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Abstract
A general multilevel carrier kinetics model is explored to simulate the saturation of absorption in the time-domain. Contrarily to approaches relying upon phenomenological descriptions, in this study, we deal with the saturation through a physics-based model that can predict the realistic temporal dynamics of the entire system. Additionally, the proposed method allows high flexibility and generality for the problems under consideration as it is built on a full-wave three-dimensional time-domain solver that can include nonlinear material dispersion, optical activity, and other effects within a joint multiphysics framework. We discretize all the equations using finite-differences combined with the auxiliary differential equation technique which allows adding polarizations driven by diverse underlying physical mechanisms accounting for multiple material dynamics. With our framework, a plethora of time-resolved spatially-dependent numerical data, which are not attainable otherwise, is becoming accessible, hence enabling a comprehensive understanding of the foundations of the materials physics and empowering accurate optimization of new nonlinear photonic devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A saturable absorber (SA) is a material with a nonlinear absorption coefficient that decreases as the incident light intensity increases. When the low-intensity light is incident on the SA, a photon is absorbed, and a carrier is excited from the ground to a higher energy atomic level causing the absorption of light, Fig. 1(a). Due to the abundance of absorbent carriers in the ground state, the material still can absorb more photons. However, if the excited-state lifetime is long enough and the intensity of light is increasing, so that the excited-state carriers are not decaying fast enough to participate in the absorption process again, while more carriers are getting excited, the saturation of absorption occurs (Fig. 1(b)). SAs find applications in passive mode-locking [1] and Q-switching [2]. A broad range of materials exhibit saturable absorption including 2D materials such as graphene [3], carbon nanotubes [4], topological insulators [5], as well as dielectric [6] and metallic nanoparticles [7]. However, full-wave analysis of saturable absorption has been limited to either phenomenological descriptions that ignore transient response or simple models that do not account for multiphysics material response. The overall objective of this study is to provide a generalized time-domain (TD) coupled-physics method for numerical simulations of the saturation of absorption in a complex (e.g., nanopatterned, dispersive, and illuminated with a structured light) three-dimensional (3D) material systems. Availability of such a modeling tool can provide means to design and optimize nanophotonic structures that can engineer the nonlinear spatiotemporal light-matter interaction, and thus, control the functionality of the entire system. Following our objective, we explore the kinetics of spatially-dependent carrier population and the coherent absorption using a set of rate equations (RE) derived for the multilevel system. The proposed technique simulates an averaged nonlinear material response as a polarization vector in a multiphysics simulation framework. Within this framework, the Maxwell equations are coupled with the RE – a set of spatially-dependent auxiliary differential equations (ADE) [8] – and share the same simulation domain, along with the same temporal and spatial discretization. A salient feature of the carrier kinetics model integrated into our simulation framework is that it offers a critical ability to predict multiple absorption peaks in the absorbing medium.
Fig. 1 Saturable absorption in a dielectric thin film modeled by a two-level system at: (a) low input fluence and (b) high input fluence where the ground state carriers are depleted and the excited-state lifetime is too long to allow for reabsorption.
Following this introduction, we organize the rest of the paper as follows: Section 2 introduces and provides details on different SA models built on two-and four-level transition topologies. Section 3 presents the analysis of thin-film dielectric SAs modeled using either two-or four-level atomic systems within our new multiphysics framework with the multilevel RE-ADE. The results of the study are summarized in Conclusions.
