Tunable Polymer/Air Bragg Optical Microcavity Configurations for Controllable Light-Matter Interaction Scenarios

Complex optical systems such as high-quality microcavities enabled by advanced lithography and processing techniques paved the way to various light-matter interactions (LMI) studies. Without lattice-matching constraints in epitaxy, coating techniques or shaky open cavity constructions, sub-micrometer-precise lithographic development of a polymer photoresist paves the way to polymer microcavity structures for various spectral regions based on the material's transparency and the geometrical sizes. We introduce a new approach based on 3D nanowriting in photoresist, which can be employed to achieve microscopic photonic Fabry-P\'erot cavity structures with mechanically-tunable resonator modes and polymer/air Bragg mirrors, directly on a chip or device substrate. We demonstrate by transfer-matrix calculations and computer-assisted modelling that open microcavities with up to two"air-Bragg"reflectors comprising alternating polymer/air mirror-pair layers enable compression-induced mode tuning that can benefit many LMI experiments, such as with 2D materials, nanoparticles and molecules.


Introduction
Optical microcavities play an important role in the investigation of a wide range of research areas, such as light-matter interaction (LMI) [1] [2][3] [4], nonlinear optics [5] [6], and quantum information processing [1][7] [8]. Various high-quality microcavities [1] [9][10] [11] have been explored for decades and enabled the hunt for ultralow-threshold nanolasers [5][12] [13], opened up fundamental cavity-QED experiments [14][15] [16], and the study of Bose-Einstein-like condensation (BEC) of polaritons in solids [17] [18] [19] [20]. Microcavities have reached popularity not only for conventional (more energyefficient) lasers, but also for polariton physics [3] [21], as well as the novel field of polariton chemistry [22], and became indispensable for optical quantum technologies [23] [24] [25]. In fact, the list of abundant research directions with confined light fields cannot be projected adequately in this short summary here. resonators. To achieve this, suitable 3D-printable micromirror structures need to be designed and developed for the incorporation of the desired active material prior to or after completion of the cavity structure, as appropriate for the given experiment. While mirror production can easily rely on metallization of surfaces, wavelength tunable on-demand deposition of (top-)mirrors can be best addressed with layered dielectric mirrors based on the distributed Bragg reflector (DBR) concept, which can conveniently be formed by polymer/air layer pairs offering considerable refractive index contrast. In case the rigidity and elasticity allow for pressure-induced thickness changes of layered polymer/air-gap structures, even mechanically tunable optical microcavities can be envisioned. Thus, research platforms for tunable Rabi splitting, BEC studies, polariton chemistry, Purcell-enhanced single photon sources and field-enhancement-benefitting nonlinear optics can arise, considering the estimated reasonably-high Q factors, wavelength or structure design flexibility, and stability of such a printed photonic microstructure.
Here in this work, two promising printed cavity system configurations are discussed utilizing simulations of optical properties of the microcavities with and without active materials based on the transfer matrix method (TMM). The here presented simulation study for different selected layer thicknesses in the polymer/air ("air-Bragg") microcavity, i.e. of air and the polymer material, is supported by additional stress analysis using a finite element analysis (FEA) for the examination of mechanical stability of the designed structures. Thereby, we demonstrate the practicality and conceptual feasibility of mode-tunable strong-coupling experiments with a van-der-Waals semiconductor monolayer incorporated theoretically in the polymer/air microcavity. In the future, laser nanoprinting of (integrated) photonic structures directly on the chip with similar and more advanced polymer optical microreflectors and microresonators promise great flexibility for optoelectronic, optomechanic, nanophotonic and nonlinear-optics applications.

