# Fabry-Perot microcavity spectra have a fine structure

M. P. van Exter1 M. Wubs2 E. Hissink1 C. Koks1 1\. Huygens-Kamerlingh Onnes
Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands
2\. Department of Photonics Engineering, Technical University of Denmark,
Ørsteds Plads 345A, DK-2800 Kgs. Lyngby, Denmark

###### Abstract

Optical cavities can support many transverse and longitudinal modes. A
paraxial scalar theory predicts that the resonance frequencies of these modes
cluster in different orders. A non-paraxial vector theory predicts that the
frequency degeneracy within these clusters is lifted, such that each order
acquires a spectral fine structure, comparable to the fine structure observed
in atomic spectra. In this paper, we calculate this fine structure for
microcavities and show how it originates from various non-paraxial effects and
is co-determined by mirror aberrations. The presented theory, which applies
perturbation theory to Maxwell’s equations with boundary conditions, proves to
be very powerful. It generalizes the effective 1-dimensional description of
Fabry-Perot cavities to a 3-dimensional multi-transverse-mode description. It
thereby provides new physical insights in several mode-shaping effects and a
detailed prediction of the fine structure in Fabry-Perot spectra.

##  I Introduction

Tunable Fabry-Perot (FP) cavities are popular tools in optics, where they are
used as spectrum analyzer Siegman (1988) and as a means to resonantly trap
light between two high-reflecting mirrors Vahala (2003). An optical
microcavity is a miniature version of a FP cavity, where the two mirrors are
now positioned at only a few wavelengths $\lambda$ from each other and at
least one of the mirrors has a radius of curvature $R_{m}\ll 100\lambda$.
Microcavities can strongly confine the optical field, boost the light-matter
interaction of intra-cavity emitters Najer  _et al._ (2019); Vallance  _et
al._ (2016); Wang  _et al._ (2019); Vogl  _et al._ (2019); Häußler  _et al._
(2021); Flatten  _et al._ (2018), and increase the collection efficiency and
emitted fraction into the zero-phonon line Ruf  _et al._ (2021).

Microcavities support compact optical modes with large opening angles. This
can push their operation beyond the paraxial regime and can require a non-
paraxial description of the optical propagation and a more thorough
description of the mirror reflections. Elements of the resulting spectral
finestructure have been reported Riedel  _et al._ (2020); Trichet  _et al._
(2018); Häußler  _et al._ (2019); Papageorge  _et al._ (2016); Mader  _et al._
(2015), but a complete description was missing.

Non-paraxial corrections to the optical propagation were already analyzed in
the seventies and eighties. Lax Lax (1975) describes a general framework that
treats these corrections as different terms in a Taylor expansion. Erickson
Erickson (1975, 1977) calculates the scalar non-paraxial correction to the
cavity resonances. Cullen Cullen (1976) and Davis Davis (1984) add a vector-
correction to this description. Yu and Luk Yu and Luk (1983, 1984) and Luk Luk
(1986) are the first to combine these corrections in a complete analysis of
the optical resonances in cavities with spherical mirrors. More recently,
Zeppenfeld and Pinkse Zeppenfeld and Pinkse (2010) performed an alternative
complete analysis of rotationally-symmetric cavities, using spheroidal wave
functions.

Additional corrections occur when the mirrors are not spherical, but have
astigmatic or more general deformations, common to microcavities Trichet  _et
al._ (2018); Benedikter  _et al._ (2015, 2019). Kleckner et al. Kleckner  _et
al._ (2010) present a general framework to describe the effect of these
deformations, but their description does not yield analytic solutions.

This paper presents a general theoretical framework for the optical resonances
in tunable FP microcavities. The description is semi-analytic and exact in the
limit of small perturbations. The description uses a roundtrip operator that
acts on field profiles and uses perturbation theory to calculate the effects
of several deviations from the standard paraxial theory with spherical
mirrors, including non-paraxial effects and deviations from the spherical
mirror shape. It thereby extends the standard 1-dimensional description of the
Fabry-Perot interferometer to a 3-dimensional multi-transverse-mode
description. The presented mathematical and physical tools can be applied to a
wide range of optical systems.

The paper then applies this theory to calculate the fine structure in Fabry-
Perot spectra. The paraxial scalar theory predicts that modes with the same
longitudinal mode number $q$ and transverse order $N$ should be frequency
degenerate. But a more complete theory shows that each $(q,N)$-group exhibits
a spectral fine structure, where modes with different radial mode number $p$,
orbital angular momentum (OAM) mode number $\ell$, and polarization $v$ have
slightly different resonance frequencies, even when they belong to the same
$(q,N)$ order. The paper analyses and classifies the different effects that
contribute to this optical fine structure and identifies which ones are
relevant under which conditions. It thus aims to present a complete
description of this intriguing phenomenon.

The fine structure in FP spectra is analogous to the fine structure observed
in atomic spectra. For atomic spectra, the simple Bohr model of hydrogenic
atoms predicts that their energy levels should only depend on the principal
quantum number $n$. But a more complete description, that among others
includes relativistic corrections and spin-orbit coupling Giffiths (2014),
shows that levels with the same $n$ are frequency split in the so-called fine
structure with additional quantum numbers, $\ell$ for the orbital angular
momentum and $s$ for the spin.

The presented theory is inspired by our own experimental observations of
intriguing fine structures in transmission spectra of FP microcavities Koks
_et al._ (2022) and the lack of an adequate theory. Similar structures must
have been observed by other groups, but hardly anything has been published.
For microwave cavities, Erickson Erickson (1975) measured the non-paraxial
frequency splittings between radial modes and Yu and Luk Yu and Luk (1983)
measured it between OAM modes. In the optical domain, Dufferwiel et al.
Dufferwiel  _et al._ (2015) reported a fine structure in the $N=1$ group of an
optical cavity filled with semiconductor quantum wells, but this TE-TM
splitting was mainly due to polariton effects. Zeppenfeld et al. reported an
observation of a spectral fine structure at the CLEO 2010 conference Pinkse
_et al._ (2011). Raman spectra of Riedel et al. Riedel  _et al._ (2020) showed
an astigmatic splitting, but did not show an additional fine structure,
presumably because their cavities had a modest finesse $F$; observable non-
paraxial fine structures require $F\lambda/R_{m}\gtrsim 10$, for plano-concave
cavities with mirror radius $R_{m}$ and wavelength $\lambda$. When more
detailed experimental spectra become available, the most challenging aspect of
their analysis will undoubtedly be the need to separate the, more fundamental,
non-paraxial effects from the, more practical, mirror-shape effects. This
paper describes how this can be done. Furthermore, we have already used the
theory to analyze our own experiments on optical microcavities.

This paper consists of three parts: Part I, in Secs. II and III, contains the
description of the general framework, and ends with a preview of the key
results in a table and a figure; Part II, in Secs. IV and V, presents the
derivations of the fine structure for cavities with spherical mirrors. This
part ends with a key equation that combines all relevant effects for plano-
concave cavities, plus a comparison to the literature; Part III, in Secs. VI
and VII, contains the analysis of several new effects for more general
cavities. These new effects are additional scalar and vector corrections,
including two astigmatic corrections. Section VIII discusses the obtained
results and addresses the potential role of residual corrections. Section IX
presents a summary and outlook. There are three Appendices.

##  II Paraxial scalar modes

We consider the propagation and reflection of light in a plano-concave Fabry-
Perot (FP) cavity. Both mirrors are highly reflective and large enough to
avoid clipping losses, such that all relevant modes are virtually lossless and
spectrally well resolved. This cavity exhibits sharp resonances, visible as
peaks in the optical transmission and dips in the optical reflection, at
particular combinations of cavity length $L$ and optical wavelength $\lambda$,
where light is resonantly trapped in the cavity. We want to derive the exact
resonance conditions and spatial profiles of the associated eigenmodes. This
problem sounds simple but is surprisingly difficult; there are no exact
solutions for the general non-paraxial case.

The propagation of light in the paraxial limit is standard material in many
textbooks on optics Siegman (1988). We consider a plano-concave cavity with
perfect rotation symmetry, use cylindrical coordinates defined by the symmetry
$z$-axis, and denoted position by $(r,\theta,z)$. This cavity supports a set
of matched Laguerre-Gaussian modes, with flat wavefronts at the flat mirror
($z=0$) and matched curved wavefronts at the concave spherical mirror ($z=L$,
radius of curvature $R_{m}$). The Rayleigh range
$z_{0}={\scriptstyle\frac{1}{2}}kw_{0}^{2}=\sqrt{L(R_{m}-L)}$ determines the
beam waist $w_{0}$ at the planar mirror, where
$E_{00}(r,\theta)\propto\exp(-r^{2}/w_{0}^{2})$, the variation of the beam
size upon propagation, via $w_{z}^{2}=w_{0}^{2}(1+z^{2}/z_{0}^{2})$, the
radius curvature of the wavefront $R(z)=R_{z}=z+z_{0}^{2}/z$, and the phase
lag $\chi(z)=\arctan(z/z_{0})$ of the fundamental mode with respect to a plane
wave. We split the intra-cavity standing wave in forward- and backward-
propagating fields and write the slowly-varying component of the forward-
propagating complex field of the Laguerre-Gaussian modes of this cavity as
$E_{p,\ell}=\psi^{+}_{p,\ell}\exp{i(kz-\omega t)}$ with

$\psi_{p,\ell}^{+}(r,\theta,z)=\frac{1}{\gamma_{z}}\Psi_{p,\ell}(\rho,\theta,\chi)\exp\left[ik\frac{r^{2}}{2R_{z}}\right]\,.$
(1)

The radial quantum number $p$ and OAM quantum number $\ell$ combine to the
transverse order $N=2p+|\ell|$. The slowly-varying backward-propagating field,
$\psi_{p,\ell}^{-}(r,\theta,z)=-(1/\gamma_{z})\Psi_{p,\ell}(\rho,\theta,-\chi)\exp[-ikr^{2}/(2R_{z})]$,
is a mirror image of the forward-propagating field and hence will not be
considered explicitly. In Eq. (1), the normalized mode functions

$\Psi_{p,\ell}(\rho,\theta,\chi)=f_{p,\ell}(\rho)e^{i\ell\theta}\exp[-i(N+1)\chi]\,$
(2)

are the eigenmodes of the two-dimensional harmonic oscillator in quantum
mechanics (QM), in terms of the normalized transverse position
$\rho=r/\gamma_{z}$, with $\gamma_{z}=w_{z}/\sqrt{2}=\gamma_{0}/\cos{\chi}$.
The amplitude functions

$f_{p,\ell}(\rho)=(-1)^{p}\sqrt{\frac{p!}{\pi(p+|\ell|)!}}\rho^{|\ell|}L^{|\ell|}_{p}\left(\rho^{2}\right)\exp[-\rho^{2}/2]\,,$
(3)

contain the generalized Laguerre polynomials $L^{|\ell|}_{p}$ and a sign
$(-1)^{p}$ to make them equal to the transverse modes generated by harmonic
oscillator ladder operators (see Appendix C). All modes are normalized via

$\displaystyle\langle\psi^{+}_{p,\ell}|\psi^{+}_{p,\ell}\rangle$
$\displaystyle\equiv$
$\displaystyle\int_{0}^{\infty}rdr\int_{0}^{2\pi}d\theta\,\,|\psi^{+}_{p,\ell}(r,\theta,z)|^{2}=1\,,$
(4a) $\displaystyle\langle\Psi_{p,\ell}|\Psi_{p,\ell}\rangle$
$\displaystyle\equiv$ $\displaystyle\int_{0}^{\infty}\rho
d\rho\int_{0}^{2\pi}d\theta\,\,|\Psi_{p,\ell}(\rho,\theta,\chi)|^{2}=1\,,$
(4b)

where $|\psi^{+}\rangle=|\psi^{+}(z)\rangle$ is the ket notation for the mode
$\psi^{+}(r,\theta,z)=\langle r,\theta|\psi^{+}(z)\rangle$. These integrals do
not depend on $z$, or $\chi$; the modes remain orthonormal because optical
propagation is a unitary operation.

The resonance condition in a cavity is determined by the requirement that the
mode reproduces itself after a roundtrip. This requirement is satisfied when
$R(L)=R_{m}$, which fixes $w_{0}$, and when the roundtrip phase $\varphi_{\rm
round}$ is a multiple of $2\pi$, such that

$\varphi_{\rm round}=\varphi-\varphi_{\rm par}-\varphi_{\rm non}=2\pi\,q\,,$
(5)

where $\varphi=2kL$ is the plane-wave roundtrip phase and $q$ is the
longitudinal quantum number. The paraxial roundtrip phase lag $\varphi_{\rm
par}$ equals $2(N+1)\chi_{0}$. The additional, typically small, roundtrip
phase lag $\varphi_{\rm non}$ combines all non-paraxial and mirror-shape
effects. Eq. (5) corresponds to resonant cavity lengths

$L_{j}=\frac{\lambda}{2}\left[q+(N+1)\frac{\chi_{0}}{\pi}+\Delta\tilde{\nu}_{j}\right]+2L_{\varphi}\,,$
(6)

where $\Delta\tilde{\nu}_{j}=\varphi_{\rm non}/(2\pi)$ and $L_{\varphi}$ is
the phase penetration depth into the (Bragg) mirrors ($L_{\varphi}=0$ in the
center of the stopband) Koks and van Exter (2020). Throughout this paper, we
will quantify the magnitude of each non-paraxial and mirror-shape effect by
its contribution to the dimensionless detuning $\Delta\tilde{\nu}_{j}$ of mode
$j$. The mode label $j=\\{p,\ell,v\\}$ combines the two transverse quantum
numbers $p$ and $\ell$ with a third vector/polarization quantum number $v$;
see Sec. V.

##  III Roundtrip operator and perturbation theory

###  III.1 The roundtrip operator ${\cal M}$

The evolution of the optical field in a cavity can be described by the
roundtrip operator $M$, which transforms the forward-propagating field
$|\psi^{+}\rangle$ into $M|\psi^{+}\rangle$ after a roundtrip Kleckner  _et
al._ (2010). If we expand the field
$|\psi^{+}\rangle=\sum_{j}c_{j}|\psi_{j}^{+}\rangle$ into a set of orthogonal
basis states $|\psi^{+}_{j}\rangle$, then we can write $|\psi^{+}\rangle$ as a
vector and $M$ as a matrix. Finding the resonance conditions and eigenmodes of
the cavity now boils down to finding the eigenvalues and eigenvectors of the
roundtrip matrix.

The roundtrip operator in any two-mirror cavity can be written as
$M=P^{+}AP^{-}B$, where $A$ and $B$ represent the reflections from the left
and right mirrors and $P^{+}$ and $P^{-}$ represent the propagation from A to
B and back (see Fig. 1). In a plano-concave cavity with a large smooth planar
mirror, $A$ is equal to unity and $M=PB$, where $P=P^{+}P^{-}$ describes the
roundtrip propagation and $B$ describes the reflection from the concave
mirror.

Figure 1: Geometry of a plano-concave cavity. Starting from the right, the
optical roundtrip $M=P^{+}AP^{-}B$ includes reflection from the concave mirror
$B$, propagation $P^{-}$ to the left, reflection from mirror $A$, and
propagation $P^{+}$ to the right. The shape of the concave mirror is specified
by its distance $z_{m}(x,y)>0$ from the plane $z=L$. The wavefront of the
intracavity mode $j$ is flat at mirror $A$ and almost spherical, and described
by $z_{j}(x,y)$, at mirror $B$.

In a typical experiment, we illuminate the optical cavity with an input field
$|\psi^{+}_{\rm in}\rangle$ through mirror A and we observe the output field
$|\psi^{+}_{\rm out}\rangle$ through mirror $B$, as a function of the cavity
length or the optical frequency. The forward-propagating intra-cavity field
$|\psi^{+}_{\rm cav}\rangle$ at mirror B can be calculated by summation over
an infinite series of reflections and repeated operations of the roundtrip
operator

$|\psi^{+}_{\rm cav}\rangle=(1+M+M^{2}+...)t_{1}P^{+}|\psi^{+}_{\rm
in}\rangle=\frac{t_{1}P^{+}}{1-M}|\psi^{+}_{\rm in}\rangle\,,$ (7)

where $t_{1}$ and $t_{2}$ are the amplitude transmissions of mirrors A and B.
Eq. (7) extends the single-mode treatment as found in many textbooks to a
multi-transverse-mode treatment of the Fabry-Perot cavity, by treating $P^{+}$
and $M$ as operators instead of scalars, and $|\psi^{+}\rangle$ as a field
profile instead of a field amplitude. The output field is derived from the
cavity field as $|\psi^{+}_{\rm out}\rangle=t_{2}|\psi^{+}_{\rm cav}\rangle$.

