## Abstract

Axial symmetry is the cornerstone for theory and applications of high-*Q* optical whispering gallery resonators (WGRs). Nevertheless, research on birefringent crystalline material persistently pushes towards breaking this symmetry. We show theoretically and experimentally that the effect of broken axial symmetry, caused by optical anisotropy, is modest for the resonant frequencies and *Q*-factors of the WGR modes. Thus, the most important equatorial whispering gallery modes can be quantitatively described and experimentally identified. At the same time, the effect of broken axial symmetry on the light field distribution of the whispering gallery modes is typically very strong. This qualitatively modifies the phase-matching for the *χ*^{(2)} nonlinear processes and enables broad-band second harmonic generation and optical parametric oscillation. The effect of weak geometric ellipticity in nominally symmetric WGRs is also considered. Altogether our findings pave the way for an extensive use of numerous birefringent (uniaxial and biaxial) crystals with broad transparency window and large *χ*^{(2)} coefficients in nonlinear optics with WGRs.

© 2016 Optical Society of America

## 1. Introduction

Optical whispering gallery resonators (WGRs) attract a strong research interest [1–6]. Their outstanding properties include very high *Q*-factors (up to 10^{10}) 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 *μ*m 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*is relevant. For the TH-modes, the light electric field

_{x}**E**lies in the

*y*,

*z*-plane, and the axial symmetry is broken. The only non-zero magnetic field component

*H*=

*H*at the light frequency

_{x}*ν*obeys the equation

*k*= 2

*πν/c*and

*c*is the speed of light. After the coordinate stretching

*ŷ*=

*yn*,

_{y}*ẑ*=

*zn*, it becomes the isotropic Helmholtz equation, but the boundary becomes elliptic. The field components

_{z}*E*are easily expressible by

_{y,z}*H*from Maxwell’s equations. From now on, we set

*n*<

_{z}*n*without loss of generality. As for the index

_{y}*n*, it can be arbitrary. The particular cases

_{x}*n*=

_{x}*n*and

_{y}*n*=

_{x}*n*correspond to the negative and positive uniaxial crystals, respectively.

_{z}The whispering gallery mode problem is solved in the elliptic coordinates *φ*, *u*, such that

*φ*≤ 2

*π*, 0 ≤

*u*≤

*u*

_{0}, and

*u*

_{0}= arctanh(

*n*) [26]. The variables

_{z}/n_{y}*φ*and

*u*are similar to the polar and radial coordinates. For

*n*−

_{y}*n*≪

_{z}*n*, the quasi-polar angle

_{z,y}*φ*is very close to the true polar angle. The coordinate line

*u*=

*u*

_{0}corresponds to the WGR boundary.

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

*ν*. The equations for Φ and

*U*are known as Mathieu and modified Mathieu equations, respectively [27]. They occur when dealing with many physical problems [26]. The physical solutions for Φ(

*φ*) are 2

*π*-periodic, and solutions for

*U*(

*u*) have to be strongly localized near

*u*

_{0}(near the rim). Both parameters, Λ and

*β*, have to be determined. If, as might be suggested, the frequency

*ν*is about

*mc/*2

*πn*

_{y,z}*R*in the leading approximation, then the ratio

*β/m*

^{2}expresses the degree of anisotropy and it has to be small. The subsequent rigorous calculations will prove this suggestion.

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 2*π*-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/m*^{2} ≪ 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* ∼ 10^{4}.

The following asymptotic relation is known for *m* ≫ 1 and *β* ≪ *m*^{2} [27]:

*i*) that ${\mathrm{\Lambda}}_{m}^{+}(\beta )={\mathrm{\Lambda}}_{m}^{-}(\beta )$ with a high accuracy and (

*ii*) that the second term on the right-hand side gives a relatively small

*β*-dependent correction to the main first term. This correction, as we have suggested, is relevant to the optical anisotropy.

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}/m^{2} versus *β/m*^{2} 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* = 10^{3} and 10^{4} 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/m*^{2} 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/m*^{2} ≲ 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*(*u*_{0}) = 0 as the boundary condition. Since *U*(*u*) is strongly localized near the rim, the linear expansion of cosh *u* near *u*_{0} in Eq. (3) is applicable. This leads to the Airy equation for *U*,

*u*

_{0}−

*u*is proportional to the Airy function:

*U*=

*const*× Ai(

*ξ*−

*ξ*

_{0}). Recalling that

*U*(

*u*

_{0}) = 0, we have furthermore

*ξ*

_{0}=

*ξ*, where

_{q}*ξ*is the

_{q}*q*-th zero of Ai(−

*ξ*):

*ξ*

_{1,2,3,...}≃ 2.34, 4.09, 5.52, etc. This is the dispersion equation for the mode frequency

*ν*=

*ν*sought for. It is simplified when we express sinh 2

_{m,q}*u*

_{0}and cosh 2

*u*

_{0}in the expression for

*ξ*

_{0}by

*n*and

_{y}*n*using the equality tanh

_{z}*u*

_{0}=

*n*. Finally, we get:

_{z}/n_{y}*g*= 2

*πνR/c*is a convenient combination of parameters and $\u3008{n}^{2}\u3009=\left({n}_{y}^{2}+{n}_{z}^{2}\right)/2$. The last term in this equation comes from Eq. (4) for Λ

*(*

_{m}*β*). All combinations of

*n*and

_{y}*n*can be expressed by

_{z}*n̄*= (

*n*+

_{y}*n*)/2 and Δ

_{z}*n*=

*n*−

_{y}*n*. Typically Δ

_{z}*n*≪

*n̄*, so that only the leading term in Δ

*n/n̄*can be kept. Using additionally the smallness of 1/

*m*, we obtain from Eq. (6) with a good accuracy:

*ν*on Δ

*n*is quadratic; changing sign of Δ

*n*=

*n*−

_{y}*n*returns us to the same

_{z}*ν*after relabeling

*y*⇄

*z*. In other words, only the square of the ellipticity affects the frequency. The second and third terms in Eq. (7) are responsible for relatively small contributions to

*ν*. These contributions can, however, strongly exceed the modal linewidths. In the limit Δ

_{m,q}*n*→ 0, when

*m*is the true azimuth number, we return to the known dispersion relation for the circular cylinder. In essence, the broken axial symmetry results in a relatively small deformation of the frequency spectrum without changing its structure.

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* ≃ 2*πn̄R/λ*. A similar estimate is in use for the axially symmetric WGRs. For *λ* = 1 *μ*m, *R* = 1 mm, and *n̄* = 2 we have *m* ≃ 1.25 × 10^{4} ≫ 1.

Let us check whether the approximations made are justified. Since *πνR/c* ≃ *m*/2*n̄* in the initial expression for *β*, we have *β/m*^{2} ≃ Δ*n*/2*n̄*. 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/m*^{2}, one can check also that the expression for the quasi-radial function acquires the form *U _{m}*(

*u*) =

*const*× Ai[2

^{1/3}

*m*

^{2/3}(

*u*

_{0}−

*u*) −

*ξ*]. For

_{q}*m*∼ 10

^{4}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 Δ*n* 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:

*m*≫ 1 and corresponds to the eikonal approximation in wave optics. The second term in the exponent describes phase modulation owing to the periodic modulation of the effective refractive index. It is controlled by the parameter

*n*= Δ

*n*(

*λ*). This phase modulation parameter can strongly exceed unity for many birefringent crystals. Figure 2(a) shows the dependence

*f*(

*λ*) for

*R*= 1 mm within the transparency window of several important birefringent crystals. All dependences are rapidly decreasing, and

*f*ranges roughly from 10 to 10

^{3}.

Note that the quasi-classical relation (8) is not commonly known, and it is expected to be valid for *m*^{2} ≫ 1, 2*β/m*^{2} ≪ 1 [29]. Numerical treatment of the Mathieu functions
${\mathrm{\Phi}}_{m}^{\pm}={\mathrm{\Phi}}_{m}^{\pm}(\beta ,\phi )$ with *m* ∼ 10^{4} is problematic. Thus, we have checked numerically the validity of Eqs. (8, 9) for *m* ≈ 200 and Δ*n/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 Δ*n/n̄*.

Strong phase modulation in Eq. (8) leads to a strong broadening of the Fourier spectrum of the 2*π*-periodic function Φ* _{m}*(

*φ*). Taking into account the equality

*J*(

_{j}*f*) is the Bessel function of the first-kind, we conclude that the Fourier spectrum of Φ

*(*

_{m}*φ*) consists not only of the fundamental harmonic

*m*, but also of a number of side harmonics

*m*± 2,

*m*± 4, . . .. The larger |

*f*|, the stronger is the broadening. When |

*f*| ≫ 1, the Bessel function

*J*(

_{j}*f*) becomes rapidly very small for |

*j*| > |

*f*| [27], so that the width of the discrete Fourier spectrum, 2

*j*

_{max}− 2

*j*

_{min}, can be estimated as ≃ 4|

*f*|. This large width is, nevertheless, much smaller than

*m*. The above features of the Fourier spectrum are illustrated by Fig. 2(b).

