Optical sum-frequency generation in whispering gallery mode resonators
Dmitry V. Strekalov, Abijith S. Kowligy, Yu-Ping Huang, Prem Kumar
aa r X i v : . [ phy s i c s . op ti c s ] J u l Optical sum-frequency generation in whispering gallery mode resonators
Dmitry V. Strekalov
Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Drive, Pasadena, California 91109-8099
Abijith S. Kowligy, Yu-Ping Huang, and Prem Kumar
Center for Photonic Communication and Computing, EECS Department,Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3118, USA (Dated: September 24, 2018)We demonstrate sum-frequency generation in a nonlinear whispering gallery mode resonator be-tween a telecom wavelength and the Rb D2 line, achieved through natural phase matching. Due tothe strong optical field confinement and ultra high Q of the cavity, we achieve a 1000-fold enhance-ment in the conversion efficiency compared to existing waveguide-based devices. The experimentaldata are in agreement with the nonlinear dynamics and phase matching theory in the sphericalgeometry employed. The experimental and theoretical results point to a new platform to manipu-late the color and quantum states of light waves toward applications such as atomic memory basedquantum networking and logic operations with optical signals. Strong optical nonlinearities have been the foundationof many applications in classical and quantum optics.Recently, the burgeoning field of high- Q nonlinear micro-and nano-cavities [1] has emerged as a new chip-scalableplatform for photonic information processing, which re-quires very low ( < χ (2) -nonlinear LithiumNiobate microresonator, it has been shown that strong,noise-free interaction can be realized among single pho-tons, thereby uncovering pathways to unprecedented ap-plications such as optical transistors and deterministicquantum logic gates [6]. Such a realization has an in-herent advantage over resonant optical interactions withmatter systems due to its compact experimental setupand a room-temperature operation.All of these proposals place exacting criteria on the res-onators, requiring a high quality factor, small mode vol-ume, and good overlap between the interacting modes.While a small mode volume is available in photonic-crystal microcavities, multiply-resonant, high- Q cavitiesare difficult to fabricate [7]. Here, we present for thefirst time, naturally phase-matched sum-frequency gen-eration (SFG) in a triply-resonant high- Q Lithium Nio-bate microresonator with strongly non-degenerate addedfrequencies. Optically nonlinear resonators thus far havebeen successfully used for either single frequency mul-tiplication (e.g., doubling [8–10], tripling [11, 12], andquadrupling [13]), or parametric down conversion [14–17]. In contrast, in our experiment we demonstrate SFGbetween a 1560 nm pump and a 780 nm signal. Suchcross band coupling opens avenues to several narrow-band frequency conversion applications that have been hitherto challenging. Indeed, SFG can be employed forefficient room-temperature detection of far-infrared, andeven sub-THz, photons [18, 19]. Since SFG does not dis-turb the quantum state [20], it also can lead to efficientmanipulation of the color and shape of single-photon sig-nals [21] for interfacing optical flying qubits with narrow-band atomic quantum memories [22]. Furthermore, thenarrow-band resonance lines in such devices can greatlysuppress incoupling of Raman noise, and potentially leadto new optical tools for mode discrimination and reshap-ing of narrowband quantum signals [23]. Our experimentis an important first step towards all of these applicationsin both the classical and quantum domains.
FIG. 1: Experimental setup. PD: Photodetector, M: Mir-ror; DM: Dichroic Mirror; PBS: Polarization Beam Splitter;HWP: Half-Wave Plate; DAQ: Data Acquisition Unit. 10XObjective Lenses were used to focus the lasers onto the prism-resonator interface and to collect the output light.
