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Abstract
Picosecond ultrasonics (PU), time-domain Brillouin scattering (TDBS), and time-domain thermo-reflectance (TDTR) are all in-situ, non-destructive, and non-contact experimental techniques based on the ultrafast laser pump-probe method, which can generate and detect coherent acoustic phonons (CAP) and thermal transport in thin metal film-glass substrate system. However, these techniques are generally considered different experimental methods to characterize the thermal or mechanical properties of metal nano-objects or transparent materials. Here we present a comprehensive characterization of the generation, propagation, and attenuation of high-frequency CAP and cross-plane thermal transport in the thin Cr film-glass substrate system by PU, TDBS, and TDTR. To investigate the key factors of characterizations, two kinds of thin Cr film-glass substrate systems were measured on the film side and substrate side. The measured thermal and mechanical properties show that boundary conditions and film thickness have significantly affected the characterization.
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1. Introduction
The ultrafast laser pump-probe method proposed by Eesley et al. [1,2] and Thomsen et al. [3,4] for detecting temperature changes and coherent acoustic phonons (CAP) in opaque thin films as the transducer, typically metallic films [1,2], relied on the coupling of the refractive index to temperature and strain. After its development for more than 30 years, according to the type of physical process studied, the method now can be classified as picosecond ultrasonics (PU), time-domain Brillouin scattering (TDBS), and time-domain thermo-reflectance (TDTR). PU [5] takes advantage of the ultrashort pulse width of the laser to generate and detect high-frequency CAP in the transducer on time durations of tens to one hundred picoseconds. TDBS [6] records the scattering of the probe laser beam by the CAP propagating in transparent materials with a time duration of hundreds to one thousand picoseconds. TDTR [7] monitors the evolution of the surface temperature of the transducer and studies the one-dimensional heat flow in the sample layered structure on nanosecond time scales.
Previous researchers have used PU, TDBS, and TDTR separately to study specific thermal or mechanical properties of metal nano-objects or transparent materials. PU has shown the advantage of using ultrathin metal films such as Ni [8] and Ru [9] to generate terahertz CAP. TDBS utilizes the propagation and attenuation of CAP to realize depth-profiling [10,11], the measurements of acoustic attenuation [12,13], and elastic properties [14] of advanced transparent materials. The thermal conductivity measurement by applying TDTR has also been well developed [7,15–18]. However, as far as we know, the comprehensive study of thin metal film-glass substrate system by ultrafast laser pump-probe method has been only reported by Ma Weigang et al. [19,20].
In this work, we comprehensively characterize the generation, propagation, and attenuation of high-frequency CAP and thermal transport in the thin Cr film-glass substrate system by PU, TDBS, and TDTR. According to the previous studies about Cr by O.B.Wright et al. [21,22], we choose the Cr film as the transducer because of its excellent adhesion and the large electron-phonon coupling constant, resulting in the potential for perfect contact with the substrate and the generation of high-frequency CAP. Two kinds of thin Cr film-glass substrate systems were measured on the film side and substrate side. The experimental results by femtosecond laser pump-probe method show that (1) the frequency of CAP is determined by the characteristic distance in Cr, which is limited by the optical absorption depth and film thickness, (2) the ability of TDBS to distinguish the kinds of the glass substrates and their spatial homogeneity has been verified, (3) both the longitudinal sound velocity measured by PU and thermal conductivity determined by TDTR are affected significantly by film thickness and boundary conditions.
2. Experimental setup
Two Cr films with thicknesses of 10 and 100 nm were deposited onto two kinds of glass substrates by electron beam evaporation. During the deposition, the chamber pressure was maintained at 5 × 10−6 Torr. For simplicity, we refer to the 10 nm Cr film-fused quartz substrate system as the 10 nm sample and the 100 nm Cr film-borosilicate glass substrate system as the 100 nm sample. The Cr film thickness of the 10 nm sample and 100 nm sample was 11.25 and 98.95 nm, measured by a scanning electron microscope. The refractive index and extinction coefficient of Cr and glass substrates were measured by standard ellipsometer (see Table 1).