The roundtrip operator $M$ determines the full dynamics of the intracavity
field, as it contains the eigenfrequencies and damping rates of all cavity
modes. To highlight this link, we write

$M=\exp(i\varphi-i{\cal H}-{\cal A})\,,$ (8)

with Hermitian operators ${\cal H}$ and ${\cal A}$. For the single-mode case,
the scalar ${\cal H}=\varphi_{j}$ describes the roundtrip phase lag of mode
$j$ with respect to the plane-wave roundtrip phase $\varphi=2kL$. The other
scalar ${\cal A}=\gamma_{j}$ describes its roundtrip modal loss, where
$\gamma_{j}=\gamma_{r}+\Delta\gamma_{j}$ combines the reflection loss
$\gamma_{r}\approx 1-r_{1}r_{2}$ (for amplitude reflectivities $r_{1,2}$ close
to unity) with potential extra loss $\Delta\gamma_{j}$, like clipping loss
from the finite-size mirrors. For the multi-mode case, the dimensionless
operator ${\cal H}$ describes the conservative dynamics, while the operator
${\cal A}$ describes the dissipative dynamics of all modes Spreeuw  _et al._
(1990); Suh  _et al._ (2004).

The dynamics of the intra-cavity field is typically dominated by the modes
that are close to resonance, i.e. modes for which
$||(1-M)|\psi^{+}_{j}\rangle||\approx 0$. We highlight this by expanding
$(1-M)\approx i({\cal H}-\varphi)\,{\rm mod}(2\pi)+{\cal A}$, where the first
term is taken modulo $2\pi$ to remove $\exp{i2\pi q}=1$. In the eigenbasis of
the dynamics operator, defined by
$M|\psi^{+}_{j}\rangle=\exp(i\varphi-i\varphi_{j}-\gamma_{j})|\psi^{+}_{j}\rangle$,
the output field of the multi-mode Fabry-Perot cavity can be written as

$|\psi^{+}_{\rm
out}\rangle=\sum_{j}\frac{t_{1}t_{2}}{\gamma_{j}-i(\varphi-\varphi_{j})}\,P^{+}\,|\psi^{+}_{j}\rangle\langle\psi^{+}_{j}|\psi^{+}_{\rm
in}\rangle\,,$ (9)

where $(\varphi-\varphi_{j})$ is again taken modulo $2\pi$. Only modes close
to resonance, with $(\varphi-\varphi_{j})\approx 0\,{\rm mod}(2\pi)$,
contribute significantly to the transmission.

The transmission spectrum described by Eq. (9) is the counterpart of the
dynamic equation

$\displaystyle T_{\rm round}\frac{d}{dt}|\psi^{+}\rangle_{X}$ $\displaystyle=$
$\displaystyle|\psi^{+}\rangle_{X+1}-|\psi^{+}\rangle_{X}$
$\displaystyle\approx$
$\displaystyle(M-1)\,|\psi^{+}\rangle_{X}+t_{1}P^{+}|\psi^{+}_{\rm
in}\rangle_{X}\,,$

which describes the evolution of the intra-cavity field $|\psi^{+}\rangle$
from roundtrip $X$ to roundtrip $X+1$, in the roundtrip time $T_{\rm round}$.
This equation again shows that resonances in optical cavities behave like
coupled harmonic oscillators, whose (complex) eigenvalues and eigenstates are
determined by the resonance condition $({\cal H}-i{\cal
A})|\psi_{j}\rangle=(\varphi_{j}-i\gamma_{j})|\psi_{j}\rangle$.

This equivalence was already discussed by Haus Haus (1984) and later extended
by Fan et al. Fan  _et al._ (2003). Suh, Wang and Fan Suh  _et al._ (2004)
expressed the dynamics of the field in any multi-mode cavity in its most
general form as

$\frac{d}{dt}\vec{a}=-(i\Omega+\Gamma)\vec{a}+\kappa^{T}|s_{+}\rangle\,,$ (11)

where the vector $\vec{a}$ combines the (complex) amplitudes of all relevant
cavity modes, the $\Omega$ and $\Gamma$ are dynamic matrices, and where the
matrix $\kappa^{T}$ couples the (multi-channel) input field $|s_{+}\rangle$ to
the cavity modes. Our Eq. (III.1) resembles Eq. (11), but also differs in
three ways: first of all, Eq. (III.1) expresses the intra-cavity field
$|\psi^{+}\rangle$ and the input field $|\psi^{+}_{\rm in}\rangle$ as two 2D
field profiles, whereas $\vec{a}$ in Eq. (11) is a 3D intra-cavity field while
$|s+\rangle$ is a 2D field profile. This difference introduces a square root
of time in the dimension of $\kappa^{T}$. Second, our Eq. (III.1) describes
the dynamics of the slowly-varying field $\psi^{+}(t)$, whereas Eq. (11)
describes the dynamics of the full field $E(t)$; this difference results in a
factor $\exp(i\varphi)$ in $M$. Finally, and most importantly, our Eq. (III.1)
links the dynamics of the intra-cavity field to the roundtrip operator $M$; we
will use this link to determine the intra-cavity modes and their properties.

###  III.2 Perturbation theory applied to ${\cal H}$

From now on, we will neglect losses, assuming ${\cal A}\approx 0$. We will
remove the + subscript and interpret $|\psi\rangle$ as the forward-propagating
field at the curved mirror, i.e. $|\psi\rangle=|\psi^{+}(z=L)\rangle$. This is
allowed when resonances are sharp and when we are not interested in their
spectral width. We split the dynamic operator as ${\cal H}={\cal H}_{\rm
par}+{\cal H}_{\rm fine}$ and associate the resonance with a hard zero in the
resonance condition

$\displaystyle(M-1)|\psi\rangle$ $\displaystyle=$
$\displaystyle(\exp{i(\varphi-{\cal H})}-1)|\psi\rangle\,\,$ (12)
$\displaystyle\approx$ $\displaystyle i(\varphi-q2\pi-{\cal H}_{\rm par}-{\cal
H}_{\rm fine})|\psi\rangle=0\,,$

where $q$ is the longitudinal mode number. The operator ${\cal H}_{\rm par}$
describes the paraxial evolution in a cavity with a spherical concave mirror.
The operator ${\cal H}_{\rm fine}$ describes the, typically small,
modifications due to non-paraxial propagation and reflection from a non-
spherical mirror.

The dynamic matrix ${\cal H}_{\rm par}$ is diagonal in the basis of the
paraxial eigenmodes presented in Sec. II. The on-diagonal elements are equal
to the roundtrip phase lag $\varphi_{j}=2(N+1)\chi_{0}$ and are thus identical
for modes with the same transverse order $N$. Each $N$-group contains $2(N+1)$
modes, divided over $(N+1)$ spatial profiles, labelled by $(p,\ell)$ for the
scalar LG-modes, times two polarizations.

The fine structure operator ${\cal H}_{\rm fine}$ can lift the frequency
degeneracy within each $N$-group and reshape the eigenmodes. We calculate
these effects by applying perturbation theory to Eq. (12). In principle, this
perturbing operator can couple and mix all paraxial modes. In practice, it
mainly mixes modes of the same order $N$. The fine structure within each
$N$-group is thus described by first-order frequency-degenerate perturbation
theory and by the transverse-mode matrix

$\varphi_{{\rm
fine},j^{\prime}j}=2\pi\Delta\tilde{\nu}_{j^{\prime}j}=\langle\psi_{j^{\prime}}|{\cal
H}_{\rm fine}|\psi_{j}\rangle\ll 1\,.$ (13)

The eigenvalues of the matrix $\Delta\tilde{\nu}$ yield the spectral shifts,
i.e. the fine structure. The eigenvectors of $\Delta\tilde{\nu}$ yield the new
eigenmodes of the cavity.

The coupling between modes from different transverse orders is far less
effective and described by second-order frequency non-degenerate perturbation
theory Sakurai and Napolitano (1985). This yields coupling rates of the form
$\Delta\tilde{\nu}_{N\neq
N^{\prime}}\approx\pi(\Delta\tilde{\nu}_{j^{\prime},j})^{2}/(N-N^{\prime})\chi_{0}$,
where $(N-N^{\prime})\chi_{0}/\pi$ is the frequency difference between the
paraxial modes in the $N$ and $N^{\prime}$ groups. Non-resonant coupling is
strongly suppressed by the denominator,
$(N-N^{\prime})\chi_{0}\gg\pi\Delta\tilde{\nu}_{j^{\prime},j}$, and can
typically be neglected. From a physics point of view, the extra field $-i{\cal
H}_{\rm fine}|\psi\rangle$ in Eq. (12) only remains trapped in the cavity when
it fits resonantly in the cavity and light scattered to other modes does not
build up resonantly, is quickly lost, and can thus be neglected. The only
exception to this rule is the situation where modes of different orders are
accidentally almost frequency-degenerate Koks and van Exter (2022); we will
not consider this case any further.

Category | Contribution | H | Form | Preferred basis |  Magnitude $\varphi_{j}=2\pi\Delta\tilde{\nu}_{j}$ | Discussed in Sec.
---|---|---|---|---|---|---
Paraxial | Paraxial | par |  $r^{2}$ and $k_{\perp}^{2}$ | no preference | $2(N+1)\arcsin{\sqrt{L/R_{m}}}$ | II
| Non-paraxial scalar |  |  |  |  |
(known) | \- Propagation/Helmholtz | prop | $k_{\perp}^{4}$ | scalar LG | magnitude prop+wave: | IV.1
non- | \- Wavefront/Spherical | wave | $r^{4}$ | scalar LG | $[g(p,\ell)+4]/(4kR_{m})$ | IV.2
paraxial | Mirror aspheric | asphere | $r^{4}$ | scalar LG | $-\tilde{p}f(p,\ell)L/[4kR_{m}(R_{m}-L)]$ | IV.3
| Non-paraxial vector | vec | $\vec{r}\otimes\vec{k}_{\perp}$ | vector LG | $[-1-\ell\cdot s]/(kR_{m})$ | V.2
| = spin-orbit coupling |  |  |  |  |
(new) | Mirror Bragg correction | bragg | $\vec{k}_{\perp}\otimes\vec{k}_{\perp}$ | vector LG | $\pm C_{0}(N+1)/(kz_{0})$ | VI
vector |  |  |  |  | hyperfine for $1A\pm$ modes |
|  |  | $k_{\perp}^{2}\,(\vec{k}_{\perp}\otimes\vec{k}_{\perp})$ |  |  $\propto C_{2}$ (not analyzed in detail) |
(new) | Mirror astigmatism | astigm | $x^{2}-y^{2}$ | scalar HG | $\eta_{\rm astigm}\sqrt{L/(R_{m}-L)}$ | VII.1
scalar |  |  |  |  | (off diagonal, $\Delta\ell=\pm 2$) |
(new) | Mirror astigmatic vector | v+a |  | x- and y-pol. | $\pm\eta_{\rm astigm}/(kR_{m})$ | VII.2
vector | = anisotropic spin-orbit |  |  |  | hyperfine for $\ell=0$ modes |
Table 1: Overview of various contributions to the transverse mode spectrum,
with their abbreviated name, operator form, preferred basis, relative strength
(= non-paraxial phase lag) $\varphi_{j}$, and associated section. The paraxial
contribution has no preferred basis; the rotation-symmetric corrections prefer
the LG-basis; mirror astigmatism prefers the HG-basis.

###  III.3 Symmetry aspects & scalar versus vector modes

The eigenmodes of the fine structure operator ${\cal H}_{\rm fine}$ can often
already be determined from the symmetry of the system. For cavities with
rotation symmetry, each scalar eigenmode has a fixed OAM, with quantum number
$\ell$, and each vector mode has a circular polarization $\sigma_{\pm}$, with
spin quantum number $s=\pm 1$. For cavities with additional mirror symmetry,
as is common, the $\ell$ and $-\ell$ eigenmodes should be frequency
degenerate. From now on, we will use this argument repeatedly and take
$\ell\geq 0$ throughout the main text. Hence, the $\ell\neq 0$ modes are
expected to form frequency-degenerate groups of four polarized modes, while
the $\ell=0$ modes are expected to form polarization pairs. Below, we will
show that spin-orbit coupling breaks each $\ell\neq 0$ group of four modes
into two pairs of vector modes. We will also show that the final pairwise
degeneracy is more difficult to break. In analogy with the atomic fine
structure, we propose to call the final pairwise break-up the hyperfine
component of the fine structure. But first, we will discuss general symmetry
aspects of ${\cal H}_{\rm fine}$, based on the distinction between scalar
versus vector effects and between rotation-symmetric versus astigmatic
cavities.

The calculation of the scalar corrections is based on the idea that the extra
field $-i{\cal H}_{\rm fine}|\psi\rangle$ results from the mismatch between
the shape of the concave mirror and the wavefront of the mode. The reflected
field is then multiplied by a factor $\exp(2ik\Delta z)\approx 1+2ik\Delta z$
and ${\cal H}_{\rm fine}=-2k\Delta z$, or actually ${\cal H}_{\rm
fine}=+2k\Delta z$ with our sign definition, where $\Delta z(x,y)=z_{\rm
mirror}-z_{\rm wave}$ and positive $z_{\rm mirror}$ and $z_{\rm wave}$ point
towards the plane mirror (see Fig. 1). Substitution of ${\cal H}_{\rm
fine}=2k\Delta z$ into Eq. (13) results in a dynamics matrix of the form

$\displaystyle\Delta\tilde{\nu}_{j^{\prime}j}$ $\displaystyle=$
$\displaystyle\frac{2}{\lambda}\langle\psi_{j^{\prime}}|\Delta
z|\psi_{j}\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\lambda}\iint\mbox{d}x\mbox{d}y\,\Delta
z(x,y)\,\psi^{*}_{j^{\prime}}(x,y)\psi_{j}(x,y)\,,$

where $\Delta z=z_{\rm mirror}-(z_{j^{\prime}}+z_{j})/2$. The complex
conjugation removes the curvature and Gouy phase from $\psi(x,y)$.

For a rotation-symmetric cavity, this matrix is diagonal in the eigenbasis of
the scalar LG-modes and the fine structure is given by

$\Delta\tilde{\nu}_{j}=\frac{2}{\lambda}\langle\psi_{j}|\Delta
z|\psi_{j}\rangle=\frac{2}{\lambda}\iint\mbox{d}x\mbox{d}y\,\Delta z(x,y)\
|\psi_{j}(x,y)|^{2}\,,$ (15)

where $\Delta z=z_{\rm mirror}-z_{j}$ is the mismatch between the shape of the
mirror $z_{\rm mirror}$ and the wavefront $z_{j}$ of the paraxial LG-mode $j$.
Equations (III.3) and (15) can also be derived by applying the theory of
Kleckner et al. Kleckner  _et al._ (2010) to a plano-concave cavity with
$\Delta z\ll\lambda/2$ (see Appendix of Ref. Koks and van Exter (2022)), but
we think the derivation presented above is easier.

Let’s consider two simple examples of Eq. (15). As the first example, we
consider a uniform displacement of the curved mirror towards the plane mirror
over $\Delta z=\alpha>0$. Substitution in Eq. (15) now yields
$\Delta\tilde{\nu}_{j}=2\alpha/\lambda$ for all transverse modes and
substitution in Eq. (12) shows that the resonant cavity length increases by
$\Delta L_{j}=\alpha$ for all modes, as expected. As the second example, we
analyze the effect of a small increase in the mirror curvature, described by
$\Delta z=\beta r^{2}$ with $\beta>0$, making $\Delta R_{m}=-2\beta
R_{m}^{2}<0$, and $|\Delta R_{m}|\ll R_{m}$. Substitution into Eq. (15) now
yields $\Delta\tilde{\nu}_{j}=(2/\lambda)\times\beta(N+1)(w_{1}^{2}/2)$.
Substitution in Eq. (6) again yields the associated change in the cavity
length, which now equals $\Delta L_{j}=\beta(N+1)(w_{1}^{2}/2)>0$. This result
can be fully attributed to the change in the Gouy phase $(N+1)\chi_{0}$ due to
the increased $R_{m}$ at fixed $L$. Using
$\chi_{0}=\arcsin{\sqrt{L/R_{m}}}\Rightarrow\Delta\chi_{0}=-\sqrt{L/(R_{m}-L)}\Delta
R_{m}/(2R_{m})$ and the expression for $w_{1}^{2}=w_{z}^{2}$ at $z=L$
presented in Sec. II, we again find full agreement.

