Microcavity resonance condition, quality factor, and mode volume are determined by different penetration depths

The penetration depth in a Distributed Bragg Reflector (DBR) co-determines the resonance condition, quality factor, and mode volume of DBR-based microcavities. Recent studies have used an incomplete description of the penetration depth and incorrect equations. We present a complete analysis that involves three different penetration depths. We also present a series of experiments on microcavities to accurately determine the frequency and modal penetration depth of our DBRs and compare these results with theoretical predictions. The obtained results are relevant for anyone who models a DBR as an effective hard mirror if lengths of the order of the wavelength are relevant, as is the case for microcavities.


Introduction
Small mode volume cavities have been used for numerous applications such as quantum information processing with individual atoms [1] and lab-on-a-chip sensors [2]. These microcavities typically consist of two highly reflective Distributed Bragg Reflectors (DBR) which can trap light in a small mode volume and thereby increase the light-matter interaction. When microcavities get smaller [3] the penetration depth in the mirrors becomes important. DBRs are also used in many other applications [4] and even exist in nature, in the form of intricate photonic crystals [5].
DBR-based microcavities are often modeled as cavities with two hard mirrors spaced by a cavity length that is extended by the penetration depths of the DBRs. This model is then used to calculate the resonance condition, quality factor and mode volume. However, the optical penetration in the DBRs is more subtle than this simple model suggests. In the literature, the penetration depth in DBRs is ambiguously defined due to this simplified model [2,[6][7][8][9][10][11]. This paper will solve these issues by introducing multiple (frequency, modal, and phase) penetration depths and by explaining when these are relevant. The first part of the paper presents a theoretical description that aims to provide physical insight in the origin of the various penetration depths. It also links them to the optical properties of a microcavity. The second part presents measurements of the penetration depth in two types of microcavities. Measurements on the frequency tuning of the modes in a planar cavity yield the frequency penetration depth. Measurements on the transverse mode splitting in a plano-concave cavity yields the modal penetration depth. We compare these two results with each other and with theoretical predictions.

Optical penetration in DBRs
We consider the reflection of light from a thick, lossless, planar DBR. The alternating layers have refractive indices and for the low and high index material and layer thicknesses and such that = = /4 for vacuum resonance wavelength . Light is incident from a medium with index (typically air with = 1). The most prominent feature of DBRs is the existence of a stopband, or bandgap, which is a frequency range where light cannot propagate and where a thick lossless DBR reflects all incident light. The full spectral width of the stopband is [12,13] where = 2 / is the resonance frequency. The approximation is valid for small to modest index contrast, where Δ ≡ − with average index = ( + )/2. [13] At resonance, in the center of the stopband, the forward-propagating field decays exponentially into the DBR, such that its amplitude decreases by a factor / per DBR pair [12,13]. The associated 1/ penetration depth of the optical intensity is The approximation again applies to the limit of small index contrast, Δ . It seems natural to call "the penetration depth" of the DBR and to model the DBR as an effective hard mirror positioned at a distance behind the front surface of the DBR. But this is wrong for several reasons. First and most important, there is no single position at which a hard mirror can mimic all reflection properties of the DBR simultaneously. Below we will argue that one actually needs three different penetration depths to mimic either (i) the reflection phase, or (ii) the time delay upon reflection, or (iii) the imaging of a focused beam upon reflection. Second, these penetration depths depend on the refractive index of the incident medium. Finally, they also depend on whether the DBR starts with a high-index layer (H-DBR) or a low-index layer (L-DBR). Only if one considers the time delay upon reflection from a H-DBR does one obtain the easy "natural" result = (see below). Figure 1 shows the calculated frequency dependence of the reflectivity | | 2 and reflection phase at normal incidence for a typical DBR, similar to the ones used in our experiments. This figure shows that the reflectivity is approximately constant inside the stopband. The interesting physics is contained in the reflection phase ( ), which is defined relative to the front facet and scales as ∝ ( − ). The insets show the physical origin of this phase change: the node of the standing wave, which resides at the DBR surface at resonance, shifts into ( > 0) or out of ( < 0) the DBR at frequencies > and < , respectively. Note the deviations from this linear behavior towards the edges of the stopband, where the maximum shift is approximately half a layer thickness for a H-BDR (see Supplement 1).