Our experiment is illustrated in Fig. 1. We observedsum-frequency generation in a MgO-doped Lithium Nio-bate z-cut microdisk (R ≈ . Q ≥ × and Q ≥ × forthe signal and pump waves, respectively. A 780nm nar-row (below 300 kHz) linewidth tunable-diode laser wasused as the signal and a DFB laser provided 1560nmpump. Two input waves are ordinarily polarized toachieve the Type-I phase-matching whereas the upcon-verted wave at 520nm is detected in the extraordinarilypolarization. The two input waves were combined on adichroic mirror and focused onto the prism-resonator in-terface by an objective lens. A lateral offset of the beamsbefore the lens allowed for optimizing of the pump andprobe in-coupling angles individually. A similar lens wasused to collect the output light, with a dichroic mirrorseparating the pump and the signal while a polariza-tion beam splitter separating the sum-frequency wave.All three optical powers were measured by photodetec-tors, whose signals were fed into a data acquisition unit.As the laser frequencies were continually swept at 50Hzacross several linewidths, the signal and pump WGMswere tracked by a software program continuously adjust-ing the lasers central wavelength to follow their respectiveWGMs. The program ensured that the pump and signalWGMs were in the center of the sweeps and that theywere pumped simultaneously. In addition, the top of theresonator was coated with silver paste and temperature-controlled to allow for electro-optic and thermal tuningfor the SFG phase matching.Having the phase matching achieved, we measured thesum-frequency output power for various input pump andsignal powers. Since the temperature stabilization of theresonator at the level of the phase matching temperaturewidth (approximately 7 mK) was deemed difficult andtime consuming, we carried out these measurements intransient by slowly varying the electro-optic bias voltageto record the peak SFG efficiencies. Each data point rep-resents the average of three consecutive measurements.The signal and pump waves were critically coupled andover-coupled, respectively. The longer wavelength pumpwave was coupled stronger than the signal due to the na-ture of the evanescent coupling. Also, due to the spatial-mode mismatch between the input Gaussian beam andthe WGM profile, in this measurement we achieved acritically-coupled contrast of 48%. We took this into ac-count by only utilizing the in-coupled powers for our the-oretical analysis.We observed efficient sum-frequency generation with amaximum in-coupled pump power of only 1.22 mW. InFig. 2 we plot the out-coupled SFG efficiency. As we var-ied the signal powers, saturation of the peak conversionefficiency was observed in all cases with sub-milliwattincoupled powers, instead of a cyclic behavior that isobserved in the traveling-wave configuration [24]. Sim-ilar saturation has been observed in frequency-doublingWGM experiments [8, 9]. At higher pump powers, anadditional nonlinear loss for the signal wave is createddue to its upconversion, leading to the reduction of itsinternal Q -factor and the coupling contrast. As a result,a smaller portion of the signal wave enters the resonatorand the SFG efficiency is reduced. This behavior is amanifestation of the “coherent” quantum Zeno effect for the signal wave, where the “potential” for the upconver-sion decouples the signal field from the cavity [4]. −3 In−coupled Pump Power (mW) N o r m a li z ed S F P o w e r Signal Power
