1. Introduction

Solid-state lighting using organic light emitting diodes (OLEDs) is a very promising new lighting technology. Normally, a typical OLED is made of four layers as shown in Fig. 1(a). A thick substrate made of glass or plastic which supports the OLED, an anode made of ITO (Indium-doped Tin Oxide) regions which removes electrons, an organic region made of organic molecules from which the light is emitted, and a metal cathode which injects electrons. The organic region can further be divided into three regions: a hole-transport layer, an emissive layer and an electron-transport layer. Figure 1(b) is a much simplified two-dimensional configuration which we will use during our simulations. It is a multi-layer system with ITO regions that may be structured. Besides the glass substrate, the thickness of each layer is several hundred nanometers.

A large portion of the light cannot escape from the OLED because of total reflection at the interfaces between different layers. In some layers there exist guided modes due to which the light is partially trapped inside the device. A multi-layer structure may act as a micro-cavity and influence the emission intensity. A review of fluorescence near interfaces is presented in Barnes’s paper [1]. On the other hand, the efficiency of the emission also depends in a complicated way on the photonic structure of the OLED, such as a grating. Rigneault’s paper [2] gives some theoretical discussion on how the diffractive structures can influence the amount of extracted light. Rigneault’s theory is confirmed by Lee’s experiments [3].

 

Fig. 1. (a). the structure of an OLED and (b). a simplified configuration used in our simulation.

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In this paper, we start from solving the Maxwell’s equation in homogeneous space, and then we introduce multi-layers and scatterers. We will investigate the optimal thickness of each layer and discuss the cause of the peaks in the far field pattern. We shall also investigate how the intensity pattern changes by using a grating structure in either the ITO or the metal cathode.

For better understanding, we have used exact and approximate analytical formulae for the radiated intensity in all cases where this was feasible. In the case of emission from a grating with strong corrugations, we used a rigorous model based on the Finite Element Method (FEM) [4] with which the radiation of a current density can be calculated in a complicated configuration. We have verified that the computed results of this model agree with the analytical formulae in all cases studied for which analytical formulae exist.

 

Fig. 2. Only the light emitted within the surface-escape cone contributes to the light extraction

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2. The optimization of the polarization

The organic layer is made of polymer molecules that can emit light after excitation by an electrical current. Each emitting molecule can be considered as an electric dipole.

Suppose we have a configuration with two regions, namely, a flat polymer layer with refractive index n and an upper half space of air. We define the extraction efficiency η_coupling as the light emitted to the external environment divided by the light generated inside the device. As shown in Fig. 2, η_coupling can be approximated as the light emitted within the critical angle θ_c, divided by the total energy emitted into the upper-half space. If we ignore the polarization of the dipole and we assume that the transmission within the critical angle is independent of the angle, then by using Snell’s law, only the fraction [5] η_coupling = l-cos θ_c of the emitted radiation is radiated into the air. Here, θ_c is the inner critical angle with the normal to the interface between the polymer and air. If the refractive index of the polymer is n=1.58, we find, η_coupling≈1/2n²=0.2.

 

Fig. 3. The polarization of the dipole can be decomposed into a) a dipole vector that is parallel to the interface and b) a vector perpendicular to the interface. The yellow area is a rough indication of the radiation pattern for the two indicated orientations of the dipole.

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Next, we will briefly show how the polarization of the dipole influences the extraction efficiency. We assume that the atomic transition which generates the photon can be described by a dipole vector p of given fixed length ∣p∣. We define our coordinate system such that the (x,y)-plane is parallel to the interfaces. Each polarization can be decomposed along the three axes. For an in-plane dipole, the proportion of light traveling perpendicular to the surface is large and thus much light is emitted at an angle less than the critical angle θ_c.

From electrodynamics, we can write the Poynting vector of a dipole polarized along the x-axis with a dipole vector p as [6],

where (x₀, y₀, z₀) is the position of the dipole, (x, y, z) is an observation point at large distance R from the dipole, k₀=2π/λ₀ with λ₀ the wavelength in vacuum, c ₀ = 1/√ε₀μ₀ with ε₀ and μ₀ the permittivity and permeability of vacuum.

By computing the integral of the Poynting vector over the escape cone, we find that, for a dipole polarized along the x-direction and n=1.58, the efficiency can be approximated by, η_coupling≈3/4n²-3/16n²=0.27. This is 35% larger than the emission of an isotropic dipole. For a dipole with dipole vector parallel to the z-axis, the absolute value of the Poynting vector is given by a similar formula as above, with x substituted for z. For n=1.58, the efficiency can be approximated by η_coupling≈3/8n⁴=0.06. We can conclude that a dipole polarized along the z-axis can emit only very little light within the surface-escape cone. In the following sections, we will therefore concentrate on in-plane dipoles.