2. Saturable absorption models
Saturable absorption is commonly modeled using a phenomenological description to account for the intensity dependence of the absorption of a material given by:
where α(I) is the intensity-dependent material absorption coefficient, α0 is the low-intensity absorption coefficient, I is the light intensity and Is is the saturation intensity of the material. This model puts forward many limitations, including the lack of a comprehensive description of the fundamental physics, the assumption of instantaneous response, which is nonphysical, lack of any direct access to electric or magnetic fields inside the medium, and the complete inability to predict the temporal dynamics of a saturable absorber. To overcome these shortcomings, more physics-based models built on the population kinetics with the REs have been introduced [9, 10]. Initially, only steady-state solutions were considered in those studies [9, 10]. An important nonlinear generalization of the Lorentz model has been applied by Varin et al. to model 2nd and 3rd-order optical nonlinearity [11, 12] with FDTD. Recently, Mock et al. made a vital contribution to modeling the saturation of absorption for a 2D material in the TD [13].In this section, we propose and discuss two TD ADE models for SAs based on two-and four-level atomic systems. The proposed four-level system can reproduce multiple absorption peaks based on the nature of the material and the application. We integrate our multi-level material model as proprietary ADE codes in the commercial 3D FDTD and FETD solvers [1415].
2.1. Two-level saturable absorption atomic model
The saturable absorbing material is modeled using a two-level system, depicted in Fig. 2(a). The transition between the two levels is governed by:
where N1 is the population density of the excited-state carriers, N0 is the population density of carriers in the ground state, with the total population density of absorbing carriers Nt being conserved, Nt = N0 + N1, τ is the lifetime of the material, ω0 is the frequency of the absorbed light defined by the energy difference between the two levels, and ħ is the reduced Planck constant. The macroscopic polarization resulting from this transition takes a Lorentzian form with a source term being dependent on the difference of the population density in the ground and excited levels: where P10 is the average macroscopic polarization vector and which is a function of time and space. The damping factor of this oscillator is γ10, and it is a summation of all sources of damping in the material such as dephasing, and the transition frequency is ω0. The coupling factor κ takes the form where ϵh is the permittivity of the host medium [16].Fig. 2 Jablonski diagrams of (a) two-level and (b) four-level atomic systems showing the allowed transitions and the corresponding lifetimes.
Using a standard FDTD Yee grid, the rate equation, the macroscopic polarization driving equation, and Maxwell’s equations are solved numerically, providing a complete spatiotemporal description of the nonlinear interplay between the population dynamics and electromagnetic fields [8]. This description offers comprehensive insights into a detailed physical picture of the system and gives access to numerous time-resolved functional dependencies of the internal dynamics that are not attainable otherwise. Following the ADE technique, the macroscopic polarization vector is coupled to Maxwell’s equations through:
where ϵ0 is the permittivity of free-space, and Pm is the polarization induced by the mth oscillator.2.2. Four-level saturable absorption atomic model
Some absorbing materials exhibit more than one absorption peak which cannot be captured by either a two-level atomic model in the TD or simpler SA models in the frequency domain. For an extended representation of the absorption spectra, we expand the two-level model to four-levels and demonstrate the appearance of three different absorption peaks. Since such an expansion is very generic, it can be adjusted for an arbitrary number of peaks with an appropriate topology of levels. Such model also has the flexibility of controlling the spectral width, position, and strength of each absorption peak individually. The Jablonski diagram of a four-level atomic system is depicted in Fig. 2(b). The RE system governing these transitions are given by:
where N1, N2, N3, and N4 are the population density of carriers in the different energy levels. The system describes the absorption of the incident light at different frequencies, ω30, ω20, and ω10, as well as the rapid decay of the excited-state population from higher levels to lower energy levels due to non-radiative relaxations. The transition lifetime between levels i and j is τij, as shown in Fig. 2(b). The macroscopic polarizations induced as a result of these transitions are: where ij ∈{30, 20, 10}, Pij is the macroscopic polarization vector formed between levels i and j. γij and κij are the ij-oscillator’s damping coefficient and the coupling factor, respectively. The Maxwell equations are solved using the FDTD method [8], along with the REs, Eq. 5, and the polarization densities, Eq. 6 within a joint multiphysics framework as mentioned earlier.3. Results and analysis
