Laser-based time-domain thermoreflectance (TDTR) and frequency-domain thermoreflectance (FDTR) techniques are widely used for investigating thermal transport at micro- and nano-scales. We demonstrate that data obtained in TDTR measurements can be represented in a frequency-domain form equivalent to FDTR, i.e., in the form of a surface temperature amplitude and phase response to time-harmonic heating. Such a representation is made possible by using a large TDTR delay time window covering the entire pulse repetition interval. We demonstrate the extraction of frequency-domain data up to 1 GHz from TDTR measurements on a sapphire sample coated with a thin layer of aluminum, and show that the frequency dependencies of both the amplitude and phase responses agree well with theory. The proposed method not only allows a direct comparison of TDTR and FDTR data, but also enables measurements at high frequencies currently not accessible to FDTR. The frequency-domain representation helps uncover aspects of the measurement physics which remain obscured in a traditional TDTR measurement, such as the importance of modeling the details of the heat transport in the metal transducer film for analyzing high frequency responses.

## I. INTRODUCTION

Non-contact optical techniques for thermal transport measurements are now well developed for many different applications. In particular, thermoreflectance-based techniques have been widely adopted for studying heat conduction over small length and time scales. These techniques use a modulated laser (pump beam) to heat the surface of a sample. The surface temperature variation leads to a change in reflectance, which is monitored by a probe laser beam. By comparing the measured response with model calculations, parameters of the sample such as the thermal conductivity can be determined. Two main versions of this general approach have been developed since the 1980s.^{1,2} In the first method,^{1,3–5} a sinusoidally modulated continuous wave (CW) pump beam (see Fig. 1(a)) produces time-harmonic heating resulting in surface temperature oscillations at the modulation frequency, which are monitored by a CW probe beam. The amplitude and phase of the surface temperature response are measured as functions of the modulation frequency, with the phase measurements oftentimes preferred due to their higher accuracy.^{5} We will refer to this method as frequency-domain thermoreflectance (FDTR). The modulation frequency in FDTR typically varies from kHz to ∼10 MHz.^{1,3,4} Recently, an extension of the frequency range up to 200 MHz was reported.^{5,6} Since the penetration depth of the temperature oscillations becomes smaller at higher frequencies, such an extension is beneficial for studying thermal transport at fine spatial scales.

In the second method known as time-domain thermoreflectance (TDTR),^{2,7–9} the pump beam is provided by a femtosecond laser operating at a high repetition rate (typically ∼80 MHz) and is additionally modulated, as illustrated in Fig. 1(b). Unlike in FDTR, the heating is not time-harmonic but comprised of many frequency components. The probe beam is derived from the same laser, and probe pulses are delayed with respect to pump pulses by a mechanical delay line. The thermoreflectance response is measured by a lock-in amplifier with the pump modulation frequency serving as a reference. The TDTR signals, i.e., the in-phase and quadrature outputs of the lock-in amplifier as functions of the delay time,^{10} do not, in general, reflect the actual time-domain dynamics of the surface temperature of the sample. However, just as in FDTR, the response can be compared to model calculations to determine the properties of the sample such as the thermal conductivity of the substrate and the thermal boundary resistance between the substrate and the metal film typically used to facilitate both laser-induced heating and thermoreflectance measurements.^{9,11}

It should be noted that typically TDTR data is interpreted by solving the heat equation in the frequency domain, to find the temperature response to time-harmonic heating, just as in FDTR measurements. In order to model a TDTR signal, many frequency responses are added together.^{9,11,27} To directly compare FDTR and TDTR data, the individual frequency components measured in TDTR need to be separated. Separation of frequency components would be useful for investigating different thermal length scales. A high heating frequency corresponds to a shallow thermal penetration depth, and for thermal length scales comparable to or smaller than phonon mean free paths in a material, non-diffusive effects may be observed.^{5,12} For example, the effective thermal conductivity of Si at room temperature measured with FDTR has been reported to decrease with increasing pump modulation frequency.^{5} In contrast, no such dependence for Si has been observed in TDTR measurements,^{12} although the pump modulation frequencies used in the latter work were lower than in the FDTR study. However, the frequency content of TDTR measurements is not restricted to the pump modulation frequency. TDTR data contain high frequency components, principally limited only by the laser pulse duration. It would be advantageous if frequency components of the TDTR response could be separated and represented in a form similar to FDTR, i.e., in terms of the amplitude and phase of the surface temperature response to time-harmonic heating.

