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Hsieh group in 2014 was operated with a diffractive- surface (Fig. 1). To this end, it is convenient to inject optic based stop-start scheme.22 The incidence was made the circularly polarized beam into the rotary optics along oblique by diffracting the original ray with a phase mask the rotation axis; see the right side of the schematic and refocusing the offset ray on the sample surface with a of Fig. 2(a). The convenience is due to the rotational reflective objective. The SHG signal was recorded while symmetry of the ; the polarization rotating the detector and the phase mask step by step. (Jones) vector for the circular polarization is stationary The system was renovated to solve the power-fluctuation seen from both the laboratory frame and the frame fixed induced low-frequency-noise issue, by using high-speed on the rotary optics. As a result, whatever the rota- (∼4 Hz) rotating phase mask based optics and position tion angle of the rotary optics may be, the polarization sensitive photodetector.23 Subsequently, the Torchinsky vector of the circularly-polarized injection is stationary group24 developed reflective optics based rotating in- when seen from the frame fixed to the rotary optics. cident beam generator for the wavelength independent measurement. However, this high speed rotation design (a) por s Stationary polarization still requires a precise alignment process to make sure the seen from Focus Polarization co-rotating frame high speed rotating optics is appropriately positioned. optics controller Offset circular optics Here, we present another design to perform the p& s por s incidence-plane rotation SHG measurement. The under- Coll. Polarization Sample Detector lying design concept is to have a compact, stable and optics selector versatile system. Our setup is based on Fresnel rhomb Rotary optics optics, a PMT detector, and a femtosecond laser source composed of optical fibers doped with ytterbium (Yb). (b) The Fresnel rhomb optics is compact, wavelength in- Lens por s linear dependent, and misalignment-tolerant in principle. Al- circular FM though we applied the stop-start-type stepping-motor Pol λ /2 method, the low frequency noise is under control with DM Pol CF Fresnel Sample the hour-long stability of the compact fiber-laser source. PMT rhomb This paper is organized as follows. After the present Rotary optics introduction (Sec. I), we describe the optics to rotate the CCD To lock-in amplifier incidence plane in Sec. II. In Sec. III, we sketch the home- made fiber laser source and its stability. In Sec. IV, we (c) describe the assembly and alignment of the entire system. Lens Pol λ In Sec. V, we show the SHG signal data of a gold mirror /2/2 Frensnel and check the operation of the SHG optics (Sec. V A), Sample rhomb RS1 and also present the SHG pattern of a few-layer WSe2, from which we infer to the crystal orientation and degree CCD of local effects (Sec. VB). The summary and outlook are RS2 provided in Sec. VI. In Appendixes A and B, we estimate DM + Pol +PMT the intensity of the faint SHG signal of gold, the order of which corresponds to 0.1 photon per pulse (Appendix A), and subsequently describe the SHG tensor analysis at the FIG. 2. Rotary optics. (a) Essential features in the rotary op- oblique incidence (Appendix B). tics. (b) Optics adopted in the present setup. (c) A snapshot of the rotary optics (Multimedia view).

