## Abstract

We derive explicit power dissipation functions for stratified anisotropic OLEDs based on a radiation model of dipole antennas inside anisotropic microcavity. The dipole field expressed by vector potential is expanded into plane waves whose coefficients are determined by scattering matrix method, and then an explicit expression is derived to calculate the energy flux through arbitrary interfaces. Taking advantage of the formulation, we can easily perform quantitative analysis on outcoupling characteristics of stratified anisotropic OLEDs, including outcoupling efficiency, normalized decay rate and angular emission profile. Simulations are carried out on a prototypic stratified OLED structure to verify the validity and capability of the proposed model. The dependencies of the outcoupling characteristics on various emission feature parameters, including dipole position, dipole orientation, and the intrinsic radiative quantum efficiency, are comprehensively evaluated and discussed. Results demonstrate that the optical anisotropy in different organic layers has nonnegligible influences on the far-field angular emission profile as well as outcoupling efficiency, and thereby highlight the necessity of our method. The proposed model can be expected to guide the optimal design of stratified anisotropic OLED devices, and help to solve the inverse outcoupling problem for determining the emission feature parameters.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Organic light emitting diodes (OLEDs) have been the subject of intensive investigation in recent years due to their applications in displays and lighting [1–4]. The high-performance demands of OLED outcoupling characteristics call for comprehensive and quantitative optical analysis of the OLED layered system, especially the effect of various emission features [5–10]. The rapid design cycles of OLED devices prefer to resort to computer models to accelerate the optimal design. Besides, during recent years, abundant organic thin films have been reported to possess a preferred orientation which leads to significant optical anisotropy in materials [11–14]. Therefore, it is necessary to develop a proper model with optical anisotropy considered to accurately analyze and evaluate the dependencies of stratified anisotropic OLEDs outcoupling characteristics on various emission feature parameters.

In most of early researches, under the implicit assumption that all layers are isotropic, various approaches have been developed to investigate the optical behavior of OLED stacks. Broadly, these methods can be roughly divided into two typical models in the OLED optical simulation: (1) one is based on the microcavity model [15–20]; (2) the other one is based on the dipole model [21–28]. The former describes the OLED structure as a simplified Fabry-Perot cavity, which can serve as a simple and intuitive quick guide to predict the irradiance spectra and angular emission. For example, by using the microcavity model, Hoang and Lee *et al*. [16,17] presented theoretical studies on the performance of the red/blue phosphorescent OLEDs constructed by epsilon negative electrodes, and they acquired the optimal luminous efficiency by varying the layer thickness of the structure. While the latter method treats the exciton emission in OLEDs as radiation from a bounded harmonically oscillating dipole antenna, and can be employed to rigorously calculate the electromagnetic field inside/outside the layer structure and provide complete information of the optical characteristics in the OLED stacks. Based on the dipole model, Meerheim *et al.* [25] analyzed the distribution of the different energy loss mechanisms in bottom and top emission OLED stacks, and obtained remarkably high external quantum efficiencies by optimizing the stack structure. Zhu *et al.* [26] analyzed the light outcoupling and angular performance of quantum dot light emitting diode, resulting in an enhancement in the outcoupling efficiency by combining optimal structure and a high refractive index substrate. However, all these mentioned approaches assumed that the materials in the OLED device are isotropic, which leads to unsuitable problems for the structure with anisotropic layers.

In recent years, as anisotropic materials being widely applied in high quality OLEDs, the dipole model has been remarkably extended to address the calculation of dipole radiation in stratified anisotropic mediums [29–35]. Chance *et al*. [29] firstly used the Hertzian vector to describe the field of the emissive dipole and applied the plane wave expansion method to determine the emissive electromagnetic field. Wasey *et al*. [30] extended the plane wave expansion method to outline a classical model for spontaneous emission decay rate within birefringent materials. Moon *et al*. [32] detailed a robust algorithm for the computation of electromagnetic fields radiated by a dipole source in the cylindrically stratified anisotropic mediums. Kim *et al*. [34] presented an optical dipole model to calculate the luminescence from emissive dipoles in a birefringent medium to describe the outcoupling efficiency. These approaches mentioned can rigorously calculate the electromagnetism in the uniaxial mediums, but they usually suffer from lack of a clear and simple way to calculate the emission from the multilayer structure with arbitrary anisotropic layers of an OLED structure. Considering that all the layers in a stack of multi-layer films are anisotropic, Penninck *et al*. [35] presented an explicit and general expression for electric field of the dipole radiation in the anisotropic medium with arbitrary optics axis orientation. However, due to the complex two-dimensional integral expression of the electric field, only an implicit representation of the Poynting vector was given in the form of a four-dimensional integral, which may lead to inconvenience in the process of numerical integration. Since the optics axis of the OLED material is generally normal to the substrate plane, the theory and method can be significant simplified to deal with this situation. Furthermore, they focused on the dipole energy of layered anisotropic systems and did not give formulations about the outcoupling characteristics of the cavity structures, such as the outcoupling efficiency, the normalized decay rate, and the far-field angular emission profile. Therefore, when dealing with more common cases in the stratified anisotropic OLED stacks, it would be more convenient if simplified and explicit formulations are derived for these outcoupling characteristics of the stratified anisotropic OLED stacks.

