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/(m^{2 }×sr)] as a function of polar angle (θ).

## Theory

For each dipole with given emission angle, stackdipole returns the radiance:

$$\text{stackdipole}(\theta) = \int_\lambda (j \cdot ef \cdot st)\times \left( \frac{rd \cdot F_{rad}(\theta, \lambda)}{rd \cdot F(\lambda) + (1-rd)} \right) \cdot \left(\text{photon probability}(\lambda) \cdot E_{ph} (\lambda)\right) d\lambda$$

where:

F(λ) |
Purcell Factor (can be calculated using stackpurcell or from FDTD simulations) |

F |
radiated power density with respect to declination angle (θ) (can be calculated using stackpurcell or from FDTD simulations) |

photon probability |
the light spectrum of the dipole normalized to 1 |

j |
current density (fixed value of 1 Ampere/m |

ef |
user-specified exciton fraction defined as #excitons/#injected np pairs (default = 1) |

st |
user-specified singlet-triplet ratio defined as #singlet excitons/#excitons (default = 0.25) |

rd |
user-specified relative decay rate in a homogenous environment defined as #photons/#singlet excitons (default = 1) |

E_{ph } |
energy of a photon \( (h×c)/λ \) where \(h\) is Planck's constant |

This quantity has units of [W/(m^{2} × sr)]. The Purcell factor (F(λ)) shows enhancement in the spontaneous emission rate of an emitter inside a microcavity and F_{rad}(λ,θ) is the power density. By defining P_{rad} as power radiated into the far field, P_{nonrad} as power lost due to absorption or captured in evanescent fields, and P_{0} 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/M^{2} and user-specified exciton fraction (ef) and singlet-triple ratio (st). It is calculated as one Coulomb in atomic units (6.241e+^{18} electrons) multiplied by the user-specified 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 user-specified 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.

### FDTD calculation of quantum yield

To calculate F_{rad}(θ) from FDTD simulations, a Frequency-domain 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 + (1-rd));

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 horizontally-oriented 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.

### Dipole polarization

The stackdipole command can calculate the radiance for randomly oriented, P_{vertical}, or P_{horizontal} orientations. Since STACK is a 1D solver we can only distinguish between horizontal and vertically oriented dipoles. Results for horizontal polarizations are normalized such that:

$$\text{random }= \frac{1}{3}P_{vertical} + \frac{2}{3} P_{horizontal} $$

If we define F_{pVertical} , F_{sHorizontal} , F_{pHorizontal} 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:

$$\text{random } = \frac{1}{3} \left( F_{pVertical} + F_{pHorizontal} + F_{sHorizontal} \right)$$

$$\text{horizontal } = \frac{1}{2} \left(F_{pHorizontal} + F_{sHorizontal} \right)$$

Note: FDTD and stackdipole results From the two equations for 'horizontal', the reader can infer that 1/2(F |

## FDTD vs stackdipole

To compare the results of stackdipole with FDTD simulations, we consider a simple case in which the dipole is located in free space, 80nm 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 f*dtd_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.

### See also

Radiated power from incoherent isotropic point dipole sources, stackdipole, stackpurcell