The flash methods, which are the most popular transient methods for measuring the thermal diffusivity of solid materials, have evolved into ultrafast laser flash methods by using picosecond or nanosecond pulse lasers as a heating source and a thermo-reflectance technique such as high-speed thermometry. In conventional ultrafast laser flash methods, thermal diffusivity is determined by fitting an analytical equation after single pulse heating to observe thermo-reflectance signals, although actual thermo-reflectance signals are observed after periodic pulse heating. This paper presents an exact analytical solution of the temperature response expressed by Fourier series for one-dimensional heat diffusion after periodic pulse heating. These Fourier coefficients are directly related to the Laplace transformation of the temperature response after single pulse heating. The signal observed for a 100 nm thick platinum thin film on a fused quartz substrate was analyzed by this Fourier expansion analysis and fitted by analytical equations with three parameters: heat diffusion time across thin film, the ratio of heat effusion of the substrate to thin film, and the amplitude of the signal over the entire range of pulse interval in the time domain. Robustness in determining the thermal diffusivity of the thin film by the ultrafast laser flash method can be improved by this new analysis approach.

## I. INTRODUCTION

In order to measure the thermal diffusivity of dense solid materials, such as metals, alloys, ceramics, semiconductors, etc., the flash method has been established as the standard method.^{1–4} It is popularly used and commercial instruments are widely available. The metrological standard of thermal diffusivity has been established in the metric convention by the Task group of thermophysical quantities of the Consultative Committee of Thermometry (CCT), International Bureau of Weight and Measures (BIPM).^{5} Document standards are established internationally by ISO, ASTM, etc.^{4}

Since the importance of thin films is rapidly growing in modern material science and industry, there are urgent needs to develop technology for measuring the thermophysical property of thin films,^{6–8} and various experimental and theoretical studies have been made.^{9–11} In order to measure the cross-plane thermophysical properties of thin films from 10 nm to several micrometers thick by thermo-reflectance methods under the configuration of front heat–front detect (FF) after impulse heated for femtoseconds, picosecond pulse lasers, which are called as time-domain thermo-reflectance (TDTR) methods, have been developed.^{12–14} These methods observe the cooling speed of the surface, which does not usually contain the heat diffusion time across definite lengths. Thus, they basically measure the thermal effusivity of the second layer under the first layer metal of about 100 nm thickness instead of measuring the thermal diffusivity of the films.^{6} In order to measure the cross-plane thermal diffusivity of thin films quantitatively, the ultrafast laser flash method has been developed under rear heat–front detect (RF) configuration.^{15–18} By introducing this configuration, the velocity of the acoustic wave can be simultaneously measured.^{19} As a challenging approach, nanoscale heat pathways have been visualized in *in situ* TEM by the combination of electron beam heating and nano-thermocouples.^{20}

In the ultrafast laser flash method, a transient change in the surface temperature of a thin film is observed, and the thermal diffusivity of thin film is determined by fitting a mathematical model to the transient change.^{16,17,21}

To observe the transient change, the thin film is heated with an ultrashort pulse laser. Ultrashort pulse lasers using mode-locking emit periodic pulse according to their own repetition rate. The electrical delay technique enabled us to observe transient changes of longer periods than the pulse interval.^{22} After developing this technique, it was recognized that the conventional model needs to be modified to represent the actual thermo-reflectance signals correctly.^{14}

The conventional model assumes that the thin film is heated by a single pulse, but actually it is heated by a periodic pulse. Because of this, the conventional model fits only in a limited time range and cannot fit in the entire range of the pulse interval. To overcome this problem, the linear correction approach is commonly taken, which assumes that the baseline decrease of the signal is linear.^{22} However, the actual baseline decrease is not linear; thus, this approximation is valid for only limited signals.

In this paper, we propose a new approach to analyze thermo-reflectance signals, which applies Fourier transform to curve fitting. We could explain the periodic temperature response by using Fourier series and achieved curve fitting to the entire range of pulse interval in the thermo-reflectance method. This approach realized the robust determination of thermal diffusivity of the thin film by the ultrafast laser flash method. In addition, this approach has expandability because we can use the transfer function in the Laplace domain for regression analysis, which is much simpler than the model function in the time domain.