To describe the reflection of a general (non-monochromatic non-planar) beam of light, we decompose the incident light in its Fourier components. We label these components by their frequency and transverse wavevector = sin , where = 0 is the wavevector in the incident medium and 0 = 2 / 0 and 0 are the wavevector and wavelength in vacuum. Each monochromatic plane-wave component will reflect with its own reflection amplitude Inside the stopband, ( , ) is approximately constant and equal to ≈ ±1. The optical field has an anti-node ( = 1) at the front facet for a L-DBR and a node ( = −1) for a H-DBR. For frequencies near and small incident angles, the reflection phase ( , ) can be approximated by [14][15][16] The approximation is a Taylor expansion, where / = 0 due to mirror symmetry. The final equation defines the frequency penetration depth and the modal penetration depth in terms of derivations of the reflection phase. We prefer to call the frequency penetration depth, whereas others have called it the phase penetration depth [14], because our name links with frequency tuning (see below).
The frequency penetration depth = /(2 ) quantifies the group delay = / that an optical pulse experiences upon reflection from a DBR when its optical spectrum fits well within the stopband. A hard mirror positioned at a distance in the incident medium will produce the same group delay and will hence mimic the time/frequency properties of the DBR.
The modal penetration depth quantifies the imaging properties of the DBR reflection. A hard mirror positioned at a distance in the incident medium will reflect light with the same angle dependence and will hence produce the same imaging as the DBR. Note that the reflection of the DBR depends on angle because the center of the stopband shifts to higher frequencies at non-zero angles of incidence as ( ) − ( = 0) ∝ 2 . This dependence results in the relation / = = 2 ( −2 + −2 )/2 ≈ ( / ) 2 [14]. This relation is intuitive, because is associated with a time delay, which scales with / , and is associated with an imaging shift, which scales with / (see Supplement 1). With the factor we can rewrite the phase penetration in Eq. (3) depth in terms of , This equation shows that small angles only have a small impact on , because is typically small for = 1. The phase penetration depth = /(2 ) that we define in Eq. (3) is new in literature. We explicitly define this quantity because determines the resonance condition of DBR-based microcavities, rather than or ; see Eqs. (6) and (8) below. also determines the locations of the anti-nodes in the microcavities, where light-matter coupling is maximal. Equation (4) shows that ( )/ = ( − )/ at normal incidence for = 1.
Babic et al. [15] have calculated the frequency penetration depth = /(2 ), using transfer matrices. Although Babic et al. only analyzed so-called "matched DBRs", where all reflections interfere constructively, their results also apply to the general case. The delay time is different for high-index and low-index DBRs due to the interference of the reflection from the first interface of the DBR with the reflections from the bulk (see Supplement 1). We have checked both equations (5) with numerical calculations based on transfer matrices (see Supplement 1). Brovelli et al. [17] have performed similar calculations, using coupled-mode theory for DBRs with small index contrast. Their results agree with the ones obtained by Babic [15] in the limit of small index contrast (see Supplement 1). We like to finish this section by noting that the theory described above is based on several assumption. First of all, the truncated Taylor expansion in Eq. (3) is valid only for small frequency detunings and small angles. Second, we have neglected polarization effects. These will play a role at larger angles where the Fresnel reflection coefficients depend on polarization. As a result, the spectral width of the stopband will increase for -polarized light and decrease for -polarized light. Finally, we have neglected dispersion effects. For our DBRs and wavelength, the ratio between the group and phase refractive index is ≈ 1.013 for the SiO 2 layers and ≈ 1.058 for the Ta 2 O 5 layers [18]. The combined effect of dispersion, as a weighted average of these values, results in a modest 3 % reduction of the spectral width of the stopband and an associated 3 % increase of the penetration depth. We have neglected this effect in our analysis.

