Metasurface-stabilized optical microcavities

Cavities concentrate light and enhance its interaction with matter. Confining to microscopic volumes is necessary for many applications but space constraints in such cavities limit the design freedom. Here we demonstrate stable optical microcavities by counteracting the phase evolution of the cavity modes using an amorphous Silicon metasurface as cavity end mirror. Careful design allows us to limit the metasurface scattering losses at telecom wavelengths to less than 2% and using a distributed Bragg reflector as metasurface substrate ensures high reflectivity. Our demonstration experimentally achieves telecom-wavelength microcavities with quality factors of up to 4600, spectral resonance linewidths below 0.4 nm, and mode volumes below \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.7{\lambda }^{3}$$\end{document}2.7λ3. The method introduces freedom to stabilize modes with arbitrary transverse intensity profiles and to design cavity-enhanced hologram modes. Our approach introduces the nanoscopic light control capabilities of dielectric metasurfaces to cavity electrodynamics and is industrially scalable using semiconductor manufacturing processes.


Introduction
Cavities can confine, shape, and enhance photons and vacuum fields and lie at the heart of achievements such as the laser 1 , gravitational wave discovery 2 , and quantum electrodynamics [3][4][5] .
Many effects can be intensified with tighter confinement, thus, optical microcavities today are diversely applied in semiconductor lasers, sensing, and nonlinear optics 6 .Waveguide-based microcavities require no alignment and can often be manufactured using available on-chip photonics or fiber-optics techniques.In contrast, open microcavities offer direct access and a large tuning range for the cavity length and the resonant wavelengths.
When aiming at achieving small mode volumes in free-space optical cavities, spherical aberrationcorrected end mirrors with radii of curvature on the order of a few to tens of micrometers are required.
One way to manufacture such mirrors currently is to dimple optical substrates using focused-ion beam milling and then coating the dimples with distributed Bragg reflectors (DBRs) 7,8 .This process enables high-quality cavities, however, strain in the DBR coatings on curved surfaces imposes limits on the realizable phase profiles 7 .Dielectric metasurfaces can control the phase of light at the nanoscale by changing the size and shape of sub-wavelength metaatoms.They have been previously used to stabilize microwave cavities 9 , to split degenerate polarization states in cavities 10 , to realize temperature sensing cavities 11 , to filter color using optical cavities 12,13 , to modify the output of laser cavities [14][15][16][17] and quantum cascade lasers 18 , and to provide feedback for semiconductor lasers 19 .However, they have not yet been implemented to stabilize open optical microcavities.Here we demonstrate they are a mass-implementable, rapidly prototypable, and flat alternative for creating optical microcavities with unprecedented design capabilities.

Hermite-Gaussian Beams in Cavities
Optical cavities have been studied extensively in literature 7,9,20,21 .When a well-defined propagation direction can be assigned to light, the paraxial wave equation applies.A set of solutions to the latter convenient for describing laser beams in free-space and cavities are Hermite-Gaussian beams.These beams' complex field evolution  "# (, , ) in cartesian coordinates (transverse directions ,  and propagation direction ) is completely characterized by their transverse mode numbers ,  ∈ ℕ $ , their minimum beam waist  $ , and their wavevector along the propagation direction .
Focused Hermite-Gaussian beams, when coupled to a planar cavity, suffer transverse spreading which strongly limits the achievable transmission and quality factor (Fig. 1a).Furthermore, a focused beam contains many components with non-negligible transverse wavevector.As the overall wavevector of light in vacuum is conserved, these transverse components decrease the wavevector along the propagation direction.Consequently, components with a non-negligible transverse wavevector are resonant in longer planar cavities.This causes unwanted broadening and asymmetry of the cavity resonances (see Fig. 1a, Fig. 3a,c,d, and Ref. 22).
In the following, we will concentrate on planar-concave cavities in which the minimum waist is always located at the flat end mirror (Fig. 1b).To form a resonant mode within a cavity, the complex field must reproduce itself after one round-trip up to a real factor 23 (a complex factor indicates a stable cavity that is off resonance).This requires the round-trip phase  78 (, ,  9:; ), which consists of the mode's propagation phase  "# (, ,  9:; ) and the mirror reflection phases  <=>>2>5/0 , to fulfill  78 (, ,  9:; ) = 2 "# (, ,  9:; where  9:; is the cavity length and  is an integer.As an example, Fig. 1c shows the phase evolution of a Hermite-Gaussian beam with ,  = 0; Fig. 1d shows its intensity distribution.Existing cavities 7,21 achieve transverse confinement by using a curved mirror surface that reverses the mode's propagation direction at one of the phase-isolines in Fig. 1c and thus reverses the transverse phase evolution 07(@ $%& ) in  "# (, ,  9:; ).When we require that the mirror's radius of curvature matches the wavefront's radius of curvature  A=>>2> = ( 9:; ) and that the Rayleigh range  7 is real, we can solve for  7 using eq.( 4) and find the stability criterion which determines for which lengths  9:; such a cavity efficiently traps light.