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* ≃ *πn*Δ*R*/2*λ*, where Δ*R* is the semi-axes difference. The phase modulation is strong for *δR* ≳ 2*λ/πn*. In practice, this condition can be fulfilled already for sub-*μ*m 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} = 2*m*_{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
$\u3008{\mathrm{\Phi}}_{{m}_{\text{p}}}^{2}{\mathrm{\Phi}}_{{m}_{\text{s}}}^{*}\u3009$. 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* = 2*m*_{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*:

*δm*and

*δf*. If

*δf*= 0, it is unity at

*δm*= 0 and zero otherwise. If |

*δf*| ≫ 1, it is clearly nonzero for |

*δm*| ≤ 2|

*δf*| and very small outside this interval. For odd values of

*δm*we have $\u3008{\mathrm{\Phi}}_{{m}_{\text{p}}}^{2}{\mathrm{\Phi}}_{{m}_{\text{s}}}^{2}\u3009=0$. Figure 3 illustrates what happens with increasing |

*δf*|. The number of significant mismatches

*δm*grows quickly, while the maximum value of $\u3008{\mathrm{\Phi}}_{{m}_{\text{p}}}^{2}{\mathrm{\Phi}}_{{m}_{\text{s}}}^{*}\u3009$ decreases slowly. Thus, many different phase-matched SHG processes with and modestly decreased efficiencies become possible for |

*δf*| ≫ 1. The maximal in

*δm*value of $\u3008{\mathrm{\Phi}}_{{m}_{\text{p}}}^{2}{\mathrm{\Phi}}_{{m}_{\text{s}}}^{*}\u3009$ can be estimated here as 0.7|

*δf*|

^{−1/3}; it follows from the properties of the Bessel function

*J*(

_{ν}*x*) with

*ν*≫ 1 and

*x*≫ 1 [27]. Note that Eq. (12) with

*δm*= ±2 is similar to the SHG phase-matching condition of [30] obtained for a microring resonator with periodically changing nonlinear susceptibility. Our case, where the presence of numerous phase-matched processes is due to the birefringence, is indeed essentially different.

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* [Δ*n*(2*λ*_{s}) − Δ*n*(*λ*_{s})]/2*λ*_{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 ∼ 10^{1} to ∼ 10^{2}. 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 3*φ*) 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} = 2*ν*_{p} rewritten in terms of *m*_{s,p}. It reads *m*_{s}/*n*_{s} = 2*m*_{p}/*n*_{p} in the leading approximation. We have neglected the weak geometrical dispersion and the quadratic in Δ*n* corrections to *ν*_{s,p}. The indices *n*_{s} and *n*_{p} are equal to *n̄*(*λ*_{s}) and *n̄*(2*λ*_{s}) for the axially asymmetric s, p-modes and to *n _{x}* (

*λ*

_{s}) and

*n*(2

_{x}*λ*

_{s}) in the case of symmetric modes. Combining the frequency condition with Eq. (12), we obtain

*δm*

_{max}= 2|

*δf*| ≫ 1. The right-hand side of this relation must be smaller than 1 in the absolute value. If this condition is fulfilled within a certain range of

*λ*

_{s}, SHG is available. Details of the spectral behavior of

*δm*are controlled by the spectral dependences of the relevant refractive indices. Since

*δm*

_{max}≫ 1, the dependence

*δm*(

*λ*

_{s}) can be treated as continuous.

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*(2

_{y}*λ*

_{s}) and

*n*(

_{z}*λ*

_{s}) −

*n*(2

_{z}*λ*

_{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*(2

_{x}*λ*

_{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}(2

*λ*

_{s}) ≤

*n*

_{o}(

*λ*

_{s}). Since the dependence

*n*

_{o}(

*λ*) is decreasing,

*n*

_{o}(2

*λ*

_{s}) stays below

*n*

_{o}(

*λ*

_{s}). At the same time,

*n*

_{o}(2

*λ*

_{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*. 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 (

_{s}*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}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*(2*λ*_{s}), *n*_{s} = *n _{x}* (

*λ*

_{s}), and Δ

*n*

_{s}= 0 in Eq. (13). Here, the broad-band SHG is possible when the condition

*n*(2

_{z}*λ*

_{s}) ≤

*n*(

_{x}*λ*

_{s}) ≤

*n*(2

_{y}*λ*

_{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*(2

_{x}*λ*) stays constant in

_{s}*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*(2

_{x}*λ*

_{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].

## 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 (Δ

*n*/4

*n̄*)

^{2}including the birefringence Δ

*n*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 (Δ

*n*)

^{2}to get the adapted dispersion relation for a birefringent disk WGR.

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 (MgF_{2}). The major and minor radii measured are *R* = 1576 ± 2 *μ*m 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 *μ*m 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 × 10^{9}, 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 MgF_{2} [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*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

_{c}*R*= 1576.3

*μ*m and

*ν*= 287.92 THz. Furthermore, eight-times repetition of the spectral experiment and the fit procedure yields

*R*= (1576 ± 1)

*μ*m 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.

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) *μ*m 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 × 10^{9}; 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 MgF_{2} 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* = 2*m*_{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.

## Funding

We gratefully acknowledge financial support from the German Federal Ministry of Education and Research (funding program Photonics Research Germany, 13N13648). The article processing charge was funded by the German Research Foundation (DFG) and the Albert Ludwigs University Freiburg in the funding program Open Access Publishing.

## Acknowledgments

The authors are grateful to A. Schiller, G. Schunk, and Ch. Marquardt for fruitful discussions.

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