Consequences for cavity resonances
The resonances of any optical cavity are determined by the condition that the round-trip phase delay is a multiple of 2 . For a cavity with two planar DBRs illuminated at normal incidence this results in 2 where is the distance between the front facets of the two DBRs and 1 and 2 are the reflection phases of the two DBRs (note that = 2 ). The longitudinal mode number counts the number of half wavelengths in the standing wave pattern between the mirrors. This number is integer when both mirrors are either H-DBR or L-DBR and hence both have an anti-node or node close to their front facet. For cavities with one H-DBR and one L-DBR we need to replace by + 1 2 to keep integer and still account for the sign difference in the reflection = ±1 of the two mirrors. Now suppose we change the cavity length and measure the resulting change in resonance frequency/wavelength at fixed . We calculate this change by substituting ( 0 ) in Eq. (6) and taking the derivative of this equation with respect to 0 to arrive at equation (7). In the process we use / 0 = 2 / in the left equation (6) or ( )/ ≈ ( − )/ in the middle equation (6). When we rewrite the end result in terms of we find a normalized slope This equation shows that the relevant penetration depth for a frequency scan is and not . The combination + 1 + 2 also determines the quality factor of the optical resonances and the associated cavity loss rate 0 / .
The resonance condition of plano-concave cavities differs from that of planar cavities by the so-called Gouy phase. For planar-concave cavities with hard mirrors, the resonant modes are Hermite-Gaussian TEM , modes with flat wavefronts at the planar mirror and matched curved wavefronts at the concave mirror. Upon propagation, these modes experience a phase lag relatively to a plane wave. This phase lag is proportional to ( + + 1) and to the Gouy phase = arcsin( √︁ / ) of the fundamental TEM 00 mode, where and are the cavity length and mirror radius.
The resonant optical modes of a planar-concave cavity with DBR mirrors are also TEM , modes. Their (round-trip) resonance condition again includes the phase penetration depth of both mirrors. It also includes a Gouy phase that is now given by = arcsin( √︁ ( + 2 )/ ). Note that the relevant penetration depth in this equation is , because the Gouy phase is associated with phase fronts and thereby linked to imaging properties. We assume that the value of for the curved mirror and the flat mirror are the same, and therefore use 2 . The radius of curvature is an effective radius, which includes all small variations of the mirror curvature in the fabrication process [19]. The Gouy phase co-determines the Rayleigh range and waist of the cavity modes and hence also the optical mode volume and attainable atom-field interaction.

Comparison with literature
A comparison of our results with literature shows that the subtleties of multiple penetration depths are often overlooked. We will give a few examples of how equations would have been different if our theory would have been used.
First, our Eqs. (6) and (8) show that the resonance condition depends on the phase penetration depth , which is typically small and zero at resonance. But Eqs. (2) and (3) in ref. [2] state that the resonance condition contains a single wavelength independent penetration depth. The authors later use the same quantity to describe the frequency tuning of the resonances, whereas our Eq. (7) shows that frequency tuning depends on the frequency penetration depth .
Second our Eqs. (6) and (7) show that the frequency spacing between consecutive longitudinal modes depends on the frequency penetration depth . But the authors in [7] determine the longitudinal mode value by taking = 2 and hence forget the contribution of . Third, our Eq. (8) shows that the frequency spacing between the transverse modes depends on the modal penetration depth . Equation (3) in ref. [2] uses the expression = arccos √︁ 1 − / for the Gouy phase and hence does not take any penetration into account. Reference [7] makes the same mistake in their Eq. (1). Reference [10] uses the frequency penetration depth in their Eq. (2) for the Gouy phase. But the correct equation should have been = arccos √︁ 1 − ( /2 + 2 )/ at = 0. Finally, incorrect use of the penetration depths also affects the Purcell effect. Our analysis shows that the Purcell factor depends primarily on the modal penetration depth , as the increase in mode volume due to the field penetration into the DBRs is compensated by a similar increase of the quality factor. The effect of on the Purcell factor is typically small but can still become important when the cavity length is order . For the = 2 fundamental mode in a cavity with one L-DBR, we predict that the modal penetration depth leads to an increase in mode area and an associated decrease of the Purcell factor ∼ 2 0 by a factor 1/ √ 2. The consequence of using instead of is an underestimation of the Purcell enhancement [9].