Table 1. Optical Constants of Materials at Different Wavelengths
In Fig. 1, we show a schematic diagram of the time-resolved pump-probe experimental setup, which was developed by ourselves. The laser system is an 80 MHz repetition rate integrated broadband Ti: Sapphire ultrafast oscillator (Vitara-T, Coherent), which generates lasers with a central wavelength of 800 nm, pulse energy of 9.375 nJ, a pulse length of 20 fs, and spectral width ∼ 30 nm [full width at half maximum (FWHM)].
The laser output is split into two beams by a polarizing beam-splitter (PBS). The pump beam transmitted the PBS is focused onto a BIBO crystal to change its wavelength to 400 nm by second-harmonic generation. Then the pump beam is modulated by an Electro-Optic-Modulator (Conoptics, Model 25A + 370-01) at a reference frequency of 1 MHz. The probe beam reflected by the PBS enters a 3× beam expander to reduce its divergence on long-length transmission at first, and then a variable optical delay path controlled by a translation stage (Thorlabs, LTS300/M). Both the pump beam and probe beam are focused at vertical incidence to the sample co-axially through a cold mirror and a 10× objective lens. The diameter of the pump beam and probe beam are 58 µm and 38 µm [1/e2 diameter for intensity]. Then the probe beam is reflected into a photodiode detector (Thorlabs, PDA36A2) through a long-pass filter, which blocks out the pump beam at 400 nm.
Lock-in detection by a lock-in amplifier (Stanford Research 844) at the reference frequency allows relative reflectivity change $\varDelta R/R\sim {10^{ - 7}}$ to be resolved. The relative reflectivity change is sensitive to the strain and temperature in the illuminated region by the probe beam, so the reflected probe beam can detect the strain caused by the CAP and temperature change [1–4].
3. Theory
When an ultrashort laser pulse is incident on a metal film, the energy of laser photons is at first given to conduction electrons that lie within the optical absorption depth (about 13.21 nm at the wavelength of pump beam 400 nm in Cr, calculated by $\xi = \lambda /4\pi \kappa $, where $\kappa \; $ is the extinction coefficient), leaving the lattice temperature almost unchanged. Then hot electrons excited by laser photons will diffuse into the Cr film interior while transferring their energy to lattices. Once the energy is transferred to lattices, it creates a displacement field and associated strain pulses over the characteristic distance. The characteristic distance over which hot electrons transfer their energy to lattices can be calculated by $L = \sqrt {{\kappa _e}/g} $, where ${\kappa _e}$ is the electronic thermal conductivity, g is the strength of the electron-phonon coupling.
For these weak electron-phonon coupling metals such as Au [23] and Cu [24], the characteristic distance is much larger than the optical absorption depth and resulting in the decreased frequency of the CAP. In the case of strong electron-phonon coupling metals, like Ni [8,22] and Cr [21,22], the characteristic distance becomes much shorter, typically in the range of the optical absorption depth itself. For example, the characteristic distance is about 15 nm in Cr at vertical incidence under pump laser pulses at 415 nm [22]. In an ultrathin film like the one studied here, the characteristic distance is limited by the film thickness and relatively short. The central frequency of the resulting CAP will be relatively high.
After a time delay, the probe beam ${A_0}\; $ enters the metal film and is partially reflected by the metal film. The reflected beam ${A_1}$ records the relative reflectivity change because of the CAP and temperature change in the metal film. If the film thickness is smaller than the optical absorption depth (about 21.12 nm at 800 nm), the probe beam ${A_0}$ will penetrate the metal film, reflected at the moving interface introduced by the CAP, and penetrate the metal film again, as the beam ${A_2}$ in Fig. 2(a).
Fig. 2. Schematic diagram of the incident and reflected probe beams in the thin metal film-glass substrate system measured on (a) the film side and (b) the substrate side. The laser beams were plotted at an angle for clarity but vertical in reality.
