Generalized Poynting vector model to calculate the spatial and spectral profiles of the electric field intensity, optical power flow, and optical absorption for all optical modes of organic light-emitting diodes

Jiyong Kim; Kyoung-Youm Kim; Jungho Kim

doi:10.1364/OE.27.0A1261

1. Introduction

Optical modeling of the organic light-emitting diode (OLED) plays an important role in determining its multilayer structure to optimize the micro-cavity effect, which results from the optical interference effect within multiple thin layers on the order of micrometer [1–3]. Various output emission characteristics such as the quantum efficiency and angular emission distribution are determined by the micro-cavity effect, which can be tuned by means of the change in the thickness or complex refractive index of the multilayer in the OLED. According to the classical electromagnetic approach, dipole emission in the OLED can be described by the classical radiation theory of an electric dipole antenna, which is equivalent to the quantum-mechanical approach to an electric dipole transition coupled to optical modes such as the air, substrate, waveguide (WG), and surface plasmon polariton (SPP) modes [4–6]. In general, three electromagnetic models have been widely used in the optical modeling of the OLED, which can be classified into the Green’s function model [6–8], the dipole radiation model [9–12], and the point dipole model (PDM) [13–16].

The classical electromagnetic approach was first presented by Chance, Prock, and Sibley (CPS), who introduced two methods of the Green’s function and dipole radiation models to calculate the lifetime of an excited molecule embedded in layered media [7]. In the CPS theory, the modification of the decay rates of the excited dipole near the stratified layer was described through integral expressions over the normalized in-plane wave vector u, which included the effect of the reflected electric field at the dipole position. In the Green’s function model, the electric field caused by an oscillating current was expressed through the dyadic Green’s function, which was described using two independent sets of eigenfunctions in cylindrical coordinates [7,8]. In the case of the dipole radiation model, the electric field, originating from the dipole source, was described based on the Hertz vectors. This could be re-written through an integral expression of Bessel functions times the normalized in-plane wave vector when the Sommerfeld expansion was applied in the cylindrical coordinate [7,9]. In both models, the application of the appropriate boundary conditions gave rise to the respective integral formula to calculate the decay rate of the excited dipole [7]. The PDM, proposed by Lukosz, was similar to the dipole radiation model except that the Hertz vectors were expanded based on a Fourier integral over the normalized in-plane wave vectors, where the Fourier components represented the propagating and evanescent plane waves [13].

Several research groups improved the above-mentioned three electromagnetic models and showed a good agreement between calculation and experimental results [8,11,12,14,15]. The three electromagnetic models have focused on the optical modeling of output emission characteristics as well as the modal analysis of the dipole emission in the OLED. To simulate the output emission characteristics, the air mode in the output ambient was calculated with respect to the wavelength, viewing angle, and dipole orientation [11,12,15]. In the case of the modal analysis, the power dissipation spectrum was calculated to quantify the excitation efficiency of the dipole emitter into the air, substrate, WG, and SPP modes [12,15,16]. However, three electromagnetic models have not yet provided the spatial distribution of the electric field intensity and optical power flow inside the OLED multilayer, which are essential to gain a complete understanding of the optical emission process occurring inside the OLED.

A combined optical model to integrate the electromagnetic and quantum-mechanical approaches can provide a comprehensive understanding of the micro-cavity effect in OLEDs [17]. According to the equivalence of the optical modeling of the OLED between the electromagnetic and quantum-mechanical approaches, the two-beam and multi-beam interference effects in the electromagnetic treatment were matched with the electric field intensity at the emitter position and the density of states of optical modes in the quantum approach [17,18]. Thus, it is important to have accurate information on the spatial profile of the electric field intensity of optical modes. The effective mode index and electric field profile of the WG and SPP modes, having a closed boundary on both sides, were calculated through the boundary eigenvalue solution of the wave equations [17,19]. In the case of the air mode with an open boundary in the air, the spatial profile of the electric field intensity was considered to sinusoidally oscillate inside the multilayers and exponentially decrease at the metal electrode [20]. Nonetheless, the above-mentioned methods to obtain the electric field profile of optical modes did not consider the effect of the dipole source such that the interaction between the OLED micro-cavity and the dipole emitter was assumed to lie in the weak-coupling regime without proof [21].

The spectral and spatial profiles of the optical absorption within the multilayer of the OLED can be used to provide a deeper understanding on the heat generation mechanism in the thermal analysis of the OLED [22,23]. According to three electromagnetic models, the spectral distribution of optical absorption in the WG and SPP modes was calculated based on the power dissipation spectrum, where the total area for the WG and SPP modes contributed to optical absorption [8,11,14,16]. In the case of the air mode, the corresponding optical absorption was calculated by subtraction of both the transmitted and reflected components of the power dissipation spectrum from the total power dissipation spectrum in the air-mode region [14,16]. Based on the dyadic Green’s function, optical absorption within the multiplayer of an OLED was calculated with respect to the longitudinal position and normalized in-plane wave vector [8]. However, both spatial and spectral profiles of the optical absorption obtained by the Green’s function model could not discriminate the polarization dependency although the dipole orientation dependency could be distinguished.

There has been another electromagnetic model of the so-called source-term method, proposed by Banisty et al. [24]. This model expressed the emission characteristic of a dipole emitter as the electric-field source term in the matrix formalism, which required only the matrix calculation in the optical modeling of OLEDs without any complicated analytical integration. Based on the source-term method, our group derived a generalized Fabry-Pérot formulation to calculate the effect of the dipole orientation and light polarization on the output emission characteristics using a simple analytical equation [18]. In addition, we mathematically proved that the theoretical expression of the spectral power density in the PDM was mathematically equivalent to that of the generalized Fabry–Pérot formulation in the air mode region. However, the application of this source-term method has been limited to the air mode, not being used in the analysis of trapped WG and SPP modes [18,25].

Recently, our group also has presented a Poynting vector model (PVM), by combining the transfer matrix method (TMM) and source-term method, to calculate the spatial profile of the electric field intensity, optical power flow, and optical absorption of the air mode both inside and outside of the top-emitting OLED [26,27]. We showed that the calculated spatial distribution of the electric field intensity of the air mode in the emission layer (EML) agreed with the emission zone profile obtained by output emission intensity based on the PDM [26]. In addition, we calculated the spatial distribution of the time-averaged Poynting vector of the air mode, where the calculated optical power and absorption loss were equal to those calculated by the PDM [27]. Nonetheless, there has not yet been any theoretical model to simultaneously calculate the spatial distribution of the electric field intensity, optical power flow, and optical absorption of WG and SPP modes with the effect of the dipole source considered.

In this paper, we propose a generalized Poynting vector model (GPVM) to simultaneously calculate the spatial and spectral profiles of the electric field intensity, optical power flow, and optical absorption together with the power dissipation spectrum for all optical modes of OLEDs. The theoretical formulation of the GPVM is derived by converting an emission angle into the normalized in-plane wave vector in the Poynting vector model. In the GPVM, the theoretical expression of the spectral power density is mathematically derived from the optical power at the EML, which proves to be identical to the theoretical formulation of the spectral power density presented by the currently-used PDM. In a bottom-emitting OLED, the power dissipation spectrum is calculated with respect to the light polarization and dipole orientation. For the WG and SPP modes confined at the bottom-emitting OLED, the spatial profiles of the electric field intensity obtained by the GPVM are compared with those calculated by the boundary modal analysis (BMA) of the wave equation. Finally, two-dimensional (2D) plots of the internal optical power flow and optical absorption are obtained in the bottom-emitting OLED with respect to the longitudinal position and normalized in-plane wave vector.

2. Theory

A schematic diagram of the multilayer structure of the OLED is described in Fig. 1. Each thin layer j (j = 1,2, ∙∙∙,n) has a thickness of d_j and a complex refractive index of ${\tilde{n}}_{j} = n_{j} + i κ_{j},$ which indicates the refractive index and extinction coefficient, respectively. All layers of the OLED are assumed to be isotropic and homogeneous with planar and parallel interfaces. For the sake of simplicity, the EML (j = W) is considered to be lossless in accordance with other optical models of OLEDs [7,8,10,14,15]. The dipole emitter assumes to be confined at the source plane, which is z_ex away from the left boundary of the EML. Forward- and backward- propagating electric field amplitudes are denoted as the + (-) superscripts while TE and TM polarizations are marked in the TE(TM) superscripts. The electromagnetic wave is assumed to be extracted into the semi-infinite transparent ambient layer on the left (j = 0) and right (j = n + 1), which has one electric-field component of $E_{0}^{-, T E (T M)} (j = 0)$ and $E_{n + 1}^{+, T E (T M)} (j = n + 1)$ respectively.

Fig. 1 Schematic diagram of the multilayer structure of the OLED enclosed with semi-infinite ambient layers. It is assumed that the dipole emitter is located at the source plane of the EML. In each layer, the forward- and backward-propagating electric field amplitudes (E) are denoted as + (-) superscripts. The TE and TM polarizations are separately represented by TE(TM) superscripts.

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Figure 2(a) shows the sign configuration of the TE- and TM-polarized electric and magnetic field amplitudes on reflection and transmission at the interface based on the Fresnel convention, where their Fresnel reflection and transmission coefficients are described in Eq. (3). When the dipole emitter in the EML radiates the electromagnetic wave with the internal emission angle of $θ_{e}$ and the total wave vector of ${\vec{k}}_{e},$ a schematic diagram of the wave vector component is shown in Fig. 2(b). The magnitudes of the in-plane and out-of-plane components are denoted as $| {\vec{k}}_{t, e} | = | {\vec{k}}_{e} | \sin θ_{e} = | {\vec{k}}_{e} | u$ and $| {\vec{k}}_{z, e} | = | {\vec{k}}_{e} | \cos θ_{e} = | {\vec{k}}_{e} | \sqrt{1 - u^{2}},$ where $u = | {\vec{k}}_{t, e} | / | {\vec{k}}_{e} | = \sin θ_{e}$ is the normalized in-plane wave vector [7,8,10,14,15]. In the case of $0 \leq u \leq 1$ , the dipole emission with the corresponding internal emission angle of $0^{\circ} \leq θ_{e} \leq 90^{\circ}$ is coupled into the air, substrate, WG modes, consecutively. On the other hand, the in-plane and out-of-plane wave vectors of the SPP mode can be described with respect to only u because the internal emission angle of $θ_{e} = \sin^{- 1} u$ cannot be defined at $u > 1.$ To describe the light emission characteristics of all optical modes, the wave vector components will be described in terms of the normalized in-plane wave vector u later [7,8,10,14,15].