In this paper, we show that the extraction of single-frequency components from the TDTR response and their representation in a form identical to FDTR are indeed possible, provided that the linear response regime applies and the delay time window is extended to include a full laser repetition period. We demonstrate an implementation of this technique, and describe the details of the measurement and data processing. Amplitude and phase responses in the frequency range up to 1 GHz are extracted from measurements on an aluminum-coated sapphire sample. While the main focus of this report is on data collection and processing rather than on modeling of thermal transport, we compare the data in the frequency-domain representation with model calculations based on the heat equation. We find that high frequency responses above ∼200 MHz are sensitive to electronic superdiffusion beyond the skin depth in the metal transducer film.

## II. THEORETICAL BASIS

We briefly review the signal formation in FDTR and TDTR and demonstrate that the frequency components in a TDTR signal can be separated and represented in a form identical to FDTR, i.e., in terms of the amplitude and phase of the surface temperature response to time-harmonic heating.

### A. FDTR and TDTR

In FDTR, the heat input supplied by the pump laser is a simple sinusoid with an angular frequency ω_{o} = 2π*f*_{o}, as illustrated in Fig. 1(a), which may be expressed as

where *q*_{o} is the pump power modulation amplitude. This periodic heat input results in surface temperature oscillations with the same periodicity,

While the frequency is set by the pump, the magnitude and phase of the surface temperature with respect to the pump are determined by sample properties. The complex function

_{o}is varied, because accurate measurements of the magnitude are more difficult.

^{5}

^{9,11,13}In FDTR, sweeping through all values of ω

_{o}is necessary to fully determine

If instead of time-harmonic excitation, the sample were heated by a short laser pulse approximating a delta function, the surface temperature would be described by the time-domain response, *h*(*t*). In the linear regime,^{14} *h* is related to the frequency-domain response,

*h*decays to zero before the next laser pulse strikes the sample. In such an experiment, which could be called “single-shot TDTR,”

*h*, and consequently

The heat input supplied by a TDTR pump beam can be approximated by a train of delta pulses modulated by a sinusoid as illustrated in Fig. 1(b), which can be expressed as

where *Q*_{o} is the absorbed pump pulse energy and the period between pulses is 2π/ω_{s}. In the frequency-domain, this becomes

The surface temperature relates to the heat input by

In TDTR measurements, the probe is derived from the same laser as the pump, and can be treated as a train of delta pulses at the laser repetition frequency, delayed relative to the pump pulses by a delay time, τ. The incident probe power is given by

which in the frequency-domain becomes

where *Q*_{i} is the probe pulse energy. The reflected probe power, which will be collected by the detector, is given by the product of the sample's thermoreflectance response and the incident probe power, *q*_{r}(*t*) = *C*_{th}θ(*t*)*q*_{i}(*t*), where *C*_{th} is the thermoreflectance coefficient which relates the change in reflectance to the change in temperature. The power of the reflected probe beam is assumed to vary proportionally to the change in surface temperature, which is a valid assumption as long as the change in surface temperature is small.^{11} In the frequency-domain, the product becomes a convolution,

Typically, TDTR employs lock-in detection in order to improve signal-to-noise. The lock-in mixes the signal from the detector with a reference signal at ω_{o} and with a reference signal with a 90° phase offset, to find in-phase and out-of-phase (quadrature) responses, which filters out all frequencies except for a narrow band^{15} around ω_{o}. The complex amplitude of the lock-in response for a given delay time is given by^{9,11}

where *g* represents gain in the detection electronics. The measured data in TDTR are in-phase, *x*, and quadrature, *y*, components of the lock-in amplitude as the delay time is varied:

### B. Frequency-domain representation of TDTR data

As evident in Eq. (8), the measured signal in a TDTR experiment is a periodic function of the delay time τ, which is represented in the form of a Fourier series. The Fourier series coefficients are given by

From Eq. (10), we can see that the magnitudes of the Fourier coefficients are proportional to

