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
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 . 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 .
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 . An experimental support for this idea is known for specially designed angle-cut WGRs . 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 .
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 . 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 , 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 nx,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 nx 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 = Hx at the light frequency ν obeys the equation
The whispering gallery mode problem is solved in the elliptic coordinates φ, u, such that26]. The variables φ and u are similar to the polar and radial coordinates. For ny − nz ≪ nz,y, the quasi-polar angle φ is very close to the true polar angle. The coordinate line u = u0 corresponds to the WGR boundary.
The variable separation H(φ, u) = Φ(φ) U(u) in Eq. (1) leads to the ordinary differential equations27]. They occur when dealing with many physical problems . The physical solutions for Φ(φ) are 2π-periodic, and solutions for U(u) have to be strongly localized near u0 (near the rim). Both parameters, Λ and β, have to be determined. If, as might be suggested, the frequency ν is about mc/2πny,z R in the leading approximation, then the ratio β/m2 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 with m = 0, 1, . . . corresponding to 2π-periodic even (+) and odd (−) real functions . 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 and the separation constant 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 :
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 Λm/m2 versus β/m2 according to Eq. (4), whereas the open and filled circles give numerical data for and , respectively; each circle is calculated independently for m = 103 and 104 with practically no difference. Also the values of and , 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,Eq. (4) for Λm(β). All combinations of ny and nz can be expressed by n̄ = (ny + nz)/2 and Δn = ny − nz. Typically Δn ≪ n̄, so that only the leading term in Δn/n̄ can be kept. Using additionally the smallness of 1/m, we obtain from Eq. (6) with a good accuracy: Eq. (7) are responsible for relatively small contributions to νm,q. These contributions can, however, strongly exceed the modal linewidths. In the limit Δn → 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 ≃ 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 Um(u) = const × Ai[21/3m2/3(u0 −u) − ξq]. 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 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 , as derived in detail in , reads: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 103.
Note that the quasi-classical relation (8) is not commonly known, and it is expected to be valid for m2 ≫ 1, 2β/m2 ≪ 1 . Numerical treatment of the Mathieu functions 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 Φm(φ). Taking into account the equality27], so that the width of the discrete Fourier spectrum, 2jmax − 2jmin, 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 ≃ 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 λp and λs = λp/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 ms = 2mp 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 .
To quantify the impact on SHG, we consider the absolute value of the φ-average . Setting aside the φ-independent transverse overlap factor, it can be treated as the ratio deff/d for the relevant nonlinear coefficient d. According to Eq. (8), this value depends on δm = 2mp − ms and also on δf = 2 fp − fs. The values of fp = fp(λp) and fs = fs(λs) 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:Figure 3 illustrates what happens with increasing |δf|. The number of significant mismatches δm grows quickly, while the maximum value of decreases slowly. Thus, many different phase-matched SHG processes with 27]. Note that Eq. (12) with δm = ±2 is similar to the SHG phase-matching condition of  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 [Δn(2λs) − Δn(λs)]/2λs. 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| = fs. Similarly, in the case when the p-mode is asymmetric, while the s-modes are axially symmetric, we get |δf| = 2 fp. Figure 2(a) enables one to judge about the behavior of |δf|(λs) 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 deff/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 νs = 2νp rewritten in terms of ms,p. It reads ms/ns = 2mp/np in the leading approximation. We have neglected the weak geometrical dispersion and the quadratic in Δn corrections to νs,p. The indices ns and np are equal to n̄(λs) and n̄(2λs) for the axially asymmetric s, p-modes and to nx (λs) and nx (2λs) 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/δmmax| ≤ 1 is fulfilled only when the index differences ny (λs) − ny (2λs) and nz (λs) − nz (2λs) are opposite in sign. Generally, the dependences nx,y,z (λ) are decreasing within the transparency window , 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/δmmax ≤ 1 corresponds to the index condition nz (λs) ≤ nx (2λs) ≤ ny (λs), and broad-band SHG is often allowed. In particular, it is typically allowed for negative uniaxial crystals. In this case, nx,y (λ) = no(λ), nz (λ) = ne(λ), and no(λ) > ne(λ); the subscripts o and e refer to the ordinary and extraordinary polarizations, respectively. The above index condition reads here: ne(λs) ≤ no(2λs) ≤ no(λs). Since the dependence no(λ) is decreasing, no(2λs) stays below no(λs). At the same time, no(2λs) can stay above ne(λs) within a certain range of λs. This means that δm/δmmax crosses the value of −1 at a certain λs, but it cannot reach the value of 1 for larger λs. This is illustrated by lines 1, 2, 3, and 4 in Fig. 4(a) for negative uniaxial crystals BBO, LB4, LN, and AGSe . For positive uniaxial crystals (ne > no), SHG is forbidden in this scheme. Employment of biaxial crystals strongly extends the possibilities for broad-band SHG. In particular, δm/δmmax can vary with λs 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 nZ > nY > nX.
