Dipole Model and Applicability

An OLED emissive layer is not a collection of tiny lamps radiating uniformly in every direction. Each transition dipole has an axis: its far-field radiation vanishes along that axis and is strongest perpendicular to it.

Planar emission simulation represents molecular, quantum-dot, or perovskite emission centers as oscillating electric dipoles, then combines their orientations, positions, and spatial distributions into an emissive layer. This article defines that source model and the conditions under which it represents a real device.

Radiation Pattern of a Point Dipole

When the spatial extent of an emission center is much smaller than the wavelength, its lowest-order radiation can be represented by an electric dipole moment $\mathbf p$. In free space, the far-field power per unit solid angle obeys

where $P$ is the total dipole radiation power, $\mathrm dP/\mathrm d\Omega$ is the power per unit solid angle, $\Omega$ is solid angle, and $\alpha$ is the angle between the observation direction and the dipole axis. Radiation is zero at $\alpha=0$ and maximal at $\alpha=90^\circ$.

In a planar device, a horizontal or parallel dipole lies in the layer plane, while a vertical or perpendicular dipole points along the layer normal. A horizontal dipole has strong radiation near the normal and couples more readily into the escape cone. For each propagation direction, the wavevector and layer normal define the plane of incidence: the TE electric field is perpendicular to that plane, while the TM electric field lies within it. A vertical dipole has no far-field radiation along the normal and produces only TM components, making coupling to large-angle guided states and surface plasmon polaritons (SPPs)—TM modes bound to metal–dielectric interfaces—more likely.

Geometric Escape-Cone Approximation

The escape cone in Figure 2 gives an efficiency estimate based only on propagation direction. For an emissive-layer index $n_{\mathrm{EML}}$ and external-medium index $n_{\mathrm{out}}$, the critical angle is

Here, $\theta_c$ is measured from the layer normal and requires $n_{\mathrm{EML}}>n_{\mathrm{out}}$. Let $\theta$ be the angle between the radiation direction and the layer normal. After integration over azimuth, the normalized angular power distributions of horizontal and vertical dipoles are

$g_h(\theta)\,\mathrm d\theta$ and $g_v(\theta)\,\mathrm d\theta$ are the fractions of total horizontal- or vertical-dipole power radiated between $\theta$ and $\theta+\mathrm d\theta$; each integrates to 1 over $0\le\theta\le\pi$. Integration over the one-sided upper escape cone gives

$f_h^{+}$ and $f_v^{+}$ are the fractions of total radiation sent directly into the upper escape cone. Figure 3 further assumes a phase-free, lossless perfect back reflector, so the identical downward-radiated fractions are redirected upward and the geometric outcoupling efficiencies become

$\eta_h^{\mathrm{geo}}=2f_h^{+}$ and $\eta_v^{\mathrm{geo}}=2f_v^{+}$ are the horizontal- and vertical-dipole light-extraction efficiencies predicted by this geometric model; light-extraction efficiency is abbreviated as LEE, and the superscript $\mathrm{geo}$ denotes the approximation. Without the ideal back reflector, both results are halved. An isotropic orientation ensemble contains two horizontal degrees of freedom and one vertical degree of freedom, so

Substitution of the preceding expressions reduces the isotropic result to

When $n_{\mathrm{out}}/n_{\mathrm{EML}}$ is small, a first-order expansion of the square root gives

For air, $n_{\mathrm{out}}\approx1$, producing the solid blue $1/(2n_{\mathrm{EML}}^2)$ curve in the figure. It is an approximation to the isotropic geometric result, not a separate law of emission. At $n_{\mathrm{EML}}=1.57$, the horizontal, vertical, exact isotropic, and $1/(2n^2)$ approximate values are about $30.7\%$, $7.3\%$, $22.9\%$, and $20.3\%$, respectively. Differences between the orientation curves arise from the dipole radiation patterns; their common decline comes from the narrowing escape cone at higher refractive index.

