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Optical whispering gallery modes (WGMs) of mm-sized axisymmetric resonators are well localized at the equator. Employing this distinctive feature, we obtain simple analytical relations for the frequencies and eigenfunctions of WGMs which include the major radius of the resonator and the curvature radius of the rim. Being compared with results of finite-element simulations, these relations show a high accuracy and practicability. High-precision free-spectral-range measurements with a millimeter-sized disc resonator made of MgF2 allow us to identify the WGMs and confirm the applicability of our analytical description.
© 2013 Optical Society of America
1. Introduction
In whispering gallery mode (WGM) resonators, light is confined near the surface owing to resonant recirculation [1]. Such resonators can be made very compact yielding mode volumes down to several hundred μm3 [2]. Owing to low bulk and surface losses, optical Q-factors as high as 1011 can be achieved [3]. Efficient methods for coupling light in and out are available [4]. They allow to enhance the pump power inside the resonator by many orders of magnitude leading to strong nonlinear effects. All this makes WGM resonators promising for a broad range of applications ranging from optical filters, lasers, and sensors [5, 6] to nonlinear devices providing, e.g., generation of optical harmonics [7], optical sum frequencies [8], tunable coherent light [9, 10], and frequency combs [11] at low pump powers.
Knowledge of the resonant frequencies and the corresponding eigenfunctions of WGMs is crucial for most of the applications and developments. Exact solutions to Maxwell’s equations are available for a spherical and cylindrical geometry of the resonator only [12, 13]. Similar to quantum mechanics, the corresponding eigenfunctions can be characterized by the azimuth, momentum, and radial quantum numbers m, l, and n. The WGM limit corresponds to very large values of the azimuth number, m ≫ 1, and, simultaneously, to very large values of the momentum number l, such that l − m is of the order of 1. For crystalline optical WGM resonators, the azimuth number m ranges roughly from 103 to 105. Approximate expressions for the WGM frequencies and eigenfunctions can be obtained in the spherical case using the asymptotic relations for the Bessel and spherical functions; these expressions possess a high accuracy owing to the large values of m and l. The WGM eigenfunctions are well localized near an equator of the sphere in both the radial and angular directions. The radiation losses are by far negligible compared to the bulk and surface losses.
Being good for illustration, the spherical geometry is not sufficient for practical purposes. Nowadays, the shapes of optical resonators are typically far from spherical. Important cases are disc resonators made of crystalline materials by mechanical turning [14, 15] or toroidal ones made of glass [11, 16]. In both of these and in many other axisymmetric cases, WGMs are strongly localized near the rim of the resonator whose shape can be well characterized by the major radius R (the maximal distance to the rotational axis) and the minor radius r (the curvature radius of the rim in polar direction), see also Fig. 1. Variation of the ratio R/r is important to achieve optimal coupling efficiencies with prism couplers [17]. Far enough from the equator, the shape of the resonator can strongly differ from spheroidal or toroidal one. However, this is of minor importance for WGMs if both radii are much larger than the resonant wavelength λ, thus R, r ≫ λ. This is the case of our interest. Note that the wedge resonators of [18] do not belong to this category.
Fig. 1 Geometry of the problem; R and r are the major and minor radii of the resonator, while u, θ, and φ are the curvilinear coordinates, and ρ = ρ(u, θ) is the distance from the observation point (shown by the red dot) to the vertical rotational axis.
Finite-element simulations [19] and also rather laborious approximate analytical methods and approaches [20–23] are in use to characterize the WGM properties. In particular, using the eikonal method the eigenfrequencies of a dielectric spheroid have been calculated with a high accuracy [22, 23]. It is not clear, however, how the corresponding relations can be applied to the case of a disk or toroidal resonator. Generally, two issues of modeling seem to be not quite satisfactory: The localized nature of WGMs is widely recognized, but the methods employed deal persistently with the global shape of the resonator. The general importance of WGMs is recognized, but there are no simple and general approximations leading to these modes.
