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Quantitative allocation of Bragg scattering effects in highly efficient OLEDs fabricated on periodically corrugated substrates

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Abstract

Bragg scattering effects in bottom-emitting organic light-emitting diodes (OLEDs) grown on corrugated aluminum-doped zinc oxide electrodes are analyzed. Periodic corrugation is introduced by structuring the oxide electrode via UV laser ablation, a process that enables flexible adjustment of the period and height of corrugation. We demonstrate that fabrication of stable and electrically efficient OLEDs on these rough substrates is feasible. Sharp spectral features are superimposed onto the broad emission spectra of the OLEDs, providing clear evidence for Bragg scattering of light from guided modes into the air cone. Theoretical analysis based on an emissive dipole model and conservation of momentum considerations allows a quantitative description of scattering and the associated dispersion relations.

© 2013 Optical Society of America

1. Introduction

One of the main priorities in the development of organic light-emitting diodes (OLEDs) is to further increase their efficiency. It has been shown that the efficiency of OLEDs is determined by electrical, optical, and quantum mechanical processes [1]. In state-of-the-art devices, the electrical and the radiative efficiency can be close to 100% [2, 3]. Losses associated with optical outcoupling, however, can be substantial. For a flat device without further outcoupling measures, outcoupling efficencies are in the range of 20–30% and thus have a major impact on the overall efficiency. In planar OLEDs, total reflection within internal layers of high refractive index inhibits outcoupling of a large fraction of the generated light [1, 4]. To mitigate these losses, index-matched macroscopic outcoupling lenses, [1, 57] embedded mesostructures, [8] or light scattering from periodic [916] or random [1719] nanoscopic structures have been proposed. Impressive efficiency enhancement factors have been reported, but in some cases the reference devices to which efficiencies are compared were not optimized, thus complicating a proper comparison. In addition, fabrication of electrically stable devices with acceptable leakage current on the aforementioned structures has proven to be highly challenging due to rough interfaces. Therefore, in most previous studies, investigation of Bragg scattering of surface plasmon polaritons (SPPs) [13, 20, 21] or index-waveguided modes [13, 22] by periodically corrugated structures has been limited to optical excitation of the organic emitters. Moreover, the layer thicknesses in these structures have not been optimized in terms of outcoupling efficiency, and only few attempts have been made to integrate phosphorescent emitters.

In this paper, we demonstrate highly efficient bottom-emitting OLEDs grown on a conductive oxide layer that is periodically corrugated in one dimension, i.e., along a single spatial direction. The electrodes are composed of aluminum doped zinc oxide (Al:ZnO) and are patterned by a laser interference ablation technique. This versatile patterning method readily enables realization of different lattice constants and grating heights. OLEDs are characterized electrically and optically, and the impact of the grating on the outcoupled light is quantified by measuring the angle resolved spectral radiant intensity. Using optical simulations of a planar reference micro-cavity, [2] we can assign the observed sharp features in the spectral radiant intensity to Bragg scattering of confined and/or guided modes into the air cone. Our analysis demonstrates that theoretical methods proposed earlier for passive microcavities [20] can successfully be applied to electrically driven OLED devices.

2. Experimental

The third harmonic of a commercial Nd:YAG laser was used as the light source to pattern 950nm thick Al:ZnO electrodes on 3mm thick glass substrates [23]. The samples were irradiated with intense UV laser pulses at a wavelength of 355nm with a pulse width of 10ns and a repetition rate of 10Hz. The 7mm diameter beam from the laser was shaped to a 25mm2 rectangle, allowing direct structuring of the substrates. To obtain an interference pattern, the p-polarized laser beam was split into two beams by a 50/50 beam splitter and then recombined at a rotation stage for holding samples with a designed interference angle. The laser power was adjusted to achieve a flux of 380mJ/cm2 at the sample surface and monitored by a power meter. The exposure time (and thus the number of laser pulses irradiating the sample) was controlled by a mechanical shutter (Uniblitz Electronic VS25S2ZMO) with a temporal resolution of 3.0ms. For two-beam interference patterning, the samples were loaded on a precision rotation stage and exposed to the pulsed interference pattern. Electrode patterning was performed in ambient atmosphere. Further information about the experimental setup has been published elsewhere [24, 25]. The interference ablation technique represents a flexible and scalable method to structure different substrates with high quality periodic patterns at reasonable troughput. Thus, the same method has been used by us to pattern PET substrates and achieve an efficiency gain in organic solar cells [26].