3.1 Picosecond ultrasonics
By solving Maxwell’s equations inside the metal film, Thomsen et al. [4] obtained the first order reflectivity change in reflected beam ${A_1}$ in Fig. 2 related to the strain as
whereIn Eqs. (1) to (4), the “sensitivity function” $f(z )$ determines how the strain at different depths below the film surface contributes to the change in the reflectivity, $\lambda $ is the probe wavelength, n and $\kappa $ are the real and imaginary components of the complex refractive index of film, ${\eta _{zz}}$ is the only nonzero component of the elastic strain tensor, $\xi $ is the optical absorption depth, and $\varphi $ is the arbitrary phase in the range of 0 to π/2 [4].
For the 10 nm sample, the thickness of Cr film is comparable to or even less than the optical absorption depth of the pump beam, leading to a homogeneous optical excitation of the whole film. Then the isotropic stress is formed in the entire film, and almost simultaneously generates two strain pulses, one starting at the surface (air-film interface) and the other starting at the film-substrate interface. As a result, the reverberations of two counter-propagating pulses inside the film will form one-dimensional breathing mode vibrations. The resulting frequency of breathing mode vibrations is given by [9]
where h is the film thickness, v is the longitudinal sound velocity.For the 100 nm sample, the film thickness is larger than the characteristic distance. Therefore, two counter-propagating strain pulses are generated over the characteristic distance. The negative-going pulse is reflected at the free surface and follows the positive-going pulse. The sum of these two pulses is regarded as the CAP we discussed previously, which propagates forth and back between the surface and film-substrate interface. As a result, several equally spaced echoes can be detected by the reflected beam ${A_1}$ in Fig. 2(a) only when the CAP enters its optical absorption depth. The intervals time between the successive echoes should be
where h is the film thickness, and v is the longitudinal sound velocity.3.2 Time-domain Brillouin scattering
For TDBS, when the probe beam enters the transparent substrate and reaches the moving interface introduced by the CAP, as shown in the diagram in Fig. 2(a) and (b), the ${A_1}$ beam reflected by the Cr film and ${A_2}$ beam scattered by the moving interface interfere with each other at the photodetector, forming Brillouin scattering oscillations in the transient reflectivity, which can be described by the function
where ${B_0}$ is the initial amplitude, $\mathrm{\Gamma \;\ }$ is the acoustic attenuation rate, and $\gamma $ is the arbitrary phase [25].The Brillouin frequency at vertical incidence is determined by [6]
where ${n_s}$ is the refractive index of the substrate at probe wavelength ${\lambda _{probe}},\; \; {v_s}$ is the longitudinal sound velocity of the substrate.3.3 Time-domain thermoreflectance
For TDTR, the temperature of the probed region in Cr film is measured by the dR/dT, or “thermoreflectance”, i.e., the change in the reflectivity with temperature [7]. This temperature-dependent change in reflectivity decays quickly on a picosecond time scale as heat diffuses through the Cr film and then decays slowly as heat moves across the film-substrate interface and diffuses into the substrate.
Considering the phonons or lattices interact with the electrons less efficiently than the electrons interact with each other, the heat transfer in the Cr film can be simplified to a first non-equilibrium electronic diffusion and a subsequent equilibrium phonon diffusion. To simplify the first non-equilibrium electronic diffusion during ∼1 ps, we assumed a part of the Cr film as an isothermal layer. The layer thickness is determined by the characteristic distance over which hot electrons transfer their energy to lattices. After a few picoseconds, the isothermal layer is thermalized through the electron-phonon interaction to a new temperature ∼10 K above the initial state. The subsequent equilibrium phonon diffusion in Cr film produces a change in reflectivity that is linearly proportional to the temperature for small deviations [1].
For the 10 nm sample, the characteristic distance is larger than the Cr film thickness. We assume the entire Cr film is an isothermal layer. This is a case where local equilibrium phonon diffusion cannot be established, which is different from the typical situation of TDTR. Therefore, our work will not discuss the fit to the thermal response of the 10 nm sample.