Motivated by the needs of experiments and the above general considerations, we propose a simple local method to describe approximately WGMs of an axisymmetric resonator characterized by the radii R and r. The idea is to use a local curvilinear coordinate system near the equator and to employ immediately the WGM localization property. This idea is not original – it is expressed in a general form in [21]; similar treatments are also known in the literature [8,24]. The final relations for the resonant frequencies and eigenfunctions are simple and informative. The calculation accuracy, as follows from comparison with the results of direct finite-element simulation, is pretty high and sufficient for practical purposes. Application of the relations for the resonant frequencies to an experiment allows us to identify WGMs of a disc resonator made of magnesium fluoride via measurements of the free spectral ranges. Furthermore, the simple analytic expressions for the eigenfunctions can be used to easily calculate the modal overlap between interacting waves which is a crucial parameter in nonlinear optics [7, 25].
A few commonly accepted features of WGMs will be used in what follows:
- – In the axisymmetric case, a WGM can be treated as a wave propagating along the rim with the frequency ω = mc/Rn̂, where c is the light speed and n̂ is the effective refractive index. The latter incorporates the geometric dispersion, depends on the type of WGM, and is close to the refractive index of the material n.
- – Vectorial effects are typically of minor importance compared to the shape effects, so that the the electromagnetic field in WGMs can be well described by the scalar Helmholtz equation for the dominating transverse light-field component (E). This is valid for optically isotropic and birefringent materials if the optical axis of the latter coincides with the resonator axis.
- – A zero (Dirichlet) boundary condition for E can be used to find the resonant frequencies of the WGMs and the corresponding eigenfunctions inside the resonator.
2. Analytical model
2.1. Local coordinate system near the equator
Near the equator, the shape of a disc resonator can be described by the major radius R and the minor radius r; the ratio R/r can be larger or smaller than 1. Next we employ the orthogonal curvilinear coordinates u, θ, φ near the equator, where u is the distance to the surface, θ is the local polar angle measured from the equatorial plane, and φ is the azimuth angle, see Fig. 1. It is assumed that u ≪ r and R, |θ| ≪ 1, 0 ≤ φ < 2π.
As evident from the drawing, the elementary volume element in our coordinate system can be written as dV = hudu × hφdφ × hθdθ with the scaling (Lame) coefficients hu = 1, hθ = r − u, and hφ = ρ, where ρ = ρ(u, θ) is the distance to the rotation axis. It is not difficult to see that
We keep only the leading terms in u and θ in the right-hand side, which are linear and quadratic, respectively. This pure geometric feature predetermines many subsequent properties of WGMs.2.2. Helmholtz equation
With the standard phase factor exp[i(mφ − ωt)] separated, the scalar amplitude of the light field E = E(u, θ) obeys the Helmholtz equation
where k = nω/c and ω is the angular frequency. We have used the general expression for the Laplace operator in the orthogonal curvilinear coordinates [26] and the fact that hu = 1. No approximations have been made so far.At this moment, we employ the WGM approximation – we neglect the derivatives of the Lame coefficients hθ and hφ in favor of derivatives of E(u, φ). The reason is clear: When differentiating E we acquire a big parameter characterizing the inverse degree of localization in u or in θ. When differentiating hθ or hφ we acquire almost nothing. The validity of this approximation will be verified after finding solutions for E. The same approximation is valid in the 2D case of a circular cylinder; the approximate results for m ≫ 1 show here a perfect agreement with the exact solution in the terms of Bessel functions.
