sapphire (Ti:Al2O3) laser, with pulse width of ~200 fs, are used to monitor the characteristics of the interface. Other type of lasers include Yb:fiber, Yb:tungstate, Er:fiber, Nd:glass. Second-harmonic generation may be utilized to achieve frequency of double or higher. The output of the laser is split into pump and probe beams by a half-wave plate followed by a polarizing beam splitter leading to a cross-polarized pump and probe. The pump beam is modulated on the order of a few megahertz by an acousto-optic or electro-optic modulator and focused onto the sample with a lens. The probe is directed into an optical delay line. The probe beam is then focused with a lens onto the same spot on the sample as the pulse. Both pump and probe have a spot size on the order of 10–50 μm. The reflected probe light is input to a high bandwidth photodetector. The output is fed into a lock-in amplifier whose reference signal has the same frequency used to modulate the pump. The voltage output from the lock-in will be proportional to ΔR. Recording this signal as the optical delay line is changed provides a measurement of ΔR as a function of optical probe-pulse time delay. Modeling materials The surface temperature of a single layer The frequency domain solution for a semi-infinite solid which is heated by a point source with angular frequency can be expressed by the following equation. where (1) (Λ: thermal conductivity of the solid, D: thermal diffusivity of the solid, r: radial coordinate) In a typical time-domain thermoreflectance experiment, the co-aligned laser beams have cylindrical symmetry, therefore the Hankel Transform can be used to simplify the computation of the convolution of equation (1) with the distributions of the laser intensities. (The Hankel transform is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel) Here g(r) is radially symmetric and by the definition of Hankel transform using Eq. (1), (2) Since the pump and probe beams used here have Gaussian distribution, the radius of the pump and probe beam are and respectively. The surface is heated by the pump laser beam with the intensity , i.e. (3) where is the amplitude of the heat absorbed by the sample at frequency . Then the Hankel transform of is . (4) Then the distributions of temperature oscillations at the surface is the inverse Hankel transforms of the product