## II. METHODS

### A. Experimental setup

Figure 1 shows the geometrical configuration of the pulse light heating thermo-reflectance apparatus used in this study. A laser beam is incident on a sample and we detect its reflection to probe its surface temperature. Another laser beam is also incident on the film to heat up its surface. The probe beam is incident on the front surface of the thin film, while the pump beam is incident to the rear face of the thin film through the transparent substrate, which is called rear heat–front detect (RF) configuration. The heat diffusion time across the film with fixed and known thicknesses is measured under the RF configuration, and the thermal diffusivity of the film is directly calculated from the heat diffusion time and the film thickness. This is the fundamental difference from the front heat–front detect (FF) configuration, which observes temperature cooling after pulse heating, which is mainly dominated by the thermal effusivity of the film and the substrate.

We measured thin films deposited on the transparent substrate with a customized picosecond pulse light heating thermo-reflectance apparatus (PicoTR, PicoTherm Corporation).^{23} Figure 2 shows a block diagram of the picosecond thermo-reflectance measurement system. There are two fiber lasers in the system, one is for the pump beam and the other is for the probe beam. The wavelength of the pump beam is 1550 nm and that of the probe beam is 775 nm, which is generated by a second harmonic generator (SHG).

A differential photodiode detects the reflected probe beam from the sample's surface. The diameter of the pump beam focused on the sample is 45 *μ*m and that of the probe beam is 25 *μ*m. Note that the detected signals are amplified by a lock-in amplifier; thus, the pump beam is modulated by a lithium niobate (LN) modulator with a modulation frequency of 200 kHz. In order to keep the linearity of signal to temperature change, only the amplitude output of the lock-in amplifier was used for analysis. In other words, the phase output of the lock-in amplifier was not used.^{24}

Repetition frequencies of two fiber lasers are synchronized at 20 MHz by electrical control. Since the modulation frequency of 200 kHz is only 1% of the pulse repetition frequency of 20 MHz, linearity of the amplitude output of the lock-in amplifier to the light intensity input is conserved without distortion. The delay time of the probe pulse train from the pump pulse train is controlled by the function generator. This electrical delay technique realized the observation of the thermo-reflectance signal over the entire interval between the periodic pulses (50 ns),^{22} whereas the conventional “optical delay technique” can observe the thermo-reflectance signal shorter than several nanoseconds, which is less than half of the repetition period of conventional mode-lock lasers of 12–13 ns.^{6,12–14} Measurements for this study were made at room temperature.

Incidentally, the development of ultrafast laser flash methods using nanosecond pulse lasers has extended the observation time longer than the microsecond range that can measure the thermal diffusivity of thin films thicker than the micrometer region.^{25,26}

### B. Mathematical model

#### 1. One-dimensionality of heat diffusion and the response function method

We can regard heat transfer in thin films as one-dimensional because the diameter of the pump beam of 45 *μ*m is far larger than the thickness of a thin film of 100 nm. In this model, the first layer thin films are confined to metals such as platinum, molybdenum, aluminum, etc., where the dominant heat carrier is free electrons. Since the mean free path of free electrons in these metals at room temperature is of the order of 10 nm,^{27} which is much smaller than film thickness, heat transfer is diffusive in the film. Thus, we can express the temperature response of the thin film in the RF configuration by solving the one-dimensional heat diffusion equation.^{16,17,21} Figure 3 shows the schematic diagram, which explains the relationship between temperature at the front face of the thin film $Tf$ and at the interface between the thin film and substrate $Ts$, and the heat flow density at the front face of the thin film $qf$ and at the interface $qs$.

By using the quadrupole matrix, the relationship is expressed as follows in Laplace domain:^{21}

where $\tau f$ is the heat diffusion time, $df$ is the thickness of the thin film, $\alpha f$ is the thermal diffusivity of the thin film, and $bf$ is the thermal effusivity of the thin film.