Metasurface-Stabilized Cavities
Metasurfaces allow to freely design an additive phase that spatially changes on the nanoscale.To realize a stable cavity, we can thus place a metasurface on the second mirror (see Fig. 1b), calculate the phase of the desired mode at its position  9:; , and design the reflection phase of the metasurface on the second DBR  BC6'<D (, ) to reverse the wavefront evolution This approach allows designing entirely planar stable cavities without the need for specially polished curved surfaces and can implement aspheric phase profiles without any added complexity.
To demonstrate this in practice, we choose a working wavelength  $ = 1550 nm due to its relevance to optical communication.We design the metasurface placed on DBRs made from alternating silica/titania quarter-wave layers optimized for high reflectivity (>99%) at the design wavelength.Using finite-difference-time-domain (FDTD) simulations (Lumerical Inc., FDTD), we calculate a reflection phase library for polarization-independent circular amorphous silicon pillars.We achieve full 2π reflection phase coverage and high reflectivities for a pillar height of 600 nm, a square metaatom cell size of 450 nm, and reasonable fabrication constraints.See Fig. 1e for a schematic of the unit cell and the pillar-diameter-dependent reflection phase.

Metasurface Cavity Modeling
To examine our metasurface-stabilized cavity, we use this design and simulate the entire cavity using FDTD modeling (see methods). .Using DBRs that are capped with the lower refractive index material (in this case silica) locates the high-intensity anti-nodes of the cavity modes on the mirror surfaces.In a future application, this choice maximizes the interaction of the cavity mode with samples -e.g., 2d-materials or nano emitters -placed on the mirrors 24,25 .Fig. 1g shows the mode's relative intensity drops below 10 %F at less than 4 um transverse distance from its center.Therefore, we expect a metasurface radius on that order sufficiently limits diffraction losses, i.e., light that is lost because it misses the metasurface due to its finite size.We then simulate the transmission of a focused beam of light through the cavity and vary the cavity length (see methods).We show results in Fig. 2a and find, as expected from eq. ( 7), cavity modes spaced by approximately

Absolute Metasurface Phase Effects
The absolute phase offset light experiences when passing through a metasurface is usually neglected.
However, in our case, we find it is crucial to obtaining an efficient cavity: adding an absolute phase offset to the metasurface phase  BC6'<D (, ) in our cavity has two immediate consequences: the abrupt boundaries on the cavity performance can be: our simulations predict up to 50 % transmission of incident light through a cavity stabilized using a metasurface with a well-chosen absolute phase.A cavity stabilized by a metasurface with the same relative phase profile but poorly chosen absolute phase achieves less than 5 % transmission -a 90 % efficiency loss.