II. THE ROTARY OPTICS The essential features in the rotary optics are illus- trated in the dashed rectangular region in Fig. 2(a). The In this section, we first sketch the essential features to injected beam is first shifted from the roto-axis by the off- rotate the incidence plane and subsequently describe the set optics. Subsequently, the polarization is set to either specific optics that we have adopted. p or s by the polarization controller. Finally, through In the incidence-plane rotation method, there are some the focus optics, the beam is directed to the sample optical components that rotate about the axis normal to surface at an oblique angle. The SHG signal from the the sample surface [Fig. 2(a)]. The role of the rotary surface is directed to the detector through the collimat- optics is to (1) rotate the incidence plane, (2) set the po- ing optics. Before reaching the detector, the signal goes larization of the incident beam, (3) focus the beam on the through the polarization selector that filtrates the p- or sample, (4) filtrate the SHG signal and its polarization s-polarization. and (5) direct the filtered signal to the detector. Figure 2(b) shows the optics adopted in our setup. During the turn of the rotary optics, the direction of For the sake of compactness, we used a Fresnel rhomb the incident polarization also has to follow the turn and (FR600QM, Thorlabs) at the entrance of the rotary op- maintain the p- or s-polarized incidence on the sample tics. The Fresnel rhomb offsets the injected beam for 3

Compressor 16 mm and simultaneously transforms the circular polar- (a) (c) FWHM (d) ization into linear over a wide range of wavelength. Note TrG 425 fs 1000 lines/mm that the beam ejected from the Fresnel rhomb need not be parallel to the roto-axis, and this degree of freedom Intensity (arb. units) (arb. Intensity 10101018 1026 -10 1 increases the tolerance of the system against alignment Isolator BPF Mirror Wavelength (nm) Delay (ps) error. PBS 8 nJ, 10 MHz (b) The subsequent polarization controller can be a half λ/4 1030 ± 5 nm λ/4 wave plate (λ/2) that rotates the linear polarization. In λ/2 WDM 1064 nm/976 nm our setup, we also added a nanoparticle-film-type polar- Collimator izer (Pol) to ensure the linear polarization. SMF For the focus and collimating optics, we used one large 20 cm LD plano-convex lens of 50 mm in diameter and a focal length 976 nm ◦ Yb:SMF of 38 mm, which sets the incidence angle to θ = 23 . In- Oscillator 20 cm stead of attaching the lens on the rotator, we fixed it (e) on the optical table, thanks to the rotational symme- 890 try of the lens about the lens axis. As a result of the 885 fixing, there is a gap between the lens and the rotary op- 880 Average: 886.1 µW ±σ = ±1.4 µW tics. This gap can be utilized for the purpose to view the 5 sample through a charge-coupled-device (CCD) camera (µW) SHG intensity 0 (Sec. IV). Moreover, this gap opens a pathway for time- 0 1 2 3 4 resolved measurements:26,27 We can introduce a beam Time (hour) splitter in the gap and direct the second beam to the FIG. 3. Femtosecond laser source. (a) Schematic of the os- sample at the normal incidence without blocking the in- cillator and compressor. (b) Snapshot of the oscillator and cident and SHG paths; here, the splitter can be held by LD. (c, d) Spectrum (c) and auto-correlation trace (d) of the a large transparent window fixed to the optical table. pulse after the compressor. The trace in (d) is overlaid with a We adopted a PMT (PMM01, Thorlabs) for the de- Gaussian (dashed curve) with FWHM of 425 fs. (e) Stability tector. The detector is protected from the strong excita- of the SHG of a BBO crystal. The error bar is ±σ about the tion beam by a dichroic mirror (DM) and a color filter average indicated by a dotted horizontal line. (CF), and a Glan-Taylor-type cubic was added to select the polarization of the SHG. The beam-induced signal is modulated in ∼2 kHz by a mechanical chop- Figure 3(a) shows the configuration of the ANDi laser. per inserted in the injection beamline (not shown in the A ∼20-m-long ring cavity is compacted on a board of 20 schematic) and the modulated current from the PMT is × 20cm2 as shown in Fig. 3(b). The Yb-doped single read with a lock-in amplifier. mode gain fiber (Yb:SMF) spliced into the single mode The rotary optics consists of two stepping-motor ro- fiber of a core diameter of 6 µm is excited by a 976-nm tary stages, RS1 and RS2 (OSMS-60YAW and OSMS- wavelength laser diode (LD; BL976-SAG300, Thorlabs) 120YAW, Sigma Koki) as shown in Fig. 2(c). RS1 holds through a wavelength division multiplex (WDM). In the a cage system on which the optical components to rotate 30-cm-long free space between two collimators, there are the incident beam are attached, and RS2 is for the SHG two quarter wave plates (λ/4), a half wave plate (λ/2) receiving optics including the PMT detector. The two and a band pass filter (BPF) of 1030 ± 5 nm. The tilt of rotary stages are computor controlled and co-rotate dur- BPF and the azimuth angle of each of the wave plates are ing the data acquisition. The rotary optics including the adjusted for the cavity to be mode locked. An isolator 2 sample stage occupied an area of 50 × 100 cm . and a polarizing beam splitter (PBS) are also inserted in the free space; the former ensures the pulse to circu- late the ring in one-way and the latter extracts a fraction III. THE FIBER LASER SOURCE of the circulating pulse from the ring cavity. The ex- tracted laser pulse from the cavity passes another isolator Optical fibers can be wound into a spool and there- that protects the oscillator from unwanted retro-reflected fore using them in a laser cavity is suited for designing beam. Subsequently, the pulse passes through a pair a compact laser source. Besides, fiber lasers are stabe of transmission gratings (TrG’s) with 1000 grooves/mm due to the all-solid nature and also cost effective ow- that compresses the pulse. ing to the economics scale of the telecommunications in- To optimize the mode-locked state of the cavity for the dustry.28 For the SHG measurement system, we adopted SHG measurement, we directed the beam after the com- an all-normal- (ANDi) femtosecond fiber laser pressor into a nonlinear crystal β-BaB2O4 (BBO), mon- source.29 As will be shown in Sec. V, the pulse with the itored the intensity of the SHG signal generated there- energy as large 8 nJ obtained directly at 10 MHz was from, and searched for the state where the signal inten- strong and frequent enough to generate detectable SHG sity was maximized. The optimal state to our best effort signals of gold and thin WSe2 samples. was found when the 976-nm input was ∼400 mW, or 4 around the maximum attained by the LD, and when the ping motor rotary stage RS1, and also detach the optical tilt of BPF was ∼9◦; the tilt can be seen in Fig. 3(b). components including the PMT detector from RS2. At The output from the cavity was 80 mW and the rep- this point, we check the co-axial alignment of RS1 and etition frequency was 10 MHz, giving 8 nJ per pulse. RS2. Then, by using some irises, we let the injection Figure 3(c) shows the spectrum of the pulse. The spec- beam path through the roto-centers of the two stepping trum was centered at 1018 nm and the bandwidth was motors, RS1 and RS2. This beam path becomes the roto- around 15 nm. The center wavelength was shifted from axis. For the purpose of alignment, we also set another the nominal value of 1030 nm for the BPF due to the laser beam that counter-propagates along the roto-axis. ∼9◦ tilt. Figure 3(d) shows the auto-correlation trace of Next, we attach a gold mirror on the sample stage and the pulse after the compressor; overlaid on the trace is adjust its orientation to retro-reflect the injected beam the Gaussian function whose full width at half the max- so that the mirror surface becomes perpendicular to the imum (FWHM) was 425 fs, which corresponds to the roto-axis. The gold mirror will serve as a reference sam- pulse FWHM of 301 fs when the temporal profile of the ple for checking the alignment of the SHG setup; see pulse is assumed to be a Gaussian. Because of the rather Sec. V. high input of ∼400 mW, the mode-locked pulse was pre- Then, we translate the plano-convex lens into the opti- sumably chirped nonlinearly during the round trip in the cal path. The tilt and position of the lens have to be ad- cavity, and therefore, the compression of the pulse to the justed so that the lens axis becomes co-axial to the roto- transform limit was difficult to be achieved; we