A spatially resolved optical method to measure thermal diffusivity

Thermal transport experiments are one of the key tools used in condensed matter physics to understand the fundamental physics of electrons, phonons, and their interactions in correlated materials over a wide range of temperatures. Thermal conductivity (κ) is considered a characteristic quantity and is widely studied to explore transport behavior and scattering mechanisms.^1–3 Combined with measurements of the electrical resistivity, it allows the construction of the Wiedemann–Franz ratio,⁴ which provides important information on electron and phonon scattering. However, thermal radiation typically limits the measurement of κ performed in cryostats to the low-temperature regime (below 100 K in most cases); measurements at higher temperatures require specifically dedicated setups in which special care must be taken to calibrate and correct for radiation losses induced by the difference in temperature between the sample and the surrounding cryostat, which is usually kept at the temperature of liquid helium (4 K).⁵ One of the consequences of this experimental limitation is that there are only a few measurements of thermal transport in the wide temperature range over which the universal “Planckian” scattering rate has been observed in electrical transport measurements.^6–10 

Thermal conductivity is usually determined by measuring a temperature difference induced by a known heat flux across a sample thermally insulated from the environment. Such an approach has many advantages, notably its easy implementation in low-temperature setups and the possibility to measure electrical transport using the same contacts. Therefore, geometric uncertainty is greatly reduced when evaluating the ratio of thermal and electrical conductivity. However, this method requires samples to be homogeneous over at least several hundred microns in length and thermally isolated from the environment. Furthermore, investigations of thermal transport anisotropy require the preparation of several samples with different contact geometries, a laborious process introducing the possibility of systematic errors. While the above limitations can be overcome in large homogeneous single crystals, they are likely to introduce errors in materials that are inhomogeneous or in symmetry-broken phases exhibiting spontaneous domain formation. In addition, the same limitations become prohibitive in thin films, which cannot be thermally insulated from substrates they were grown on, and in van der Waals hetero-structures, which are typically only several tens of microns in size.

The above considerations motivate a different approach to studying thermal transport. We were particularly influenced by recent work,^11–14 suggesting that thermal diffusivity (D), instead of thermal conductivity (κ), can be a more informative quantity to describe thermal transport in the high-temperature regime, at least in transient conditions. Traditionally, thermal diffusivity is obtained from separate measurements of κ and heat capacity (c) (D = κ/c); therefore, the limitations of the thermal conductivity measurements as well as the acquired uncertainties related to the two separate measurements are inherited. Real samples also frequently contain inhomogeneities, either chemical or due to the formation of domains in ordered states of interest, so a technique with a high spatial resolution is desirable. Extensive work in several fields of research has shown that this can be achieved optically using two laser beams, one to locally heat the sample and a second to probe the response to that heating. If the heat is applied at a non-zero frequency, the relative phase of the probe and heat signals contain information on thermal diffusivity. Techniques based on this principle, known as modulated thermal reflectance microscopy or photothermal reflectance microscopy, are extensively used in materials science and have been refined to produce compact bench-top instruments for the room-temperature characterization of materials.^15–20 For the physical problems in condensed matter physics outlined above, however, room-temperature measurements are not sufficient. It is necessary to combine the optics with cryogenics, producing an instrument enabling measurement over a large temperature range. Furthermore, it is desirable to incorporate micron-scale spatial resolution, for example, to enable work with inherently inhomogeneous samples. In this paper, we describe our approach to the design of such an instrument and its use.

Direct measurement of D using lasers started from a flash method at high temperatures in the 1960s, which records the temperature vs time over a centimeter-scale sample after heating it up with a light pulse.^21,22 This method relied on a direct measurement of the value of temperature increase due to the laser heating, making it difficult to account for thermal losses and impedance between interfaces, which has limitations similar to those experienced using contact methods. Furthermore, the measurement of semi-transparent materials can lead to other problems that can be solvable with special tricks.^23,24 From the 1990s, a photo-deflection method has been applied to measure D without direct temperature measurement, by monitoring changes to the refractive index of a gas above the sample surface, but the technique still requires an mm scale sample and is unsuitable for cryogenic applications extending below the boiling point of nitrogen at 77 K.^25–27 The required sample size was further reduced to ∼50 μm by employing two laser spots, one as a source of heat and one as a probe of sample reflectivity.²⁸ This principle was also adopted in materials science, leading to the development of the room-temperature microscopies mentioned above. For condensed matter physics applications, cryostats can be incorporated, enabling its application to quantum materials.^13,14,28,29

