Calculating the Fine Structure of a Fabry-Perot Resonator using Spheroidal Wave Functions

A new set of vector solutions to Maxwell's equations based on solutions to the wave equation in spheroidal coordinates allows laser beams to be described beyond the paraxial approximation. Using these solutions allows us to calculate the complete first-order corrections in the short-wavelength limit to eigenmodes and eigenfrequencies in a Fabry-Perot resonator with perfectly conducting mirrors. Experimentally relevant effects are predicted. Modes which are degenerate according to the paraxial approximation are split according to their total angular momentum. This includes a splitting due to coupling between orbital angular momentum and spin angular momentum.


Introduction
From cavity quantum electrodynamics to deterministic single-photon generation, a host of exciting recent experiments are made possible due to the precise control of light achievable inside optical resonators [1,2]. In the case of Fabry-Perot resonators, the detailed understanding of the resonator modes necessary for these experiments is generally based on solutions to the paraxial wave equation. Despite being approximate in nature, these solutions allow many crucial features of the resonator modes to be explained, from the Gaussian profile of the lowest-order TEM 0,0 mode to the regularly spaced eigenmode spectrum.
Despite its success, novel experiments allow the boundary of the validity of the paraxial approximation to be probed. This includes microresonators, where the small mode volume leads to a violation of the assumed paraxiality, and ultra-high-finesse resonators, where the high spectral resolution allows even minute details to be resolved. Understanding all observable features in such experiments requires the electromagnetic field to be treated more precisely than the paraxial approximation allows.
In the past, several approaches have been used to obtain corrections to the paraxial approximation for resonators [3,4,5,6]. Lazutkin [3] chose an analytic expansion of the wavefunction to satisfy the wave equation with appropriate boundary conditions. Laabs et al. [4] calculated a resonator round-trip propagation matrix using the method of Lax et al. [7] of calculating corrections to the paraxial equation. The effect of spherical aberrations was calculated by Visser et al. [5] based on perturbation to Gaussian propagation inside a resonator. Zomer et al. [6] used a diffraction integral to propagate light around a resonator.
None of these papers take into account all effects needed to calculate the complete first-order correction beyond the paraxial approximation of the electromagnetic eigenfrequencies of a resonator. Lazutkin  An alternative approach for treatment of the resonator eigenmode problem is through the use of spheroidal coordinates. The spheroidal coordinate system, depicted in Fig. 1, is ideally suited for calculations with Gaussian beams due to the match between surfaces of constant phase of a beam and surfaces of constant ξ as well as between the variation in beam diameter along a beam and surfaces of constant η. With appropriate approximations, these coordinates have been applied in the past to resonators to obtain results in agreement with paraxial theory [8,9].
In this paper, spheroidal coordinates are used to calculate the eigenfrequencies of a Fabry-Perot resonator to first order beyond the paraxial approximation in the short-wavelength limit. The electric field inside the resonator is expanded in terms of vector spheroidal wave functions. Requiring the boundary conditions for perfectly conducting mirrors to be satisfied allows the expansion coefficients of the outgoing wave at each mirror to be expressed in terms of the expansion coefficients of the incoming wave. The resulting round-trip propagation matrix for the expansion coefficients is solved via perturbation theory, resulting in a compact expression for the resonator round-trip phase shift, Eqs. (40) and (41). The paper concludes with a discussion on the experimental implications of this result.

Mathematical Foundations
This paper relies heavily on the theory of spheroidal wave functions as found elsewhere [10,11]. In particular, the theory of spheroidal wave functions in the short-wavelength limit has recently been significantly expanded in Ref. [11]. Here we briefly summarize the most important definitions and results from this paper. The mapping from spheroidal coordinates (ξ , η, φ ) to cylindrical coordinates (r, z, φ ) is given by the φ -coordinate being the same in both coordinate systems and d being the interfocal distance as shown in Fig. 1. The scalar wave equation is separable in spheroidal coordinates, allowing the solutions to be written as the product of three functions depending only on ξ , η and φ , respectively. This leads to so-called scalar spheroidal wave functions ψ mν = R mν (ξ )S mν (η)e imφ with and labeled by the indices m and ν. Note that m is simply the integer orbital angular momentum due to the e imφ φ -dependence. The variable x = 2c(1 − η), not to be confused with the Cartesian coordinate, is introduced to simplify calculations. The parameterc = kd/2, with wavevector k, quantifies the scaling of the coordinate system relative to the wavelength. For short wavelengths compared to d,c is a large number, and the functions r mν (ξ ) and s mν (x) can be expanded as asymptotic series in 1 c . To lowest order in 1 c , r mν (ξ ) = 1 and s mν ( Based on the scalar spheroidal wave functions, vector spheroidal wave functions E ± Jσ ν propagating in the ±ξ direction are defined which satisfy the wave equation as well as ∇ · E ± Jσ ν = 0. The E ± Jσ ν can therefore be regarded as the electric field of a solution to Maxwell's equations in free space. With spin angular momentum σ = σ ± corresponding to left and right circular polarization, J = m ± 1 denotes the total integer angular momentum of the field about the symmetry axis.