Fig. 2 (a) Sign configuration of the TE- and TM-polarized electric and magnetic field amplitudes on reflection and transmission at the interface based on the Fresnel convention. (b) Schematic diagram of the wave vector component in the EML. When ${\vec{k}}_{e}$ denotes the total wave vector in the EML, ${\vec{k}}_{t, e}$ and ${\vec{k}}_{z, e}$ represent the in-plane and out-of-plane components. The normalized in-plane wave vector is given by $u = | {\vec{k}}_{t, e} | / | {\vec{k}}_{e} | = \sin θ_{e},$ where the internal emission angle of θ_e can be defined only for $0 \leq u \leq 1.$

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2.1 Determination of the electric field amplitudes at the left and right boundaries of the source plane

According to the TMM, the layer matrix $(\bar{L})$ and the interface matrix $(\bar{I})$ are used to describe the light propagation in multiple thin layers. The propagation of the electric field amplitude over a distance d_j between the left and right boundaries of the j-th layer is given by

[\begin{matrix} E_{j, L}^{+, T E (T M)} \\ E_{j, L}^{-, T E (T M)} \end{matrix}] = {\bar{L}}^{j} (d_{j}) [\begin{matrix} E_{j, R}^{+, T E (T M)} \\ E_{j, R}^{-, T E (T M)} \end{matrix}] = [\begin{matrix} \exp (- i k_{z, j} d_{j}) & 0 \\ 0 & \exp (i k_{z, j} d_{j}) \end{matrix}] [\begin{matrix} E_{j, R}^{+, T E (T M)} \\ E_{j, R}^{-, T E (T M)} \end{matrix}] .

Here, the time dependence of exp(iωt) is assumed while L and R denote the left and right boundaries of the each layer. The electric field amplitudes at the interface between the j-th and (j + 1)-th layer are

[\begin{matrix} E_{j, R}^{+, T E (T M)} \\ E_{j, R}^{-, T E (T M)} \end{matrix}] = {\bar{I}}^{j / (j + 1)} [\begin{matrix} E_{(j + 1), L}^{+, T E (T M)} \\ E_{(j + 1), L}^{-, T E (T M)} \end{matrix}] = \frac{1}{t_{j (j + 1)}^{T E (T M)}} [\begin{matrix} 1 & r_{j (j + 1)}^{T E (T M)} \\ r_{j (j + 1)}^{T E (T M)} & 1 \end{matrix}] [\begin{matrix} E_{(j + 1), L}^{+, T E (T M)} \\ E_{(j + 1), L}^{-, T E (T M)} \end{matrix}],

where

r_{j / (j + 1)}^{T E (T M)}

and

t_{j / (j + 1)}^{T E (T M)}

represent the complex Fresnel reflection and transmission coefficients at the interface from the j-th to (j + 1)-th layers. On the basis of the Fresnel convention, they are expressed as

\begin{array}{l} r_{j / (j + 1)}^{T E} = \frac{\sqrt{{\tilde{n}}_{j}^{2} - {\tilde{n}}_{e}^{2} u^{2}} - \sqrt{{\tilde{n}}_{(j + 1)}^{2} - {\tilde{n}}_{e}^{2} u^{2}}}{\sqrt{{\tilde{n}}_{j}^{2} - {\tilde{n}}_{e}^{2} u^{2}} + \sqrt{{\tilde{n}}_{(j + 1)}^{2} - {\tilde{n}}_{e}^{2} u^{2}}}, r_{j / (j + 1)}^{T M} = - \frac{\sqrt{{\tilde{n}}_{j}^{2} - {\tilde{n}}_{e}^{2} u^{2}} / {\tilde{n}}_{j}^{2} - \sqrt{{\tilde{n}}_{(j + 1)}^{2} - {\tilde{n}}_{e}^{2} u^{2}} / {\tilde{n}}_{(j + 1)}^{2}}{\sqrt{{\tilde{n}}_{j}^{2} - {\tilde{n}}_{e}^{2} u^{2}} / {\tilde{n}}_{j}^{2} + \sqrt{{\tilde{n}}_{(j + 1)}^{2} - {\tilde{n}}_{e}^{2} u^{2}} / {\tilde{n}}_{(j + 1)}^{2}}, \\ t_{j / (j + 1)}^{T E} = \frac{2 \sqrt{{\tilde{n}}_{j}^{2} - {\tilde{n}}_{e}^{2} u^{2}}}{\sqrt{{\tilde{n}}_{j}^{2} - {\tilde{n}}_{e}^{2} u^{2}} + \sqrt{{\tilde{n}}_{(j + 1)}^{2} - {\tilde{n}}_{e}^{2} u^{2}}}, t_{j / (j + 1)}^{T M} = \frac{2 \sqrt{{\tilde{n}}_{j}^{2} - {\tilde{n}}_{e}^{2} u^{2}}}{\frac{{\tilde{n}}_{(j + 1)}}{{\tilde{n}}_{j}} \sqrt{{\tilde{n}}_{j}^{2} - {\tilde{n}}_{e}^{2} u^{2}} + \frac{{\tilde{n}}_{j}}{{\tilde{n}}_{(j + 1)}} \sqrt{{\tilde{n}}_{(j + 1)}^{2} - {\tilde{n}}_{e}^{2} u^{2}}} . \end{array}

Here,

{\tilde{n}}_{e}

is the refractive index of the lossless EML, which has no extinction coefficient.

When $E_{a}^{+ (-), T E (T M)}$ and $E_{b}^{+ (-), T E (T M)}$ represent the electric field amplitude at the left and right boundaries of the source plane, the relationship between them is written as

[\begin{matrix} E_{a}^{+, T E (T M)} \\ E_{a}^{-, T E (T M)} \end{matrix}] + [\begin{matrix} A_{h (v)}^{+, T E (T M)} \\ A_{h (v)}^{-, T E (T M)} \end{matrix}] = [\begin{matrix} E_{b}^{+, T E (T M)} \\ E_{b}^{-, T E (T M)} \end{matrix}] .

In the above equation,

A_{h (v)}^{+ (-), T E (T M)}

represents the normalized power density of the oscillating dipole current per unit solid angle, which induces additional discontinuous boundary conditions at the source plane [20]. The source term can be given by [20]

\begin{array}{l} A_{h}^{+ (-), T E} = - (+) \sqrt{\frac{3}{16 π}}, A_{h}^{+ (-), T M} = - (+) \sqrt{\frac{3}{16 π}} \sqrt{1 - u^{2}}, \\ A_{v}^{+ (-), T E} = 0, A_{v}^{+ (-), T M} = \sqrt{\frac{3}{8 π}} u, \end{array}

where h and v stand for the horizontal and vertical orientations of the dipole emitter, respectively. In Fig. 1, the electric field amplitudes of

E_{0}^{-, T E (T M)}

and

E_{n + 1}^{+, T E (T M)}

in the transparent ambient layers are related to the electric field amplitudes at the left and right boundaries of the source plane, which are expressed as

[\begin{matrix} 0 \\ E_{0}^{-, T E (T M)} \end{matrix}] = {\bar{S}}^{A} [\begin{matrix} E_{a}^{+, T E (T M)} \\ E_{a}^{-, T E (T M)} \end{matrix}] = [\begin{matrix} s_{11}^{A, T E (T M)} & s_{12}^{A, T E (T M)} \\ s_{21}^{A, T E (T M)} & s_{22}^{A, T E (T M)} \end{matrix}] [\begin{matrix} E_{a}^{+, T E (T M)} \\ E_{a}^{-, T E (T M)} \end{matrix}],

[\begin{matrix} E_{b}^{+, T E (T M)} \\ E_{b}^{-, T E (T M)} \end{matrix}] = {\bar{S}}^{B} [\begin{matrix} E_{n + 1}^{+, T E (T M)} \\ 0 \end{matrix}] = [\begin{matrix} s_{11}^{B, T E (T M)} & s_{12}^{B, T E (T M)} \\ s_{21}^{B, T E (T M)} & s_{22}^{B, T E (T M)} \end{matrix}] [\begin{matrix} E_{n + 1}^{+, T E (T M)} \\ 0 \end{matrix}] .

Here, the 2 × 2 system matrices on the left and right side of the source plane can be defined as

{\bar{S}}^{A} = {\bar{I}}^{0 / 1} {\bar{L}}^{1} (d_{1}) \dots {\bar{I}}^{(W - 1) / W} {\bar{L}}^{W} (z_{e x})

and

{\bar{S}}^{B} = {\bar{L}}^{W} (d_{e} - z_{e x}) {\bar{I}}^{W / (W + 1)} \dots {\bar{L}}^{n} (d_{n}) {\bar{I}}^{n / (n + 1)} .