The calculation of vector corrections requires an extension from scalar modes
$|\psi_{j}\rangle$ to vector modes $|\vec{\psi}_{j}\rangle$ and from scalar
operators ${\cal H}_{\rm fine}$ to $2\times 2$ tensor operators. We will
discuss two different vector corrections in Secs. V and VI and show that: (i)
the spin-orbit coupling is relatively strong and present for all $\ell\neq 0$
modes and (ii) the Bragg correction is typically weak and mainly observable
for some $\ell=1$ modes.

In Sec. VII.1, we will analyze astigmatic cavities, without rotation symmetry.
We will show how astigmatism modifies the eigenvalues and eigenmodes, by
coupling modes with different $\ell$, and how it retains the two-fold
degeneracy of the $\ell\neq 0$ modes while creating a small (second-order)
frequency splitting of the $\ell=0$ pair.

###  III.4 Contributions to the fine structure (Table I)

Before discussing in detail the announced vector corrections and effects of
astigmatism etcetera, we present a brief overview in a table of the various
contributions to the transverse mode spectrum that are covered in this paper.
The optical fine structure has many contributions, which are linked to
different physical processes and described by different contributions to the
fine structure operator ${\cal H}_{\rm fine}$. Table I lists the most relevant
contributions and compares their properties and approximate strengths. The
contributions are divided in four categories/blocks: The first block describes
the effect of the paraxial operator ${\cal H}_{\rm par}$. The second block
describes effects that occur in cavities with rotational symmetry. These
effects are divided in two non-paraxial scalar effects ${\cal H}_{\rm
scalar}={\cal H}_{\rm prop}+{\cal H}_{\rm wave}$, an aspherical mirror effect
${\cal H}_{\rm asphere}$, and a non-paraxial vector effect ${\cal H}_{\rm
vec}$. The third block describes a vector effect ${\cal H}_{\rm Bragg}$ that
occurs in rotation-symmetric cavities with Bragg mirrors. The fourth block
describes the effects of astigmatic mirrors, divided in the dominant effect
${\cal H}_{\rm astigm}$ and a second-order effect ${\cal H}_{\rm v+a}$. The
effects in Block Two have been discussed in the literature, albeit often as
individual effects and in different notations; the analysis of the effects in
Blocks Three and Four is new. All mentioned effects are typically small and
hence simply add up, albeit as matrices if they prefer different bases.

The first three columns in Table I show the names of the various effects and
their history. The fourth column shows the functional form of the associated
operator ${\cal H}_{\rm fine}$. The fifth column shows the preferred
eigenmodes, which are Laguerre-Gauss (LG) scalar or vector modes for rotation-
symmetric cavities and Hermite-Gauss (HG) modes for astigmatic cavities. The
sixth column quantifies the relative strengths of the expected effects. The
final column refers to the sections in which each effect is discussed.
Sections IV and V describe three scalar corrections and the spin-orbit vector
correction for rotation-symmetric cavities. Together, they present the “known”
non-paraxial rotation-symmetric corrections discussed in part II of this
paper. Section VI describes the vector Bragg correction for rotation-symmetric
cavities with Bragg mirrors, while Sec. VII analyzes two scalar corrections in
astigmatic cavities. Together, they present the “new” non-paraxial effects
that form the basis of part III of this paper.

Figure 2 shows a sketch of the changes expected for the different
contributions. The first two columns show how the paraxial resonances in a
planar and a plano-concave cavity cluster in groups with the same quantum
numbers $q$ and $N$. The third column shows how non-paraxial scalar and vector
corrections split these clusters into pairs of modes with additional quantum
numbers $\ell$ and $v$. The right column shows how deviations from rotation
symmetry, due to astigmatic mirrors, will modify both the fine structure and
the character of the eigenmodes, from LG-modes to HG-modes. The third and
final column depict the fine structure mentioned in the title. A detail that
is not visible in the figure is that each line consists of two, typically
frequency degenerate, modes that sometimes exhibit a tiny hyper-fine
splitting. Below we will show that these hyperfine splitting originate from
mixing of optical polarizations and that the $\ell=0$ and $\ell=1,A$ modes are
most susceptible to hyperfine splittings.

We end this section by addressing the completeness of our list of non-paraxial
effects. We first note that operators that are 3rd-order in $\vec{r}$ and
$\vec{k}_{\perp}$ are irrelevant for cavities with inversion symmetry, as
their effects average to zero. But why did we single out the listed effects as
the dominant ones, and why did we choose not to include other $2^{\rm nd}$ and
$4^{\rm th}$-order contributions to ${\cal H}_{\rm fine}$ that are also
allowed by symmetry? Our reasoning is as follows: A potential contribution of
the $4^{\rm th}$-order operator $r^{2}k_{\perp}^{2}$, whatever its physical
mechanism, is probably much weaker than that of the related 2nd-order operator
$\vec{r}\otimes\vec{k}_{\perp}$ and has hence been neglected. The same
argument applies to the operator $x^{4}-y^{4}$, which describes the non-
paraxial contribution to the astigmatism. Furthermore, we find it hard to
envision a physical mechanism for the $\vec{r}\otimes\vec{r}$ operator. And
individual $k_{x}^{2}-k_{y}^{2}$ and $k_{x}k_{y}$ operators are probably only
relevant in birefringent cavities. Hence, we think our list is complete for
most practical purposes.

Figure 2: Sketch of the expected resonance frequencies of an optical cavity
(vertical axis) under the influence of various perturbations (horizontal
axis). Left column: modes in a planar cavity. Second column: paraxial modes in
a plano-concave cavity. Third column: fine structure in a rotational-symmetric
cavity. Right column: modified fine structure in an astigmatic cavity.

##  IV Non-paraxial scalar corrections

###  IV.1 Helmholtz correction ${\cal H}_{\rm prop}$

The Helmholtz correction ${\cal H}_{\rm prop}$ originates from a non-paraxial
contribution to the propagation. Non-paraxial propagation has been studied
extensively; Ref. Sheppard and Saghafi (1999) gives a brief historic overview.

We base the first part of our analysis on the work of Lax Lax (1975), who
starts by noting that the intra-cavity optical field $\vec{E}(\vec{r})$ must
satisfy Maxwell’s equations, which for a monochromatic field reduce to the
vector Helmholtz equation $(\nabla^{2}+k^{2})\vec{E}(\vec{r})=\vec{0}$ and the
divergence condition $\vec{\nabla}\cdot\vec{E}(\vec{r})=0$. He then introduces
the slowly-varying forward-propagating field, which we write as
$\vec{E}=(\vec{\psi}_{\perp}+\psi_{z}\vec{e}_{z})\exp{i(kz-\omega t)}$, where
$\vec{\psi}_{\perp}$ combines the transverse components $\psi_{x}$ and
$\psi_{y}$ and $\psi_{z}$ is the axial/longitudinal component. And he
expresses this field as a Taylor expansion of the paraxial field and a series
of non-paraxial corrections, with amplitudes that decay as a power series in
the expansion parameter $f=1/(kw_{0})=\Theta_{0}/2\ll 1$, where $w_{0}$ is the
waist and $\Theta_{0}$ is the opening angle of the fundamental paraxial mode.
The resulting equations are

$\displaystyle\left(\Delta_{\perp}+2ik\frac{d}{dz}\right)\vec{\psi}_{\perp}^{(0)}(\vec{r})$
$\displaystyle=$ $\displaystyle 0\,,$ (16)
$\displaystyle\psi_{z}^{(1)}(\vec{r})$ $\displaystyle\approx$
$\displaystyle\frac{i}{k}\left(\frac{d\psi_{x}^{(0)}}{dx}+\frac{d\psi_{y}^{(0)}}{dy}\right)\,,$
(17)
$\displaystyle\left(\Delta_{\perp}+2ik\frac{d}{dz}\right)\vec{\psi}_{\perp}^{(2)}(\vec{r})$
$\displaystyle=$
$\displaystyle-\frac{d^{2}}{dz^{2}}\vec{\psi}_{\perp}^{(0)}\approx\frac{\Delta_{\perp}^{2}}{4k^{2}}\,\vec{\psi}_{\perp}^{(0)}\,,$
(18)

where $\vec{\nabla}_{\perp}=\partial_{x}\vec{e}_{x}+\partial_{y}\vec{e}_{y}$
is the transverse nabla operator and
$\Delta_{\perp}=\vec{\nabla}_{\perp}^{2}=\partial_{x}^{2}+\partial_{y}^{2}$.
These three equations describe, respectively, the evolution of the paraxial
field $\vec{\psi}^{(0)}_{\perp}$, the axial field $\psi_{z}^{(1)}$, and the
extra (non-paraxial) transverse field $\vec{\psi}^{(2)}_{\perp}$.

The scaling/expansion argument of Lax Lax (1975) is as follows: For Gaussian
beams with a waist $w_{0}$, the ‘transverse’ derivatives $d/dx$ and $d/dy$ in
Eq. (17) will generate factors of the order $1/w_{0}$. Hence, the axial field
$\psi_{z}^{(1)}$ is approximately a factor $f=1/(kw_{0})$ smaller than the
transverse field $\vec{\psi}_{\perp}^{(0)}$. And the extra ‘axial’ derivative
$d/dz$ in Eq. (18) will generate a factor of the order $1/z_{0}$. Hence, the
non-paraxial Helmholtz correction to the transverse field,
$\vec{\psi}^{(2)}_{\perp}$, will be about a factor
$f^{2}/4=1/(2kw_{0})^{2}=1/(8kz_{0})$ smaller than the original paraxial
transverse field. By ending the Taylor expansion after the $f^{2}$-term, we
neglect an even smaller $f^{3}$ correction on $\psi_{z}$ and an $f^{4}$
correction on $\vec{\psi}_{\perp}$.

To calculate the non-paraxial scalar correction, we use a procedure introduced
by Erickson Erickson (1975). We consider the propagation of a paraxial LG-mode
$j$ from the flat to the concave mirror and expand the propagating field in
the basis of LG-modes as Erickson (1975)

$|\psi(z)\rangle=|\psi_{j}\rangle+\sum_{j^{\prime}}c_{j^{\prime}}(z)|\psi_{j^{\prime}}(z)\rangle\,,$
(19)

with $c_{j^{\prime}}(0)=0$ and $c_{j^{\prime}}(z)\ll 1$. Substitution in Eq.
(18) and projection on $|\psi_{j^{\prime}}(z)\rangle$ shows that non-paraxial
propagation modifies the paraxial modes by

$\frac{dc_{j^{\prime}}(z)}{dz}=\frac{-i}{8k^{3}}\,\langle\psi_{j^{\prime}}|\Delta_{\perp}^{2}|\psi_{j}(z)\rangle\,.$
(20)

Erickson Erickson (1975) has shown that the action of the $\Delta_{\perp}^{2}$
operator, or the related $\frac{d^{2}}{dz^{2}}$ operator, on LG-modes changes
the radial index $p$ to $p^{\prime}=p-2,p-1,p,p+1,p+2$, while leaving $\ell$
unchanged on account of the rotation symmetry. But we are only interested in
the $p^{\prime}=p$ term, as the coupling to modes with different order $N$ is
non-resonant.

Equation (20) becomes intuitive when we write
$\langle\psi_{j}|\Delta_{\perp}^{2}|\psi_{j}\rangle=\langle
k_{\perp}^{4}\rangle$. We then find that the correction $dc_{j}(z)/dz$ to the
propagation $ik_{z}$ originates from the third term in the Taylor expansion
$k_{z}=\sqrt{k^{2}-k_{\perp}^{2}}=k-k_{\perp}^{2}/(2k)-k_{\perp}^{4}/(8k^{3})$.
As LG-modes retain their functional form under Fourier transformation, one
easily finds
$\langle\psi_{p,\ell}|\Delta^{2}_{\perp}|\psi_{p,\ell}\rangle=\langle
k_{\perp}^{4}\rangle=4f(p,\ell)/w_{0}^{4}$, where $f(p,\ell)$ is defined in
Eq. (22). This result, and many others, can also be derived with the operator
algebra described in Appendix C. Substitution in Eq. (20) yields the relative
frequency shift of the cavity resonances $\Delta\nu_{j}/\nu=-\Delta
k_{j}/k=f(p,\ell)/(2k^{4}w_{0}^{4})>0$. Conversion to a normalized frequency
$\Delta\tilde{\nu}_{j}=(2L/\lambda)\Delta\nu/\nu$, yields the Helmholtz
contribution to the fine structure

$\Delta\tilde{\nu}_{{\rm prop},j}=\frac{1}{8\pi
k}\frac{L}{z_{0}^{2}}f(p,\ell)=\frac{1}{8\pi
kR_{m}}\frac{w_{1}^{2}}{w_{0}^{2}}f(p,\ell)\,,$ (21)

in terms of the polynomial Yu and Luk (1984)

$\displaystyle f(p,\ell)$ $\displaystyle=$ $\displaystyle
6p^{2}+6p\ell+\ell^{2}+6p+3\ell+2$ (22) $\displaystyle=$
$\displaystyle\frac{3}{2}(N+1)^{2}-\frac{1}{2}(\ell^{2}-1)\,,$

with $\ell\geq 0$. This result is consistent with earlier results of Erickson
Erickson (1975, 1977), Yu and Luk Yu and Luk (1983, 1984), and Luk Luk (1986).

###  IV.2 Wave-front correction ${\cal H}_{\rm wave}$

The wave-front correction ${\cal H}_{\rm wave}$ originates from the difference
between the optical wavefront and a reference surface. On first sight, one
might think that the paraxial wavefront should be parabolic, because the
optical phase $\phi(r,z)=kz-(N+1)\chi+kr^{2}/(2R)$ in Eq. (1) increases
quadratically with $r$ in any $z$-plane. But wavefronts are defined by
surfaces of fixed phase, and both $\chi$ and $R$ are functions of $z$. To find
the true wavefronts, we use the pragmatic approach of Yu and Luk Yu and Luk
(1984) by expanding $\chi(z)$ and $R(z)$ around the paraboloidal reference
surface $z=L-r^{2}/(2R_{m})$ that one would naively expect, to find Yu and Luk
(1984)

$z_{\rm
wave}(r)=\frac{r^{2}}{2R_{m}}+\frac{2(N+1)}{k^{2}w_{1}^{2}}\frac{r^{2}}{2R_{m}}-\frac{r^{4}}{4R_{m}^{2}L}\left(1-\frac{2L}{R_{m}}\right)\,,$
(23)

where $z_{\rm wave}(z)>0$ for displacements from the $z=L$ plane towards the
flat mirror; see Fig. 1. Incidentally, an alternative Taylor expansion of
$\chi(z)$ and $R(z)$ around a spherical, instead of a paraboloidal, surface
would yield a similar result, as the extra terms are relatively small for
$L\ll R_{m}$. The first term in Eq. (23) yields the curvature that we started
from. The second term shows that the central parts of the wavefronts are
actually more curved, by a relative amount $2(N+1)/(kw_{1})^{2}$. The final
$r^{4}$-term makes the outer regions of the paraxial wavefronts “flatter than
paraboloidal” for the typical case $L<R_{m}/2$, and even further away from
spherical.