In this paper, we derive explicit power dissipation functions for the general stratified anisotropic OLED structure based on a radiation model of dipole antennas inside anisotropic microcavity. This proposed formulation brings us much convenience to quantitively analyze the dependencies of the outcoupling characteristics of stratified anisotropic OLEDs on various emissive properties. Simulations are performed on a prototypic OLED structure to investigate the influence of emission features on outcoupling characteristics of the OLED device to verify the effectiveness and necessity of the proposed method.

## 2. Theory and methods

A multilayer OLED made of various functional layers is superior in terms of efficiency and lifetime. A typical stack layout of such an OLED is displayed in Fig. 1(a), and from top to bottom are encapsulation, cathode, electron injection layer (EIL), electron transport layer (ETL), emission layer (EML), hole transport layer (HTL), hole injection layer (HIL), anode and substrate. According to the investigation of Chance, Prock and Silbey (CPS) [29], the exciton formation in OLEDs can be treated as a bounded harmonically oscillating dipole. Therefore, the formulation model is simplified as the dipole source embedded in stratified anisotropic layers illustrated in Fig. 1(b). Then the problem becomes to calculate the energy flux through any interface *z* = *z _{i}* with a dipole source embedded in the

*s*-th layer with each layer characterized by dielectric tensor

**ε**

**I**is the unit dyad,

*ε*

_{h}and

*ε*

_{v}are corresponding to the dielectric eigenvalues in the planes which parallel and perpendicular to the

*z*axis, respectively.

#### 2.1 Power dissipation functions for a stratified anisotropic OLED

Considering a dipole located at ** R**′ = (

*x′*,

*y′*,

*z′*) and assuming a harmonic time component

*e*

^{j}

*common to all field terms, the magnetic vector potential*

^{ωt}

*A**by*

^{x}*x*orientated dipole and

*A**by*

^{z}*z*orientated dipole can be written by [36]

*x*and

*z*directions respectively, j is the imaginary unit,

*μ*is the magnetic permeability,

*k*is the surface-parallel wavevector, and

_{ρ}*J*

_{0}is the zero-order Bessel function. The sign (sgn) function denotes the correspondence of the positive (negative) sign to the wave propagating in the +

*z*(−

*z*) direction, and

*k*is the wavevector in vacuum, and for the convenience, the position coordinates (

_{0}*ρ*,

*z*) are omitted in the following.

Applying the relationship between magnetic vector potential and electromagnetism filed $E=\text{j}\omega (A+1/{k}^{2}\nabla \nabla \cdot A)$and $H=1/{\mu}_{0}(\nabla \times A)$, we can obtain the *z* components of the source electric and magnetic fields as

*z*component of the electric filed corresponds to TM waves while those of the magnetic field corresponds to TE waves.

Using the previous expression as a source term within the microcavity, and considering additional terms due to reflection and transmission at the interfaces, the integral expression of the previous equations can be reconstructed as the following forms

*A′*,

_{l}*B′*,

_{l}*C′*and

_{l}*A*,

_{l}*B*,

_{l}*C*correspond to the forward and backward traveling waves in the

_{l}*l*-th layer respectively. The source terms with Dirac delta function

*δ*are omitted hereinafter to achieve a simplified formulation. The boundary conditions between

*l*-th layer and (

*l*+ 1)-th layer are [37]

*A′*

_{1}=

*B′*

_{1}=

*C′*

_{1}= 0 and

*A*

_{N}=

*B*

_{N}=

*C*

_{N}= 0, we can calculate the ratios of the coefficients starting at the outer layers by the scattering matrix method [38,39]. Then the source term is added into the field to calculate coefficients of the emission layer from the calculated ratios. Once again recursively utilizing the interface equations, the coefficients for each layer can be determined, then the

