A Reflection and transmission at a birefringent interface

Dielectric tuning and coupling of whispering gallery modes using an anisotropic prism

Abstract

Optical whispering gallery mode (WGM) resonators are a powerful and versatile tool used in many branches of science. Fine tuning of the central frequency and line width of individual resonances is however desirable in a number of applications including frequency conversion, optical communications and efficient light-matter coupling. To this end we present a detailed theoretical analysis of dielectric tuning of WGMs supported in axisymmetric resonators. Using the Bethe-Schwinger equation and adopting an angular spectrum field representation we study the resonance shift and mode broadening of high WGMs when a planar dielectric substrate is brought close to the resonator. Particular focus is given to use of a uniaxial substrate with an arbitrarily aligned optic axis. Competing red and blue resonance shifts ( MHz), deriving from generation of a near field material polarisation and back action from the radiation continuum respectively, are found. Anomalous resonance shifts can hence be observed depending on the substrate material, whereas mode broadening on the order of  MHz can also be simply realised. Furthermore, polarisation selective coupling with extinction ratios of can be achieved when the resonator and substrate are of the same composition and their optic axes are chosen correctly. Double refraction and properties of out-coupled beams are also discussed.

I Introduction

Whispering gallery mode (WGM) resonators are a powerful and versatile tool in modern day optics and have found great employ as novel light sources (1); (2); (3); (4), in spectroscopic studies (5), frequency comb generation (6); (7), quantum electrodynamics (8); (9), sensing (10), nonlinear optics (11); (12) and optomechanics (13); (14). Such extensive usage derives from the narrow bandwidths, high field strengths and small modal volumes boasted by WGMs. Primarily, the resonance structure of WGM resonators is dictated by their geometry and material composition, however, tunability of resonance properties, such as central frequency and line width, is desirable in a number of applications. For example, WGM microcavities can be used as tunable filters or switches, which play a central role in optical signal processing and classical communication networks (15). Alternatively, matching the frequency and bandwidth of WGMs to those of atomic transitions (16) enables efficient light-matter coupling required in quantum communications (17) and information processing (18). Controlled coupling and differential tuning of WGM resonances can, moreover, greatly improve the efficiency of polarisation conversion (19) and nonlinear optical processes such as second harmonic generation or parametric oscillation (20); (21) and is therefore important for realisation of novel light sources and frequency converters.

A number of strategies for tuning WGM resonances based on: external temperature control (9); electro-optical (22) or thermo-optical effects (23); (24); (25); and variation of an applied strain (26); (27); (28) or pressure (29); (30), have previously been reported in the literature. So-called dielectric tuning, in which a dielectric substrate is brought into close proximity to a WGM resonator, has also been proposed and demonstrated as a route to continuous fine tuning (16); (31). In this work, we present a detailed theoretical formalism describing dielectric frequency tuning of WGM resonances using planar substrates, in addition to quantifying substrate induced line width changes. Emphasis is especially placed on the use of arbitrary uniaxial dielectric planar substrates, which we show can enable differential tuning and selective coupling of transverse electric (TE) and transverse magnetic (TM) WGMs (32). The structure of this article is therefore as follows. In Section II we first generalise the analytic WGM profiles given in (33) to account for the open nature of WGM resonators and detail the properties of WGMs in an isolated uniaxial axisymmetric resonator required in subsequent derivations. After establishing a suitable angular spectrum representation of WGMs in Section III we continue to determine the interface induced coupling and resulting mode distributions in the presence of a uniaxial dielectric prism (note we shall use the terms prism and substrate interchangeably throughout this text). Resonance shifts and mode broadening induced by introduction of a planar substrate are subsequently derived by exploiting the Bethe-Schwinger cavity perturbation formula in Section IV. Critically, far-field contributions, which can give rise to anomalous radiative shifts in open resonators (34), are incorporated into the Bethe-Schwinger equation. Section V proceeds to present a number of illustrative numerical results, before we finally conclude in Section VI.

Ii Whispering gallery modes in open axisymmetric resonators

Figure 1: WGM resonator geometry: (a) Definition of the coordinate system and geometrical parameters of an axisymmetric resonator perturbed by a planar substrate. (b) Coordinate system in the far field defining the angular dependence of a plane wave component with wavevector (c) Schematic of integration volume and surface used to evaluate the resonance shift and mode broadening upon an infinitesimal shift of the dielectric substrate away from the resonator.