We compare the paraxial wavefronts with the surface of a spherical mirror

$z_{\rm
mirror}(r)=R_{m}-\sqrt{R_{m}^{2}-r^{2}}\approx\frac{r^{2}}{2R_{m}}+\frac{r^{4}}{8R^{3}_{m}}\,.$
(24)

The mismatch $\Delta z=z_{\rm mirror}-z_{\rm wave}$ results in a shift of the
resonance by an amount $\Delta\tilde{\nu}_{\rm
wave}=(2/\lambda)\langle\psi_{j}|\Delta z|\psi_{j}\rangle$. We substitute Eqs.
(23) and (24) into Eq. (15), to find

$\displaystyle\Delta\tilde{\nu}_{{\rm wave},j}$ $\displaystyle=$
$\displaystyle-\frac{(N+1)^{2}}{2\pi kR_{m}}+\frac{f(p,\ell)}{8\pi
kR_{m}}\left(3-\frac{w_{1}^{2}}{w_{0}^{2}}\right)$ (25) $\displaystyle=$
$\displaystyle\frac{1}{8\pi
kR_{m}}\left[g(p,\ell)+4-\frac{w_{1}^{2}}{w_{0}^{2}}f(p,\ell)\right]\,,$

where the combination Yu and Luk (1984)

$\displaystyle g(p,\ell)+4$ $\displaystyle=$ $\displaystyle
2p^{2}+2p\ell-\ell^{2}+2p+\ell+2$ (26) $\displaystyle=$
$\displaystyle\frac{1}{2}(N+1)^{2}-\frac{3}{2}(\ell^{2}-1)\,.$

To obtain this result we used $\langle r^{2}\rangle=(N+1)w_{1}^{2}/2$,
$\langle r^{4}\rangle=f(p,\ell)w_{1}^{4}/4$, and
$w_{1}^{2}/w_{0}^{2}=R_{m}/(R_{m}-L)$, and wrote some combinations of $L$ and
$R_{m}$ in terms of the beam waists $w_{0}$ and $w_{1}$ at the two mirrors.
Our Eq. (25) is identical to Eq. (27) in Ref. Luk (1986) and consistent with
Eq. (19) in Ref. Yu and Luk (1984).

The combination of Eqs. (21) and (25) finally yields the total non-paraxial
scalar correction for a plano-concave cavity with a spherical mirror

$\Delta\tilde{\nu}_{{\rm scalar},j}=\Delta\tilde{\nu}_{{\rm
prop},j}+\Delta\tilde{\nu}_{{\rm wave},j}=\frac{1}{8\pi
kR_{m}}\left[g(p,\ell)+4\right]\,.$ (27)

That the sum of the two scalar corrections does not depend on the cavity
length suggests an underlying physical reason, but we have not found it yet.

###  IV.3 Aspherical correction ${\cal H}_{\rm asphere}$

In the previous section we calculated the wave-front correction by comparing
the shape of the wavefront with a spherical mirror. We will now calculate the
effect of a deviation from this spherical mirror shape by an amount

$\displaystyle z_{\rm mirror}(x,y)-z_{\rm
sphere}(x,y)=-\tilde{p}\frac{r^{4}}{8R_{m}^{3}}\,,$ (28)

where $\tilde{p}=0$ for a sphere (our reference) and $\tilde{p}=1$ for a
paraboloid. This aspherical correction modifies the third term in the Taylor
expansion $z_{m}(r)=a+br^{2}+cr^{4}+...$ of a rotation-symmetric mirror.
Substitution of Eq. (28) in the generic Eq. (15) yields the aspherical
correction

$\Delta\tilde{\nu}_{\rm asphere,j}=-\frac{f(p,\ell)}{8\pi
kR_{m}}\,\tilde{p}\,\frac{L}{R_{m}-L}\,.$ (29)

This result is consistent with the result of Zeppenfeld and Pinkse Zeppenfeld
and Pinkse (2010), who chose the paraboloidal mirror as their reference shape
instead, and used a different notation; see Appendix A.

A comparison between Eq. (29) with Eq. (27) shows that the aspherical
correction contains an extra factor $L/(R_{m}-L)$. The aspherical correction
is thus relatively small for short cavities, basically because the modes in
these cavities are relatively compact and hence less sensitive to mirror
deformations.

##  V Vector correction & L-S coupling

###  V.1 Vector-LG modes

The analysis presented above used the scalar LG-modes $|\psi_{p,\ell}\rangle$
as basis set. This section extends the analysis to vector fields, by including
the optical polarization. It starts by introducing the vector LG-modes
$|\vec{\psi}_{p,\ell,v}\rangle$, with their additional vector quantum number
$v$.

In a cavity with mirror and rotation symmetry, the paraxial scalar modes
$\psi_{p,\ell}$ and $\psi_{p,-\ell}$, with $\ell\geq 0$ are frequency
degenerate. For vector fields, one also expects $x$\- and $y$-polarized vector
versions of these modes, such that the $\ell=0$ mode is two-fold degenerate
and the $\ell\neq 0$ modes are four-fold degenerate. But in reality, the four
$\ell\neq 0$ modes couple and split into two frequency-degenerate pairs which
differ in the orientation of the photon spin $s$, or circular polarization
$\sigma_{\pm}$, with respect to the orbital angular momentum $\ell$ due to a
form of L-S coupling (see Sec. V.2).

We will use the notation of Yu and Luk Yu and Luk (1983) and label the
resulting vector LG-modes as: (i) series A modes with total angular momentum
$J=\ell-1$ and (ii) series B modes with total angular momentum $J=\ell+1$.
Each A and B mode is a superposition of $(\ell,s)$ and $(-\ell,-s)$
circularly-polarized modes, where $\ell>0$ and $s=-1$ for A modes and $s=+1$
for B modes. To distinguish between the + and - superposition within each set,
we add a second component to the polarization label which we denote by $+$ and
$-$, depending on the symmetry of the state under mirror action in the $xz$
plane. With these polarization and symmetry aspects in mind, we thus specify
the vector quantum number as $v=\\{y,x\\}=\\{+,-\\}$ for the two $p,\ell=0$
modes and as $v=\\{A+,A-,B+,B-\\}$ for the four $p,\ell\geq 1$ modes.

As a vector generalization of the scalar paraxial modes of Eqs. (1)-(3), we
write the transverse field of the vector-LG modes as

$\vec{\psi}_{p,\ell,v}^{+}(r,\theta,z)=\frac{1}{\gamma_{z}}\,\vec{e}_{\ell,v}(\theta)f_{p,\ell}(\rho)\exp\left[ik\frac{r^{2}}{2R_{z}}-i(N+1)\chi\right]\,.$
(30)

For $\ell=0$ modes: $\vec{e}_{0,+}(\theta)=\vec{e}_{x}$ and
$\vec{e}_{0,-}(\theta)=\vec{e}_{y}$. For $\ell\geq 1$ modes, the vector fields
are also linearly polarized, but the orientation of this linear polarization
depends on $\theta$ as Yu and Luk (1983)

$\begin{array}[]{cccccc}\vec{e}_{\ell,v}(\theta)&=&\cos{(\ell\theta)}\vec{e}_{x}+\sin{(\ell\theta)}\vec{e}_{y}&=&\cos{[(\ell-1)\theta]}\vec{e}_{r}+\sin{[(\ell-1)\theta]}\vec{e}_{\theta}&(\rm{for}\,\,v=A+)\,,\\\
\vec{e}_{\ell,v}(\theta)&=&-\sin{(\ell\theta)}\vec{e}_{x}+\cos{(\ell\theta)}\vec{e}_{y}&=&-\sin{[(\ell-1)\theta]}\vec{e}_{r}+\cos{[(\ell-1)\theta]}\vec{e}_{\theta}&(\rm{for}\,\,v=A-)\,,\\\
\vec{e}_{\ell,v}(\theta)&=&\cos{(\ell\theta)}\vec{e}_{x}-\sin{(\ell\theta)}\vec{e}_{y}&=&\cos{[(\ell+1)\theta]}\vec{e}_{r}-\sin{[(\ell+1)\theta]}\vec{e}_{\theta}&(\rm{for}\,\,v=B+)\,,\\\
\vec{e}_{\ell,v}(\theta)&=&\sin{(\ell\theta)}\vec{e}_{x}+\cos{(\ell\theta)}\vec{e}_{y}&=&\sin{[(\ell+1)\theta]}\vec{e}_{r}+\cos{[(\ell+1)\theta]}\vec{e}_{\theta}&(\rm{for}\,\,v=B-)\,.\end{array}$

Figure 3 shows the polarization profiles of the A and B modes for $\ell=0-3$.
Note that all + modes have $\vec{e}(\theta=0)=\vec{e}_{x}$ and all - modes
have $\vec{e}(\theta=0)=\vec{e}_{y}$. Our short-hand notation for these modes
is $|0X+\rangle$ and $|0Y-\rangle$ for the $\ell=0$ modes and $|\ell
A\pm\rangle$ and $|\ell B\pm\rangle$ for the $\ell\geq 1$ $A\pm$ and $B\pm$
modes. The radial dependence $f_{p\ell}(\rho)$ is not included in this
labeling, but can be easily added with an extra quantum number $p$ or
$N=2p+\ell$.

The twofold frequency degeneracy expected in a cavity with rotation and mirror
symmetry has two consequences. First, it makes it difficult in experiments to
find the true eigenmodes of the cavity, as the output field $|\vec{\psi}_{\rm
out}\rangle$ will be a superposition of the two degenerate modes with relative
amplitudes that are determined by the input field $|\vec{\psi}_{\rm
in}\rangle$. Second, in the theory it leaves room for an alternative labeling
of the vector modes. Zeppenfeld and Pinkse Zeppenfeld and Pinkse (2010) chose
the total angular moment $J=\ell+s$ and the circular polarization
$\sigma_{\pm}$ as labels, instead of our $\ell,v$ labels. The vector profiles
of their $(J,\sigma_{\pm})$ modes, which are of the form
$\vec{e}_{J,\sigma_{\pm}}(\theta)=\vec{e}_{\pm}\exp{\pm i\ell\theta}$ with
$\vec{e}_{\pm}=(\vec{e}_{x}\pm i\vec{e}_{y})$ as circular polarizations, are
linear superpositions of our A and B modes (see Appendix A and C).

Figure 3: Sketches of the polarization profiles, indicated as lines, of the
vector LG-modes for $\ell=0-3$. The two $\ell=0$ modes are $x$-polarized and
$y$-polarized. Each of the four $\ell\geq 1$ modes split in two A and two B
modes; the 1A+ mode is radially polarized; the 1A- mode is azimuthally
polarized; the other modes have a mixed radial/azimuthal polarization. Most
modes occur in two versions, with field patterns that are rotated over half a
lobe/finger with respect to each other. The +/- labels indicate whether the
vector field is symmetric (+) or anti-symmetric (-) upon reflection in the $x$
axis.

###  V.2 Non-paraxial vector correction ${\cal H}_{\rm vec}$

The non-paraxial vector correction ${\cal H}_{\rm vec}$ originates from the
vector character of the optical field, and in particular from the small axial
component of the optical field. At the curved mirror, part of this axial field
transforms into a reflected transverse field because the boundary condition is
not decribed by $\vec{E}_{\perp}=\vec{0}$ but by the requirement that
“$\vec{E}$ is directed along the surface normal”. Cullen Cullen (1976); Cullen
and Yu (1979) was one of the first to mention this vector correction. Davis
Davis (1984) quantified it for a linearly-polarized fundamental Gaussian mode,
using geometric arguments. The vector correction to the reflection produces an
effective spin-orbit coupling that is similar to the one observed under strong
focusing Bliokh  _et al._ (2010, 2015).

Yu and Luk Yu and Luk (1984) and Luk Luk (1986) generalized the analysis to
any vector LG-modes. They derived the vector corrections of these modes with
the so-called action theorem, which is based on thermodynamic arguments and
relates the relative frequency shift to the relative change in stored energy,
via $\Delta f/f=\Delta W/W$ Davis (1984); Yu and Luk (1984); Luk (1986);
Uphoff  _et al._ (2015). In this subsection, we will instead use the
roundtrip-matrix formalism to generalize the calculation to the vector
coupling between any pair of modes. The roundtrip formalism is more general
because it yields a coupling matrix, whereas the action theorem only yields
the on-diagonal elements of this matrix.

We start our analysis with the earlier statement that every non-uniform
transverse field has an axial component
$\psi_{z}\approx(i/k)\vec{\nabla}_{\perp}\cdot\vec{\psi}_{\perp}$, see Eq.
(17). For the transverse vector field described by Eq. (30), the operator
$\vec{\nabla}_{\perp}$ yields three contributions: The derivative of the phase
factor in Eq. (30) yields the in-phase field $\psi_{z}=-(r/R)\psi_{\perp}$
needed to orient the vector field of the traveling wave along the curved
wavefront; this field has no further consequences. By contrast, the
derivatives of the two other factors in the right-hand side of Eq. (30), which
together form the normalized derivative
$\vec{\nabla}_{\perp}\cdot\vec{\Psi}(\rho,\theta)$, yield a small out-of-phase
longitudinal field that projects into an additional radially-polarized
transverse field that does modify the resonance. This projection includes a
geometric factor $-\vec{r}/R_{m}$ and a factor 2 to account for the standing-
wave character of the field, similar the factor 2 in the phase lag $2k\Delta
z$ that described the mirror shape. The resulting additional transverse field
is

$\vec{\Psi}_{\perp,{\rm
project}}=-i\frac{2\vec{\rho}}{kR_{m}}\vec{\nabla}_{\perp}\cdot\vec{\Psi}_{\perp}(\vec{r})\,,$
(31)

where $\vec{\rho}=\vec{r}/\gamma_{z}$ and
$\vec{\nabla}_{\perp}=\vec{e}_{x}(\partial/\partial\rho_{x})+\vec{e}_{y}(\partial/\partial\rho_{y})=(\vec{e}_{r}/\rho)(\partial/\partial\rho)\rho+(\vec{e}_{\theta}/\rho)\partial/\partial\theta$
is the transverse derivative vector operator in normalized coordinates. By
comparing Eq. (31) with the generic equations in Sec. III, we find the non-
paraxial vector correction

$\Delta\tilde{\nu}_{{\rm vec},j^{\prime},j}=\frac{1}{\pi
kR_{m}}\langle\vec{\Psi}_{j^{\prime}}|\,\vec{\rho}\otimes\vec{\nabla}_{\perp}|\vec{\Psi}_{j}\rangle\,,$
(32)

where the tensor product symbol $\otimes$ indicates that
$\vec{\nabla}_{\perp}$ operates on $|\vec{\Psi}_{j}\rangle$ and $\vec{\rho}$
operates on $\langle\vec{\Psi}_{j^{\prime}}|$.

The rotation symmetry of the $\vec{\rho}\otimes\vec{\nabla}_{\perp}$ operator
imposes conservation of total angular momentum $J^{\prime}=J$. In Appendix C
we will show that vector coupling is even diagonal, both in the basis of the
vector-LG modes and in the basis of the $(J,\sigma_{\pm})$ modes of Zeppenfeld
of order $N$. The corresponding (normalized) frequency shifts are

$\Delta\tilde{\nu}_{{\rm vec},j}=\frac{\pm\ell-1}{2\pi
kR_{m}}=\frac{-1-\ell\cdot s}{2\pi kR_{m}}\,,$ (33)

where the + sign applies to $A$-modes, with total angular momentum $J=\ell-1$,
and the - sign applies to $B$-modes, with $J=\ell+1$.

Equation (33) agrees with earlier results presented in Refs. Yu and Luk
(1984); Luk (1986); Zeppenfeld and Pinkse (2010). For $\ell\geq 1$ modes, the
vector correction acts as an effective $L-S$ coupling, denoted by $\ell.s$ in
Eq. (33), which splits each set of four $\ell\geq 1$ modes into two pairs of
frequency-degenerate vector modes. For the two, $x$ and $y$-polarized,
$\ell=0$ modes, Eq. (33) yields equal shifts. These results can be derived by
the operator algebra described in Appendix C or by partial integration over
vector fields of the form
$\vec{\Psi}_{j}(\rho,\theta)=\vec{e}_{j}(\theta)f_{j}(\rho)$. For the
$x$-polarized $\ell=0$ mode, $|\vec{\psi}\rangle=\vec{e}_{x}|\psi\rangle$, the
result simplifies to
$\langle\vec{\Psi}|\vec{r}\otimes\vec{\nabla}_{\perp}|\vec{\Psi}\rangle=\langle\Psi|x\partial_{x}|\Psi\rangle=-1/2$.