Applying the WGM approximation and inserting Eq. (1) into Eq. (2) we acquire:
This equation allows the separation of variables. Assuming the field to be E = U(u)Θ(θ), we get where Λ = const is the separation constant to be found. By solving these ordinary differential equations with proper boundary conditions, one can find Λ, k, U(u), and Θ(θ).2.3. Eigenvalues and eigenfunctions
Consider first the polar angular function Θ(θ). Scaling θ to ξ = θ/θm with θm = (R/r)3/4m−1/2, we find from Eq. (4) that
where Λ̃ = Λr1/2R3/2/m does not depend on θ. This equation has to be supplemented by the requirement Θ(θ) → 0 for ξ → ±∞ to ensure the localization in |θ|. After that, the eigenproblem to be solved becomes equivalent to the quantum-mechanical eigenproblem for the linear oscillator [26]. The corresponding solutions are well known: where p = 0, 1,... and Hp(ξ) is the Herimitian polynomial of the order p. The first three polynomials are H0 = 1, H1 = 2ξ, and H2 = 4ξ2 − 2. Generally, p gives the number of zeros of the angular function Θ(θ) and determines its spatial parity. As soon as θm = (R/r)3/4m−1/2 ≪ 1, the requirement of localization in |θ| is fulfilled. At m ∼ 104 it is very soft for the ratio R/r. Note that the localization in |θ| is Gaussian, i.e. very strong. It is also worth mentioning that the angular function Θp = Θp(θ/θm) and the separation constant Λ = Λmp = −m(2p+1)/r1/3R3/2 depend on the azimuth number m.Now, we turn to the radial function U(u) that obeys Eq. (4). Taking into account the found value Λ = Λmp and transferring from u to the variable ζ = (u − δ)/um with
we arrive at the Airy equation and the Airy-function solution for U: Next we employ the zero (Dirichlet) boundary condition U(u = 0) = 0, which means negligible radiation losses. In accordance with Eq. (8), it reads Ai(−δ/um) = 0 and allows us to determine δ and, consequently, k and ω. It is known that the equation Ai(−ζ) = 0 has an infinite sequence of positive solutions ζq with q = 1, 2, . . . , such that ζ1 ≃ 2.338, ζ2 ≃ 4.088, and ζ3 ≃ 5.521. Therefore we get δq = umζq and the following explicit relation for the radial function: It also depends on two mode numbers. Since Ai(x) ∝ exp(−2x3/2/3) for x ≫ 1 [27], strong localization of Umq(u) for not very large q is ensured by the inequality um ≪ r. This is equivalent to the requirement R/r ≪ m2/3 which is very soft for m ∼ 104.Collecting all angular and radial factors, we have for the WGM eigenfunction Em,p,q(φ, θ, u):
where θm = (R/r)3/4/m1/2, um = R/21/3m2/3, Hp(x) is the Herimitian polynomial of degree p, and ζq > 0 is the q-th root of the Airy-function equation Ai(−ζ) = 0. The normalization constant C can be chosen as convenient.Now we turn to the determination of the resonant angular frequency ω. Using Eq. (7) and recalling that δq = umζq and k = ωn/c, we obtain:
Generally, the material dispersion n(ω) cannot be neglected. Being modeled properly, using, e.g., a Sellmeier equation, it allows to determine the frequency ω for each particular set of the three mode numbers m (azimuthal), p (polar), and q (radial). Thus, we have ω = ωm,p,q. Retaining only the first term in the square bracket brings us to the simplified view of WGMs, when the effects of geometric dispersion are ignored and ω = cm/nR. The last two terms give the effects of geometric dispersion; they both are positive. The radial correction (the second term) coincides with that calculated for the circular cylinder of radius R. For the lowest near-equatorial modes with p = 0, 1,... and the ratio of the order of one, the second (radial) correction noticeably exceeds the third (polar) one. Correspondingly, the effect of the ratio R/r on the geometric dispersion is relatively small.By setting ω = cm/Rn̂, we obtain for the effective refractive index n̂:
Obviously, we have n̂ < n. This is equivalent to R̂ < R if we define the effective major radius R̂ by the relation ω = cm/nR̂. Both these inequalities are intuitively clear.One of the most important characteristics for the experiment is the free spectral range (FSR), i.e., the frequency difference Δωm,p,q/2π = (ωm+1,p,q − ωm,p,q)/2π between the neighboring (in m) values of ω/2π for fixed p and q. While the FSR differences are very small in m and p, they can considerably exceed the frequency width of individual WGM resonances. This allows for identification of the mode number q, see also below.
3. Comparison with numerical simulations
To estimate the accuracy of our analytical method, we have compared the eigenvalues and eigenfunctions calculated analytically with those found by a finite-element simulation. We utilized an axisymmetric formulation of [28] and implemented it into the standard commercial COMSOL finite-element package. For the bulk refractive index we set n = 1.37. As we are interested in well localized WGMs, the calculation area of a few tens of wavelengths around the equator of the resonator was used. In order to allow for higher order p, q-modes, this area was chosen to be 30μm wide and 70μm high. The surrounding of the resonator was air with perfectly conducting boundaries a few wavelengths away from the equator. The mesh consisted typically of 5811 elements with a higher density of elements inside the dielectric compared to that in air. The model provides the resonant WGM frequencies and the corresponding eigenfunctions including the vectorial effects. The relative accuracy of the calculations is about 10−5. Two polarization types of WGMs, which we refer to as v- and h-modes, correspond to predominantly vertical and horizontal polarization of the electric field vector, see also Fig. 1.