After some mathematical manipulations of Eqs. (4), (6) and (7), the four electric field amplitudes at the source plane can be expressed in terms of the source term in Eq. (5) and the components of system matrices

{\bar{S}}^{A}

and

{\bar{S}}^{B}

. They are written as [26]

E_{a}^{+, T E (T M)} = \frac{A_{h (v)}^{+, T E (T M)} r_{A}^{T E (T M)} r_{B}^{T E (T M)} \exp (i k_{z, e} 2 d_{e}) - A_{h (v)}^{-, T E (T M)} r_{A}^{T E (T M)} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T E (T M)} r_{B}^{T E (T M)} \exp (i k_{z, e} 2 d_{e})},

E_{a}^{-, T E (T M)} = \frac{A_{h (v)}^{+, T E (T M)} r_{B}^{T E (T M)} \exp (i k_{z, e} 2 (d_{e} - z_{e x})) - A_{h (v)}^{-, T E (T M)}}{1 - r_{A}^{T E (T M)} r_{B}^{T E (T M)} \exp (i k_{z, e} 2 d_{e})},

E_{b}^{+, T E (T M)} = \frac{A_{h (v)}^{+, T E (T M)} - A_{h (v)}^{-, T E (T M)} r_{A}^{T E (T M)} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T E (T M)} r_{B}^{T E (T M)} \exp (i k_{z, e} 2 d_{e})},

E_{b}^{-, T E (T M)} = \frac{A_{h (v)}^{+, T E (T M)} r_{B}^{T E (T M)} \exp (i k_{z, e} 2 (d_{e} - z_{e x})) - A_{h (v)}^{-, T E (T M)} r_{A}^{T E (T M)} r_{B}^{T E (T M)} \exp (i k_{z, e} 2 d_{e})}{1 - r_{A}^{T E (T M)} r_{B}^{T E (T M)} \exp (i k_{z, e} 2 d_{e})} .

The term

r_{A (B)}^{T E (T M)}

indicates the back (front) reflection coefficient, and

t_{A (B)}^{T E (T M)}

denotes the back (front) transmission coefficient from the source plane to the transparent ambient, as shown in Fig. 1. Using the elements of the system matrices

{\bar{S}}^{A}

and

{\bar{S}}^{B},

these coefficients can be expressed as [26]

\begin{array}{l} r_{A}^{T E (T M)} = - \frac{s_{12}^{A, T E (T M)}}{s_{11}^{A, T E (T M)}}, r_{B}^{T E (T M)} = \frac{s_{21}^{B, T E (T M)}}{s_{11}^{B, T E (T M)}}, \\ t_{A}^{T E (T M)} = \frac{s_{11}^{A, T E (T M)} s_{22}^{A, T E (T M)} - s_{12}^{A, T E (T M)} s_{21}^{A, T E (T M)}}{s_{11}^{A, T E (T M)}}, t_{B}^{T E (T M)} = \frac{1}{s_{11}^{B, T E (T M)}} . \end{array}

2.2 Derivation of the spectral power density per normalized in-plane wave vector from the normalized time-average Poynting vector at the EML

Inside the absorption-free EML, the electric field amplitude on the left-hand side of the source plane can be expressed as

[\begin{matrix} E_{e, l h}^{+, T E (T M)} (z) \\ E_{e, l h}^{-, T E (T M)} (z) \end{matrix}] = {\bar{L}}^{W} (z_{e x} - z) [\begin{matrix} E_{a}^{+, T E (T M)} \\ E_{a}^{-, T E (T M)} \end{matrix}] = [\begin{matrix} \exp (- i k_{z, e} (z_{e x} - z)) & 0 \\ 0 & \exp (i k_{z, e} (z_{e x} - z)) \end{matrix}] [\begin{matrix} E_{a}^{+, T E (T M)} \\ E_{a}^{-, T E (T M)} \end{matrix}] .

According to the sign convention of the TE- and TM-polarized reflected electric field amplitudes shown in Fig. 2(a), the electric field amplitude within the EML can be written as

E_{e, l h}^{T E} (z) = [E_{e, l h}^{+, T E} (z) + E_{e, l h}^{-, T E} (z)] {\hat{a}}_{y},

E_{e, l h}^{T M} (z) = [E_{e, l h}^{+, T M} (z) + E_{e, l h}^{-, T M} (z)] \sqrt{1 - u^{2}} {\hat{a}}_{x} + [- E_{e, l h}^{+, T M} (z) + E_{e, l h}^{-, T M} (z)] u {\hat{a}}_{z},

where

{\hat{a}}_{x}

,

{\hat{a}}_{y}

, and

{\hat{a}}_{z}

denote unit vectors of the corresponding components of the electric field amplitude. In the case of TE polarization, the y component of the electric field amplitude at the left-hand side of the source plane is given by [26,27]

\begin{array}{l} E_{e, l h}^{y, T E} = [E_{e, l h}^{+, T E} + E_{e, l h}^{-, T E}] e^{i k_{x, 0} x} \\ = [\exp (- i k_{z, e} (z_{e x} - z)) E_{a}^{+, T E} + \exp (i k_{z, e} (z_{e x} - z)) E_{a}^{-, T E}] \exp (i k_{x, 0} x) \\ = [\exp (- i k_{z, e} (z_{e x} - z)) r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + \exp (i k_{z, e} (z_{e x} - z))] E_{a}^{-, T E} \exp (i k_{x, 0} x), \end{array}

where a relation of

E_{a}^{+, T E} = r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) E_{a}^{-, T E}

is used based on Eqs. (8) and (9). Corresponding x and z components of the magnetic field amplitude are written as [27]

\begin{array}{l} H_{e, l h}^{x, T E} = \frac{1}{i ω μ_{0}} (\frac{\partial E_{e, l h}^{z, T E}}{\partial y} - \frac{\partial E_{e, l h}^{y, T E}}{\partial z}) \\ = \frac{- k_{z, e}}{ω μ_{0}} [\exp (- i k_{z, e} (z_{e x} - z)) r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) - \exp (i k_{z, e} (z_{e x} - z))] E_{a}^{-, T E} \exp (i k_{x, 0} x), \end{array}

\begin{array}{l} H_{e, l h}^{z, T E} = \frac{1}{i ω μ_{0}} (\frac{\partial E_{e, l h}^{y, T E}}{\partial x} - \frac{\partial E_{e, l h}^{x, T E}}{\partial y}) \\ = \frac{k_{x, 0}}{ω μ_{0}} [\exp (- i k_{z, e} (z_{e x} - z)) r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + \exp (i k_{z, e} (z_{e x} - z))] E_{a}^{-, T E} \exp (i k_{x, 0} x), \end{array}

where μ₀ is a permeability in free space. According to Eq. (5), only the horizontally-oriented dipole contributes to the radiation of the TE-polarized light. Thus, the time-averaged Poynting vector in the z direction for TE polarization, which corresponds to the optical power radiated from the horizontally-oriented dipole in the EML, can be expressed as

\begin{array}{l} S_{e, h, l h}^{z, T E} = - \frac{1}{2} Re {E_{e, l h}^{y, T E} {(H_{e, l h}^{x, T E})}^{*}} \\ = - \frac{1}{2 ω μ_{0}} Re {k_{z, e} (1 - \exp (- i k_{z, e} 2 z) {(r_{A}^{T E})}^{*} + \exp (i k_{z, e} 2 z) r_{A}^{T E} - {| r_{A}^{T E} |}^{2})} {| E_{a}^{-, T E} |}^{2} \\ = - \frac{k_{z, e}}{2 ω μ_{0}} \frac{3}{16 π} {| \frac{1 + r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))}{1 - r_{A}^{T E} r_{B}^{T E} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{A}^{T E}), \end{array}

where

R_{A}^{T E} = {| r_{A}^{T E} |}^{2}

indicates the optical power reflectance at the left boundary of the EML, as shown in Fig. 1.

Next, we repeat the above procedure to the case of right-hand side of the source plane. The corresponding electric field amplitude can be written as

[\begin{matrix} E_{b}^{+, T E (T M)} \\ E_{b}^{+, T E (T M)} \end{matrix}] = {\bar{L}}^{W} (z - z_{e x}) [\begin{matrix} E_{e, r h}^{+, T E (T M)} (z) \\ E_{e, r h}^{-, T E (T M)} (z) \end{matrix}] = [\begin{matrix} \exp (- i k_{z, e} (z - z_{e x})) & 0 \\ 0 & \exp (i k_{z, e} (z - z_{e x})) \end{matrix}] [\begin{matrix} E_{e, r h}^{+, T E (T M)} (z) \\ E_{e, r h}^{-, T E (T M)} (z) \end{matrix}] .

The y-component electric field amplitude at the right-hand side of the source plane is given by

\begin{array}{l} E_{e, r h}^{y, T E} = [E_{e, r h}^{+, T E} + E_{e, r h}^{-, T E}] \exp (i k_{x, 0} x) \\ = [\exp (i k_{z, e} (z - z_{e x})) E_{b}^{+, T E} + \exp (- i k_{z, e} (z - z_{e x})) E_{b}^{-, T E}] \exp (i k_{x, 0} x) \\ = [\exp (i k_{z, e} (z - z_{e x})) + \exp (- i k_{z, e} (z - z_{e x})) r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))] E_{b}^{+, T E} \exp (i k_{x, 0} x), \end{array}

where the relation of

E_{b}^{-, T E} = r_{B}^{T E (T M)} \exp (i k_{z, e} 2 (d_{e} - z_{e x})) E_{b}^{+, T E}

is applied based on Eqs. (10) and (11). The corresponding x and z components of the magnetic field amplitude are

\begin{array}{l} H_{e, r h}^{x, T E} = \frac{1}{i ω μ_{0}} (\frac{\partial E_{e, r h}^{z, T E}}{\partial y} - \frac{\partial E_{e, r h}^{y, T E}}{\partial z}) \\ = \frac{- k_{z, e}}{ω μ_{0}} [\exp (i k_{z, e} (z - z_{e x})) - \exp (- i k_{z, e} (z - z_{e x})) r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))] E_{b}^{+, T E} \exp (i k_{x, 0} x), \end{array}

\begin{array}{l} H_{e, r h}^{z, T E} = \frac{1}{i ω μ_{0}} (\frac{\partial E_{e, r h}^{y, T E}}{\partial x} - \frac{\partial E_{e, r h}^{x, T E}}{\partial y}) \\ = \frac{k_{x, 0}}{ω μ_{0}} [\exp (i k_{z, e} (z - z_{e x})) + \exp (- i k_{z, e} (z - z_{e x})) r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))] E_{b}^{+, T E} \exp (i k_{x, 0} x) . \end{array}

Consequently, the time-averaged Poynting vector in the z direction can be expressed as

S_{e, h, r h}^{z, T E} = - \frac{1}{2} Re {E_{e, r h}^{y, T E} {(H_{e, r h}^{x, T E})}^{*}} = \frac{k_{z, e}}{2 ω μ_{0}} \frac{3}{16 π} {| \frac{1 + r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T E} r_{B}^{T E} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{B}^{T E}) .