In this section we derive a number of WGM properties that will be required in later sections. We consider a WGM with vacuum wavelength supported in a uniaxial axisymmetric resonator in air with major radius and for which the radius of curvature of the outer surface in the polar direction (rim radius) is (see Figure 1(a)). We denote the ordinary and extraordinary refractive index of the resonator by and respectively. Throughout this work we assume that and and that the optic axis of the resonator is parallel to the symmetry axis (-cut). With these assumptions the WGM is well described using a scalar Helmholtz equation (33). Within the resonator interior () the associated electric field distribution can then be expressed in the form

(1)

where, with reference to Figure 1(a), is the position vector relative to the centre of the resonator, is the distance from the centre of curvature of the resonator rim, is the toroidal polar angle, is the azimuthal angle, are the Hermite polynomials of degree , jointly describes the azimuthal, polar and radial indices, or TM is the polarisation index and is an arbitrary normalisation constant which is henceforth taken to be unity for simplicity. Eq. (1) was originally derived assuming a Dirchlet boundary condition at the resonator surface (33), however, we account for the openness of the resonator through use of the effective radius where describes the penetration depth of the WGM field into the host medium, , for TM modes and , for TE modes (35); (36), such that

(2)
(3)
(4)

where is the -th root of the Airy function . The unit polarisation vectors in Eq. (1) are given by and , where we have here chosen to neglect the azimuthal polarisation component of the TM mode within the resonator due to its small magnitude relative to the radial component (note that we shall use the caret notation exclusively to denote unit vectors for which ).

Due to the strongly confined nature of WGMs the total energy stored within the resonator volume, , can be found by integrating over a toroidal volume defined by the local curvature of the resonator as shown in Figure 1, yielding

(5)

where we have also introduced the vector containing all four mode indices. Throughout this work we shall freely switch between using the combined mode index and separating the polarisation and scalar indices as and depending on whether the polarisation index must be emphasised. For large and small the mode is highly localised near the outer surface and at the equator of the resonator whereby we can make the small angle approximation . Combining Eqs. (1) and (5) hence gives

(6)

Use of the orthogonality relation of the Hermite functions where is the Kronecker delta, then yields

(7)

where and the prime notation denotes differentiation with respect to the argument. For large we note that and .

Exterior to the resonator () the WGM exhibits an evanescent decay (35) viz.

(8)

where is a normalisation constant required to match the fields at the resonator surface, , and as follows from application of the Fresnel coefficients for light undergoing total internal reflection at a glancing angle to the resonator surface (37). From applying the usual Maxwell boundary conditions it follows that . Care must be taken when matching TM modes however since in this case the magnitude of the azimuthal field component becomes times greater than that of the radial component at the (interior) resonator surface and thus must also be included. With this in mind it can be shown that . Moreover, given Eq. (8) it follows that the relative contribution to the mode energy from the field outside the resonator is

(9)

from which it quickly follows that such that the majority of the mode energy is seen to lie within the resonator. Eq. (9) holds within the approximation that the field outside the resonator decays exponential, however, similar expressions can also be derived through use of the complete mode profile (5); (38). Finally, we note that although expressions for the WGM mode profile and energy given in this section are approximate in nature, they have been found to agree well with rigorous finite element calculations (39).

Iii Mode distributions in the presence of an anisotropic interface

Interaction of a WGM with a dielectric perturbation, gives rise to a redistribution of the mode profile. This phenomenon has been well studied for the case of perturbing dielectric and plasmonic nanoparticles, in addition to consideration of layered structures (40); (41); (42); (43). Changes in the properties of morphological dependent resonances of a dielectric sphere near a conducting plane have also been considered using a multipolar expansion (44); (45). Here we consider the mode distribution, , resulting from interaction of a WGM with a semi-infinite dielectric substrate whose interface is placed at a distance from the surface of a -cut birefringent resonator and with its normal directed along as depicted in Figure 1(a). The mode profiles defined in Eqs. (1) and (8) represent the limiting case . Moreover, we consider a uniaxial dielectric substrate with arbitrary optic axis and (extra-)ordinary refractive index () . To account for potential back coupling into the resonator from reflection of the WGM from the interface, we first consider surface dressing of the WGM amplitude. With this in hand, we then proceed to derive the mode distribution in both the resonator interior and exterior, in addition to that in the infinite half space of the anisotropic dielectric, using an angular spectrum approach.

iii.1 Surface dressed scattering amplitudes

To describe excitation of WGMs in an isolated resonator structure, we can consider an arbitrary incident field, which we represent as a superposition of modes viz.