For the 100 nm sample, the characteristic distance is smaller than the Cr film thickness, and we divide the Cr film into an isothermal layer and a non-isothermal layer. Then the thermal transport in the 100 nm sample can be modeled as the heat flow in the four-layered structure composed of an isothermal layer, a non-isothermal layer, an interface layer, and the glass substrate. By multi-dimensional least square fitting, see Appendix for more detail, we can obtain the best fit to the thermal response of 100 nm sample and determine the unknown parameters in the model.
4. Results and discussion
In this section, measurements by PU, TDBS, and TDTR for two samples have been made. For each sample, measurements were made on both the film side and substrate side as shown in Fig. 2. PU records the behaviors of CAP trapped in Cr films, which are the breathing mode vibrations in the 10 nm Cr film and the periodic echoes in the 100 nm Cr film. TDBS records the propagation of CAP in the glass substrate. TDTR monitors the thermal transport in the 100 nm Cr film-glass substrate system.
4.1 Breathing mode vibrations in 10 nm Cr film
Figure 3(a) and (b) show the relative reflectivity change $\varDelta R/R$, as a function of the time delay between the pump beam and probe beam, in the time range of -5 to 35 ps with a resolution of 0.125 ps. The time zero of pump-probe delay is the point at which the curve starts to increase. After a strong rise, the curve reaches its peak at t = 5 ps. Then the measured curves are truncated at t = 7.25 and 6.75 ps, indicated by the vertical black dashed lines. Truncation is required to eliminate the peak affected by hot electrons essentially. Afterward, we observe a slow thermal decay with several sinusoidal oscillations. To better show the oscillations, the slow thermal decay was removed by subtracting an exponentially decaying function from fitting the experimental data. The single-frequency oscillations shown in Fig. 3(c) and 3(d) should be the breathing mode vibrations, which is the behavior of CAP reverberated in the 10 nm Cr film.
Fig. 3. The relative reflectivity change of the 10 nm sample measured on the (a) film side and (b) substrate side. The breathing mode vibrations of the (c) film side and (d) substrate side.
Figure 4 shows the fast Fourier transform (FFT) spectra of the breathing mode vibrations in Fig. 3(c) and (d). The inset of Fig. 4 shows the central frequencies of breathing mode vibrations are ${f_B} = \textrm{}$281.2 GHz on the film side and ${f_B} = \textrm{}$275.2 GHz on the substrate side. According to the film thickness and Eq. (1), the calculated values of longitudinal sound velocity of the 10 nm Cr film are 6.33 km/s on the film side and 6.19 km/s on the substrate side. This slight differences in the central frequency and sound velocity may come from the illuminated region's different boundary conditions or thickness uncertainty. In addition, the vibrations of two breathing modes in Fig. 3(c) and (d) show similar attenuation behaviors, regardless of being measured on the film or substrate side.
4.2 Periodic echoes in 100 nm Cr film
In Fig. 5(a), we plot the relative reflectivity change measured on the film side of the 100 nm sample, in the time range of -50 to 200 ps with a resolution of 0.25 ps. As in the previous subsection, we truncate the curve at t = 15 ps to eliminate the sharp peak affected by hot electrons. After similarly removing the slow thermal decay, several pulses considered as successive echoes are shown in Fig. 5(b). The detected echoes are not the CAP itself but a measure of its impact on the refractive index of the Cr film within optical absorption depth.
Fig. 5. (a) The relative reflectivity change of the 100 nm sample measured on the film side. (b) The successive echoes between 15 and 165 ps. (c) The extracted 1st, 2nd, and 3rd echoes. (d) The FFT spectra of these three echoes.