*k*ω

_{s}+ ω

_{o}, where

*k*is an integer from −∞ to ∞. Since the impulse response,

*h*(

*t*), is a real function, we can invoke the complex conjugate relation

where *n* is a positive integer. By determining the Fourier coefficients from a measured TDTR response represented as a complex function of the delay time, one is able to determine the frequency response

*n*ω

_{s}± ω

_{o}, which is equivalent to determining

In some implementations of TDTR, the pump pulses are delayed with respect to the probe.^{8} If the delay line is placed after the modulator used to modulate the pump beam at ω_{o}, an additional phase lag^{11} given by exp (*i*ω_{o}τ) is introduced. In this case, the lock-in response is no longer a periodic function of τ. Multiplying the response by exp ( − *i*ω_{o}τ) removes the additional phase lag and yields a periodic function described by Eq. (8), after which Eq. (11) can be used to find frequency-domain responses.

A similar problem of finding frequency responses from a lock-in output arises in analyzing acoustic waves measured in a femtosecond pump-probe experiment with a high repetition rate, and an analogous methodology for extracting acoustic frequency responses from the lock-in output has been developed concurrently.^{16,17}

## III. EXPERIMENTAL METHODS

The typical experimental arrangement used in TDTR has been described in numerous works.^{2,7–9} Our setup,^{9} illustrated in Fig. 2, uses a pulsed laser oscillator operating at a center wavelength of 800 nm, with a pulse width of ∼200 fs, and a repetition rate of *f*_{s} = 81 MHz. We frequency-double the pump beam to allow the pump to be easily prevented from reaching the detector. An electro-optic modulator (EOM) sinusoidally modulates the pump beam at a frequency *f*_{o}, which we vary from 2 to 12 MHz.

The sample used for the measurement was fabricated by depositing a 110 nm layer of Al on a crystalline substrate of (0001) Al_{2}O_{3} using electron beam evaporation. The Al transducer layer thickness was verified with atomic force microscopy by scratching away a small area of Al from the substrate and measuring the step height.

The Gaussian pump and probe 1/*e*^{2} radii were 28 μm and 5 μm, respectively. Measurements with larger pump radii, up to 55 μm, produced the same results, but with lower signal-to-noise. The pump radius is chosen to be much larger than the probe radius to mitigate overlap alignment errors.

### A. Data collection

To determine the frequency responses of the surface temperature given by Eq. (11), we need a full period of delay-time-domain data from τ = 0 ns to τ = 1/*f*_{s} ≈ 12.3 ns. We use a motorized mechanical delay stage with a 0.5 m travel distance, and pass the probe beam through this delay four times, resulting in a maximum probe delay time of around 7 ns. To obtain an additional 6 ns of delay time, we introduce an additional fixed delay. Thus, the full period of delay time data is collected in two sets. The necessary delay length in the experiment is some amount longer than 1/*f*_{s}, because measuring the reflectance signal peak at zero delay for both data sets is essential for “stitching” the data sets, as described in Sec. III B. To mitigate optical alignment errors and minimize divergence issues, we expand the beam diameter by 4× before the optical delay line, to ∼8 mm.

### B. Data stitching

The short and long delay data sets are stitched to form the full period of delay time data in Fig. 3, which shows both the in-phase, *x*, and quadrature, *y*, parts of the complex lock-in amplitude signal as a function of delay time, (see Eq. (9)). To combine the delay time data sets, several post-processing steps are used. A peak in the thermoreflectance signal occurs when the pump and probe beams arrive at the sample simultaneously, because at that delay time the probe measures the maximum temperature rise. The long delay time data set is shifted so that the peak in the signal magnitude occurs at τ = 1/*f*_{s}. The short delay time data set is similarly shifted so that the peak occurs at τ = 0. Each data set is independently phase-corrected by requiring that the quadrature lock-in signal component, *y*, does not experience a jump^{11} at τ = 0 or τ = 1/*f*_{s}. This procedure removes any phase that may be added by the electronics.

The data sets are scaled so that the magnitudes at τ = 0 and τ = 1/*f*_{s} match. The same scaling factor is used on both *x* and *y* to preserve the phase information. This scaling is performed to compensate for the reduction in signal magnitude caused by the added optics in the long delay data set. After scaling, the data sets are shifted so that the values of *x* and *y* just before τ = 0 and τ = 1/*f*_{s} match.

