Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.Comment: 35 pages. 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.0288

A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\varepsilon\in(0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{\tau^2}{\varepsilon^2}$ and $h^{m_0}+\tau^2+\varepsilon^2$, where $h$ is the mesh size, $\tau$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\lesssim \tau$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its limiting models when $\varepsilon\to0^+$.Comment: 25 pages, 1 figur

Thermal analysis of high-bandwidth and energy-efficient 980 nm VCSELs with optimized quantum well gain peak-to-cavity resonance wavelength offset

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Appl. Phys. Lett. 111, 243508 (2017) and may be found at https://doi.org/10.1063/1.5003288.The static and dynamic performance of vertical-cavity surface-emitting lasers (VCSELs) used as light-sources for optical interconnects is highly influenced by temperature. We study the effect of temperature on the performance of high-speed energy-efficient 980 nm VCSELs with a peak wavelength of the quantum well offset to the wavelength of the fundamental longitudinal device cavity mode so that they are aligned at around 60 °C. A simple method to obtain the thermal resistance of the VCSELs as a function of ambient temperature is described, allowing us to extract the active region temperature and the temperature dependence of the dynamic and static parameters. At low bias currents, we can see an increase of the −3 dB modulation bandwidth f−3dB with increasing active region temperature, which is different from the classically known situation. From the detailed analysis of f−3dB versus the active region temperature, we obtain a better understanding of the thermal limitations of VCSELs, giving a basis for next generation device designs with improved temperature stability

Ultra-bright Photoactivatable Fluorophores Created by Reductive Caging

Sub-diffraction-limit imaging can be achieved by sequential localization of photoactivatable fluorophores, where the image resolution depends on the number of photons detected per localization. Here, we report a strategy for fluorophore caging that creates photoactivatable probes with high photon yields. Upon photoactivation, these probes can provide 104–106 photons per localization and allow imaging of fixed samples with resolutions of several nanometers. This strategy can be applied to many fluorophores across the visible spectrum

Isotropic 3D Super-resolution Imaging with a Self-bending Point Spread Function

Airy beams maintain their intensity profiles over a large propagation distance without substantial diffraction and exhibit lateral bending during propagation1-5. This unique property has been exploited for micromanipulation of particles6, generation of plasma channels7 and guidance of plasmonic waves8, but has not been explored for high-resolution optical microscopy. Here, we introduce a self-bending point spread function (SB-PSF) based on Airy beams for three-dimensional (3D) super-resolution fluorescence imaging. We designed a side-lobe-free SB-PSF and implemented a two-channel detection scheme to enable unambiguous 3D localization of fluorescent molecules. The lack of diffraction and the propagation-dependent lateral bending make the SB-PSF well suited for precise 3D localization of molecules over a large imaging depth. Using this method, we obtained super-resolution imaging with isotropic 3D localization precision of 10-15 nm over a 3 μm imaging depth from ∼2000 photons per localization