Experimental Results
Using top-down processing (see methods), we fabricated such metasurfaces on top of a commercially available silica/titania DBR terminated with a silica layer (reflectivity >99.5%).Fig. 3e shows a scanning electron microscopy picture of a final device after measurement.We then placed the manufactured device opposite to a planar DBR and measured the wavelength and cavity length-dependent transmission of the resulting cavity for a focused incident light beam (numerical aperture NA ≈ 0.3, wavelength 1520 − 1580 nm, see methods for details).
Without a metasurface (Fig. 1a), we observe broad and strongly asymmetric transmission peaks (see Fig. 3d) which is the expected behavior of a Fabry-Perot cavity.The resonance lengths, i.e., the cavity lengths at which the transmission peaks appear, shift linearly when changing the wavelength of the incident light (see Fig. 3a).For the same focusing conditions with a metasurface (Fig. 1b), we observe much narrower symmetric Lorentzian line shapes (compare Fig. 3g, with Fig. 3d), signifying a stable cavity and efficient longitudinal and transverse trapping of the incident light.A full measurement and a lineout at wavelength  = 1550 nm are presented in Figs.3b, f.We can see resonances with longitudinal mode numbers down to  = 2, indicating good parallel alignment of the two DBRs.In Fig. 3h we show the longitudinal mode index-dependent resonance transmission and the length tuning bandwidth (i.e., the width of the resonance peak indicated by the black arrows in Fig. 3g).At the working wavelength, we find finesses up to 157 ± 7 (see Fig. 3i), which set the upper limit for the round-trip loss to 4 %, indicating less than 2 % scattering losses per pass through the metasurface.
We now compare the behavior of a resonance position in the metasurface-stabilized cavity (Fig. 3b) with that of a resonance with the same longitudinal mode index q in the Fabry-Perot cavity (Fig. 3a) when changing the wavelength of the incident light.The metasurface-stabilized resonance position shifts non-linearly and faster (increased slope

GE
in Fig. 3b).We find that the increased caused by the dispersion of the metasurface, i.e., the additional group delay light experiences when it transmits through the metasurface and reflects from the metasurface cavity end mirror (

GE
we observe here decreases the spectral linewidth of the  = 2 longitudinal mode by more than a factor of 14 compared to a resonator without metasurface 12 .This leads to narrow spectral linewidths down to below 0.4 nm, see Fig. 3j, and highlights the application of metasurface microcavities as, e.g., narrowband spectral filters.
Furthermore, this leads to large quality factors of up to (4.6 ± 0.4) × 10 L (see Fig. 3k) at the design wavelength.
The measured cavity length-dependent resonance transmission of the Fabry-Perot cavity (Fig. 3c) shows a monotonic decrease with increasing cavity length.Conversely, the metasurface-stabilized cavity shows local dips for the longitudinal modes with the indices  = 4 and  = 9, 10,11, see Figs.
3f, h.Our simulations reproduce these transmission dips for the longitudinal modes with the indices  = 4 and  = 8,9, 10 (see Figs. 2a, c).We attribute the small offset of the longitudinal mode numbers to fabrication effects that cause a slightly decreased metasurface effective radius of curvature (see below).Two main factors determine the maximum transmission through the cavity for a resonant mode: the coupling of the incoming light with the cavity mode (i.e., the overlap of the incoming transverse beam profile with the cavity mode's transverse profile) and the round-trip loss of the mode itself.Whereas the first only modifies the transmission, the latter modifies the transmission and the resonance bandwidth at the same time.In our measurements, we observe the resonance linewidths increasing concurrently with the decreased transmission, see Fig. 3h, which identifies intra-cavity losses rooted in the finite aperture of the metasurface as the origin of the transmission dips.This is corroborated by our simulations showing the cavity length-dependent resonance transmission behavior independent of the incoming beam waist size and the inverse correspondence of the resonance transmission and the diffraction loss in Fig. 2c.

Transverse Confinement and Modified Stability Criterion
We examine the transverse confinement of light in the metasurface-stabilized cavity by comparing the resonant lengths of different transverse modes (see Supplementary Method 1 and Supplementary Fig. 2).We find that the manufactured metasurface achieves a mode with the target minimum mode waist of  $ = (2.00 ± 0.03) um close to the design cavity length  9:; = 4.0 um ( A=>>2>%A=>>2> = 4.6 um) albeit having a slightly smaller than targeted effective radius of curvature of  <D = (16.3± 0.5) um.
Furthermore, we find that the penetration of light into the planar DBR and the metasurface-covered DBR makes the cavity appear longer than  9:; when calculating the Gouy phase and the cavity mode's radius of curvature (see Supplementary Method 1).We account for this by introducing a modal penetration depth 24  M BC6 +  M BC6'<D = (2.8 ± 0.2) um.Because this affects the reproduction of the modes after one cavity round trip, the modal penetration depth modifies the cavity stability criterion.