The last coupling scheme to consider corresponds to axially symmetric (TE) s-mode and axially asymmetric (TH) p-mode, i.e., to np = n̄(2λs), ns = nx (λs), and Δns = 0 in Eq. (13). Here, the broad-band SHG is possible when the condition nz (2λs) ≤ nx (λs) ≤ ny (2λs) 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 , and it is illustrated by Fig. 5. For the scheme with axially symmetric p-mode, the refractive index np = nx (2λs) stays constant in φ, whereas the effective index of the signal wave varies periodically between ny (λs) and nz (λs), see Fig. 5(a). This provides locally fulfillment of the equality neff(φ) = nx (2λs). In the case of negative uniaxial crystals, we have: nx,y = no and nz = ne < no; for any λs, the horizontal red line stays here below the upper dotted line providing the inequality −1 < δm/δmmax ≤ 1. For positive uniaxial crystals, we have instead: nx,z = no and ny = ne > no; 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 .
3. Experimental identification of WGR modes with broken axial symmetry
Equation (7) shows that the resonant frequencies νm,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  and experimentally verified . 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.
To verify it experimentally, we basically follow the procedure presented in Ref.  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 νc 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 . 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)  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 , 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 νc 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 νc 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.
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.
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 = 2mp − ms and δm = mp − ms − mi 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 . 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.
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.
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.
The authors are grateful to A. Schiller, G. Schunk, and Ch. Marquardt for fruitful discussions.
References and links
2. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006). [CrossRef] [PubMed]
6. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014). [CrossRef]
7. V. Braginsky, M. Gorodetsky, and V. Il’chenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989). [CrossRef]
8. A. Matsko and V. Ilchenko, “Optical resonators with whispering-gallery modes, part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006). [CrossRef]
10. J. Fürst, D. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Quantum light from a whispering-gallery-mode disk resonator,” Phys. Rev. Lett. 106, 113901 (2011). [CrossRef] [PubMed]
11. T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett. 106, 143903 (2011). [CrossRef] [PubMed]
12. G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008). [CrossRef]
13. G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun. 2, 254 (2011). [CrossRef] [PubMed]
14. M. Gorodetsky and A. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006). [CrossRef]
15. M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whipering-gallery spheroidal resonators,” Proc. SPIE 8236, 823623 (2012). [CrossRef]
16. I. Breunig, B. Sturman, F. Sedlmeir, H. G. L. Schwefel, and K. Buse, “Whispering gallery modes at the rim of an axisymmetric optical resonator: analytical versus numerical description and comparison with experiment,” Opt. Express 21, 30683–30692 (2013). [CrossRef]
17. M. Oxborrow, “Traceable 2-D Finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theory Tech. 55, 1209–1218 (2007). [CrossRef]
18. G. Schunk, J. U. Fürst, M. Förtsch, D. V. Strekalov, U. Vogl, F. Sedlmeir, H. G. L. Schwefel, G. Leuchs, and C. Marquardt, “Identifying modes of large whispering-gallery mode resonators from the spectrum and emission pattern,” Opt. Express 22, 30795–30806 (2014). [CrossRef]
19. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant cavities,” Nature 385, 45–47 (1997). [CrossRef]
21. G. Lin, J. Fürst, D. V. Strekalov, I. S. Grudinin, and N. Yu, “High-Q UV whispering gallery mode resonators made of angle-cut BBO crystals,” Opt. Express 20, 21372–21378 (2012). [CrossRef] [PubMed]
22. F. Sedlmeir, M. Hauer, J. Fürst, G. Leuchs, and H. Schwefel, “Experimental characterization of an uniaxial angle cut whispering gallery mode resonator,” Opt. Express 21, 23942–23949 (2013). [CrossRef] [PubMed]
23. J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015). [CrossRef]
24. G. Lin, J. U. Fürst, D. V. Strekalov, and N. Yu, “Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators,” Appl. Phys. Lett. 103, 181107 (2013). [CrossRef]
25. D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).
26. G. B. Arfken and H. G. Weber, Mathematical Methods for Physicists (Academic, 2001).
27. M. Abramowitz and A. Stergun, Handbook in Mathematical Functions (Dover Publications, 1972).
28. C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992). [CrossRef]
29. D. Richards, Advanced Mathematical Methods with Maple (Cambridge University, 2002).
30. Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. 32, 826–828 (2007). [CrossRef] [PubMed]
31. P. S. Kuo, J. Bravo-Abad, and G. S. Solomon, “Second-harmonic generation using 4̄-quasi-phasematching in a GaAs whispering-gallery-mode microcavity,” Nat. Commun. 5, 3109 (2014). [CrossRef]
32. B. E. A. Saleh and M. C. Teich, “Fundamentals of photonics,” (Wiley, 1991). [CrossRef]
33. Y. Dumeige, S. Trebanol, L. Ghişa, T. K. N. Nguyên, H. Tavernier, and P. Féron, “Determination of coupling regime of high-Q resonators and optical gain of highly selective amplifiers,” J. Opt. Soc. Am. B 25, 2073–2080 (2008). [CrossRef]