Methods
Our planar and patterned mirrors were produced by Oxford High-Q. The patterned substrate is fabricated with a focused-ion-beam-etching technique that creates a series of high-quality concave structures with typical radii of curvature of 2-20 m [2,8,19]. The SiO 2 substrates were coated with 31 and 35 alternating layers of SiO 2 and Ta 2 O 5 , to produce two DBRs that both end Planar Cavity Plano-concave Cavity Feature depth with high-index material and hence have virtually identical reflection properties. These DBRs have a stopband with a width Δ ≈ 150 nm centered around = 640 nm, as expected for a DBR with ≈ 1.46 and ≈ 2.09. The transmission of the patterned mirror is (3.4 ± 0.2) × 10 −5 and the transmission of the flat mirror is (1.1 ± 0.1) × 10 −4 inside the stopband. The transmission through the plano-concave cavity is only ≈ 1%, due to scattering losses on the mirrors. These losses are not relevant for the analysis presented in this paper.
One of the mirrors is fixed while the other mirror can be moved with 6 degrees of freedom on a hexapod system. We align the mirrors to the point where they are parallel and touch each other. This point is referred to as 'touch down'. We scan the mirror position from touch down to over >2 m distance with sub-nm precision.
The non-linearity of the piezo scan and the point of touch down are determined by measuring the microcavity transmission outside the stopband with a green laser ( = 520 nm). A CCD image of the microcavity confirms the parallelism when no fringes are visible. The point of touch down is determined from the part in the scan where the transmitted intensity is constant.
In the next section, we will present accurate measurements of penetration depths on a planar and a plano-concave microcavity, as indicated in the section titles. Figure 2 shows that we use the same mirrors in both experiment, but we focus light on different parts. The planar microcavity transmission spectra presented in Fig. 3 are obtained with a spatially-filtered Xenon lamp and a fiber-coupled spectrometer. The experimental results on the transverse mode splitting of the plano-concave microcavity, presented in Figs. 4 and 5, are obtained by measuring the microcavity transmission of a HeNe laser ( = 633 nm) with a photo-multiplier tube. The light was coupled in with an = 7.5 mm lens and coupled out with an = 8 mm lens.

Frequency penetration depth (planar cavity)
In the first experiment, we measure the transmission spectrum ( ; ) of the planar cavity. For each wavelength , the transmitted power varies between ( ) and ( ) with . We use these extrema to normalize the transmission spectrum between 0 and 1 and show the results as false-color plot in Fig. 3. Due to this normalization, this figure does not show the 4 orders of magnitude difference between the very low transmission (10 −4 ) for wavelengths inside the stopband and the order unity transmission outside the stopband.
The cavity length is varied from just below touch down (red dashed line indicated by ) to a mirror position ≈ 3.  Fig. 3 is the observation that the = 0 mode is also slanted or, equivalently, that the point does not correspond to = 0. The distance between these points, indicated by the arrow in Fig. 3, yields the frequency penetration depth = 0.28 ± 0.02 m; see Eqs. (6) and (7) for theory. The uncertainty estimate is based on a comparison between results from different measurement series and different methods of analysis, both manually and by computer. Note that the analysis presented above was based on the reasonable assumption that the first mode after touch down is the = 1 mode. This assumption yields a distance 0.14 ± 0.02 m between touch down and zero cavity length. If the first mode would have been = 2, this would have led to a much larger distance of 0.46±0.02 m and an unrealistically low value of = 0.12 ± 0.02 m (see discussion below).