FIG. 2: The out-coupled SF emission is measured and nor-malized to the various input signal powers. Symbols representthe experimentally measured data, and the solid lines are the-oretical fits. Note that the theory predictions for 2.5 µW and5 µW are identical because the nonlinear loss is negligible atsuch powers. A theoretical description of the SFG in WGMRs iswarranted because manifestly different behavior is ob-served at high pump powers compared to a traditionaltraveling-wave geometry. Neglecting Rayleigh backscat-tering and assuming linearly polarized fields, we find thescalar equations of motion in the cavity are: ∂c p ∂t = − κ p c p + i r ω p Q cp a p + i Ω c ∗ s c f (1) ∂c s ∂t = − κ s c s + i r ω s Q cs a s + i Ω c f c ∗ p (2) ∂c f ∂t = − κ f c f + i Ω ∗ c s c p (3)where Q cµ and Q iµ denote the coupling and intrinsic Q -factors, κ µ = ( ω µ Q iµ + ω µ Q cµ − i ∆ µ ) for µ = s, p, f indicatingrespectively signal, pump and the sum-frequency, andΩ = ǫ ¯ h d R dV E f E ∗ p E ∗ s is the internal conversion effi-ciency of the SFG process. The input field operators, a µ ,are related to the output fields by b µ = √ T a µ + i q ωQ cµ c µ .Using quasi-static analysis we solve Eqns. (1)-(3) for theout-coupled sum-frequency field, | b f | = | i q ω f Q cf Ω κ f c s c p | .Before we provide the solution, a few salient points areto be noted. In this formulation, Ω is the internal con-version efficiency of the SFG process, whereas only theSF out-coupled power is measured experimentally. Since Q cµ is determined by the distance from the prism to theresonator d , we need only to fit to Ω and the intrisic Q -factor for the SF [25]. Moreover, these equations aredifficult to solve in general for c s , c p , and c f analytically.To guide us, however, we make the undepleted pump ap-proximation and then use numerical methods to acquirethe generic solution. This approximation yields: c p = i q ω p Q cp a p κ p , c s = i q ω s Q cs a s κ s + | Ω | ω p | a p | κ f Q cp | κ p | , (4) | b f | = ω s ω p ω f Q cs Q cp Q cf | Ω | / | κ f κ p | | κ s + | Ω | ω p | a p | κ f Q cp | κ p | | | a p | | a s | . (5)Note that the expressions for c p and c s are asymmetricdue to the nature of the undepleted pump approxima-tion. These solutions indicate that we do not observe anoscillatory behavior in the frequency-conversion dynam-ics. For low pump and signal energies, the SFG outputbehaves linearly, | b f | ∝ | a p | | a s | (see Fig. 2), whereasat higher energies, the dynamics are different, i.e., theupconversion and subsequent down conversion processesare asymmetric in this geometry [4].Using the in-coupled pump and signal powers, andmeasured Q iµ , Q cµ for µ = s, p , we can provide a theo-retical fit for the measured data in Fig. 2 with the fit-ting parameters Ω and Q if . Fitting our efficiency mea-surements at 2.5 µ W signal inputs (where the undepletedpump approximation is valid) with Eq. (5), we can esti-mate Ω, which does not vary as the same WGM tripletis studied throughout the experiment. Throughout thisprocedure, we assumed that the SFG peak occurredwhen the three waves were exactly on resonance, i.e.,∆ s = ∆ p = ∆ f = 0.We then solved for Eqs. (1)-(3) numerically, which pro-vided the efficiency curves for data where the undepletedpump approximation is invalid. The total Q -factors forthe pump and signal waves were not constant throughthe experiment, which we account for by using Q if asa fitting parameter in Fig. 2. We attribute this varia-tion in part to the unaccounted photorefractive effects inthe resonator, caused by the SF. These effects were mostevident for the highest signal power used, P s = 60 µW .Indeed, this particular data set in Fig. 2 also presents thelargest data scatter and the worst agreement with theory.In spite of these theoretically unaccounted backgroundprocesses, by using just the Ω-parameter, which deter-mines the shape of the efficiency curves, we are able toaccurately model the signal