Glass-forming materials are often used to create optical elements and components for optical integrated circuits. In the last two decades, the technology of femtosecond laser microfabrication of various optical structures inside glass-forming materials has been intensively developed. Due to the extremely high peak intensities at the focus of femtosecond laser pulses, nonlinear processes such as multiphoton absorption can lead to significant energy absorption even within transparent materials such as glass. Thus, femtosecond laser-induced microstructuring of glass-forming materials opens up wide possibilities for various applications. Laser radiation can cause irreversible local phenomena at the beam focus, such as phase transitions [1], the formation of bubbles or microfluidic channels [2,3,4,5], changes in chemical composition or refractive index [6,7], for example, diffusion and aggregation of silver ions [6] or photo-oxidation [7]. Spatially selective laser-induced crystallization or structure changes of glasses allows direct recording of channel waveguides [1,8,9,10,11,12,13,14]. In fact, laser-written waveguides (with improved mode structure of guided light) can be fabricated by local laser heating inside a glass matrix [8,9]. Thus, optical integrated circuits can be fabricated using a laser beam that induces local structural changes in a glass [15,16,17,18]. Photopolymerization and photodamage by highly focused laser pulses can be used in microchemistry and stereolithography [6,19]. Moreover, femtosecond laser structuring of glass-forming materials can be used for optical long-term storage of information [3,20,21,22,23,24,25,26,27,28,29,30], photonics [31], fabrication of phase gratings [32], nano-gratings [33], and quantum dots that can be used in various devices [34,35,36]. Notably, optical storage based on glass-forming materials has the potential to replace magnetic storage in the quest to provide high-speed, high-capacity, low-power, low-cost, highly secure, and long-term data storage [25,26,27,28,29,30]. However, for the development of these technologies, theoretical models of thermal processes occurring during spatially selective laser-induced structuring of glass-forming materials are required. For example, understanding the dynamics of the laser crystallization process of amorphous Ge films is important for transistor technology, photovoltaic devices, particle detectors, and photodetectors [37]. In fact, the laser crystallization method makes it possible to control the local temperature inside the material and avoid random nucleation [37]. The ability to form and control the dynamics of changes in the temperature distribution $T(t,r)$ inside the material is very important for the opportunity to control the morphology of the laser-written structure inside the glass matrix [22,33,38]. Thus, a deep understanding of the thermal processes inside glass-forming materials under fast laser thermal perturbations is required. In this paper, we focus on local temperature changes, especially the local cooling rate $-\mathcal{R}(t,r)$ in the focal region during laser-writing processes in a glass matrix (where $\mathcal{R}\left(t,r\right)$ is the rate of temperature change).

Femtosecond laser-induced microstructuring is a promising tool because the structures induced by femtosecond laser pulses can be even smaller than the optical diffraction limit of the beam focusing, since multiphoton absorption is most effective in the central part of the beam focus. Typically, in laser-induced microstructuring, the laser pulse duration ${\tau }_{laser}$ is of the order of 100 fs and the radiation wavelength is about 1 μm. When a transparent material is irradiated with a powerful femtosecond laser pulse focused into a micro-sized focusing zone, multiphoton absorption of the laser radiation energy by electrons of the irradiated material occurs. The energy of the electrons in the laser focus increases sharply during ${\tau }_{laser}$ due to electron-photon interaction. The electrons then transfer their energy to the lattice through electron-phonon interaction. The thermalization process takes approximately tens or hundreds of picoseconds [6,10,39,40]. Consequently, the duration of heating pulses ${\tau }_p$ that heat the material can be on the order of tens to hundreds of picoseconds. Thus, the material can be locally heated above the glass transition temperature $T_g$ and melting temperature $T_m$. Heat then spreads through the material and the hot focal zone is rapidly cooled below $T_g$.

In this article, we will consider laser pulses of moderate energy $E_p$ (in the range 10 – 100 nJ), sufficient to locally heat the material well above the glass transition temperature $T_g$, but not sufficient to destroy the material with the formation of voids. Thus, the intensity of the laser pulses considered in this article is below the material damage threshold, which, for example, for silica glass is about 10¹⁶ W/cm² [41]. We focus on micrometer-scale local thermal perturbations caused by laser pulses near or below the threshold ionization intensity of a material. The threshold ionization intensity of most transparent solids ranges from 10¹³ to 10¹⁴ W/cm² for light with a wavelength of about 1 μm [42,43]. For example, the threshold intensity in silica glass and similar glass-forming materials is about 10¹³ W/cm² for light pulses with a wavelength of about 1 μm and a duration of 100 fs [42].

