1. Introduction

Optical whispering gallery resonators (WGRs) attract a strong research interest [1–6]. Their outstanding properties include very high Q-factors (up to 10¹⁰) of WGR modes, ultra-small mode volumes, and flexible tools to couple light in and out [7,8]. Applications of WGRs span from sensors of single atoms and optomechanics [2–4] to quantum and nonlinear optics [1,5,6,9–11]. The nonlinear-optical applications benefit from huge intensity enhancement in WGRs providing efficient frequency transformation processes already in the milliwatt power range available from conventional continuous-wave (CW) lasers [9–11]. The sizes of WGRs range from tens of μm to a few mm, and the optical materials can be both isotropic (glasses) and anisotropic (crystals). The phase-matching issues lie often at the center of nonlinear-optical studies with WGRs [9,11–13].

The theoretical basis for the use of WGRs is heavily based on the assumption of axial symmetry. The only situation with a full analytical solution for the WGR modes is the case of an isotropic sphere. Most of WGRs, especially made of crystals, are non-spherical. Different asymptotic methods and approximations employing the axial symmetry are known for analysis of light field distributions and resonant frequencies, i.e. the modal structure of the WGR [14–16]. The results depend on the WGR shape – spheroidal, toroidal, etc. Numerical simulations incorporating a 3D–to–2D reduction of the spatial dimensions owing to the axial symmetry have become available only recently [17]. Broken axial symmetry makes accurate 3D simulations highly problematic. Experimental identification of the modal structure in mm-sized WGRs, relevant to crystalline host materials, is still a topical issue [18].

Despite the absence of a firm basis for assessment of the WGR modal structure in the case of broken axial symmetry, some theoretical and experimental work has been conducted. It is known within geometric optics that a part of the light ray trajectories may become chaotic for sufficiently large deviations from the axial symmetry leading to a deterioration of the WGM structure [19]. An experimental support for this idea is known for specially designed angle-cut WGRs [20]. Moreover, high-Q modes have been demonstrated in WGRs with the optic axis tilted with respect to the resonator axis [21–23]. Notably, using a heuristic idea about the phase matching under broken axial symmetry, broad-band second harmonic generation has been realized experimentally in asymmetric WGRs made of beta barium borate (BBO) crystals [24].

Advent of crystal-based WGRs for low-power nonlinear optics necessitates search for materials with broad IR–to–UV transparency windows and large χ⁽²⁾ coefficients. However, many important crystals possessing these properties, such as, e.g., lithium triborate (LBO), potassium titanyle phosphate (KTP), and bismuth triborate (BIBO), are biaxial [25]. WGRs made of such crystals have generally no axial symmetry even when being shape-symmetric. Also uniaxial crystals, such as lithium niobate (LN), lithium tetraborate (LB4), zinc germanium phosphide (ZGP), silver selenogallate (AGSe), and BBO [25], can be used to produce WGRs with broken axial symmetry.

In this article, we provide a basis for the use of WGRs made of birefringent crystals and possessing no axial symmetry. We show that the relative smallness of the optical anisotropy, inherent in optical crystals, makes its influence on the resonant frequencies fairly weak and accessible. At the same time, the impact of the broken axial symmetry on the modal eigenfunctions can be very strong leading to well defined predictions with regard to χ⁽²⁾ nonlinear processes, such as second harmonic generation (SHG) and optical parametric oscillation (OPO).

2. Theoretical treatments

2.1. Basic equations

Full three-dimensional (3D) treatment of WGRs with broken axial symmetry seems to be non-realistic so far. However, the experience in investigation of the modal structure in the axially-symmetric case says that 2D analysis of the modes of a circular cylinder provides important information about the frequencies and field distributions for the most important equatorial WGR modes in the 3D case. Therefore, this model is taken as a basis for WGRs also with broken axial symmetry and we expect that experiment confirms the theoretical predictions.