Modal Penetration depth (plano-concave cavity)
In the second experiment, we measure the transmission of a HeNe laser through a planoconvave cavity while scanning the cavity length. Each group of transmission peaks contains the fundamental TEM 00 mode and multiple high-order TEM modes. The wavelength of the HeNe ( = 633 nm) is close enough to the center wavelength ( = 640 nm) to neglect the phase penetration depth (theory predicts 2 ≈ −0.01 m). Figure 4 shows the measured splitting Δ between each transverse higher-order mode (indicated by + > 0) and the associated fundamental mode ( + = 0) as a function of mirror position. We measured these splittings for 7 groups of modes, of which the first three ( = 2, 3, 4) are indicated by dashed black lines. Below, we will explain why we start counting from = 2.
The solid curves are based on a simultaneous fit of all measurements using two fit parameters: the mirror radius and the position of full degeneracy of the transverse modes. Our fit yields be a deviation of the transverse mode splitting, and the associated Gouy phase, from the simple paraxial theory [20]. In the absence of an alternative theory, we cannot estimate the size of this systematic errors. From an experimental point of view, we can only determine statistical errors to find that errors of multiple measurements on multiple cavities agree with each other (see below). In Fig. 4, we have added an extra (black dotted) curve for the virtual + = −1 modes. A comparison of Eqs. (6) and (8) shows that these virtual modes should have the same resonances as the planar cavity modes. By extrapolation of these virtual planar modes to = 0 we find the point = 0, indicated as the middle dashed line. The key result in Fig. 4 is the modal penetration depth. Equation (8) shows how this value can be obtained from the distance 2 between the leftmost vertical lines, assuming identical penetration depth in the flat and curved mirror. From the analysis of Fig. 4 we thus obtain a measured modal penetration depth = 0.211 ± 0.015 m. Finally, we note that the distance between touch down and = 0 in Fig. 4 is 0.53 ± 0.03 m. This value is larger than for the planar cavity because it contains the feature depth of the concave mirror (see Fig. 2). Trichet et al. [19] report feature depths of 0.30 m and 0.32 m for curved mirror radii of 23 m and 12 m, respectively. By comparing transmission spectra of planar and plano-concave cavities, we find 0.34 ± 0.02 m and 0.35 ± 0.02 m for our mirrors. With a feature depth of ≈ /2, the lowest mode of the plano-concave cavity will be = 2, while the planar cavity has = 1. After subtraction of the measured feature depth, we determine the spacing between the planar parts of the mirrors to be 0.18 ± 0.03 m at touch down in this new alignment. This mode splitting is expressed as the displacement Δ between the resonance of the high-order mode ( + = 1 − 6) and the corresponding fundamental mode ( + = 0). The combined fit of these data, depicted as a set of black curves, yield the radius of curvature and a fictitious mirror position where all transverse modes are frequency degenerate. We add the fictitious cavity mode ( + = −1) to compare with the planar cavity and to find = 0 (see text).
We performed the analysis depicted in Fig. 4 on 9 data sets, obtained from 6 different cavities on 2 different days (3 cavities were measured on both days). We only analyzed data sets that contained at least four clearly visible transverse modes. The solid point in Fig. 5 shows the fit parameters obtained from Fig. 4. The colors of the points indicate measurement series on different days.
The data points in Fig. 5 are divided in two groups, corresponding to cavities with = 10 − 11 m and = 21 − 23 m. The distribution of the data points shows that the modal penetration depth is approximately the same for all cavities and does not depend on mirror radius over the studied range.
The horizontal lines show the weighted average of the modal penetration depth with its intrinsic error = 0.22 ± 0.02 m. This intrinsic error is based on the spread in the measurements, which is slightly larger than the error bars estimated for individual measurements. This estimate only contains statistical errors.

Discussion
We start the discussion by comparing experiment with theory. We have measured a frequency penetration depth = 0.28 ± 0.02 m and modal penetration depth = 0.22 ± 0.02 m. For our H-DBRs, with = 1.46 and = 2.09, we predict = 0.25 m and = 0.09 m. The measured frequency penetration depth is in reasonable agreement with theory. The measured modal penetration depth is smaller than , also as expected. But the measured value of is not as small as theory predicts.