Thus, we will consider thermal processes in dielectric glasses below the threshold intensity of ionization of the material and in the time interval after electron-phonon thermalization, when there is no electron-hole plasma. Thus, we are interested in changes in lattice temperature on a time scale outside the time interval when a multi-temperature model is usually considered [12,13,14]. In fact, we are interested in changes in lattice temperature that affect the structure of the material at moderate laser pulse energies and time $t$ exceeding the electron-phonon thermalization time.

The initial structure of glass-forming materials is not completely restored after the rapid heating-cooling cycle, which leads to a change in the local properties of the material in rapidly cooled areas. The local structure, specific volume, density, Rayleigh scattering loss and refractive index of glass-forming materials significantly depend on the local cooling rate [19,44,45], see Figure 1.

Thus, we focus on the dynamics of the temperature distribution $T(t,r)$ and study the local cooling rate $-\mathcal{R}(t,r)$ near the focal region during laser-induced microstructuring. Measurements of the temperature distribution at the focus of the laser beam can be carried out using micro-Raman spectroscopy [46,47,48]. However, the time resolution of Raman spectroscopy (with a measuring pulse duration of about 10 ns and a repetition rate of 1 kHz) is not sufficient to detect ultrafast changes in the local temperature $T(t,r)$. We found that the local cooling rate $-\mathcal{R}(t,r)$ after the end of the heating pulse can reach more than 10¹¹ K/s in a thin layer around the focus of the laser beam. In fact, to resolve such ultrafast temperature changes at the periphery of the hot focal zone requires a temporal and spatial resolution of at least about 0.1 ns and 10 nm, respectively, since the cooling rate $-\mathcal{R}(t,r)$ reaches about 600 GK/s in a narrow layer around the hot zone at the laser focus (see below). Thus, the maximum change in the glass structure occurs at the periphery of the focal region (where the material undergoes ultrafast quenching). In fact, laser microstructuring of glasses makes it possible to form micron-sized domains that are close to spherical or ring-shaped, with a modified glass structure at the periphery of the domains [19,34,36,40,46,49,50,51,52,53]. For example, local modification of silver-doped phosphate glasses with the laser pulse energies $E_p$ in the range of 10 – 100 nJ led to the formation of micron-sized ring-shaped domains due to the aggregation of silver nanoclusters at the periphery of the domains [49]. Laser pulse energy $E_p$ and pulse repetition rate play an important role in the microstructuring of glass-forming materials. In fact, laser pulses with a sufficient repetition rate can provide cumulative heating near the focus of the laser beam. In this article we will focus on the effect of single laser pulses (the effect of repetition rate will be discussed in a separate article).

