arXiv subjects

J. Gärtner

Publications and source records attributed to J. Gärtner.

3 recordsLinked to original sources

arXiv2020

Analysis of Kerr comb generation in silicon microresonators under the influence of two-photon absorption and free-carrier absorption

Kerr frequency comb generation relies on dedicated waveguide platforms that are optimized towards ultralow loss while offering comparatively limited functionality restricted to passive building blocks. In contrast to that, the silicon-photonic platform offers a highly developed portfolio of high-performance devices, but is deemed to be inherently unsuited for Kerr comb generation at near-infrared (NIR) telecommunication wavelengths due to strong two-photon absorption (TPA) and subsequent free-carrier absorption (FCA). Here we present a theoretical investigation that quantifies the impact of TPA and FCA on Kerr comb formation and that is based on a modified version of the Lugiato-Lefever equation (LLE). We find that silicon microresonators may be used for Kerr comb generation in the NIR, provided that the dwell time of the TPA-generated free-carriers in the waveguide core is reduced by a reverse-biased p-i-njunction and that the pump parameters are chosen appropriately. We validate our analytical predictions with time integrations of the LLE, and we present a specific design of a silicon microresonator that may even support formation of dissipative Kerr soliton combs.

physics.optics↗

arXiv2010

Intermittency on catalysts: Voter model

In this paper we study intermittency for the parabolic Anderson equation $\partial u/\partial t=κΔu+γξu$ with $u:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$, where $κ\in[0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, $γ\in(0,\infty)$ is the coupling constant, and $ξ:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium. The solution of this equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $ξ$. We focus on the case where $ξ$ is the voter model with opinions 0 and 1 that are updated according to a random walk transition kernel, starting from either the Bernoulli measure $ν_ρ$ or the equilibrium measure $μ_ρ$, where $ρ\in(0,1)$ is the density of 1's. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$. We show that if the random walk transition kernel has zero mean and finite variance, then these exponents are trivial for $1\leq d\leq4$, but display an interesting dependence on the diffusion constant $κ$ for $d\geq 5$, with qualitatively different behavior in different dimensions. In earlier work we considered the case where $ξ$ is a field of independent simple random walks in a Poisson equilibrium, respectively, a symmetric exclusion process in a Bernoulli equilibrium, which are both reversible dynamics. In the present work a main obstacle is the nonreversibility of the voter model dynamics, since this precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.

math.PR↗

arXiv2007

Intermittency in a catalytic random medium

In this paper, we study intermittency for the parabolic Anderson equation $\partial u/\partial t=κΔu+ξu$, where $u:\mathbb{Z}^d\times [0,\infty)\to\mathbb{R}$, $κ$ is the diffusion constant, $Δ$ is the discrete Laplacian and $ξ:\mathbb{Z}^d\times[0,\infty)\to\mathbb {R}$ is a space-time random medium. We focus on the case where $ξ$ is $γ$ times the random medium that is obtained by running independent simple random walks with diffusion constant $ρ$ starting from a Poisson random field with intensity $ν$. Throughout the paper, we assume that $κ,γ,ρ,ν\in (0,\infty)$. The solution of the equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $ξ$. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$, and show that they display an interesting dependence on the dimension $d$ and on the parameters $κ,γ,ρ,ν$, with qualitatively different intermittency behavior in $d=1,2$, in $d=3$ and in $d\geq4$. Special attention is given to the asymptotics of these Lyapunov exponents for $κ\downarrow0$ and $κ\to\infty$.

math.PR↗