Different from the breathing mode vibrations in the 10 nm Cr film, these successive echoes are isolated from each other rather than forming a continuous vibration. Using the average time interval between 1st, 2nd, and 3rd echoes and Eq. (6), the longitudinal sound velocity of the 100 nm Cr film is 6.60 km/s, which is in good agreement with the literature [22]. However, the sound velocity of the 10 nm Cr film measured on the film side is 6.33 km/s, which is smaller than the 6.60 km/s. This should attribute to the thin-film softening effect. A similar phenomenon that the sound velocity in thin metal films decreases with decreasing thickness has been reported for Ni [8], and Ru [9].
In order to further analyze the variation of each echo, the extracted 1st, 2nd, and 3rd echoes are plotted in Fig. 5(c). The time t = 0 ps corresponds to the peak of each echo. The amplitude decrease of the 1st, 2nd, and 3rd echoes is partially due to the ultrasonic attenuation of the CAP as it propagates in the film, and partly attributed to the transmission of CAP to the substrate. The amplitude reflection coefficient for the CAP reflected from the substrate is given by [4]
where $\rho$ is the density, v is the longitudinal sound velocity, and the subscripts $f$ and $s$ refer to the film and substrate.Using values in Table. 1 and Eq. (9), the calculated value for r is -0.60 for 100 nm sample. Figure 5(d) shows the FFT spectra of the three echoes in Fig. 5(c). For the 1st, 2nd, and 3rd echoes, their spectra show frequency components up to 200 GHz with central frequencies of 69.4, 56.2, and 49.6 GHz, respectively. The ratio of these spectra for the 100 nm Cr film indicates an ultrasonic attenuation rising with frequency. In Fig. 5(d), the frequency- dependent attenuation will slightly distort and broaden the CAP shapes by the stronger attenuation at higher frequencies.
4.3 Propagation of CAP in the glass substrate
In Fig. 6(a) to (c), we plot the time-dependent relative reflectivity change from -100 to 1400 ps with a resolution of 4 ps. In Fig. 6(a)-(c), the curves consist of a peak affected by hot electrons and slow thermal decay with single-frequency oscillations. Similarly, we truncate the curvet at t = 100 ps to eliminate the peak affected by hot electrons. Then we employ a central difference function with an approximate period of the oscillation to remove the slow thermal decay [12]. After we get the best fit for the period of the oscillations, the single-frequency damping oscillating curves shown in the insets of Fig. 6(a) to (c) are the Brillouin oscillations, which demonstrate the propagation of CAP in the glass substrates.
Fig. 6. The relative reflectivity change of the 10 nm sample measured on the (a) film side and (b) substrate side, and (c) the 100 nm sample measured on the substrate side. (d) The FFT spectra of the Brillouin oscillations in the insets of (a) to (c).
The Brillouin oscillations are damped by the energy dissipation into the medium, and the ultrasonics attenuation coefficient $\alpha $ can be evaluated by [26]
where the acoustic attenuation rate $\mathrm{\Gamma \;\ }$ is the FWHM of the Brillouin peak.The spectra in Fig. 6(d) give ${f_{BS}} = $ 22.20 GHz with a damping rate of $\mathrm{\Gamma \;\ =\ }$1.52 GHz for the fused quartz substrate and ${f_{BS}} = $ 20.87 GHz with a damping rate of $\mathrm{\Gamma \;\ =\ }$1.47 GHz for the borosilicate glass substrate. Using the refractive index in Table 1 and Eq. (10), the calculated values of longitudinal sound velocity and ultrasonic attenuation coefficients are $v = \; $6.14 km/s and $\alpha = $0.77 µm-1 for the fused quartz substrate of the 10 nm sample, and $v = \; $5.61 km/s and $\alpha = $0.82 µm-1 for the borosilicate glass substrate of the 100 nm sample. The sound velocity and ultrasonic attenuation coefficients are close to the known values of glasses determined in previous studies [12,26].
The frequencies and longitudinal sound velocities of the CAP propagating in the Cr film and glass substrate, summarized in Table 2, were obtained by averaging the results from multiple measurements on each sample.