The Optical Microcavity Configurations
The optical mirrors in the form of DBRs, i.e. alternating refractive index layers with thicknesses of , where 0 is the wavelength in vacuum, is the refractive index of the material used and = 1,3,5 and so forth (odd integer numbers), solely rely on the resist material (and air) and are designed to work well with WS2 monolayer excitonic resonances. In the following, = ( 0 ⁄ ).  The first configuration (Fig. 1a) is composed of a conventional dielectric mirror, comprising 6 pairs of SiO2 and Ti3O5, as the bottom mirror as well as substrate for the active medium, and the polymer/air-Bragg structure as the top mirror (in the following "air-Bragg" reflector). The second configuration ( Fig. 1b)  form. This is a unique approach as far as spectrally-tunable microcavities are concerned.
The refractive index of the IP-DIP photoresist in the visible spectral range is 1.52 at room temperature which exhibits minor changes at low temperature [46]. This allows one to obtain a reasonable refractive index contrast of 1.52/1 between the two materials of the dielectric mirror. The refractive index contrast is not very high, but compared to typical III/V DBRs made of GaAs/AlAs with index contrast in the range of 3.5/3, the dielectric mirror with air and polymer still yields an improvement.
Nonetheless, with a large number of mirror pairs, an overall high reflectivity is achievable.
For the TMM calculations, considering the active material to be thin layered semiconducting materials (TMDCs), the design wavelength of the air-Bragg reflector is targeted to be in the range of 600 nm to 800 nm (for the most popular TMDCs with their A-excitons in that range). Note that longer wavelengths, e.g. for near-infrared to infrared intralayer or interlayer excitonic species in TMDC monolayers or 2D heterostructures, respectively, provide more favorable printing conditions than for the here chosen WS2 A-exciton resonance. For detailed explanations regarding the TMM, we refer to the supplementary information of our previous work by Wall et al. [42].  (Fig. 2e). A large number of mirror pairs typically results in a high reflectivity, as evidenced in the calculated reflectivity spectra for all four configurations.
In a fully air-Bragg-based cavity system, the bottom reflector consists of 7 or 8 pairs and the top one of 6 pairs for better out-coupling. The air-Braggs with 7 and 6 pairs possess maximum reflectivities of 98.5 % and 95%, respectively (Fig. 2e).
Next, the influence of the layer thickness on the stopband width is briefly summarized. It can be clearly seen that the stopband width changes drastically as a function of the layer thickness. The stopband width for the /4, 3 /4, 5 /4 and 7 /4 air-Bragg reflector amounts to approximately 100 nm, 60 nm,