and pump transmission spec-tra as well as the SF emission spectrum [25]. Moreover,we were able to calculate [25] the fundamental channelsmodes overlap to give Ω theor = 253 kHz, whereas theempirical value gives us Ω expt = 5 kHz. In calculatingthis latter value, we used Q if = 3 . × , the intrinsicsum-frequency Q -factor and that it was strongly under-coupled to the resonator, Q cf Q if = 164, see [25].Not every pair of signal and pump WGMs can generate sum-frequency. SFG in a triple-resonant system requiresthe phase matching between these modes, which can beviewed as conservation of the integrals of motion deter-mined by the system’s symmetry.Usually WGMs have nearly perfect spherical symme-try, so their eigenfunctions inside the resonator can bewell approximated byΨ Lmq ( r, θ, ϕ ) = Y mL ( θ, ϕ ) j L ( k q r ) , (6)where r, θ, ϕ are spherical coordinates, L, m, q are az-imuthal, polar and radial mode numbers, respectively, Y mL ( θ, ϕ ) is a spherical harmonic and j L ( k q r ) is a spheri-cal Bessel function. The radial wave number k q is deter-mined from the boundary conditions.In [25] we show that many combinations of WGMtriplets (called channels in the following) may lead toSFG albeit with different conversion efficiencies, whereasmost efficient channel is the one that couples the fun-damental modes, i.e. such that q p = q s = q f = 1, L p − m p = L s − m s = L f − m f = 0.Achieving a triple resonance for a selected channel re-quires tuning of each WGM’s frequency such that theenergy conservation ω p ( L p , m p , q p ) + ω s ( L s , m s , q s ) = ω f ( L f , m s + m p , q f ) is fulfilled to better than a WGMlinewidth. In lithium niobate resonators this can beachieved for ordinary pump and signal and extraordinarysum-frequency WGMs due to different temperature de-pendencies of the ordinary and extraordinary refractionindices. We find the phase matching temperatures [25] byiteratively solving the energy conservation condition us-ing the WGM dispersion equation [26] and temperature-dependent Sellmeier equations [27]. Let us point outthat for sub-mm resonators the resulting temperaturesmay significantly (by tens of degrees) differ from the bulkphase matching temperature, due to the geometrical, orwaveguide, part of the WGM dispersion.In the experiment, identifying a WGM’s q may presenta considerable difficulty. Fortunately, the WGM’s freespectral range (FSR) depends on its q much stronger thanon L and m , because q affects the effective length of theresonator. Therefore a WGM’s q can be inferred fromthe FSR measurements. We carried out the FSR mea-surements for the pump laser by frequency-modulationtechnique [28]. The results of such measurements car-ried out with 11 best-coupled modes within one FSR areshown in Fig. 3. The theoretical FSR values shown inFig. 3 were derived from the WGM dispersion equation[26]. We fit the theory value for q = 1 to the smallestmeasured FSR of 32.362 GHz by varying the resonatorradius from the initially measured 0 . ± .
01 mm to0 . q values.The theoretical FSR value for q = 1 WGMs at the sig-nal wavelength is 31.049 GHz. We found a high-contrastmode with a very close FSR value of 31 . ± .
004 GHz[29]. Coupling the pump and signal lasers to these WGMswe slowly varied the resonator temperature while moni-toring the SFG signal, and acquired the data for Fig. 2.
Free spectral range (GHz)q = 1 2 3 4 5 6 7 8
FIG. 3: Theoretical FSR values for q = 1 ... W a v e l eng t h ( n m ) T e m pe r a t u r e ( o C ) W a v e l eng t h ( n m ) FIG. 4: Numeric simulation result for the fundamentalSFG channel (black dots) and experimental observations (redstars). Projections emphasize a good agreement between thetheory and experiment.