Cavity-Enhanced Hologram Modes
Due to the large design freedom offered by dielectric metasurfaces, the presented concept can be adapted to create cavity modes with arbitrary transverse intensity profiles, see Fig. 4b.We start with the desired mode intensity profile (, ,  = 0) on the planar cavity mirror.We can then calculate the mode profile in the planar cavity mirror plane (, ,  = 0) = e(, ,  = 0) and its evolution into another plane (, ,  =  9:; ) along the propagation direction from the Rayleigh-Sommerfeld diffraction integral 28 : The propagation phase from the  = 0 to the  =  9:; plane is given by argu(, ,  =  9:; )v.
Even if we illuminate the resulting cavity with a beam that does not have the desired mode profile, e.g., a Gaussian beam or a plane wave, the metasurface designed using the above method will only build up the desired mode in the cavity.
To examine the viability of this approach we choose a cavity mode with an H-shaped intensity profile (see Fig. 4a) and a cavity length  9:; = 10.3 um.Using eq. ( 8) and eq.( 9), we calculate  BC6'<D (, ) and create a metasurface design by matching this phase with metaatoms.In this step, we fine-tune  9:; so the absolute phase of the metasurface causes no abrupt pillar diameter changes in highintensity regions of the cavity mode (see Absolute Metasurface Phase Effects).We expect scattering losses due to sharp features in our desired mode profile (Fig. 4a), therefore we reduce the reflectivity of our DBRs and metasurface to 95% by decreasing the number of silica/titania layers.Fig 4c displays the final metasurface design.
We then perform FDTD simulations of an entire cavity consisting of a planar DBR on one side and an opposing DBR with the metasurface stabilizer placed on it (see Fig. 4b).Fig 4d shows the cavity mode intensity profile at the mirror without metasurface.Even though we illuminate with a plane wave, the metasurface stabilizes and enhances only the desired mode profile, therefore the intra-cavity intensity has the desired H-shape with sharp edges.We find an average electric field enhancement of 6 in the desired mode compared to the incoming plane wave.
To highlight that this approach is not limited to binary or symmetric mode profiles, we choose an asymmetric and greyscale dot-pattern (see Fig. 4f).Our simulations show that a metasurface cavity designed following the previously detailed method, again illuminated with an incoming plane wave, also enhances the desired asymmetric dot pattern (see Fig. 4g).
For Hermite-Gaussian modes, a single metasurface design with a fixed radius of curvature can stabilize many longitudinal modes (with the cavity length limited by the stability criterion).This works because a mode's propagation length-dependent radius of curvature can be counteracted by a change in its minimum beam waist, such that its wavefront fits the metasurface's effective radius of curvature.In this more general case, the cavity length-dependent resonance transmission (see Fig. 4e) shows a clear local enhancement of the longitudinal mode occurring close to the design cavity length.This happens because other longitudinal modes cannot easily adapt to match the complicated metasurface profile.
The presented technique thus also offers control over the evolution of the longitudinal cavity modes.

Conclusion
In summary, we combined commercially available DBRs with metasurfaces to realize stable microcavities with classical and designed mode profiles.The approach offers unprecedented design freedom, is rapidly prototypable, and at the same time directly manufacturable on the industrial scale as it is fully compatible with widely available semiconductor fabrication techniques.Especially the ability to implement complicated phase profiles, design chromaticity and achromaticity 29,30 , and control the polarization state 31 of light down to the ultraviolet spectral region 32 will offer unprecedented control of light in microcavities.

Longitudinal Mode Counting
We use the following convention when assigning the longitudinal mode index q: our DBRs are terminated by a low-refractive-index material, therefore, the cavity mode has intensity maxima at the mirror facets.Therefore, the Fabry-Perot cavity without metasurface supports a cavity mode at  9:; ≈ 0 um (because of the light penetration into the DBRs), to which we assign the mode number  = 0.
Accordingly, the longitudinal mode with index  = 1 occurs at a free space cavity length  9:; ≈ E 0 and so on.When comparing the Fabry-Perot cavity with the metasurface-stabilized cavity, at the same distance between the DBRs  A=>>2>%A=>>2> , the metasurface-stabilized cavity length is 0.6 um smaller because the metasurface is 0.6 um high (compare Fig. 1b and Supplementary Fig. 1).Therefore, at the same  A=>>2>%A=>>2> , the longitudinal mode numbers assigned to modes in the metasurface-stabilized cavity are roughly one smaller than modes that occur in the Fabry-Perot cavity.