Satisfying the Boundary Conditions
The electric field inside the resonator is expanded in terms of vector spheroidal wave functions as with expansion coefficients b ± Jσ ν . For perfectly conducting mirrors, the component of the electric field parallel to a mirror surface S must vanish at S, i.e. E| S,|| = 0.
The mirror surfaces are matched to the spheroidal coordinate system as exemplified in Fig. 1 as follows. For a resonator of length L with spherical mirrors with radius of curvature R − and R + , the spheroidal coordinate system is appropriately scaled by choosing the distance d between the focal points to be [8] In this case, the two surfaces of constant ξ = ξ ± with are separated by a distance L and have a radius of curvature of R ± on the resonator axis. For a spherical mirror, a correction to the spheroidal surface of constant ξ is generally necessary. Switching briefly to cylindrical coordinates (z, r, φ ), the mirror surfaces are specified as the distance z =z ± (r, φ ) above the plane z = 0. For a smooth, cylindrically symmetric mirror, z ± (r, φ ) can be expanded in powers of r 2 as Here, z ± and c 4± are expansion coefficients and R ± is the radius of curvature of the surface at r = 0. For a spherical surface, c 4± = 1/(8R 3 ± ). Other values of c 4± can be chosen to describe cylindrically-symmetric mirror aberrations. For example, for a spheroidal surface of constant ξ = ξ ± we have z ± = dξ ± /2 and c 4± = 1/(4d ξ ± R 2 ± ). For a parabolic mirror, c 4± = 0 by definition.
Switching back to spheroidal coordinates, the mirror surfaces can equivalently be specified as the value of ξ as a function of x = 2c(1 − η) and φ . Eq. (7) is transformed into S ± being the mirror surface at ξ ± . Note that for small r, x is proportional toc r 2 . Additionally, due to the factor e −x/2 in the definition of S mν (η), Eq. (3), the spheroidal wave functions vanish for x ≫ 1, motivating the use of x in Eq. (8). The term linear in x is missing due to the specific choice of d and ξ ± in Eqs. (5) and (6). As will be seen, the x 2 term in Eq. (8) is the highestorder term in x which must be retained to calculate first-order corrections to the resonator eigenfrequencies.
The mirror surfaces can now be parameterized in terms of the transverse coordinates η and φ as where x(ξ , η, φ ) is the vector in R 3 from the origin to the point denoted in spheroidal coordinates by (ξ , η, φ ). A basis for the tangent space to the mirror surface at a point on the mirror surface, needed to calculate the component of E parallel to the mirror surface, is given by where theê u i are unit vectors and the h u i are scale factors in spheroidal coordinates [12]. The component of the electric field parallel to the mirror surface is proportional to the inner product of the electric field with these basis vectors.
We now show that as long as we are only interested in corrections to the resonator eigenfrequencies to first order in 1 c , we can use {ê φ ,ê η } as an approximation for the basis (11). For x = O(1), we have the following order of magnitudes, as can be seen from the corresponding definitions in Ref. [11]. As a result, the contribution to ∂ S ± ∂ η · E of theê ξ · E term is of orderc −2 compared to the contribution of theê η · E term and can therefore be neglected, justifying our simplified basis.
To calculateê η · E ± Jσ ν andê φ · E ± Jσ ν we additionally need the following expressions, For first-order corrections, the expressions withê ξ are not needed, but have been included for completeness. Using previous expressions and with a significant amount of algebra, one obtainŝ The L ν (x) with the dependence on x being implicit. Heavy use of recursion relations among the Laguerre polynomials has been made. Note that we have evaluated the spheroidal wave functions at the mirror surfaces S ± . For E − Jσ ν , the field running in the other direction, the results are the same except that ξ ± is replaced by −ξ ± and f 4± is replaced by − f 4± .
Combining Eqs. (14-17) with Eq. (4), the boundary condition for E for the mirror at ξ ± can be reformulated as a matrix equation for the coefficients b ∓ Jσ ν of the outgoing wave in terms of the coefficients b ± Jσ ν of the incoming wave. We begin by defining the set of vector functions v Jσ ν as and and by defining the functions u 1Jσ ν and u 2Jσ ν according to and The parallel component of the electric field of E + Jσ ν at the mirror surfaces can then be written in terms of the v Jσ ν as and For E − Jσ ν we must again replace ξ ± everywhere by −ξ ± and f 4± by − f 4± . The A s Jν are expansion coefficients for the angular spheroidal functions [11]. We have A ν±1 J+1,ν = O( 1 c ) so that to lowest order in 1 c , E ± Jσ ν | S ± ,|| is simply proportional to v Jσ ν , motivating the choice of v Jσ ν . Eqs. (24) and (25) have the form with appropriate coefficients a ± 1 ,± 2 ,J,σ ′ ,σ ν ′ ν . The subscripts on the ± signs denote two independent choices of + or −. Eqs. (24) and (25) contain factors of x as part of the coefficients of the functions v Jσ ν . These can be removed using the relations and x making the coefficients a ± 1 ,± 2 ,J,σ ′ ,σ ν ′ ν independent of x and φ . The boundary condition for E can now be written as Due to cylindrical symmetry, modes of different J do not couple, so we have restricted our attention to a single J. The v Jσ ν , considered as vector functions of x and φ , are linearly independent. As a result, the expression in the square bracket in Eq. (29) must be zero for each value of σ ′ and ν ′ . We define the following matrices and vectors, The matrix A ± 1 ,± 2 maps the traveling wave described by the vector b ± 1 σ of coefficients of wavefunctions E ± 1 Jσ ν onto a set of coefficients of surface functions v Jσ ν describing the electric field component parallel to the surface at S ± 2 . This finally allows us to write the boundary conditions as matrix equations, Solving for b ∓ σ in terms b ± σ using the boundary condition imposed by the mirror at ξ ± , we find the round-trip matrix for the resonator to be given by The eigenvectors of A are vectors of coefficients b + Jσ ν of resonator eigenmodes, the corresponding eigenvalues are the round-trip phase shifts. To lowest order in 1 c , the matrix A is diagonal. The resonator eigenmodes are therefore of the form b + Jσ ± ν E + Jσ ± ν + b − Jσ ± ν E − Jσ ± ν and the corresponding round-trip phase shift is equal to This result is equivalent to the one obtained by considering a resonator in the framework of the paraxial approximation.