Specifically, we consider whispering gallery modes in a circular cylinder of radius R made of a birefringent (biaxial or uniaxial) crystal with the refractive indices n_x,y,z along the principle dielectric axes assuming that x is the cylinder axis, see also Fig. 1. Unless the crystal symmetry is too low (monoclinic or triclinic), each principal axis coincides with one of the crystallographic axes X, Y, Z. In the case considered, the x-polarized TE-modes are axially symmetric and well investigated. For these modes, only the refractive index n_x is relevant. For the TH-modes, the light electric field E lies in the y, z-plane, and the axial symmetry is broken. The only non-zero magnetic field component H = H_x at the light frequency ν obeys the equation

 

Fig. 1 a) Geometric scheme of a WGR with broken axial symmetry. The x, y, z-axes are directed along the principal dielectric axes and n_y > n_z; the TE- and TH-modes are axially symmetric and asymmetric, respectively. b) Normalized dependence of Λ_m on β: The circles are obtained numerically for m = 10³ without approximations; the open and filled circles correspond to ${\mathrm{\Lambda }}_m^{+}$ and ${\mathrm{\Lambda }}_m^{-}$. The circles calculated for m = 10⁴ and the same β/m² practically coincide with the shown ones. The solid line corresponds to Eq. (4).

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The whispering gallery mode problem is solved in the elliptic coordinates φ, u, such that

The variable separation H(φ, u) = Φ(φ) U(u) in Eq. (1) leads to the ordinary differential equations

It is known that for any value of β there are two infinite sets ${\mathrm{\Lambda }}_m^{\pm }(\beta )$ with m = 0, 1, . . . corresponding to 2π-periodic even (+) and odd (−) real functions ${\mathrm{\Phi }}_m^{\pm }(\beta ,\phi )$ [27]. The integer m gives the number of zeros; to a great extend it is analogous to the azimuth number in the axisymmetric case. We call m the quasi-azimuth number.

2.2. Whispering gallery mode frequencies and eigenfunctions

Handling of the Mathieu functions ${\mathrm{\Phi }}_m^{\pm }(\beta ,\phi )$ and the separation constant ${\mathrm{\Lambda }}_m^{\pm }(\beta )$ is generally complicated, both with analytical and numerical tools. However, the situation is greatly simplified for m ≫ 1 and β/m² ≪ 1. This limiting case, as we will see in detail, is fully sufficient for our purposes. In practice, the azimuth number for mm-sized WGRs can roughly be estimated as m ∼ 10⁴.

The following asymptotic relation is known for m ≫ 1 and β ≪ m² [27]:

In order to verify the accuracy and the field of applicability of the asymptotic relation, we have compared its prediction with virtually exact numerical data evaluated using Wolfram Mathematica. The solid line in Fig. 1 shows Λ_m/m² versus β/m² according to Eq. (4), whereas the open and filled circles give numerical data for ${\mathrm{\Lambda }}_m^{+}/m^2$ and ${\mathrm{\Lambda }}_m^{-}/m^2$, respectively; each circle is calculated independently for m = 10³ and 10⁴ with practically no difference. Also the values of ${\mathrm{\Lambda }}_m^{+}$ and ${\mathrm{\Lambda }}_m^{-}$, calculated for the same m and β, coincide with each other to the fifth decimal place. Thus, increasing the ratio β/m² is the main source for inaccuracy of the asymptotic relation. As follows from Fig. 1, the circles lie practically on the solid line for β/m² ≲ 0.15. Within this range, we can substitute Eq. (4) into Eqs. (3) and be sure that this procedure is well justified.

Next we must employ a boundary condition for the quasi-radial function U(u). Experience of the WGR modeling says that a zero boundary condition for the radial functions gives a sufficiently high accuracy for determination of the modal spectrum, see, e.g., [15,28] and references therein. Thus, we set U(u₀) = 0 as the boundary condition. Since U(u) is strongly localized near the rim, the linear expansion of cosh u near u₀ in Eq. (3) is applicable. This leads to the Airy equation for U,

Domination of the first term in the square bracket of Eq. (7) allows us to express approximately the quasi-azimuth number m by the vacuum wavelength λ = c/ν, the radius R, and the average refractive index n̄: m ≃ 2πn̄R/λ. A similar estimate is in use for the axially symmetric WGRs. For λ = 1 μm, R = 1 mm, and n̄ = 2 we have m ≃ 1.25 × 10⁴ ≫ 1.