To maximize the suitability of the laser-based technique to quantum materials research, it is desirable to simultaneously maximize spatial resolution and spatial range (the area over which scanning can be performed) over a wide range of temperatures. In order to achieve this, we build on the previous work discussed above, introducing a number of modifications based on technologies recently developed for cryogenic optics experiments. We operate in vacuum in a dedicated optical cryostat (Montana Instruments Cryostation S50) and obtain high range and resolution by using a piezo-driven scanning stage to move the sample. Although we can also move the laser spots, this occurs over a much more limited range, and most experiments are done in a moving-sample-fixed-optics mode. To profit from the high spatial resolution offered by the scanning stage, we mathematically and experimentally demonstrate that the in-plane diffusivity can be measured by overlapping the source and probe beams. Combining these approaches yields an instrument capable of scanning regions ∼4000 × 4000 μm², with ∼1 μm spatial resolution at temperatures ranging from 5 to 330 K. Furthermore, we describe in detail how to minimize artifacts arising from thermal contraction of the setup and detail the parameter range over which reliable quantitative measurements of diffusivity are possible.

Our optical setup for thermal diffusivity measurements is shown in Fig. 1(a). Two laser beams are used, one as a source (780 nm) to locally increase the sample temperature, and the other as a probe (633 nm) of reflectivity, which is a function of sample temperature. Both source and probe beams are focused by the same objective onto the surface of the sample as spots of ∼2 μm radius. The source beam is modulated by a mechanical chopper, causing periodic heating at the frequency selected by the chopper. Sample temperature, detected by the probe beam, changes at the same frequency. However, the temperature change is not instantaneous; the time scale at which it occurs depends on thermal diffusivity, which can, therefore, be obtained from the phase delay between source and probe signals. In order to measure the phase difference between the source and probe, two lock-in amplifiers are used. One records the source and the other one measures the probe signal, both sharing the same modulation frequency of the optical chopper. Measurements are performed in two configurations, which we refer to as separated and overlapped: in the former (used in the previous work in the field that we reviewed in Sec. I above), the two laser spots are separated by a distance r [Fig. 1(b)], while in the latter, they spatially overlap [Fig. 1(c)]. In Figs. 1(b) and 1(c)  R_s and R_p represent the radius of the source and probe, r is the distance between the center of the two spots (a finite value for the separated case and 0 for overlapped case), and z_s and z_p represent the penetration depths of the source and probe lasers, respectively.

Each of the two measurement configurations (overlapped and separated) has advantages and disadvantages and may, therefore, be appropriate under different conditions. The larger relevant length scale in the separated case results in a larger phase delay, making it less sensitive to systematic errors, such as laser fluctuations and small focal shifts. Furthermore, by controlling the direction of a separation between the two spots with respect to crystalline axes, it is possible to utilize the separated configuration to measure diffusivity anisotropy. In contrast, the overlapped case offers a better spatial resolution but is more sensitive to imperfections in beam overlap and changes the focus. The latter is particularly problematic when measuring the temperature dependence of diffusivity: due to the unavoidable thermal expansion of the sample holder, a 20-μm-shift in the vertical position of the sample is observed during temperature changes from 330 to 50 K, which brings it out of focus along the z-axis. We were able to precisely calibrate and compensate for that temperature-dependent shift ( Appendix C) by adjusting the vertical position of the sample using piezo-stages. In addition, we designed two ways of switching between the two measurement configurations: one using a mirror and another requiring coordination between two mirrors. The former is easier to operate and is applicable to single-temperature experiments, while the latter is more stable against z-shifts of the sample stage during the temperature-dependent experiments. Details can be found in  Appendix D. To summarize, the overlapped configuration offers better spatial resolution but potentially suffers from more experimental artifacts, which must be accounted for by careful experimental design and calibration, which we describe in this paper.