Figure 2 compares the analytical data with numerical ones for eigenfrequencies at the representative parameters R = 1 mm, r = 0.3 mm, and m = 104, which correspond to the vacuum wavelength λ about 0.860μm and the frequency ω/2π about 350 THz. The subfigures 2(a) and 2(b) refer to the radial and polar numbers q = 1 to 5 and p = 0 to 4, respectively. The analytical data coincide with the numerical ones with a relative accuracy better than 10−4, which is not far from the accuracy of numerical calculations. The frequency shifts owing to increasing mode numbers p and q are indeed fully described by Eq. (11). The influence of vectorial (polarization) effects on the WGM frequencies is very small.
Fig. 2 Resonant frequencies ωm,p,q/2π for a WGM resonator with R = 1 and r = 0.3 mm for m = 104. The circles correspond to Eq. (11), while the horizontal and vertical dashes represent the simulations for the h- and v-modes. Note different vertical scales in a) and b).
Now we turn to the comparison of the eigenfunctions. To present the analytical results, the transverse field distributions Em,p,q(x, z), taken at φ = 0 in the cartesian x, z coordinates, are convenient. Figure 3(a) shows two such distributions near the equator – one for the fundamental mode with p = 0, q = 1 and another for the higher-order mode with p = q = 2. The vertical and horizontal scales are the same. Normalization is chosen such that Emax = 1. As earlier, we set R = 1 mm, r = 0.3 mm, and m = 104. Figure 3(b) represents the corresponding numerical data for the h- and v-modes. Only the largest field components (h and v) are shown. The other components are smaller by at least two orders of magnitude, which quantifies the accuracy of our scalar analytical approach. Visual inspection of each mode shows that there are no differences between the three spatial distributions – the positions of the minima and maxima as well as their widths and heights are the same. To get a closer look, we have compared the field dependences E(x) taken at two representative values of z, see Fig. 3(c) and the white solid lines in Fig. 3(a,b). They show practically exact coincidence of the results inside the resonator. The numerical solution provides tiny evanescent tails, which can not be resolved by the analytic expression. These tails are of importance for coupling and sensing applications, but do not fundamentally change the internal field distribution.
Fig. 3 The normalized transverse field distributions for R = 1 mm, r = 0.3 mm, and m = 104. The upper and lower rows correspond to the modes with p = 0, q = 1 and p = 2, q = 2, respectively. The panel a) gives the analytical results in accordance with Eq. (10), the panel b) represents the numerical data for the largest field component of the v- and h-modes, and the panel c) shows the dependences E(x) along the white solid lines in a) and b). The dashed lines indicate the resonator rim.
4. Comparison with experiment
In our experiment, sketched in Fig. 4(a), we used a disc resonator made of magnesium fluoride (MgF2). The optically measured radii R and r were 2.96 and 1.63 mm, respectively. The tunable light source was a Toptica DL pro design laser operating at λ ≃ 1550 nm corresponding to an angular frequency ω ≃ 2π × 193 THz. Thus, the corresponding extraordinary refractive index is n ≃ 1.38 [30] and the azimuthal number m ≃ 16500. A glass prism was used to couple vertically polarized light in and out of the resonator. A photodiode detects the transmitted light.
Fig. 4 a) Schematic of the experimental setup including the laser emitting at the angular frequency ω, the electro-optic modulator (EOM) driven at the angular frequency Ω, the coupling prism, the whispering gallery resonator (WGR), and the detector. b) Typical detector signal versus the frequency ω showing two side peaks with the distance |Ω − Δω| with respect to the main resonance. Here, Δω is the free spectral range.
Within a free spectral range (FSR) Δωm,p,q/2π ≃ 11.6 GHz there are usually multiple resonances. Its dependence on q is typically the strongest. This can be used to identify different WGMs by employing the above model. Our experimental procedure is similar to that used earlier for silica wedge resonators [29]. Its essence is as follows. The frequency ω of the input beam is swept across a resonance at ω0 and the transmission spectrum is recorded. The input beam is modulated using an electro-optic modulator with a variable frequency Ω to generate the side bands at ω ± Ω. If Ω is close to Δω, but |Ω − Δω| is larger than the linewidth, then the ω-spectrum consists of a main dip at ω = ω0 and two symmetric dips at ω = ω0 ± |Ω − Δω|, see Fig. 4(b). When Ω approaches Δω, the side dips move towards each other. At Ω = Δω, the side dips disappear and the depth of the main dip is sharply increasing. This method allows for FSR measurements with a precision of up to a tenth of the WGM linewidth which leads to an accuracy of about 50 kHz for our resonator exhibiting linewidths ≈ 500 kHz (finesse ℱ ≈ 2 × 104).