###  V.3 Combined result for rotation-symmetric cavities

Let’s combine the effects discussed up to now, into a single equation for the
fine structure of a plano-concave cavity with a simple spherical mirror. By
combining Eqs. (27) and (33), we find that the non-paraxial contribution to
the roundtrip phase lag is

$\varphi_{\rm
non}=2\pi\Delta\tilde{\nu}=\frac{1}{kR_{m}}[\frac{1}{8}(N^{2}+2N-4)-\frac{3}{8}\ell^{2}-\ell\cdot
s]\,,\\\ $ (34)

This result is identical to the results of Refs. Luk (1986) and Zeppenfeld and
Pinkse (2010). For a plano-concave cavity with a non-spherical rotation-
symmetric mirror, the factor $3/8$ in the top equation is replaced by
$[3/8+\tilde{p}L/8(R_{m}-L)]$, due to spherical aberration and the polynomial
in $N$ is also slightly modified; see Eqs. (29) and (50).

##  VI Bragg (vector) correction ${\cal H}_{\rm Bragg}$

If we replace the ideal mirror by a more realistic Bragg mirror, the optical
field will penetrate in the mirror and the reflection amplitude will change
from a steady $r=-1$ to $r=-\exp[i\varphi(\omega,k_{\perp})]$ Babic and
Corzine (1992); Babic  _et al._ (1993); Koks and van Exter (2020). The
reflection phase $\varphi(\omega,k_{\perp})=2kL_{\varphi}(\omega,k_{\perp})$
will now depend on the optical frequency $\omega$ and the angle of incidence,
which we express via its transverse momentum $k_{\perp}$ to avoid confusion
with the orientation angle $\theta$. The reflection phase is typically
different for TE ($s$-polarized) light than for TM ($p$-polarized) light,
because TE-light typically reflects better and thus yields a wider spectral
stopband. The resulting phase difference
$\varphi_{s}-\varphi_{p}=C(k_{\perp}/k)^{2}$ can impose an additional fine
structure on the cavity modes and push the intra-cavity field towards
radially- and azimuthally-polarized eigenmodes. Foster et al. Foster  _et al._
(2009) have described this Bragg effect and compared its strength relative to
the non-paraxial vector correction. We will briefly add our thoughts to their
treatment.

First of all, we note that the phase difference
$\varphi_{s}-\varphi_{p}=C(k_{\perp}/k)^{2}$, effectively also contains a
$(k_{\perp}/k)^{4}$-term because $C$ depends on $(k_{\perp}/k)^{2}$ as
$C(\vec{k}_{\perp})=C_{0}+C_{2}(k_{\perp}/k)^{2}$ and $C_{0}=0$ in the center
of the stopband Koks and van Exter (2020). Keeping this in mind, we write the
perturbation imposed by the Bragg effect in the two mirrors as

${\cal H}_{\rm
Bragg}=\frac{2C(k_{\perp})}{k^{2}}\,\left(\vec{k}_{\perp}\otimes\vec{k}_{\perp}-\frac{1}{2}k^{2}_{\perp}\right)\,,$
(35)

where the final $k^{2}_{\perp}/2$ balances the effect to zero for unpolarized
light. Appendix B calculates $C_{0}$ and $C_{2}$ from the properties of the
DBR. The $(k_{\perp}/k)^{4}$-term was not mentioned by Föster et al. Foster
_et al._ (2009), but could be relevant in experiments.

The rotation symmetry of the Bragg operator matches the rotation symmetry of
the vector LG-modes. As a result, many matrix elements
$\langle\psi_{j^{\prime}}|{\cal H}_{\rm Bragg}|\psi_{j}\rangle$ are zero. The
only non-zero on-diagonal elements of the Bragg operator are $J^{\prime}=J=0$.
The Bragg operator will thereby split each pair of $J=0$ modes in a radially-
polarized 1A+ mode and an azimuthally-polarized 1A- mode with opposite
frequency shifts equal to

$\Delta\tilde{\nu}_{\rm Bragg,1A\pm}\approx\pm\frac{C_{0}}{kz_{0}}(N+1)\,.$
(36)

To arrive at this result, we only included the $C_{0}$ term of the Bragg
effect and used the mean-square opening angle $\langle
k_{\perp}^{2}/k^{2}\rangle=(N+1)/kz_{0}$ (see Appendix B). The relative
strength of the predicted Bragg splitting, compared to the common factor
$1/(8\pi kR_{m})$ for non-paraxial effects, is $Y\equiv(8\pi
kR_{m})\Delta\tilde{\nu}_{\rm Bragg,1A+}\approx 16\pi C_{0}\sqrt{R_{m}/L}$ for
$L\ll R_{m}$ and $N=1$. Bragg effects become more prominent for cavities with
$L\ll R_{m}$ because they scale with the mean-square modal opening angle,
which increases when $L$ decreases. The observation of a 1A+/1A- splitting is
a hallmark for the Bragg effect.

Dufferwiel et al. Dufferwiel  _et al._ (2015) have observed a TE-TM splitting
between the $1A+$ and $1A-$ mode for a cavity filled with an active
semiconductor. They attributed the observed effect to TE-TM splitting of the
polariton eigenstates associated with two different branches in the
semiconductor band structure. But a remnant of the Bragg effect might also
have been present.

As an aside, we note that the $\ell=1$ spectra shown in Ref. Dufferwiel  _et
al._ (2015) actually comprises three peaks. The frequency difference between
the outer $1A+$ and $1A-$ is due to the mentioned TE-TM splitting. But the
average frequency of these peaks doesn’t coincide with the frequency of the
inner peak, which must originate from the degenerate $1B+$ and $1B-$ modes. We
think that this additional frequency difference is due to the spin-orbit
coupling described in our Sec. V.2.

The Bragg operator also has non-zero off-diagonal elements that couple
$(p-1,\ell=J+1,A)$ modes with $(p,\ell=J-1,B)$ modes, both of order $N=2p+J-1$
(see Appendices C and D). The effect of these off-diagonal elements on the
fine structure is limited in rotational-symmetric cavities, because the on-
diagonal elements typically differ a lot and dominate. We thus expect hardly
any Bragg-related hyperfine splitting for $J\neq 0$ modes in these cavities.
From a physics perspective, the $J\neq 0$ vector modes are (to first order)
insensitive to the Bragg effect because they contain equal amounts of radial
and azimuthal polarization. But even $J\neq 0$-type vector modes can exhibit
some Bragg-induced hyperfine splitting when the cavity is sufficiently
astigmatic to mix LG-modes, c.q. modify the eigenmodes, and thereby make them
sensitive to off-diagonal matrix elements and the Bragg effect (see Appendix
D).

##  VII Mirror-astigmatic corrections

###  VII.1 Astigmatic correction ${\cal H}_{\rm astigm}$

In the previous sections we analyzed the resonances in a plano-concave cavity
with a rotationally symmetric and almost spherical mirror. In this section we
will analyze mirror deformations that lack rotation symmetry, which are known
as astigmatic deformations. Many concave mirrors are not rotationally
symmetric but have slightly different curvatures in two orthogonal directions,
which we will call $x$ and $y$. We describe the (paraboloidal component of
the) astigmatic mirror shape as

$z_{\rm
mirror}(x,y)=\frac{x^{2}}{2R_{x}}+\frac{y^{2}}{2R_{y}}\approx\frac{x^{2}+y^{2}}{2\overline{R}}+\eta_{\rm
astigm}\frac{x^{2}-y^{2}}{2\overline{R}}\,,$ (37)

where $\overline{R}=(R_{x}+R_{y})/2=R_{m}$. The parameter $\eta_{\rm
astigm}=(R_{y}-R_{x})/(2\overline{R})\ll 1$ quantifies the strength of the
astigmatism.

Astigmatism breaks the rotation symmetry and prefers Hermite-Gaussian modes
over Laguerre-Gaussian modes. A combined treatment of astigmatism and
rotationally-symmetric perturbation thus requires a matrix description that
includes all relevant modes. As astigmatism also has mirror symmetry in the
(just-defined) $x$ axis, it only couples modes with the same +/- vector
character. Each $N$ group is thus expected to split in two subgroups, the $N+$
group and $N-$ group.

If astigmatism would be the only effect, then we would simply use the HG-modes
as a basis instead of the LG-modes. Based on the factorization of the HG
profiles, we would then conclude that (i) the HG-modes are the eigenmodes of
the astigmatic cavity and (ii) the modes in each $N$-group split into (pairs
of) vector HG-modes with an equidistant spacing $\propto\eta_{\rm astigm}$.
But this scalar analysis does not take spin-orbit coupling into account. To
include this effect, we will instead analyze astigmatism in the basis of the
vector LG-modes.

We start with the $N=1$ group. This group contains four modes and splits into
two pairs: the $(A+,B+)$ set and the $(A-,B-)$ set. The astigmatic coupling
matrix for each of these sets is (see Appendix C)

$\displaystyle\Delta\tilde{\nu}_{{\rm
astigm},(N=1)}=\begin{pmatrix}0&\tilde{X}\\\ \tilde{X}&0\end{pmatrix}\,,$ (38)
$\displaystyle\tilde{X}\equiv\frac{2}{\lambda}\langle 1A+|\Delta
z(x,y)|1B+\rangle=\eta_{\rm astigm}\frac{\tan{\chi_{0}}}{2\pi}\,.$ (39)

This result was checked via integration, using the relation
$\langle\psi|r^{2}|\psi\rangle=(w_{1}^{2}/2)(N+1)$ with $w_{1}^{2}=(\lambda
R/\pi)\tan{\chi_{0}}$. The eigenvalues of this astigmatic matrix are
$\lambda_{\pm}=\pm\tilde{X}$. The associated eigenmodes, $(A+)\pm(B+)$, have a
$\cos{\theta}\vec{e}_{x}$ and $-\sin{\theta}\vec{e}_{y}$ angular dependence.
They are the mirror-symmetric vector $HG_{10}$ and $HG_{01}$ modes that one
expects in the absence of spin-orbit coupling.

Next, we add the non-paraxial corrections as on-diagonal elements
$\Delta\tilde{\nu}_{j}=[g(p,\ell)-4\ell\cdot s]/(8\pi kR_{m})$ with
$g(0,1)=-2$. We also introduce the relative strength of the astigmatism as
$X=(8\pi kR_{m})\tilde{X}=2\eta_{\rm astigm}k^{2}w_{1}^{2}$ to remove a common
factor. The final result in the $(1B+,1A+)$ basis

$(8\pi kR_{m})\,\Delta\tilde{\nu}_{(N=1)}=\begin{pmatrix}-6&X\\\
X&2\end{pmatrix}$ (40)

combines astigmatism with non-paraxial corrections. The eigenvalues of this
complete matrix are $\lambda_{\pm}=-2\pm\sqrt{16+X^{2}}$. The new eigenmodes
are of the form $[\cos{\beta}\psi_{A}+\sin{\beta}\psi_{B}]$ and
$[-\sin{\beta}\psi_{A}+\cos{\beta}\psi_{B}]$, with the mode-mixing angle
$\beta=\arctan{(X/4)}$. This shows that astigmatism increases the splitting
between the non-paraxial modes by a factor $\sqrt{1+(X/4)^{2}}$, while
gradually changing the $A+$ and $B+$ eigenmodes that are visible at $X=0$ into
the $x$-polarized $HG_{10}$ and $y$-polarized $HG_{01}$ eigenmodes for $X\gg
1$.

To calculate the astigmatic matrix for any $N\geq 2$ group, we need to find
the associated matrix elements. The $x^{2}-y^{2}=r^{2}\cos{2\theta}$ angular
dependence of the perturbation, shows that the astigmatic coupling obeys the
selection rule $\Delta\ell=\pm 2$. Furthermore, astigmatism only couples
$\ell\leftrightarrow(\ell+2)$ modes with the same vector label
$v=\\{A+,A-,B+,B-\\}$. But astigmatism can also couple $1A$ and $1B$ modes,
because the $\ell=1$ modes implicitly also contain $-1$ modes. Hence, the only
non-zero matrix elements of the astigmatic contribution are

$\displaystyle\frac{2}{\lambda}\langle\psi_{p,\ell=1,A+}|\Delta
z(x,y)|\psi_{p,\ell=1,B+}\rangle$ $\displaystyle=$
$\displaystyle\tilde{X}(p+1)\,,$ (41)
$\displaystyle\frac{2}{\lambda}\langle\psi_{p+1,\ell-2,v}|\Delta
z(x,y)|\psi_{p,\ell,v}\rangle$ $\displaystyle=$
$\displaystyle\tilde{X}h(N,\frac{N-\ell}{2})\,,$ (42)
$\displaystyle\frac{2}{\lambda}\langle\psi_{p-1,\ell+2,v}|\Delta
z(x,y)|\psi_{p,\ell,v}\rangle$ $\displaystyle=$
$\displaystyle\tilde{X}h(N,\frac{N+\ell}{2})\,,$ (43)

were $v$ can be any vector label. The first equation has an identical
counterpart for the $-$ modes. The second equation assumes $\ell\geq 2$ and
introduces the function $h(N,n_{s})\equiv\sqrt{(n_{s}+1)(N-n_{s})}$, which
obeys the symmetry $h(N,N-n_{s})=h(N,n_{s}-1)$. These results were again
obtained with the operator algebra described in Appendix C.

Figure 4: Sketches of the polarisation profile for the ”+” modes of the $N=2$
group, as a function of the normalized astigmatism $X$. The eigenmodes on the
right correspond to $X=0,2,4,10$.

Using the results presented above, we can now calculate the coupling matrix
for any non-paraxial astigmatic cavity. We will show the result only for the
$N=2$ and $N=3$ groups and leave the general analysis to the reader. The $N=2$
group contains $2(N+1)=6$ members, 3 with a $+$ mirror symmetry and 3 with a
$-$ mirror symmetry. The three LG-vector modes with a $+$ character are the
$(0+,2A+,2B+)$. The on-diagonal elements of the spectral matrix are determined
by the non-paraxial correction $\left[g(p,\ell)\pm 4\ell\right]=2$ for the
$0$-mode, $4$ for the $2A$-mode and $-12$ for the $2B$-mode. The off-diagonal
elements are determined by the astigmatism, which couples the $\ell=0$ mode
with the two $\ell=2$ modes with an equal normalized coupling $\sqrt{2}X$.
This makes the combined spectral matrix in the $(2B+,0+,2A+)$ basis equal to

$(8\pi kR_{m})\,\Delta\tilde{\nu}_{(N=2)}=\begin{pmatrix}-12&\sqrt{2}X&0\\\
\sqrt{2}X&2&\sqrt{2}X\\\ 0&\sqrt{2}X&4\end{pmatrix}\,.$ (44)

The three $-$ modes are coupled by an identical matrix that now operates in
the $(2B-,0-,2A-)$ basis. A similar exercise for the $N=3$ group, where the
set of $+$ modes are $(3B+,1B+,1A+,3A+)$, yields

$(8\pi kR_{m})\,\Delta\tilde{\nu}_{(N=3)}=\begin{pmatrix}-20&\sqrt{3}X&0&0\\\
\sqrt{3}X&0&2X&0\\\ 0&2X&8&\sqrt{3}X\\\ 0&0&\sqrt{3}X&4\end{pmatrix}\,.$ (45)

Figure 4 shows the fine structure in the $N=2$ group as a function of the
normalized astigmatism $X$ and the associated eigenmodes. In the absence of
astigmatism, at $X=0$, the vector-LG $2A+$ and $2B+$ modes have
eigenfrequencies $(8\pi kR_{m})\tilde{\nu}=4$ and -12, respectively, while the
$0A+$ mode has eigenfrequency 2. At non-zero astigmatism, these three modes
mix and gradually transform from LG- to HG-modes, while their eigenfrequencies
also change. At $X=10$, where astigmatism dominates over non-paraxial effects,
the eigenmodes strongly resemble the HG-modes and the distance between the
eigenvalues becomes approximately equal. The three asymptotes show the
eigenfrequencies -4 and $-1\pm 2X$ that are reached at $X\gg 1$. The depicted
transition from dominant non-paraxial effects to dominant astigmatic effects
in an optical cavity resembles the transition from dominant spin-orbit
coupling to a dominant Zeeman effect in atomic physics.