In both configurations, the inverse diffusivity is proportional to the square of the phase shift between the local temperature change δT and the power source causing the temperature change. The fundamental signal measured in our experiments is the change of reflectivity caused by the temperature change, δρ, which is proportional to δT if the heating is weak enough to keep the system in the linear response regime. In order to ensure that the measurements are performed under these conditions, it is important to measure δρ as a function of the source power ( Appendix E) and choose a suitable power to remain in the linear regime. Special care needs to be taken in the vicinity of phase transitions, where both the reflectivity and the diffusivity might be changing rapidly with temperature; typically, a lower power will need to be used compared with deep within a phase. Furthermore, different power levels may be appropriate at temperatures below and above a phase transition.

This measurement scheme requires reflectivity to have a measurable temperature dependence. Since our modulation technique allows the detection of relative reflectivity changes in the order of 10⁻⁶, and the signal can be further increased by increasing the source power (while staying in the linear regime), we expect that, for the large majority of materials, the thermal diffusivity can be measured via this optical method over extended temperature ranges. One exception is the unusual case in which the temperature derivative of reflectivity vanishes, e.g., the reflectivity undergoes a maximum or minimum. Even in such a situation, a measurement at a different wavelength may yield a finite temperature derivative. Crucially, as long as the measurements are taken in the linear regime, as captured by our mathematical description of the experiment ( Appendix A), the phase shift is independent of the value of reflectivity and its temperature derivative. However, the noise in the phase shift is proportional to the signal-to-noise ratio of the reflectivity change δρ; so better data quality is obtained for larger δρ.

One of the advantages of optics-based diffusivity measurement over methods based on traditional measurements of thermal conductivity and heat capacity is the ability to perform spatially resolved measurements, which is enabled in our setup by mounting the sample on a three-axis piezo-stage located in the cryostat. This allows for the measurement of diffusivity in samples, which are not spatially uniform, whether due to disorder or formation of domains.

We tested our techniques on two stoichiometric bilayer ruthenates, Ca₃Ru₂O₇ and Sr₃Ru₂O₇, as well as Ti-doped Ca₃Ru₂O₇. Ca₃Ru₂O₇ belongs to the orthorhombic space group Bb2₁m, with lattice constants of a = 0.5396 nm, b = 0.5545 nm, and c = 1.961 nm.^32,33 Single crystals exhibit twin domains and in-plane anisotropy, allowing us to use the spatial resolution to identify the domains, and measure the diffusion anisotropy, which has not been previously reported. In contrast, Sr₃Ru₂O₇ has lattice constants a = b = 0.3890 nm and c = 2.0732 nm, and an electrical resistivity that is isotropic in the ab plane, within the experimental resolution, for all temperatures above 2 K.^34–36 We performed temperature-dependent measurements on both Ca₃Ru₂O₇ and Sr₃Ru₂O₇ to demonstrate that the two measurement configurations provide consistent results for the average in-plane diffusivity.

The enhanced spatial resolution provided by the overlapped configuration offers opportunities to investigate the local properties, especially in doped systems that exhibit inherent disorder. To confirm this, we studied Ti-doped Ca₃Ru₂O₇. Ti doping into Ru sites in Ca₃Ru₂O₇ has been used to tune the magnetic and Mott transitions.^37,38 The Ti-doped Ca₃Ru₂O₇ exhibits a metal–insulator transition (MIT), whose temperature is strongly dependent on the Ti doping level. By changing the doping level from 3 to 10%, the MIT temperature varies from 50 to 110 K. Therefore, by identifying the MIT temperature locally, we are able to reveal the local doping level and check the in-plane chemical homogeneity of the sample.

To establish that the in-plane diffusivity can be measured in both configurations, it is necessary to confirm that (a) the low-frequency regime described above can be reached in both configurations and (b) both configurations yield the same diffusivity. In Figs. 2(a) and 2(b), we show the frequency dependence of phase delay at several temperatures ranging from 300 to 50 K in Ca₃Ru₂O₇ as a function of the square root of the modulation frequency for separated and overlapped configurations, respectively, together with the linear fits (dashed lines). The upper limit of the linear regime, marked by a gray dotted line, is 4 × 10⁴ rad/s for separated and 3.4 × 10⁴ rad/s for overlapped beams. We note a spurious intercept of a few degrees at zero frequency for both configurations. The intercept is a temperature-independent setup artifact, which is easily accounted for in the analysis: if the measured frequency dependence varies as $A+Bω$⁠, the intercept A can be subtracted. This observation emphasizes the importance of measuring, rather than assuming, the frequency dependence of the phase delay. The intercept subtraction is not a perfect procedure and is likely prone to introduce errors in cases where the absolute value of the phase delay is small, as we discuss below. However, as long as the intercept is small compared to the frequency-dependent part of the signal, the procedure works well.