Figure 5(a) shows a fraction of the transmission spectrum of the whispering gallery resonator. It comprises multiple modes that can be considered for the FSR measurement. The red filled circles in Fig. 5(b) indicate our experimental FSR data for 31 different modes. They can be divided into 5 groups with almost the same FSR values. These groups are expected to correspond to different values of q. In order to verify this, we compare the experimental data with the model predictions according to Eq. (11). At this stage it is important to employ the frequency dependency n(ω) given by the relevant Sellmeier equation [30]. The theoretical predictions for m = 16500, p = 0, and q = 1 to 5 are shown by the horizontal blue lines in Fig. 5(b). We kept the m number fixed and used the radius R as a fitting parameter leading to a value of 2.9525 mm. One can see that our model nicely reproduces the experimental observations. The FSR dependence on the polar number p is very weak; the corresponding changes are all within the linewidth.
Fig. 5 a) Fraction of the transmission spectrum of our MgF2 resonator; b) Free spectral ranges determined for 31 considered modes (red filled circles). The blue lines indicate the model predictions for m = 16500, p = 0, and q = 1 to 5.
5. Discussion
We have presented a simple perturbation method to describe the resonant frequencies and the eigenfunctions of WGMs localized near the equator of an axisymmetric optical resonator characterized by the major and minor radii R and r. As follows from this method and comparison with numerical simulations, it gives a relative accuracy 10−3 – 10−4 for R, r ≳ 100λ, i.e., R, r ≳ 100 μm in the optical range. This accuracy grows with increasing R, and it seems to be sufficient for practical purposes, such as identification of different types of WGMs in FSR experiments and characterization of nonlinear processes via calculation of the overlap integrals. The global shape of the resonator does not affect the WGM properties within the above accuracy.
The well known alternative to our approach is the characterization of WGMs by the eikonal method [22,23]. This method is more complicated, it is also essentially scalar, and it experiences difficulties with boundary conditions. Nevertheless, being applied to the spheroidal geometry it gives very precise asymptotic expressions for the eigenfrequencies. Our Eq. (11) corresponds to the leading terms of Eq. (19) of [23]. The latter includes also higher-order corrections in m. For millimeter-sized resonators and m ∼ 104, both expressions coincide with a relative accuracy of 10−6. Thus, we have got simplicity in expense of a very high accuracy.
Three issues relevant for the calculation accuracy of the above methods have to be mentioned:
- – Ignorance of vectorial effects leads to errors in determination of the WGM frequencies. A very high accuracy of solving the scalar Helmholtz equation does not ensure the same accuracy for the true vectorial case.
- – It is not clear from the results of [22, 23], what kind of requirements to the shape of the resonator are needed to fit the very high calculation accuracy. Even the simplest shape parameters R and r as well as the refractive index n cannot be known precisely enough. For this reason, even our modest calculation accuracy (∼ 10−4) can be excessive in many cases.
- – In the case of ultra-small resonators with radii R, r considerably smaller than 100λ, the relative accuracy of our method drops perhaps to ∼ 10−2. This can be sufficient for many WGM applications but, quite possible, insufficient for the comb applications [11]. Here the use of more precise calculation methods can be necessary.
- – The evanescent fields and the radiation losses can be incorporated into our model by changing the boundary condition for the radial function U(u). This complicates the description without significant changes of the resonant frequencies and the field distributions inside mm-sized resonators. For ultra-small resonators, including the bottle resonators [24], such a generalization would be useful.
6. Conclusions
We have derived simple analytic expressions for the eigenfrequencies and the corresponding field distributions relevant to optical whispering gallery resonators with two size parameters –the major radius R and the minor radius r. They are in very good agreement with the results of our numerical finite-element simulations and also with our experimental data for a MgF2 disc resonator. These expressions are useful for calculation of the effective refractive indices, the mode cross-sections, the mode volumes, the modal overlaps, and identification of WGMs in experiment.
Acknowledgments
We gratefully thank the Deutsche Forschungsgemeinschaft (DFG) for financial support.
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