The observed discrepancies in are most likely due to simplifications in the theory. We will mention four and start with the least likely. The most crucial simplification seems to be the Taylor expansion of the reflection phase in Eq. (3), which is based on the assumption that all Fourier components fit well within the stopband. At large angles of incidence the blue shift of the resonance frequency could bring us into the non-linear regime of the reflection phase, where an additional cubic term increases the effects (see refs. [21,22]). This scenario sounds reasonable, as the opening angle of the = 2 modes is as large as ≈ 0.25 rad ( −2 ) for the fundamental mode and much larger for the high-order modes. However, even opening angles as large as 0.6 rad will shift the resonance frequency only Δ / ≈ (1/2) 2 ≈ 5 % or ≈ 30 nm, which does seem to be enough to reach the non-linear regime. A second simplification lies in the treatment of the curved DBR as a mirror with a single curvature [19] that mimics the angle-dependent reflection of a flat DBR and hence has the same optical penetration. Again, this argument is not convincing as the range of angles of incidence on the curved and flat DBR are not very different, given the large opening angle of the cavity modes.
A simplification that could be crucial is the scalar treatment of the optical field. Computer simulations show that polarization effects will become important at larger angles, even far below the Brewster angle. Hence, one might have to distinguish transverse modes based on their optical polarization. Finally, we didn't take potential coating inhomogeneities and thickness distortions of the mirrors into account [10,19]. Small-scale distortions are likely to average out over the mode profile. But large-scale distortions can deform the transverse modes away from the ideal spherical case and hence result in an incorrect assignment of the radius and the modal penetration depth . Furthermore, non-paraxial propagation effects could also lead to an incorrect assignment and introduce systematic errors in the analysis. A discussion of all these complications is beyond the scope of the paper.
In conclusion, we have presented an analysis of optical penetration in DBRs and accurate measurements thereof. Our analysis shows that there are actually three penetration depths which are relevant in different experiments. We have measured the frequency penetration depth to find that it agrees with theory. We have also measured the modal penetration depth to find that it is smaller than , but not as small as expected. We attribute the observed deviations to theoretical simplifications. Maybe most important, we have argued that the effect of optical penetration on microcavity resonances is often misinterpreted. The absolute resonance conditions depend on the reflection phase and hence on the phase penetration depth . The frequency spacing between the longitudinal modes and their quality factor depend on the frequency penetration depth . The frequency spacing between the transverse modes and their area/cross sections depend on the modal penetration depth . The Purcell factor depends primarily on the modal penetration depth, as the increase in mode volume due to the field penetration into the DBRs is compensated by an increase of the quality factor.

Microcavity resonance condition, quality factor, and mode volume are determined by different penetration depths: supplemental document
This document provides the supplemental information to the paper entitled "Measurement of the frequency and modal penetration depth of Bragg mirrors and their effect on microcavity resonances". It elaborates on three statements from the main text. First, it discusses the shift of the nodes in the DBR (Distributed Bragg reflector) with frequency tuning and the limited role of dispersion therein. Second, it discusses the effect of the incident medium on the group delay τ and the associated frequency penetration depth L τ . Finally, it presents intuitive arguments for the relation L D = βL τ between the modal and frequency penetration depth.

COUPLED-MODE THEORY AND SHIFT OF (ANTI-)NODES
The main text states that 'the node of the standing wave, which resides at the DBR surface at resonance, shifts into (ϕ > 0) or out of (ϕ < 0) the DBR at frequencies ω > ω c and ω < ω c , respectively. Note the deviations from this linear behavior towards the edges of the stopband, where the maximum shift is approximately half a layer thickness for a H-BDR (see Supplement 1).' In this section we will proof that statement. We will do so by using the coupled-mode theory formulated e.g. in [1] and applied to DBRs in [1,2].
We consider the reflection of light from a thick lossless planar DBR illuminated at normal incidence. The alternating layers have refractive indices n L and n H , and layer thicknesses d L and d H such that n L d L = n H d H = λ c /4 for vacuum resonance wavelength λ c . The average index n is defined via nΛ = n L d L + n H d H , where Λ = d L + d H is the periodicity (= thickness per pair). The DBR is embedded in a medium with index n and ∆n = (n H − n L ) n (small index contrast). The DBR starts with the high-index material at position z = 0.
