidth of 2.53 MHzat 1546 nm, corresponding to Q = 7.67 × 107. We mea-sured the Q-factor in different wavelengths regions andrecorded comparably high-Q values for other resonantmodes. The obtained Q, which exceeded 100 million,is the highest value recorded in a crystalline WGM mi-croresonator fabricated solely by ultra-precision machin-ing without a conventional polishing process. In otherwords, our approach has overcome the manufacturinglimitation, namely the need for skilled manual techniquesthroughout the fabrication process to obtain ultrahigh-Qcrystalline microresonators.

Figures 5(f) and 5(g) show the measured integrateddispersion, defined as Dint = ωµ−ω0−D1µ = D2µ

2/2 +D3µ

3/6 + · · · , where ωµ/2π is the resonance frequencyof the µ-th mode (µ = 0 designates the center mode),D1/2π is the equidistant free-spectral range (FSR),D2/2π is the second-order dispersion linked to group ve-locity dispersion, and the above D3/2π terms correspondto higher-order dispersion. The microresonator disper-sion measurement was performed assisted by a fiber lasercomb and a wavelength meter [31]. The measured dis-persion agrees well with the theoretical dispersion cal-culated with the finite element method (FEM) by using(COMSOL Multiphysics), and these results indicate thatultra-precision turning enables us to obtain the designedresonator shape. Measured results and the fabrication

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flow of a WGM crystalline resonator with a sophisticatedcross-sectional shape, i.e., a triangular shape, are pro-vided in Supplement 1.

In fact, we realized a Kerr frequency comb in thefabricated MgF2 crystalline microresonator by machin-ing alone where the dispersion was designed to gener-ate octave-wide parametric oscillation [32]. Hence, fullycomputer-controlled machining could be a great advan-tage as regards extending the potential of crystalline mi-croresonators for optical frequency comb generation fromthe standpoint of dispersion engineering.

V. DISCUSSION

A. Outlook: Towards further Q-factorimprovement

The total (loaded) Q-factor is determined by foldingthe several loss contributions as,

Q−1tot = Q−1

mat +Q−1surf +Q−1

scatt +Q−1rad +Q−1

ext, (1)

where Q−1mat is determined by material absorption, Q−1

surf

andQ−1scatt are determined by surface absorption and scat-

tering loss, respectively. The radiation (tunneling) lossis given by Q−1

rad, and Q−1ext is related to the coupling rate

between the resonator and the external waveguide (e.g.,tapered fiber, prism). Since the total Q-factor can readilyreach 109 by polishing in fluoride crystalline resonators,the effect of Q−1

mat, Q−1rad, and Q−1

ext should be negligible.

Q−1surf is one possible reason for this, whereas single crys-

tals such as MgF2 and CaF2 inhibit the diffusion of waterinto the crystal lattice, which makes Q−1

surf negligible inour case [2].

Then, we highlight Q−1scatt as a fundamental limitation

of ultra-precision machining. Since the surface rough-ness of the polished resonator reaches a sub-nanometerscale [2], it is reasonable to consider that the surface scat-tering limits Q in diamond turning (Fig. 3(g) shows a sur-face roughness of a few nanometer scale). The maximumQ-factor as regards surface roughness can be estimatedas [2, 33]:

Qscatt ≈3λ3R

8nπ2B2σ2(2)

where R is the resonator radius, n is the refractive index,B is the correlation length, and σ is the surface roughness(RMS). The maximum Q-factor versus surface roughnessand correlation length of MgF2 resonator is plotted inFig. 6. Theoretically achievable values for ultra-precisionmachining correspond to Q values of 107−109, which areconsistent with measured Q-factors. The plot indicatesthat the roughness of the machined surface could limitthe present Q. A possible way to improve the surfaceroughness and correlation length is to optimize the cut-ting parameters, for example by using a smaller depth of

cut and a lower feed rate. Specifically, ideal conditionsare believed to realize an ultrasmooth surface for the en-tire cylindrical position, and consequently eliminate theeffect of crystal anisotropy, as seen in Fig. 3(g).

In addition, we should take the effect of subsurfacedamage into account since it causes the degeneration ofthe inner structure of the material. Subsurface damageoccurs when machining a single crystal, and so it hasbeen intensively studied in the field of micromachiningand material science [34]. Such underlayer damage coulddegrade Q in the same way as surface scattering; there-fore, we investigated the surface and subsurface damageby using a SEM and a transmission electron microscope(TEM) in comparison with the results for polishing. As aresult, we found that the damaged subsurface layers werearound several tens of nanometers with single-crystal pre-cise turning. (Details and results are presented in Sup-plement 1.) It is generally known that the subsurfacedamage mechanism strongly depends on the crystal prop-erties and cutting condition, and the efforts to reduce thesubsurface damage are described elsewhere [35]. The re-duction of underlayer damage could also help to improvethe present Q.