Let us check whether the approximations made are justified. Since πνR/c ≃ m/2n̄ in the initial expression for β, we have β/m² ≃ Δn/2n̄. This quantity is well below 0.15 for the known birefringent optical materials, so that the asymptotic relation Eq. (4) is applicable with a big margin of safety. Using our estimate of β/m², one can check also that the expression for the quasi-radial function acquires the form U_m(u) = const × Ai[2^1/3m^2/3(u₀ −u) − ξ_q]. For m ∼ 10⁴ and not very large values of q it is well localized near the rim, which justifies the linear expansion of cosh u near the rim.

Now we consider in some detail the properties of the quasi-azimuth functions ${\mathrm{\Phi }}_m^{\pm }(\phi )$ that express the broken axial symmetry. Importantly, the effect of Δn on these functions is much stronger than the effect on ν. The relevant quasi-classical expression for the complex eigenfunction ${\mathrm{\Phi }}_m(\phi )={\mathrm{\Phi }}_m^{+}(\phi )+\text{i}{\mathrm{\Phi }}_m^{-}(\phi )$, as derived in detail in [29], reads:

 

Fig. 2 a) Dependence f(λ) for several birefringent crystals and R = 1 mm within the transparency window on a semi-logarithmic scale; the letters in the parentheses indicate the crystallographic orientation of the WGR x-axis. b) Illustration of the discrete Fourier spectrum of the function Φ_m (φ) for f = 100.

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Note that the quasi-classical relation (8) is not commonly known, and it is expected to be valid for m² ≫ 1, 2β/m² ≪ 1 [29]. Numerical treatment of the Mathieu functions ${\mathrm{\Phi }}_m^{\pm }={\mathrm{\Phi }}_m^{\pm }(\beta ,\phi )$ with m ∼ 10⁴ is problematic. Thus, we have checked numerically the validity of Eqs. (8, 9) for m ≈ 200 and Δn/n̄ < 0.1. The standard deviation between the numerical and analytical functions is below 1% for m = 200, and it decreases with increasing m and decreasing Δn/n̄.

Strong phase modulation in Eq. (8) leads to a strong broadening of the Fourier spectrum of the 2π-periodic function Φ_m(φ). Taking into account the equality

In a similar manner, one can find substantial phase distortions in elliptically-shaped resonators made of isotropic materials. In this case, the phase distortion parameter is f ≃ πnΔR/2λ, where ΔR is the semi-axes difference. The phase modulation is strong for δR ≳ 2λ/πn. In practice, this condition can be fulfilled already for sub-μm deviations from the ideal circular shape.

2.3. Impact on the phase-matched nonlinear processes

Strong phase modulation of the azimuth functions for the modes with broken axial symmetry leads to important physical consequences for the phase-matched nonlinear processes like second-harmonic generation and optical parametric oscillation. For definiteness, we restrict ourselves below to the impact on second-harmonic generation (SHG), when only the pump (p) and signal (s) modes at the vacuum wavelengths λ_p and λ_s = λ_p/2, respectively, are present. At least one of the p,s-modes is expected to be axially asymmetric. Different polarization combinations correspond to different coupling schemes for the p- and s-modes, see also below. In any case, the former strict selection rule m_s = 2m_p for SHG, caused by the axial symmetry, is not valid any more. This enriches the choice of the quasi-azimuth numbers for the p- and s-modes. In a sense, the situation is similar to the quasi-phase-matching in radially poled WGRs [11].

To quantify the impact on SHG, we consider the absolute value of the φ-average $〈{\mathrm{\Phi }}_{m_{\text{p}}}^2{\mathrm{\Phi }}_{m_{\text{s}}}^{*}〉$. Setting aside the φ-independent transverse overlap factor, it can be treated as the ratio d_eff/d for the relevant nonlinear coefficient d. According to Eq. (8), this value depends on δm = 2m_p − m_s and also on δf = 2 f_p − f_s. The values of f_p = f_p(λ_p) and f_s = f_s(λ_s) correspond to the p- and s-modes. If one of these modes is axially symmetric (TE), the corresponding value of f is zero. Using Eq. (10), we obtain easily for even δm:

 

Fig. 3 Factor $d_{\text{eff}}/d=\left|〈{\mathrm{\Phi }}_{m_{\text{p}}}^2{\mathrm{\Phi }}_{m_{\text{s}}}^{*}〉\right|$ versus δm for different values of δf. Note the difference in the horizontal scales in the subfigures.