According to the electromagnetic models of the OLED, the optical power radiated from a dipole emitter with the micro-cavity effect has been normalized in reference to the optical power generated by the dipole emitter with no cavity effect, which was related with the Purcell effect [11–15]. If we assume that the dipole emitter is immersed in a boundless EML, the time-averaged Poynting vectors on the left- and right-hand side of the source plane are given by

S_{e, l h, \infty}^{z, T E} = - \frac{1}{2 η} {| E_{e, r h, \infty}^{y, T E} |}^{2} = - \frac{k_{z, e}}{2 ω μ_{0}}, S_{e, r h, \infty}^{z, T E} = - \frac{1}{2 η} {| E_{e, l h, \infty}^{y, T E} |}^{2} = \frac{k_{z, e}}{2 ω μ_{0}} .

Here,

η = k_{z, e} / ω μ_{0}

is an impedance of the electromagnetic wave in the EML. In addition, we assume that

{| E_{e, r h, \infty}^{y, T E} |}^{2} = 1

because the power densities of the dipole emitter are normalized in Eq. (5). Based on Eqs. (19), (24), and (25), the spectral power density per unit solid angle for TE polarization is obtained by

\begin{array}{l} P_{e, h}^{T E} = \frac{S_{e, h, t o t a l}^{z, T E}}{S_{e, \infty}^{z, T E}} = \frac{S_{e, h, r h}^{z, T E}}{S_{e, r h, \infty}^{z, T E}} + \frac{S_{e, h, l h}^{z, T E}}{S_{e, l h, \infty}^{z, T E}} \\ = \frac{3}{16 π} {{| \frac{1 + r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T E} r_{B}^{T E} \exp (i k_{z, e} 2 d e)} |}^{2} (1 - R_{B}^{T E}) + {| \frac{1 + r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))}{1 - r_{A}^{T E} r_{B}^{T E} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{A}^{T E})} . \end{array}

For TM polarization, we take the same mathematical procedure as we do for TE polarization except that we use Eq. (15) to obtain the x and z components of the electric field amplitude. After some mathematical manipulations, the spectral power density per unit solid angle of horizontally- and vertically-oriented dipole emitter for TM polarization can be written as

\begin{array}{l} P_{e, h}^{T M} = \frac{S_{e, h, t o t a l}^{z, T M}}{S_{e, \infty}^{z, T M}} \\ = \frac{3}{16 π} (1 - u^{2}) {{| \frac{1 + r_{A}^{T M} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T M} r_{B}^{T M} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{B}^{T M}) + {| \frac{1 + r_{B}^{T M} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))}{1 - r_{A}^{T M} r_{B}^{T M} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{A}^{T M})}, \end{array}

\begin{array}{l} P_{e, v}^{T M} = \frac{S_{e, v, t o t a l}^{z, T M}}{S_{e, \infty}^{z, T M}} \\ = \frac{3}{8 π} u^{2} {{| \frac{1 - r_{A}^{T M} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T M} r_{B}^{T M} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{B}^{T M}) + {| \frac{1 - r_{B}^{T M} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))}{1 - r_{A}^{T M} r_{B}^{T M} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{A}^{T M})}, \end{array}

where we assume that the TM-polarized time-averaged Poynting vectors generated by the dipole emitter without the cavity effect have the same expression as the TE-polarized ones.

To quantify the spectral profile of power dissipation for all optical modes in the OLED, the spectral power density per normalized in-plane wave vector $K (u),$ which is dimensionless, has been widely used [15]. When it comes to the air mode, it was demonstrated that the mathematical formulation of the spectral power density per unit solid angle, derived from the source-term method together with the TMM, was identical to the mathematical expression of the spectral power density per normalized in-plane wave vector $K (u)$ obtained by the PDM [18]. In this case, the spectral power density per unit solid angle of the air mode was calculated based on the output optical power in the ambient layer. Here, we extend this proof to the WG and SPP modes using the normalized time-average Poynting vector per unit solid angle generated by the dipole emitter at the EML. The spectral power density per unit solid angle $P_{e} (θ_{e})$ can be converted into the spectral power density per normalized in-plane wave vector $K (u)$ as [14,15]

2 π P_{e} (θ_{e}) \sin θ_{e} d θ_{e} = K (u) d u^{2} .

Using the relation of

\sin θ_{e} = u

and

d θ_{e} = 1 / \sqrt{1 - u^{2}} d u

, we have

K (u) = \frac{π}{\sqrt{1 - u^{2}}} P_{e} (u),

where the spectral densities per unit solid angle are shown in Eqs. (26)-(28). Substituting these equations into

P_{e} (u)

in Eq. (30), the spectral power densities per normalized in-plane wave vector obtained by the GPVM, which are denoted as K’, can be expressed as

\begin{array}{l} {K^{'}}_{e, h}^{T E} = \frac{π}{\sqrt{1 - u^{2}}} \frac{S_{e, h, t o t a l}^{z, T E}}{S_{e, h, \infty}^{z, T E}} \\ = \frac{3}{16} \frac{1}{\sqrt{1 - u^{2}}} {{| \frac{1 + r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T E} r_{B}^{T E} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{B}^{T E}) + {| \frac{1 + r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))}{1 - r_{A}^{T E} r_{B}^{T E} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{A}^{T E})}, \end{array}

\begin{array}{l} {K^{'}}_{e, h}^{T M} = \frac{π}{\sqrt{1 - u^{2}}} \frac{S_{e, h, t o t a l}^{z, T M}}{S_{e, h, \infty}^{z, T M}} \\ = \frac{3}{16} \sqrt{1 - u^{2}} {{| \frac{1 + r_{A}^{T M} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T M} r_{B}^{T M} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{B}^{T M}) + {| \frac{1 + r_{B}^{T M} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))}{1 - r_{A}^{T M} r_{B}^{T M} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{A}^{T M})}, \end{array}

\begin{array}{l} {K^{'}}_{e, v}^{T M} = \frac{π}{\sqrt{1 - u^{2}}} \frac{S_{e, v, t o t a l}^{z, T M}}{S_{e, v, \infty}^{z, T M}} \\ = \frac{3}{8} \frac{u^{2}}{\sqrt{1 - u^{2}}} {{| \frac{1 - r_{A}^{T M} \exp (i k_{z, e} 2 z_{e x})}{1 - r_{A}^{T M} r_{B}^{T M} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{B}^{T M}) + {| \frac{1 - r_{B}^{T M} \exp (i k_{z, e} 2 (d_{e} - z_{e x}))}{1 - r_{A}^{T M} r_{B}^{T M} \exp (i k_{z, e} 2 d_{e})} |}^{2} (1 - R_{A}^{T M})} . \end{array}

According to the PDM [14,15], the power spectral densities per normalized in-plane wave vector, designated as K, are expressed as

K_{e, h}^{T E} = \frac{3}{16} \frac{1}{\sqrt{1 - u^{2}}} {\frac{{| 1 + a_{T E}^{-} |}^{2}}{{| 1 - a_{T E} |}^{2}} (1 - R_{B}^{T E}) + \frac{{| 1 + a_{T E}^{+} |}^{2}}{{| 1 - a_{T E} |}^{2}} (1 - R_{A}^{T E})},

K_{e, h}^{T M} = \frac{3}{16} \sqrt{1 - u^{2}} {\frac{{| 1 - a_{T M}^{-} |}^{2}}{{| 1 - a_{T M} |}^{2}} (1 - R_{B}^{T M}) + \frac{{| 1 - a_{T M}^{+} |}^{2}}{{| 1 - a_{T M} |}^{2}} (1 - R_{A}^{T M})},

K_{e, v}^{T M} = \frac{3}{8} \frac{u^{2}}{\sqrt{1 - u^{2}}} {\frac{{| 1 + a_{T M}^{-} |}^{2}}{{| 1 - a_{T M} |}^{2}} (1 - R_{B}^{T M}) + \frac{{| 1 + a_{T M}^{+} |}^{2}}{{| 1 - a_{T M} |}^{2}} (1 - R_{A}^{T M})} .

Here, the coefficient are

a_{T E (T M)}^{+} = r_{B}^{T E (T M)} \exp (2 j k_{z, e} (d_{e} - z_{e x})), a_{T E (T M)}^{-} = r_{A}^{T E (T M)} \exp (2 j k_{z, e} z_{e x}),

and

a_{T E (T M)} = a_{T E (T M)}^{+} a_{T E (T M)}^{-}

. In the literatures [14,15], the Fresnel reflection coefficients of

r_{A}^{T E (T M)}

and

r_{B}^{T E (T M)}

are expressed based on the Verdet convention, where the + (-) sign of the TM-polarized reflection coefficient is opposite to that based on the Fresnel convention in this study. Thus, + (-) sign of the reflection coefficient in the numerator for TM polarization in Eqs. (35) and (36) is opposite to that in Eqs. (32) and (33). However, the formulas of the spectral power densities per normalized in-plane wave vector derived by both models are equivalent in the WG and SPP modes as well as the air and substrate modes.