*z*components of electric and magnetic fields can be obtained. Finally, the

*x*and

*y*components can be calculated by the following equations directly derived from Maxwell equations

The energy flux through the any infinite plane *σ* then can be calculated as

**is the unit normal vector of the plane, the superscript symbol * denotes the complex conjugation, and Re indicates taking the real part. After applying Bessel closure equation [40,41] into the integrand and divided by the total energy of the dipole in unbounded emission materials, the normalized energy flux through any**

*n**z*plane in the

*l*-th layer of the

*z*orientated dipole can be calculated by

*u*is defined as${k}_{\rho}=u{k}_{0}\mathrm{Re}\sqrt{{\epsilon}_{s,v}}$,

*q*is the intrinsic radiative quantum efficiency, and the subscript

*s*denotes the emission layer index. Similarly, the normalized energy flux through any

*z*plane in the

*l*-th layer of the

*x*orientated dipole is

#### 2.2 Outcoupling characteristics of OLED with a dipole in anisotropic microcavity

Outcoupling characteristics, including the outcoupling efficiency, the normalized decay rate, as well as the angular emission profile, have a pronounced influence on the quantum efficiency, and further can be applied to extract the parameters that cannot be measured directly. Upon the proposed model we derived, the outcoupling characteristics could be easily calculated as following.

Based on Kuhn model [29] and by using the reflected filed at the emitter’s position in Eqs. (9) and (10), we obtain an expression for the normalized decay rate in the cavity system as

*N*is the outmost layer index of the OLED structure, and

*S*is the normalized energy flux which can be calculated through Eqs. (15)–(17). The

*x*and

*z*in the superscript denote the

*x*oriented and

*z*oriented dipole respectively.

In the case of plane waves propagating through multilayers, the emitting angle *θ*_{air} in the air region is associated with *u* and wavevector *k*_{0} = 2π*n*_{air}/*λ*, therefore, the angular emission profile can be easily calculated by Eqs. (15)–(17) just by replacing *u* with *θ*_{air} in the outmost layer

*S*is the power emitted in Eqs. (15)–(17) as a function of

*u*, and

*α*denotes the dipole orientation:

*α*= 1 denotes that the dipoles are orientated perpendicular to the substrate plane,

*α*= 0 denotes that the dipoles are orientated parallel to the substrate plane, and

*α*= 1/3 means a random distribution of the dipole orientation.

## 3. Simulation and discussion

Based on the proposed model, comprehensive analysis can be performed for the OLED stack outcoupling characteristic, including the outcoupling efficiency, the normalized decay rate and the far-field angular emission profile. Here we focus on the effect of the emission features, such as the relative emissive dipole position, the dipole orientation and the intrinsic radiative quantum efficiency *q*. Besides, in order to highlight the necessity and effectiveness of the proposed method, effects of the organic layer birefringence on the far-field angular emission profile as well as outcoupling efficiency are meanwhile quantitatively predicted and analyzed in this section. The optical simulations are performed by custom-made MATLAB codes.

The state-of-the-art OLED stack under investigation and the intrinsic emitter spectrum of the emitting materials are shown in Fig. 2. This structure can be described as: the substrate is glass precoated with 160 nm thick layer of indium tin oxide (ITO). Next, a 30 nm thick layer of poly(3,4-ethylenedioxythiophene):poly(4-styrenesulphonate) (PEDOT:PSS) is spun on. Then 50 nm of *N*,*N`*-diphenyl-*N*,*N`*-bis(3-methylphenyl)-[1,1*`*-biphenyl]4,4*`*-diamine (TPD) is evaporated as HTL layer, 30 nm of tris(8-hydroxyquinoline) aluminum (Alq3) is as EML layer and 50 nm of 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline (BCP) is as ETL layer. The cathode is a 100nm thick Mg:Ag layer with 60:1 Mg:Ag ratio with a 20 nm thick Ag cap on top. For more information about the OLED device structure or fabrication, we refer to [24]. Figure 2(b) shows the intrinsic luminance spectrum with a peak emission of about 539 nm. Applied Eqs. (18)–(21) for this structure stack, we can obtain the outcoupling efficiency, the normalized decay rate as well as the far-field angular emission profile.