Since the thickness of the substrate can be regarded as semi-infinite, the temperature of the interface between the thin film and the substrate $Ts(t)$ and its Laplace transform $Ts~(\xi )$ are expressed as follows:

where $bs$ is the thermal effusivity of the substrate. If the rear face of the thin film is heated through the substrate by a pulse beam, we need to introduce the Dirac delta function $\delta (t)$ into Eq. (1) as follows:

Since the Laplace transform of $\delta (t)$ is 1 and the negative sign is added because of the definition of heat flow direction at rear face, we can substitute 0 to $qf~(\xi )$ because the sample's surface is adiabatic to the environment. By solving Eq. (3) with given conditions, we can obtain $Tf~(\xi )$ as follows:

where $\gamma $ is the ratio of virtual heat sources of the mirror image method.^{21}

As we can regard Eq. (4) as a geometric series, the equation can be transformed as follows:

Thus, its inverse Laplace transform $Tf(t)$ is finally expressed as follows:

As we detect the surface of the thin film, the temperature response in the RF configuration corresponds to $Tf(t)$.^{21} According to the reciprocity of the heat transfer,^{28} this temperature response of Eq. (5) under the RF configuration is equal to the temperature response under the front heat–rear detect (FR) configuration.

We can regard the temperature of the material as proportional to its reflectance unless the range of temperature change is significantly wide. Thus, the intensity of reflected probe beam (thermo-reflectance signal) can be expressed as follows:

where *k* is the proportionality constant. If we redefine the proportionality constant as $k\u2032=2k/(bf+bs)$, Eq. (4) is expressed as follows:

In the conventional model that analyzes the signal after a single pulse heating, the model function in the time domain is derived from Eq. (7) by the inverse Laplace transform,^{21}

#### 2. Periodic pulse heating and Fourier expansion

As mentioned in the Introduction, this conventional model function in time domain assumes single pulse heating. However, the surface of the thin film is heated with a periodic pulse by the apparatus of this paper. If a pump laser has a repetition rate $frep$, its temperature response obeys the periodic function with period $\Delta T=1/frep$. This means that we can express temperature responses as a summation of the Fourier series because the periodic function can be expressed as a summation of the Fourier series.

If a thermo-reflectance signal $ym$ is obtained, its Fourier coefficient $Yn$ derived from the discrete Fourier transform (DFT) is given by

where *N* is the number of samplings, $\Delta t$ is the sampling interval, and $\nu n$ is the frequency. Frequency $\nu n$ is defined as multiples of the sampling rate $1/\Delta t$. On the other hand, its inverse discrete Fourier transform (IDFT) is given by

According to the sampling theorem, any frequency components above the Nyquist rate $1/2\Delta t$ are aliases. Thus, we can transform Eq. (10) as follows:

where $Yn\xaf$ is the conjugate of $Yn$. We can regard Eq. (11) as the Fourier series of the periodic function with period $N\Delta t$. The period $N\Delta t$ corresponds to the interval of the periodic pulse $\Delta T$.

We already know that the thermo-reflectance signal can be explained by the transfer function $Y~(\xi )$ in Eq. (7). Thus, the Fourier coefficient $Yn$ obeys the transfer function $Y~(\xi )$ in the frequency domain. We will discuss this point in Sec. IV.

If we define complex number frequency as $\xi n=i2\pi \nu n$, we can express the regression model as follows:

where $k^\u2032$, $\tau f^$, and $\gamma ^$ are estimates of the proportionality constant $k\u2032$, the thermal diffusion time $\tau f$, and the ratio of the virtual heat source $\gamma $, respectively. $\epsilon $ is an error term that explains residuals.

This regression model is valid in the frequency domain below the Nyquist rate, and the coefficient at frequency zero $Y0$ is omitted. Note that four parameters $\nu n,k^\u2032,\tau f^,and\gamma ^$ are real numbers, whereas $Yn$, $Y~(\nu n,k^\u2032,\tau f^,\gamma ^)$, and $\epsilon $ are complex numbers.

After determining the model function by regression analysis in the frequency domain, we can determine the thermal diffusivity of the thin film $\alpha f$, the thermal effusivity of the thin film $bf$, and the thermal effusivity of the substrate $bs$ as follows:

where $df$ is the thickness of the thin film, $cf$ is the specific heat capacity of the thin film, and $\rho f$ is the density of the thin film. Note that an estimate of proportionality constant $k\u2032$ is not significant and only serves as a fitting parameter.

Now, we can calculate the periodic response in time domain as Fourier series by using the Fourier coefficient $Yn^$ substituting the determined parameters as follows:

Note that the square roots $i2\pi \nu n$ and $i2\pi \nu n\tau f^$ in Eq. (14) are multivalued since they include the imaginary unit *i*. In this paper, the square root of complex number $i$ is always calculated as $(1+i)/2$ to keep consistency of branch cuts for a complex function.

