Usable Analytical Expressions for Temperature Distribution Induced by Ultrafast Laser Pulses in Dielectric Solids

This paper focuses on the critical role of temperature in ultrafast direct laser writing processes, where temperature changes can trigger or exclusively drive certain transformations, such as phase transitions. It is important to consider both the temporal dynamics and spatial temperature distribution for the effective control of material modifications. We present analytical expressions for temperature variations induced by multi-pulse absorption, applicable to pulse durations significantly shorter than nanoseconds within a spherical energy source. The objective is to provide easy-to-use expressions to facilitate engineering tasks. Specifically, the expressions are shown to depend on just two parameters: the initial temperature at the center denoted as T₀₀ and a factor R_τ representing the ratio of the pulse period τ_p to the diffusion time τ_d. We show that temperature, oscillating between T_max and T_min, reaches a steady state and we calculate the least number of pulses required to reach the steady state. The paper defines the occurrence of heat accumulation precisely and elucidates that a temperature increase does not accompany systematically heat accumulation but depends on a set of laser parameters. It also highlights the temporal differences in temperature at the focus compared to areas outside the focus. Furthermore, the study suggests circumstances under which averaging the temperature over the pulse period can provide an even simpler approach. This work is instrumental in comprehending the diverse temperature effects observed in various experiments and in preparing for experimental setup. It also aids in determining whether temperature plays a role in the processes of direct laser writing. Toward the end of the paper, several application examples are provided.