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Timestamp: 2019-04-25 18:44:45+00:00

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We present a numerical model that describes the propagation of a single femtosecond laser pulse in a medium of which the optical properties dynamically change within the duration of the pulse. We use a finite-difference time-domain method to solve the Maxwell’s equations coupled to equations describing the changes in the material properties. We use the model to simulate the self-reflectivity of strongly focused femtosecond laser pulses on silicon and gold under laser ablation conditions. We compare the simulations to experimental results and find excellent agreement.
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Fig. 1. Diagrammatic view of the model. The iteration starts at Maxwell’s equations, which solve the propagation of the laser pulse in an initially unexcited material. Once every optical half-cycle, the optical intensity inside the material can be derived using the electric field obtained with Maxwell’s equations. This optical intensity is used as an input to the material equations that describe the response of the material to the intensity. By solving the material equations, parameters of the excited material, such as electron density and electron temperature, are obtained. Subsequently, a new susceptibility can be deduced from the parameters of the excited material. This new susceptibility and thus the dielectric function are further inserted into Maxwell’s equations to complete a single iteration. The iterations are conducted once every optical half-cycle until the end of the pulse.
Fig. 2. Layout of the 2D-FDTD simulation box. The FDTD grid is excited by a soft source that is located 200 nm above the silicon–air interface. Scattered E and H near-field values are recorded at the detector plane to extract the reflectivity.
Fig. 3. 2D-FDTD calculations and experimental measurements of self-reflectivity for bulk silicon and SOI samples. Open and closed symbols indicate results of two independent experimental runs for bulk silicon (circles), SOI 1 (squares), and SOI 2 (diamonds). The solid lines show the reflectivity calculated by 2D-FDTD simulations with the TM and TE modes combined for bulk (green), SOI 1 (blue), and SOI 2 (red). The dashed lines show the results obtained from a 1D-FDTD simulation. The blue dashed–dotted and dotted lines show the reflectivity calculated for SOI 1 using either the TE or the TM mode, respectively. The left inset shows the reflectivity calculated using a dynamically changing collision rate with a T dependence, and the right inset shows the reflectivity calculated using a collision rate with a T 2 dependence.
Fig. 4. FDTD calculations and experimental measurements of self-reflectivity on the SOI 1 , SOI 2 , and bulk samples. For the calculations, an impact ionization coefficient θ = 0 cm 2 / J is used. The lines and symbols have the same meaning as in Fig. 3.
Fig. 5. Calculated (a) carrier density, (b) carrier temperature, and (c) optical effective mass on the surface of the sample after the end of the pulse. In each plot, the green line shows the data for the bulk silicon sample, the blue line for the SOI 1 sample, and the red line for the SOI 2 sample.
Fig. 6. (a) FDTD calculation and experimental measurements of self-reflectivity on a 400 nm gold film on glass. Open and closed squares are the experimental data from two independent runs. The solid line shows the calculated self-reflectivity taking both the TM and the TE mode into account. The dotted and dashed–dotted lines show the self-reflectivity taking only the TE mode or only the TM mode into account, respectively. The dashed line shows the self-reflectivity calculated with the conduction electron density at room temperature. (b) Calculated electron temperature on the surface of the sample. (c) The corresponding Drude damping time.
Fig. 7. Error of the FDTD method. Results with a range of PML thicknesses are presented.

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