#### 3.1 Verification of the validity of the proposed model

Since the calculated outcoupling efficiency and normalized decay rate of Celebi *et al*. [24] have been accurately verified by the measurement in the experiment using a reverse bias technique, here we compare the simulation results using our methods with those using Celebi’s methods to verify the validity of the model we proposed. In simulations, the dipole is located at the middle of the EML layer, the intrinsic radiative quantum efficiency is assumed as *q* = 1, and wavelength is set as *λ* = 539 nm corresponding to the peak in intrinsic emission spectrum. The simulated outcoupling efficiency and normalized decay rate as a function of the ETL layer are shown in the top subfigure of Fig. 3. For simplicity, dipoles orientated perpendicular to the substrate plane are denoted as VED and those orientated parallel to the substrate plane are denoted as HED. The phrase “My” and “Re” in the legend respectively denote the simulation results from our model and the referent model of Celebi *et al*. [24]. It can be easily observed that the curves are completely consistent not only for the outcoupling efficiency but also the normalized decay rate. In order to further quantitatively analyze the results of the two models, the differences between these results are presented in the bottom subfigures of Fig. 3, where “δVED” and “δHED” in the legend denote the difference between their model and our proposed model for VED and HED respectively. It can be seen from Fig. 3 that the difference between two models is of the level of 10^{−3}, and therefore the model we proposed can be verified with the simulation results of stratified isotropic OLED structure.

In order to further verify the validity of the proposed model, we also compare the simulation results using our methods with those using the commercial software Fluxim Setfos [42]. The forward normalized emission spectrums under different ETL layer thickness is shown in Fig. 4. The configuration in this simulation is the same with previous one, except that the dipole orientation is assumed as HED here, since the radiation contribution of VED is neglectable in the direction normal to the substrate. The thickness of the ETL layer ranges from 40 nm to 160 nm with a step of 40 nm, and the results using our model are denoted by solid lines while those using Setfos are denoted by open circles as shown in Fig. 4. It can be easily observed that these results by our model and Fluxim Setfos are completely consistent with each other, which further highlights the validity of the model we proposed.

#### 3.2 Influence of the emission features on the outcoupling characteristics

Based on the proposed model, we can easily predict and analyze the influence of the emission features, including the dipole position, the dipole orientation, and intrinsic radiative quantum efficiency, on the outcoupling characteristics of the OLED stacks. The simulated outcoupling efficiency for the VED and HED as a function of the ETL layer thickness and wavelength are shown in Figs. 5(a) and 5(b), respectively. The dipole is located at the middle of the EML layer, and the intrinsic radiative quantum efficiency is assumed as *q* = 1. The ETL layer thickness ranges from 5 to 200 nm, and as the parameter to tune the distance from the dipole position to the metal, it is the main determining factor for the weak microcavity interference effects in this OLED stack. As we can obtain from the figure, the outcoupling efficiency for VED can reach the peak value 10% at the wavelength *λ* = 539 nm and 140 nm thick ETL layer. While the outcoupling efficiency for HED can reach its extremum as high as 34.3% at the wavelength *λ* = 541 nm combined with 30 nm thick ETL layer, and 36.4% at the wavelength *λ* = 535 nm combined with 200 nm thick ETL layer. From the simulation results we found that, compared with the VED, the HED significantly increases the maximum of the outcoupling efficiency, and therefore, the control of the dipole orientation has been identified as a particularly powerful handle for outcoupling efficiency enhancement.

In order to analyze of the influence of various emission features on the outcoupling efficiency more straightforward, without loss of generality, the simulation are carried on mono-chromatic wave and the wavelength is assumed as *λ* = 539 nm which corresponds to the maximum value of the intrinsic emission spectrum. The optical constants of OLED layers within this wavelength are shown in the Table 1. The dielectric eigenvalues in the Eq. (1) under the isotropic assumption then can be related to the refractive indices *n* and the extinction coefficient *k* as *ε _{h} = ε_{v}* = (

*n*+ j

*k*)

^{2}.

Under the wavelength *λ* = 539 nm, the outcoupling efficiency of the OLED stack as a function of the ETL layer thickness under various intrinsic radiative quantum efficiency *q* are shown in Figs. 6(a) and 6(b) for VED and HED, respectively. And Fig. 7 demonstrates the behavior of the normalized decay rate as a function of the ETL layer thickness under various *q* for VED and HED. Both curves exhibit a qualitatively similar trend with two maxima separated by a pronounced minimum, and the pace of the oscillations between the outcoupling efficiency and the normalized decay rate curves is almost the same, which means that the higher outcoupling efficiency is produced mainly due to the stronger normalized decay rate. Furthermore, it can be clearly seen from the figures that the oscillations of the outcoupling efficiency and the normalized decay rate are both damped for decreasing radiative quantum efficiency *q*. And the first order optical cavity is more efficient owing to a higher outcoupling and a stronger normalized decay rate when compared to the second order in the OLED structure under investigation. Therefore, for simulation-based optimization of OLEDs in this section, the emissive dipole should be placed in the first cavity maximum for all the radiative quantum efficiency *q* ranged from 0.2 to 1.