We write the optical field as E(z, t) = [A(z) exp [i(π/Λ)z] + B(z) exp [−i(π/Λ)z]] exp (−iωt), where A(z) and B(z) are the slowly-varying amplitudes of the forward and backward traveling wave. Reflection at the periodic interfaces couples forward and backward traveling waves with a coupling rate κ that satisfies the relation κΛ = ∆n/n [1,2]. We define the resonance frequency as ω c and the resonance wavevector as k c = π/Λ. We also define a wavevector detuning δ = k − k c = n(ω − ω c )/c for light with frequency ω, where we neglect dispersion. Upon propagation in the DBR the amplitudes of the waves evolve as [1] d dz Within the stopband, where |δ| < κ, the eigenvalues of the propagation matrix are ±γ = ± √ κ 2 − δ 2 . When we express the detuning as δ = κ sin φ, this corresponds to γ = κ cos φ. In the center of the stopband, at δ = 0, the eigenvalue ±γ = ±κ indicates that the optical field decays at a rate κ = ∆n/(nΛ) = 2∆n/λ c into the DBR. Off-center, the field penetrates deeper as the decay rate γ decreases by a factor cos φ. The full spectral width of the stopband, from δ = −κ to κ, is equal to ∆ω gap /ω c = 2κ/k c = 2∆n/(πn), as already mentioned in the main text.
The two eigenmodes of the propagation matrix have B/A = ± exp (±iφ). A decomposition of the incident and transmitted field in these eigenmodes shows that the reflection amplitude of an infinitely thick DBR can be written as for the reflection at the low-to-high-index interface (at z = 0). The reflection at the high-to-lowindex interface only differs in sign and is Γ 0 = exp iφ. At resonance, the reflection amplitude is Γ 0 = ±1, where the plus sign applies to L-DBRs and the minus sign applies to H-DBRs. For off-center frequencies in the stopband, the reflection phase where τ 0 = 2/∆ω gap is the group delay in the center of the stopband. At the edges of the stopband, where δ = ±κ, the reflection phase reaches its limiting values φ = ±π/2. This phase shift corresponds to a shift of the node in the standing wave over ± 1/8 wavelength or half the thickness of a DBR layer. The role of dispersion, i.e. the frequency dependence of the refractive indices, is as follows. In the presence of dispersion, the wavevector detuning δ = dk/dω(ω − ω c ) = n gr (ω − ω c )/c, where n gr is the weighted average of the group refractive indices of the DBR layers. For our DBRs and wavelength, the ratio n gr /n = 1.013 for SiO 2 and 1.058 for Ta 2 O 5 [3], which results in a most correction of order 3 %.
We finish this section with two side notes. First, we note that the arcsin-dependence in Eq. (S3) shows that the phase varies linear with detuning in the center of the stopband but changes faster towards the edges of the stopband. This is agreement with our observations and simulations. Second, we note that Eqs. (S2) and (S3) describe the reflection of a DBR embedded in a medium with index n. The reflection phase will change when the DBR is embedded in a medium with index n in = n (see next section).

EFFECT OF INCIDENT MEDIUM ON FREQUENCY PENETRATION DEPTH L τ
The main text states that "the reflection phase ϕ also depends on the ratio n in /n" and that the "time delay τ = dϕ/dω is different for high-index and low-index DBRs due to interference of the reflection from the first interface of the DBR with the reflections form the bulk". In this section we will proof that statement.