The Temperature response can also be expressed by the discrete expression considering the sampling interval as follows:

The coefficient at frequency zero $Y0$ has information about the baseline only; thus, we can use the same value as we already derived from DFT.

Figures 4(a) and 4(b) show the theoretical temperature response curves in the RF configuration after periodic pulse heating calculated based on Eq. (16). The time scale is nondimensionalized by the interval of the periodic pulse, and the temperature scale is arbitrary normalized. Figure 4(a) shows the temperature response curves with different heat diffusion times across the film with $\gamma =1$, when the thermal effusivity of substrate is 0, which is equivalent to the adiabatic boundary condition to the thin film. The heat diffusion time $\tau f$ is normalized by $\Delta T$, and a dimensionless parameter $\Phi =\tau f/\Delta T$ is introduced. Figure 4(b) shows the temperature response curves with different $\gamma $ with $\Phi =0.1$.

## III. RESULTS

The platinum thin film deposited on the fused quartz substrate was measured with apparatus under the RF configuration. The film was deposited by sputtering, and its thickness is 100 nm. The black line in Fig. 5 is the thermo-reflectance signal observed from the film. The sampling interval $\Delta t$ is 10 ps. The nominal repetition rate $frep$ of the pump laser is 20 MHz; thus, the interval of the periodic pulse $\Delta T$ is 50 ns.

We calculated the Fourier coefficient $Yn$ from the thermo-reflectance signal by DFT in the range of 0 s–50 ns. $Yn$ is a complex number that consists of a real part and an imaginary part. We can also express the complex number with absolute value and argument. Figure 6 shows the absolute value of the Fourier coefficient $|Yn|$. Note that the Fourier coefficients exhibit symmetry around the Nyquist rate of 50 GHz. This plot can be regarded as a frequency response.

Since this sample is modeled as a film on a semi-infinitely thick substrate as shown in Fig. 3, the Laplace transform of the temperature response after single pulse heating is expressed by Eq. (7).

We implemented curve fitting by using the non-linear least-squares method. This time, we applied regression analysis only to the absolute value of the Fourier coefficient $|Yn|$ as follows:

Figure 7 shows the regression curve in the frequency domain. As we mentioned before, we implemented curve fitting in the range below the Nyquist rate. This time, we omitted high frequency components above 5 GHz. We determined $\tau f^$ at $5.96\xd710\u221210s$ and $\gamma ^$ at 0.700 by fitting Eq. (17) to the absolute of Fourier coefficients as shown in Fig. 6.

The red line in Fig. 5 is the theoretical curve in the time domain derived from Eq. (15). Note that the observed signal and the theoretical curve agree with each other over the entire repetition period by introducing three fitting parameters $k^\u2032,\tau f^,and\gamma ^$.

Since the thickness of the platinum layer is 100 nm, the thermal diffusivity of the platinum thin film was calculated at $1.68\xd710\u22125m2/s$ by using Eq. (13). Assuming the literature value of specific heat capacity as $cf=133J/(kgK)$ and density as $\rho f=21500kg/m3$ of platinum,^{29} the thermal effusivity of the fused quartz substrate $bs$ is calculated as $2070J/(s0.5m2K)$.

## IV. DISCUSSION

### A. Fourier coefficient and Laplace transformation of the analytical solution

Figure 8 shows the relationship between the temperature response of single pulse heating $Tsingle$ and periodic pulse heating $Tperiodic$. $Tperiodic$ is explained as an accumulation of $Tsingle$, where we consider $Tsingle$ and $Tperiodic$ as continuous functions of time *t* instead of a discrete value,

If we consider the range of pulse interval only, the Fourier coefficient of degree *n*$T^periodic(\nu n)$ is expressed as follows:

Note that $exp(\u2212i2\pi nm)$ equals 1.

If we change valuable *t* into $t\u2032=t+m\Delta T$, the equation is expressed as follows:

This means the Fourier coefficients of the temperature response of periodic pulse heating correspond to the Fourier transform of the temperature response of single pulse heating. If we redefine complex values as $\xi n=i2\pi \nu n$, $T^periodic(\nu n)$ can be expressed by $Tsingle(t)$ as follows:

Note that $\xi n$ is a complex number, whereas $\nu n$ is a real number.

As shown in Eq. (21), $T^periodic(\nu n)$ is finally expressed by the Laplace transform of the temperature response of single pulse heating $T~single(\xi )$ as follows:

Note that $Tsingle(t)$ corresponds to the model function in the time-domain Eq. (8).

