Identifying modes of large whispering-gallery mode resonators from the spectrum and emission pattern

: Identifying the mode numbers in whispering-gallery mode resonators (WGMRs) is important for tailoring them to experimental needs. Here we report on a novel experimental mode analysis technique based on the combination of frequency analysis and far-ﬁeld imaging for high mode numbers of large WGMRs. The radial mode numbers q and the angular mode numbers p = (cid:2) -m are identiﬁed and labeled via far-ﬁeld imaging. The polar mode numbers (cid:2) are determined unambiguously by ﬁtting the frequency differences between individual whispering gallery modes (WGMs). This allows for the accurate determination of the geometry and the refractive index at different temperatures of the WGMR. For future applications in classical and quantum optics, this mode analysis enables one to control the narrow-band phase-matching conditions in nonlinear processes such as second-harmonic generation or parametric down-conversion.

WGMRs support a discrete set of eigenmodes, i.e. the whispering-gallery modes (WGMs).Most WGMR experiments could benefit from the knowledge of mode numbers for an exact quantification of their parameters.In sensing experiments with WGMRs, for instance, exact information on the spatial overlap of the WGM and the probe particle is advantageous [3].In nonlinear optics, information on spectral properties, in addition to the spatial overlaps is needed to address and quantify conversion channels [12,19].
Several experimental methods for mode analysis have been studied in the past, such as nearfield probing [20,21,22,23], far field imaging [24,25,26,27,28] and investigating the spectral response of these resonators [23,29,30,31].Regarding the breakthrough experiments mentioned earlier in the text, an experimental characterization of a macroscopic-size WGMR mode structure has only been achieved in the context of optical sum-frequency generation [14] on the basis of sideband spectroscopy [31].This technique requires an optical probe coupled to the mode of interest at a wavelength that may not be available.Furthermore, finding the mode of interest, e.g. the signal or idler down-converted mode, with the probe laser is experimentally challenging.
The spatial structure and the resonance frequency of each WGM is characterized by a unique set of numbers, i.e. the polar, the azimuthal, and the radial mode number ℓ, m, and q, respectively.The full information on these mode numbers is in principle contained in the far-field images of the outcoupled modes [24], and in the frequency spectrum [29].The exact identification of WGMs can be extremely difficult in practice using only one of these approaches.
Here we present a combination of these techniques that allows for a complete identification of all significantly coupled WGMs within the investigated frequency range.The radial mode numbers q and the angular mode numbers p=ℓ-m are found via an evaluation of the measured far-field emission patterns.The polar mode numbers ℓ are unambiguously determined by fitting the frequency spectrum.This fitting results in a very accurate determination of the product of the refractive index n times the major radius R of the resonator.Prior knowledge of either n or R allows the determination of the other parameter.This detailed characterization of the properties of the WGMR and the mode structure is important for WGMRs to become a versatile standard for classical and quantum optics.
The manuscript is structured as follows: section one theoretically describes far-field emission patterns based on the electric field distributions of WGMs.Section two explains the mode spectrum of WGMRs via the dispersion relation.After a description of the experimental setup in section three, experimental results on the characterization of the WGMR, in particular mode analysis, are presented on the basis of measured emission patterns and frequency spectra in section four.