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The value of δf depends on the coupling scheme for the p- and s-modes. If they both are subjected to the axial symmetry breaking (i.e., of the TH-type), then we have from Eq. (9): δf ≃ πR [Δn(2λ_s) − Δn(λ_s)]/2λ_s. Because of a partial cancelation of the contributions, the parameter |δf| is substantially smaller than f at the same wavelength, see Fig. 2(a); it ranges roughly from ∼ 10¹ to ∼ 10². In the case when the p-mode is axially symmetric (TE), while the s-mode is asymmetric (TH), we have |δf| = f_s. Similarly, in the case when the p-mode is asymmetric, while the s-modes are axially symmetric, we get |δf| = 2 f_p. Figure 2(a) enables one to judge about the behavior of |δf|(λ_s) in the last two cases for different crystals.

Note that SHG is controlled generally by different elements of the nonlinear susceptibility tensor d̂ in the above polarization schemes. For some elements, often the weakest, additional φ-dependent factors (like cos φ or cos 3φ) appear in the expression for d_eff/d [25, 31]. They result in shifts in m in Eq. (12), which are of minor importance for |δf| ≫ 1.

What is the impact of the modified azimuth phase matching condition (12) on SHG? To see this impact, it is necessary to take into account the frequency condition ν_s = 2ν_p rewritten in terms of m_s,p. It reads m_s/n_s = 2m_p/n_p in the leading approximation. We have neglected the weak geometrical dispersion and the quadratic in Δn corrections to ν_s,p. The indices n_s and n_p are equal to n̄(λ_s) and n̄(2λ_s) for the axially asymmetric s, p-modes and to n_x (λ_s) and n_x (2λ_s) in the case of symmetric modes. Combining the frequency condition with Eq. (12), we obtain

Let us apply Eq. (13) to the different coupling schemes. In the case when both s, p-modes are axially asymmetric (TH), the condition |δm/δm_max| ≤ 1 is fulfilled only when the index differences n_y (λ_s) − n_y (2λ_s) and n_z (λ_s) − n_z (2λ_s) are opposite in sign. Generally, the dependences n_x,y,z (λ) are decreasing within the transparency window [32], so that both above differences are positive and the modified phase matching does not help to reach SHG. For the other two schemes, the situation is different.

Consider the scheme with an axially symmetric (TE) p-mode and an asymmetric (TH) s-mode. Here the range −1 ≤ δm/δm_max ≤ 1 corresponds to the index condition n_z (λ_s) ≤ n_x (2λ_s) ≤ n_y (λ_s), and broad-band SHG is often allowed. In particular, it is typically allowed for negative uniaxial crystals. In this case, n_x,y (λ) = n_o(λ), n_z (λ) = n_e(λ), and n_o(λ) > n_e(λ); the subscripts o and e refer to the ordinary and extraordinary polarizations, respectively. The above index condition reads here: n_e(λ_s) ≤ n_o(2λ_s) ≤ n_o(λ_s). Since the dependence n_o(λ) is decreasing, n_o(2λ_s) stays below n_o(λ_s). At the same time, n_o(2λ_s) can stay above n_e(λ_s) within a certain range of λ_s. This means that δm/δm_max crosses the value of −1 at a certain λ_s, but it cannot reach the value of 1 for larger λ_s. This is illustrated by lines 1, 2, 3, and 4 in Fig. 4(a) for negative uniaxial crystals BBO, LB4, LN, and AGSe [25]. For positive uniaxial crystals (n_e > n_o), SHG is forbidden in this scheme. Employment of biaxial crystals strongly extends the possibilities for broad-band SHG. In particular, δm/δm_max can vary with λ_s over the whole interval [−1, 1] and the choice of the x-axis is not unique. This is illustrated by lines 1 and 2 in Fig. 4(b) for biaxial LBO crystal possessing mutually orthogonal crystallographic axes and the property n_Z > n_Y > n_X.

