# Purcell factor calculation: power ratio vs formula

Hi,

Regarding to the calculation methods of determining Purcell factor, we got the “power ratio” method suggested by the following link https://kb.lumerical.com/en/diffractive_optics_cavity_purcell_factor.html and the typical formula which states that Purcell factor is proportional to the ratio of quality factor and effective mode volume in textbooks. I encountered difficulties that these two methods are not consistent for the same simulation. In regard to my simulation, the power-ratio method gives Purcell factor 2.3 while the other gives 27. Which one should I trust or is their any way to make judgement ?

A lot of thanks.

Hi,

The quality factor method requires an accurate mode volume. What is the volume that you considered for your calculation?

Best Wishes,

Vivek

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In addition, there are different definitions for the mode volume. In our analysis group mode_volume we provide 3 popular methods however this does not exclude other definitions.

The difference of 2.3 and 2.7 is not significant and you can use either of them by mentioning your calculation method.

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Hi,

Thank you for your inputs. I’m currently working on a simulation of micropillar cavity, trying to decide its Purcell factor by calculation of its Q and V. I think I found the point where simulations went inconsistent. The problem may comes from the result of the Qanalysis group. Normally a broadband source should be used first to locate the resonance in spectrum followed by another simulation with single wavelength which is at resonance to calculate the Q factor. I got my Q factor directly from the broadband simulation result. I guess that’s the problem and I’m working on it.

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Hi all,

Regarding to this topic of Purcell factor calculation, I got a simulation problem to ask for help. Back to the official page on deciding Purcell factor, https://kb.lumerical.com/en/diffractive_optics_cavity_purcell_factor.html . Therein a two-step process is suggested: first enable a preliminary simulation using a broadband source to determine the resonant frequency and followed by a second simulation with a dipole source enabled at the resonance frequency. My problem is , in some optical systems where broadband enhanced spontaneous emission are expected such as waveguide, plasmonic and metameterials…etc. How can I calculate the Purcell factor in these systems (which our members are currently working on)? The tutorial seems only valid for typical narrow peak Purcell enhancement. I would try the power ratio method here with broadband source used only. Deeply appreciate.

for broadband, the power method is more efficient.

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Hi all experts:
I still find inconsistent of Purcell factor among official power method and typical formula. Here I provide my simulation file to help you get into my problem.
Purcell_factor.fsp (281.9 KB)
calc_purcell.lsf (650 Bytes)

The simulation process closely follows the steps suggested by Lumerical tutorial ( https://kb.lumerical.com/en/diffractive_optics_cavity_purcell_factor.html ) and I just briefly list my work flow below: (My attached file has already done step 1 and 2. So just run my script “calc_purcell” to see final result).

1. First run a preliminary simulation using a broadband source(0.3um) to determine the resonant frequency.
2. By plotting the spectrum result from the Qanaysis group, I find that the resonant frequency of the mode is at around 1575 nm.
3. I then set the dipole source frequency to the resonant frequency and rerun the simulation.
4. Running the script in attachment, it shows:
calc_purcell;
Purcell factor by power ratio: 5.57409
Purcell factor by official method: 5.57409
Purcell factor by formula: 77.7994

There is an order of difference !
Notes:
The mode volume is calculated by type 2.
The simulation was tested on 2016 version.

I appreciate your help and technical support !

I still believe the problem is from the modal volume calculation.

The monitor you used is too small to catch the whole mode. Most importantly, when we say modal volume, we know it is for a specific mode. The monitor may record more than one mode so how do you justify that it is the correct mode you want to calculate?

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Thank you for your inputs. But I still get lost on following issues,

1. Why my monitor of modal volume is too small ? It covers almost the whole pillar. The uncovered substrate however just has limited leakage field.
2. The monitor has been set to record the mode with specific resonant frequency from single frequency source. I’m not sure if I catch your point. Would you be more specific on “the monitor may record more than one mode” ?
3. Based on your advises, are you suggesting that Purcell factor of 5.57409 obtained from the result data of dipole source is probably right or trustworthy due to ruling out the issue of justifying modal volume?

Thank you.

I guess that there are different methods to define the modal volume. You may need to use them and compare the results.

Although you only used a single frequency at source, it can excite more than one spatial mode! to verify that the recorded mode is the mode you want, please use a modal expansion monitor to check if it is true.

Since now the dipole is located inside a non-dispersive material and the mesh is not very fine, I would assume the dipolepower should give the correct result.

We first need to figure out the mode issue and mode volume.

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Hi Dr. Sun