Wave-Optical Model of an OLED

A real OLED violates all of the assumptions above: its layer thicknesses are comparable to the wavelength; reflection has wavelength-dependent amplitude and phase; electrodes and functional layers absorb; guided and SPP states exist outside the escape cone; and the reflected field returns to the dipole and changes its total electromagnetic decay rate through the Purcell effect.

Dreapex TMM decomposes a dipole of orientation $o$ over normalized in-plane wavevector $u$ and calculates the complex amplitude of every component through the complete stack. When power is normalized to total radiation in a homogeneous reference environment, the total electromagnetic dissipation and target outcoupling efficiency are

Here, $o\in\{h,v\}$ denotes horizontal or vertical orientation; $\lambda$ is vacuum wavelength; $z$ is the dipole position in the EML; $u=k_{\parallel}/(n_{\mathrm{EML}}k_0)$, where $k_{\parallel}$ is in-plane wavevector and $k_0=2\pi/\lambda$ is vacuum wavenumber; $K_{o,\mathrm{tot}}$ is normalized total dissipated-power density per unit $u$, while $K_{o,\mathrm{out}}$ is the power density transmitted through the complete stack and across the target external boundary; $F_o$ is the orientation-dependent Purcell factor, or total electromagnetic decay relative to a homogeneous reference environment; $\mathcal U_{\mathrm{out}}$ is the interval associated with the target external-medium light cone; and $\eta_{\mathrm{out},o}$ is the fraction of electromagnetic-channel power that actually crosses the target external boundary.

Both power densities come from the same layered dipole field, which includes microcavity interference, Fresnel reflection and transmission, material absorption, waveguiding, and SPP coupling. $F_o$ is no longer fixed at 1, showing that the layered environment changes both energy partition and total dipole decay. Figure 3 contains no $\lambda$, $z$, layer thickness, complex refractive index, or reflection phase, so it cannot model an OLED quantitatively.

Orientation Distribution

An isotropic orientation ensemble contains two orthogonal horizontal degrees of freedom and one vertical degree of freedom, giving

Here, $K_{\mathrm{iso}}$ is the power density of an isotropic dipole ensemble, $K_h$ is the mean power density of horizontal dipoles, and $K_v$ is the power density of vertical dipoles. The weights $2/3$ and $1/3$ arise from the three-dimensional orientation degrees of freedom; they do not mean that each molecule has three simultaneous dipole axes.

Using the vertical-dipole fraction $a$ for an arbitrary uniaxial orientation distribution,

Here, $K(a)$ is the ensemble power density; $a=0$ is fully horizontal, $a=1$ is fully vertical, and $a=1/3$ is equivalent to isotropic orientation. The commonly reported horizontal-dipole ratio is $1-a$. Molecular shape, substrate temperature, and deposition kinetics can all change this ratio.

Emission Position and Spatial Distribution

Dipoles with the same orientation see different reflection phases and near-field environments at different depths in the EML. A single position suits a narrow recombination zone; a broader zone requires a normalized distribution $\rho(z)$:

Here, $z_1$ and $z_2$ are the EML boundaries, $\rho(z)$ is the dipole probability density through the layer thickness, $X(z)$ is target-channel power, $F_o$, or another single-dipole result at position $z$, and $\langle X\rangle$ is the spatial average. Point, exponential, Gaussian, and custom-file sources are different forms of $\rho(z)$.

Approaching a reflector changes the interference phase; approaching a metal also strengthens evanescent coupling. EML thickness alone therefore does not fully define the source—the location of the recombination zone inside that layer matters as well.

Applicability Conditions

Finite metals and other absorbing layers may remain inside the device; their energy is assigned to absorption or evanescent channels. A bottom-outcoupled channel exists when the bottom external medium is transparent; it is zero for an absorbing bottom boundary.

Next Step

Continue with Microcavity Effect to see how reflecting interfaces transform the same dipole source into different spectra, colors, and angular distributions.

Reference

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