(10)

Assuming the modes form a complete orthogonal basis over the surface, , of the resonator it follows that

(11)

The incident field gives rise to a scattered field, which can again be represented as a superposition of modes as

(12)

where the additional parametric dependence on is introduced for later convenience. Note that we consider different modes and due to differing physical requirements at infinity. Specifically, the scattered modes must satisfy the Sommerfeld radiation condition, whereas the incident modes have a zero net energy flow through a closed surface and physically cannot possess a singularity in the volume of interest. Through application of the Maxwell boundary conditions at the surface of the resonator, the incident and scattered mode coefficients can be related through the matrix equation , where is the scattering matrix and () is a vector formed by stacking all the incident (scattered) mode coefficients. For a spherical resonator, the modes and would correspond to the vector multipole modes with a radial dependence described by the spherical Bessel functions and Hankel functions (of the first kind) respectively. Accordingly, would be a diagonal matrix with non-zero elements given by the usual Mie scattering amplitudes (46). Modal properties of resonances in isolated optical resonators, such as the associated resonance frequencies and line widths, can then be determined through analysis of the poles of the scattering matrix (47). The field in the interior of the resonator can similarly be written

(13)

where the internal mode coefficients can also be related to the illumination coefficients through .

Introduction of a dielectric inhomogeneity in the resonator surroundings, e.g. a dielectric interface or a perturbing nanoparticle, produces two additional contributions to the total field exterior to the resonator. Firstly, the incident field is reflected from the inhomogeneity giving rise to a field . Similarly, the field scattered from the resonator is reflected giving rise to a field . Each contribution can also be decomposed according to 1

(14)

The total field incident upon the resonator is that formed from the superposition of the incident and the reflected fields, i.e. , such that the scattered field is

(15)

where and , and are perturbed coefficient vectors defined analogously to above. Noting further, that the reflected incident field originates from the incident field, we may write and similarly for the reflected scattered field we have . Solving for the scattered coefficients in terms of the illumination coefficients , thus yields

(16)

where is the identity matrix and is an effective, or dressed, scattering matrix. Through an analogous argument it also follows that

(17)

Excitation of WGMs is typically achieved through evanescent coupling using a prism or waveguide structure (48), such that the excitation field is non-negligible over only a small extent of the resonator surface. Practically, dielectric tuning is also achieved using an independent dielectric substrate (31); (32), such that the excitation field does not contribute to the total field at the substrate interface. Accordingly we can safely neglect the contribution of for our purposes (i.e. we assume , where is the null matrix). We briefly note that the effective scattering coefficient derived above is equivalent to that which would be found by considering multiple reflections of the WGM from the dielectric substrate and summing over all reflected orders.

iii.2 Prism induced mode coupling

Notably, evaluation of the effective scattering and transmission matrices described by Eq. (16) and Eq. (17) requires determination of the complex coupling coefficients, , which comprise , which we consider in in detail in this section. Our derivation consists of three steps: firstly we determine the angular spectrum of the WGM with mode index at the dielectric interface; secondly, each constituent plane wave component is reflected from the interface by means of generalised Fresnel reflection coefficients; before finally, calculating the mode overlap between the reflected field with the WGM of order (defined analogously to ).

The first step of our derivation requires the angular spectrum of a given WGM at the dielectric interface located at . Restricting our attention to large WGMs we can legitimately make a small angle approximation, and with reference to Figure 1(a), it follows that and . The symmetry of the TE and TM WGMs therefore implies the angular spectrum of the field is of the form and , where , and, and are the (complex) polar and azimuthal angles in space (analogous to and ) as depicted in Figure 1(b). We note that the radial polarisation component is absent in the far field so as to ensure transversality of the field. The scalar amplitudes, evaluated in a given plane, are given by

(18)

where is the complex amplitude of and we parameterise the amplitudes in terms of the distance of the plane from the resonator rim for later convenience. At the dielectric interface we have , which for small and gives . Together with Eq. (8) we thus find

(19)

where , and we have also used the fact that and . Eq. (19) represents the product of the WGM mode with a Gaussian coupling window, such that use of the convolution theorem yields

(20)

where , and . From Eq. (20) we see that as a consequence of the small angle approximation the angular spectrum is separable in and . Moreover, in the direction the angular spectrum is centred on (i.e. the propagation constant of the associated WGM) and has a width of , whereas in the direction the spectrum is centred around and has a width .

