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

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.

💡 Research Summary

This paper addresses the thermal response of human skin when exposed to an electromagnetic (EM) beam, focusing on the combined effects of the beam’s incident angle and lateral heat conduction. The authors build upon their previous work, which derived only the leading‑order (ε⁰) asymptotic solution, by extending the asymptotic expansion to include the first‑order (ε¹) and second‑order (ε²) terms, where ε is the ratio of the EM penetration depth to the lateral beam size.

The physical setting is a millimeter‑wave (MMW) beam (30–300 GHz) incident on a flat skin surface at an arbitrary angle θ₁. The beam is described in its intrinsic coordinate system (x′,y′,z′) and then projected onto the skin coordinate system (x,y,z). Two geometric effects of the incident angle are identified: (1) the projected spot on the skin is elongated by 1/cos θ₁, reducing the surface power density by cos θ₁, and (2) the refracted beam propagates inside the tissue at angle θ₂ (given by Snell’s law), which stretches the effective propagation distance to z cos θ₂ and thus modifies the exponential absorption profile. In addition, the refracted beam shifts laterally by x = z tan θ₂ as it penetrates, creating a depth‑dependent lateral displacement of the heating region.

The governing heat equation, after nondimensionalization, reads

∂T/∂t = ε²(∂²T/∂x² + ∂²T/∂y²) + ∂²T/∂z² + Pₐ f(x + ε z tan θ₂, y) e^{‑z/λ},

with λ = cos θ₂, ε = zₛ/rₛ (penetration depth over beam radius), and f representing the normalized beam profile after the cos θ₁ scaling. Boundary conditions are a zero heat flux at the surface (∂T/∂z = 0 at z = 0) and an initially uniform temperature (T = 0 at t = 0).

The temperature is expanded as

T = T⁽⁰⁾ + ε T⁽¹⁾ + ε² T⁽²⁾ + …,

and each coefficient satisfies its own initial‑boundary‑value problem (IBVP).

• Zero‑order (ε⁰) problem: Lateral diffusion terms vanish, leaving a one‑dimensional heat equation with a source term that depends on the incident‑angle‑modified profile f(x,y). The solution, denoted W⁽⁰⁾(z,t), is obtained analytically by separation of variables and satisfies the surface Neumann condition.

• First‑order (ε¹) problem: The source now contains the derivative of the profile with respect to x multiplied by the lateral shift factor tan θ₂ · z. This term captures the depth‑dependent lateral translation of the heating region caused by refraction. Its solution, W⁽¹⁾(z,t), is again a closed‑form expression involving exponential integrals.

• Second‑order (ε²) problem: Two distinct contributions appear.

  • A‑term: ε²∇⊥²T⁽⁰⁾, representing pure lateral heat conduction acting on the leading‑order temperature field.
  • B‑term: ε² tan θ₂ ∂ₓT⁽¹⁾, representing the interaction between the incident‑angle‑induced lateral shift and the first‑order temperature correction. Both sub‑problems reduce to one‑dimensional equations for W⁽²ᴬ⁾(z,t) and W⁽²ᴮ⁾(z,t), which are solved analytically. The total second‑order correction is T⁽²⁾ = W⁽²ᴬ⁾ + W⁽²ᴮ⁾.

The final asymptotic approximation is

T ≈ W⁽⁰⁾ + ε W⁽¹⁾ + ε²(W⁽²ᴬ⁾ + W⁽²ᴮ⁾).

To assess accuracy, the authors performed high‑resolution finite‑difference simulations of the full three‑dimensional heat equation for several test cases: (i) no lateral conduction, normal incidence; (ii) no lateral conduction, θ₁ = 30°; (iii) positive lateral conduction, normal incidence; (iv) positive lateral conduction, θ₁ = 30°. With ε = 0.1 (a moderate scale separation), the leading‑order solution alone yields temperature errors exceeding 20 % in peak magnitude and spatial location. Adding the ε¹ term reduces errors to about 5 % but still fails to capture the broadened temperature profile when lateral conduction is present. Incorporating the ε² terms brings the error down to below 2 %, accurately reproducing both the peak temperature and its lateral spread. A detailed error decomposition shows that the A‑term primarily widens the temperature distribution (lateral diffusion), while the B‑term shifts the peak laterally in accordance with the incident angle.

Key contributions:

1. A systematic asymptotic framework that retains the full geometric influence of the incident angle (both power‑density dilution and refracted‑path stretching) in every order of ε.
2. Closed‑form analytical expressions for the first‑ and second‑order temperature corrections, enabling rapid evaluation without resorting to full 3‑D numerical solvers.
3. Quantitative demonstration that the second‑order term, despite being formally O(ε²), is essential for ε ≈ 0.1 and for realistic beam sizes, especially when lateral heat conduction cannot be neglected.

Limitations and future work: The model assumes homogeneous skin properties, neglects surface heat loss, and treats the tissue as semi‑infinite. For larger ε (≥ 0.2) or for multi‑layered skin (epidermis, dermis, subcutaneous), higher‑order terms or a different asymptotic scaling may be required. The authors suggest extending the analysis to layered media, incorporating nonlinear temperature‑dependent material properties, and validating the predictions against experimental measurements of surface and subsurface temperatures in vivo.

Overall, the paper provides a valuable analytical tool for rapid assessment of skin heating in MMW applications, with potential uses in safety standards, device design, and inverse problems such as reconstructing internal temperature fields from surface thermography.