We also found the neighboring SFG channels by tun-ing the pump and signal lasers across an integer numberof FSRs. The wavelengths and phase matching temper-atures for these channels are shown in Fig. 4 togetherwith the numeric simulation for the fundamental chan-nel. A good agreement was achieved by using the MgOconcentration as a fitting parameter in the simulation.This parameter affects the phase matching temperatureby entering Lithium Niobate dispersion [27]. Unfortu-nately its value is not precisely known for our congruentwafer, except that it should slightly exceed the thresholdvalue of approximately 5%. The fitting yielded a veryplausible value of 5.63%. It should be pointed out thatno other efficient SFG channels can fit the observationswith any reasonable MgO concentration. However, manyless efficient equatorial channels exist for the same tem-peratures [25]. The lower observed Ω suggests that theWGM triplet may not have been fundamental.To summarize, we have demonstrated triple-resonantSFG in a WGM resonator. We have extended the the-oretical analysis for finding the phase-matched WGMsinside the resonator and understood the nonlinear dy-namics of frequency conversion in the strong pump-signalcoupling regime. The efficiency of this process in the res-onator is much higher than in a traveling-wave geometry,requiring sub-milliwatt powers for saturation. We expectto find applications of this interaction in the fields of spec-troscopy, optical communications and data processing,both at classical and at quantum levels. Other conceiv-able applications are for fundamental tests of quantumtheory, e.g. in cavity optomechanics and quantum non-demolition measurements.This work was supported by the DARPA Zeno-basedOpto-Electronics program (Grant No. W31P4Q-09-1-0014). It was partly carried out at the Jet PropulsionLaboratory, California Institute of Technology under acontract with the National Aeronautics and Space Ad-ministration. We thank J. U. F¨urst and T. Beckmannfor useful discussions. [1] K. J. Vahala,
Optical Microcavities , World Scientific(2005).[2] Nozaki et al. , Nat. Photon. , 477-483 (2010).[3] Y.-P. Huang and P. Kumar, Opt. Lett. , 2376-2378(2010).[4] Y.-P. Huang and P. Kumar, IEEE J. Sel. Top. QuantumElectron. , 60011 (2012).[5] D. A. B. Miller, Nat. Photon. , 3 (2010).[6] Y.-Z. Sun et al. , Phys. Rev. Lett. , 223901 (2013).[7] A. Majumbar and D. Gerace, Phys. Rev. B , 235319(2013).[8] V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L.Maleki, Phys. Rev. Lett., , 043903 (2004).[9] J. U. F¨urst et al. , Phys. Rev. Lett. , 153901 (2010).[10] P.S. Kuo, and G.S. Solomon, Opt. Exp., , 16898-16918(2011).[11] T. Carmon and K. J. Vahala, Nature Physics , 430 (2007).[12] K. Sasagawa and M. Tsuchiya, Appl. Phys. Exp. ,122401 (2009).[13] J. Moore et al. , Opt. Exp. , 24139-24146 (2011).[14] A. A. Savchenkov et al. , Opt. Lett., , 157-159 (2007).[15] J. U. F¨urst et al. , Phys. Rev. Lett. , 263904 (2010).[16] T. Beckmann et al. , Phys. Rev. Lett. , 143903 (2011).[17] Ch.S. Werner et al. , Opt. Lett. , 4224-4226 (2012).[18] D.V. Strekalov et al. , Phys. Rev. A., , 033810 (2009).[19] L. Ma, O. Slattery, and X. Tang, Physics Reports, ,6994 (2012).[20] P. Kumar, Opt. Lett. , 1476 (1990)[21] M. G. Raymer and K. Srinivasan, Phys. Today , 32(2012)[22] M. S. Shahriar et al. , J. Phys. B: At. Mol. Opt. Phys. ,124018 (2012)[23] B. Brecht et al. , New J. Phys., , 065029 (2011). [24] R.W. Boyd, Nonlinear Optics (New York: Academic, 3rdEdition, 2008)[25] See Supplementary material.[26] M.L. Gorodetsky and A.E. Fomin, IEEE J. Sel. Topicsin Quantum Electronics , 33-39 (2006).[27] U. Schlarb and K. Betzler, Phys. Rev. B , 751 (1994).[28] J. Li, H. Lee, K.Y. Yang, and K.J. Vahala, Opt. Expr., , 26337-44 (2012). [29] The majority of the signal modes had smaller FSRs anddid not follow the trend of Fig. 3. This contradicts boththe assumption that the selected probe mode has q = 1and the resonator radius measurement. Localized pho-torefractive damage induced by the 780 nm light is onepossible explanation of this discrepancy. r X i v : . [ phy s i c s . op ti c s ] J u l Optical sum-frequency generation in whispering gallery mode resonators -Supplementary Material
Dmitry V. Strekalov
Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Drive, Pasadena, California 91109-8099
Abijith S. Kowligy, Yu-Ping Huang, and Prem Kumar
Center for Photonic Communication and Computing, EECS Department,Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3118, USA (Dated: September 24, 2018)
I. ACQUIRING Ω For low signal powers, the sum-frequency efficiency canbe understood in the undepleted pump regimes (Fig. 1).