The temperature change associated with the heating pulse Is of the order of ${∆T=E}_p/C_{loc}$, where $C_{loc}=\rho c_p{V}_0$ and $V_0$ are the heat capacity and volume of the heating zone; $\rho$ and $c_p$ are the density and specific heat capacity of the material. The thermal effect caused by elastic deformations is negligible compared to $∆T$, at least at moderate $E_p$, say in the range of 10 – 100 nJ for a focal region of about 1 μm radius. Indeed, the thermoelastic pressure in the hot zone is less than the maximum pressure response ${p_{max}=K}_B{\alpha }_V∆T$ to the temperature change $∆T$ at constant volume, where $K_B$ and ${\alpha }_V$ are the bulk modulus and the volumetric thermal expansion coefficient. However, the volume of the hot zone is not rigidly fixed by the material surrounding this zone. In fact, a pressure wave is created around the hot zone [54,55]. In this case, the thermoelastic pressure is even less than $p_{max}$. Thus, the elastic energy associated with the thermoelastic pressure in the hot zone of volume $V_0$ is less than $E_{TE}=\frac{1}{2}{K}_B{({\alpha }_V∆T)}^2{V}_0$. Therefore, the relative effect associated with thermoelastic deformations does not exceed $E_{TE}/E_p=\frac{1}{2}{K}_B{\left({\alpha }_V\right)}^2∆T/\rho c_p$. For example, for sodium-lime-silicate glasses with the composition Na₂Oꞏ2CaOꞏ3SiO₂, $E_{TE}/E_p$ is about 1% at $∆T=$ 2000 K, $K_B=$ 55 Gpa, $\rho =$ 2.8 g/cm³, $c_p=$ 1.14 J/gK, ${\alpha }_V=3\alpha$, and linear thermal expansion $\alpha =$ 7.7ꞏ10^-6 1/K [56,57,58,59,60]. This ratio is even lower for borosilicate glasses with a low coefficient of thermal expansion [61]. For example, for borosilicate glasses such as Pyrex, the $E_{TE}/E_p$ ratio is about 0.1% at $∆T=$ 2000 K, $K_B=$ 33 Gpa, $\rho =$ 2.23 g/cm³, $c_p=$ 1.1 J/gK, and $\alpha =$ 3.3ꞏ10^-6 1/K [61,62,63,64]. However, $E_{TE}$ becomes comparable with $E_p$ at laser pulse energies approximately 2 orders of magnitude higher than those considered in this work. We are interested in local temperature changes in a micron-sized hot zone at ${t>\tau }_p$, when the pressure waves created around the hot zone are already at a distance of more than 1 – 2 μm from the hot zone. However, at the same time, the front of the temperature change extends only about ten nanometers from the periphery of the hot zone. Thus, we consider the change in local temperature after the local pressure in the hot zone has almost stabilized. For example, in silicate glasses at $t=$ 0.5 ns, the distance ${tV}_L$ is about 3 µm at the longitudinal speed of sound $V_L\approx$ 6ꞏ10³ m/s [59], and $\sqrt{{tD}_0}\approx$ 10 nm at the thermal diffusion coefficient $D_0$ of about 3ꞏ10^-7 m²/s [57,63,65].

It Is noteworthy that an Important property of glass-forming materials is the long-term relaxation of the specific heat capacity with rapid changes in temperature. Thus, the heat capacity of glass-forming materials must be considered as a function of time $c_{dyn}\left(t\right)$ [66,67,68]. In fact, if a glass-forming material is heated to a liquid state with heat capacity $c_L$ from a solid glassy state with heat capacity $c_S$, then the heat capacity of the material does not change immediately with the change in temperature, but slowly relaxes from $c_S$ to $c_L$, see below. Long-term relaxation of the dynamic heat capacity $c_{dyn}\left(t\right)$ of glasses is due to the slow exchange of energy between different degrees of freedom in glasses. Thus, the thermal response of glass-forming materials to a thermal perturbation at time $t$ depends on the temperature at earlier times. The effect of long-term relaxation of dynamic heat capacity significantly affects the dynamics of local temperature changes $T(t,r)$ [69,70]. In turn, the rate of change in local temperature $T(t,r)$ can significantly affect the micro-structuring of glass-forming materials [19,38,44,45,71]. An experimental study of the dynamics of the temperature distribution $T(t,r)$ inside a glass matrix under the action of fast laser thermal perturbations is very difficult. Therefore, it is necessary to develop theoretical models of thermal processes occurring during laser-induced structuring of glasses. In this paper, we will focus on modeling such thermal processes, taking into account the relaxation effect of the dynamic heat capacity $c_{dyn}\left(t\right)$. The dynamic behavior of glass-forming materials under fast local thermal perturbations can be described using the integro-differential heat equation with “memory” [69,70]. This equation has an analytical solution, at least in spherical, cylindrical and planar geometries [69,70,72]. This work aims to determine the dynamics of local temperature changes, especially the local rate of the temperature change $\mathcal{R}(t,r)$, associated with laser-induced thermal excitations during laser processing of glasses. An analytical method for determining $T(t,r)$ and $\mathcal{R}(t,r)$ has been developed. The knowledge obtained can be useful for various technologies related to laser-induced microstructuring.

In the first part of the article, the heat equation with dynamic heat capacity $c_{dyn}\left(t\right)$ is considered. An analysis of the dynamic heat capacity $c_{dyn}\left(t\right)$ of glass-forming materials has been conducted and an analytical solution of the heat equation with dynamic heat capacity for a spherically symmetric problem has been constructed. Then, the temperature distribution $T(t,r)$ is calculated for glasses with dynamic heat capacity $c_{dyn}\left(t\right)$ under local fast laser excitations. Examples with borosilicate and sodium-lime-silicate glasses are considered at different heating pulses and dynamic heat capacity parameters. Finally, the local distribution of the cooling rate $-\mathcal{R}(t,r)$ and its influence on the microstructuring processes of glasses is discussed.