Reflection and transmission of each angular component can be described through use of generalised Fresnel reflection and transmission coefficients, a derivation of which can be found in Appendix A. We note, however, that the transmission coefficients act on the and field components, i.e. those perpendicular and parallel to the plane of incidence. It is therefore necessary to express the and field components in terms of the and basis. Upon making the usual small angle approximation we find and . The reflected angular spectrum in each case, denoted , is hence

(21)
(22)

where is the reflected wavevector, , and the reflection coefficients, , are defined in Appendix A. Accordingly the reflected field at a general position exterior to the resonator is given by

(23)
(24)

as follows from reciprocity and where , , and similarly for and . The amplitude factors are given by

(25)
(26)
(27)
(28)

where is the position vector relative to the plane of the prism interface. Determination of the reflected field over the resonator surface requires evaluation of these integrals for . To do so we make a number of further approximations. We once more note that , , and , such that . Furthermore given the small range of and , we assert that the variation of the reflection coefficients is small, such that for whereby the reflection coefficient can be factored outside of the integral. With these approximations Eqs. (25)–(28) essentially reduce to the 2D inverse Fourier transform of the angular spectra with some additional Gaussian factors due to the additional evanescent decay of the mode. We thus find on the resonator surface, i.e. , that

(29)

where () for (TM) and similarly for . With this association understood we henceforth adopt the shorthand notation . The coupling constant can then be evaluated viz. (c.f. Eq. (11))

(30)

We first evaluate the integral which follows easily from the orthogonality of the Hermite functions as

(31)

Turning attention to evaluation of , we first consider the polarisation dependence. Within our small angle approximation it can be shown that for and for and zero otherwise. Moreover, the narrow windowing function described by the factor means that we need only perform the integration over a small region of the resonator surface whereby . Use of Eqs. (19) and (29) then allows us to express in the form

(32)

where . For large we have , such that the second exponential in Eq. (32) can be safely replaced by the Kronecker delta whereby . The remaining integral term, , in Eq. (32) can then be written as

(33)

with . It is evident that is identically zero when and are of opposite parity, i.e. polar modes of differing symmetry do not couple as would be expected. When is even, however, can be evaluated analytically (49) yielding

(34)

where and are the Gamma and Gauss hypergeometric functions respectively. Hence, we ultimately arrive at the desired coupling coefficients

(35)

A heat map depicting the interface induced coupling strength between modes of different polar orders ( and ) as described by Eq. (35) is shown in Figure 2. Note that for ease of comparison we have also included a further normalisation factor of in the data shown in Figure 2, which arises from the scaling of the initial mode energy (Eq. (5)). Inspection of Eq. (35) and Figure 2 reveals that a WGM of order couples most strongly to the lowest order WGM of the same symmetry upon reflection, i.e. the fundamental mode for even or the mode for odd . This is a direct consequence of the finite width coupling window. We reiterate that coupling between modes of differing symmetry is not possible. Moreover, as follows from the conservation of angular momentum, coupling between modes with different azimuthal indices ( and ) is also forbidden. Eq. (35), however, does not forbid coupling between modes of differing polarisation, albeit, within our small angle approximation, such polarisation mixing stems from the anisotropy of the substrate and would be absent for an isotropic prism.

Figure 2: Interface induced coupling strengths: Heat map of the coupling coefficients between a WGM of polar order to a WGM of polar order , as given by . A value of was assumed. Mode profiles of differing polar orders are shown in the lower inset from which different odd-even symmetry classes are apparent. Strongest coupling of a given WGM order is to the lowest order mode of the same symmetry as highlighted by the green boxes, however, coupling to modes of a different symmetry class is not possible.

iii.3 Mode distributions

Having described the reflection of a WGM by a dielectric substrate, we now determine the complete perturbed mode distributions. WGMs with differing mode indices are typically spectrally distinct in most resonators (with the exception of a perfectly spherical resonator for which the polar modes are degenerate) as is reflected in the amplitude of the scattering coefficients, and , contained in and respectively. Consequently, assuming , we can make a single mode approximation whereby Eq. (16), simplifies to

(36)

Close to resonance and initially neglecting possible material absorption in the resonator, the scattering coefficient can be approximated by the Breit–Wigner line shape (50) viz.