2.3 Derivation of the formulation for the spatial profile of the electric field intensity, optical power flow, and optical absorption

We will derive the mathematical expressions of the spatial profiles of the electric field intensity, optical power flow, and optical absorption both inside and outside the multilayer of the OLED. For convenience, we show the derivation steps for TE polarization, but the theoretical formula for TM polarization can be also derived in the similar manner. The internal power flow located at an arbitrary position in the layer k $(k = 0, 1, \dots, n, n + 1)$ is obtained by the corresponding system matrix and the electric field amplitudes at the left and right boundaries of the source plane [26,27]. As shown in Fig. 1, the electric field amplitudes at the left-hand side of the source plane are written as

\begin{array}{l} [\begin{matrix} E_{k, l h}^{+, T E (T M)} (z) \\ E_{k, l h}^{-, T E (T M)} (z) \end{matrix}] = {\bar{L}}^{k} (d_{k} - z) {\bar{S}}^{k / W} [\begin{matrix} E_{a}^{+, T E (T M)} \\ E_{a}^{-, T E (T M)} \end{matrix}] \\ = [\begin{matrix} \exp (- i k_{z, k} (d_{k} - z)) & 0 \\ 0 & \exp (i k_{z, k} (d_{k} - z)) \end{matrix}] [\begin{matrix} s_{11}^{k / W, T E (T M)} & s_{12}^{k / W, T E (T M)} \\ s_{21}^{k / W, T E (T M)} & s_{22}^{k / W, T E (T M)} \end{matrix}] [\begin{matrix} E_{a}^{+, T E (T M)} \\ E_{a}^{-, T E (T M)} \end{matrix}], \end{array}

where

{\bar{S}}^{k / W} = {\bar{I}}^{k / (k + 1)} {\bar{L}}^{(k + 1)} \dots {\bar{I}}^{(W - 1) / W} \bar{L} (z_{e x})

is the corresponding system matrix. Thus, the TE-polarized electric field amplitude at an arbitrary position z in the layer k can be given by

\begin{array}{l} E_{k, l h}^{T E} (z) = (E_{k, l h}^{+, T E} (z) + E_{k, l h}^{-, T E} (z)) {\hat{a}}_{y} \\ = [\begin{array}{l} (s_{11}^{k / W, T E} E_{a}^{+, T E} + s_{12}^{k / W, T E} E_{a}^{-, T E}) \exp (- i k_{z, k} (d_{k} - z)) \\ + (s_{21}^{k / W, T E} E_{a}^{+, T E} + s_{22}^{k / W, T E} E_{a}^{-, T E}) \exp (i k_{z, k} (d_{k} - z)) \end{array}] {\hat{a}}_{y}, \end{array}

where the TE-polarized electric field amplitude has only y component. Therefore, the total electric field amplitude for TE polarization is represented by

E_{k, l h}^{y, T E} (z) = [\begin{array}{l} (s_{11}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{12}^{k / W, T E}) \exp (- i k_{z, k} (d_{k} - z)) \\ + (s_{21}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{22}^{k / W, T E}) \exp (i k_{z, k} (d_{k} - z)) \end{array}] E_{a}^{-, T E} \exp (i k_{x, 0} x) .

The corresponding x and z components of the magnetic field amplitude are

\begin{array}{l} H_{k, l h}^{x, T E} (z) = \frac{1}{i ω μ_{0}} (\frac{\partial E_{k, l h}^{z, T E} (z)}{\partial y} - \frac{\partial E_{k, l h}^{y, T E} (z)}{\partial z}) \\ = \frac{- k_{z, k}}{ω μ_{0}} [\begin{array}{l} (s_{11}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{12}^{k / W, T E}) \exp (- i k_{z, k} (d_{k} - z)) \\ - (s_{21}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{22}^{k / W, T E}) \exp (i k_{z, k} (d_{k} - z)) \end{array}] E_{a}^{-, T E} \exp (i k_{x, 0} x), \end{array}

\begin{array}{l} H_{k, l h}^{z, T E} (z) = \frac{1}{i ω μ_{0}} (\frac{\partial E_{k, l h}^{y, T E} (z)}{\partial x} - \frac{\partial E_{k, l h}^{x, T E} (z)}{\partial y}) \\ = \frac{k_{x, 0}}{ω μ_{0}} [\begin{array}{l} (s_{11}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{12}^{k / W, T E}) \exp (- i k_{z, k} (d_{k} - z)) \\ + (s_{21}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{22}^{k / W, T E}) \exp (i k_{z, k} (d_{k} - z)) \end{array}] E_{a}^{-, T E} (i k_{x, 0} x) . \end{array}

As a result, the time-averaged Poynting vector in the z direction for TE-polarized light on the left-hand side of the source plane can be written as

\begin{array}{l} S_{k, l h}^{z, T E} (z) = - \frac{1}{2} Re {E_{k, l h}^{y, T E} (z) {(H_{k, l h}^{x, T E} (z))}^{*}} \\ = \frac{k_{z, k}}{2 ω μ} Re {\begin{cases} [\begin{array}{l} (s_{11}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{12}^{k / W, T E}) \exp (- i k_{z, k} (d_{k} - z)) \\ + (s_{21}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{22}^{k / W, T E}) \exp (i k_{z, k} (d_{k} - z)) \end{array}] \\ \times [\begin{array}{l} {(s_{11}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{12}^{k / W, T E})}^{*} \exp (i k_{z, k} (d_{k} - z)) \\ - {(s_{21}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{22}^{k / W, T E})}^{*} \exp (- i k_{z, k} (d_{k} - z)) \end{array}] \end{cases}} {| E_{a}^{-, T E} |}^{2} . \end{array}

By differentiating the time-averaged Poynting vector with the z direction, the optical absorption can be give by

\begin{array}{l} Q_{k, l h}^{z, T E} (z) = - \frac{d S_{k, l h}^{z, T E}}{d z} = \frac{k_{0}^{2} p_{k} q_{k}}{ω μ_{0}} {\begin{cases} [\begin{array}{l} {| s_{11}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{12}^{k / W, T E} |}^{2} \exp (k_{0} q_{k} 2 (d_{k} - z)) \\ + {| s_{21}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{22}^{k / W, T E} |}^{2} \exp (- k_{0} q_{k} 2 (d_{k} - z)) \end{array}] \\ + 2 | ρ_{k}^{T E, L} | \cos (δ_{k}^{T E, L} - k_{0} p_{k} 2 (d_{k} - z)) \end{cases}} {| E_{a}^{-, T E} |}^{2} \\ = \frac{k_{0}^{2} p_{k} q_{k}}{ω μ_{0}} {| E_{k, l h}^{y, T E} (z) |}^{2} . \end{array}

Here, the coefficients of

p_{k}

,

q_{k}

,

ρ_{k}^{T E, L}

,

δ_{k}^{T E, L}

can be obtained by

\begin{array}{l} k_{z, k} = k_{0} (p_{k} + i q_{k}), \\ | ρ_{k}^{T E, L} | \exp (i δ_{k}^{T E, L}) = (s_{11}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{12}^{k / W, T E}) {(s_{21}^{k / W, T E} r_{A}^{T E} \exp (i k_{z, e} 2 z_{e x}) + s_{22}^{k / W, T E})}^{*} . \end{array}

In the next, the electric field amplitudes at the right-hand side of the source plane can be written by

\begin{array}{l} [\begin{matrix} E_{b}^{+, T E (T M)} \\ E_{b}^{-, T E (T M)} \end{matrix}] = {\bar{S}}^{W / k} {\bar{L}}^{k} (z) [\begin{matrix} E_{k, r h}^{+, T E (T M)} (z) \\ E_{k, r h}^{-, T E (T M)} (z) \end{matrix}] \\ = [\begin{matrix} s_{11}^{W / k, T E (T M)} & s_{12}^{W / k, T E (T M)} \\ s_{21}^{W / k, T E (T M)} & s_{22}^{W / k, T E (T M)} \end{matrix}] [\begin{matrix} \exp (- i k_{z, k} z) & 0 \\ 0 & \exp (i k_{z, k} z) \end{matrix}] [\begin{matrix} E_{k, r h}^{+, T E (T M)} (z) \\ E_{k, r h}^{-, T E (T M)} (z) \end{matrix}], \end{array}

where

{\bar{S}}^{W / k} = {\bar{L}}^{W} (d_{e} - z_{e x}) {\bar{I}}^{W / (W + 1)} \dots {\bar{L}}^{(k - 1)} {\bar{I}}^{(k - 1) / k}

indicates the corresponding system matrix. The TE-polarized electric field amplitude at an arbitrary position z in the layer k can be written as

\begin{array}{l} E_{k, r h}^{T E} (z) = (E_{k, r h}^{+, T E} (z) + E_{k, r h}^{-, T E} (z)) {\hat{a}}_{y} \\ = \frac{1}{\det ({\bar{S}}^{W / k})} [\begin{array}{l} (s_{22}^{W / k, T E} - s_{21}^{W / k, T E}) \exp (i k_{z, k} z) \\ - (s_{12}^{W / k, T E} + s_{11}^{W / k, T E}) r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x})) \exp (- i k_{z, k} z) \end{array}] E_{b}^{+, T E} \exp (i k_{x, 0} x) {\hat{a}}_{y} . \end{array}

Then, the total electric field amplitude for TE polarization is expressed as

E_{k, r h}^{y, T E} (z) = \frac{1}{\det ({\bar{S}}^{W / k})} [\begin{array}{l} (s_{22}^{W / k, T E} - s_{21}^{W / k, T E}) \exp (i k_{z, k} z) \\ - (s_{12}^{W / k, T E} + s_{11}^{W / k, T E}) r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x})) \exp (- i k_{z, k} z) \end{array}] E_{b}^{+, T E} \exp (i k_{x, 0} x) .