Since the pioneering work of Birge and Nagel [66], the dynamic heat capacity $c_{dyn}\left(t\right)$ of glass-forming materials as a function of frequency (or time $t$) has been intensively studied. For such studies, heat capacity spectroscopy can be used [67,68]. The time dispersion of the dynamic heat capacity $c_{dyn}\left(t\right)$ can be described within the framework of linear response theory [73], similar to the time dispersion of the dielectric constant [74]. Then, heat transfer in the glass-forming material can be described by an integro-differential heat equation [69,70]. In the case of spherical geometry and zero initial conditions, this heat equation can be represented in the following form [69,70]: where $\Phi (t,r)$ is the volumetric heat flux density, $T^{\prime }\left(t,r\right)=\frac{\partial }{\partial t}T(t,r)$, $\Delta$ – Laplacian, $\lambda$ and $\rho$ are the thermal conductivity and density of the material. Eq.(1) has an analytical solution for homogeneous boundary conditions, see below.

For example, let us consider the two most common types of silicate glasses: sodium-lime-silicate glasses (Na₂O-CaO-SiO₂), as the most widely used of all industrial glasses, known for their low cost and availability, and borosilicate glasses (B₂O₃-SiO₂), which are widely used due to their low coefficient of thermal expansion and high resistance to chemical attack. Sodium-lime-silicate glass with the composition Na₂Oꞏ2CaOꞏ3SiO₂ and borosilicate glass of the Pyrex type were chosen as model systems due to the availability of the necessary parameters [57,58,61,62,63,64,65]. The thermal parameters of these glasses are collected in Table 1.

Now consider the spatial distribution of the rate of temperature change $\mathcal{R}\left(t,r\right)$. Due to thermal expansion, the size of the heating zone changes with temperature by about 10 nm or less. We neglect these changes with respect to $r_0$. Let us consider the rate of temperature change $\mathcal{R}\left(t,r\right)=\frac{\partial T\left(t,r\right)}{\partial t}$ due to thermal diffusion. This temperature change affects the physical properties of the material. We will focus on very fast (about 10⁹ K/s or more) local temperature changes that have the greatest impact on the microstructuring process, and will study the effect associated with the time dispersion of the dynamic heat capacity. It is worth noting that this effect is significant even well above the glass transition temperature $T_g$.

For example, the rate of temperature change $\mathcal{R}\left(t,r\right)$ for borosilicate glass at $r_0=$ 2 μm, ${\tau }_p=$ 300 ps, $E_p=$ 120 nJ and various ${\tau }_0$ is presented in Figures10 – 11. Note that the rate of temperature change $\mathcal{R}\left(t,r\right)$ is greatest near the periphery of the hot zone, see Figure10(a). The cooling rate $-\mathcal{R}(t,r)$ is about 600 GK/s at $r_1=$ 1.98 μm and $t_1=$ 0.36 ns, see Figure11(a). However, in the center of the heating zone, the cooling rate $-\mathcal{R}(t,0)$ does not exceed 0.6 GK/s, i.e. $-\mathcal{R}(t,0)$ is three orders of magnitude less than the cooling rate at the periphery, see Figure11. Therefore, the local structure of glass in the center of the hot zone should differ significantly from the structure at the periphery of the heating zone after laser modification. This conclusion is consistent with experiments [19,34,36,40,46,49,50,51,52,53]. However, the effect associated with dynamic heat capacity (at ${\tau }_0\ne$ 0) is significant both in the center and at the periphery of the heating zone. Indeed, the maximum cooling rate is approximately 2.5 times greater at ${\tau }_0\ne$ 0 than at ${\tau }_0=$ 0 both at the periphery and in the center of the heating zone, see Figure11.