In−coupled Pump Power (mW) E ff i c i en cy ( a . u . ) Undepleted Pump Approx.Exact Numerical Result
FIG. 1: The dotted line is the exact numerical solution for2.5 µW signal in-coupled power, and the solid line is the unde-pleted pump approximation. Their good agreement enablesus to estimate Ω and utilize it for higher signal powers, wherethe approximation is invalid and exact analytical quasi-staticexpressions do not exist. Since Eq. (5) in the main text is valid only in the un-depleted pump approximation, we can infer Ω only fordatasets where the signal powers are very small. Addi-tionally, we assume that the inherent nonlinearity is soweak that for both pump and signal powers P ≤ µW ,there is no noticeable conversion (which is justified by allof our data sets, wherein the efficiency is nearly zero inthis region). Since we use the same WGM triplet for theentire experiment, and Ω is an overlap measure betweenthe excited WGMs, we operate under the condition thatΩ is the same for arbitrary input powers. However, whatwe can acquire from the measured data is Ω and κ f ,where κ f = ω f ( Q if + Q cf ). The coupling- Q factors, Q cµ , are given by [1] for a TM WGM in a spherical resonator: Q cµ = π √ n s ( n s − p n c − n s ( Rk ) / exp(2 γd ) (1) µ = s, p , where n s , n c are the indices of refraction in theresonator and prism respectively, R is the resonator ra-dius, γ = p k ( n s − d is the distance between theprism and resonator. We will use (1) as an order of mag-nitude estimate for our spheroidal resonator, and multi-ply it by n s when computing Q cf for the extraordinarilypolarized (TE) 520 nm sum-frequency wave, which is dueto the different boundary conditions. Since we operateunder critical coupling conditions, i.e., Q cs = Q is , for the λ s = 780 nm signal wave, we can acquire an empiricalvalue for d . In our experiment, for Q cs = Q is = 4 . × leads to d = 70 . Q cf = 5 . × . From data fittingwe find Q if = 3 . × , which leads to the undercou-pling factor Q cf /Q if = 164. This estimated intrinsic Q isvery close to an earlier reported [2] value of Q i = 4 × in the same material at a close wavelength of 532 nm. Letus point out that in lithium niobate a lower Q is indeedexpected at a shorter visible wavelength.Therefore, the internally generated green light was cou-pled out very inefficiently. This leads us to believe thatthe photorefractive damage induced by the green lightcould have been significant. Moreover, from Eq. (1),we see that any changes in the index of refraction dueto thermo-optic effects, light-induced charge transport,or even the electro-optic effect due to charge movementon the periphery of the resonator can significantly affect Q cµ . The exponential dependence on d places stringentstability criteria, which may have been breached due tothe mechanical hysteresis in the brass-oven that was em-ployed to allow for temperature-tuning, leading to a time-varying Q cµ . In fact, by studying the transmission curvesover the duration of the experiment, we have seen thatthe total Q of the resonator does change. Therefore, weconclude that the Ω we acquire by assuming that ∆ f = 0and by also neglecting these concurrent effects, might bedifferent slightly from the value acquired when these ef-fects are accounted for. This may further explain whythe empirically measured internal conversion efficiency islower by a factor of 50. II. OBSERVING AND FITTING THE WGMSPECTRA
As discussed in the main part, our measurement pro-duced the signal and pump transmission spectra, and theSF emission spectra, such as shown in Fig. 2. Having esti-mated the Ω , κ f parameters from the experimental data,we were able to acquire parameter-free fits for the trans-mission spectra of the resonator, also shown in Fig. 2. Time (ms)
TheoryPump TransmissionTheorySignal TransmissionTheorySF Emission
FIG. 2: Dotted lines are the observed signal, pump and SFspectra, and solid lines are theoretical fits.