Next, we compared the diffusivity obtained from the two configurations in Ca₃Ru₂O₇ and Sr₃Ru₂O₇. In Ca₃Ru₂O₇, the diffusivity anisotropy can be measured by separating the two laser spots along two principal axes, while the overlapped measurement is sensitive to the average diffusivity (see  Appendix A for more details). In Fig. 2(c), we plot the diffusivity along the two principal directions measured in the separated configuration, their average value, and the result of the measurement in the overlapped configuration. The average value obtained from the separated configuration [according to Eq. (A15) in  Appendix A] agrees well with the value obtained from the overlapped configuration. Furthermore, we confirm that the temperature-dependent diffusivity obtained from the two configurations in the isotropic Sr₃Ru₂O₇ is consistent [Fig. 2(d)]. Taken together, these results unambiguously show that the average in-plane diffusivity can be obtained by measuring the phase difference between overlapped source and probe laser beams, enabling the measurement of thermal diffusivity with diffraction-limited spatial resolution.

Although the overall agreement between the two methods is good, we note that the discrepancy increases at low temperatures, as shown in the deviation plot in the inset of Fig. 2(d). As discussed above, the error associated with the intercept subtraction procedure becomes more important as the intrinsic delay phase becomes smaller. This is most serious for the overlapped configuration at low temperatures and likely accounts for the larger deviation from the results in the separated configuration in that temperature range. In most circumstances, the delay phase is at its smallest for any given configuration of the pump and probe pulses at low temperatures. However, it is precisely in this low-temperature region that other methods for obtaining the diffusivity (for example a direct combination of thermal conductivity and heat capacity) are most accurate. The full power of the optical technique for our purposes comes in the region 100–300 K where standard thermal conductivity measurements are subjected to quite large systematic errors. Sometimes, however, even qualitative low-temperature information is useful, particularly when combined with high spatial resolution. We will discuss such a case in Sec. III D of detecting diffusivity inhomogeneity.

By using polarized light microscopy, one can clearly see the two-level contrast of the domain structures of Ca₃Ru₂O₇ [as shown in Fig. 3(a)] arising from optical birefringence.³⁹ In our optical setup with a micron-scale resolution, we are able to measure spatially resolved birefringence in the same sample by employing a polarization analysis, as shown in Fig. 3(b). Since the domain boundaries are known to be along the (110) direction in Ca₃Ru₂O₇,³⁹ the observation of a diagonal domain boundary allows us to identify the principal crystalline axes as vertical and horizontal in our measurement configuration. In order to confirm that the thermal diffusivity is anisotropic, we fix the separation between two laser spots and move both of them across the domain boundary. Such one-dimensional line scans are conducted under different directions of the separation between the source and probe spots. If the spot separation is kept fixed and vertical (horizontal), the measurement is sensitive to diffusivity D_b (D_a) in domain 1 and D_a (D_b) in domain 2. Performing such a measurement as a function of position along a one-dimensional line intersecting the domain boundary reveals a step at the boundary, shown in Fig. 3(c) for the vertical (purple) and horizontal (orange) separation. Additionally, we can continuously rotate the spots about each other and obtain the phase difference as a function of the angle θ between the spot separation and the principal axes [Fig. 3(d)]. The sinusoidal variation of the experimental results (open circles) offers additional proof that the diffusivity is indeed anisotropic. The purple curve is a fit to Eq. (6).

Furthermore, our scanning setup allows us to map diffusivity over large two-dimensional regions. In Figs. 4(a)–4(c), we show the diffusivity mapping across a 300 × 250 μm² area at three selected temperatures. Two main observations can be made. First, the two-domain structure is clearly identified, confirming that the thermal diffusivity is anisotropic; second, the anisotropy decreases with decreasing temperature. In addition, increased values of the phase delay near the domain boundary (Fig. 3) are evidence of enhanced scattering at the boundary, emphasizing the power of spatially resolved measurements to access the intrinsic behavior of homogeneous domains.