Spatial analysis of whispering-gallery modes
Far-field emission patterns from WGMRs carry information on the internal mode structure.According to Maxwell's equations, WGMs in spherically-symmetric resonators are described by spherical harmonic functions and spherical Bessel functions [32].For WGMs close to the equatorial plane and the surface of a spheroidal WGMR, the electric field amplitudes are approximated by [33,34] (see Fig. 1): where R is the major radius of the WGMR and ρ the curvature.Analogous to the solution of Maxwell's equations in a box with Dirichlet boundary conditions, the WGMs in this spheroidal geometry are characterized by three integer mode numbers: the polar, the azimuthal, and the radial mode numbers ℓ ≫ 1, m ≫ 1 and q ≥ 1, respectively (see Fig. 1).The angular mode number p = ℓ − m = 0, 1, 2, ... of the WGMs gives the degree of the Hermite polynomials and therefore the number of field oscillations in θ -direction.θ m is the angular size parameter.The radial part of the electric field in Eq.( 1b) is described by the radial size parameter u m and the q-th root α q of the Airy-function Ai(−α).
ρ ϕ θ u p=2 q=1 Fig. 1.Spatial structure of WGMs.The coordinate system for the electric field distributions defined by Eq.( 1) is based on the radial distance u of the observation point to the surface of the WGMR, the polar angle θ , and the azimuthal angle φ .The intensity distributions on the right-hand side are shown for various radial mode numbers q and angular mode numbers p=ℓ-m.
The spatial structure of a WGM with open boundaries, where evanescent fields are present in the close vicinity of the WGMR, can be probed by placing a prism with refractive index n prism > n next to the WGMR.The far-field emission pattern from such a prism is described as a Fourier transform of the evanescent WGM field at the prism surface.This near-field at the prism surface can be derived by applying a coupling window [24], i.e. a finite aperture, to the angular part of Eq.( 1).This operation is equivalent to a low-pass filtering for the far-field.
According to Eq.( 1), the mode profile in θ -direction contains information on the angular mode number p.For equatorial WGMs (p=0) (see Fig. 1), the near-fields and therefore the farfields show exactly one maximum.For p > 0, the far-fields in θ -direction show two distinct lobes symmetric around the equatorial plane for large spheres [24].This is not generally the case for aspherical resonators.For oblate resonators (R > ρ), these far-field distributions can also be modeled with two main maxima that are symmetric about this plane.For our resonator (R/ρ ≈ 3.8) and p=2, these maxima overlap strongly.For p=4, they can be resolved.For prolate resonators (R < ρ), the effect of the coupling window is reduced.In the extreme case of R/ρ ≪ 1, the width of the WGM in the near-field is much smaller than the width of the coupling window.Hence, the coupling window does not affect the out-coupling and more detailed features of the respective WGM mode structure are visible in the far-field.
For equatorial WGMs, the equatorial coupling angle Φ and the divergence ∆Φ are given by [35]: The equatorial angle Φ is in good approximation equivalent to the critical angle of total internal reflection of the resonator and the prism material and is independent of the minor radius ρ.A measurement of the equatorial angle Φ reveals the polar mode number ℓ, which is equal to the azimuthal mode number m for equatorial modes.

Spectral analysis of whispering-gallery modes
An understanding of the mode spectrum requires a discussion of the dispersion relation [32,34,36] on the basis of frequency differences between individual WGMs.Using the dispersion relation [32,36], we connect the mode numbers ℓ, q, and p = ℓ − m to the resonant optical frequency ν ℓ,q,p : with p and R/ρ of the order of one.Depending on the polarization, the parameter χ is 1 for TE modes and 1/n 2 for TM modes.The wavelength-dependent refractive index of the WGMR is n.This renders the right-hand side of Eq.(4) frequency dependent.The q-th root of the Airy function α q > 0 can be approximated as α q = [3π/2(q − 1/4)] 2/3 .The scaling factor x = (2πnR)/c appears as a refractive index dependent and thus frequency-dependent normalization factor.The spectrum around a certain WGM at frequency ν ℓ,q,p is found by evaluating frequency differences: to other WGMs at frequencies ν ℓ+∆ℓ,q+∆q,p+∆p .Due to the material dispersion, Eq.( 4) is an implicit function of the frequency, hence, the correct description of the frequency differences requires a full numerical approach.For a better qualitative understanding of the relevant quantities for the formation of the frequency spectrum, we can give analytic expressions using first order approximations.For this, we define a dispersive scaling factor x d by including the slope of the refractive index ∂ n ∂ ν at the frequency ν ℓ,q,p as: As for all resonators, the free spectral range is the characteristic mode spacing for one mode family.Here, we show two mode families (dashed and solid lines), whose respective free spectral ranges (FSR ℓ,1 = FSR ℓ ′ ,2 , see Eq.( 7)) are determined by geometric dispersion.In addition to equatorial WGMs {q ≥ 1, p = 0}, each mode family contains non-equatorial WGMs {q ≥ 1, p = 0} at an offset frequency given by Eq.( 9).Within a given spectral observation window, the frequency differences of the WGMs belonging to different mode families allow to unambiguously identify the polar mode number ℓ.
Small contribution of the polarization dependent term χ • n/ √ n 2 − 1 in Eq.( 4) are omitted.In the following, we model the frequency spectrum on the basis of changes in the respective mode numbers ℓ, q, and p.This discussion is illustrated in Fig. 2. The free spectral ranges FSR ℓ,q originate from steps in the ℓ number and depend only on ℓ and q: ∆ν ℓ,q,p (∆ℓ = 1, 0, 0) For increasing polar mode numbers ℓ, found for example at higher optical frequencies, WGMs are located closer to the WGMR surface.This increases the effective radius for these WGMs and decreases the FSR q,ℓ .In contrast, an increase in the radial mode number q leads to the field shifting away from the surface, which decreases the effective radius and increases the FSR ℓ,q .The spacings between different radial WGMs: can exceed the FSR ℓ,q by orders of magnitude and depend only on ℓ and q.Within the spectral observation window, modes with a different radial number q will also have a different polar number ℓ ′ (see Fig. 2).
The spacings between WGMs with different angular mode numbers p: are determined solely by the ratio of the radii R/ρ and independent on q and ℓ.Eq.( 9) has been used extensively in the context of frequency comb generation [16].For R/ρ ≈ N 2 where N = 1, 2, 3..., the spacings can match multiples of the FSR ℓ,q given by Eq.( 7), which can lead to degenerate frequencies ν ℓ,q,p = ν ℓ+p•(N−1),q,0 .1The frequency spectrum depends on a variety of parameters of the WGMR and the environment, such as temperature, material, pressure, and geometry.Each of these parameters can be used independently to tailor the frequency spectrum.
In principle, accurate knowledge about the scaling factor x, i.e. the fundamental value n • R, together with an absolute frequency measurement of one WGM can provide a way to determine mode numbers ℓ, q, and p of this particular WGM.This can be experimentally challenging due to the limited knowledge of the geometry and the refractive index of the WGMR.In contrast, the experimental study presented in the following two sections is based on relative frequency measurements.First, we obtain the mode numbers q and p from a spatial mode analysis.Using this knowledge, we show a complementary analysis of the frequency differences ∆ν ℓ,1,0 (∆ℓ, ∆q, ∆p) from the fundamental WGM to higher-order WGMs to obtain the WGMR properties, such as the scaling factor x, the major radius R, and minor radius ρ.In this analysis, ℓ is exactly determined by a fitting procedure.