The knowledge of the intrinsic radiative quantum efficiency is paramount important for OLED performance, and this information is usually not available in the directly measurement. Based on the relationship of the normalized decay rate and the lifetime *τ*:

*q*and lifetime

*τ*in the absence of the cavity can be easily determined by comparing measured lifetime data with simulation results. Due to that the normalized decay rate is critically depends on

_{0}*q*and the difference significantly increases at the cavity maximum, the thickness of ETL layer should be the peak maximum if one wants to determinate the

*q*by fitting the measured excited states’ lifetimes. As shown in Figs. 6 and 7, the optimal ETL thickness is 145 nm for VED, while 48 nm or 220 nm for HED.

#### 3.3 Influence of optical anisotropy on the angular emission profile

These simulation results demonstrate that the proposed model can be utilized simply to predict and evaluate the dependencies of the stratified OLEDs outcoupling characteristics on various emission feature parameters. Nevertheless, in order to highlight the necessity and effectiveness of the proposed method, we also need to quantitatively study the impact of anisotropic organic layers on these outcoupling characteristics. Note that the angular emission profile not only allows us to calibrate the layer thicknesses but also to predict or determine the emission zone profile as well as the dipole orientation, and the accuracy of the investigations on the angular emission profile could be improved with the anisotropy considered in the organic layers. In this section we focus on analyzing influence of the anisotropy on the far-field angular emission profile.

The radiative quantum efficiency hereinafter is assumed to be *q* = 1, and the dipole is located at the middle of the EML layer. The extraordinary refractive indices *n*_{e} of the functional layers are set as *n*_{e} = *n* + Δ*n*, where *n* denotes the ordinary refractive index shown in the Table 1 and the Δ*n* is the birefringence. Then the dielectric eigenvalues in the Eq. (1) becomes

Figures 8 and 9 present the normalized far-field angular emission intensity versus the emission angle under various ETL and HTL birefringence respectively. From the Figs. 8(a) and 9(a), we found that for VED, the birefringence of the ETL layer leads to a peak shift while the influence of the HTL birefringence on the far-field angular emission profile can be neglectable. As the birefringence Δ*n* increasing from −0.4 to 0.4, the intensity maximum peak shifts from 43.6° to 49.2° under anisotropic ETL layer. The influence of the anisotropy in organic layer on HED is quite different compared with VED. As shown in Figs. 8(b) and 9(b), the low angle range emission intensity rises obviously with the birefringence Δ*n* increasing from −0.4 to 0.4 for both the ETL layer and the HTL layer. The simulation results therefore demonstrate that the anisotropy in ETL and HTL layer of OLED structure has non-neglectable influence on the far-field angular emission profile. Moreover, when the angular emission profile is applied to fit the emitter properties, including dipole position and dipole orientation, the proposed formulation with the capability of considering the optical anisotropy in arbitrary layer can be expected to provide more accurate analysis in the fitting process.

#### 3.4 Influence of optical anisotropy on the outcoupling efficiency

To further perform analysis about the effect of optical anisotropy in organic functional layers on the outcoupling efficiency of the OLED, power dissipations in different channels, including absorption in top contact, waveguiding in organic layers, waveguiding in ITO layer, waveguiding in glass substrate, and emission into air, are studied by using the proposed optical simulation model. The proportion of power emission into the air to the total power emission is usually defined as the outcoupling efficiency. Substituting the organic function layer birefringence which has been defined in Eq. (23) into Eqs. (15)–(17), we can calculate the dipole energy coupling into *l*-th layer by (*S _{l}* −

*S*

_{l}_{−1})

*b*

_{0}/

*b*, and then the power dissipations of different optical channels can be obtained.