The greatest advantage of the overlapped configuration is enhanced spatial resolution, limited by the beam radii, which are 1.9 μm for the probe and 2.5 μm for the source in our measurements ( Appendix F), but could be as small as 0.5 μm in a diffraction-limited experiment. Spatial resolution is invaluable in the study of inhomogeneous systems, such as doped samples, in which the doping level may vary across the crystal, leading to variations in physical properties, including thermal diffusivity. In Fig. 5, we show the temperature-dependent phase delay for three different positions on the same Ti-doped sample, $Ca3(Ru1−xTix)2O7$⁠, with the nominal doping level of x = 5% (the corresponding sample positions are marked in the birefringence inset of Fig. 5). Sharp jumps in phase indicate the metal–insulator transition, which varies from 50 to 80 K for sample positions that are only ∼70 µm apart. Such variations of the transition temperature across the macroscopic sample originate from the inherent doping inhomogeneity, which, for this sample, is comparable in magnitude to the nominal doping itself. The clear observation of the variations in transition temperature is enabled by the high spatial resolution of the overlapped configuration. It is a valuable improvement compared with traditional bulk measurements or even with the separated spot optical measurements, as both such methods unavoidably exhibit a significant broadening in the transition due to the need to average over the separation distance from the pump to probe beams. In addition, the fact that the diffusivity increases rather than decreases below the metal–insulator transition when the electrons are frozen out indicates that the diffusivity of the Ti-doped sample is dominated by phonons at high temperatures. This observation supports the aforementioned statement that even qualitative knowledge of low-temperature diffusivity can be very useful.

Our results demonstrate the potential of thermal diffusivity measurement by optical methods in a cryogenic microscope setup. Separated and overlapped beam configurations provide consistent results for thermal diffusivity and exhibit their own advantages and disadvantages: the separated configuration can be used to identify the in-plane anisotropy, while the overlapped configuration provides a higher spatial resolution, which is especially helpful when investigating potentially inhomogeneous systems. Thermal diffusivity within domains and across domain boundaries can be studied. When working with metals, our fundamental signal, the phase shift, becomes smaller as the temperature decreases and the diffusivity increases. This means that the accuracy of the optical method decreases at low temperatures. However, this fact provides a useful complementarity with traditional methods of deducing the diffusivity from measurements of thermal conductivity and heat capacity because the accuracy of traditional thermal conductivity measurements is the poorest at high temperatures. We believe that the combination of the two methods will pave the way to measuring over several decades of temperature, yielding higher quality information than is currently available. More broadly, our setup gives us the possibility of studying probes of symmetry breaking, such as birefringence, Kerr rotation, and second harmonic generation, with the same spatial resolution as we have demonstrated here for thermal diffusivity. The combination will allow us to explore and visualize thermal properties in broken symmetry phases of correlated materials.

We wish to acknowledge Aharon Kapitulnik and Jiecheng Zhang for their useful discussions and for bringing the independent work of Ref. 14 to our attention. We also acknowledge Elena Hassinger and Ulrike Stockert for their useful discussions. V.S. is supported by the Miller Institute for Basic Research in Science, University of California, Berkeley. N.K. is supported by a KAKENHI Grants-in-Aid for Scientific Research (Grant Nos. 17H06136, 18K04715, and 21H01033), a Core-to-Core Program (Grant No. JPJSCCA20170002) from the Japan Society for the Promotion of Science (JSPS), and a JST-Mirai Program (Grant No. JPMJMI18A3). A.P.M. and S.M. acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—TRR 288-422213477 (project A10). Research in Dresden benefits from the environment provided by the DFG Cluster of Excellence ct.qmat (EXC 2147, project ID 390858940).

The authors have no conflicts to disclose.

F. Sun: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). S. Mishra: Conceptualization (supporting); Formal analysis (equal); Investigation (equal); Writing – original draft (supporting); Writing – review & editing (supporting). P. H. McGuinness: Investigation (supporting); Writing – review & editing (supporting). Z. H. Filipiak: Investigation (supporting); Writing – review & editing (supporting). I. Marković: Investigation (supporting); Writing – review & editing (supporting). D. A. Sokolov: Investigation (supporting); Writing – review & editing (supporting). N. Kikugawa: Investigation (supporting); Writing – review & editing (supporting). J. W. Orenstein: Conceptualization (supporting); Writing – review & editing (supporting). S. A. Hartnoll: Conceptualization (equal); Investigation (supporting); Writing – review & editing (supporting). A. P. Mackenzie: Conceptualization (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). V. Sunko: Conceptualization (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Since heat capacity is not a directional quantity, thermal diffusivity obeys the same angular dependence as thermal conductivity, leading to expression Eq. (6) of the main text for diffusivity anisotropy in 2D.