###  VII.2 Shape birefringence

In the previous section, we stated that the coupling matrices of the $+$ and
$-$ modes are identical. We will now show that these matrices can be slightly
different on account of a second-order effect that combines astigmatism with
the non-paraxial vector correction. The associated anisotropic spin-orbit
coupling results in shape birefringence, i.e. it induces a frequency
difference between $x$\- and $y$-polarized light, just as birefringence would,
but only because of the $x/y$ difference in the shape of a mirror. The story
is as follows:

The vector correction in Eq. (31) uses the transverse derivative
$\vec{\nabla}_{\perp}.\vec{\Psi}_{\perp}(\vec{r})$ to calculate a small
additional field, which is then projected onto a radial transverse field by
multiplying it with $\vec{\rho}/R_{m}$. But in a cavity with an astigmatic
mirror, the projection should instead be on

$\frac{\vec{\rho}_{x}}{R_{x}}+\frac{\vec{\rho}_{y}}{R_{y}}=\frac{\rho}{\overline{R}}\vec{e}_{r}+\eta_{\rm
astigm}\frac{\rho}{\overline{R}}(\cos{\theta}\vec{e}_{x}-\sin{\theta}\vec{e}_{y})\,.$
(46)

The first term in this equation describes the vector correction discussed in
Sec. V. The second term described the astigmatic component of this vector
correction.

The anisotropic vector correction can again be calculated via integration or
operator algebra. For $\ell=0$ modes, the resulting on-diagonal matrix
elements yield

$\Delta\tilde{\nu}_{v+a}=\mp\frac{\eta_{\rm astigm}}{2\pi k\overline{R}}\,,$
(47)

where the $-$ sign applies to the $+$ mode and vice versa and where Eq. (47)
applies to all $\ell=0$ modes, irrespective of $p$. The same result can be
obtained by interpreting the vector correction of Eq. (33) as
$\Delta\tilde{\nu}=-1/(2\pi kR_{x})$ for the $x$-polarized mode with
$R_{x}=\overline{R}(1+\eta_{\rm astigm})$. For $\ell\geq 1$ modes, the
anisotropic spin-orbit coupling only has non-zero off-diagonal elements (see
Appendices C and D). As a result, these modes experience typically hardly any
shape birefringence and retain their $+/-$ degeneracy in cavities with small
astigmatism. Some hyperfine splitting might, however, still be present in
cavities where the astigmatism is strong enough to mix the vector LG-modes.

##  VIII Discussion & residual ${\cal H}_{\rm rest}$

The analysis presented above describes the most common perturbations in
optical cavities, but is unavoidably incomplete. We included the quartic
scalar corrections $k_{\perp}^{4}$ and $r^{4}$, the spin-orbit and Bragg
vector correction, and astigmatic deformations of the form $(x^{2}-y^{2})$,
but the concave mirror might also be deformed in different ways. We will
combine all residual mirror deformations in the scalar operator ${\cal H}_{\rm
rest}=2k\Delta z_{\rm rest}\ll 1$.

The residual operator ${\cal H}_{\rm rest}$ will scatter light and couple
modes, just like the other operators do. And this coupling will again mainly
be effective between modes with the same transverse order $N$, if the cavity
is operated far from frequency degenerate points. The effect of ${\cal H}_{\rm
rest}$ on the spectral fine structure and the associated eigenmodes is then
completely described by the residual spectral matrix
$\Delta\tilde{\nu}_{j^{\prime},j}=\langle\psi_{j^{\prime}}|{\cal H}_{\rm
rest}|\psi_{j}\rangle$. In principle, knowledge of the spectrum and eigenmodes
of a specific order $N$ allows one to reconstruct the full coupling matrix
${\cal H}_{\rm fine}$ of that order and disentangle its contributions. The
accuracy of such an analysis is only limited by the cavity finesse, which
makes it an extremely sensitive probe of the actual mirror shape. Benedikter
et al. Benedikter  _et al._ (2019) have previously used the resonances around
frequency degenerate points as a similar sensitive probe for the topography of
their planar mirror.

The eigenmodes of the spectral matrix will provide a better match with the
deformed mirror than the original LG-modes, but the match is typically not
perfect. The resulting modal loss per roundtrip can be calculated from the
next term in the Taylor expansion of $\exp(-2ik\Delta z)\approx 1-i2k\Delta
z-2k^{2}\Delta z^{2}$. The calculated amplitude loss

$\gamma_{{\rm extra}}\approx 2k^{2}\iint\mbox{d}x\mbox{d}y\,[\Delta
z(x,y)]^{2}\,|\psi(x,y)|^{2}\,,$ (48)

with $\Delta z=z_{\rm mirror}-z_{\rm mode}$, yields the famous expression
$(4\pi\sigma/\lambda)^{2}$ with $\sigma^{2}=\langle\Delta z^{2}\rangle$ for
the intensity loss Harvey  _et al._ (2012). A finite mirror radius $r_{\rm
max}$ will add clipping amplitude loss

$\gamma_{{\rm clipping}}\approx\int_{r_{\rm max}}^{\infty}\pi
dr^{2}\,|\psi(r)|^{2}\,,$ (49)

where $|\psi(r)|^{2}$ is the rotational-averaged intensity profile.

At first sight, it might be surprising that height variations introduce modal
loss in a system that was assumed to be lossless. But any submatrix ${\cal H}$
of the system is bound to be non-Hermitian even when the full dynamic matrix
is Hermitian. The residual loss originates from coupling between modes with
different order $N$, which will increase the mode size beyond reasonable
bounds and thereby result in clipping loss. Even when the coupling between
individual modes is inefficient, the multitude of available coupling channels
creates the residual loss described by Eq. (48).

##  IX Summary & outlook

This paper calculates the resonance frequencies and eigenmodes of a planar-
concave cavity beyond the common paraxial limit and beyond the spherical
mirror shape. It does so by describing the roundtrip dynamics of the intra-
cavity field in a 3-dimensional Fabry-Perot (FP) cavity in a general operator
formalism, which reduces to a modest-size matrix description after the
application of perturbation theory. It then shows that the $2(N+1)$ modes with
the same longitudinal $q$ and transverse order $N$ are not frequency
degenerate, as predicted by a paraxial theory, but split. The associated
optical fine structure has many contributions, which are listed in Table I and
calculated in parts II and III of the paper.

The paper presents a complete theoretical framework for the expected optical
fine structure, by systematically analyzing all contributions that can
realistically be expected. For cavities with rotation and mirror symmetry we
basically recover the results of Refs. Luk (1986) and Zeppenfeld and Pinkse
(2010), albeit often in easier forms. This analysis is presented in part II,
Secs. IV and V. In part III, we analyze the fine structure of more general
cavities. This results in four additional contributions: (i) a Bragg
correction, to quantify the role of the Bragg mirror, (ii) an astigmatic
correction to quantify the effect or an astigmatic mirror, (iii) an isotropic
spin-orbit coupling in astigmatic cavities, and (iv) a residual correction,
which was only discussed in general terms. In experiments, the astigmatic
correction is expected to be an important technical complication that can
easily dominate the more fundamental non-paraxial corrections. The paper
introduces a dimensionless parameter $X$ to compare these effects and predicts
how the fundamental effects are more likely to dominate in short cavities with
mirrors with small radii of curvature. A second dimensionless parameter $Y$
compares the strength of the Bragg effect relative to the more fundamental
non-paraxial corrections.

The optical fine structure in FP spectra resembles the fine structure in
atomic spectra. The energy levels in atomic physics depend primarily on the
principal quantum number $n$, but exhibit a fine structure that is an order
$\alpha^{2}$ smaller, where $\alpha\approx 1/137$ is the fine structure
constant Giffiths (2014). In comparison, the optical resonances in FP spectra
are primarily determined by the two principal quantum numbers $(q,N)$. The
relative strength of the optical fine structure $\Delta\tilde{\nu}$ is of the
order $1/(8\pi kR_{m})\propto\lambda/R_{m}$, where $R_{m}$ is the radius of
curvature of the mirror.

The results presented in Table I describe the non-paraxial effects in a plano-
concave cavity with mirror spacing $L$ and mirror radius $R_{m}$. This result
can easily be generalized to an arbitrary cavity, with two mirrors with radii
$R_{1}$ and $R_{2}$, using the following procedure: (i) use paraxial optics to
find the waist and the distances $L_{1}$ and $L_{2}$ from the mirrors to this
waist, (ii) use Table I to determine the various contributions to
$\varphi_{\rm non,1}$ for the optical path from the waist to Mirror 1 and
back, (iii) repeat this step for Mirror 2 to find $\varphi_{\rm non,2}$, and
(iv) add the two results to find the non-paraxial roundtrip phase lag
$\varphi_{\rm non}=\varphi_{\rm non,1}+\varphi_{\rm non,2}$. For a symmetric
bi-concave cavity of length $2L$, one thus finds that (i) $\varphi_{\rm non}$
is twice as large as in a plano-concave cavity of length $L$, (ii) the
relative frequency shift $\Delta\tilde{\nu}$ is also twice as large, but (iii)
the absolute frequency shifts $\Delta\nu$ are equal, as expected.

The analysis presented in this paper neglects losses. This assumption is valid
for cavities with large high-reflective mirrors, such that the cavity
resonances are clearly resolved in the optical spectrum. The influence of loss
on the spectral resonances is probably limited to detunings
$\Delta\tilde{\nu}<1/F$ and therefore small for finesses $F\gg 10R/\lambda$,
where the factor 10 was added to compensate for a factor $\ll 10$ in Eq. (34).
A measurement of the modal finesses, and the associated clipping losses at
finite-size mirrors, can however be useful to further characterize the
individual modes.

The analysis predicts that most modes appear in frequency-degenerate pairs,
with polarization patterns of the form $A\pm$ or $B\pm$ for the $\ell\geq 1$
modes and $x$ and $y$ polarization for the $\ell=0$ modes. It also predicts
that this pairwise degeneracy will be slightly broken for some pairs by
effects that one might thus call hyperfine splitting. This paper quantifies
two effects and shows that: (i) the $\ell=1,A$ pairs split in modes with
radial and azimuthal polarization in cavities with Bragg mirrors and (ii) the
$\ell=0$ mode pairs exhibit a small second-order splitting in astigmatic
cavities. It also argued how the degeneracy of other mode pairs could be
slightly broken in strongly-astigmatic cavities, due to admixture of HG-
character in the vector LG-modes that are preferred by spin-orbit coupling.
The pairwise degeneracy can probably also be broken when the mirror symmetry
is broken, for instance when one mirror has a twist Blows and Forbes (1998);
Kasyutich (2009), or a higher-order astigmatism of the form $x^{\prime
4}-y^{\prime 4}$ with an $x^{\prime}y^{\prime}$ orientation different from the
$xy$-orientation of the prime astigmatism. And mirror symmetry is obviously
broken in cavities with chiral stuctures, like the ones recently reported in
Ref. Gautier  _et al._

As a further outlook, we note that the optical fine structure contains
information on the mirror shape down to sub-nm precision. It can thus in
principle be used to inspect these shapes, without the need to dismount the
mirrors and inspect them by AFM or optical interference.

In future work, the analysis could be extended by including the other, $C_{2}$
or $k_{\perp}^{4}$, Bragg effect. This effect was neglected in most of the
analysis, but the example presented in Appendix B shows that this
simplification is not always correct.

The analysis could also be extended by including the coupling between modes of
different $N$-groups. The latter coupling is typically small but will become
important at so-called frequency-degenerate cavity lengths, where the Gouy
phase $\chi_{0}$ is a rational fraction of $\pi$ and modes with different
$(q,N)$ numbers become frequency degenerate. The resulting modified eigenmodes
can potentially lead to a reduction in mode area and an increase the light-
matter interaction Koks and van Exter (2022). The analysis could also be
extended to optical cavities with different geometries, beyond the two-mirror
plano-concave type. The theoretical framework developed in this paper is
general enough to also analyze these related geometries in a perturbative way.

Finally, it might be interesting to compare the presented analysis with the
geometric approach to cavity aberration presented in a recent publication of
Jaffe et al. Jaffe  _et al._ (2021). Or to compare the presented analysis of
mode formation in open optical cavities with the analysis of mode formation in
rotational-symmetric graded-index optical waveguides/fibers presented in ref.
Petrov (2013). We leave these topics as challenges to the reader.

###### Acknowledgements.

The authors would like to thank Sean van der Meer and Martin Bijl for
experiments that stimulated our ideas on the optical fine structure. They also
acknowledge Gerard Nienhuis for his suggestion to investigate the operator
formalism Nienhuis and Visser (2004); Nienhuis  _et al._ (2021) and Pepijn
Pinkse for sharing his knowledge on prior experiments on the cavity fine
structure Pinkse  _et al._ (2011). MW acknowledges support by the Danish
National Research Foundation through NanoPhoton - Center for Nanophotonics,
grant number DNRF147 and Center for Nanostructured Graphene, grant number
DNRF103, and by the Independent Research Fund Denmark - Natural Sciences
(project no. 0135-00403B).

##  Appendix A Comparison with Zeppenfeld-Pinkse

This Appendix summarizes the key results of the article entitled ’Calculating
the fine structure of a Fabry-Perot resonator using spheroidal wave functions’
by Zeppenfeld and Pinkse Zeppenfeld and Pinkse (2010), and compares them with
the associated non-paraxial effects listed in Table I. We will transform their
results to our notation for the specific case of a plano-concave cavity.

Zeppenfeld and Pinkse Zeppenfeld and Pinkse (2010) label their vector-LG modes
with quantum numbers $(\nu,J,\sigma_{\pm})$. Zeppenfeld’s quantum number $\nu$
is equal to our radial quantum number $p$. Zeppenfeld’s quantum number $J$
denotes the total angular momentum. As such, it combines our radial quantum
number $\ell$ with a vector quantum number $s$ that is $\pm 1$ for circular-
polarized $\sigma_{\pm}$ light. Their $v=\\{J,\sigma_{+}\\}$ mode is a
superposition of our $B_{+}$ and $B_{-}$ mode with $\ell=J-1$. Their
$v=\\{J,\sigma_{-}\\}$ mode is a superposition of our $A_{+}$ and $A_{-}$ mode
with $\ell=J+1$.

Equation (40) of Ref. Zeppenfeld and Pinkse (2010) states that the roundtrip
phase of the $(\nu,J,\sigma^{+})$ mode is

$\varphi_{\nu,J,\sigma^{+}}=2kL-2(2\nu+J)\arctan{\xi_{+}}-\frac{2}{kR_{m}}\nu(\nu+J)+\frac{\xi_{+}}{kz_{0}}(\frac{1}{4}-\frac{L}{4R_{m}}\tilde{c}_{4})[6\nu(\nu+J)+J(J+1)]\,,\\\
$ (50)

To arrive at Eq. (50), we combined Zeppenfeld’s expansion parameter
$1/\overline{c}=2/(kd)$, with $d=2z_{0}$, with equations that are specific for
plano-concave cavities, like $\xi_{+}=2L/d$,
$\xi/[\overline{c}(1+\xi^{2})]=1/(kR_{m})$,
$f_{4+}=(2R_{m}^{2}L\tilde{c}_{4}-\frac{1}{4})\xi$, and Rayleigh range
$z_{0}=\sqrt{L(R_{m}-L)}$. The parameter $\tilde{c}_{4}$ describes the
deviation from a paraboloidal mirror shape and is related to the parameter
$\tilde{p}$ in the main text via $\tilde{c}_{4}=1-\tilde{p}$.

The first term on the right-hand side of Eq. (50) is the plane-wave roundtrip
phase. The second term is the phase lag predicted by paraxial theory. The
third and fourth terms in Eqs. (50) describe the spectral fine structure in
Zeppenfeld’s notation. We transform this to our notation by writing $J=\ell+s$
for $\ell>0$, with $J=\ell+1$ for $\sigma_{+}$ polarization, $2\nu+J=N+1$ with
transverse order $N=2p+\ell$, and by introducing the fundamental Gouy phase
$\chi_{0}=\arctan{\xi}=\arctan{\sqrt{L/(R_{m}-L)}}$. With this rewrite, we
arrive at the equivalent equation in our notation

$\displaystyle\varphi_{p,\ell,\sigma^{\pm}}$ $\displaystyle=$ $\displaystyle
2kL-2(N+1)\chi_{0}-\varphi_{\rm non}\,,\quad\mbox{with}$
$\displaystyle\varphi_{\rm non}$ $\displaystyle=$
$\displaystyle\frac{1}{kR_{m}}[\frac{1}{8}(N^{2}+2N-4)-\frac{3}{8}\ell^{2}-\ell\cdot
s]\,.$ (51)

This final result includes a rewrite of Eq. (41) of Ref. Zeppenfeld and Pinkse
(2010) and is therefore valid for modes with both types of polarization. It
only applies to a plano-concave cavity with spherical mirrors
($\tilde{c}_{4}=1$ in Zeppenfeld’s notation), but can be extended to
aspherical mirrors by replacing the factor $3/8$, in front of the
$\ell^{2}$-term, by $3/8+\tilde{p}L/[8(R_{m}-L)]$ and by slightly modifying
the function of $N$. Equation (A) is identical to Eq. (34) in the main text,
and to the results of Luk Luk (1986).