Figures 1(a)–1(c) shows the results of an atomic force microscopy (AFM) characterization of three Al:ZnO coated samples patterned at different interference angles (AFM from AIST-NT Co). The scans reveal deep and narrow grooves between broad and rather flat mesa structures, with corrugation depths approximately proportional to the grating period. Besides AFM, we have also used laser diffraction and the emission spectra of the OLEDs (see below) for measuring the lattice constant Λ of the different gratings. The results are summarized in Table 1.

 figure: Fig. 1

Fig. 1 a–c) Atomic force microscopy (AFM) images of the substrates for Devices A to C, with periodicity as indicated on each panel. d) Schematic layer structure of investigated devices.

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Tables Icon

Table 1. Comparision of the lattice constants Λ for Devices A, B and C from measurements of the OLEDs emission spectra, from diffraction experiments with a red laser, and from AFM measurements.

To investigate the impact of the electrode corrugation on the characteristics of OLEDs fabricated on top of them, we compared OLEDs comprising electrodes with three different grating periods shown in Figs. 1(a)–1(c) against reference devices with unstructured, planar electrodes. The OLED stack shown in Fig. 1(d) was used for all devices. This stack is optically optimized for the planar reference device. Following the “pin”-concept, we use moleculary doped charge transport layers to reduce injection barriers at the electrodes and the voltage drop across these layers, thus minimizing the driving voltage of the OLED [27]. Adjusting the thickness of these doped transport layers allows us to place the recombination zone into a resonance of the optical microcavity, hence maximizing the outcoupling efficiency [28, 29]. Charge carriers are injected from an aluminum cathode and from the corrugated Al:ZnO anode. MeO-TPD (N,N,N’,N’-tetrakis(4-methoxyphenyl)-benzidine) doped with 2 wt.% F6-TCNNQ (2,2’-(perfluoronaphthalene-2,6-diylidene)dimalononitrile) forms the p-doped hole transport layer, BPhen (4,7-diphenyl-1,10-phenanthroline) doped with cesium is the n-doped electron transport material. The emission layer consists of NPB (N,N’-di(naphthalen-1-yl)-N,N’-diphenyl-benzidine) doped with 10 wt.% Ir(MDQ)2(acac) (Iridium(III)bis(2-methyldibenzo-[f,h]chinoxalin)(acetylacetonat)). It is enclosed by blocking layers which are chosen such that they confine the charge carriers within the emission layer and avoid diffusion of excitons away from the emission layer. We use Spiro-TAD (2,2’,7,7’-tetrakis-(N,N-diphenylamino)-9,9’-spirobifluorene) as electron blocker and BAlq2 (Aluminum (III) bis(2-methyl-8-quninolinato)-4-phenylphenolate) as hole blocker. All organic layers of the devices were fabricated by physical vapor deposition in an UHV tool (Kurt J. Lesker Co.) at a base pressure of 10−7–10−8 mbar. The devices were encapsulated with a glass lid under inert nitrogen atmosphere immediately after processing. Further details of the OLED stack have been described elsewhere [29].

The current-voltage-luminance (IVL) characteristics of the different devices were determined by a source measure unit (Keithley 2400) and a calibrated spectrometer (Instrument Systems CAS140) providing the spectral radiant intensity. Angle-dependent measurements for emission angles θ (0° ≤ θ < 90°) were performed with a custom-build spectro-goniometer equipped with a polarization filter (for s- and p-polarization) and a calibrated miniature spectrometer (USB 4000, Ocean Optics Co.). From the IVL characteristics and the angle dependent emission spectra of the OLEDs, the external quantum efficiency (EQE) is calculated. Figure 2(a) shows the IVL characteristics of OLEDs comprising either corrugated electrodes or planar contacts.

 figure: Fig. 2

Fig. 2 a) Current-voltage and voltage-luminance characteristics of the planar and corrugated phosphorescent red emitting OLEDs. b) External quantum efficiency with respect to the current density.

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For positive bias voltages, the electrical characteristics of the OLEDs are hardly affected by the corrugation, but for negative bias, the leakage current increases slightly with the highest corrugation depth corresponding to the largest leakage currents. However, in all cases the current density at −5V remains below an absolute current density of 1mA/cm2.