 

Fig. 4 a) δm/δm_max versus λ_s within the transparency windows of different uniaxial crystals. Lines 1, 2, 3, and 4 correspond to negative crystals BBO, LB4, LN, and AGSe; line 5 refers to positive ZGP crystal. The symbols indicate the crystallographic orientation of the x-axis and polarization of the p-mode. b) The same for biaxial LBO crystal. Curves 1 to 4 correspond to different choices of the x-axis and polarization of the p-mode.

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The last coupling scheme to consider corresponds to axially symmetric (TE) s-mode and axially asymmetric (TH) p-mode, i.e., to n_p = n̄(2λ_s), n_s = n_x (λ_s), and Δn_s = 0 in Eq. (13). Here, the broad-band SHG is possible when the condition n_z (2λ_s) ≤ n_x (λ_s) ≤ n_y (2λ_s) is fulfilled. It is is accessible for positive uniaxial crystals and also for biaxial crystals. The first possibility is shown by line 5 in Fig. 4(a) for ZGP crystal, while the second one is illustrated by lines 3 and 4 in Fig. 4(b) for LBO crystal. The same biaxial crystal LBO can thus be employed for different polarization schemes. Looking at Figs. 4(a) and 4(b), we see that the spectral ranges for SHG vary strongly for different crystals and schemes, but these ranges are really broad.

Remarkably, the index conditions, necessary for broad-band SHG in the above two coupling schemes, admit an intuitively accessible representation. This representation is relevant to the so-called cycling phase matching [24], and it is illustrated by Fig. 5. For the scheme with axially symmetric p-mode, the refractive index n_p = n_x (2λ_s) stays constant in φ, whereas the effective index of the signal wave $n_s^{\text{eff}}(\phi )$ varies periodically between n_y (λ_s) and n_z (λ_s), see Fig. 5(a). This provides locally fulfillment of the equality n_eff(φ) = n_x (2λ_s). In the case of negative uniaxial crystals, we have: n_x,y = n_o and n_z = n_e < n_o; for any λ_s, the horizontal red line stays here below the upper dotted line providing the inequality −1 < δm/δm_max ≤ 1. For positive uniaxial crystals, we have instead: n_x,z = n_o and n_y = n_e > n_o; here the red line stays below the lower dotted line and SHG is impossible. Similarly, one can consider the scheme with axially symmetric s-mode, see Fig. 5(b). In this case, SHG is forbidden for negative uniaxial crystals and allowed for positive ones. In such a way, there is one-to-one correspondence between Eq. (13) and the heuristic idea of cycling phase matching [24].

 

Fig. 5 Local in φ fulfillment of the phase matching conditions $n_{\text{p}}=n_{\text{s}}^{\text{eff}}(\phi )$ (subfigure a) and $n_{\text{s}}=n_{\text{p}}^{\text{eff}}(\phi )$ (subfigure b) for the polarization schemes with TE and TH pump modes, respectively. Position of each horizontal solid line relative to its cos-like counterpart depends on λ_s. In the case of negative uniaxial crystals, n_x,y = n_o and n_z = n_e < n_o, while for positive uniaxial crystals n_x,z = n_o and n_y = n_e > n_o.

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3. Experimental identification of WGR modes with broken axial symmetry

Equation (7) shows that the resonant frequencies ν_m,q of a birefringent cylindrical WGR are determined by modifying the dispersion relation of a non-birefringent cylindrical WGR. Here, the refractive index is substituted with the average value n̄ and the term (Δn/4n̄)² including the birefringence Δn is added. Now, we want to investigate if such a simple modification is also valid for birefringent disk WGRs possessing the major radius R and the minor radius r, see Fig. 6(a). Since the WGR modes are strongly localized near the rim, the resonant frequencies of such a disk resonator are very close to the ones of a spheroidal resonator with the same major and minor radii. Here, the dispersion relation for the non-birefringent case is known [14] and experimentally verified [18]. Thus, we take this relation and substitute the refractive index by n̄. Furthermore, we replace the polarization dependent term with the one proportional to (Δn)² to get the adapted dispersion relation for a birefringent disk WGR.

 

Fig. 6 a) Geometry of a disk WGR with the major and minor radii R and r; the optic axis (red line) lies in the equatorial (horizontal) plane. b) Experimental setup comprising a tunable external-cavity diode laser (ECDL), a Fabry-Pérot interferometer (FPI), a coupling prism (P), the WGR, and two detectors (D). c) Transmission spectrum of the WGR ±15 GHz around the center frequency.