##  Appendix B Bragg correction in detail

This Appendix analyzes the polarization dependence of the reflection phase of
a DBR and the resulting Bragg correction ${\cal H}_{\rm Bragg}$. The
reflection phase of a DBR

$\varphi_{s,p}(\omega,\phi)=2kL_{\varphi}(\omega,\phi)=[\omega-\omega_{c}(\phi)]\tau_{s,p}(\phi)$
(52)

depends on the detuning $[\omega-\omega_{c}(\phi)]$ between the optical
frequency and the center of the stopband and on the (polarization-dependent)
phase penetration depth $L_{s,p}(\phi)=c\tau_{s,p}(\phi)/2$ in the DBR. Both
quantities depend on the angle of incidence $\phi\approx k_{\perp}/k\ll 1$ as
Babic and Corzine (1992); Babic  _et al._ (1993)

$\displaystyle\omega_{c}(\phi)$ $\displaystyle\approx$
$\displaystyle\omega(0)(1+B\phi^{2})\,\,;\,B=\frac{1}{4}(\frac{1}{n_{L}^{2}}+\frac{1}{n_{H}^{2}})\,,$
(53) $\displaystyle\tau_{s,p}(\phi)$ $\displaystyle\approx$
$\displaystyle\tau(0)(1\pm
A\phi^{2})\,\,;\,A=\frac{1}{2}(1+\frac{1}{n_{H}n_{L}})\,,$ (54)

where the + sign in Eq. (54) applies to $p$-polarized light, with its reduced
Fresnel reflection and reduced stopband, and the - sign applies to
$s$-polarized light. Equation (53) is the generic result of a Taylor expansion
of Snell’s law. Equation (54) is valid only for H-DBRs, i.e. DBRs that start
with the high-index $n_{H}>n_{L}$ layer on the air side $(n_{\rm in}=1)$. By
combining these expansions with the H-DBR result
$\omega_{c}(0)\tau(0)=\pi/(n_{H}-n_{L})$ Babic and Corzine (1992); Babic  _et
al._ (1993); Koks and van Exter (2020), we find

$\varphi_{p}(\phi)-\varphi_{s}(\phi)=\frac{2\pi}{n_{H}-n_{L}}\,A\phi^{2}\,\left[\frac{(\omega-\omega_{c}(0))}{\omega_{c}(0)}-B\phi^{2}\right]\,.$
(55)

This polarization-dependent reflection at $k_{\perp}\neq 0$ creates the Bragg
correction introduced in Eq. (35) in the main text, which reads

${\cal H}_{\rm
Bragg}=\frac{2C(k_{\perp})}{k^{2}}\,\left(\vec{k}_{\perp}\otimes\vec{k}_{\perp}-\frac{1}{2}k^{2}_{\perp}\right)\,,$
(56)

with $C(\vec{k}_{\perp})=C_{0}+C_{2}(k_{\perp}/k)^{2}$ and

$C_{0}=\frac{2\pi
A}{n_{H}-n_{L}}\frac{\omega-\omega_{c}(0)}{\omega_{c}(0)}\,\,;\,\,C_{2}=\frac{-2\pi
AB}{n_{H}-n_{L}}\,.$ (57)

The factor 2 in Eq. (56), which indicates that there are two reflections, is
only approximately 2, because the angle-dependent reflection from the curved
mirror and flat mirror are only similar in the short-cavity limit $L\ll
R_{m}$. The “quadratic Bragg effect”, quantified by $C_{0}$, depends
critically on the frequency detuning. The “quartic Bragg effect”, quantified
by $C_{2}$, does not and could thus even become dominant around the center of
the stopband. The “quadratic Bragg effect” has on-diagonal elements only for
the $j=j^{\prime}=1A\pm$ modes, where

$\langle\psi_{j}|{\cal H}_{\rm
Bragg}|\psi_{j}\rangle=\pm\frac{C_{0}}{kz_{0}}(N+1)\,,$ (58)

and where we used $\langle\phi^{2}\rangle=\langle
k_{\perp}^{2}/k^{2}\rangle=(N+1)/kz_{0}$.

For a typical DBR coating of SiO2 ($n_{L}\approx 1.46$) and Ta2O5
($n_{H}\approx 2.09$), and a typical relative detuning
$[\omega-\omega_{c}(0)]/\omega_{c}(0)$ of 1%, we find $C_{0}\approx-0.13$,
while $C_{2}\approx 2.3$ at any detuning. To calculate the associated
polarization shifts, these values should be multiplied by
$\langle\phi^{2}\rangle=\langle k_{\perp}^{2}/k^{2}\rangle=(N+1)/kz_{0}$ and
$\langle\phi^{4}\rangle=f(p,\ell)/(kz_{0})^{2}$. For a typical microcavity
with $L=2~{}\mu$m, $R=20~{}\mu$m and $\lambda=0.63~{}\mu$m, the rms opening
angle of the fundamental mode
$\sqrt{\langle\phi^{2}\rangle}\approx\sqrt{0.016}\approx 0.13$ rad. For the
$1A+$ and $1A-$ modes, this results in frequency shifts $\Delta\tilde{\nu}$ of
$\pm C_{0}\langle\phi^{2}\rangle/(2\pi)=\mp 6.6\times 10^{-4}$ due to the 1%
detuning and $C_{2}\langle\phi^{4}\rangle/(2\pi)=\pm 5.6\times 10^{-4}$ due to
the quartic correction. These numbers show that the Bragg correction is
typically small at small detuning, where the quartic effect typically also
plays a role. But the Bragg correction should be observable, in particular at
larger frequency detuning.

For L-DBRs, i.e. DBRs that start with an $n_{L}$ layer on the air side, two
parameters are different Babic and Corzine (1992); Babic  _et al._ (1993);
Koks and van Exter (2020). The product
$\omega_{c}(0)\tau(0)=n_{L}n_{H}\pi/(n_{H}-n_{L})$ is larger than for H-DBRs,
but the polarization factor in Eq. (54) is now

$A_{L}=\frac{1}{2}(\frac{1}{n_{H}^{2}}+\frac{1}{n_{L}^{2}}+\frac{1}{n_{H}n_{L}}-1)\,,$
(59)

and is typically smaller. For L-DBRs we thus find the modified equations,

$C_{0}=\frac{2\pi\tilde{A}}{n_{H}-n_{L}}\frac{\omega-\omega_{c}(0)}{\omega_{c}(0)}\,\,;\,\,C_{2}=\frac{-2\pi\tilde{A}B}{n_{H}-n_{L}}\,,$
(60)

with

$\tilde{A}=n_{H}n_{L}A_{L}=\frac{1}{2}(1+\frac{n_{L}}{n_{H}}+\frac{n_{H}}{n_{L}}-n_{H}n_{L})\,.$
(61)

For the SiO2 / Ta2O5 example discussed above, the L-DBR is expected to show a
smaller Bragg effect as it has $\tilde{A}=0.08$, while the H-DBR has $A=0.66$.

##  Appendix C Operator algebra

This appendix introduces ladder operators for the scalar LG-modes and shows
how they can be used to calculate the matrix elements of the perturbing
operators. It also shows how these concepts can be applied to vector LG-modes,
including the $X/Y/A/B$ modes introduced in the main text. We will only
consider coupling between modes with the same transverse order $N$, such that
the wave-front curvature and Gouy phase drop out of the problem.

In Cartesian $(x,y)$ coordinates, with normalized coordinates
$(\xi,\eta)=(x,y)/\gamma_{z}$, we define the creation and annihilation
operators in the $\xi$ direction as Nienhuis  _et al._ (2021)

$\hat{a}_{\xi}(\chi)=\frac{1}{\sqrt{2}}(\xi+\partial_{\xi})\quad;\quad\hat{a}^{\dagger}_{\xi}(\chi)=\frac{1}{\sqrt{2}}(\xi-\partial_{\xi})$
(62)

and likewise for the $\eta$ direction. These ladder operators allow one to
ladder through the set of scalar HG-modes. As our system is approximately
rotational symmetric, it is more convenient to work with cylindrical
coordinates $(\rho,\theta)$ and the circular ladder operators

$\hat{a}_{\pm}=\frac{1}{\sqrt{2}}(\hat{a}_{\xi}\mp
i\hat{a}_{\eta})\quad;\quad\hat{a}^{\dagger}_{\pm}=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{\xi}\pm
i\hat{a}^{\dagger}_{\eta})\,.$ (63)

These ladder operators satisfy the commutation relation
$[\hat{a}_{i},\hat{a}_{j}^{\dagger}]=\delta_{ij}$ and combine into number
operators $\hat{n}_{+}=\hat{a}_{+}^{\dagger}\hat{a}_{+}$ and
$\hat{n}_{-}=\hat{a}_{-}^{\dagger}\hat{a}_{-}$. They allow one to ladder
through the set of scalar LG-modes, using the relation

$\displaystyle\sqrt{n_{+}!n_{-}!}|\tilde{\Psi}_{n_{+},n_{-}}\rangle$
$\displaystyle=$
$\displaystyle\left(\hat{a}_{+}^{\dagger}\right)^{n_{+}}\left(\hat{a}_{-}^{\dagger}\right)^{n_{-}}|\tilde{\Psi}_{0,0}\rangle\,,$
(64) $\displaystyle\langle\xi,\eta|\tilde{\Psi}_{n_{+},n_{-}}\rangle$
$\displaystyle=$ $\displaystyle e^{i(n_{+}-n_{-})\theta}f_{p\ell}(\rho)\,,$
(65)

where $|\tilde{\Psi}_{0,0}\rangle$ is the fundamental mode. The final equation
provides the link to the modes used in the main text. We have added a tilde to
the notation to indicate that these LG-modes are labeled with quantum numbers
$n_{+}$ and $n_{-}$. The relation with the quantum numbers used in the main
text is $p={\rm min}(n_{+},n_{-})$ and $\ell=n_{+}-n_{-}$, where the later can
still be positive or negative.

Next, we will express each perturbing operator as combinations of ladder
operators and calculate the matrix representation of these operators in the
basis of the LG-modes of transverse order $N$. Using the ladder operators
introduced above, we can for instance rewrite the paraxial form
$\rho^{2}=\xi^{2}+\eta^{2}$ and
$\Delta_{\perp}=\partial_{\xi}^{2}+\partial_{\eta}^{2}$ as

$\displaystyle\hat{\rho}^{2}$ $\displaystyle=$
$\displaystyle(\hat{N}+1)+\hat{a}_{+}\hat{a}_{-}+\hat{a}^{\dagger}_{+}\hat{a}^{\dagger}_{-}\,,$
(66) $\displaystyle\hat{\Delta}_{\perp}$ $\displaystyle=$
$\displaystyle-(\hat{N}+1)+\hat{a}_{+}\hat{a}_{-}+\hat{a}^{\dagger}_{+}\hat{a}^{\dagger}_{-}\,,$
(67)

where $\hat{N}=\hat{n}_{+}+\hat{n}_{-}$. When we sandwich these operators
between two LG-modes of order $N$, the first term yields the familiar
expression
$\langle\tilde{\Psi}_{j^{\prime}}|\hat{\rho}^{2}|\tilde{\Psi}_{j}\rangle=(N+1)\delta_{j^{\prime}j}$,
while the second and third term do not contribute as they only couple modes
with different order.

The quartic non-paraxial operators yield expressions with more terms. When we
only keep the operator combinations that couple modes of the same order $N$,
we find that the quartic non-paraxial operators are diagonal in the LG-basis
with

$\langle\tilde{\Psi}_{j}|\hat{\rho}^{4}|\tilde{\Psi}_{j}\rangle=(N+1)^{2}+N+1+2n_{+}n_{-}=f(p,\ell)\,,$
(68)

where $j=(n_{+},n_{-})$, and an identical result for
$\langle\tilde{\Psi}_{j}|\hat{\Delta}_{\perp}^{2}|\tilde{\Psi}_{j}\rangle$.
The final expression shows the link to the quadratic polynomial $f(p,\ell)$,
used in the main text. The astigmatic operator
$\hat{\xi}^{2}-\hat{\eta}^{2}=\hat{a}^{\dagger}_{+}\hat{a}_{-}+\hat{a}^{\dagger}_{-}\hat{a}_{+}$
simultaneously lowers $n_{+}$ and raises $n_{-}$ by one, or visa versa, and
can therefore couple modes of the same order $N$ with $\Delta\ell=\pm 2$. More
precisely

$\displaystyle(\hat{\xi}^{2}-\hat{\eta}^{2})|\tilde{\Psi}_{n_{+},n_{-}}\rangle$
$\displaystyle=$
$\displaystyle\sqrt{(n_{+}+1)n_{-}}|\tilde{\Psi}_{n_{+}+1,n_{-}-1}\rangle$
(69) $\displaystyle+$
$\displaystyle\sqrt{n_{+}(n_{-}+1)}|\tilde{\Psi}_{n_{+}-1,n_{+}+1}\rangle\,.$

When we label the scalar modes with a single quantum number $n_{s}=n_{+}$,
such that $n_{-}=N-n_{s}$, the only non-zero elements of the astigmatic
operator are

$(\hat{\xi}^{2}-\hat{\eta}^{2})_{n_{s}+1,n_{s}}=(\hat{\xi}^{2}-\hat{\eta}^{2})_{n_{s},n_{s}+1}=h(N,n_{s})\,,$
(70)

where the function $h(N,n_{s})\equiv\sqrt{(n_{s}+1)(N-n_{s})}$ obeys the
symmetry $h(N,N-n_{s})=h(N,n_{s}-1)$.

Next, we introduce the vector modes via

$|\vec{\Psi}\rangle=\vec{e}_{+}|\Psi_{+}\rangle+\vec{e}_{-}|\Psi_{-}\rangle\,,$
(71)

where $\vec{e}_{\pm}=(\vec{e}_{x}\pm i\vec{e}_{y})/\sqrt{2}$ are the circular-
polarized unit vectors and where $|\Psi_{\pm}\rangle$ are scalar mode
profiles. We write this vector mode as a vector of two scalar modes and
describe the action of any tensor operator ${\cal H}$ by the associated
$2\times 2$ tensor that acts via

$\begin{pmatrix}{\cal H}_{++}&{\cal H}_{+-}\\\ {\cal H}_{-+}&{\cal
H}_{--}\end{pmatrix}\begin{pmatrix}|\Psi_{+}\rangle\\\
|\Psi_{-}\rangle\end{pmatrix}\,.$ (72)

We will consider three vector corrections: (i) the isotropic spin-orbit
coupling, (ii) the anisotropic spin-orbit coupling, and (iii) the Bragg
effect. The most prominent vector correction is the isotropic spin-orbit
coupling described by the operator ${\cal H}_{\rm
vec}=2/(kR_{m})\,\vec{\rho}\otimes\vec{\nabla}_{\rho}$. The $2\times 2$ matrix
representation of the operator $\vec{\rho}\otimes\vec{\nabla}_{\rho}$ contains
the combinations
$\\{\hat{\xi}\partial_{\xi},\hat{\xi}\partial_{\eta},\hat{\eta}\partial_{\xi},\hat{\eta}\partial_{\eta}\\}$
in the linear-polarized $(\xi,\eta)$ basis. Conversion of these expression to
the circular ladder operators and to the circular-polarized form defined in
Eqs. (71) and (72) yields

${\cal H}_{\rm vec}=\frac{-1}{kR_{m}}\begin{pmatrix}1+n_{+}-n_{-}&0\\\
0&1-n_{+}+n_{-}\end{pmatrix}\,.$ (73)

In this conversion, we removed combinations of operators that only projects to
modes of different order $N$, like the operator
$\hat{\xi}\partial_{\xi}+1=\frac{1}{2}[\hat{a}_{\xi}^{2}-(\hat{a}^{\dagger}_{\xi})^{2}]$
and combinations of the form $\hat{a}_{\xi}\hat{a}_{\eta}$ and
$\hat{a}^{\dagger}_{\xi}\hat{a}^{\dagger}_{\eta}$.