note that axis. To adjust the tilt, we retract the gold mirror and the estimated limit of the auto-correlation FWHM was illuminate the planer side of the lens with the counter- 203 fs for the spectrum shown in Fig. 3(c). At input pow- propagating beam that has been aligned co-axial to the ers lower than 400 mW, there were mode-locked states prinicipal axis. We adjust the tilt until the faint reflec- that resulted in sub-300-fs pulses although the SHG sig- tion from the planer surface of the lens is retro-reflected. nal intensity from BBO was weaker than the state found To adjust the position, we re-insert the gold mirror and at 400 mW. adjust the position of the lens until the reflection from To show the power stability, we monitored the in- the gold mirror matches the injection beam path. tensity of the SHG signal generated by the BBO crys- Next, we attach the incidence beam optics, namely, the tal. The intensity monitored for 4 hours was averaged Fresnel rhomb, half wave plate, and thin-film polarizer, as 886.1 µW with one standard deviation σ = 1.4 µW on the cage held by the rotary stage RS1. We also attach [Fig. 3(e)]; namely, the power fluctuation was less than a small power meter on the cage and monitor the power of 0.2 %. The low fluctuation helps to balance the vulnera- the offset and polarization-controlled beam while RS1 is bility of the minute-long stop-start-type measurement to in motion. We adjust the quarter wave plate in the injec- low-frequency noise. tion beamline [located outside the area seen in Fig. 2(c)] until the variation of the monitored power during the ro- tation of RS1 becomes less than ±1 %. This procedure is IV. ASSEMBLY AND ALIGNMENT OF THE SYSTEM to ensure the circular polarization of the injection beam. After setting the circular polarization, we remove the The SHG detection system consists of the rotary optics small power meter from the cage and direct the beam and laser source. The entire system including the sample on the sample surface at an oblique angle. There is a stage can be packed in the table area of 1 × 1 m2. flip mirror (FM) in the gap between the lens and the There are four essential points when assembling and rotary optics; the FM opens a path to view the sample aligning the system: (1) The beam injected into the ro- through the CCD camera [Figs. 2(b) and 2(c)]. Note that tary optics has to be highly circularly polarized. (2) The the incidence beam spot on the sample can be viewed injection beam has to be co-axial to the roto-axis of the with the CCD because of its sensitivity in the infra-reds. rotary optics. (3) The incident beam has to be directed The beam spot generally draws a circle as the incidence to the point on the sample surface where the roto-axis plane is rotated. We shift the sample along the roto-axis intersects. (4) The sample surface has to be perpendicu- until the beam spot stays still on a point. This point lar to the roto-axis. Once the requirements (2) to (4) are is identified as the intersection of the roto-axis and the met, the SHG signal will have a fixed trajectory when sample surface. seen from the frame on the rotary optics. In our setup, Finally, we flip out the FM and attach the SHG receiv- the rotary optics consists of two stepping-motor rotary ing optics on the rotary stage RS2. stages and one plano-convex lens, as explained in Sec. II. Therefore, aligning these three separate parts to form a firm roto-axis is the prerequisite to the four criteria men- V. DEMONSTRATION OF THE SHG MEASUREMENT tioned above. Here, we describe some of the practical procedures to A. The SHG of a gold mirror as a reference assemble and align the SHG measurement system. First, we retract the plano-convex lens facing the sam- When the SHG measurement system is perfectly ple stage, remove the optics on the cage held by the step- aligned, the SHG signal from the gold mirror would be 5 constant over the 360◦ rotation of the incidence plane through an optical microscope. The area of ∼100 × because the isotropic polycrystalline mirror surface ef- 50 µm2 was identified to have a few mono-layers; see fectively creates rotational isotropy characterized by the the arrows in the upper image of Fig. 5(a) indicating the point group symmetry of C∞v. The degree of isotropy of