III. WGMS OVERLAP AND SFG CHANNELSEFFICIENCY
Spherical symmetry may be a good approximation fordetermining the eigenfunction shape (but not always theeigenfrequencies) in a spheroid WGM resonator whoseaspect ratio is not too large, because the variation of theboundary from a true sphere in the region of significantWGM field is minute. Due to the spherical symmetrythe phase matching in WGM resonators corresponds toconservation of photons’ orbital momenta [2–5]. If L ≫ L − m and latter on the q for all three modes. Each partalso weakly depends on wavelengths and the three valuesof L . The angular parts give rise to the SFG selectionrules corresponding to the Clebsch-Gordan coefficients.In particular, they enforce m p + m s = m f , (2) L p + L s ≥ L f , (3) L p + L s + L f = 2 N, (4) where N is an integer. The radial parts lead to no partic-ular selection rules, however as we will see they stronglyfavor the cases when q p + q s ≈ q f .For mm-size resonators the orbital numbers L, m arelarge, and evaluating the overlap integrals with Legendrepolynomials and Bessel functions of such orders is im-practical. Appropriate asymptotic approximations needto be made. Let us introduceΨ( r, θ, ϕ ) ≈ Ψ ang ( θ )Ψ rad ( r ) e imϕ (5)inside the resonator, and Ψ( r, θ, ϕ ) ≡ ang ( θ ) = N a √ π H L − m ( √ L cos( θ )) exp (cid:26) − L cos( θ ) (cid:27) , (6)and radial asymptotic isΨ rad ( r ) = N r √ r Ai (cid:18) α q L − k r ( L + 1 / , q ) r/RL − k r ( L + 1 / , q ) (cid:19) . (7)In (6) H L − m is the Hermite polynomial of the order L − m , and N a is the normalization factor, N − a = Z − H L − m ( √ Lx ) exp (cid:8) − Lx (cid:9) dx. (8)In (7), R is the resonator radius, Ai is the Airy function, α q is its q th root (positive value), dimensionless wavenumber can be approximated [7] as k r ( L, q ) ≈ L + α q ( L/ / , (9)and the normalization factor is N − r = R Z Ai (cid:18) α q L − k r ( L + 1 / , q ) xL − k r ( L + 1 / , q ) (cid:19) xdx. (10)The factors (8) and (10) provide the asymptotic wave-function (5) normalization: R dV | Ψ( r, θ, ϕ ) | = 1.Approximation (5) allows us to evaluate the radial andangular overlap integrals separately. These overlap fac-tors are shown in Fig. 3 in the normalized form. Todetermine the overlap integral for a particular channelone needs to take the appropriate angular part fromthe top of Fig. 3 and multiply it by the appropriateradial part from the bottom of Fig. 3. For the fun-damental modes, i.e. such that q p = q s = q f = 1, L p − m p = L s − m s = L f − m f = 0 (the most ef-ficient SFG channel for our wavelengths and resonatorsize) | R dV Ψ f Ψ ∗ p Ψ ∗ s | ≈ . × cm − . For compar-ison, the fundamental WGM mode volume in our res-onator ranges from approximately 1 × − cm for theSF to 4 × − cm for the pump.This analysis is practically useful only if we can identifythe mode numbers for a given mode from a WGM spec-trum. As we have discussed in the main part of the paper,identification of the q ’s is particularly important. Since L - m L - m L - m q q q FIG. 3: The absolute-square of the angular (top) and radial(bottom) parts of the WGMs overlap integral are representedby the dot size as a function of the respective mode numbers.In both plots, the largest dot is normalized to unity. the technique described therein provided only a limitednumber of data points at the pump wavelength for ourworking resonator, and had even more limited success atthe signal wavelength, we feel compelled to demonstrateits robustness in a separate measurement with a differ-ent test resonator. The test resonator was made fromthe same material but had a larger radius, R ≈ µ m,and a denser WGM spectrum. The denser spectrum hasallowed us to measure the FSR for 50 modes within asingle FSR. The results of this measurement carried outwith the pump laser are shown in Fig. 4. Here again,we have forced the q = 1 theoretical value to match thetwo lowest-FSR modes by correcting the resonator radiusvalue from 700 µ m to 699 . µ m. This has achieved a re-markable agreement up to the q = 9. At higher q ’s theWGM coupling contrast is considerably reduced, and themeasurements become unreliable. Free spectral range (GHz)
