## Photonics

#### Purcell factor calculation

• Sangyeon_Cho
Subscriber

Hi Lumerical,

I am Sangyeon Cho, a post-doc researcher at Harvard Medical School.

I found that the dipole source generates the result of the Purcell factor after the simulation under the option of recording the local field.

I am trying to use this option to calculate spatial Purcell factor distribution by putting the dipole in the random position.

However, I cannot find the details of how FDTD calculates the Purcell factor using this option on the website.

Would you explain the details of it?

For example, FDTD generates a box with a certain size to measure emitting power? What would be the size of the box?

• Guilin Sun
Ansys Employee
Inside the dipole source, you can directly extract the Purcell factor, which is simply the ratio of actual radiated power and the power in the uniform material. Please refer this article
https://support.lumerical.com/hc/en-us/articles/360041612594-Purcell-factor-of-a-microdisk
and here https://support.lumerical.com/hc/en-us/articles/360034382794-Dipole-source-Simulation-object
In lossy material, the dipolepower function may not work well: dipolepower - Script command
This is why the box monitor group is used. In highly lossy material, it can be very challenge to get accurate radiated power. There is no strict requirement for the box size: if it is too small, the power integration may not be accurate; if it is too large, some power may be absorbed inside the box. So there is always a trade-off.
Another thing to take care is: dipole radiated power may be very sensitive to its location in the Yee cell. This is because the 6 components of E and H internally are interleaved:

ideally, the polarized dipole should be located on the correspondent location, eg, Ex polarized dipole should be located at Ex point in Yee cell. In your case, if it is randomly located, the Purcell factor map may not be very smooth.

• stefan.appel
Subscriber
Hi gsun I do have the same question as Sangyeon_Cho.
How is the Purcell factor calculation implemented for a dipole source, without any additional monitors placed?
So what happens in detail when running dipolpower() or while retrieving the "purcell" or "dipolepower" results from a dipole source?
How can I replicate these results using a point monitor to record electric and magnetic fields at the dipole source position?

I am asking because I would be interested to calculate the emitted power from a dipole (or the purcell enhancement from said dipole) while the dipole is illuminated by a mode source.
The simulation is 2d, the dipole sits in a dielectric slab (no loss, no gain, no metal), and one of the eigenmodes of the slab with known amplitude, power, phase and mode profile hits the dipole from left or right during emission.
As the phase and amplitude/power of the mode will later be changing to arbitrary values, I would prefer NOT to run a FDTD simulation every time, but do an analytic calculation.
Since for the dipole source, a local recording of the fields is sufficient for purcell enhancement calculation, I hope there is a way to calculate how this enhancement changes due to mode fields?

Best, Stefan
• Guilin Sun
Ansys Employee
"How is the Purcell factor calculation implemented for a dipole source, without any additional monitors placed?"
It uses analytical dipolepower : dipolepower - Script command
"How can I replicate these results using a point monitor to record electric and magnetic fields at the dipole source position?"
You cannot use a point monitor to get the power. You have to use box monitor group found in object library : 6 (3) or 4(2D0 plane monitors to enclose the dipole to measure the actual radiate power.
"one of the eigenmodes of the slab with known amplitude, power, phase and mode profile hits the dipole from left or right during emission."
do you mean there are two sources in the simulation? this can be tough, as the initial phases can affect the result, since they can constructively/destructively interact with each other. I would suggest to first simulate dipole only. Once the box monitors work fine, try your own way to simulate under the 2nd source. Please note that there is theoretical complications when two sources are in FDTD region.

• stefan.appel
Subscriber
Hi gsun thank you for you quick answer!
Yes, it is clear to me that one can use a box of monitors to determine the power emitted from a dipole, and one can use this to calculate the Purcell enhancement. I was just interested in how the dipolepower() command does the same without such a box. I hoped that I could use the same method for some analytical calculations.
I know that the relative phase and amplitude between two sources will affect results quite drastically. As I want to add a third source (left propagating mode, right propagating mode and the dipole), this will lead to a 4 dimensional parameter space (2 relative phases and 2 amplitudes) which I would need to sweep through to see the complete behavior.
That's why I had hoped for an analytical solution, such that I do not need those parameter sweeps. It is linear optics after all, and the docs even mention a method based on Green's function for Purcell factor calculation (https://support.lumerical.com/hc/en-us/articles/360042216834-Greens-function-and-local-density-of-states-of-a-dipole-source). However, they do not really go into detail about how to implement it with Lumerical while not using the dipolepower() command, and I fear that there are also some typos on this page.
If you have an idea how to solve this problem, I would be thankful to hear about it!
Best, Stefan
• Guilin Sun
Ansys Employee
from that page:

The dipolepower is calculated by the above formula analytically.
From what you described, I think you will still need to do the sweeps, as we do not know the effects of other sources to the dipole radiated power.