While FDTD can be used to simulate OLED/LED designs of arbitrary geometries, these simulations tend to be very time consuming. Often, a more efficient approach is to start by using analytical methods to analyze the planar structure first. In this section we will explain how the stackdipole command (included in Lumerical's Stack optical solver) can be used to analytically calculate the farfield power density [W/(m2 ×sr)] as a function of polar angle (θ).

For each dipole with given emission angle, stackdipole returns the radiance:
$$stackdipole(\theta) = \int_\lambda (j \times ef \times st)\times \left( \frac{rd \times F_{rad}(\theta, \lambda)}{rd \times F(\lambda) + (1rd)} \right)\times \left(photonprobability(\lambda) \times E_{ph} (\lambda)\right) d\lambda$$
where:
F(λ) 
Purcell Factor (can be calculated using stackpurcell or from FDTD simulations) 
Frad(λ,θ) 
radiated power density with respect to declination angle (θ) (can be calculated using stackpurcell or from FDTD simulations) 
photon probability 
dipole spectrum 
j 
current density (fixed value of 1 Ampere/m2) 
ef 
userspecified exciton fraction defined as #excitons/#injected np pairs (default = 1) 
st 
userspecified singlettriplet ratio defined as #singlet excitons/#excitons (default = 0.25) 
rd 
userspecified relative decay rate in a homogenous environment defined as #photons/#singlet excitons (default = 1) 
Eph 
energy of a photon (h×c/λ) where h is Planck's constant 
This quantity has units of [W/(m2 × sr)]. The Purcell factor (F(λ)) shows enhancement in the spontaneous emission rate of an emitter inside a microcavity and Frad(λ,θ) is the power density. By defining Prad as power radiated into the far field, Pnonrad as power lost due to absorption or captured in evanescent fields, and P0 as the power that would be emitted in an infinite uniform medium, we have:
$$\mathrm{Purcell\ factor} = F(\lambda) = \frac{P_{rad} + P_{nonrad}}{P_0}$$
$$\mathrm{Power\ density} = F_{rad}(\theta, \lambda) = \frac{P_{rad}}{P_0}$$
$$\mathrm{Extraction\ efficiency} = \frac{F_{rad}(\theta, \lambda)}{F(\lambda)} = \frac{P_{rad}}{P_{rad} + P_{nonrad}}$$
The first factor in the integrand, j × ef × st, is the rate of singlet exciton decays, calculated for a fixed current density of 1 A/M2 and userspecified exciton fraction (ef) and singlettriple ratio (st). It is calculated as one Coulomb in atomic units (6.241e+18 electrons) multiplied by the userspecified constants. This quantity outputs the total number of generated photons.
The second is the angular density of the Quantum Yield. Calculation of this quantity in FDTD is described in the next section. The Purcell factor is calculated analytically from the stack geometry using dipole illumination. The relative decay rate (rd) denotes the proportion of singlet decays that produce a photon.
The third is derived from the userspecified intensity spectrum, normalized according to midpoint integration in wavelength. This quantity accounts for the dipole emission spectrum. The stackdipole command will automatically normalize the intensity spectrum.
To calculate Frad(θ) from FDTD simulations, a Frequencydomain field and power monitor is employed to capture the near field profile. Then farfield commands are employed to project near to far field profile. The post processing step for calculating quantum yield is summarized below:
T = transmission(monitorname);
E2 = farfield3d(monitorname);
ux = farfieldux(monitorname);
uy = farfielduy(monitorname);
angDistrib = E2/farfield3dintegrate(E2,ux,uy); # normalized angular distribution
angDistrib = farfieldspherical(angDistrib,ux,uy,θ,0); # mapped into phi=0 to eliminate azimuthal angle
Frad = T * angDistrib;
F = dipolepower(c/λ) / sourcepower(c/λ); # Purcell factor
quantumYieldDensity = rd*Frad/(rd*F + (1rd));
Due to azimuthal symmetry, far field profile is mapped onto the φ=0 plane to eliminate the azimuthal angle. Note that the value φ=0 is suitable for a vertically or randomly oriented dipole. For horizontallyoriented dipoles, either φ=0, φ=90, or average of those results can be used.
The angular distribution is normalized to one. One can check this by integrating it over a half space in spherical coordinates:
?integrate(angDistrib * sin(θ),1,θ)*2*pi; # output should be 1
where θ is an array of values from 0 to π/2.
The stackdipole command can calculate the radiance for randomly oriented, Pvertical, Shorizontal, or Phorizontal orientations. Results for horizontal polarizations are normalized such that:
$$random = \frac{1}{3}P_{vertical} + \frac{2}{3}\left(P_{horizontal} + S_{horizontal} \right)$$
$$horizontal = P_{horizontal} + S_{horizontal}$$
If we define FpVertical , FsHorizontal , FpHorizontal as the farfield radiance calculated from FDTD simulations for vertically oriented dipole, horizontally oriented dipole with φ=90, and horizontally oriented dipole with φ=0, then the farfield radiance of randomly oriented and horizontal randomly oriented dipole are given by:
$$random = \frac{1}{3} \left( F_{pVertical} + F_{pHorizontal} + F_{sHorizontal} \right)$$
$$horizontal = \frac{1}{2} \left(F_{pHorizontal} + F_{sHorizontal} \right)$$
NOTE: FDTD and stackdipole results
From the two equations for 'horizontal', the reader can infer that 1/2(FpHorizontal) is the proper quantity to compare with the Phorizontal result of stackdipole. The same analogy holds for comparing FsHorizontal with Shorizontal i.e. Shorizontal = FsHorizontal /2.
To compare the results of stackdipole with FDTD simulations, we consider a simple case in which the dipole is located in free space, 80 nm above a medium with dielectric constant of 1.5. The FDTD simulation is performed in 3D and the power monitor is located above the dipole (upward) to capture light traveling in the +z direction. To create the plots below, download fdtd_dipole_halfspace.fsp and run fdtd_dipole_halfspace.lsf script. The script will save three different FDTD simulation files for different dipole polarizations along each Cartesian axes, and will compare it with the analytical results calculated directly from the stackdipole command. Since these simulations are for a single frequency source, the photon probability is 1 assuming that all the light is injected at the central frequency.