Transient Radiative Transfer in a Graded Index Slab with Fresnel Surfaces

Abstract

One-dimensional transient radiative transfer in a graded index slab with Fresnel surfaces subject to a short-pulse laser irradiation is investigated by the lattice Boltzmann method. In the scattering medium having Fresnel surfaces, the radiative intensity includes two parts: the collimated intensity, and scattered or diffuse intensity. The contribution of the collimated pulse to the transient process is considered as a source term in the transient radiative transfer equation. First, lattice Boltzmann method solutions for time-resolved signals are validated by comparison with results obtained by Monte Carlo method. The investigations are mainly focused on the medium with linearly changing graded index, and the commonalities and differences of the time-resolved signals resulting from the slab with diffuse reflected surfaces or Fresnel surfaces are discussed. The effects of the gradient of the linearly varying refractive index on transmittance and reflectance signals are examined. Then, transient radiative transfer with two graded index distributions with a singular point is analyzed. Finally, by setting the right boundary of the slab as a diffusely reflecting surface, the transient radiative transfer in the graded index medium under the combined optical boundary conditions is examined. Nomenclature a = anisotropy factor c 0 = speed of light in vacuum, m · s −1 e = velocity of propagation of the particle distribution function along a lattice link, m · s −1 H = Heaviside function I = intensity of radiation, particle distribution function in lattice Boltzmann method for radiation transfer, W · m −2 · sr −1 I eq = equilibrium particle distribution function in lattice Boltzmann method for radiation, W · m −2 · sr −1 L = geometrical thickness of the scattering slab, m M = total number of discrete directions n = refractive index q = time-resolved reflectance or transmittance signal S = source term for radiation, W · m −2 t = time, s t p = pulsewidth, s β = extinction coefficient; κ a σ s , m −1 θ i , θ r = incident angle and refracted angle κ a , σ s = absorption coefficient and scattering coefficient,cosine of the polar angle μ c = direction cosine of critical angle for total reflection ρ = reflection coefficient σ = Stefan-Boltzmann constant; 5.67 × 10 −8 , W · m −2 · K −4 τ m = relaxation time in lattice Boltzmann method, s ϕ = azimuthal angle Φ = scattering phase function ω = scattering albedo; σ s ∕β Subscripts c = collimated radiation d = diffuse radiation R = reflectance T = transmittance 0 = vacuum 1 = left surface 2 = right surface Superscripts m, m 0 = index for direction = nondimensional quantity , − = positive direction and negative direction of x axis