This page reviews how one can calculate Green's function components, local density of states and spontaneous decay rate of a point source and discusses how the aforementioned quantities relate to the dipolepower command in FDTD.
The derivation of the equations are for the most part taken from Novotny's "Principles of Nano-Optics".
Green's function calculations
The dyadic Green’s function \(\mathbf{G}\), is defined by the electric field at point \(\mathbf{r}\), generated by a point source at point \(\mathbf{r}_0\) with dipole moment \(\boldsymbol{\mu}\):
$$\mathbf{E}(\mathbf{r}) =\frac{\omega^2}{ \varepsilon_0 \varepsilon_r c^2} \mathbf{G}\left(\mathbf{r}, \mathbf{r}_{0}\right) \cdot \boldsymbol{\mu}$$
\(\mathbf{G}\) is a 3x3 symmetric matrix and so each component of \(\mathbf{G}\) can be calculated using the corresponding dipole orientation and electric field component:
$$\mathbf{G}=\left[\begin{array}{lll}{G_{x x}} & {G_{x y}} & {G_{x z}} \\ {G_{x y}} & {G_{y y}} & {G_{y z}} \\ {G_{x z}} & {G_{y z}} & {G_{z z}}\end{array}\right]$$
For example, \(G_{zz}\) can be calculated from a dipole oriented along the z direction:
$$G_{z z}=\frac{\varepsilon_{0} \varepsilon_{r} c^{2} }{\mu \omega^{2}} E_{z} $$
Open and run dipole_gf.fsp and once the simulation is run, run the script dipole_gf.lsf . This first part of the script will calculate the real and imaginary parts of the Green's function for a range of frequencies. The real part of the Green's function approaches infinity at the location of the dipole.
To observe the behaviour of the real part of the electric field as a function of position, open and run dipole_field.fsp and dipole_field.lsf. Both the real and imaginary parts are plotted. The real part approaches infinity at the location of the dipole. However, when it is averaged over one mesh cell, it is a finite negative value, as shown in the figure below. To achieve closer agreement between the FDTD result and the theoretical average, special care must be taken to set-up the simulation appropriately. For more information, see the note below for advanced users.
|
Note: For advanced users The real part of the electric field at the location of the dipole is carefully simulated by placing the dipole at the Ez point on the Yee Cell illustrated in the image below.
This ensures that the dipole is implemented at a single point. To observe its behaviour, open and run dipole_field_advanced.fsp and dipole_field_advanced.lsf. The following changes are applied in this simulation file:
The script file is modified to account for the half mesh step shift in the dipole position. The resulting plot is shown below. When the mesh size is reduced, the relative error between the FDTD real part of the field and theoretical average at the dipole location decreases, however the absolute error increases.
|
Local Density of States ( LDOS) and the decay rate
The partial local density of states along one direction, say z, can be obtained from the imaginary part of Green's function:
$$\rho_{z}\left(r_{0}, \omega\right)=\frac{6 \omega}{\pi c^{2}}\left[\operatorname{Im}\left\{G_{z z}\left(r_{0}, r_{0} , \omega\right)\right\}\right]$$
From this, the spontaneous decay rate for a two-level quantum system at r0 and w0 can be calculated:
$$\gamma=\frac{2 \omega_0}{3 \hbar \varepsilon_{0}}|\mu|^{2} \rho_{z}\left(r_{0},\omega_0\right)$$
The total local density of states is obtained by averaging over the different orientations which translates to averaging the three diagonal G components:
$$\begin{eqnarray}
\rho(r_{0}, \omega) &=& \frac{1}{3}\left[\frac{6 \omega}{\pi c^{2}} \operatorname{Im}\{G_{xx}(r_{0}, r_{0} ; \omega)\}+\frac{6 \omega}{\pi c^{2}} \operatorname{Im}\{G_{yy}(r_{0}, r_{0} ; \omega)\}+\frac{6 \omega}{\pi c^{2}} \operatorname{Im}\{G_{zz}(r_{0}, r_{0} ; \omega)\}\right] \nonumber \\
&=& \frac{2 \omega}{\pi c^{2}} \operatorname{Im}\left\{T r\left[\bar{G}\left(r_{0}, r_{0} ; \omega\right)\right]\right\} \nonumber
\end{eqnarray}$$
Rate of energy dissipation and the built-in dipolepower command
Starting from the poynting theorem, the rate of energy dissipation can be written in terms of the Green's function:
$$\frac{d W}{d t}=\frac{\omega^{3}|\mu|^{2}}{2 c^{2} \varepsilon_{0} \varepsilon_{r}}\operatorname{Im}\left\{G\left(r_{0}, r_{0} ; \omega\right)\right\}$$
In FDTD, there is a built-in command , dipolepower, which will calculate the radiated power of a dipole and is readily available over the range of injected frequencies. See Dipolepower for more details.
The dipolepower command returns a quantity that is related to the imaginary part of the Green's function:
$$\text {dipolepower} = \frac{d W}{d t} | \text {sourcenorm}\left.\right|^{2}=\frac{\omega^{3}|\mu|^{2}}{2 c^{2} \varepsilon_{0} \varepsilon_{r}} \operatorname{Im}\left\{G\left(r_{0}, r_{0}; \omega\right)\right\}$$
sourcenorm is the source normalization spectrum used to normalize data in the cwnorm state for standard Fourier transform quantities. See sourcenorm for more details.
The second part of the script, dipole_gf.lsf, calculates the power radiated by the dipole next to a dielectric sphere normalized to the power that would be radiated by a dipole in a homogenous medium.
This calculation is done using the dipolepower function, the Green's function formulation as well as a box of monitors enclosing the dipole. The box of monitors is useful for dispersive or lossy media where the results of dipolepower are not reliable. All results are in close agreement as shown in the figure below:
Given the close agreement of the dipolepower command with the green's function formulation, when we are interested in calculating LDOS or the decay rate, we can directly use the dipolepower function instead of calculating the imaginary part of Green's function first:
$$\rho_{z}\left(r_{0}, \omega_{0}\right)=\frac{12 \varepsilon_{0} \varepsilon_{r} \text { dipolepower}(f) \text { |sourcenorm}(f) |^2} {\pi \omega^{2}|\mu|^{2}}$$
$$\gamma(\omega)=\frac{8 \varepsilon_{r} \text { dipolepower}(f) \text { |sourcenorm}(f) |^2}{\pi \omega \hbar}$$
The last part of the script dipole_gf.lsf, calculates and plots these. Note that both the decay rate and the LDOS quantities are for the zz direction only. One would need to sum up the results in all three orientations for the total rates.
Related publications
- L. Novotny and B. Hecht, Principles of Nano-Optics, Cambridge (2006).
See also
dipolepower, sourcepower, Transmission box monitor, Dipole source, Fluorescence enhancement example