Temperature and microwave near field imaging by thermo-elastic optical indicator microscopy

A high resolution imaging of the temperature and microwave near field can be a powerful tool for the non-destructive testing of materials and devices. However, it is presently a very challenging issue due to the lack of a practical measurement pathway. In this work, we propose and demonstrate experimentally a practical method resolving the issue by using a conventional CCD-based optical indicator microscope system. The present method utilizes the heat caused by an interaction between the material and an electromagnetic wave, and visualizes the heat source distribution from the measured photoelastic images. By using a slide glass coated by a metal thin film as the indicator, we obtain optically resolved temperature, electric, and magnetic microwave near field images selectively with a comparable sensitivity, response time, and bandwidth of existing methods. The present method provides a practical way to characterize the thermal and electromagnetic properties of materials and devices under various environments.

where, β is a linear birefringence induced by thermal stress in the sample, and θ is a angle where the E i is the amplitude of electric field of the incident light, and A and B are: φ π δ π φ π δ π φ π δ π φ π δ π β β θ 2, 2 2, 2 1 sin 2 2 I I I I φ π δ π φ π δ π φ π δ π φ π δ π β β θ The β is related to a difference of stress between the two principle axes, and assuming the thermal stress can be described as the plane stress, the stress tensor can be expressed as : where, σ 1 and σ 2 are the two principal axis. From equation (S2.9) and (S2.10), and from the stress-optic law, the equation (S2.9) can be related to the stress as: where S is the stress optical constant, λ is the wavelength of incident light and d is the thickness of the medium.

S3. Constructing the heat source distribution
For plane strain condition in a rectangular Cartesian coordinate system, the thermal stress can be expressed by introducing the stress function 3 : where, σ x , σ y , and σ xy are the stress components of the stress tensor, T is the temperature distribution, α, ν, and E are the thermal expansion coefficient, the Poisson's ratio and the elastic modulus of a material, and Ф is the stress function satisfying 3 : The stress function is related to the stress components from (S3.1) as: and it is related to the LB measurement results from (S2.11) and (S3.3) as: From the stationary heat equation with a heat source, the heat source distribution can be calculated as: where, q is the heat source density, and k is the effective thermal conductivity of the platinum coated glass substrate. Finally, from equations (S3.4) and (S3.5), the heat source distribution can be expressed as: Figure S2 shows the measurement process for stationary (a) and time resolved (b) imaging. For each measurement steps, the 100-images were captured by the CCD camera for data and background, and wait steps of~5 seconds were introduced to saturate and cool down the temperature of the indicator. The time resolved measurement was conducted by capturing continuously with a frame rate of~13 fps and an exposure time of~50 ms. The averaging process was conducted by repeating the measurement processes 10~100 times, and therefore, 1,000~10,000 images were averaged. Figure S3c The smoothed images were then differentiated along the horizontal and vertical direction by calculating the intensity difference between pixels (b). Table S1 shows standard deviation of intensity (N rms ) and temperature sensitivity (TS) obtained for varying averaging times (AVE), moving average cell size (MAC) and times (MAT).

S5. Electromagnetic heating mechanisms at microwave frequency
In general, there are three different mechanisms of electromagnetic energy conversion to the heat at microwave frequency: magnetic, dielectric, and resistive losses 4 . For non-magnetic materials, the heating mainly comes from the dielectric and resistive loss.
When a loss material is coated on a glass substrate, the temperature change of the glass substrate can be expressed as: where, ρ, k, and C p are the density, thermal conductivity, and heat capacity of the glass substrate, and q is the heat flux density (or heat source density) of the loss material. For a steady state, one can simplify the equation as: where it is assumed that the thermal conductivity of the indicator is homogenous and isotropic.
The generated heat by the microwave depends on the loss property of the material coated on the glass substrate. For a case that when the glass substrate is coated by a dielectric loss material, one can express the generated heat by the dielectric loss per unit volume as: where and ω is the microwave frequency, ε" is the imaginary part of the dielectric permittivity of the loss material, E is the electric field strength of the microwave. On the other hand, for a case that a highly conductive metal thin film is coated on the glass substrate (σ » ωε), the resistive losses (or ohmic losses) is responsible mechanism for the heat generation.
Then, one can express the absorbed microwave power by the metal thin film as: where, H t is the microwave magnetic field tangential to the surface of the metal thin film, R s and δ s are the surface resistivity and skin depth of the metal thin film, respectively. Then, one can express the heat source density by the resistive losses induced by microwave magnetic field as: where t is the thickness of the metal thin film.

S7. LB measurement results of the indicators
where the AlNP and PMMA layers were prepared by thermal evaporation and spin coating techniques, respectively. The samples were placed on the stepped impedance low pass filter (SILPF), and an air gap of 0.5mm was introduced between the samples and SILPF to prevent direct heat conduction between them. The applied microwave frequency and power were 5GHz and 30mW, and 10,000 images were captured and averaged.

S9. Spatial resolution
The spatial resolution of an optical microscopy system is determined by the wavelength of the probing light and pixel size of the CCD camera, where the intensity of a pixel corresponds to the information of a physical property at the position. However, because the TEOIM is based on the plane stress analysis, where a stress for a given position is determined by the overall stress distribution in the plane, the information is in the overall intensity relation between pixels rather than an intensity of single pixel.
To verify the resolution limit of the TEOIM, we conducted simulations (COMSOL Multiphysics) for a case that two disk-shaped heat sources (radius=d) are separated with a distance of d exist in the OI. The first column in Fig. S7 shows simulated optical images as a function of pixel size, and second column shows calculated heat source distribution from the simulated photoelastic images. From the results, one can see that a reasonable resolution that enables to determine the two heat sources distinctly is achieved when the size ratio of the pixel and disks (d/px) was larger than 1.25. By considering an ideal case that the dimension of single pixel corresponds to the optical resolution limit, it can be concluded that the minimum detectable dimension of the TEOIM is comparable to that of a typical optical microscope system. The resolution of the TEOIM can be further enhanced by a proper interpolation process. The third column of Fig. S7 shows calculated heat source distribution images with an interpolation process. The interpolation process was conducted as follows: each pixel was divided by a small virtual pixel having a same intensity to real one, and then, the image was smoothed by taking a moving average. As shown in the calculation result with the interpolation process, a reasonable resolution was achieved when the d/px is around 0.75.