Exceptional point engineered glass slide for microscopic thermal mapping

Thermal sensing with fine spatial resolution is important to the study of many scientific areas. While modern microscopy systems allow optical detection at high spatial resolution, their intrinsic functions are mainly focused on imaging but limited in detecting other physical parameters, for example, mapping thermal variations. Here, with a coating of an optical exceptional point structure, we demonstrate a low-cost but efficient multifunctional microscope slide, supporting real-time monitoring and mapping of temperature distribution and heat transport in addition to conventional microscopic imaging. The square-root dependency associated with an exceptional point leads to enhanced thermal sensitivity for precise temperature measurement. With a microscale resolution, real-time thermal mapping is conducted, showing dynamic temperature variation in a spatially defined area. Our strategy of integrating low-cost and efficient optical sensing technologies on a conventional glass slide enables simultaneous detection of multiple environmental parameters, producing improved experimental control at the microscale in various scientific disciplines.

Supplementary Note 1: Calculation of the scattering matrix by the optical transfer matrix method. We utilize the optical transfer matrix method to determine the scattering matrix of the multi-layer glass slide 1 . The light propagation in each layer of the glass slide can be characterized by where the indices 3 2, 1,  j denote the Au, PMMA, and Au layers, j k and j L are the corresponding wavenumber and thickness of each layer (see Supplementary Table 1 for the refractive indices of the materials). The total transfer matrices of the layered structure for light propagation in the forward and backward directions can be derived as The corresponding reflection and transmission coefficients which compose the scattering matrix then read where L is the total length of the three-layer structure, l k is the wave number in thick silica glass as the entering medium for forward light propagation, and r k is the wave number of the air.

Supplementary Note 2: Comparison of sensitivity between EP and metal-induced
reflection enhancement. We note the degeneracy (|rf| = 0) of our multilayer scattering system with non-vanishing Au layers is inherently a non-Hermitian exceptional point (EP), while a symmetric Hermitian configuration with metal-enhanced reflectivity cannot reach the diabolic point (DP). To justify this, we note the prerequisite of a DP is a Hermitian scattering system (|rf| = |rb|) where the polymer layer is sandwiched by two symmetric Au layers of the same thickness. We plot in Supplementary Fig. 1 the calculated reflection of the Hermitian system with variable thicknesses of the polymer layer and the Au layers, which shows the resonance of such a Hermitian system can reach the DP degeneracy (|rf| = 0 and |rb| = 0) only when the Au layers completely vanish. Hence, it is obvious that: 1) if one likes to compare our EP scheme with a DP scheme, one has to reach a control sample of a PMMA anti-reflection film; 2) if one likes to compare our scheme with a Hermitian multilayer structure of finite Au, the Au layers enhance reflectivity such that the reflection resonance can never reach 0. In fact, in a Hermitian system, a larger Au layers leads to increasing reflection at resonance. In contrast, the judiciously designed Au layers in our non-Hermitian multilayer structure vanish the reflection in one direction. While we have proved the superiority of our EP scheme over the DP structure in terms of sensitivity in our manuscript, here we show that in the latter case the increase of reflection as a response of thermal perturbation is still significantly lower than the increase of forward reflection in our EP structure. Supplementary Fig. 2 depicts such a comparison between our EP scheme and a Hermitian multilayer structure with similar parameters and a reflection resonance at the probe wavelength. Although large reflection can be observed away from the resonance in both structures, the increase of forward reflection in our EP structure is almost 5 times the increase of the forward or backward reflection in the Hermitian structure as a response to the same thermal perturbation. One may still raise the concern that the slope of reflection in the Hermitian multilayer structure is limited because the reflection does not reach 0. However, we stress a Hermitian structure with both Auenhanced reflectivity and yet a DP resonance does not exist in the multilayer normal incidence scheme, and the ultimate slope of forward reflection can only be optimized in an asymmetric non-Hermitian structure, i.e., the EP structure. To further proof the superiority of our EP structure over the Hermitian multilayer structures, we provide in Supplementary d Thermal responses of our EP structure and reflection of the Hermitian multilayer structure. 6 To conclude, we emphasize that the enhanced response of reflection under thermal perturbation arises from the topology of the non-Hermitian EP. Neither a DP structure nor a Hermitian multilayer structure with Au-enhanced reflection can possess the comparable response.  Supplementary Fig. 4a, so all the settings remain the same (such as the same spot). The generalized reflection is then obtained by subtracting the transmission term, followed by renormalization against the incident power.

