Calculation of the emission power distribution of microstructured OLEDs using the reciprocity theorem
Introduction
OLEDs which contain lateral photonic microstructures are of interest for a number of different applications. Most commonly they have been used to improve the OLED external quantum efficiency, by out-coupling the light trapped in substrate modes, waveguide modes and surface plasmon polaritons [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. To use such devices in displays or lighting applications, such microstructured OLEDs must be designed to provide good angular colour stability. Photonic microstructures have also been studied for other applications, such as developing OLEDs with directional emission [12], [13], where a strong contrast of emission intensity across different viewing angles is required.
For a typical microstructured OLED, the overall far field emission pattern is composed of the out-coupled emission of the trapped modes superimposed on a spatially broad background emission. The broad background is due to the light which was directly emitted from the OLED with wavevectors within the light escape cone and thus not affected by the photonic microstructures. Currently the most common method to model the spatial emission pattern of a microstructured OLED is simply to analyse the photonic dispersion of a grating by the Bragg condition [14], [15], [16], [17]. This calculates the out-coupling elevation and azimuthal angles of trapped modes, but does not provide information about the relative power distribution of the far field emission. Alternative optical modelling methods which calculate quantitatively the far field spatial power distribution from microstructured OLEDs are therefore needed.
A full quantitative analysis of the far field emission pattern requires a rigorous solution to Maxwell's equations. The radiating molecules within the OLED can be described as classical forced electric dipole oscillators distributed in the active layer [18], [19]. Such dipoles can have different orientations and emission frequencies and incoherently contribute to the far field emission pattern. The electromagnetic (EM) waves of the dipoles generated inside the OLED structure are affected by the multiple reflections of the OLED cavity, complicating the optical modelling of OLEDs. In practice, a large number of dipoles need to be used to accurately calculate the spatial emission pattern in such a method, which consumes extensive computational resources. Rigorous methods such as finite-difference time-domain (FDTD), finite element method (FEM) and rigorous coupled-wave analysis (RCWA) have been used to investigate the enhancement of the light out-coupling efficiencies of microstructured OLEDs by distributing a large number of dipoles in the active layer and integrating the energy extracted into air [20], [21], [22], [23], [24], [25]. Apart from such forward methods, Zhang et al. investigated the modes in the microstructured OLEDs in a reverse way by sending an EM plane wave into the OLED at normal incidence and investigating how efficiently it can be coupled into a waveguide [26]. This reported modelling focused on the improvement in light out-coupling efficiencies, however, they did not address the calculation of spatial amplitude of the emission pattern from microstructured OLEDs in the far field, and the need to address such calculations is increasing in the OLED community [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].
In this paper, we develop an efficient optical modelling method based on the Lorentz reciprocity theorem [27], [28] to quantitatively characterise the spatial emission pattern of microstructured OLEDs. This theorem has not been applied to calculate the spatial emission pattern of OLEDs before. The key idea of the reciprocity theorem is to convert a light out-coupling problem of an OLED into a light in-coupling problem [29], [30], [31], [32]. It significantly simplifies the simulation and provides the information of the spatial emission pattern in a computationally efficient calculation. In Section 2, the theory of the Lorentz reciprocity theorem and the calculation methods used are introduced. The experimental details of device fabrication and testing are described in Section 3. A test of the model is presented in Section 4. The calculation results are compared with experimental results to verify the accuracy of the modelling.
Section snippets
Theory and calculation
The Lorentz reciprocity theorem states that the relationship between a localised oscillating current and the resulting electric field is unchanged if the positions where the current is placed and where the field is measured are swapped [32], [33].
is the localised time-harmonic current density oscillating at an angular frequency of ω, and is the resulting electric field produced by the current density, with subscripts as defined in Fig. 1a. The theorem is valid
Experimental methods
In order to verify the accuracy of the model, microstructured OLEDs were fabricated and tested. A 240 nm-thick nanoimprint lithography (NIL) resist was spin-coated on the glass substrate and then brought in contact with a UV-cured polymer stamp which previously replicated the photonic microstructure pattern from a silicon master grating. After being cured by high dose UV exposure, the pattern was imprinted into the NIL resist and the resist film formed a square array pillar grating with a
Results and discussion
We first used the reciprocity theorem to investigate the emission intensity at a fixed azimuthal angle (φ = 0°) with varied elevation angles (θ = 0°–90°). The adaptive sampling method was also applied. The simulation was calculated in both s- and p-polarisation, as shown in Fig. 3. and are the integration intensities over the active layer region in s- and p-polarisation for a case where the refractive indices of all layers are set to 1.52, the refractive index of the substrate at 612 nm.
Conclusions
In this paper, we demonstrate a method to quantitatively analyse the spatial emission pattern of microstructured OLEDs. Using the reciprocity theorem, the calculation of the emission in each direction in the far field can be simplified into two simple calculations of an incident plane wave propagating into a single cell of the periodic structure. This makes the modelling highly parallelisable and saves a large amount of computational resources compared to forward modelling methods. An adaptive
Acknowledgement
S. Zhang, E.R. Martins, G.A. Turnbull and I.D.W. Samuel are grateful to the Scottish Universities Physics Alliance (SUPA) and the Engineering and Physical Sciences Research Council (EPSRC) for financial support.
References (37)
- et al.
Org. Electron.
(2006) - et al.
Org. Electron.
(2014) - et al.
Org. Electron.
(2012) - et al.
Appl. Phys. Lett.
(2000) - et al.
Adv. Mater.
(2001) - et al.
Adv. Funct. Mater.
(2004) - et al.
Appl. Phys. Lett.
(2006) - et al.
Nat. Photonics
(2010) - et al.
Adv. Funct. Mater.
(2012) - et al.
Opt. Express
(2012)
Adv. Mater.
Adv. Funct. Mater.
Appl. Phys. Lett.
Adv. Opt. Mater.
Opt. Express
Phys. Rev. B
Opt. Lett.
Opt. Express
Cited by (30)
Inkjet-Printed and Nanopatterned Photonic Phosphor Motifs with Strongly Polarized and Directional Light-Emission
2023, Advanced Functional MaterialsSurface lattice resonances for beaming and outcoupling green μ LEDs emission
2023, NanophotonicsSpontaneous emission enhancement and directional emission by an optical nanonatenna array on a metallic mirror
2023, Wuli Xuebao/Acta Physica Sinica