Table 2. Mechanical properties of 10 nm and 100 nm samples measured by PU and TDBS
4.4 Thermal transport in 100 nm Cr film-glass substrate system
We model the thermal transport in the 100 nm sample as the heat flow in the four-layered structure. As shown in Fig. 7(a), the 100 nm sample measured on the film side is thought to be a four-layered structure [7,19]: a 15-nm-thick isothermal layer whose thickness is determined by the characteristic distance in Cr [22], an 83-nm-thick non-isothermal layer, a 1-nm-thick interface layer, and a 5-mm-thick glass substrate. For the 100 nm sample measured on the substrate side in Fig. 7(b), the location of the isothermal layer and non-isothermal layer is exchanged.
Fig. 7. Schematic diagram of the four-layered structure used in the thermal model of the 100 nm sample measured on the (a) film side and (b) substrate side.
In Fig. 8(a) and (b), we plot the experimental results of TDTR in the time range of -100 to 3100 ps with a resolution of 4 ps. Experimental data between 100 and 3100 ps are selected to fit the thermal response properly because the equilibrium thermal diffusion is dominant in this time range. The insets of Fig. 8(a) and (b) present the detailed curves between 200 and 1200 ps. As discussed in the above section, the continuous oscillations in the inset of Fig. 8(b) correspond to the Brillouin oscillations in Fig. 6(c).
Fig. 8. The phase signal and best-fit curves of the 100 nm sample measured on the (a) film side and (b) substrate side. Their insets show the detailed curves between the 100 and 1500 ps. The sensitivity of the phase signal to the unknown parameters in the thermal model of the 100 nm sample measured on the (c) film side and (d) substrate side.
In order to determine whether an unknown thermal property can be measured with good accuracy, we define the sensitivity of the measured phase signal to the unknown parameter $\beta $ as that of Gundrum et al. [27]
In Fig. 8(c) and (d), we show the sensitivity of three interested unknown parameters, including the thermal conductivity of the Cr film and the glass substrate, and the interface thermal conductance. It can be seen that the sensitivity of interface thermal conductance is always close to zero with little change. Therefore, the interface thermal conductance can not be determined with good accuracy.
For the thermal conductivity of the Cr film, its sensitivity is close to 0.1 at first, and gradually decrease to zero within 500 ps. In contrast, for the thermal conductivity of the substrate, the value of its negative sensitivity is greater than 0.1 in the entire time range. The tendency of these two sensitivities shows that the thermal conductivity of the Cr film has a significant effect on the measured phase signal within 500 ps, whereas the thermal conductivity of the glass substrate has a greater opposite effect on the phase signals in the entire time range. As a result, the two unknown parameters of the thermal model are set as the cross-plane thermal conductivity of the Cr film and the glass substrate. By two-dimensional least square fitting, see Appendix for more detail, we obtained the best-fit curves in Fig. 8(a) and (b) and determined the unknown parameters in Table 3.
Table 3. Thermal properties of 100 nm sample measured by TDTR
As shown in Table 3, the thermal conductivity of the 100 nm Cr film is 31.33 Wm-1K-1for the film side measurement, whereas the value is 27.61 Wm-1K-1 for the substrate side measurement. The difference in values may come from the different boundary conditions of the film side and substrate side. Considering a typical bulk Cr value of 94 Wm-1K-1[22], the obtained thermal conductivity is significantly decreased. The decreased thermal conductivity presents a significant size effect, which is attributed to the restriction on the heat flow via hot electrons at grain boundaries. The reduction of film thickness results in a decrease in grain size and characteristic distance. The determined thermal conductivity of the borosilicate glass substrate is in the range of 1.5-1.6 Wm-1K-1.