Fabrication
Using plasma-enhanced chemical vapor deposition, we deposit a 600 nm-thick amorphous silicon layer on commercially available DBRs (Eksma 031-1550-i0).On top, we spin-coat a layer of negative electron beam resist (Micro Resist Technology ma-N 2403) and subsequently a conductive polymer (Showa Denko ESPACER 300) to avoid charging effects.We then write the metasurface mask patterns using electron beam lithography (Elionix HS-50).After developing, (MicroChemicals MIF 726) we remove amorphous silicon in non-exposed areas using inductively coupled plasma-reactive ion etching (ICP-RIE using SF6 and C4F8).Finally, we remove the remaining electron beam resist using piranha solution.

Experimental Setup
To characterize our metasurface-stabilized cavities, we use coherent light from a tunable semiconductor laser (Santec TSL-550), which we focus using an aspheric lens (NA ≈ 0.3) through the 3 mm thick substrate of a planar DBR mirror (Eksma 031-1550-i0, reflectivity >99.5% for wavelengths between 1520 nm and 1570 nm).Excluding a 2x2 mm area in its center, we grind down the front facet of this DBR to allow small cavity lengths even for imperfect angle alignment regarding a second, opposing, planar DBR.This second DBR has the metasurface on its surface and is mounted on a threeaxis stage (Thorlabs NanoMax).This mirror order avoids changing the position of the incoming beam waist with respect to the beam waist of the cavity modes when varying the cavity length with the stage's piezo actuators.Transmitted light is collected using a lens (NA ≈ 0.3) and detected using an amplified InGaAs detector (Thorlabs PDA10CS2).

Length Calibration
To calibrate the length of our microcavity, we move the minimum beam waist of the incoming beam only along the propagation direction.The resulting increase of the beam waist on the planar cavity end mirror leads to strong excitation of the planar-planar Fabry-Perot cavity modes around the metasurface, shown their asymmetric resonance profile, see Fig. 3a.The slope G@ /011213/01121 GE then reveals the sum of the current cavity length  A=>>2>%A=>>2> and the frequency penetration depth into the two planar DBRs  O BC6 ( To correct for the DBR penetration, we analytically calculate and simulate (see below)  O BC6 = 1.5 um.
Results from both methods coincide.We then subtract it from  G@ /011213/01121

GE
to obtain  A=>>2>%A=>>2> .The metasurface height is 0.6 um, thus the length of the stabilized cavity is  9:; =  A=>>2>%A=>>2> − 0.6 um.Furthermore, as the metasurface height is constant, Calculation of the Finesse and the Quality Factor We determine the finesse  ≈   for the optimal design in Fig. 1h.d) finite difference time domain modeling of the longitudinal mode index-dependent mode volumes (blue dots) for the metasurface-stabilized microcavity using the optimal design in Fig. 1h.Combining (S2) with ∆ T =  T,"'#U5 −  T,"'#U$ leads to ∆ T = E 0P D2 tan %5 5 @ =,",#?@ + 9,=,",#?' 8 − tan %5 5 @ =,",#?' + 9,=,",#?' 8E.We then replace @ =,",#?' + 9,=,",#?' and @ =,",#?@ + 9,=,",#?' and introduce  <D using eq.(S1).The penetration of the mode into the cavity mirrors adds an additional Gouy phase to the phase accumulated by the mode in between the cavity mirrors.Therefore, we introduce the modal penetration depths for a planar DBR  M BC6 and the metasurface on the DBR  M BC6'<D , and replace  T,",# →  T,",# +  M BC6 +  M BC6'<D in the Gouy phase to obtain the result: Fig. 1f and 1g show the calculated phase evolution and intensity distribution of the  = 5 mode.While the wavefront curvature and transverse intensity distribution (compare with Figs.1c, d) of the Hermite Gaussian mode is retained in the cavity, the flat metasurface planarizes the mode's wavefront within its 0.6 um pillar height by providing a phase shift 2 & !'( !07(+) which suppresses the local beam curvature.Furthermore, the intensity evolution along the propagation direction shows the expected standing wave pattern with maxima spaced by approximately E ' 0 cavity lengths at which resonances appear change and the metasurface design will consist of phaseshifted metaatoms (compare the insets of Figs.2a, bfor two examples).The latter controls the zone boundaries at which the diameters of the metaatoms change abruptly due to the 2 phase jumps (see the red dashed lines in the insets in Figs.2a, b).These abrupt changes can lead to scattering losses due to inter-element coupling (i.e., even if a small and a large adjacent pillar yield the same overall transmission phase -light coupled to those different pillars can become out of phase during its propagation along the pillars).Thus, abrupt changes should -if at all -occur in areas where the cavity mode has low intensity.The calculations in Figs.2a, bhighlight how profound the influence of these