The EQE data of the devices investigated are summarized in Fig 2(b). Our bottom emitting OLEDs on corrugated substrates are highly efficient, achieving an EQE of 14.1% at a luminance of 1000cd/m2 (see Fig. 2(b), Device A). These efficiencies are particularly notable when taking into account the nearly one micron thick semi-transparent Al:ZnO electrode that introduces parasitic transmission losses compared to a standard indium tin oxide (ITO) electrode. Figure 3 compares the transmission of flat and corrugated ZnO:Al layers deposited on glass with a 90 nm thick ITO layer, including also the photoluminescence spectrum of the emitter material. More specifically, in the region of interest, the direct transmission of about 87 % for 90 nm ITO is reduced to an average around 80 % across the flat ZnO:Al, dropping further to about 73 % for corrugated doped zinc oxide. Hence, already the bare corrugated oxide shows a total Bragg scattering efficiency of the order of 10 %. The lower transmission of the unstructured zinc oxide can be assigned to a higher refractive index than for ITO [3032], whereas absorption and scattering losses at the rough surface proportional to the fourth power of the photon energy [33] occur mainly far above the emitter spectrum. For the transmission across the bare and corrugated ZnO:Al substrates, optical simulations assign the transmission maximum at 2.04 eV to the 7th cavity waveguide mode. In the completed OLED, the optical thickness increases even further, so that the observed waveguided modes carry higher indices.

 figure: Fig. 3

Fig. 3 Transmission spectra of an ITO reference electrode, and the unstructured and structured ZnO:Al electrodes deposited on glass substrate. The normalized photoluminescence spectrum of the emitter Ir(MDQ)2(acac) highlights the energy interval of interest.

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The device with the smallest grating period (Λ ≈ 0.7μm) enhances the EQE by a factor of about 1.26 with respect to the planar reference whereas Device C with the largest grating period (Λ ≈ 1.9μm) performs similar to the uncorrugated reference. The intermediate grating period of Λ ≈ 1.3μm achieves considerably lower luminance and EQE. Our results prove that the corrugated Al:ZnO electrode is fully compatible with highly efficient state-of-the-art OLEDs, and therefore this electrode material represents a promising and cost-efficient alternative to ITO.

In Fig. 2(a), the electrical performance of all devices remains similar in the forward direction. Hence, in order to explain the efficiency differences between Devices A–C, the angle resolved emission spectra for both polarization directions are shown in Figs. 4(a)–4(f), where the emission angles were converted to their respective in-plane wavenumbers kx = sinθk. We observe the contributions from the 8th and 9th cavity mode, where the shape of the phosphorescence spectrum of the Ir(MDQ)2(acac) shifts the lower of these modes to higher energy. Moreover, Bragg scattering of waveguided modes into the air light cone produces additional sharp features within the spectra which are discussed in detail in the next Section. The forward luminance characteristics of the devices shown in Fig. 2(a) can be understood from the emission at normal emission angle θ = 0 corresponding to kx = 0. In Figs. 4(e) and 4(f), we report enhanced emission at energies E ∼ 2.04eV for Device C compared to Devices A and B (Figs. 4(a)–4(d)). At large in-plane wavevectors kx, Device A shows the strongest emission intensity (Figs. 4(a) and 4(b)), contributing substantially to the particularly large EQE in Fig. 2(b). Device B shows neither high intensity along the substrate normal nor at large in-plane wavevectors, so that the overall performance in terms of EQE remains poor.

 figure: Fig. 4

Fig. 4 Measured angle resolved emission spectra for the OLEDs on corrugated electrodes. Figures 4(a) and 4(b) show the s- and p-polarized emission for Device A, Figs. 4(c) and 4(d) for Device B, and Figs. 4(e) and 4(f) for Device C.

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3. Optical simulation

The emission spectra of planar OLEDs with the stack architecture described above are modeled by using the full dipole model including the Purcell effect [2, 34] and taking into account the non-isotropic dipole orientation of the emitter material [35]. From the obtained spectral power density IK (summarized in Fig. 5), we extract the dispersion relations of the optical modes supported by the organic microcavity [2].

 figure: Fig. 5

Fig. 5 Simulated power dissipation spectrum K for a planar OLED with stack geometry as shown in Fig. 1(a) for s-polarization, 1(b) for p-polarisation. Dash-dotted lines representing the light lines seperate the regions with coupling of power to air, into the glass substrate, index-guided within the organic material (org), the transparent electrode material (AZO), and the evanescent regime.