The corresponding x and z components of the magnetic field amplitude are expressed as

\begin{array}{l} H_{k, r h}^{x, T E} (z) = \frac{1}{i ω μ_{0}} (\frac{\partial E_{k, r h}^{z, T E} (z)}{\partial y} - \frac{\partial E_{k, r h}^{y, T E} (z)}{\partial z}) \\ = \frac{- k_{z, k}}{ω μ_{0}} \frac{1}{\det ({\bar{S}}^{W / k})} [{\begin{cases} (s_{22}^{W / k, T E} - s_{21}^{W / k, T E}) \exp (i k_{z, k} z) \\ + (s_{12}^{W / k, T E} + s_{11}^{W / k, T E}) r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x})) \exp (- i k_{z, k} z) \end{cases}}] E_{b}^{+, T E} \exp (i k_{x, 0} x), \end{array}

\begin{array}{l} H_{k, r h}^{z, T E} (z) = \frac{1}{i ω μ_{0}} (\frac{\partial E_{k, r h}^{y, T E} (z)}{\partial x} - \frac{\partial E_{k, r h}^{x, T E} (z)}{\partial y}) \\ = \frac{k_{x, 0}}{ω μ_{0}} \frac{1}{\det ({\bar{S}}^{W / k})} [\begin{array}{l} (s_{22}^{W / k, T E} - s_{21}^{W / k, T E}) \exp (i k_{z, k} z) \\ - (s_{12}^{W / k, T E} + s_{11}^{W / k, T E}) r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x})) \exp (- i k_{z, k} z) \end{array}] E_{b}^{+, T E} \exp (i k_{x, 0} x) . \end{array}

Then, the time-averaged Poynting vector in the z direction for TE-polarized light on the right-hand side of the source plane can be written as

\begin{array}{l} S_{k, r h}^{z, T E} (z) = - \frac{1}{2} Re {E_{k, r h}^{y, T E} (z) {(H_{k, r h}^{x, T E} (z))}^{*}} \\ = \frac{k_{z, k}}{2 ω μ_{0}} \frac{1}{{| \det ({\bar{S}}^{W / k}) |}^{2}} Re {\begin{cases} [\begin{array}{l} (s_{22}^{W / k, T E} - s_{21}^{W / k, T E}) \exp (i k_{z, k} z) \\ - (s_{12}^{W / k, T E} + s_{11}^{W / k, T E}) r_{B}^{T E} \exp (i k_{z, e} 2 (d_{e} - z_{e x})) \exp (- i k_{z, k} z) \end{array}] \\ \times [{\begin{cases} {(s_{22}^{W / k, T E} - s_{21}^{W / k, T E})}^{*} \exp (- i k_{z, k} z) \\ + {(s_{12}^{W / k, T E} + s_{11}^{W / k, T E})}^{*} {(r_{B}^{T E})}^{*} \exp (- i k_{z, e} 2 (d_{e} - z_{e x})) \exp (i k_{z, k} z) \end{cases}}] \end{cases}} {| E_{b}^{+, T E} |}^{2} . \end{array}

The corresponding optical absorption leads to

Q_{k, r h}^{z, T E} (z) = - \frac{d S_{k, r h}^{z, T E}}{d z} = \frac{1}{ω μ_{0}} \frac{k_{0}^{2} q_{k} p_{k}}{{| \det ({\bar{S}}^{W / k}) |}^{2}} {\begin{cases} [\begin{array}{l} {| s_{22}^{W / k, T E} - s_{21}^{W / k, T E} |}^{2} e^{- k_{0} q_{k} 2 z} \\ + {| s_{12}^{W / k, T E} + s_{11}^{W / k, T E} |}^{2} {| r_{B}^{T E} |}^{2} e^{k_{0} q_{k} 2 z} \end{array}] \\ - 2 | ρ_{k}^{T E, R} | \cos (δ_{k}^{T E, R} - k_{0} p_{k} 2 z) \end{cases}} {| E_{b}^{+, T E} |}^{2} = \frac{k_{0}^{2} p_{k} q_{k}}{ω μ_{0}} {| E_{k, r h}^{y, T E} (z) |}^{2},

where the coefficients of

ρ_{k}^{T E, R}

,

δ_{k}^{T E, R}

are defined as

| ρ_{k}^{T E, R} | \exp (i δ_{k}^{T E, R}) = (s_{22}^{W / k, T E} - s_{21}^{W / k, T E}) {(s_{12}^{W / k, T E} + s_{11}^{W / k, T E})}^{*} {(r_{B}^{T E})}^{*} \exp (- i k_{z, e} 2 (d_{e} - z_{e x})) .

It is noticeable that the optical absorption in Eqs. (43) and (51) depends on the y component of the electric field intensity. Applying the same procedure, we can derive the mathematical expressions for TM polarization, where x and z components of the electric field amplitudes together with the corresponding y component of the magnetic field amplitude are determined.

3. Calculation results

The structure of the bottom-emitting OLED used in this study is taken from Ref. 17. As shown in Fig. 3, it consists of a thick glass substrate, a 140-nm indium tin oxide (ITO) as a bottom anode, 30-nm poly(3,4)-ethylendioxythiophene doped with poly(styrene sulfonate) (PEDOT:PSS) as a hole injection layer, 80-nm N,N’-diphenyl-N,N’-bis(3-methylphenyl)-1,1’-biphenyl-4,4-diamine (TPD) as a hole transporting layer, 342-nm tris-(8-hydroxyquinoline) aluminum (Alq3) as both the electron-transport layer and EML, 15-nm calcium as an electron injecting layer, and 100-nm aluminum (Al) as a top cathode. The thickness of the electron transport plus the EML can be as much as 300 nm for an efficient third-order micro-cavity OLED. This can be very important for large TV applications, where surface particle and roughness must not be insensitive to high production yields [28]. Thus, the OLED device having an Alq₃ thickness of about 350 nm can be a feasible structure in this optical simulation study [17]. In Fig. 1, the hole injection layer of PEDOT:PSS is spin-coated and the remaining TPD/Alq₃/Ca/Al stacks are thermally evaporated. Although this mixture of spin-coated PEDOT:PSS and thermally-evaporated organic multilayer is not often used in the OLED manufacturing, it has been applied to several OLED devices with high-efficiency output performance [29,30].

Fig. 3 The structure of a bottom-emitting OLED with corresponding layer thicknesses and complex refractive indices at the wavelength of 520 nm.

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All the calculation results are obtained at the wavelength of λ = 520 nm, which corresponds to the photoluminescence peak of Alq3 in the literature [18]. The complex refractive indices of the materials shown in Fig. 3 are obtained from the literature [31] or experimental date measured by the ellipsometry. The EML thickness of 342 nm is determined to maximize the multi-beam interference of the micro-cavity effect [18]. The dipole emitter is assumed to be isotropic and have the delta function distribution. We place the emitter 50 and 198 nm away from the interface between the Ag reflective bottom anode and the EML, which corresponds to the first and second resonance condition of the two-beam interference at the emission angle of $θ_{e} = 0^{\circ}$ [17]. In Fig. 3, the 2-mm glass substrate should be modeled as optically incoherent because its thickness is larger than the coherence length of the light emitted by the dipole emitter in the OLED [15,18]. The incoherent multiple reflections inside the thick substrate cause the reduction of the optical power extracted into air [15,18]. In this calculation, the ambient layer on the right-hand side in Fig. 1 (j = n + 1) is assumed to be glass. If the output optical characteristic of the air mode in the air needs to be calculated, a fraction of incoherent power transmission from the incoherent substrate to air can be multiplied by the optical characteristic of the air mode in the glass substrate [15,18].

3.1 Power dissipation spectrum

Figure 4 shows the calculation results of the TE- and TM-polarized power dissipation spectrum at the two dipole positions of the bottom-emitting OLED. Because we assume the dipole orientation to be isotropic, the polarization-dependent spectral power densities per normalized in-plane wave vector are obtained by

K_{t o t a l}^{T E} (u, λ) = 2 K_{h}^{T E} (u, λ) / 3, K_{t o t a l}^{T M} = [2 K_{h}^{T E} (u, λ) + K_{v}^{T M} (u, λ)] / 3,

where Eqs. (31)-(33) based on the GPVM, equivalent to Eqs. (34)-(36) based on the PDM, are used. When the refractive indices of the EML and the glass substrate are

n_{e} = 1.769

and

n_{s u b} = 1.547

, the four optical modes for TM polarization are classified into the air mode (

0 \leq u < 1 / n_{e} = 0.565

), the substrate mode (

1 / n_{e} = 0.565 \leq u < n_{s u b} / n_{e} = 0.874

), the WG mode (

n_{s u b} / n_{e} = 0.874 \leq u < 1

), and the SPP mode (

1 \leq u

). Because the SPP mode is only generated for TM polarization, the spectral region of the TE-polarized WG lies at

n_{s u b} / n_{e} = 0.874 \leq u .

In Fig. 4, these four regions of the air, substrate, WG, and SPP modes are marked in the different shaded colors of sky blue, gray, yellow, and pink, respectively. According to the denominators of Eqs. (31)-(33), the singularities in the power dissipation spectrum take place at the roots of the equation

1 - r_{A}^{T E} r_{B}^{T E} \exp (i k_{z, e} 2 d_{e}) = 0,

which corresponds to the resonance condition of the trapped WG and SPP modes [32]. It is noticeable that the left-hand side of Eq. (54) can be approximated as a polynomial having the same complex roots as Eq. (54) [32]. Then, each singularity in the power dissipation spectrum can be approximately expressed as a single Lorentzian lineshape function, which is given by [17,32]

K_{m} (u) \approx \frac{σ_{m}}{{(u - u_{m})}^{2} + {(α_{m})}^{2}} .

Here,

u_{m}

and

α_{m}

represent the real and imaginary parts of the complex root for the m-th trapped WG/SPP mode. Finally,

σ_{m}

is related with the excitation efficiency of the m-th WG/SPP trapped mode.

Fig. 4 Calculation results of TE- and TM-polarized power dissipation spectrum of the bottom-emitting OLED when the two dipole positions are (a) z_ex = 50 nm and (b) z_ex = 198 nm. The four regions of the air, substrate, WG, and SPP modes are marked in the different shaded colors of sky blue, gray, yellow, and pink. Five spectral peaks correspond to the TE0 (u = 0.946), TE1 (u = 1.038), TM0 (u = 0.908), and TM1 (u = 0.998), WG modes as well as the TM2 (u = 1.085) SPP mode.

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In Fig. 4, five spectral peaks correspond to the TE0 (u = 0.946), TE1 (u = 1.038), TM0 (u = 0.908), and TM1 (u = 0.998) WG modes as well as the TM2 (u = 1.085) SPP mode, whose notations are determined in reference to the polarization-dependent mode number obtained by our previous BMA in the same OLED structure [17]. The relative intensities of the five spectral peaks with respect to the dipole position result from the variation of the electric field intensity of the corresponding WG or SPP mode at the emitter position [17]. Thus, the spectral peak of the TM1 WG mode in Fig. 4(a) is not observed at the emitter position of z_ex = 50 nm because the dissipated power of the TM1 mode is much smaller than the other two TM modes [17]. In the same reason, the spectral peak of the TM2 SPP mode in Fig. 4(b) is not observed when the emitter position is z_ex = 198 nm.