Experimental setup
The experimental setup shown in Fig. 3 allows spatial and spectral characterization of WGMs.The macroscopic WGMR is manufactured from a congruent 5.3% MgO-doped lithium niobate wafer, such that the optic axis is aligned with the symmetry axis of the resonator (z-cut [12]).The major and the minor radius of the disk are measured to be R = 1.594 ± 0.006 mm and ρ = 0.423 ± 0.006 mm with a microscope leading to a scaling factor of x ≈ 1/(13.4GHz).
The WGMR is mounted in an oven whose temperature is stabilized to a millikelvin level.The light source for probing the whispering-gallery resonator is a continuous wave laser at 532 nm (Nd:YAG Prometheus, Innolight) with a Hermite-Gauss TEM 00 mode.Its polarization is parallel to the optic axis of the WGMR (TE mode).Evanescent coupling to the disk is achieved by focusing the beam onto the inner surface of a diamond prism under an angle of total internal reflection (see Fig. 3).The transverse coupling angle was adjusted by maximizing coupling to equatorial modes.All light coming from the coupling point is collimated and sent to a photo detector.At the resonance frequencies of individual WGMs, light can be coupled to the WGMR.Tuning the laser frequency over more than an FSR, this configuration can be used for the spectral analysis.
At the in-coupling prism, the emission pattern from the WGMR destructively interferes with the directly reflected spatial mode of the laser source.We use a second diamond prism for out-coupling to investigate the modes independently from this interference.The light emerging from the second prism is sent to a CCD camera placed at a distance of 1.4 cm from the coupling point, which gives us in a good approximation the far-field emission pattern.