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From the maxima of the calculated s-polarized power dissipation spectrum (Fig. 5(a)) for light originating from horizontally oriented dipoles, we observe five prominent dispersion branches trapped within the organic material. Their effective refractive indices ranging from neff = ∂kx/∂k0 = 2.14 for the first to neff = 2.34 for the fifth index-waveguided mode. The dispersion branches of the index-waveguided modes extend into the evanescent regime due to the high refractive index of the electrode material. We emphasize that all modes on the right-hand side of the light line of the organic emitter material are evanescent index-waveguided modes with respect to the emitting dipoles in the organic material, so that they are only excited by the dipoles due to a finite extent of the evanescent envelope into the emissive region of the device. The large number of guided modes is due to the very thick transparent ZnO:Al layer as already explained from Fig. 3. In total we are able to determine nine dispersion branches within the power dissipation spectra. Due to the power which is radiated from the emitting dipoles into the many trapped modes, this is expected to lead to a decrease in outcoupling efficiency compared to the standard ITO electrode (thickness ∼ 100nm) [36]. For p-polarized light (Fig. 5(b)) originating from vertically and horizontally oriented dipoles with respect to the substrate normal, there are five dispersion branches with effective refractive indices between neff = 2.14 and neff = 2.31 associated to the index-waveguided modes for p-polarized light in the organic material and the Al:ZnO. Besides, there is one additional dispersion branch close to the right hand side of the organic emitter material light-line which has an effective refractive index of neff = 2.03, corresponding to the SPP mode at the Al/BPhen interface [37]. Power dissipation from emissive dipoles to this branch is weak in comparison to the other modes because the coupling is largly suppressed by the large distance (about 3/4 λ) between emitter and metal contact [2].

For the shallow gratings embedded in our three devices, the ratio r = h/d between the height h of the grating and the thickness of the microcavity d takes the values of rA = 0.056, rB = 0.073, and rC = 0.152, i.e. far below unity. Hence, the change in the shape of the dispersion relations kx(E) with respect to the planar reference OLED is expected to be negligible [38]. This enables us to base our analysis on the modes propagating in planar cavities. By applying a wave vector model, we are able to predict the position of the scattered dispersion relations kx,scatt in reciprocal space via the equation of conservation of momentum along the grating [38, 39]

kx,scatt=ksinθEML=kx+mG,
where kx is the in-plane component of the wave vector k⃗, G = 2π/Λ is the reciprocal lattice constant for a spatial period Λ, and m is an integer describing the scattering order. The wavenumber k is defined with respect to the emitter material as |k|2=n2k02=kx2+ky2+kz2 where n is the refractive index of the emitter material, and k0 is the vacuum wavenumber. In the special case of ky = 0, θEML represents the angle of propagation with respect to the surface normal inside the cavity. In the abscence of Bragg scattering, kx/k = sinθEML < 1 corresponds to propagating modes inside the cavity with real components kx and kz. In the opposite case, i.e. kx/k > 1, the component kz along the normal becomes complex, resulting in exponentially decaying evanescent envelopes in the direction normal to the substrate, but propagation along the x-direction. According to Eq. (1), Bragg scattering of sufficiently high order m can diffract intensity from modes outside of the outcoupling cone with kx > k0 into components inside the outcoupling cone (0 ≤ θ < 90°), i.e. into components with kx,scatt/k0 = sinθ < 1.

Figures 6(a)–6(c) visualize the measured spectral radiant intensity of the OLEDs comprising gratings with different periodicity (Devices A–C). Emission angles θ were converted into the corresponding kx value and emission wavelengths into photon energy E = h̄ck0. To clarify the influence of the grating structure on the angle dependent spectral radiant intensity, we computed the difference, ΔInorm, between the normalized spectral radiant intensity of the corrugated devices and the reference device, i.e.

