We present an analytical method for evaluating the first and second moments of the effective exciton spatial distribution in organic light-emitting diodes (OLED) from measured emission patterns. Specifically, the suggested algorithm estimates the emission zone mean position and width, respectively, from two distinct features of the pattern produced by interference between the emission sources and their images (induced by the reflective cathode): the angles in which interference extrema are observed, and the prominence of interference fringes. The relations between these parameters are derived rigorously for a general OLED structure, indicating that extrema angles are related to the mean position of the radiating excitons via Bragg's condition, and the spatial broadening is related to the attenuation of the image-source interference prominence due to an averaging effect. The method is applied successfully both on simulated emission patterns and on experimental data, exhibiting a very good agreement with the results obtained by numerical techniques. We investigate the method performance in detail, showing that it is capable of producing accurate estimations for a wide range of source-cathode separation distances, provided that the measured spectral interval is large enough; guidelines for achieving reliable evaluations are deduced from these results as well. As opposed to numerical fitting tools employed to perform similar tasks to date, our approximate method explicitly utilizes physical intuition and requires far less computational effort (no fitting is involved). Hence, applications that do not require highly resolved estimations, e.g., preliminary design and production-line verification, can benefit substantially from the analytical algorithm, when applicable. This introduces a novel set of efficient tools for OLED engineering, highly important in the view of the crucial role the exciton distribution plays in determining the device performance.
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The difference in definitions is due to the fact that for n > 0, the nth interface dn separates the nth and (n + 1)th layers, while for n < 0 the nth interface separates the nth and (n − 1)th layers (Fig. 1).
The difference between the procedures is that now the orientation factor of the VEDs (Eq. (8)) should be considered in the evaluation of IS contribution to the emission pattern (Eq. (12)), and the phase induced by the reflection from the cathode (αimg − α0), which appears in Eq. (23) should be suitable for a TM reflection coefficient.24 Moreover, in Eqs. (24)–(27), the side-lobe emission should be compared to the emission to an angle θ ≠ 0, as VEDs do not emit in the forward direction.