Let us consider the cooling rate $-\mathcal{R}(t,r)$ in a thin shell in the region from 0.95$r_0$ to 0.99$r_0$. As we have established for borosilicate and sodium-lime-silicate glasses, ultra-fast cooling of the material occurs in this region at a rate of more than 10¹¹ K/s, see Figures10 – 13. It is noteworthy that at the beginning of the cooling process (during the first few nanoseconds), the rate of temperature change $\mathcal{R}\left(t,r\right)$ is the same for small and large relaxation times ${\tau }_0$, see Figures10(a) and 12(a). This means that at the beginning of the cooling process at the periphery of the hot zone, the shape of the distribution function $H\left({\tau }_0\right)$ does not affect the cooling rate $-\mathcal{R}(t,r)$. Therefore, in this case, even at very small relaxation times ${\tau }_0$, the influence of the time dispersion of the dynamic heat capacity is significant.

The cooling rate $-\mathcal{R}(t_1,r)$ reaches a maximum at $r_{max}$ for a given $t_1$, see Figures10 and 12. The rate $-\mathcal{R}\left(t,r_{max}\right)$ decreases over time, and the distance $r_{max}$, at which the rate $-\mathcal{R}\left(t,r_{max}\right)$ is maximum, shifts slightly towards the center of the heating zone. However, $-\mathcal{R}\left(t,r_{max}\right)$ still exceeds 2 – 3 GK/s at $t_1=$ 100 ns, see Figures10(a) and 12(b). The change in cooling rate $-\mathcal{R}\left(t,r_{max}\right)$ over time for sodium-lime-silicate glass at $r_0=$ 1 μm, ${\tau }_0=$ 10 μs and the same conditions as in Figure3 is presented in Table 3. The rate $\mathcal{R}(t_1,r)$ depending on the distance $r$ at the periphery of the hot zone for different $t_1$ is shown in Figure13.

Thus, since the local structure of the material strongly depends on the local cooling rate $-\mathcal{R}(t,r)$, the local structure of the material at the periphery of the hot zone changes significantly relative to the original glass matrix. This modification of glass practically stops as soon as the local region is cooled below the glass transition temperature $T_g$. Thus, the modification process occurs until the temperature $T(t,r_1)$ in the local region at $r=r_1$ drops below $T_g$. For example, consider the cooling curves calculated at the periphery of the hot zone for borosilicate ($T_g$= 900 K [63]) and sodium-lime-silicate glasses ($T_g$= 840 K [57]), see Figure10(b) and Figure14, respectively. It should be noted that the effect associated with the time dispersion of the dynamic heat capacity is significant well above the glass transition temperature $T_g$. Indeed, in the case of $r_0=$ 1 μm, the modification process occurs within approximately 80 ns at $r_1=$ 0.95 μm and ${\tau }_0\ge$ 1 μs, see Figure14(a). However, without taking into account the time dispersion of the dynamic heat capacity (at ${\tau }_0=$ 0), this process occurs in less than 20 ns, see Figure14(a). In the case of $r_0=$ 2 μm, the modification process occurs during a longer period of about 300 ns at $r_1=$ 1.98 μm and ${\tau }_0>$ 1 μs, see Figure10(b). However, without taking into account the time dispersion of the dynamic heat capacity (at ${\tau }_0=$ 0), this process occurs in less than 50 ns, see Figure10(b).

To summarize, we can conclude that the maximum cooling rate exists in the outer regions of the heating zone, in the shell of about $(0.97\pm 0.02)r_0$. The cooling rate reaches several hundred GK/s in the first nanosecond after the laser pulse. Then the cooling rate decreases. However, even 100 ns after the laser pulse, $-\mathcal{R}(t,r)$ in the outer shell of the hot zone still exceeds 1 GK/s. It is noteworthy that in the center of the heating zone the cooling rate is three orders of magnitude less than $-\mathcal{R}(t,r)$ at the periphery. Since regions quenched at different cooling rates have different physical properties, strong gradients of physical properties should exist in the material predominantly at a distance of about $0.9r_0$ from the center of the heating zone. This effect is significantly enhanced by the time dispersion of the dynamic heat capacity.