Supplementary
By replacing the broadband incidence with the He-Ne laser, the thermal response of the generalized reflection at the probe wavelength can also be characterized. Despite the distinct measurement approaches, we observed approximately the same characteristics of the resonant spectrum as well as the thermal response. This is because the multilayer structure in our case is a completely linear system, which means the responses (reflection and transmission) to either forward or backward incidence are independent and satisfy the principle of superposition 2 . Hence, the measurement of | | b f r r in its bilinear form does not contain more information than the combination of two measurements of rf and rb. In other words, . From another perspective, the transmission and reflection results in theory are obtained in a source-free condition, which means they remain unchanged no matter how the incidence is applied, either single-side incidence (used in the separate measurements) or double-side incidence (the scheme in Supplementary Fig. 4). In our work, therefore, we can apply the separate measurements of the forward and backward reflection to characterize the thermal response for the much simplified experimental setup and its compatibility with a microscope system.
Last but not least, we should note that the square-root derivation of the experimentally observed reflectance or transmittance is always necessary to correctly characterize the scattering matrix (whose entries are reflection and transmission). Such mathematical derivation should not discredit the validity of the experimental data since the enhancement of the sensitivity, as we clarified in the following, does not come from the simple square-root operation. Rather, it arises from the zero forward reflection and large backward reflection (i.e. the highly asymmetric reflection at the probe wavelength), which form the EP together. 13 Supplementary Note 4: Spatial resolution of thermal mapping. The spatial resolution of thermal mapping is limited by the heat transfer process on the glass slide [3][4][5][6] , which spreads the heat flux generated from the pulsed laser incidence in the vicinity of the heat source. To solve the temperature distribution and directly visualize the spatial resolution in our multilayer structure, we simulate the temperature distribution where the effect of laser heating is considered as a heat flux resulted from the absorbed laser power density.

Supplementary Figure 8. Simulation of the temperature distribution under the pulsed laser incidence.
Each of the two heat sources is 30 µm wide, mimicking the size of the laser spots. With increasing separation distance from 4 µm to 10 µm (a-e), the high temperature region splits to two parts from the merging state.
As shown in Supplementary Fig. 8, the temperature exponentially decays within the approximately 4 µm range outside of the heat source. When the separation of two laser spots is as small as 4-7 µm, the high temperature region overlaps leading to spatially a nondistinguishable feature. With increasing of the distance (above 8 µm), the high temperature region starts to separate into two discernible parts. At a further separation of 10 µm, the two heating laser spots can be clearly resolved in thermal mapping. Therefore, we estimate the ultimate spatial resolution of our thermal mapping on the engineered glass slide is at a scale of 10 µm. M PLAN APO NIR). Meanwhile, the temperature sensing is realized by the forward reflection of a collimated He-Ne laser incidence. The transmission of the pulsed laser beam is blocked by a filter with a passing band from 500 to 700 nm in order to eliminate its impact on the temperature reading. Supplementary Fig. 9a shows  Fig. 10a). However, the parameters deviate from the EP as the exit medium changes from air to water ( 0 r 33 the glass slide is bonded with the water reservoir ( Supplementary Fig. 10b). Therefore, to assure the EP condition at room temperature under a solvent environment, we slightly vary the parameters of two Au layers to 21.4 nm and 25.9 nm while maintain the thickness of the PMMA layer ( Supplementary Fig. 10c). Due to the almost equivalent refractive indices between water and PDMS, the region bonded with PDMS also reaches the optimized EP condition after the reconfiguration ( Supplementary Fig. 10d).

Supplementary Figure 10. Design modification of the exceptional point due to exit medium change. a
Spectrum of the optimized optical exceptional point (EP) when the exit medium is air. b Deviation from EP when the exit medium changes to water. c The recovery of the EP condition after the reconfiguration when exit medium changes to water. d The EP condition almost preserves after the reconfiguration when the exit medium changes to PDMS.

Supplementary Note 7: Recording of the thermal conduction process.
To detail the process of thermal conduction on the glass slide after the hot water injection, we constantly monitored the forward reflection of the He-Ne laser incidence, which indicates the instantaneous temperature distribution. As presented in Supplementary Fig. 11, the thermal maps are successively captured with a time interval of 10 seconds time interval. This detailed real-time recording further confirms the gradual heat dissipation from the hot water into the ambient PDMS.
Supplementary Figure 11. Detailed temporal sequence of thermal imaging after hot water injection.
White numbers indicate the time when the corresponding images are captured after the hot water injection at an interval of 10 s. Dashed lines represent the boundary between water and PDMS. Scale bars in all panels, 100 μm.