1 2 3 4 5 6 7 8 9
10 11
12 13 14 15 q =
FIG. 4: Validation of the q -identification approach based onthe FSR measurement: vertical lines are theoretical values ofFSR for the q ’s as labeled, data points are the measurementresults. IV. PHASE MATCHING TEMPERATURE
Based on the previous section analysis, we can rankvarious SFG channels by their efficiency. We then nu-merically find the dependence of the frequency detuning∆ ω = ω f − ω s − ω p on temperature and determine thephase matching (∆ ω = 0) temperature as well as theWGM frequencies ω f , ω s , ω p at that temperature, foreach channel from the ranked list.In Fig. 5 we show the theoretical prediction for phasematching temperatures in the range between 120 and 150 ◦ C. The pump modes in this simulation have been as-sumed equatorial ( L p − m p = L s − m s = 0) and with q s,p <
8. Both assumptions are justified by the highcontrast of the observed WGMs and by the FSR mea-surements discussed in the main part of the paper. Noassumptions have been made about possible SFG WGMs.From Fig. 5 we see that the most efficient [1 , ,
1] SFGchannels have the phase matching temperature near 129 ◦ C, far from other channels of significant efficiency. How-ever, there are also many “minor” SFG channels at thistemperature. In our experiment, we indeed observed sev-eral other SFG channels with the efficiency at least anorder of magnitude below than the one we have beenusing.The phase matching temperature width is determinedby the relative drifts of the pump, signal and sum-frequency WGMs with varying temperature. These inturn are determined by the thermal expansion and ther-mal refractivity of Lithium niobate, and by the mechan-ical aspects of the resonator mounting. To measure thetemperature width of the phase matching, we first cali- [1,3,3][2,4,5][2,3,4][1,2,2] P ha s e m a t c h i ng t e m pe r a t u r e ( o C ) W a v e l eng t h ( n m ) W a v e l eng t h ( n m ) [1,1,1] FIG. 5: Main equatorial SFG channels [ q s , q p , q f ] form dis-tinct “layers” in the temperature-wavelength space. Withineach layer, the channels form the rows of constant L s and L p .Dot sizes represent the channels efficiency. brated the temperature variation in units of the electro-optical bias voltage variation by a compensation tech-nique, and then recorded the SFG signal vs. the biasvoltage sweep. The result of this measurement is shown in Fig. 6. We wee that the phase matching is achievedwithin a rather narrow temperature range of approxi-mately 7 mK.
30 40 50 60 70 80 90 100 1100.00.10.20.30.4
Equivalent temperature ( o C) P ea k s i gna l ( W ) Bias (V)
FIG. 6: The SFG phase matching temperature width mea-sured by a bias voltage sweep is found from a Lorentzian fitto be approximately 7 milidegrees.[1] M. L. Gorodetsky and V. S. Ilchenko, J. Opt. Soc. Am. B, , 147-154 (1999).[2] J. U. F¨urst et al. , Phys. Rev. Lett. , 153901 (2010).[3] P.S. Kuo, and G.S. Solomon, Opt. Exp., , 16898-16918(2011).[4] J. U. F¨urst et al. , Phys. Rev. Lett. , 263904 (2010). [5] G. Kozyreff et al. , Phys. Rev. A , 043817 (2008).[6] Y. Louyer, D. Meschede, and A. Rauschenbeutel, Phys.Rev. A, , 31801 (2005).[7] M.L. Gorodetsky and A.E. Fomin, IEEE J. Sel. Topics inQuantum Electronics12