5. Conclusion
In this paper, we demonstrated the ability of the femtosecond laser pump-probe method to comprehensively characterize the generation, propagation, and attenuation of high-frequency CAP and thermal transport in the thin metal film-glass substrate system. Two kinds of thin Cr film-glass substrate systems have been measured on the film side and substrate side by PU, TDBS, and TDTR. For PU, the behaviors of CAP can be classified as the high-frequency breathing mode vibrations of the 10 nm Cr film and periodic echoes of the 100 nm Cr film. The measured longitudinal sound velocity of Cr films shows a slight trend of thin-film softening effect. For TDBS, the Brillouin oscillations signal records the propagation and attenuation of CAP in glass substrates, which can be used to distinguish the kinds of glasses and their spatial homogeneity. For TDTR, the model of heat flow in four-layered structures matches well with the thermal response. The determined thermal conductivity of the 100 nm Cr film is affected significantly by size effect and boundary conditions.
Appendix: thermal model and fit procedure
In order to fit the thermal response of the 100 nm sample, it is important to establish a model of heat flow in the four-layered structure.
At the reference frequency ${\omega _0}$ of pump beam, the output of lock-in amplifier for a reference wave ${e^{i{\omega _0}t}}$ is given by
where $Z({{\omega_0}} )$ is the “transfer function”, which contain the thermal response as well as the physical properties of pump and probe beams, and can be given in terms of the sample frequency response, $H({{\omega_0}} )$ [7,18]The output of the lock-in amplifier consists of four signals, including the in-phase signal $\textrm{X}$, the out-of-phase signal $\textrm{Y}$, the amplitude signal $A = \sqrt {{X^2} + {Y^2}} $ and the phase signal $\phi = {\tan ^{ - 1}}(Y/X)$ . The in-phase signal X and out-of-phase signal Y are also given by the real and imaginary parts of Eq. (13)
All the four signals can be used for fitting. However, the phase signal $\phi $ has the advantage over other three signals as it does not require normalization and has less noise, so it is mainly used in the TDTR experiments to fit the thermal model [7,17,28].
One-dimensional thermal conduction in a single layer can be expressed as
It is noteworthy that thermal interface conductance should be considered. Obtained from Eq. (16) by taking the limit as the heat capacity of the interface goes to zero and considering G = ${\lambda _z}$/d, the thermal interface conductance can be expressed as
For the four-layered structure in Fig. 7(a), the relationship between heat flow and matrixes of layers can be written as:
The heat flow term at the surface side is given by the Hankel transform of a Gaussian spot with power ${Q_{\textrm{pump}}}$ and 1/e2 radius of the pump beam. The surface temperature in Eq. (20) then becomes
By applying the inverse Hankel transform and then weighing the result by the probe intensity distribution, the sample frequency response $H(\omega )$ in real space can be expressed as [18]
So far, we established a correlation between the phase signal and transfer function in Eqs. (12) to (22). By adjusting the two unknown parameters in the thermal model, we minimize the square error between the phase signal and fit data. As a result, we obtained the best-fit curve in Fig. 8(a) and determined unknown parameters in Table 3.
For the four-layered structure in Fig. 7(b), the heat would flow up to the substrate and down to the isothermal layer. To obtain the temperature on the upper side of the isothermal layer, the heat flow is divided into two parts
where ${f_{1t}}$ and ${f_{2t}}$ are the upward heat flow and downward heat flow.For the upward heat flow, there are two layers, including the interface layer and substrate. Thus
Taking adiabatic boundary conditions into account
There are two layers for downward heat flow: the isothermal layer and the non-isothermal layer. Thus
Taking adiabatic boundary conditions into account as well,
Since both ${\theta _{1t}}$ and ${\theta _{2t}}$ represent the temperature on the upper side of the isothermal layer, they can be expressed as
where ${\theta _t}$ is the temperature of the upper side of the isothermal layer.According to the above Eqs. (24) to (28), the temperature on the upper side of the isothermal layer can be obtained as
Replacing Eq. (20) in Eq. (22) with Eq. (29), we can fit the heat flow in the four-layered structure in Fig. 7(b). As shown in Fig. 8(b), the fit curve and the phase signal are in good agreement.
Funding
National Natural Science Foundation of China (12041201, 12172251).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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