# Application Gallery

 Purcell factor

The calculation method for the Purcell factor is discussed and the Purcell factor for a microdisk resonator is calculated.

FDTD

### cavity_purcell_factor.fsp

dipolepower

sourcepower

Quality factor calculations

Mode volume

apodization

### Related publications

Ultrasmall Mode Volume Plasmonic Nanodisk Resonators Martin Kuttge, F. Javier García de Abajo and Albert Polman Nano Lett., 2010, 10 (5), pp 1537–1541 DOI: 10.1021/nl902546r

L. Novotny and B. Hecht, "Principles of Nano-Optics", Cambridge University Press, Cambridge, (2006).

Philip Kristensen, Cole Van Vlack, Stephen Hughes, Generalized mode volume for leaky optical cavities, Published in Optics Letters (May, 2012)

A. Faraon et al., "Resonant enhancment of the zero-phonon emission from a colour centre in a diamond cavity", Nature Photonics 5, 301-305 (2011)

## Calculation method

The Purcell factor is the emission rate enhancement of a spontaneous emitter inside a cavity or resonator. Dipoles in FDTD automatically return the Purcell factor as a result that you can directly visualize or access from the Result View window after a simulation is run.

The Purcell factor result is equivalent to dividing the power emitted by a dipole source in the environment by the power emitted by the dipole in a homogeneous environment (bulk material) since the emission rate is proportional to the local density of optical states (LDOS), and the LDOS is proportional to the power emitted by the source. You can use the following line of code to verify the result.

purcell_factor=dipolepower(f)/sourcepower(f);

The command dipolepower returns the power actually radiated in the environment, and sourcepower returns the power that would be radiated by a dipole in a homogeneous medium. This ratio of power is equal to the decay rate enhancement, or Purcell factor (please see Fluorescence enhancement where this approach is used to calculate the decay rate enhancement of fluorphores near metallic particles and the Novotny, "Principles of Nano-Optics" reference). The same result can be obtained from the "purcell" result of the dipole source after the simulation has been run.

It’s important to note that this analysis method depends on having the simulation run until the fields have completely decayed. As well, the dipolepower command only works if your dipole is in a lossless dielectric medium. If the dipole is embedded in a dispersive medium then instead of using the dipolepower command, you can get the power emitted by the source by measuring the power flow out of a small box of monitors surrounding the source. The transmission box analysis object from the Object library can be added for this purpose.

Another method for measuring the Purcell factor is by using the following formula:

$$F_{P}=\frac{3}{4 \pi^{2}}\left(\frac{\lambda}{n}\right)^{3} \frac{Q}{V}$$

Q is the quality factor and V is the mode volume calculated using:

$$V=\frac{\int_{V} \varepsilon(r)|E(r)|^{2} d^{3} r}{\max \left(\varepsilon(r)|E(r)|^{2}\right)}$$

Where ε(r) is the material permittivity at position r.

Based on the discussion in the paper referenced above by A. Faraon, this assumes that the dipole is ideally positioned and oriented to give the maximum Purcell factor.

## Example

In this example we calculate the Purcell factor of a microdisk resonator structure using the first method of measuring the emitted power.

The Purcell factor will depend on location and frequency so we first run a preliminary simulation using a broadband source to determine the resonant frequency and location of the strongest modal fields. For the preliminary simulation, make sure the broadband source is enabled, and the dipole source is disabled. The time monitor is set up to start recording fields after 20 fs when the transient fields have decayed so that only resonant fields of the structure remain. By plotting the spectrum result from the time monitor, we find that the resonant frequency of the mode is at 714 nm.

Using this information we can set up the profile monitor to record data at 714 nm. The profile monitor is also set up to use apodization to exclude the effect of the source.

The following shows the mode profile at 714 nm.

We then set the dipole source frequency to the resonant frequency and place the dipole in the location where the modal fields are maximized. With the dipole source enabled, and the broadband source disabled, re-run the simulation. We find a Purcell factor of about 2354 which is similar to the Purcell factor calculated using the boundary element method from the reference. Note that we only used one simulation with one dipole orientation to determine the Purcell factor since this orientation results in the maximum Purcell factor, and the maximum Purcell factor is what we are looking for. In the case where you want to Purcell factor of an isotropic emitter, you can average the results from three orthogonal dipole orientations.

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