# Dipole Power/Purcell Factor in a Thin Solid Film

#1

Hello,

From the lumerical website: “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).”
So I ran three very basic simulations with - 1) an electric dipole emitting 1550 nm polarized in either X, Y or Z direction, 2) dipole is inside a 140nm thick film with n = 2.7.
The purcell factor’s value I got is - 0.94 for both X and Y directional dipoles, but 0.024 for Z directional dipole.

My question is, why the purcell factor is so low for Z-directional dipole? Does it imply that the Z-dipole is radiating low power, thus have low number of available local density of optical states? If that’s the case, why is it only happening for Z-dipole?

I have attached my simulation files as well.

Thank you in advance for the help.Dipole power_1550nm xdipole.fsp (279.8 KB)
Dipole power_1550nm ydipole.fsp (279.8 KB)
Dipole power_1550nm zdipole.fsp (279.8 KB)

#2

I did some convergence testing and the results were converging but I do not have a clear answer for you. Were you expecting something else or do you have any publication that proves these results are right or wrong?

Thanks

#3

My goal was to get the radiative transition rate of a dipole (mimicking an ion) in a dielectric slab compared to homogeneous medium.
As the website ( https://apps.lumerical.com/diffractive_optics_cavity_purcell_factor.html) said- " _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."
So,

• Why z-directional dipole’s purcell factor value is so low?
• Does this means, for z-dipole there are not may states/modes available?
• Do you have theory/ref to help me to explain this result?

#4

I checked the calculations with stackpurcell command (which provides a theoretical results and you need a license for it to use it), and obtained same results as FDTD. The X and Y polarizations are essentially behave the same due to symmetry, but there is less density of states for Z-polarized dipole. As a results the value is lower. Unfortunately I do not have a paper to refer, but the stackdpurcell results are quite reliable. Below is the script that I used to check the results with stackpurcell command:
https://kb.lumerical.com/en/ref_scripts_stackpurcell.html

# frequency range

clear;

Nfreqs = 101;

lambda = linspace(1500e-9,1600e-9,Nfreqs); # 380nm to 780nm

f = c/lambda;

# multilayer geometry

n = matrix(3,Nfreqs); d = matrix(3,1);

n(1,1:Nfreqs) = 1; d(1) = 0; # bottom substrate

# Note: this reads the material properties from FDTD Solutions material database. If you are using another product, please entire (n,k) explicitly

n(2,1:Nfreqs) = 2.7; d(2) = 140e-9;

n(3,1:Nfreqs) = 1; d(3) = 0;

# dipole positions/orientations

Ndipole= 1; # number of dipoles

z = d(2)/2; # consider positions only within middle dielectric layer

orientation = Ndipole * 1; # consider only randomly oriented dipoles

# angular_res: resolution for emission angle (farfield angle)

res = 198;

result = stackpurcell(n,d,f,z,orientation,res); # result is a struct

# plot Purcell factor for dipole located at the middle of the layer

purcell = pinch(result.power.purcell_factor); # size is Ndipole by Nfreqs

plot(lambda*1e9, purcell,'wavelength (nm)','Purcell factor');

# plot far field power density at center frequency

#theta = result.density.theta;

#density = pinch(result.density.upward_into_air); # size is res by Ndipole by Nfreqs

#image(theta, z*1e+9, pinch(density,3,round(Nfreqs/2)), &quot;far-field angle (degrees)&quot;, &quot;dipole position (nm)&quot;, num2str(f(round(Nfreqs/2))*1e-12)+&quot;THz into Air&quot;);