As discussed in the main text, it is not immediately obvious whether an experiment performed under the overlapped configuration dominantly probes the in-plane or out-of-plane diffusion. Here, we give numerical solutions to Eq. (A11), based on realistic parameters: R_s = R_p = 2.3 μm, D = D_x = D_y = 5 × 10⁻⁶ m²/s, and D_z = 0.01D. In Fig. 6(a), we show the phase delay as a function of the square root of the frequency. The solid green curve is the numerical solution with a finite penetration depth z_s = z_p = 50 nm, and the dashed curve is the solution with zero penetration depth. Small deviations can be seen lower, especially at lower frequencies. In Fig. 6(b), we plot the relative error between the numerical solution and the 2-dimensional approximation and find that the maximum error is 4%–5%. The numerical solutions indicate that even under the overlapped configuration, the phase delay between the source and the probe is dominated by the in-plane diffusion.

When performing the temperature-dependent measurements, special attention should be paid to the shift of the sample height due to the thermal expansion. In the overlapped configuration, a small z-shift changes the size of the laser spots, leading to a noticeable change in the phase delay (as shown in Fig. 7(a)]. To compensate for a such shift, we first scan the stage along the z-direction and record the temperature-modulated amplitude of the reflectivity, as well as the phase delay. As shown in Fig. 7(b), at the focal position, the amplitude is maximized, and the phase is minimized. By performing such z-scans, we are able to determine the focus with an accuracy of 1 μm. Furthermore, by measuring the z-scan as a function of temperature, we are able to quantify the change of the focal position due to the thermal expansion of the stage [Fig. 7(c)]. Once measured, such a shift can be compensated by using the piezo stage, allowing us to obtain a reliable measurement of the phase delay.

In the separated configuration, a small change in the laser spot size does not affect the phase because the diameter of both the source and probe laser is much smaller than the separation between the two spots. However, if the laser paths are not parallel to each other, the separation between the two spots can vary with the shift in the z-direction. In practice, for the convenience of the detection, we keep the probe beam path fixed and normal to the sample surface. The source beam can be moved in two ways (Fig. 8). The first method [Fig. 8(a)] is by adjusting the dichroic mirror (DM), which allows us to continuously move the source position on the sample. It is useful and convenient when we want to change the separation or switch the configuration between overlapped and separated cases at a fixed temperature. In addition, we can also use a piezo-electrical mirror mount to control the dichroic mirror, realizing an automatic scan of the separation. However, the disadvantage is that since the source path is tilted, the separation between spots changes during a temperature-dependent measurement due to the shift of the sample along the z-direction. Thus, we utilized a second way to control the beam separation [Fig. 8(b)], by coordinating two mirrors (M and DM), in order to control both the beam position and the incident angle. The separation is now more stable with respect to small changes in z.

As discussed in the main text, source laser power has to be carefully chosen. The temperature change δT induced by it should be large enough to yield a measurable δρ signal, but small enough to ensure that the measurement is taken in the linear regime, where δρ ∝ δT. In order to establish the latter, the dependence of δρ on the source power should be measured under the experimental conditions (material, configuration, frequency, and temperature) before each diffusivity measurement. Results of such measurements for both separated and overlapped configurations in Ca₃Ru₂O₇ are shown in Fig. 9, where perfect linear dependence of δρ on source power is observed for both configurations. The source powers we used in the experiments are marked by arrows and are both within the linear regime.

In order to precisely determine the spot sizes and the separation, a micro-structured calibration sample has been made by focused ion beam (FIB), which has a series of gold coated (bright) and non-gold coated (dark) stripes with the interval of 5 μm, as shown in Fig. 10(a). By taking a CCD image of the calibration stripes and the laser spots simultaneously, the distance can be directly measured. To be more precise, a Gaussian function fitting can be done to analyze the profile of the laser beams. One example has been shown in Fig. 10(b), where the radii of the source and probe beams are determined to be 2.5 and 1.9 μm, respectively, and the separation is 20.6 μm.