Figure 10 shows power dissipations in different channels versus the birefringence in the ETL and HTL. In the simulations, a monochromatic wave with the wavelength *λ* = 539 nm is used. The radiative quantum efficiency here is assumed as *q* = 1, the dipole is in the middle of the EML layer, and the birefringence ranges from −0.4 to 0.4. The simulation results for VED independently with birefringence in the ETL and HTL layer are shown in Figs. 10(a) and 10(b), respectively. It can be obviously observed from Fig. 10(a) that the power dissipation into glass substrate waveguiding increases with negative birefringence in ETL layer. Therefore, if the enhancement tools are applied to extract light from the glass substrate of the OLED under investigation, ETL layer with negative birefringence could be beneficial to improve the outcoupling efficiency when the dipole orientation are VED domain in the OLED device. The trend of the power dissipation for HED is completely different compared with that for VED. As shown in Fig. 10(c), the outcoupling efficiency could benefit from negative birefringence in the ETL, and it can be improved from 26.2% to 37.2% when the birefringence in ETL layer reduces from 0.4 to −0.4. By comparison as shown in Fig. 10(d), negative birefringence HTL layer leads to a decrease in outcoupling efficiency, which denotes that for HED, the HTL and the ETL have opposing birefringence requirements. Important to note here is that the effect of the HTL layer is notably smaller than the effect of the ETL layer, since the outcoupling efficiency increases only from 29.5% to 32.9% as birefringence in HTL layer raises from −0.4 to 0.4.

Results in Fig. 10 demonstrate that the optical anisotropy in organic layers of the OLED structure has non-neglectable influence on the power dissipation into different optical channels and the final outcoupling efficiency. Therefore, the proposed optical formulation model with the capability of considering the optical anisotropy in arbitrary layer can be expected to provide accurate analysis and prediction on the detailed power dissipations and to guide the optimal design for stratified anisotropic OLEDs with high efficiency.

Since the OLED layers are almost thin films in panel display such as smartphones and tablets, the proposed work focuses on the simulation and quantitative analysis on this kind of OLEDs straightforwardly. Whereas, in applications of general lighting, periodical structures or random patterns are usually employed to improve the light extraction and enhance the final outcoupling efficiency of OLEDs [43–48]. Therefore, how to simulate structured or patterned OLEDs is currently a key issue in this field. It should be pointed out that although the formulations in this work cannot deal with structures or patterns in current version because the optical refractive index of each layer is a tensor with constants in the formulation, the proposed model could be well extended to structured or patterned OLEDs with periodicity structures by simply introducing the mature algorithms, such as the rigorous coupled wave analysis (RCWA) method [49–51].

## 4. Conclusion

In this paper, explicit power dissipation functions are derived for stratified anisotropic OLED. In construction of the model, the dipole field expressed by the vector potential is expanded into a superposition of plane waves, whose coefficients are determined by the scattering matrix method. Then explicit power dissipation functions are derived to calculate the Poynting vector through an arbitrary interface in the anisotropic multi-layer stack. Based on the proposed methods, we can define the calculation of the outcoupling characteristics of a stratified anisotropic OLED, including the outcoupling efficiency, the normalized decay rate and the far-field angular emission profile. Simulations are carried out on a prototypic stratified OLED structure to verify the proposed method and to evaluate the dependencies of the outcoupling characteristics on various emission feature parameters. Results demonstrate the capability and validity of the proposed model, and the necessity of our method is also highlighted due to the non-neglect influence of the layer anisotropy on the far-field angular emission profile and the outcoupling efficiency. Therefore, the proposed formulation model with the capability of anisotropy considered in arbitrary layer can be expected to guide the structure optimization by providing more accurate analysis on the outcoupling characteristics and even prediction on emission features of stratified anisotropic OLEDs.

It is worth noting that, some existing commercial softwares, such as FDTD solution and Fluxim Setfos have been applied to perform the simulation and optimization of stratified anisotropic OLEDs. However, to the best of our knowledge, the FDTD solution is based on the OLED layer discretization which works at the expense of more computation resources, and the Setfos complies with the assumption that the emitting layer must be transparent. In this paper, the proposed model without such constraints is universal and therefore can be flexibly employed to quantitatively analyze or even optimize the performance of stratified anisotropic OLED structures. Moreover, the formulation proposed could be well extended to the thin patterned OLEDs with periodicity structures if the rigorous coupled wave analysis methods are combined.

## Funding

National Natural Science Foundation of China (51727809, 51805193, 51525502, and 51775217), China Postdoctoral Science Foundation (2016M602288 and 2017T100546), National Science and Technology Major Project of China (2017ZX02101006-004), and the Natural Science Foundation of Hubei Province of China (2018CFB559 and 2018CFA057).

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