mono-, few- and thick-layer areas. After the characteri- the SHG signal of the gold mirror, thus, can serve as the zation by using the microscope, the sample was mounted measure of the alignment accuracy. on the sample stage. The lower image of Fig. 5(a) is the mounted sample seen via the CCD camera. We adjusted the sample position so that the incidence beam spot stayed on the few-layer area while the rotary stage was in motion. The SHG measurement was performed at room temperature, and the incident fluence was 1 mJ/cm2 as in the gold mirror measurement. The incidence polariza- tion was set to p (pin) and both the p- and s-polarized SHG (pout and sout) were recorded. Figure 5(b) shows the SHG profile of the few-layer WSe2 sample, and Fig. 5(c) is its polar plot. The six-fold SHG pattern is observed for all the polarization configu- rations. First, the symmetric pattern indicates that the beam spot stayed on the ∼100 × 50 µm2 area on the FIG. 4. SHG of a gold mirror. (a) Normalized SHG intensity sample surface during the turn of the rotary optics. as a function of the rotation angle ϕ. The error bar indicates The standard analysis of the SHG pattern relies on the one standard deviation σ = 0.016 of the 121 data points. (b) symmetry of the second-order nonlinear optical polariz- The polar plot of (a). (2) ability, χijk. When the crystal is symmetric with the D3h point-group operations, there are only four non-zero (2) 2 The measurement was performed at room tempera- components of χijk which are related to each other as ture. The beam spot diameter on the sample surface (2) (2) (2) (2) was 20 µm as estimated from the beam seen through the χyyy = −χyxx = −χxxy = −χxyx ≡ χ. (1) CCD camera. The incident beam was p-polarized with ◦ (2) 36 the incidence angle of 23 ; its wattage was 30 mW and Because χyyy is finite, yz is the mirror plane in the the spot area on the sample was 300 µm2 so that the flu- Cartesian indexing adopted in Eq. (1), and thus, y is ence was 1 mJ/cm2; the p-polarized SHG was detected. along the armchair direction while x is along the zig-zag The acquisition time for the 121 data points with the direction of the hexagonal WSe2 crystal; see the middle ◦ azimuth-angular step of 3 was ∼6 min. inset of Fig. 5. We also note that the z component of Figure 4(a) shows the SHG signal intensity as a func- (2) the polarization is not induced because χijk vanishes for tion of the rotation angle ϕ of the incidence plane. First i = z. The SHG signal intensity Iαβ for the αin-βout con- of all, the SHG intensity under the setup was intense figuration (α, β = p or s) can be calculated as functions enough to be detected by the PMT read with the lock- of the rotation angle ϕ − ϕ0 and incidence angle θ as (see in amplifier. The estimated wattage of the SHG was Fig. 1 and Appendix B) 0.5 pW, which is the order of 0.1 photon per pulse; for the 6 2 detection scheme of the small wattage, see Appendix A. Ipp = App cos θ sin {3(ϕ − ϕ0)}, (2) 4 2 Second, the intensity profile after the alignment was fairly Ips = Aps cos θ cos {3(ϕ − ϕ0)}, (3) isotropic in the radial plot [Fig. 4(b)]. The intensity is I = A cos2 θ sin2{3(ϕ − ϕ )}, (4) constant with one standard deviation σ less than 2 % of sp sp 0 0 2 the average value. The degree of the isotropy is the mea- Iss = Ass cos θ cos {3(ϕ − ϕ0)}. (5) sure of the integrity of the setup, and we routinely check Here, Aαβ is a proportionality constant which includes the isotropy before measuring the samples. the effects of local fields33,37 and that can, in principle, depend on the configuration of the light (Appendix B). In Fig. 5(b), we overlay the sinusoidal curves of Eqs. (2) B. The SHG of a few-layer WSe ◦ ◦ 2 to (5) with θ = 23 and ϕ0 = 21.6 . The curves nicely reproduce the angular pattern. The relative amplitudes As a demonstration, we measured the SHG profile of Aαβ/Ass derived from the fit are plotted in Fig. 5(d). a few-layer WSe2 sample. The atomically thin transition More than a factor of 2 difference was observed among metal dichalcogenides including WSe2 with D3h symme- the four amplitudes. The amplitude can depend on the try are known to exhibit strong SHG emissions.30–35 polarizatoin through the Fresnel and local-field effects The flakes of WSe2 were mechanically exfoliated on (Appendix B). Because the Fresnel effect, or the differ- polydimethylsiloxane (PDMS) and subsequently trans- ence in the reflection and transmission between p- and ◦ ferred to a 300-nm SiO2/Si substrate. The upper im- s-polarizations, is negligible at θ as small as 23 , we at- age of Fig. 5(a) shows one of the transferred flakes seen tribute the strong dependence to the local-field effects. In 6