(37)

where and denote the resonance frequency and radiative line width respectively. The pre-factor follows from imposing field continuity at the resonator surface. As shown in Appendix B material absorption in the resonator reduces the magnitude of the peak scattering amplitude such that

(38)

where is the absorptive line width (51). Similarly Eq. (17) simplifies to

(39)

Combination of Eqs. (36), (38) and (39) shows that on resonance the scattered and internal mode coefficients and are reduced by a factor of relative to the unperturbed () case. For absorption limited resonators for which we find and hence and . Within the resonant mode approximation, it therefore immediately follows that the WGM field distribution within the resonator () in the presence of a dielectric interface is given simply by , where is given by Eq. (1). Recalling results from above we also find the field exterior to the resonator (, ) is given by , where now is given by Eq. (8) and follows from Eqs. (23)-(28). Following the same line of arguments used to derive Eq. (29) we can, however, write

(40)

The field transmitted into the volume of the anisotropic substrate can be found similarly to the reflected field using the generalised Fresnel transmission coefficients (Appendix A). From the preceding analysis in Section III.2 we note that for an initially unperturbed WGM of order , the component of the perturbed field incident on the prism interface is given by , with a corresponding angular spectrum . Upon transmission, each constituent plane wave generates an ordinary and extraordinary wave in the prism, with associated wavevectors and , such that the transmitted field takes the form

(41)

where the subscript for respectively, () are the generalised Fresnel transmission coefficients and and are the unit polarisation vectors for the ordinary and extraordinary waves respectively. The explicit dependence of , and on the incident wavevector is given in Appendix A. To simplify Eq. (41) further we note that exponential terms vary rapidly with and . In comparison the Fresnel coefficients and polarisation terms vary weakly within the limited range of and over which is non-negligible. As such we make the approximations and , where and are the central ordinary and extraordinary wavevectors found from Eqs. (77)–(81) with and . We hence obtain

(42)

where and is defined analogously to above. Once more taking advantage of the small angular spread in and we expand the exponent around and which to leading order yields

(43)
(44)

where , and follow from Eqs. (79)–(81) in Appendix A. The constant terms in Eqs. (43) and (44) can be factored out of the integrals of Eq. (42) such that the integrals are now of the form of a simple Fourier transform with respect to the variables and . Accordingly, the terms linear in can be simply accounted for by recalling the shift theorem, whereby Eq. (42) becomes

(45)

where and are defined by Eqs. (43) and (44). We note that near the critical angle, the variation of the transmission coefficients can vary strongly with such that the approximations leading to Eq. (45) are not valid. In this case the full integral expressions of Eq. (41) must be used to account for Fresnel filtering effects (52); (53). Similar restrictions also apply to calculation of the reflected field and mode coupling as discussed in Section III.2.

Iv Prism induced resonance perturbations

Shifts in the resonance frequency of modes in a closed cavity induced by local dielectric perturbations can be described by the Bethe-Schwinger equation (54). Moreover, it has recently been shown that this formula can also be used to account for mode broadening and radiative shifts in open resonators if the far-field components are incorporated (34). In Appendix C we briefly present a derivation of the Bethe-Schwinger equation for open cavities in the presence of anisotropic dielectric perturbations. Although ultimately it is our goal to determine the total resonance shift and line width broadening induced by the presence of a uniaxial substrate relative to the case in which no substrate is present, the perturbed and unperturbed modes required to evaluate the Bethe-Schwinger equation (Eq. (113)) can not be taken as and given above. This can intuitively be seen, since the second term in Eq. (113) can be associated with radiative losses in the far field, however, this term evaluates to zero if the unperturbed mode is chosen because in the far field . Physically such a scenario is incorrect since radiative losses give rise to a mode broadening which is not accounted for. Instead, we consider the change of the WGM frequency and lifetime assuming that the dielectric is initially located at a distance from the resonator and is then displaced by an infinitesimally small distance, away from the resonator. Accordingly, the Bethe-Schwinger equation takes the form

(46)

where we have defined the integral terms (assuming )