3.1

Material response function

Material’s energy band diagram is often simplified and simulated by using coupled rate equations:

ω_K – Keldysh ionization rate which includes both multiphoton and tunneling ionizations,^10,11 ω_II – impact ionization rate and ω_CB→STE – trapping rate, which couples self-trapped exciton state to the system:

where N_STE is density of self-trapped excitons, N_STE,max – maximum density of self-trapped excitons density, τ_CB→STE – trapping time and N_CB density of conduction band electrons. Relaxation to valence band and ionization of trapped states is omitted, because these processes take place on a longer time scale, but can easily be incorporated if multiple pulse regime is of interest. The full graphical representation of the simulated energy band system is shown in Figure 4.

Figure 4.

Simplified diagram of energy levels model.

The material model is described through the polarization vector. In our case, the polarization is affected by Kerr effect, conduction band electrons and self-trapped exciton state:

The response of conduction band electrons is taken into account by using the Drude model for free electrons¹² with ω_p,CB as plasma frequency:

The response of self-trapped exciton states is taken into account by assuming the Lorentz model for bound states¹² with ω_p,STE as resonance frequency:

m_CB is effective mass of conduction band electrons, m_STE – effective mass of self-trapped excitons, e – elementary charge, ϵ₀ – vacuum permittivity.

Electric displacement vector D and complex relative permittivity ϵ_r can then be expressed as:

The real part of complex dielectric permittivity can be directly used in the FDTD scheme with real fields:

However, the inclusion of imaginary part of complex relative permittivity (and thus absorption) is not as straightforward. In order to use it with real fields in the FDTD scheme, we must express it through the conductivity term:¹²

where the imaginary part of complex relative permittivity is:

The derived conductivity term (14) only includes absorption by conduction band electrons and self-trapped excitons. An additional absorption term is needed to include Keldysh ionization. This can be accomplished by first equating the loss of laser energy J_K E (where J_K is current density) to energy required to ionize electrons through the Keldysh mechanism E_gapω_K :^13,14

The nonlinear conductivity term that accounts for Keldysh ionization σ_K and effective conductivity σ_{ef f} of the whole system can then be expressed as:

3.2

Electromagnetic field in layered structures

In order to simulate the propagation of femtosecond laser pulse in layered dynamical system the most fundamental method of solving Maxwell’s equations was chosen. The electromagnetic field simulation is based on Maxwell’s curl equations.¹⁵ In the one dimensional case (assuming propagation along the z axis) these equations can be written as:

where E_x denotes electric field, H_y – magnetizing field, ϵ – permittivity, σ – conductivity, µ - permeability

The finite-difference time-domain method can be used to discretize these equations using Yee’s method,¹⁶ which separates E_x and H_y both in time and space in order to achieve second-order accuracy:¹⁵

Δ_z is spatial step, Δ_t – temporal step, m – spatial step index, q – temporal step index. Permittivity ϵ is calculated using the real part of previously derived relative permittivity (12) and effective conductivity from (18) is used as σ. At every temporal time step, ϵ and σ are adjusted by calculating the densities of conduction band electrons and self-trapped excitons using rate equations (1) and (2).

Since the pump pulse is more than 10 times longer than the probe pulse, semi-analytical methods can be used to evaluate relative transmission and phase shift of the probe beam. Transmission can be evaluated by using S-matrix method by taking FDTD grid of complex refractive indices as an input.¹⁷ Transmittance ratio can then be evaluated by comparing transmittance with the linear case. Phase shift can be evaluated directly from B-integral:¹⁸

4.1

Fitting of 1D time-resolved data

All of the one-dimensional curves (transmittance ratio and phase shift at different intensities) were collectively fitted (SLSQP optimization¹⁹) using the theoretical model described in the previous section. The collective fit was performed instead of many separate fits in order to constrain the model as much as possible. Since the nonlinear response of the coating is dominated by Nb₂O₅ layers (Kerr effect is 10 times stronger² and nonlinear ionization is at least 10¹¹ times higher in Nb₂O₅ than in SiO₂) only this material’s parameters were varied during the fitting procedure. Fitted values of the model’s parameters are provided in Table 2 and corresponding transmission ratio and phase shift curves are shown as solid curves in Figure 5.

Table 2.

Fitted model’s material parameters for Nb2O5 layers.

As can be seen in Figure 5, fitted model is in good agreement with experimental data: it accurately describes both transmittance ratio and phase shift changes for all measured intensity levels. Model predicts 150 fs trapping time for self-trapped excitons which is in good agreement with the same parameter measured for SiO₂.²⁰

4.2

Contribution of each layer

Since the model has spatial resolution perpendicularly to the coating, information about the contribution of each Nb₂O₅ layer can also be extracted. Evolution of conduction band electrons and self-trapped excitons for 145% LIDT intensity level is provided in Figure 6 with a corresponding material response in Figure 7 for the top 2 µm of the coating. It can clearly be seen that the majority of nonlinear ionization takes place in only second and third Nb₂O₅ layers, while the contribution from the Kerr effect is more evenly spread out (Figure 7 (a)). The generation of electron plasma begins at the peak of the pulse (Figure 6 (a)), however, electrons are quickly trapped and at the end of the laser pulse only the self-trapped excitons remain (Figure 6 (b)) which are responsible for the absorbtion within the coating (Figure 7 (b)).

Figure 6.

Evolution of conduction band electrons (a) and self-trapped excitons (b) densities during the 145% LIDT pulse in the first 2 µm of the coating.

Figure 7.

Evolution of real (a) and imaginary (b) parts of complex refractive index during the 145% LIDT pulse in the first 2 µm of the coating.

4.3

Simulation of time-resolved reflectance

Fitting procedure provided valuable information about the Nb₂O₅ coating which can be used for future investigations of such nonlinear coatings. For example, lets assume a 1030 nm, 300 fs laser pulse is used to exponate the F502 coating at 145% LIDT intensity (0 degree angle of incidence). Our model can be used to evaluate the nonlinear evolution of both angular and spectral reflectance during the laser pulse (Figure 8). We see that nonlinear reflectance is highly dynamical for intensity values higher than LIDT, however, any further analysis of such behaviour is outside the scope of this paper and is left for future work.

Figure 8.

Evolution of angular (a) and spectral (b) reflectance during the 145% LIDT pulse. Bottom figures show the linear case for respective nonlinear cases.