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FIG. 5. SHG of a few-layers WSe2 film. (a) Sample seen through an optical microscope (upper image) and the identical sample on the sample stage seen through the CCD camera (lower image). Mono-, few-, and thick-layer areas on the sample are indicated by arrows. (b) SHG signal-intensity profiles for the four polarization configurations as functions of the rotation angle. Theoretical curves are overlaid on the data points. (c) Polar plot of (b). (d) Amplitude of the SHG pattern normalized to Ass.The dashed horizontal line is the expected level when there is no local-field effects. The middle inset shows the hexagonal crystal surface. The y axis is set along the arm-chair direction, which is in the mirror plane; here, the setting of the Cartsian (2) 2 coordinate system follows the indexing adopted for χijk tabulated in the textbook. other words, the relative amplitudes of the SHG signal tation and degree of local effects in the atomically thin among the four polarization configurations fingerprint sample (Fig. 5 and Appendix B). whether the degree of the local-field effects is substan- Because the sample is fixed in the laboratory frame, tial or not, the information of which is uniquely derived there are more degrees of freedom to subject the sam- using the off-normal incidence-plane rotation method. ple to various conditions than the case where the sam- ple rotates. In our setup, the working distance between the plano-convex objective lens and the sample is around 38 mm, and therefore, cryostats can readily be installed VI. SUMMARY AND OUTLOOK to cool down the sample. In addition, because we have set the objective lens fixed instead of rotating, there is In summary, we have constructed a compact and sta- also a gap between the lens and the rotary optics; this ble incidence-plane-rotating second-harmonics detector gap can be utilized for the pump-probe measurement, based on Fresnel-rhomb optics, a PMT detector, and as described in Sec. IV. Thus, our incidence-plane rota- a home-made Yb:fiber laser source. We described the tion detection design can be extended to investigating design principle and schematics (Sec. II and III), align- the samples under various conditions such as cryogenic, ment procedure (Sec. IV), and experimental realization strained, ultrafast non-equilibrium, and high magnetic (Sec. V) of the setup. The entire system fits into the op- fields. tical table area of 1 × 1 m2 and can produce harmonic signal with the intensity deviation less than ±0.2 % in 4 hours owing to the stability of the laser source (Fig. 3). The data acquisition for a gold reference sample took ACKNOWLEDGMENTS ∼300 s for 3◦-step measurement and provided <±2 % reliable data for the 0.5 pW output, which corresponds This work was conducted under the ISSP-CCES Col- to ∼0.1 photon per pulse at the 10-MHz repetition (Fig. 4 laborative Program and was supported by the Institute and Appendix A). The SHG data of an exfoliated WSe2 for Basic Science in Republic of Korea (Grant Num- film were also presented; the six-fold pattern and the bers IBS-R009-Y2 and IBS-R009-G2) and by JSPS KAK- relative strength for the four polarization configurations ENHI (Grant Numbers 17K18749, 18H01148, 19K22140 provided us with the information of the crystal orien- and 19KK0350). 7

Appendix A: Measuring the faint SHG wattage with the Cartesian coordinate basis (¯ex, ¯ey, ¯ez) as