Figure 3 illustrates the analysis of a 200-nm SA with a dielectric constant of 2.25 using the two-level model. We use a lifetime (τS1) of 1 ps, a transition wavelength of 532 nm, a damping rate (γ10) of 1014 rad/s, and a concentration of 10 mM. The absorption of the film decreases as the input fluence increase as depicted in Fig. 3(a). The saturation fluence of the SA, defined as the fluence required to reduce the absorption to 1/e of its linear value, is 87 µJ/cm2. Figure 3(b-d) illustrates the time dynamics of the system. We plot the internal system dynamics at different input fluences: 6 µJ/cm2 where the system is unsaturated and still linear and 94 µJ/cm2 in the nonlinear regime. Figure 3(b) illustrates a time-dependent behavior of macroscopic polarization P10 in both linear and nonlinear cases showing the relative increase in its magnitude. Figure 3(c) shows the time evolution of unsaturated normalized carrier population densities in the upper and lower energy levels, N1 and N0. Figure 3(d) shows a significant difference in the carrier population dynamics. Once the incoming light elevates more carriers to the upper energy levels, it causes a decrease in the available population pool and hence the saturation of absorption. Therefore, the two-level system gives us access to the numerous details of internal time-dynamics that help understand the physics and further engineer the response of more complex systems. Next, we study the dependence of the absorption on the SA film thickness and the concentration of absorbing carriers. Figure 4(a) shows the increase of the linear absorption with increasing the film thickness. The absorption increases from 0.46 for a 100-nm film to a 0.94 for a 500-nm one. The saturation intensity shifts with the thickness increase from 81 µJ/cm2 for the 100 nm film to 126 µJ/cm2 for the 500 nm film. The effect of the carrier concentration on absorption is also explored. As anticipated, as the carrier concentration increases, the absorption increases. Figure 4(b) shows the increase of the linear absorption from 0.145 for 1 mM concentration to 0.87 for 20 mM, and Is increase from 51 µJ/cm2 for 1 mM concentration to 115 µJ/cm2 for 20 mM.
Fig. 3 Dynamics of a dielectric SA modeled using the two-level system. (a) Absorption vs. input fluence showing the saturation of absorption. (b) Time-dependent normalized macroscopic polarization P10 in both linear and nonlinear cases. Time evolution of carrier population density in (c) linear (6 µJ/cm2) and (d) nonlinear (94 µJ/cm2) regimes.
Using the four-level atomic system, we simulate the absorption in a 100-nm SA with a carrier concentration of 0.1 mM and the absorption central wavelengths being, λ30 =450nm, λ20 = 550 nm, and λ10 = 650 nm. The damping coefficients for each Lorentzian are γ30 = γ20 = γ10 = 2π × 1014 rad/s, while the life-times are τ30 = τ20 = τ10 = 10 ps, τ32 = 1 ps = τ31 =1 ps and τ21 = 10 fs. Figure 5(a) depicts the simulated absorption spectrum showing three peaks at the selected wavelengths. The bandwidth of each peak along with its amplitude can be controlled using the model parameters. At 1 µJ/cm2, the time-dependent carrier population in all four levels is given in Fig. 5(b) showing the decrease in the ground state population density N0 which accounts for the absorption reduction. The normalized macroscopic polarizations as a function of time are shown in Fig. 5(c) indicating the relative amplitudes of the material response.
Fig. 5 An SA dielectric film modeled using the four-level system. (a) The absorption spectrum shows three peaks at the selected wavelengths. Time-dependent (b) carrier population densities and (c) normalized polarizations at an input fluence of 1 µJ/cm2
4. Conclusions
We proposed a hybrid FDTD-RE modeling approach to explore the saturation of absorption as the interplay of the carrier population kinetics in a multilevel atomic system with classical electromagnetic fields. This versatile modeling approach is built upon the ADE technique that allows adding polarizations driven by diverse underlying physics and hence accounting for multiple material dynamics. Our multiphysics framework opens up access to a complete spatiotemporal picture of the entire nonlinear system, which is not attainable otherwise. This framework could enable further comprehensive understanding of the foundations of the materials physics and empower accurate optimization of nonlinear photonic devices.
Funding
DARPA/DSO Extreme Optics and Imaging (EXTREME) Program (HR00111720032).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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