Meep Tutorial/Local density of states

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:<math>\operatorname{resonant LDOS} \approx \frac{2}{\pi\omega^{(n)}} \frac{Q^{(n)}}{V^{(n)}}</math> :<math>\operatorname{resonant LDOS} \approx \frac{2}{\pi\omega^{(n)}} \frac{Q^{(n)}}{V^{(n)}}</math>
-where <math>Q^{(n)}=\omega^{(n)}/2\gamma^{(n)}</math> is the dimensionless quality factor and <math>V^{(n)}</math> is the modal volume.+where <math>Q^{(n)}=\omega^{(n)}/2\gamma^{(n)}</math> is the dimensionless quality factor and <math>V^{(n)}</math> is the modal volume. This represents another way to compute the LDOS and in this tutorial, we will verify this expression by comparing it to the earlier expression which is a built-in feature. Here, we consider the simple example of a two-dimensional perfect-metal <math>a</math>x<math>a</math> cavity of finite thickness 0.1<math>a</math>, with a small notch of width <math>w</math> on one side that allows the modes to escape. The nice thing about this example is that in the absence of the notch, the lowest-frequency TM-polarized mode is known analytically to be <math>E_z^{(1)}=\frac{4}{a^2}\sin(\pi x/a)\sin(\pi y/a)</math>, with a frequency <math>\omega^{(1)}=\sqrt{2}c\pi/a</math> and modal volume <math>V^{(1)}=a^2/4</math>. The notch slightly perturbs this solution, but more importantly the notch allows it to radiate out into the surrounding air, yielding a finite <math>Q</math>. For <math>\omega \ll a</math>, this radiative escape occurs via an evanescent (sub-cutoff) mode of the channel waveguide formed by the notch, and it follows from inspection of the evanescent decay rate <math>\sqrt{(\pi/\omega)^2-(\omega^{(1)})^2}/c</math> that the lifetime scales asymptotically as <math>Q^{(1)} \sim e^{\#/\omega}</math> for some coefficient #. We will validate both this prediction and the LDOS calculations above by computing the LDOS at the center of the cavity (the point of peak <math>|\vec{E}|</math>) in two ways.

Revision as of 04:21, 24 July 2012

In this example, we will investigate the Purcell enhancement of the local density of states (LDOS), computed in two different ways, for a metallic microcavity. The LDOS is a measure of how much the harmonic modes of a system overlap with the source point (proportional, as well, to the radiation resistance of a dipole antenna) and is a key quantity in electromagnetism due to the fact that the same current radiates a different amount of power depending on the surrounding geometry. The LDOS is of central importance not only for understanding classical dipole sources, but also in many physical phenomena that can be understood semiclassically in terms of dipole currents -- for example, the spontaneous emission rate of atoms (key to fluorescence and lasing phenomena) is proportional to the LDOS. As can be shown analytically, the per-polarization LDOS is exactly proportional to the power radiated by an \ell-oriented point-dipole current at a given position in space. For a more mathematical treatment of the theory behind the LDOS, we refer you to the relevant discussion in chapter 4 of our upcoming book, but for now we simply give the result:

\operatorname{LDOS}_{\ell}(\vec{x}_0,\omega)=-\frac{2}{\pi}\varepsilon(\vec{x}_0)\frac{\operatorname{Re}[\hat{E}_{\ell}(\vec{x}_0,\omega)\hat{p}(\omega)^*]}{|\hat{p}(\omega)|^2}

where the |\hat{p}(\omega)|^2 normalization is necessary for obtaining the power exerted by a unit-amplitude dipole (assuming linear materials). In FDTD, computing the LDOS is straightforward: excite a point dipole source and accumulate the Fourier transforms of the field at a given point in space to obtain the entire LDOS spectrum in a single calculation.

A lossless localized mode yields a δ-function spike in the LDOS, whereas a lossy (i.e. arising from either small absorption or radiation) localized mode -- a resonant cavity mode -- leads to a Lorentzian peak. The large enhancement in the LDOS at the resonant peak is known as a Purcell effect, named after Purcell's proposal for enhancing spontaneous emission of an atom in a cavity (analogous to a microwave antenna resonating in a metal box). In this case, the resonant mode's contribution to the LDOS at ω(n) can be shown to be:

\operatorname{resonant LDOS} \approx \frac{2}{\pi\omega^{(n)}} \frac{Q^{(n)}}{V^{(n)}}

where Q(n) = ω(n) / 2γ(n) is the dimensionless quality factor and V(n) is the modal volume. This represents another way to compute the LDOS and in this tutorial, we will verify this expression by comparing it to the earlier expression which is a built-in feature. Here, we consider the simple example of a two-dimensional perfect-metal axa cavity of finite thickness 0.1a, with a small notch of width w on one side that allows the modes to escape. The nice thing about this example is that in the absence of the notch, the lowest-frequency TM-polarized mode is known analytically to be E_z^{(1)}=\frac{4}{a^2}\sin(\pi x/a)\sin(\pi y/a), with a frequency \omega^{(1)}=\sqrt{2}c\pi/a and modal volume V(1) = a2 / 4. The notch slightly perturbs this solution, but more importantly the notch allows it to radiate out into the surrounding air, yielding a finite Q. For \omega \ll a, this radiative escape occurs via an evanescent (sub-cutoff) mode of the channel waveguide formed by the notch, and it follows from inspection of the evanescent decay rate \sqrt{(\pi/\omega)^2-(\omega^{(1)})^2}/c that the lifetime scales asymptotically as Q^{(1)} \sim e^{\#/\omega} for some coefficient #. We will validate both this prediction and the LDOS calculations above by computing the LDOS at the center of the cavity (the point of peak |\vec{E}|) in two ways.

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