VI. CONCLUSION

In conclusion, we demonstrated the fabrication of anultrahigh-Q crystalline microresonator by using ultra-precision turning alone. For the first time, we achieveda Q value exceeding 100 million without polishing andthereby managed both an ultrahigh-Q and dispersion en-gineering simultaneously. We revealed the critical depthof cut needed to sustain ductile mode cutting, and thisinformation contributes significantly to reducing surfaceroughness. Moreover, we proposed an optimal cuttingcondition for cylindrical turning for realizing an ultra-smooth surface throughout an entire cylindrical surface.This result provides the path towards the fabrication of ahigh-Q microresonator without the need for skilled man-ual work. Furthermore, we discussed the possibility offurther improving the Q-factor from the standpoint ofthe theoretical limitation imposed by surface roughness.The described fabrication and cleaning procedure can beapplied to various single-crystal materials and will raisethe potential for realizing crystalline microresonators.

FUNDING INFORMATION

Japan Society for the Promotion of Science (JSPS)(JP18J21797) Grant-in-Aid for JSPS Fellow; JSPS KAK-ENHI (JP18K19036); Strategic Information and Com-munications RD Promotion Programme (191603001)from MIC.

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ACKNOWLEDGMENTS

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

See Supplement 1 for supporting content.

DISCLOSURES

The authors declare no conflicts of interest.

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[30] S. Azami, H. Kudo, Y. Mizumoto, T. Tanabe, J. Yan,and Y. Kakinuma, Precision Engineering 40, 172 (2015).

[31] S. Fujii and T. Tanabe, Nanophotonics 9, 1087 (2020).[32] S. Fujii, S. Tanaka, M. Fuchida, H. Amano, Y. Hayama,

R. Suzuki, Y. Kakinuma, and T. Tanabe, Opt. Lett. 44,3146 (2019).

[33] M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko,J. Opt. Soc. Am. B 17, 1051 (2000).

[34] J. Yan, T. Asami, H. Harada, and T. Kuriyagawa, Pre-cision Engineering 33, 378 (2009).

[35] Y. Mizumoto, H. Amano, H. Kangawa, K. Harano,H. Sumiya, and Y. Kakinuma, Precision Engineering52, 73 (2018).

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7

a

b

c

a

b

c c

a

b

[100]

(100)(010)

[010](001)

[001]

(111)

[111]

(110)

(002)

[110][110]

(001)

(101)

(111)

(011)

(110)

(111)

(111)

(111)

ab

c(100) (010)

(101)

(110)

(100)

(010)

(001)

(011)

(110)

CaF2 MgF2

(a)

(b)

FIG. 1. (a) Examples of the Miller index, where (h k l) and[h k l] indicate the corresponding plane and direction, respec-tively. (b) Crystallographic images of CaF2 and MgF2 crystal.In contrast to the cubic symmetry system of CaF2, MgF2 ischaracterized by a more complex rutile structure.

Page 8: Ultrahigh-Q crystalline microresonator fabrication with ...

8

Single crystal substrate

Diamond tool Ductile modeBrittle mode

Crack (black point)

Top view Cutting direction

(a) (b)

(c) (d)

0 30 60 90 120 150 180 210 240 270 300 330Cutting direction (deg)

0

100

200

300

400

500

600

Crit

ial d

epth

of c

ut (n

m)

Cutting direction [hkl](f)(e)[100][010] [010][100]

Cutting directionSubstrate

Cutting slope D/L

L D Groove

Side viewDiamond tool

20 µm

20 µm

(010)

(010)

[001]

[100]

(001)

(110)

[001]Cutting direction

Slip system Cleavage

(110)Brittle fracture

90°

180°

270°

30°60° 120°

150°

330°300°

210°240°

Vacuume chuck

0 30 60 90 120 150 180 210 240 270 300 330Cutting direction (deg)

Cutting direction [hkl]

0

100

200

300

400

500

600

Crit

ial d

epth

of c

ut (n

m)

(g)[001][100] [100][001]

(010)

FIG. 2. (a) Experimental setup for orthogonal cutting to investigate the critical depth of cut in MgF2 single crystal. (b)Schematic illustration of the diamond tool to substrate motion. The diamond tool cut a V-shaped groove with a slope of D/L.(c) Reconstructed image of machined surface using scanning white light interferometer. The critical depth of cut is given bythe depth where the first brittle fracture appeared (black point in (c)). (d) Cutting along the slip plane (110)[001] promotesductile mode cutting (left panel). Cleavage and subsequent brittle fractures are induced by the cutting force against cleavageplane (right panel). (e) Scanning electron micrographs showing machined surfaces of a (010) plane with [001] direction (upperpanel) and [100] direction (lower panel). The yellow shaded area corresponds to the original uncut surfaces. The differencebetween the machined surface conditions is attributed to the crystal anisotropy of the MgF2 crystal. (f), (g) The measuredcritical 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. To perform ductile mode cutting, the depth ofcut must be kept below the critical depth of cut.