3.2 Spatial and spectral profiles of the electric field intensity with the effect of the dipole source included

The proposed GPVM can provide the spatial profiles of the electric field intensities of all the optical modes in the OLED, which include the traveling-wave-like air or substrate mode as well as the standing-wave-like WG or SPP mode. Figures 5 and 6 show the 2D contour plot of the polarization-dependent electric field intensity at the two dipole positions, where are marked with a white dashed line. In Figs. 5 and 6, the electric field intensities of the air and substrate modes are spatially distributed at all the spatial regions while those of the WG and SPP modes are not positioned at the glass substrate. This difference is ascribed to the reflectivity of the TPD/PEDOT:PSS/ITO multilayer between the EML and the glass substrate. In the case of the air and substrate modes, the reflectivity of the TPD/PEDOT:PSS/ITO multilayer has a non-zero value at its respective internal emission angle. Thus, a large amount of optical power generated by the air and substrate modes passes through the TPD/PEDOT:PSS/ITO multilayer and is distributed at the glass substrate. On the other hand, the internal emission angle of the WG and SPP modes satisfies the total internal reflection condition for the reflectivity of the TPD/PEDOT:PSS/ITO multilayer. Correspondingly, no optical power of the WG and SPP modes penetrated into the glass substrate.

Fig. 5 Calculated 2D contour plot of the electric field intensity for (a) TE- and (b) TM-polarized optical modes with respect to the longitudinal position and normalized in-plane wave vector when the dipole emitter, which is 50 nm away from the Ca/Alq₃ interface, is marked with a white dashed line.

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Fig. 6 Calculated 2D contour plot of the electric field intensity for (a) TE- and (b) TM-polarized optical modes with respect to the longitudinal position and normalized in-plane wave vector when the dipole emitter, which is 198 nm away from the Ca/Alq₃ interface, is marked with a white dashed line.

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To investigate the detailed variation of the electric field intensity with respect to the type of the optical mode, the polarization-dependent electric field intensities of several optical modes are plotted in Fig. 7. For TE polarization, the four optical modes are chosen as the air mode (u = 0), the substrate mode (u = 0.7), the TE0 WG mode (u = 0.946), and the TE1 WG mode (u = 1.038), respectively. In the case of TM polarization, the normalized in-plane wave vector corresponds to the air mode (u = 0), the substrate mode (u = 0.7), the TM0 WG mode (u = 0.908), the TM1 WG mode (u = 0.998), and the TM2 SPP mode (u = 1.085). The relative magnitude of the electric field intensity of the respective optical mode depends on the micro-cavity effect, which is related with the power dissipation spectrum in Fig. 4. Thus, the electric field intensity of the TE0 mode is very large and separately plotted. The spatial profile of the electric field intensity of the air mode sinusoidally oscillates inside the OLED multilayer and has the constant value at the glass substrate [26]. Regarding the WG and SPP modes, the spatial profiles of the electric field intensity decay in the glass substrate.

Fig. 7 The polarization-dependent electric field intensities of several optical modes when the dipole position is (a) z_ex = 50 and (b) z_ex = 198 nm. For TE polarization, the four optical modes are chosen as the air mode (u = 0), the substrate mode (u = 0.7), the TE0 WG mode (u = 0.946), and the TE1 WG mode (u = 1.038). In the case of TM polarization, the normalized in-plane wave vector corresponds to the air mode (u = 0), the substrate mode (u = 0.7), the TM0 WG mode (u = 0.908), the TM1 WG mode (u = 0.998), and the TM2 SPP mode (u = 1.085). The relative magnitude of the electric field intensity of the respective optical mode depends on its micro-cavity effect, which is related with the power dissipation spectrum in Fig. 4.

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The spatial profile of the normalized electric field intensity of the WG and SPP modes in the OLED can be also calculated through the boundary eigenvalue solution of the wave equations, where the effect of the dipole source is not considered with the assumption of the weak-coupling regime [17,19]. To investigate the validity of the spatial profile of the electric field profile calculated by the GPVM, the spatial profiles of the normalized electric field intensity of the WG and SPP modes obtained by the GPVM are compared with those calculated by the BMA, whose results are shown in Fig. 8. The calculation results of the normalized electric field intensity based on the BMA are obtained by the COMSOL MULTIPHYSICS [33]. In the GPVM, the spatial profiles of the normalized electric field intensity are calculated at the spectral peaks of the power dissipation spectrum in Fig. 4. Thus, the electric field profile of the TM1 WG mode cannot be obtained at z_ex = 50 nm because the dissipated power of the TM1 WG mode is observed. Because of the same reason, the electric field profile of the TM2 SPP mode cannot be calculated at z_ex = 198 nm in the GPVM. In Fig. 8, the normalized spatial profiles of the electric field intensity obtained by the GPVM are in a good agreement with those calculated by the BMA except the slight difference at the position of the dipole emitter, which is noticeable for the TE1 and TM2 modes in Fig. 7(a) because their spectral peaks are not clearly distinguished due to relatively small excitation efficiency. This difference was also observed at the air mode when the electric field intensity at the source plane was smaller than the absolute magnitude of the source term [26]. In addition, the discontinuity of the normalized electric field intensity at the interface is ascribed to the different boundary conditions of the x- and z-directional components of the TM-polarized electric field intensities [26].

Fig. 8 Comparison of the polarization-dependent spatial profiles of the normalized electric field intensity of the WG and SPP modes obtained by the GPVM and the BMA when the dipole position is (a) z_ex = 50 and (b) z_ex = 198 nm. The electric field profiles of the TM1 WG mode in (a) and TM2 SPP mode in (b) based on the GPVM cannot be obtained because their spectral peaks are not distinguished in the power dissipation spectrum, shown in Fig. 4.

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3.3 Spatial and spectral profiles of the internal optical power flow and optical absorption

The proposed GPVM can simultaneously calculate spatial and spectral profiles of the internal optical power flow and optical absorption with respect to the longitudinal position and normalized in-plane wave vector. Figures 9 and 10 show the polarization-dependent 2D contour plot of the optical power flow at two different dipole positions, which are 50 and 198 nm away from the Ca/Alq₃ interface and marked with a white dashed line. In Figs. 9 and 10, a positive value of the optical power, which is obtained by the time-average Poynting vector, represents the power propagation in the forward direction (from the left to right directions). On the other hand, a negative value of the optical power corresponds to the backward-propagating power flow (from the right to left directions) [27]. Because the 342-nm Alq₃ EML and 80-nm TPD are assumed to be lossless, the spatial profiles of all the optical modes are flat in these two regions. On the other hand, the optical power flow decreases at the absorptive layer made of Al, Ca, PEDOT:PSS, and ITO.

Fig. 9 Calculated 2D contour plot of the optical power flow for (a) TE- and (b) TM-polarized optical modes with respect to the longitudinal position and normalized in-plane wave vector when the dipole emitter, which is 50 nm away from the Ca/Alq₃ interface, is marked with a white dashed line. A positive (negative) value of the optical power flow, which is obtained by the time-average Poynting vector, represents the power propagation in the forward (backward) direction.

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Fig. 10 Calculated 2D contour plot of the optical power flow for (a) TE- and (b) TM-polarized optical modes with respect to the longitudinal position and normalized in-plane wave vector when the dipole emitter, which is 198 nm away from the Ca/Alq₃ interface, is marked with a white dashed line. A positive (negative) value of the optical power flow, which is obtained by the time-average Poynting vector, represents the power propagation in the forward (backward) direction.

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Figure 11 shows the TE- and TM-polarized optical power profiles of four optical modes when the dipole position is set to be z_ex = 50 and 198 nm. For TE polarization, the four optical modes are chosen as the air mode (u = 0), the substrate mode (u = 0.7), the TE0 WG mode (u = 0.946), and the TE1 WG mode (u = 1.038). In the case of TM polarization, the normalized in-plane wave vector corresponds to the air mode (u = 0), the substrate mode (u = 0.7), the TM0 WG mode (u = 0.908), the TM1 WG mode (u = 0.998), and the TM2 SPP mode (u = 1.085). The magnitude of the optical power at the EML is proportional to its micro-cavity effect, which is related with the power dissipation spectrum in Fig. 4. In reference to the source plane located within the absorption-free EML, the optical power propagates without reduction in the backward and forward directions. Because of the high extinction coefficients, the optical power profiles of all the optical modes exponentially decreases inside the Ca/Al metal layers. In the case of the air (u = 0) and substrate (u = 0.7) modes, a large amount of optical power passes through the PEDOT:PSS/ITO layers with small absorption and reaches the glass substrate. However, the optical power generated by the WG and SPP modes experiences huge absorption at the PEDOT:PSS/ITO layers so that there is no penetrating optical power at the glass substrate.

Fig. 11 The polarization-dependent optical power profiles of several optical modes when the dipole position is (a) z_ex = 50 and (b) z_ex = 198 nm. For TE polarization, the four optical modes are chosen as the air mode (u = 0), the substrate mode (u = 0.7), the TE0 WG mode (u = 0.946), and the TE1 WG mode (u = 1.038). In the case of TM polarization, the normalized in-plane wave vector corresponds to the air mode (u = 0), the substrate mode (u = 0.7), the TM0 WG mode (u = 0.908), the TM1 WG mode (u = 0.998), and the TM2 SPP mode (u = 1.085).