The quality of the data set stitching can be evaluated by observing the delay time region around 6 ns, where the data sets overlap, as shown in Fig. 3. Good overlap provides confidence in the data stitching procedure, and in the alignment of the optics during the experiment, demonstrating that the probe does not walk or diverge significantly as the mechanical delay stage is swept.

### C. Fourier series representation

A numerical fast Fourier transform operation on *z*(τ), from τ = 0 to τ = 1/*f*_{s}, produces Fourier series coefficients, *a*_{k}, which are proportional to

*f*=

*nf*

_{s}±

*f*

_{o}, according to Eq. (11). We present our frequency-domain data in terms of the amplitude and phase of the surface temperature frequency response, where

*R*has a relative magnitude with arbitrary units, while ϕ is the absolute phase with units of angle. ϕ(

*f*) and

*R*(

*f*) can be directly compared to an equivalent FDTR measurement.

Figure 4(a) presents *R* and ϕ obtained from the lock-in output data shown in Fig. 4(b), which were collected at three modulation frequencies: 4, 8, and 12 MHz. A continuous frequency dependence could be obtained^{16} if *f*_{o} could be varied up to *f*_{s}/2. Signal-to-noise issues with our existing TDTR system limit our maximum pump modulation frequency to ∼12 MHz, so the frequency-domain data curves in Fig. 4(a) have frequency gaps. However, since thermal responses typically lack sharp resonant features, filling in the gaps is not crucial for practical purposes. The high time resolution inherent in TDTR (typically limited by the pulse width) enables the extraction of very high frequency data components. However, at high frequencies, the Fourier coefficients are reduced, resulting in a poorer signal-to-noise ratio.

Representing TDTR data in the frequency-domain leads more readily to a physical interpretation than a conventional delay-time-domain representation such as that of Fig. 4(b). In the delay-time-domain, each different pump modulation frequency results in a different curve which must be evaluated separately or using global fitting strategies, while in the frequency-domain, all data from different *f*_{o} measurements collapse into a single curve. The amplitude data depend on the amplitude factor that generally varies as we change *f*_{o} because of the variations in the modulation efficiency of the EOM and the sensitivity of the detection electronics, as well as drift of the laser energy. The delay-time-domain signal magnitude jump at τ = 0 should be independent of the pump modulation frequency. We normalize each delay-time-domain curve to the magnitude jump at τ = 0, using the same normalization factor on both *x* and *y* to preserve absolute phase information. Thus, amplitude data from multiple *f*_{o} measurements form a single curve seen in the top panel of Fig. 4(a). The phase data shown in the bottom panel form a single curve without any calibration effort.

## IV. DATA ANALYSIS

Modeling TDTR and FDTR responses with the thermal diffusion equation is well documented in the literature.^{9,11,13} We follow the methodology of Ref. 9 to model heat transport using a thermal quadrupole^{18,19} solution to the heat equation for a multilayer, semi-infinite solid, which accounts for radial, anisotropic conduction. Our model outputs the frequency-domain response of the surface temperature,

_{2}O

_{3}substrate thermal conductivity,

_{2}O

_{3}thermal interface conductance, G, using a least squares fitting routine. All other material parameters are set to literature values. Either the relative amplitude,

*R*(

*f*), or absolute phase, ϕ(

*f*), data sets may be used for fitting.

The top part of the Al film is modeled as an isothermal layer to mimic energy deposition into a finite depth. The isothermal layer is modeled as having no radial thermal conductivity and a high cross-plane thermal conductivity. We begin by choosing an isothermal layer thickness of 10 nm, as was done in Ref. 11, which is comparable to the optical skin depth in Al. By fitting the data up to 200 MHz, as shown by the dotted lines in Fig. 5, we find best fit values of

^{2}K from

*R*(

*f*), and

^{2}K from ϕ(

*f*). The literature value

^{20}for

In spite of recovering close to the literature value of Al_{2}O_{3} thermal conductivity, the model fails to capture the phase behavior at high frequencies, suggesting the need to properly model the transport processes in the Al. Fast non-equilibrium electronic diffusion during ∼1 ps following short-pulse excitation deposits the pump energy over a much larger depth than the optical skin depth of ∼7 nm.^{21} For the purposes of this work, with the main focus on the experimental methodology rather than on non-equilibrium dynamics in a metal following a femtosecond excitation,^{22,23} we proceed by finding the isothermal layer thickness that most closely matches our phase data at high frequencies. We find that an isothermal layer thickness of 25 nm produces a good fit to our high frequency data, as shown by the solid lines in Fig. 5. A fit with a 25 nm isothermal layer yields best fit values of

^{2}K from

*R*(

*f*), and

^{2}K from ϕ(

*f*). The choice of isothermal layer thickness is important for modeling the high frequency responses, but yields nearly the same thermal conductivity values, because low frequency data are more sensitive to the substrate thermal conductivity, as will be shown below. The values of

*f*).