We consider an infinitely thick DBR and write the reflection amplitude from the bulk as Γ 0 = ± exp [iτ 0 (ω − ω c )] for frequencies in the central (linear) region of the stopband. For the general case n in = n, we add the reflection amplitude r 0 from the first interface c.q. front facet, which is given by the well-known expression r 0 = (n in − n)/(n in + n). The interference between the reflection from the front facet and the bulk yields a combined reflection amplitude [4] It is easy to show that r = ±1 when Γ 0 = ±1 and that |r| = 1 because |Γ 0 | = 1. Hence, we can write r = ± exp [iτ(ω − ω c )] in the central region of the stopband. We calculate the ratio τ/τ 0 by taking the frequency derivative of equation (S4) at ω = ω c to find For H-DBRs, where the minus sign applies as Γ 0 (ω c ) = −1, this results in τ/τ 0 = n in /n. For L-DBRs, where the plus sign applies as Γ 0 (ω c ) = +1, this results in τ/τ 0 = n/n in . These final expressions proof Eq. (3) in the main text, which states that They quantify the influence of the environment on the group delay τ, and the associated penetration depth L τ . And they show that the penetration in H-DBRs and L-DBRs differs. For completeness, we note that the above equations were derived from coupled-mode theory and assumed the refractive index contrast in the DBR to be relatively small. Babic et al. [5] have used a transfer matrix approach to analyze the general case of larger index contrast to find τ = n in n H n H n H − n L π ω c (H-DBR) or τ = n L n in n H n H − n L π ω c (L-DBR) (S7) To verify the above prediction, we have performed numerical calculations using the transfer matrix approach. Figure S1 shows that τ indeed increases linearly with n in for the H-DBR, but inversely proportional for the L-DBR. The two lines cross at the point where n in ≈ n, where τω c ≈ 8.7, in agreement with the predicted value π √ n L n H /(n H − n L )) = 8.7.

RATIO L τ AND L D
The main text stated that the relation between the modal and frequency penetration depth is L D /L τ = β = n 2 in (n −2 L + n −2 H )/2 ≈ (n in /n) 2 [5]. We claimed that "this relation is intuitive, because L τ is associated with a time delay, which scales with n/n in , and L D is associated with an imaging shift, which scales with n in /n ". In this section, we will show this analytically and graphically.
The analytic result is derived from the blueshift of the center frequency of the DBR. The reflection r = exp[i2L τ (k − k c (θ in ))] changes with incident angle θ in (in medium n in ) because the resonance wavevector k c (θ in ) changes. This resonance wavevector is determined by the relation n L d L cos θ L + n H d H cos θ H = π/k c (θ) where θ L and θ H are the angles in both DBR layers. A Taylor expansion of cos θ L ≈ 1 − θ 2 L /2 ≈ 1 − θ 2 in (n in /n L ) 2 /2 and similar for cos θ H yields a blue shift of the form k c (θ in ) ≈ k c (1 + βθ 2 in /2), where β = n 2 in (n −2 L + n −2 H )/2. A hard mirror positioned at a distance L D in medium n in has an angle dependent reflection of the form exp [i2kL D θ 2 in /2]. A comparison with the DBR shows that this hard mirror mimics the angle dependent reflection from the DBR when L D = βL τ . Fig. S2 shows the relation between L τ and L D graphically. The green lines show the reflection of an incident ray at some penetration depth in the DBR. This ray refracts at the front facet on account of Snell's law, where it bends towards the surface normal for the considered case n > n in . The red lines/rays show how the apparent position of the reflecting layer, as observed from the n in medium, moves towards the surface normal. This figure thus shows that the modal penetration depth L D scales with n in /n. On the other hand, light propagates slower through the high-index material of the DBR than through the low-index environment. Therefore, the frequency penetration depth L τ of the equivalent hard mirror is larger than the actual penetration depth in the DBR by a factor n/n in . A comparison of these two prima facie results yields L D /L τ ≈ (n in /n) 2 , in agreement with the earlier expression for β for small ∆n. L L D Fig. S2. Sketch of the reflection of light (green ray) incident at an angle on a DBR for n/n in = 1.94), which reflect at some penetration depth. The yellow region indicates the environment with n in , the blue and red regions indicate the high (n H ) and low (n L ) index region of the DBR. The red rays indicate the equivalent point of focus if the DBR is replaced by a hard mirror in the refractive index of the environment. The point L τ is located at L D /β.