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To verify it experimentally, we basically follow the procedure presented in Ref. [18] for a non-birefringent disk WGR. We have fabricated a disk WGR out of uniaxial birefringent magnesium fluoride (MgF₂). The major and minor radii measured are R = 1576 ± 2 μm and r = 0.6 mm, and the optic axis lies in the equatorial plane, see also Fig. 6(a). Only the major radius was measured with μm accuracy, as it dominates the spectral characteristics of the WGR.

The transmission spectrum and hence the WGR resonant frequencies were investigated with the experimental setup depicted in Fig. 6(b). A CW external-cavity diode laser (DLpro from Toptica) possessing a center frequency ν_c of about 288.2 THz, a linewidth of ≈ 100 kHz, and a mode-hop-free tuning range of ≈ 30 GHz was employed as a tunable light source. The optical powers used were below 1 mW. The focused laser beam was evanescently coupled to the WGR via a sapphire prism placed in close proximity to the resonator rim. The optic axis of the prism was parallel to the rotation axis of the WGR. The spacing between the rim and the prism was controlled with a micrometer screw. The outcoupled light beam was monitored with a photo-detector. The setup was placed in a temperature stabilized environment.

Additionally, a Fabry-Pérot interferometer (FPI) with a free spectral range (FSR) of (1500 ± 1) MHz was employed. Scanning the laser frequency, one gets optical transmission peaks every FSR of this interferometer. This frequency difference was used as a frequency ruler, applying linear interpolation between consecutive peaks. An independent calibration of the laser frequency with a wavemeter has shown a standard deviation of the frequency ruler to the actual laser frequency in the scan of approximately 50 MHz.

The frequency spectrum of our WGR was measured by scanning the laser frequency over about 30 GHz. All measurements were performed in the undercoupled regime [8]. The light was polarized normal to the resonators axis, which corresponds to the excitation of axially asymmetric modes. The result is shown in Fig. 6(c) by the downward peaks. Numerous very narrow resonances are clearly seen. The intrinsic Q factors of the resonances are about 3.3 × 10⁹, and ringing (peaks with transmission > 1) [33] could be observed.

The experimental spectrum of Fig. 6(c) carries a great body of information about the frequencies of the axially asymmetric modes. Using the adapted dispersion relation and the refractive index data of MgF₂ [34], one can also calculate the theoretical spectrum of discrete frequencies for any values of R and r. In particular, we can find the frequency spectrum within a certain window of 100 GHz (fully covering the 30 GHz experimental range) for the most important low-order radial modes. Superimposing both frequency spectra and slightly shifting them from each other, the resonances coincide (or almost coincide) if the theoretical spectrum is calculated correctly. Thus, the spectral fingerprint of the WGR can be identified. The necessity of the the frequency adjustment of the spectra stems from the fact that only relative (but not absolute) values of the frequencies are known in experiment. A quantitative measure for the similarity of the spectra is provided by their cross-correlation. Its maximum value (MCC) corresponds to the best correlation.

Importantly, slight uncertainties in the center frequency ν_c and in the major radius R affect the MCC. Thus, we consider them as variable fit parameters. As the above FPI frequency calibration leads to errors of about 50 MHz, this flexibility has to be put on the theoretical spectrum. This is done by introducing an artificial linewidth of 50 MHz for all resonant frequencies. The result of calculation of the MCC as a function of R and ν_c is presented in Fig. 7(a). One sees that only one dominant correlation peak is present over a large parameter window. Correspondingly, we can estimate the major radius and the center frequency very precisely. The best fit gives here R = 1576.3 μm and ν = 287.92 THz. Furthermore, eight-times repetition of the spectral experiment and the fit procedure yields R = (1576 ± 1) μm and ν = (288.0 ± 0.2) THz. Both values match well with the independently measured major WGR radius and the center frequency of the laser.

 

Fig. 7 a) Maximum cross correlation (MCC) versus the center frequency ν_c and the major radius R. The absolute maximum occurs at ν_c = 287.92 THz and R = 1576.3 μm. b) The resonant frequencies for q = 1 . . . 10 (red) at R = 1576.3 μm around ν_c = 287.92 THz compared to the experimentally recorded transmission spectrum (blue).