### B. Absolute value and argument of Fourier coefficients

This time, we used Eq. (17), which is the equation for absolute value, instead of using Eq. (12), which is the equation of the complex function. This means that we did not use the argument of Fourier coefficient $ArgYn$ in regression analysis. In this subsection, we discuss the reason why only the fitting of the absolute value is enough although, in general, the original function cannot be reproduced by the complex Fourier series without information of argument of complex Fourier coefficients.

The black line in Fig. 9 is the argument of Fourier coefficients $ArgYn$, which is calculated from the observed temperature response. It should be noted that there is uncertainty of pulse heating moment of the apparatus in Fig. 2. Therefore, it is necessary to introduce a parameter $\tau d$ that expresses the difference between the real pulse incident moment and the tentative time origin set by the apparatus. Then, Eq. (14) can be replaced by the following equation:

The red line in Fig. 8 is the argument of Fourier coefficient $ArgYn^$ fitted by Eq. (23) with a parameter $\tau d$, with substituting fixed values of $k^\u2032,\tau f^,and\gamma ^$, which are determined by fitting to absolute values using Eq. (17). Note that this exponential function only affects arguments of Fourier coefficient and does not affect absolute values since $|exp(\u2212i2\pi \nu n\tau d)|$equals 1. The time delay $\tau d$ was determined as 540 ps. This means that fitting to the absolute value by Eq. (17), which is invariant to $\tau d$, can be performed without knowledge of the pulse heating moment of the apparatus.

### C. Sensitivity and uncertainty analyses

Figures 10(a) and 10(b) show the magnification of Fig. 5 in time scale in order to visualize the sensitivity analysis for the heat diffusion time $\tau f$ and the ratio of virtual heat sources $\gamma $, respectively. The red curves in Figs. 10(a) and 10(b) correspond to the regression curve in Fig. 5. Note that the temperature starts to rise late because of the time delay $\tau d$. When $\tau f$ and $\gamma $ are deviated by ±10% from the fitted value, these theoretical curves are fluctuated as green curves in Figs. 10(a) and 10(b). The signal and the curves are normalized by their peak values. Figures 10(c) and 10(d) show the residual expression of sensitivity analyses in Figs. 10(a) and 10(b). It can be seen that the deviation of the signal can roughly fit into the range of ±10% deviation of fitting parameters $\tau f$ and $\gamma $. The standard deviation of the signal is around 0.024. According to Figs. 10(c) and 10(d), this value is less than half of the maximum fluctuation caused by the ±10% deviation of fitting parameters. Thus, we can estimate that the uncertainty is less than 5%. We can also see that $\tau f$ is mainly determined by the rising part of the temperature response and $\gamma $ is mainly determined by the cooling part of the temperature response.

## V. CONCLUSION

In this paper, we applied the Fourier expansion analysis to the thermo-reflectance signals by taking into account the periodicity of pulse laser heating, which is not possible by conventional methods. We proved that the thermo-reflectance signal after periodic pulse heating could be expanded to the Fourier series. We also proved that the complex Fourier coefficient can be calculated from the Laplace transform of the temperature response by single pulse heating. To analyze the experimentally observed thermo-reflectance signals, the platinum thin film on the fused quartz substrate was measured by an electrical delay-type picosecond pulse heating thermo-reflectance apparatus under the RF configuration. The observed signal over the entire range of the periodic pulse interval was fitted with this analytical solution of the Fourier series. We fitted the absolute value of complex Fourier coefficients by the Laplace transform of the analytical equation by single pulse heating. The proposed analysis achieves a robust determination of thermal diffusivity of the thin film by utilizing the periodic nature of the thermo-reflectance signals and excluding the effect of time delay. We expect that this approach can be applied to a variety of temperature signals after periodic pulse heating, including more complicated samples like multi-layered thin films.

## ACKNOWLEDGMENTS

The authors acknowledge support from the JST Mirai Program (Grant No. JPMJMI19A1).

## DATA AVAILABILITY

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

## AUTHOR DECLARATIONS

### Conflict of Interest

Takahiro Baba and Tetsuya Baba have filed one Japanese patent application (Appl. No. 2018-516599, Patent No. 6399329), one PCT patent application (PCT/JP2018/012324), and one U.S. patent application (16/620.294) related to the work described here.