Application of ${\cal H}_{\rm vec}$ to the righthand circular-polarized modes
yields

${\cal H}_{\rm
vec}\,\vec{e}_{+}|\tilde{\Psi}_{n_{+},n_{-}}\rangle=-\frac{1+\tilde{\ell}}{kR_{m}}\,\,\vec{e}_{+}|\tilde{\Psi}_{n_{+},n_{-}}\rangle\,,$
(74)

where $\tilde{\ell}=n_{+}-n_{-}$, with associated matrix elements

$\displaystyle\left(\left({\cal H}_{\rm
vec}\right)_{++}\right)_{n_{s},n_{s}}=-\frac{1+\tilde{\ell}}{kR_{m}}=-\frac{1+2n_{s}-N}{kR_{m}}\,.$
(75)

The middle part of Eq. (75) is identical to Eq. (33) in the main text, as the
signed $\tilde{\ell}=\pm\ell$ depending $n_{+}\lessgtr n_{-}$. The right side
of Eq. (75) introduces $n_{s}$ as the circular quantum number along the spin
direction, such that $n_{s}=n_{+}$ for $s=1$ and $n_{s}=n_{-}$ for $s=-1$.
With these definitions, the only non-zero matrix elements of $\left({\cal
H}_{\rm vec}\right)_{--}$ are

$\displaystyle\left(\left({\cal H}_{\rm
vec}\right)_{--}\right)_{n_{s},n_{s}}=-\frac{1-\tilde{\ell}}{kR_{m}}=-\frac{1+2n_{s}-N}{kR_{m}}\,.$
(76)

The anisotropic component of the spin-orbit coupling follows from the Taylor
expansion

$\displaystyle(\frac{x\vec{e}_{x}}{kR_{x}}+\frac{y\vec{e}_{y}}{kR_{y}})\otimes\vec{\nabla}_{\perp}$
$\displaystyle\approx$
$\displaystyle(\frac{x\vec{e}_{x}+y\vec{e}_{y}}{k\overline{R}})\otimes\vec{\nabla}_{\perp}$
$\displaystyle+$ $\displaystyle\eta_{\rm
astigm}(\frac{x\vec{e}_{x}-y\vec{e}_{y}}{k\overline{R}})\otimes\vec{\nabla}_{\perp}\,,$

where $\overline{R}=(R_{x}+R_{y})/2$, $\eta_{\rm
astigm}=(R_{y}-R_{x})/\overline{R}$. The first operator on the right-hand side
is ${\cal H}_{\rm vec}/2$. The second operator describes the anisotropic spin-
orbit coupling ${\cal H}_{\rm v+a}/2$. The $2\times 2$ matrix representation
of this operator in $(\xi,\eta)$ coordinates and polarization contains the
combinations
$\\{\hat{\xi}\partial_{\xi},\hat{\xi}\partial_{\eta},-\hat{\eta}\partial_{\xi},-\hat{\eta}\partial_{\eta}\\}$.
Conversion to the circular-polarized vector basis yields

${\cal H}_{\rm v+a}=\frac{-\eta_{\rm
astigm}}{kR_{m}}\begin{pmatrix}0&1-n_{+}+n_{-}\\\
1+n_{+}-n_{-}&0\end{pmatrix}\,.$ (78)

The off-diagonal elements show how this operator converts $s=+1\Leftrightarrow
s=-1$ circular-polarized light. Application of ${\cal H}_{\rm v+a}$ to the
righthand circularly-polarized modes yields

${\cal H}_{\rm
v+a}\,\vec{e}_{+}|\tilde{\Psi}_{n_{+},n_{-}}\rangle=-\frac{\eta_{\rm
astigm}}{kR_{m}}\,(1+\tilde{\ell})\,\,\vec{e}_{-}|\tilde{\Psi}_{n_{+},n_{-}}\rangle\,.$
(79)

This equation differs in two ways from Eq. (74). First of all, the extra
factor $\eta_{\rm astigm}$ shows that anisotropic spin-orbit coupling is
linked to astigmatism. Second, the ${\cal H}_{\rm v+a}$ operator changes the
handedness of the circular polarization. As a result the projected circular
quantum number changes from $n_{s,in}=n_{+}$ to $n_{s,out}=n_{-}=N-n_{s}$. The
associated matrix in the circular-polarized basis therefore only has anti-
diagonal elements

$\left(\left({\cal H}_{\rm
v+a}\right)_{-+}\right)_{N-n_{s},n_{s}}=-\frac{\eta_{\rm
astigm}}{kR_{m}}\,(1+2n_{s}-N)\,,$ (80)

where the first mode label refers to $n_{-}=N-n_{s}$ and the second mode label
refers to $n_{+}=n_{s}$. A similar analysis for the lefthand circular
polarized modes yields the identical, Hermitian-conjugated, result

$\left(\left({\cal H}_{\rm
v+a}\right)_{+-}\right)_{N-n_{s},n_{s}}=-\frac{\eta_{\rm
astigm}}{kR_{m}}\,(1+2n_{s}-N)\,.$ (81)

The final vector correction originates from the Bragg effect described by Eq.
(35). We will only consider the $k_{\perp}^{2}$ contribution to the Bragg
effect and neglect the $k_{\perp}^{4}$ contribution, which is typically weaker
but can still be relevant at small frequency detuning. In the linearly-
polarized basis and $x,y$ units used in the main text, this part of the Bragg
operator has the form

${\cal H}_{\rm
Bragg}=\frac{C_{0}}{k^{2}}\begin{pmatrix}k_{x}^{2}-k_{y}^{2}&2k_{x}k_{y}\\\
2k_{x}k_{y}&k_{y}^{2}-k_{x}^{2}\end{pmatrix}\,,$ (82)

where $k_{x}=i\partial_{x}$ and $k_{y}=i\partial_{y}$. Conversion to
normalized coordinates and to the preferred circular-polarized vector basis
yields

${\cal H}_{\rm
Bragg}=\frac{2C_{0}}{kz_{0}}\begin{pmatrix}0&\hat{a}^{\dagger}_{-}\hat{a}_{+}\\\
\hat{a}^{\dagger}_{+}\hat{a}_{-}&0\end{pmatrix}\,.$ (83)

The off-diagonal elements show how this operator also converts
$s=+1\leftrightarrow s=-1$ circular-polarized light. It does so under
conservation of total angular momentum $J=l+s$, such that $\Delta\ell=\Delta
n_{+}-\Delta n_{-}=\pm 2$.

Application of ${\cal H}_{\rm Bragg}$ to the righthand circular-polarized
modes yields

${\cal H}_{\rm
Bragg}\,\vec{e}_{+}|\tilde{\Psi}_{n_{+},n_{-}}\rangle=\frac{2C_{0}}{kz_{0}}\,h(N,n_{s})\,\,\vec{e}_{-}|\tilde{\Psi}_{n_{+}+1,n_{-}-1}\rangle\,.$
(84)

where $h(N,n_{s})=\sqrt{(n_{s}+1)(N-n_{s})}$ as before. The projected circular
quantum number now changes from $n_{s,in}=n_{+}$ to
$n_{s,out}=n_{-}-1=N-n_{s}-1$ and the associated matrix only has elements one
row below the anti-diagonal, with

$\displaystyle\left(\left({\cal H}_{\rm
Bragg}\right)_{-+}\right)_{N-n_{s}-1,n_{s}}=\frac{2C_{0}}{kz_{0}}\,h(N,n_{s})\,.$
(85)

A similar analysis for the lefthand circular polarized modes again yields
identical matrix elements for $\left({\cal H}_{\rm Bragg}\right)_{+-}$, due to
our use of projected indices.

##  Appendix D Hyperfine splittings

Appendix C showed that two vector corrections can change the handedness of the
light, i.e. have non-zero operators ${\cal H}_{+-}$ and ${\cal H}_{-+}$. As a
result, the circular polarized modes become coupled. In this Appendix we will
argue that the vector LG-modes introduced in the main text are the new
eigenmodes of this coupled system. The coupling can lift the original two-fold
degeneracy of some vector LG-modes, though, and create a hyperfine splitting
between the $+$ and $-$ versions of some $\ell A$ or $\ell B$ modes.

In the main text we introduced a special set of vector LG-modes that we
labeled by their absolute OAM $\ell\geq 0$, their X/Y/A/B character, and their
$\pm$ polarity under $x$-mirror reflection. The link between these vector LG-
modes and the scalar LG-modes in Eq. (64) is

$\displaystyle|\ell A+\rangle$ $\displaystyle=$ $\displaystyle\sqrt{2}\,{\rm
Re}[\,\vec{e}_{+}|\tilde{\Psi}_{\frac{N-\ell}{2},\frac{N+\ell}{2}}\rangle\,]\,,$
(86) $\displaystyle|\ell A-\rangle$ $\displaystyle=$
$\displaystyle\sqrt{2}\,{\rm
Im}[\,\vec{e}_{+}|\tilde{\Psi}_{\frac{N-\ell}{2},\frac{N+\ell}{2}}\rangle\,]\,,$
(87) $\displaystyle|\ell B+\rangle$ $\displaystyle=$
$\displaystyle\sqrt{2}\,{\rm
Re}[\,\vec{e}_{+}|\tilde{\Psi}_{\frac{N+\ell}{2},\frac{N-\ell}{2}}\rangle\,]\,,$
(88) $\displaystyle|\ell B-\rangle$ $\displaystyle=$
$\displaystyle\sqrt{2}\,{\rm
Im}[\,\vec{e}_{+}|\tilde{\Psi}_{\frac{N+\ell}{2},\frac{N-\ell}{2}}\rangle\,]\,,$
(89)

where ${\rm Re}$ and ${\rm Im}$ denote the real- and imaginary part, with
$\vec{e}_{+}^{\,*}=\vec{e}_{-}$ and
$|\tilde{\Psi}_{\frac{N-\ell}{2},\frac{N+\ell}{2}}\rangle^{*}=|\tilde{\Psi}_{\frac{N+\ell}{2},\frac{N-\ell}{2}}\rangle$.
These vector LG-modes are the true eigenmodes of the perturbed cavity, as the
two vector corrections that we consider are symmetric under $x$-mirror
reflection and hence cannot couple $+$ and $-$ modes. Equations (86)-(89) show
that A modes are like B modes with signed OAM $\tilde{\ell}=-\ell$ instead of
$\tilde{\ell}=\ell$ in the $\vec{e}_{+}$ component of their vector field. The
$\ell=0$ modes obey the relations $|0X+\rangle=|0A+\rangle=|0B+\rangle$ and
$|0Y-\rangle=|0A-\rangle=|0B-\rangle$.

Application of ${\cal H}_{v+a}$ to the vector LG-modes yields

$\displaystyle{\cal H}_{\rm v+a}\,|\ell A\pm\rangle$ $\displaystyle=$
$\displaystyle\pm\frac{\eta_{\rm astigm}}{kR_{m}}\,(\ell-1)\,\,|\ell
B\pm\rangle\,,$ (90) $\displaystyle{\cal H}_{\rm v+a}\,|\ell B\pm\rangle$
$\displaystyle=$ $\displaystyle\pm\frac{\eta_{\rm
astigm}}{kR_{m}}\,(-\ell-1)\,\,|\ell A\pm\rangle\,.$ (91)

For $\ell=0$, the ${\cal H}_{\rm v+a}$ operator has on-diagonal element
$-\eta_{\rm astigm}/(kR_{m})$ for the $|0X+\rangle$ modes and $\eta_{\rm
astigm}/(kR_{m})$ for the $|0Y-\rangle$ mode. This difference creates the
hyperfine splitting of the $\ell=0$ modes described in the main text. For
$\ell\geq 1$, the ${\cal H}_{\rm v+a}$ operator has off-diagonal elements that
couple $|\ell A\pm\rangle$ to $|\ell B\pm\rangle$ in an asymmetric way,
related to the $\pm\ell$ alignment of the OAM.

Application of ${\cal H}_{\rm Bragg}$ to the vector LG-modes yields

$\displaystyle{\cal H}_{\rm Bragg}|\ell B\pm\rangle$ $\displaystyle=$
$\displaystyle\pm\frac{2C_{0}}{kz_{0}}h(N,\frac{N+\ell}{2})\,\,|(\ell+2)A\pm\rangle\,,$
(92) $\displaystyle{\cal H}_{\rm Bragg}|\ell A\pm\rangle$ $\displaystyle=$
$\displaystyle\pm\frac{2C_{0}}{kz_{0}}h(N,\frac{N-\ell}{2})\,\,|(\ell-2)B\pm\rangle\,.$
(93)

The $\ell=1$ case of Eq. (93) corresponds to ${\cal H}_{\rm Bragg}|\ell
A\pm\rangle=\pm C_{0}(N+1)/(kz_{0})|\ell A\pm\rangle$, if we interpret the
mode $|-1,B,\pm\rangle=|1A\pm\rangle$. This creates the hyperfine splitting
between the $1A+$ and $1A-$ mode described in the main text. The $\ell=0$ case
of Eq. (93) is effectively described by Eq. (92), as
$|0X+\rangle=|0A+\rangle=|0B+\rangle$.

To visualize the obtained results, we end by deriving the full ${\cal H}_{++}$
and ${\cal H}_{--}$ matrices in the vector LG-mode basis for the $N=1$ and
$N=2$ subspace. We quantify the relative strength of the astigmatism with the
parameter $X=(8\pi kR_{m})\eta_{\rm astigm}\tan{\chi_{0}}/(2\pi)$ used in Sec.
VII.1 and add the two polarization-changing effects mentioned above. In the
normalized units used below, the on-diagonal elements of the spin-orbit
coupling are $-4(\tilde{\ell}+1)$, where $\tilde{\ell}=\ell$ for $B$ modes and
$\tilde{\ell}=-\ell$ for $A$ modes. In the same units, the anti-diagonal
elements associated with the anisotropic spin-orbit coupling are $\mp
4(\tilde{\ell}+1)\eta_{\rm astigm}$, see Eqs. (90) and (91). In the same
units, the matrix elements of the Bragg correction are of the form $\pm
Yh(N,n_{s})$, see Eqs. (92) and (eq:Bragg-A). Combination of these
contributions for the $N=1$ group changes Eq. (40) into

$(8\pi kR_{m})\,\Delta\tilde{\nu}_{(N=1)}=\begin{pmatrix}-6&X\\\ X\mp
8\eta_{\rm astigm}&2\pm Y\end{pmatrix}$ (94)

where the upper signs describe the spectral matrix of the $(1B+,1A+)$ modes
and the lower signs that of the $(1B-,1A-)$ modes. The $\pm Y$ on-diagonal
element describes the hyperfine splitting of the $1A\pm$ modes. The $\mp
8\eta_{\rm astigm}$ off-diagonal element can also produce some hyperfine
splitting in strongly astigmatic cavities, but this effect is typically very
small as $\eta_{\rm astigm}\ll 1$.

For the $N=2$ group the hyperfine splittings change Eq. (44) into

$(8\pi kR_{m})\,\Delta\tilde{\nu}_{(N=2)}=\begin{pmatrix}-12&\sqrt{2}X&\pm
4\eta_{\rm astigm}\\\ \sqrt{2}X&2\mp 4\eta_{\rm
astigm}&\sqrt{2}X\pm\sqrt{2}Y\\\ \mp 12\eta_{\rm
astigm}&\sqrt{2}X\pm\sqrt{2}Y&4\end{pmatrix}\,.$ (95)

where the upper/lower signs refer to the $(2B+,0X+,2A+)$ and $(2B-,0Y-,2A-)$
basis, respectively. The $\mp 4\eta_{\rm astigm}$ on-diagonal element
describes the (typically small) hyperfine splitting between the $0X$ and $0Y$
due to shape birefringence. The $\pm\sqrt{2}Y$ off-diagonal elements are
typically also small and will only produce a measurable hyperfine splitting in
strongly astigmatic cavities, where the mode mixing induced by the off-
diagonal $\sqrt{2}X$ element makes the new eigenmodes sensitive to the Bragg
effect. The spectral matrix shows that the resulting hyperfine splittings are
stronger for the $0$ and $2A$ modes than for the $2B$ modes.

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