Page 9: Ultrahigh-Q crystalline microresonator fabrication with ...

9

(a)

(e)

Diamond tool

Lubricant

Cylindrical workpiece

200 µm

50 µm

50 µm

(b) 135°

180°

Rough turning region

Finish turning region

(d)

90°

180°

270°

45°

135°

315°

175°

Orientation flat

Cylindrical workpieceEndface orientation

Observation point (15° intervals) 030

0

120

150180

210

240

270

300

33010

5

Surfa

ce ro

ughn

ess

(nm

)

(f) (g)0

30

0

120

150180

210

240

270

300

330

100

1000

10Su

rface

roug

hnes

s (n

m) [100]

[010]

[100]

[010]

[100]

[010]

[100]

[010]

(c)

Finish turning

Rough turning

FIG. 3. Ultra-precise cylindrical turning and surface roughness measurement of MgF2 single crystal. (a) Experimental setupof a ultra-precision lathe for the cylindrical turning of a single crystal. (b) Micrograph showing a machined surface, whereclear boundaries are observed between the rough turning and finish turning regions. The horizontal boundary in the roughturning region is evidence of the dependence of the cutting direction on the crystal anisotropy in MgF2 single crystal. (c), (d)Magnified views of the finish turning region in 135◦ and 180◦, respectively. The machined surface of 135◦ is smoother thanthat of 180◦, which agrees with the result of the surface roughness (RMS) measurement. (e) Schematic of surface roughnessmeasurement. The yellow line and dot correspond to orientation flat [100] with an endface orientation of (001). The surfaceroughness 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 dueto crystal anisotropy. (g) Magnified plot on linear scale of finish turning in (f).

1. Rough turning 2. (Pre-) Finish turning

3. Pre-shaping I4. Pre-shaping II5. Final shaping

500 µm

FIG. 4. Fabrication flow of WGM microresonator when us-ing ultra-precision turning. First, a rough turning determinesthe approximate diameter of resonator. Next, pre-finish andfinish turning with ductile mode cutting are used to realize acylindrical surface that is smooth and entirely crack-free. Fi-nally, fully-programmed shaping steps are performed to fab-ricate the designed resonator structure.

Page 10: Ultrahigh-Q crystalline microresonator fabrication with ...

10

(a)

100 um

-10 -8 -6 -4 -2 0 2 4 6 8 10Detuning (MHz)

0.2

0.4

0.6

0.8

1

Tran

smitt

ance

-20 -15 -10 -5 0 5 10 15 20Detuning (MHz)

0

0.2

0.4

0.6

0.8

1

Tran

smitt

ance

180 185 190 195 200 205Frequency (THz)

-1.5

-1

-0.5

0

0.5

1

1.5

180 185 190 195 200 205Frequency (THz)

-1.5

-1

-0.5

0

0.5

1

1.5

Din

t/2π

(GH

z)

Din

t/2π

(GH

z)

Q = 1.39×108 Q = 7.67×107

MgF2 crystalline CaF2 crystalline

D1/2π = 136.9 (GHz)D2/2π = -84.7 (kHz)D3/2π = -1.0 (kHz)

D1/2π = 129.8 (GHz)D2/2π = -269 (kHz)

10 um20 um

(b) (c)

(d) (e)

(f) (g)

FIG. 5. Q-factor and dispersion measurement of crystalline microresonators fabricated by ultra-precise turning. (a) SEM imageof a fabricated MgF2 microresonator with a diameter of 508 µm and a curvature radius of 36 µm. (b), (c) Magnified views ofthe resonator. (d) Normalized transmission spectra of the fabricated MgF2 microresonator. The Lorentzian fitting (red line)yield loaded a Q value of 139 million. (e) Normalized transmission spectra of the fabricated CaF2 microresonator. The fittingcurves give a loaded Q for the fundamental mode of 76.7 million. (f), (g) Measured dispersion Dint versus frequency. The redcurve indicates the calculated dispersion of the fundamental TM mode, which agrees well with the experimental result.

0.11101001000

100

10

1

Surface roughness σ (nm)

Cor

rela

tion

leng

th B

(nm

)

Q-factor lim

itation by surface roughness

1015

1014

1013

1012

1011

1010

109

108

107

106

105

104

103

102

101

1000

Q-factor limitation by abresive polishing

Q = 10⁹Q = 10⁸

Measured Q by abresive polishing

Measured Q by ultra-precision machining

Q = 10⁷

Conventionalmachining

Ultra-presicionmachining(this work)

FIG. 6. Q-factor limitation caused by surface scattering lossin MgF2 crystalline microresonators derived from Eq. (2)(λ=1550 nm and R = 250 µm). The dashed contours showthe estimated value from our measurement and previous stud-ies [2, 33].


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