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Figures 12 and 13 show the polarization-dependent 2D contour plot of the optical absorption at two dipole positions, which correspond to 50 and 198 nm away from the Ca/Alq₃ interface and designated with a white dashed line. No absorption is observed at the lossless regions of the 342-nm Alq₃ EML and 80-nm TPD layers. Optical absorption takes place at the absorptive layer made of Al, Ca, PEDOT:PSS, and ITO. Figure 14 shows the TE- and TM-polarized optical absorption profiles of four optical modes when the dipole position is z_ex = 50 and 198 nm. For TE polarization, the four optical modes are chosen as the air mode (u = 0), the substrate mode (u = 0.7), the TE0 WG mode (u = 0.946), and the TE1 WG mode (u = 1.038). In the case of TM polarization, the normalized in-plane wave vector corresponds to the air mode (u = 0), the substrate mode (u = 0.7), the TM0 WG mode (u = 0.908), the TM1 WG mode (u = 0.998), and the TM2 SPP mode (u = 1.085). To increase the visibility in Fig. 14, the absorption profiles are only plotted near the Ca/Al metal and PEDOT:PSS/ITO/glass layers while the zero-valued absorption profiles at the loss-free Alq₃/TPD layers are omitted. In Fig. 14, the spatial profiles of optical absorption at the Ca/Al metal show the exponentially decaying shape because only backward-propagating electric and magnetic field components are existent inside the Ca/Al layer. The discontinuity of the optical absorption at the Ca/Al interface is ascribed to the difference of the material’s extinction coefficient, as shown in Eqs. (43)-(44). On the other hand, the spatial distribution of optical absorption shows the oscillating behavior due to the optical interference effect between the forward- and backward propagating waves within the PEDOT:PSS/ITO layers [34].

Fig. 12 Calculated 2D contour plot of the optical absorption for (a) TE- and (b) TM-polarized optical modes with respect to the longitudinal position and normalized in-plane wave vector when the dipole emitter, which is 50 nm away from the Ca/Alq₃ interface, is marked with a white dashed line.

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Fig. 13 Calculated 2D contour plot of the optical absorption for (a) TE- and (b) TM-polarized optical modes with respect to the longitudinal position and normalized in-plane wave vector when the dipole emitter, which is 198 nm away from the Ca/Alq₃ interface, is marked with a white dashed line.

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Fig. 14 The polarization-dependent optical absorption profiles of several optical modes when the dipole position is (a) z_ex = 50 and (b) z_ex = 198 nm. To increase the visibility in Fig. 11, the absorption profiles are only plotted near the Ca/Al metal and PEDOT:PSS/ITO/glass layers while the zero-valued absorption profiles at the loss-free Alq₃/TPD layers are omitted. For TE polarization, the four optical modes are chosen as the air mode (u = 0), the substrate mode (u = 0.7), the TE0 WG mode (u = 0.946), and the TE1 WG mode (u = 1.038). In the case of TM polarization, the normalized in-plane wave vector corresponds to the air mode (u = 0), the substrate mode (u = 0.7), the TM0 WG mode (u = 0.908), the TM1 WG mode (u = 0.998), and the TM2 SPP mode (u = 1.038).

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4. Discussion

Generally, the optical modeling of OLEDs has been performed based on either the analytical or numerical method. It is noticeable that the internal Poynting vectors and electric field distributions were simulated in OLED devices by using the conventional numerical methods such as the finite-difference time-domain [35–37], the finite element method [38,39], and the rigorous coupled wave analysis [40]. Because the numerical method approximately solves the Maxwell’s equations in discrete tiny meshes, it requires a relatively large computation complexity and time. In addition, the accuracy of the calculation results obtained by the numerical method can depend on the size of the mesh and the type of the boundary condition. In contrast, the proposed GPVM is classified into the analytical method based on the exact mathematical solutions of the governing equations. This has the advantage of a relatively small computation power and the easiness to gain a physical insight. Compared with the numerical methods, the proposed GPVM applied the OLED device has the advantage that it can provide the exact distributions of the electric field intensity, Poynting vector, and optical absorption in both spatial and spectral domains without huge computation power and time.

The calculated internal profiles of the optical field intensity and absorption loss can be used in the optimization process of the OLED performance as follows. The internal spatial profile of the electric field intensity of the air mode calculated by the GPVM can be used as an efficient calculation method of the emission zone profile. According to the quantum-mechanical approach to the optical modeling of the OLED, the output intensity of the air mode is determined by the electric field amplitude at the emitter position, which corresponds to the spatial profile of the electric field intensity of the air mode inside the EML. Classically, the spatial profile of the emission zone has been theoretically predicted by means of the output intensities into air with respective to different dipole positions, where the PDM is used [41]. Figure 15 shows the comparison of the normalized emission zone profiles between the output intensities calculated by the PDM and the internal electric field profile obtained by the GPVM at the extraction angles of 0°, 30°, and 60° for TE- and TM-polarized light emitted by the horizontally-oriented dipole. Two normalized profiles in the EML are in a very good agreement. To calculate the emission zone profile based on the PDM, the output intensities for each extraction angle are calculated 342 times at the emitter-position interval of 1 nm. Regarding the GPVM, only one calculation of the internal electric field profile for each extraction angle is required when the dipole emitter is assumed to be located at the EML/TPD interface.

Fig. 15 Comparison of the normalized emission zone profile between the output intensities calculated by the PDM and the internal electric field profile obtained by the GPVM at the extraction angle of 0°, 30°, and 60° when the horizontally-oriented dipole emits (a) TE- and (b) TM-polarized light. To calculate the emission zone profile based on the PDM, the output intensities for each extraction angle are calculated 342 times when the emitter-position interval is 1 nm. Regarding the GPVM, only one calculation of the internal electric field profile for each extraction angle is required.

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Figure 16 shows the calculated ratio of the optical absorption distributed to each absorptive layer as a function of the emitter position for three dipole orientations, which are the TE-polarized horizontal, TM-polarized horizontal, and TM-polarized vertical orientations, respectively. The calculated ratios of the optical absorption are obtained by summing the optical absorption over the spatial region of each absorptive layer for whole spectral regions at a given dipole position. There is no optical absorption at the transparent layers of the EML, TPD, and glass substrate. Although the 15-nm thickness of the Ca layer is small, a very large portion of the optical absorption occurs at the Ca layer. The calculation results in Fig. 16 can be used as a detailed source of heat generation in the thermal analysis of the OLED.

Fig. 16 Calculated ratio of the optical absorption distributed to each absorptive layer as a function of the emitter position for three dipole orientations, which are the TE-polarized horizontal, TM-polarized horizontal, and TM-polarized vertical orientations. The calculated ratios are obtained by summing the optical absorption over the spatial region of each absorptive layer for whole spectral regions at a given dipole position.

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In this paper, the validity of the proposed GPVM is demonstrated through the comparison of the calculated results between the proposed GPVM and the currently-used PDM. Firstly, we show that the mathematical formulation of the power dissipation spectrum based on the GPVM is identical to that used by the PDM. In addition, we demonstrated that the viewing angle dependence of the light output power calculated by the PVM, identical to the GPVM in the air mode, was well matched with that obtained by the PDM in a top-emitting OLED [27]. Because there have been many reports to show a good agreement between experiment and PDM-based calculation results [15,42,43], the equivalence of the calculation results between the proposed GPVM and the currently-used PDM indicates that the calculation results obtained by the GPVM will be also in a good agreement with the experimental results.

5. Conclusion

We presented the theoretical formulation and calculation results of the GPVM to simultaneously calculate the spatial and spectral profiles of the electric field intensity, optical power flow, and optical absorption as well as the power dissipation spectrum for all optical modes of OLEDs. The mathematical formulation of the GPVM was derived by combining the dipole source term and TMM as a function of the normalized in-plane wave vector u. In the GPVM, the spectral power density per unit solid angle at the EML P_e(u) could be converted into the spectral power density per unit normalized in-plane wave vector K(u), which proved to be equivalent to that presented by the currently-used PDM.

The proposed GPVM provided the spatial profile of the electric field intensity of all the optical modes in the OLED, which was required to evaluate the two-beam interference effect in the electromagnetic model that corresponded to the electric field intensity at the emitter position in the quantum-mechanical treatment. In particular, the GPVM could calculate the spatial profile of the electric field intensity of both the traveling-wave-like air and standing-wave-like WG and SPP modes. The normalized electric field profiles of the WG and SPP modes calculated by the GPVM were close to those obtained by the BMA except the slight difference at the dipole position, which was only noticeable in the case that the dissipated power of the WG or SPP mode was very small.

Finally, 2D plots of the internal optical power flow and optical absorption were calculated as a function of the longitudinal position and normalized in-plane wave vector, which provided physical and intuitive information on the internal emission process as well as the absorption loss of all the optical modes. The optical power flows of all the modes were flat within the lossless EML and exponentially decreased in the highly absorptive Ca/Al layer. In the PEDOT:PSS/ITO/glass multilayer, a large amount of optical power generated by the air and substrate modes passed through the PEDOT:PSS/ITO layer, but no optical power of the WG and SPP modes penetrated into the glass substrate. Compared with the currently-used electromagnetic methods such as the Green’s function model, dipole radiation model, and PDM, the proposed GPVM has the advantage that it can calculate all the spatial and spectral profiles of the electric field intensity, optical power flow, and optical absorption depending on the dipole orientation and light polarization. We expect the proposed GPVM to be used as a comprehensive optical modeling and analysis tool to enhance the understanding of the internal emission process as well as to provide more efficient optical design of OLEDs.

Funding

National Research Foundation of Korea (NRF) (2018R1D1A1B07047249).

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Generalized Poynting vector model to calculate the spatial and spectral profiles of the electric field intensity, optical power flow, and optical absorption for all optical modes of organic light-emitting diodes

Abstract

1. Introduction

2. Theory

2.1 Determination of the electric field amplitudes at the left and right boundaries of the source plane

2.2 Derivation of the spectral power density per normalized in-plane wave vector from the normalized time-average Poynting vector at the EML

2.3 Derivation of the formulation for the spatial profile of the electric field intensity, optical power flow, and optical absorption

3. Calculation results

3.1 Power dissipation spectrum

3.2 Spatial and spectral profiles of the electric field intensity with the effect of the dipole source included

3.3 Spatial and spectral profiles of the internal optical power flow and optical absorption

4. Discussion

5. Conclusion

Funding

References

Cited By

Figures (16)

Equations (55)

Optics Express