Asymptotic Effects of Incident Angle and Lateral Conduction in Electromagnetic Skin Heating

TL;DR

This study develops a high-order asymptotic framework for 3D electromagnetic skin heating in which the depth-to-lateral-scale ratio $\varepsilon$ is small. By expanding the temperature as $T^{(0)} + \varepsilon T^{(1)} + \varepsilon^2 T^{(2,A)} + \varepsilon^2 T^{(2,B)} + \cdots$, the authors derive closed-form expressions for the first two orders, capturing both the incident-angle effects (present at all orders) and lateral heat conduction (entering at $O(\varepsilon^2)$). The work provides explicit, separable solutions $T(x,y,z,t)=f(x,y)W(z,t)$ with reduced IBVPs for $W$, including scalable forms for $W^{(0)},W^{(1)},W^{(2,A)},W^{(2,B)}$, which are then put into perspective via an extended formulation that isolates the two main sources of $O(\varepsilon^2)$ behavior. Numerical comparisons against a full 3D solver show that the complete second-order solution remains accurate even at moderately small $\varepsilon$ (around $0.1$), making it a fast and effective tool for predicting temperature distributions and enabling inverse problems related to internal skin temperatures.

Abstract

Previously we derived the leading term asymptotic solution of temperature distribution in skin heating by an electromagnetic beam at an arbitrary incident angle. The asymptotic analysis is based on that the penetration depth of the beam into skin is much smaller than the size of beam cross-section. It allows arbitrary incident angle. We expand the temperature in powers of the small depth to lateral scale ratio. The incident angle affects all terms in the expansion while the lateral heat conduction appears only in terms of positive even powers. The previously obtained leading term solution captures only the main effect of incident angle. The main effect of lateral heat conduction is contained in the second order term, which is mathematically negligible in the limit of small depth to lateral scale ratio. At a moderate length scale ratio (e.g., 0.1), however, the contribution from lateral conduction is quite significant and needs to be included in a meaningful approximate solution. In this study, we derive closed form analytical expressions for the first order and the second order terms in the asymptotic expansion. The resulting asymptotic solution is capable of predicting the temperature distribution accurately including the effects of both incident angle and lateral heat conduction even at a moderate length scale ratio.

Figures (6)

• Figure 1: Schematics of the skin, the incident beam, the skin coordinate system $(x, y, z)$, and the beam intrinsic coordinate system $(x', y', z')$.
• Figure 2: The refracted direction describes the propagation distance of beam inside skin vs depth and the lateral shift of the high power density region vs depth.
• Figure 3: Case 1: no lateral conduction ($\varepsilon_1 = 0$), zero incident angle ($\theta_1 = \theta_2 = 0$). Left: estimated error $\frac{1}{3}| T_\text{coarse}-T_\text{fine}|$. Right: exact error $| T_\text{fine}-T_\text{exact}|$.
• Figure 4: Case 2: no lateral conduction ($\varepsilon_1 =0$), incident angle ($\theta_1 = 30^\circ$, $\theta_2 = 20^\circ$) modulated by $\varepsilon_2 =0.1$. Top left: estimated error in numerical solution. Top right: error in $T^{(0)}$. Bottom left: error in $T^{(0+1)}$. Bottom right: error in $T^{(0+1+2B)}$.
• Figure 5: Case 3: positive lateral conduction ($\varepsilon_1 =0.1$), zero incident angle ($\theta_1 = \theta_2 = 0$). Top left: estimated error in numerical solution. Top right: error in $T^{(0)}$. Bottom: error in $T^{(0+2A)}$. Here $T^{(1)} = T^{(2,B)} = 0$ because $\tan\theta_2 = 0$.