Stress Analysis of Tunable Polymer/air-Bragg Microcavities
The crucial stress analysis based on FEA was performed on aforementioned polymer/air-Bragg reflector designs, using a computer-aided design (CAD) tool (see methods section). Here, Autodesk Inventor allows simulating the practical pressure-affected cavity configurations, which employ air-Bragg reflectors towards their applications in tunable open-microresonator devices. This is possible due to material-specific mechanical properties allocated to the structure's CAD model. The physical and mechanical properties of the photoresist IP-DIP are summarized in Tab. 1. The CAD of air-Braggs for various layer thickness configurations is shown in Fig. 3 along with the stress analysis simulation. The air-Bragg reflectors consist of 8 layers with = 1,3,5,7 quarter-wavelength layer thickness (Fig. 3a, b, c,   Our simulation-based mechanical analysis indicates that the 4 ⁄ air-Bragg structure is unstable as can be evidenced by the strong bending and bunching of the photoresist layers. Besides, the air-Bragg with layer thickness 3 4 ⁄ (Fig. 3b), 5 4 ⁄ (Fig. 3c) and 7 4 ⁄ (Fig. 3d) exhibit a clearly more stable behavior under influence of gravity and external pressure due to the improved rigidity. As can be deduced from TMM calculations based on extracted structural information, the effective deformation (in terms of thickness reduction) of the air layers in air-Bragg structures influences the overall stopband of the microcavity. The stopband of the individual air-Bragg reflector experiences an increasing blue-shift in its spectral position when the external pressure is gradually increased as applied on the upper surface of the structure (indicated by double arrows in Fig. 3).
The stress analysis method used in this work can also be utilized to study the behavior of the cavity structures upon application of mechanical pressure. To address the targeted effect of cavity-mode tunability by external pressure, two different examples of (printable) cavity configurations under a variation of the applied pressure are discussed, that are (a) the air-Bragg/DBR microcavity and (b) the air-Bragg/air-Bragg microcavity (see Fig. 1a and b, respectively).
To begin with, Fig. 4a demonstrates the tunability induced by applied pressure in the range of 0 to 50 MPa (corresponding to a uniform force of maximally 0.1 N on the two side bars) on the 5 /4 air-Bragg/DBR microcavity which causes the quality factor = /∆ (that is mode energy over linewidth, i.e. full-width-at-half-maximum FWHM) to change drastically. The cavity length of 2.6 µm, whereas being 620 nm, allows us to determine the cavity mode (C) q = 8. The cavity mode q = 8 clearly changes  its spectral position upon application of pressure (Fig. 4b), which compresses the overall multilayered air-Bragg structure. In the waterfall diagram of Fig. 4b, the TMM calculated reflectivities for the microcavity with arbitrary increment of pressure in steps of 10 MPa are displayed. The resonance of the cavity mode was adjusted to be resonant with the 617 nm A-exciton mode in WS2 at room temperature by fine-tuning the cavity spacer thickness and performing the TMM calculations for this configuration in the absence of external pressure. This ensures one to obtain directly resonance conditions at elevated temperatures around 300 K and also reach a red photon-exciton detuning at low temperature, which is beneficial for pressure-based cavity mode tuning in strong-coupling measurements in cryogenic experiments. An example of the calculated strong-coupling situation in a WS2-air-Bragg microcavity for different pressure levels is displayed in Fig. 5. structure visually delivered by the FEA simulation. On the other hand, the expected blue-shift of the mode is attributed to the compressed structure with reduced air-gap thicknesses which is seemingly sufficient in order to obtain the tuning effect. Nonetheless, owing to the increased thickness unproportionality in the polymer/air configuration for increased pressure levels, the resonance conditions in the cavity suffer and a gradually reduced Q factor is obtained.
In principle, the SiO2/Ti3O5 DBR stopband width (150 nm) is much larger than that of the 5 /4 air-Bragg (40 nm), which allows one to practically shift the stopband of the air-Bragg over a wider spectral region by external pressure. However, a large structural compression may lead to a relatively strong deformation of the air-Bragg configuration from the actual design, which influences not only the spectral position of the cavity mode and stopband, but also the Q factor of the cavity mode (as indicated in Fig. 4).
The second approach pursued utilizes a fully air-Bragg-based microcavity design (Fig. 1b), and, in this study, the homogeneous external pressure is applied on the whole cavity structure (on its side bars).
This particular scenario provides the unique opportunity to investigate light-matter coupling scenarios due to the flexibility with respect to mode detunings, i.e. the energy difference between cavity and emitter modes, in a substrate-independent fashion. Moreover, it can be used to insert suspended 2Dmaterial sheets with support structures, if appropriately designed, as an additional tool to tweak the LMI by changing the position of the ML with respect to the standing electromagnetic (EM) fields inside the cavity. As a consequence of suspension, also the substrate-induced impurities and all other consequent effects get automatically eliminated.
In this configuration, the cavity spacer is similar to the first example proportionally reduced together with air gaps in the Bragg sections. Again, a gradual reduction of the Q factor is obtained (Fig. 6a) and the overall cavity stopband experiences a comparable shift. Figure 6b demonstrates

Conclusion
To summarize, two spectrally-tunable microcavity configurations based on polymer/air Bragg This implies that different light-matter coupling scenarios with flexibility in the cavity-emitter resonance detuning, even after production, can be realized with the help of such polymer/air-based optical microcavities.
In the future, precise laser nanoprinting of (integrated) photonic structures directly on the chip will be a common practice, and an approach to achieve optical microreflectors and microresonators such as the here presented one can be very useful in many specific scenarios, including for LEDs and photovoltaics, nanolasers and nonclassical light sources, optical filters or couplers and nonlinear optical elements, as well as optical sensing through LMI.

Stress analysis of CAD structures: The stress analysis function provided by Autodesk Inventor
Professional 2019 was used to determine the impact of vertical external pressure (force per area) on different polymer/air-Bragg structures using CAD models. The stress analysis function based on a finite element analysis (FEA) allows one to assign the applied force on the surface of the CAD model and to obtain mechanical deformation for every setting of externally applied force (represented by the double arrow in the shown figure) and under influence of gravity (single arrow). The hereby obtained prediction of the overall structural response was later used to model reflectivity spectra (using manually extracted position changes of the layered structure for given pressure level).

Calculation of spectra:
The targeted microreflector and microresonator systems were modeled using the transfer matrix method (TMM) regarding their standing-wave light-field profile and reflectivity spectra. Based on theoretical data and considerations with chosen design parameters, reflectivity spectra towards strong light-matter coupling with a virtual 2D semiconductor inside the cavity were equally obtained. The optical properties such as reflection, transmission and angle-resolved spectra of multi-layered open cavity structures were calculated using the TMM-based simulation code as used in Wall et. al [42].  Step 2