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Figure 7(b) gives comparison of the experimental and theoretical spectra for the fit parameters of Fig. 7(a). A strong similarity of the spectra is evident. We can identify unambiguously the first seven radial modes. And even the remaining three radial modes have counterparts. Thus, we deduce that our theoretical model describes very well the frequencies of our non-axial resonator.

To gain more insights into the modal structure, we have repeated our experiment and the cross-correlation analysis for light polarized parallel to the WGR symmetry axis, i.e., for the axisymmetric case. Four independent measurements yield R = (1577 ± 2) μm and ν = (287.9 ± 0.3) THz, and the mode assignment works perfectly well again despite of entirely different individual resonant frequencies. The intrinsic quality factor of WGR modes is here about 3.4 × 10⁹; it is almost the same as the Q factor in the non-axial case. Similar Q-factors are observed also in a z-cut axisymmetric MgF₂ based WGR for both polarizations. Degradation of the Q factor in the non-axial case cannot thus be deduced within the measurement accuracy.

4. Discussion

Combining theory and experiment, we have proven that high-Q whispering gallery modes exist in mm-sized resonators with broken axial symmetry, caused by birefringence of the host material. The most important and well recognizable equatorial modes are structurally similar to the modes of axisymmetric resonators, so that the notion of azimuth and radial modal numbers m and q can still be applied. Non-equatorial high-Q modes also exist, but they cannot be described and identified so far. A similar assertion is valid also with regard to the modes of WGRs possessing a weak shape ellipticity, caused, e.g., by noncontrollable lashing misalignments.

Importantly, the effect of broken axial symmetry on the modal functions is much stronger than the effect on the modal frequencies. Moreover, this effect cannot be considered as weak for most of birefringent optical materials. This circumstance is crucial for the phase-matched nonlinear processes in WGRs, such as SHG and OPO: Large azimuth mismatches δm = 2m_p − m_s and δm = m_p − m_s − m_i become possible between the pump (p), signal (s), and idler (i) modes. This can lead to broad-band SHG and OPO with only a modest lowering of the effective nonlinear susceptibility. The latter can be easily compensated by high quality factors of the relevant WGR modes.

Within our approach, we have considered in some detail the properties of the broad-band SHG as applied to different WGR polarization schemes and different types of birefringent crystals – negative and positive uniaxial crystals and biaxial crystals. We predict that many known birefringent χ⁽²⁾ materials, including LN, LB4, AGSe, ZGP, LBO, and BIBO, can be employed in different schemes to realize broad-band SHG in mm-sized WGRs within a broad spectral range spanning from near infrared to ultra-violet. Realization of broad-band OPO with WGRs is also very likely.

The above broad-band SHG has been convincingly realized in non-axial WGRs made of uniaxial BBO crystals [24]. It was interpreted on the basis of a heuristic idea of “cycling phase matching” assuming a continuous changing refractive index of the signal wave along the WGR circumference. In essence, the theoretical part of our paper provides a solid theoretical basis for this idea. It gives an adequate description of the modal structure, of the modified phase matching, and of the effective nonlinear coefficient controlling the process.

Thus, the sum of our findings paves the way for a quantitative analysis of experimental results obtained with whispering gallery resonators made of birefringent optical materials (both uniaxial and biaxial) in nonlinear optics. It is clear that broken axial symmetry cannot not be an obstacle for employment of broad transparency windows and large nonlinear coefficients of such materials. Moreover, the use of asymmetric WGRs promises substantial advantages for broad-band phase matching.

5. Conclusions

It is shown theoretically and experimentally that the frequencies of the equatorial modes in WGRs with broken axial symmetry, caused by in-plane optical anisotropy, can be quantitatively described and identified in terms of quasi-azimuth and quasi-radial modal numbers. The effect of broken axial symmetry on the modal functions is much stronger than the effect on the frequencies. This leads to an essential modification of the phase-matching conditions for the χ⁽²⁾ nonlinear processes and enables one to realize broad-band second-harmonic generation and optical parametric oscillation. The results obtained pave the way for an extensive use of numerous birefringent optical materials in nonlinear optics with WGRs.