First-Order Corrections to the Round-Trip Phase Shift
First-order corrections to the round-trip phase shift of Eq. (33) can be obtained via degenerate perturbation theory. We begin by determining those eigenmodes of the round-trip matrix A which are degenerate to lowest order in 1 c . As can be seen from Eq. (33), the lowest-order eigenvalues corresponding to the modes E Jσ + ν and E Jσ − ,ν−1 are the same for arbitrary J and ν. This reflects the fact that, within the paraxial approximation, the resonator eigenfrequency is independent of the polarization. Additionally, for a cavity geometry such that arctan(ξ 1 )−arctan(ξ 2 ) π/2 is a rational number, say p n with relatively prime integers p and n, the lowest-order eigenvalues corresponding to the modes E Jσ ν and E Jσ ,ν+n are the same for arbitrary J, ν and σ .
From this analysis, we see that the degeneracy between the modes E Jσ + ν and E Jσ − ,ν−1 must be taken into account. On the other hand, a degeneracy between the modes E Jσ + ν and E Jσ + ,ν+n due to resonator geometry is only relevant for n = 1 and for n = 2, f 4± = 0. For these two special cases, an infinite set of degenerate modes are coupled by first-order off-diagonal matrix elements, and finding the first-order corrections to the eigenvalues of A is significantly more difficult. We therefore exclude these two cases from further analysis.