≤ 1 .
With this modified criterion, our measured effective radius of curvature and modal penetration depth predict cavity stability up to a longitudinal mode index  ≤ 17 ± 1. Indeed, we observe a steep increase of the cavity modes' bandwidths for longitudinal mode indices  > 16, see Fig.3h.Mode VolumeAs our experimental results are well reproduced by FDTD simulations, we model our cavity's mode volumes and show their evolution in Fig.2d.For the experimental mode with longitudinal mode index  = 2, we find a volume of  < 2.7 L ( < 22 b is comparable to the values reported for traditionally fabricated open access microcavities 7 .Our simulations predict that this can be reduced to  < 1.5 L ( < 12 b the minimum cavity length (currently limited to larger than 1.5 um by technical constraints in our setup) and even further by employing high-refractive-indexterminated DBRs.The measured quality factors and calculated mode volumes suggest achieving Purcell enhancement26,27 of 250 is already possible using this first demonstrator device.

24 (=
567-from the full-width-at-half-maximum length tuning bandwidth δ 9:; IJK< , an expression which we derived from the full-width-at-halfmaximum phase linewidth δ IJK< = 2δ 9:; IJK< , see Refs.7,21.Subsequently, we calculate the quality factor using  = angular frequency , the angular frequency spectral linewidth δ IJK< , and the wavelength spectral linewidth δ IJK< ).As the quality factor measures the dissipated energy per oscillation period, we use the sum of the cavity length  9:; and the frequency-penetration-depths  O BC6 +  O BC6'<D into the cavity mirrors 7,21, BC6 + BC6'<D ).To determine  9:; +  O BC6 +  O BC6'<D , we measure the slope of the resonance conditionG@ /011213/01121 GE and use the modified relation (M1):  G@ /011213/01121 GE =  9:; +  O BC6 +  O BC6'<D .(M2) Finite Difference Time Domain Simulations A full length ( 9:; = 0 − 15 um) and nanometer-resolution sweep would require excessive computational resources.To remedy that, we again use the relation (M2) to project the wavelengthdependent results of our simulations in a 10 nm bandwidth around the working wavelength on the distance axis.We determine the frequency-penetration-depths of our DBR  O BC6 = 1.5 um and the metasurface-covered DBR  O BC6'<D = 7.0 um by comparing their simulated reflection phases to that of a perfect electrical conductor placed at the front facet of the DBR/metasurface.The mapping allows us to cover the entire distance sweep in 250 simulations.

Figure 1 Concept
Figure 1

Figure 4 Metasurface
Figure 4 T,"'#U5 −  T,"'#U$ = length-dependent transmission data is presented in Supplementary Fig.2a.We observe wavelength-dependent resonance length differences ∆ T () which are explainable by a wavelength-dependent effective radius of curvature and a wavelength-dependent modal penetration depth.Fitting ∆ T () allows us to extract both the effective radius of curvature of the metasurface and the sum of the modal penetration depths into the planar and metasurface covered DBR  M BC6 +  M BC6'<D , which we plot in Supplementary Fig.2b.Because the effective radius of curvature of the metasurface corresponds to that of the cavity mode's wavefront, this measurement also gives the minimum waist diameters of the cavity modes, which we display in Supplementary Fig.2c.