Eα(ω)= E ¯eα = Ei ¯e . (B1) The SHG wattage of the gold mirror I2ω = 0.5 pW is ω ω X α i at the scale of 0.1 photon per pulse: one SHG photon of i=x,y,z 509 nm emitted at 10 MHz corresponds to 3.9 pW. The faint wattage was measured by utilizing an artificial SHG Here, α (= p or s) indicates the polarization, Eω is the i emission of a BBO crystal as described below. signed amplitude and Eα is the Cartesian coordinate. The PMT sensitivity (signal voltage per a fixed pho- The unit polarization vectors can be expanded with the ton count/s) is controlled by the gain control voltage: Cartesian coordinate basis as follows (see Fig. 1): When we increase it, the gain becomes approximately ¯ep ¯e ¯e ¯e 10 times higher for every 0.10 V. We first tune the gain ω = −cθcϕ x −cθsϕ y +sθ z, (B2) ¯es ¯e ¯e to measure the SHG of the gold mirror; namely, we set ω = sϕ x −cϕ y. (B3) the value large enough to detect faint SHG emission and small enough to prevent any unwanted electronic signal The components on the right-hand side of Eqs. (B2) and i i saturation. The lock-in amplifier modulated signal was (B3) are the direction cosines Ep/Eω and Es/Eω of the 0.45 mV under the gain control voltage 0.80 V. We then incident fields, respectively; see Eq. (B1). Thus, install the PMT (with same color filter to block the fun- x x damental beam) in front of the fundamental pulse laser Ep –cθcϕ Es sϕ  y    y   beam and BBO crystal to make the artificial SHG emis- Ep = Eω –cθsϕ , Es = Eω –cϕ .(B4) z z sion. The beam need not be focused since what we want Ep   sθ  Es   0  is the SHG emission of ∼µW at most. We control the in- tensity of the fundamental beam with strong continuous The incident field Eα(ω) induces the polarization P of revolving neutral-density (ND) filter of optical density 4 the medium in the substrate. The 2ω component of the until the lock-in amplifier modulated signal becomes the induced polarization through the second-order nonlinear same value of the gold film SHG, and measure the funda- optical process can be described as mental beam intensity with a conventional power meter. We then take out the ND filter to obtain the maximum P(2)(2ω)= P (2)¯e = ε χ(2) Ej Ek¯e , (B5) α X α,i i X 0 ijk α α i value of fundamental beam intensity with a power me- i ijk ter, and its resulting SHG wattage with the power meter covered with the same color filter. Note that the SHG where ε0 is the vacuum permittivity. induced by the maximum power is strong enough to be The intensity of the SHG signal emitted from the in- detected by the conventional power meter, and even vis- duced polarization of the medium can be described as ible as a green spot on a paper by naked eyes. ¯eβ P(2) 2 Here, we use the fact that the SHG wattage is pro- Iαβ = aαβ| 2ω · α (2ω)| . (B6) portional to the square of fundamental beam intensity: max max 2 β I2ω/I2ω = (Iω/Iω ) . The symbols and measured val- ¯e Here, 2ω (β = p or s) is the unit polarization vector ues are the followings: the faint fundamental wattage of the outgoing SHG field in the reflection geometry and Iω = 64 µW; the maximum fundamental beam wattage can be described as follows (see Fig. 1): max Iω = 40 mW; and the maximum-power-induced SHG max p wattage I2ω = 0.25 µW. Thus, we obtain I2ω = 0.5 pW, ¯e ¯e ¯e ¯e 2ω = cθcϕ x +cθsϕ y +sθ z, (B7) which is the faint SHG wattage of the gold mirror. ¯es ¯e ¯e 2ω = sϕ x −cϕ y. (B8)

Meanwhile, aαβ is the proportionality constant which in- Appendix B: SHG tensor analysis at an oblique incidence cludes the effects of local fields as well as the Fresnel correction33,37 and that can, in principle, depend on the polarization of the fields; thus, a is indexed with α and The standard analysis of the SHG pattern of non- β. By inserting Eqs. (B4), (B5), (B7) and (B8) into 2 centrosymmetric crystals relies on the symmetry argu- Eq. (B6), the SHG signal intensity Iαβ can be obtained ments of the second-order nonlinear optical polarizabil- as a function of θ, ϕ and |Eω|. (2) ity, χijk. Here, we calculate the SHG signal intensity as When the crystal is symmetric with respect to the D3h a function of the rotation angle ϕ − ϕ0 when the inci- point-group operations, the elements of χ(2) obey Eq. (1) dence angle θ is finite. We hereafter set ϕ = 0 without ijk 0 P(2) the loss of generality and also abbreviate sine and cosine and the coordinates of α (2ω) are simplified as functions as sin θ → sθ and cos θ → cθ, respectively. (2) x y The polarized incidence fields Ep(ω) and Es(ω) can be Pα,x –2EαEα  (2)  y 2 x 2 ¯ep ¯es = ε0χ (E ) – (E ) . described using the unit vectors ω and ω directed along Pα,y α α  (2) 0 the polarization (see Fig. 1) and further be expanded Pα,z    8

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