Experimental results
In our mode analysis technique, we combine the information from far-field emission patterns and from the frequency spectrum.Mode numbers q and p are obtained from an analysis of the far-field emission patterns.A complementary analysis of the frequency differences ∆ν ℓ,1,0 (∆ℓ, ∆q, ∆p) from the fundamental WGM to higher-order WGMs is performed to obtain the WGMR properties and the polar mode numbers ℓ.
As a first step, we tune the pump laser frequency over 14 GHz and record the WGMR spectrum with a photodetector.It is known from sensing with WGMRs that objects within the evanescent fields, in this case the coupling prism, can induce a shift of the resonance frequencies [3].To avoid this effect, the experiment is carried out in the under-coupled regime of the fundamental WGM.The measured frequency spectrum is shown in Fig. 4a).The mode numbers {q,p} shown in this figure are now yet determined at this stage of the experiment.
As a second step, we take far-field images (see Fig. 5a)) of all WGMs with a reasonably strong coupling within one FSR.With our coupling optimized for equatorial modes using a TEM 00 laser beam, the excitation of modes with odd p is strongly suppressed (see also [23]).Consequently, we expect to see only even numbers of p, whose coupling decreases rapidly for higher p.The variation of the coupling efficiency among the equatorial modes is relatively small.Coupling of modes with different q can be optimized via the coupling angle Φ. Exciting the WGMR with a polar coupling angle Θ c = 0, one excites an increasing number of nonequatorial modes.The spectrum becomes more dense and less pronounced features of wellcoupled modes are visible.
WGMs with different p numbers are distinguished according to the cross section of the WGMs, which is depicted in Fig. 5c) as an average for all WGMs with the same radial mode number q. Equatorial WGMs with p=0 show a single-lobe emission pattern.WGMs with p=2 exhibit a broader angular distribution in comparison to WGMs with p=0.For WGMs with p=4, a distinct two-lobe structure appears.
The measured emission patterns of the equatorial WGMs were fitted with Gaussian functions.This reveals the information on the central positions (see in Fig. 5b)) and 1/e-values of the Gaussian functions, and hence the divergences of the emission patterns.
The larger the radial mode number q of a WGM in Fig. 5b), the smaller is ℓ.This can be understood with Eq.( 4).The resonance frequencies ν ℓ,q,p of all WGMs within the spectral ob- servation window (see Fig. 2) are approximately equal to the pump laser frequency.Since α q is a monotonically increasing function with q, an increase in the radial mode number q is equivalent to an decrease in the polar mode number ℓ.Hence, an increase in the radial mode number q results in a decrease in the out-coupling angle Φ for equatorial WGMs according to Eq.( 2).This corresponds to the horizontal shift of the center of the spatial profiles depicted in Fig. 5a).
The analysis of the far-field emission patterns have now allowed an assignment of radial mode number q and angular mode number p in the mode spectrum shown in Fig. 4. The polar mode numbers ℓ are obtained from an analysis of frequency differences in the mode spectrum, Fig. 6.Evaluation of the measured frequency spectrum (see Fig. 4a)).a) The mismatch between measured and computed frequency differences (see Eq.( 5)) over a broad range of ℓ numbers of the fundamental WGM {q=1,p=0}.The frequency mismatches of a selection of equatorial WGMs are shown.The individual mismatches reach zero at multiple values of ℓ, however, all together they reach zero at the unique value of ℓ = 41903 in b).n = 2.2244 ± 0.0011, we estimate the large radius to R = (1.5982± 0.0008) mm at 70 • C.This is consistent with the a prior value but has a precision improved by an order of magnitude.Additionally, knowledge of the mode numbers and the frequency spacing between non-equatorial modes described by Eq.( 5) can be used to fit the ratio R/ρ.Together with the knowledge of R, the fit yields ρ = (0.4223 ± 0.0011) mm.The measurements of the radii agree well with the microscopically measured value 2 .
In a final step, the spectral characterization is carried out again at a different temperature of the WGM resonator within the same spectral observation window (see Fig. 4c)).This results in a shift of the relative frequencies for all the modes within the spectrum (see Fig. 4b)).A change in temperature effectively changes the resonator optical length, both through thermal expansion and the thermal change in the refractive index of lithium niobate.This changes the size parameter x of the resonator.The change in the measured mode spectrum in the fixed spectral observation window can thus be directly related to a change in the ℓ number for each mode in the observation window by ∆ℓ = 45.In principle, temperature tuning can also be used to obtain accurate information on the thermo refractive and thermal expansion coefficient.Due to experimental uncertainties in the temperature determination, this is not possible here with a reasonable accuracy.In contrast, we use temperature tuning to demonstrate an efficient method for tailoring the frequency spectrum by choosing the proper ℓ number at a certain temperature.

Conclusions
In summary, we have demonstrated a practical experimental technique to identify mode numbers of all significantly coupled WGMs within the spectral observation window by evaluating far-field emission patterns and frequency differences.This exact determination is even applicable for the case of high mode numbers in large resonators.Conversely, the understanding of the frequency spectra can be used to specifically tailor WGM spectra.In addition, this measure-ment technique allows for accurate measurements of the resonator radii R and ρ times refractive index n.Experimental limits for the determination of the optical circumference n • R are mainly the uncertainty in laser frequency.This can be increased experimentally by orders of magnitude to allow very accurate measurements of either resonator geometry or refractive index.The large set of accurately characterized WGMs can be employed to select WGMs with specific spatial or spectral properties on demand for a huge variety of applications in classical and quantum optics.

Fig. 2 .
Fig.2.Illustration of a WGMR mode spectrum.As for all resonators, the free spectral range is the characteristic mode spacing for one mode family.Here, we show two mode families (dashed and solid lines), whose respective free spectral ranges (FSR ℓ,1 = FSR ℓ ′ ,2 , see Eq.(7)) are determined by geometric dispersion.In addition to equatorial WGMs {q ≥ 1, p = 0}, each mode family contains non-equatorial WGMs {q ≥ 1, p = 0} at an offset frequency given by Eq.(9).Within a given spectral observation window, the frequency differences of the WGMs belonging to different mode families allow to unambiguously identify the polar mode number ℓ.