ΔInorm=Icorr(E,kx)/max(Icorr(E,0))Iplanar(E,kx)/max(Iplanar(E,0)).
Here, the normalization was done independently for each spectrum with respect to the maximum value of emission along the sample normal, i.e. the maximum for the emission angle θ = 0° and the in-plane wave vector kx = 0.

 figure: Fig. 6

Fig. 6 Difference ΔInorm of normalized Spectral Radiant Intensities between periodically corrugated and planar devices Inorm inside the air cone (colored scale) and simulated power dissipation spectra K outside the air cone (grey scale): Fig. 6(a) Device A in s-polarization, Fig. 6(b) Device B in p-polarization, Fig. 6(c) Device C in p-polarization. In all panels the fitted reciprocal lattice vector G is visualized by double ended arrows. Different Bragg-scattered modes are indicated by dashed lines: 0th order (black), 1st order (red), 2nd order (blue), 3rd order (yellow), 4th order (magenta), 5th (light-blue). The black dash-dotted lines seperate air cone, substrate, organic index-waveguides, transparent electrode index-waveguides and evanescent modes.

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In all three devices, we observe several approximately linear and relatively sharp features superimposed to broad parabolic structures. The main color contrast along the parabolic shapes arises from a small shift of the cavity modes upwards for all corrugated substrates, reducing in turn the emission intensity below about 2 eV and enhancing it above. This shift of the cavity modes results from a small reduction of the average thickness of the oxide layer during the laser patterning process.

Outside the air cone in Fig. 6, experimental data are not available, allowing us to report the simulated power dissipation spectra at large in-plane wavevector of a planar device alongside the observed spectra. In this region, the calculated dispersion relations of the different modes supported by the cavity are highlighted by dashed black lines. Scattering of these modes corresponds to a displacement in Fig. 6 by m · G along the kx-axis. Increasing orders of Bragg scattering are visualized in different colors.

Figure 6(a) shows the measured data of Device A for s-polarized emission and the corresponding calculated s-polarized power dissipation. Figure 6(b) and 6(c) show p-polarized spectra of Devices B and C, respectively, as well as the p-polarized power dissipation. For the three devices, the simulated dispersion relations of the confined modes are replicated into different regions of the air cone, because of the different corrugation periodicities Λ and the resulting difference in Bragg vectors G.

The sharp approximately linear features inside the air-cone of Device A–C are attributed to wave guided modes that are Bragg scattered by integer multiples of the reciprocal lattice vectors. The distance (in k-space) between the different scattering orders and between the original mode and the scattered modes provides an independent measure of the reciprocal lattice vector G and the corresponding periodicity Λ in real space. We obtain reciprocal lattice vectors of GA = 8.78μm−1 (Λ = 0.72μm) for Device A, GB = 4.70μm−1 (Λ = 1.34μm) for Device B, and GC = 3.23μm−1 (Λ = 1.94μm) for Device C. Table 1 compares lattice constants derived from Bragg scattering of confined modes to the values obtained from light diffraction with a red laser and from AFM scans measured over 100 periods of the substrate corrugation. All measures of the lattice constant agree within the expected error margins. The good agreement of these values with the AFM measurements that provide a local measure of the periodicity confirms that the observed features can indeed be described by Bragg scattering using dispersion relations calculated for the planar reference OLEDs. For the s-polarized emission in Fig. 6(a), we identify scattered dispersion branches of first and second order. The dispersion branches with negative slope arise from solutions of the wave equation for negative wave numbers. The scattering is fully symmetric around the kx = 0 plane. Within the accuracy of our measurement, we cannot identify any cross-coupling effect of different waveguide modes at their intersections. For Device B in Fig. 6(b), first order scattering is not sufficient to reach the air cone and therefore remains confined to the cavity. As a result, only second and third order Bragg scattered modes contribute to the observed emission spectrum. The spectral radiant intensity pattern of Device C (Fig. 6(c)) reveals contributions from scattered modes of second to fifth order.

4. Conclusion

In conclusion, we have demonstrated highly efficient OLEDs on a periodically corrugated Al:ZnO electrode. Direct laser interference patterning of this transparent electrode on the sub-micron length scale proves to be a versatile and fast technique allowing to improve the performance of OLEDs by enhancing the optical outcoupling through the corrugated electrode. The structuring method is compatible with roll-to-roll processing on flexible substrates, so that it can easily be integrated into the production of organic optoelectronic devices at a large industrial scale. Moreover, the nanostructured transparent electrodes can easily be combined with arrays of macroscopic lenses on the rear side of the substrate, a technique which would combine a nanoscopic and a macroscopic strategy for the enhancement of the outcoupling efficiency.

