In this example, we will demonstrate the local density of states (LDOS) feature by investigating the Purcell enhancement phenomena in a metallic cavity. The simulation script is metal-cavity-ldos.ctl. The LDOS, in general, has many important uses 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. The LDOS is equivalent to the power radiated by a unit dipole,
$$\operatorname{LDOS}_{\ell}(\vec{x}_0,\omega)=-\frac{2}{\pi}\varepsilon(\vec{x}0)\frac{\operatorname{Re}[\hat{E}{\ell}(\vec{x}_0,\omega)\hat{p}(\omega)^*]}{|\hat{p}(\omega)|^2}$$
where the dft-ldos
feature which is the subject of this tutorial.
A lossless localized mode yields a δ-function spike in the LDOS, whereas a lossy, 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. This is analogous to a microwave antenna resonating in a metal box. In this case, the resonant mode's contribution to the LDOS at
where
We consider the simple example of a 2d perfect-metal $a$x$a$ cavity of finite thickness 0.1$a$, with a small notch of width
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 dft-ldos
command. Second, we compute the cavity mode and its lifetime harminv
and then compute the LDOS by the Purcell formula shown above. The latter technique is much more efficient for high Q (small
We'll first set up the 2d simulation with the metal cavity and PML absorbing boundary layers:
(set-param! resolution 200)
(define-param sxy 2)
(define-param dpml 1)
(set! sxy (+ sxy (* 2 dpml)))
(set! geometry-lattice (make lattice (size sxy sxy no-size)))
(set! pml-layers (list (make pml (thickness dpml))))
(define-param a 1)
(define-param t 0.1)
(set! geometry (list
(make block (center 0 0) (size (+ a (* 2 t)) (+ a (* 2 t)) infinity) (material metal))
(make block (center 0 0) (size a a infinity) (material air))))
Next we'll create a notch opening in the cavity so that the field can radiate away:
(define-param w 0)
(if (> w 0)
(set! geometry
(append geometry
(list (make block (center (/ a 2) 0) (size (* 2 t) w infinity)
(material air))))))
We can now set up the
(define-param fcen (/ (sqrt 0.5) a))
(define-param df 0.2)
(set! sources (list (make source
(src (make gaussian-src (frequency fcen) (fwidth df))) (component Ez) (center 0 0))))
As both the structure and sources have a mirror symmetry in the
(set! symmetries (list (make mirror-sym (direction Y))))
In the first part of the calculation, we compute the Purcell enhancement. This requires the mode frequency and quality factor:
(define-param Th 500)
(run-sources+ Th (after-sources (harminv Ez (vector3 0) fcen df)))
(define f (harminv-freq-re (car harminv-results)))
(define Q (harminv-Q (car harminv-results)))
(define Vmode (* 0.25 a a))
(print "ldos0:, " (/ Q Vmode (* 2 pi f pi 0.5)))
Next, we rerun the same simulation and compute the LDOS using Meep's dft-ldos
feature at the mode frequency:
(reset-meep)
(define-param T (* 2 Q (/ f)))
(run-sources+ T (dft-ldos f 0 1))
We need to run for a sufficiently long time to ensure that the Fourier-transformed fields have converged. A suitable time interval is, due to the Fourier Uncertainty Principle, just one period of the decay which we can determine using the
We run several simulations spanning a number of different notch sizes and plot the result in the following figure which shows good agreement between the two methods.
![](../images/Metalcavity_ldos.png)