We can quantify the thermal model sensitivity to a particular parameter, *x*, such as the substrate thermal conductivity or the interface conductance, by considering the logarithmic derivative of the model's response with respect to that parameter,^{24}

Sensitivity curves for our Al coated Al_{2}O_{3} sample are plotted in Fig. 6, showing the thermal model sensitivity to

*d*

_{iso}. Figure 6 shows that the thermal model is reasonably sensitive to

*R*and ∼200 MHz in ϕ, and at high frequencies, the thermal interface conductance has a more dominant contribution to the signal than the Al

_{2}O

_{3}thermal conductivity.

In addition to model sensitivity, we can evaluate the quality of the model fit by considering the summed squares of the residuals between the model and the data, χ^{2}, for a range of

^{2}value. The χ

^{2}contours indicate that

*R*(

*f*) data is more sensitive to G, while ϕ(

*f*) data is more sensitive to

The influence of how the metal layer is modeled is apparent in the high frequency phase data. As shown in Fig. 8(a), the effect of reducing the isothermal layer thickness to 10 nm or increasing it to 40 nm is negligible below 100 MHz, but becomes increasingly important at high frequencies. In fact, isothermal layers of 10 and 40 nm thicknesses fail to describe the high frequency data even if we allow both

^{25}would be the next logical step for improving the accuracy of modeling high frequency responses. The importance of such analysis in interpreting high-frequency FDTR responses in terms of possible non-diffusive effects has been pointed out in a concurrent study.

^{26}

## V. SUMMARY

We have detailed a modified TDTR technique that allows for transforming TDTR data collected in the delay-time-domain into the frequency-domain, a representation equivalent to that of FDTR techniques. A single TDTR measurement provides the same information as sweeping through many different modulation frequencies in FDTR. The high time resolution inherent in TDTR measurements enables the extraction of very high frequency content, up to 1 GHz or more with improvements to signal-to-noise, which goes well beyond the current capabilities of FDTR techniques.^{5,6} The method only requires a small modification of a conventional TDTR experiment, i.e., the extension of the optical delay range up to a full repetition rate period, and can be easily implemented in any laboratory possessing a standard femtosecond pump-probe apparatus with a high repetition rate. The frequency-domain representation has revealed that while the standard heat equation model works well at frequencies below ∼200 MHz, higher frequency responses are affected by electron superdiffusion in the metal transducer film. This effect will be even more pronounced for metals with weaker electron-phonon coupling such as gold.^{26} The described methodology not only allows a direct comparison of TDTR and FDTR data and yields frequency responses at hitherto unattainable high frequencies, but also provides a physically intuitive way of analyzing TDTR measurements.

## ACKNOWLEDGMENTS

The authors gratefully acknowledge helpful discussions with Aaron Schmidt, Austin Minnich, and Oliver Wright, as well as critical reading of the manuscript by Vazrik Chiloyan and Richard Wilson. AFM measurements by Poetro Sambegoro were greatly appreciated. Funding support was provided by the “Solid State Solar-Thermal Energy Conversion Center (S3TEC),” an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0001299/DE-FG02-09ER46577.

## REFERENCES

^{27}This is different from the definition of FDTR adopted in this paper which assumes a time-harmonic heating.

^{9,11,13}is valid as long as temperature variations are small compared to the background temperature. In TDTR, this assumption may be inaccurate at early times (typically < 1 ps) when the non-equilibrium electron temperature rise in the metal film may be significant even for a moderate excitation fluence.

^{22,23}We assume that the linear response model holds for slower dynamics determining frequency components of the thermoreflectance response below 1 GHz. However, establishing the domain of validity of the linear response model in TDTR requires further investigation.

_{2}O

_{3}Single Crystal