In conclusion, let us briefly discuss laser micro-processing in the cumulative heating mode. Typically, cumulative heating is achieved at a laser pulse repetition rate more than hundreds of kHz. Usually, this mode operates at frequencies from 200 kHz to 10 MHz [3]. Indeed, the temperature $T(t,0)$ in the center of the hot zone relaxes in a time of the order of $r_0^2/D_0$, which for the glasses under consideration is about 4 μs at $r_0=$ 1 μm, see inset in Figure14(a). Thus, in this case, the cumulative effect can be achieved at a frequency of about 250 kHz or more. Note that the cumulative effect leads to stable average heating of the material in the heating zone. Thus, the effect of laser pulse repetition can be approximately simulated by shifting upward the local temperature distribution $T(t,r)$ by a certain temperature difference $∆T$, which increases with increasing pulse repetition frequency. It is clear that with increasing $∆T$ the modification process will occur over a longer time interval, see Figure14, where $T(t,r_1)$ should be shifted upward by the amount $∆T$. Then $T(t,r_1)$ in the local region at $r=r_1$ will fall below $T_g$ at a cooling rate lower than at zero repetition frequency (at $∆T=$ 0). Consequently, local gradients in the physical properties of locally modified glass will be smoothed out due to cumulative heating. The effect of the repetition rate will be discussed in more detail in a separate article.

The local temperature distribution $T(t,r)$ and the local cooling rate $-\mathcal{R}(t,r)$ determine the local structure of the glass matrix during laser-induced microstructuring. Thus, knowledge of the dynamics of the local temperature distribution $T(t,r)$ is very important for applications associated with local laser heating, for example, for femtosecond laser microstructuring of glasses. However, as shown in this article, the dynamics of the local temperature distribution $T(t,r)$ significantly depends on the time dispersion of the dynamic heat capacity $c_{dyn}\left(t\right)$ of the glass matrix. This work proposes a method for analytical calculation of $T(t,r)$ and $\mathcal{R}(t,r)$ for glass-forming materials under local fast laser excitations. It is shown that the dynamics of the local temperature distribution $T(t,r)$ caused by a laser pulse can be described by the heat equation with dynamic heat capacity $c_{dyn}\left(t\right)$. This equation has an analytical solution to a spherically symmetric boundary value problem. Using this analytical solution, we obtained the temperature distribution $T(t,r)$ and local cooling rate $-\mathcal{R}(t,r)$ for the thermal parameters of borosilicate and sodium-lime-silicate glasses. In fact, this analytical solution depends on the following relaxation parameters of the dynamic heat capacity $c_{dyn}\left(t\right)$ of the material: relaxation time ${\tau }_0$ and the distribution function $H\left({\tau }_0\right)$, which can be obtained from broadband heat capacity spectroscopy. However, as shown in this article, the shape of the distribution function $H\left({\tau }_0\right)$ has little effect on $T(t,r)$ and $\mathcal{R}(t,r)$, since with increasing ${\tau }_0$ the influence of the time dispersion of the dynamic heat capacity $c_{dyn}\left(t\right)$ reaches saturation. It should be noted that the influence of the time dispersion of the dynamic heat capacity is most pronounced at the beginning of the process, when $t$ changes on a nanosecond time scale. We found that this effect is significant already at ${\tau }_0$ about 0.1 μs, increases with increasing ${\tau }_0$ and reaches saturation at ${\tau }_0$ above ${r_0}^2/D_0$. Thus, the influence of the time dispersion of dynamic heat capacity on the temperature distribution $T\left(t,r\right)$ can be calculated for a fixed, sufficiently large ${\tau }_0$ (in fact, for ${\tau }_0$ of the order of ten or several tens of microseconds, depending on the radius of the heating zone $r_0$). As expected, the results obtained are very similar for different $r_0$. However, as the size of the heating zone increases, the laser pulse energy $E_p$ must increase in proportion to the volume $V_0$ of the heating zone in order to obtain the same amplitude of the local thermal response.

It has been established that the temperature distribution $T(t,r)$ and local cooling rate $-\mathcal{R}(t,r)$ are not affected by the heating pulse duration ${\tau }_p$ and the shape of the heating pulse. It should be noted that the temperature perturbation $\delta T\left(t\right)=T\left(t,0\right)-T_{in}$ in the center of the heating zone relaxes not as $\delta T~t^{-3/2}$, in contrast to the often-used estimate based on the fundamental solution of the Fourier heat equation. In fact, the heating pulse cannot be considered as an instantaneous point source of heat. Thus, it is necessary to integrate the thermal response over the time interval $(0,{\tau }_p)$ and the volume of the heating zone, then the resulting correct solution $T_{FS}\left(t,r\right)$ completely coincides with the temperature distribution $T(t,r)$ found in this article for ${\tau }_0=$ 0. In addition, the solution found in this article can be used for materials with dynamic heat capacity $c_{dyn}\left(t\right)$ at ${\tau }_0\ne$ 0.