Despite the large thickness of the electrode layer and the associated large mode volume that contains multiple guided modes, we observed remarkably high OLED efficiencies. Our theoretical analysis demonstrates that the approximately linear features in the angle resolved emission spectra can be quantitatively assigned to Bragg scattered confined modes, hence contributing to the improved performance.

Using the flexibility and versatility of the laser interference patterning process, we intend to analyze a wider range of grating constants, grating heights and electrode thicknesses, in particular with regard to further substantial efficiency enhancement. For thinner electron transport layers, coupling of emissive dipoles to surface plasmon polaritons propagating along the electrode-organic interface will become more prominent so that the respective OLEDs should unveil pronounced Bragg scattering of these modes in the outcoupled spectra. This provides the potential for dramatic efficiency improvements. Nevertheless, to guide a rational device design and to capitalize the full potential of this approach, we need to further develop the theoretical understanding of Bragg scattering in OLEDs, including a quantitative description of the efficiency of the scattering process.

Acknowledgments

We thank the von Ardenne Anlagentechnik for granting the unstructured substrates and BMBF for partially funding this work under contract No. 13N11060, project acronym “R2FLEX”. We also like to thank the Fraunhofer Society facilitating this work by the FhG Internal Programs under Grant No. Attract 692 174. Additionally we thank the Fraunhofer Society, the European Union, and the Free State of Saxony (Dresdner Innovationszentrum Energieeffizienz - DIZE-eff) for financial support. This work was supported by the European Social Fund and the free state of Saxony through the OrganoMechanics project. Moreover we received funding from the AIF/BMWi in the framework IGF under contract no. GF-11/05 resp. AiF-Nr. 16784 BR.

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Figures (6)

Fig. 1
Fig. 1 a–c) Atomic force microscopy (AFM) images of the substrates for Devices A to C, with periodicity as indicated on each panel. d) Schematic layer structure of investigated devices.
Fig. 2
Fig. 2 a) Current-voltage and voltage-luminance characteristics of the planar and corrugated phosphorescent red emitting OLEDs. b) External quantum efficiency with respect to the current density.
Fig. 3
Fig. 3 Transmission spectra of an ITO reference electrode, and the unstructured and structured ZnO:Al electrodes deposited on glass substrate. The normalized photoluminescence spectrum of the emitter Ir(MDQ)2(acac) highlights the energy interval of interest.
Fig. 4
Fig. 4 Measured angle resolved emission spectra for the OLEDs on corrugated electrodes. Figures 4(a) and 4(b) show the s- and p-polarized emission for Device A, Figs. 4(c) and 4(d) for Device B, and Figs. 4(e) and 4(f) for Device C.
Fig. 5
Fig. 5 Simulated power dissipation spectrum K for a planar OLED with stack geometry as shown in Fig. 1(a) for s-polarization, 1(b) for p-polarisation. Dash-dotted lines representing the light lines seperate the regions with coupling of power to air, into the glass substrate, index-guided within the organic material (org), the transparent electrode material (AZO), and the evanescent regime.
Fig. 6
Fig. 6 Difference ΔInorm of normalized Spectral Radiant Intensities between periodically corrugated and planar devices Inorm inside the air cone (colored scale) and simulated power dissipation spectra K outside the air cone (grey scale): Fig. 6(a) Device A in s-polarization, Fig. 6(b) Device B in p-polarization, Fig. 6(c) Device C in p-polarization. In all panels the fitted reciprocal lattice vector G is visualized by double ended arrows. Different Bragg-scattered modes are indicated by dashed lines: 0th order (black), 1st order (red), 2nd order (blue), 3rd order (yellow), 4th order (magenta), 5th (light-blue). The black dash-dotted lines seperate air cone, substrate, organic index-waveguides, transparent electrode index-waveguides and evanescent modes.

Tables (1)

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Table 1 Comparision of the lattice constants Λ for Devices A, B and C from measurements of the OLEDs emission spectra, from diffraction experiments with a red laser, and from AFM measurements.

Equations (2)

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k x , scatt = k sin θ EML = k x + m G ,
Δ I norm = I corr ( E , k x ) / max ( I corr ( E , 0 ) ) I planar ( E , k x ) / max ( I planar ( E , 0 ) ) .
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