3. TE Waves in Homogeneous space and in a multi-layer system

3.1 TE waves in homogeneous space

Besides the polarization, η_coupling is also seriously influenced by the distance between the metal and the dipole, the thickness of the various layers and the refractive index in each layer.

To simplify the analysis we shall consider a 2-D model with a line current parallel to the y-axis. One can verify that the radiated far field intensity by a current line with strength I inside a homogeneous dielectric with refractive index n₀, is identical to the radiated far field intensity of a line of incoherent dipoles, with dipole vectors parallel to the line and dipole strength $\left|\mathbf{p}\right|=I\sqrt{3}/\omega \sqrt{k_0{n}_0}$ . Hence we can use the current line to simulate a row of incoherent dipoles. When the refractive index increases, we have to increase the current by a factor $\sqrt{n_0}$ to get the correct emission pattern.

Consider a line current parallel to the y-axis given by: J=Iδ(x-x₀)δ(z-z₀)i_y, The dipole moment density per unit of length in the y-direction is given by: p=Ii_y/(-iω), where ω>0 is the frequency of the light and the time dependency of all fields is given by the implicit factor exp(-iωt). Figure 4(a) shows the electric field emitted when the current is at x₀=y₀=0 in air with refractive index n₀=l. The wavelength is λ₀=550nm. It is clear that in the near field, the electric field only has a y component which is given by E_y=-(I/4)ωμ₀H₀⁽¹⁾(k₀n₀R), where H₀⁽¹⁾(k₀n₀R) is Hankel function of the first kind of 0th order and R is the distance between the source and the observation point.

We call this case TE-polarized because the electric field is perpendicular to the (x,z)-plane. Fig. 4(b) shows the absolute value of the Poynting vector (i.e. the intensity) in the far field. The intensity in the far field is observed along a circle with large radius and with center at the current line. The observation points are supposed to be far away from the source, but always inside our (x,z)-plane. In all subsequent figures of the far field intensity, the abcissa is not the angle with the z-axis itself, but the sine of that angle: sinθ=k_x/k₀n₀.

 

Fig. 4. A current source along y-direction in air a) Electric field b) Intensity in the far field

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In this article, we use arbitrary units for the ordinate, but the relative values from different intensity patterns are comparable. It is seen from Fig. 4(b) that the intensity as function of the emission angle is constant, as should be. (The peaks at k_x=±k₀n₀ are due to rounding errors and should be ignored.) We can prove Fig. 4(b) by the analytic calculation of the Poynting vector in the far field ∣S (x,z)∣ =ωμ₀I²/16πR.

To analyze the total radiated energy, we integrate the intensity over the upper-half space. The result is always indicated in the upper-left corner of the intensity pattern.

3.2 A current wire above a flat metal surface

 

Fig. 5. (a). A current source located in a half space of refractive index n=1.58 above a perfectly conducting metal plane and b) the intensity of the far field for source-metal distances at 50nm, 100nm ,150nm and 200nm

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We now add a metal surface below the current source as shown in Fig. 5(a) and consider the metal to be a perfect conductor. The half space above the metal consists of a polymer with refractive index n₀=1.58 and the wavelength emitted by the source is 348nm which is equivalent to 550nm in air. Fig. 5(b) shows the far field intensity pattern for different metal-source distances. The peaks come from the interference between the light emitted by the source and the light reflected by the metal.

 

Fig. 6. The total energy radiated as a function of the distance between the source and the metal plane. The red line indicates the radiated energy into the upper-half space in a homogeneous polymer space without metal.

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In Fig. 6, the total radiated energy into the upper half space is shown as function of the distance of the current line to the metal. The red line indicates the radiated energy into the upper-half space in a homogeneous polymer space without metal. We see that at a distance of 110nm from the metal surface, the emitted energy reaches its maximum, even more than the energy radiated in a homogeneous polymer space.

3.3 A current wire in a one-layer system

 

Fig. 7. (a). A one-layer system with flat surface. (b). The far field intensity pattern is influenced by the metal-source distance z₀ and the thickness of the polymer layer d.

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Next, we add a half-space of air (with refractive index n₁=1) above the polymer layer (with refractive index n₂=1.58) as shown in Fig. 7(a). The far field intensity patterns shown in Fig. 7(b) are determined by the metal-source distance and the thickness of the polymer layer. They follow from the Fourier transform of the near field, multiplied by the factor k_z⁽¹⁾/k₀n₁, which in the present case is given by:

where r₂₁ and t₂₁ are the reflection and transmission coefficients for a TE-polarized plane wave respectively and $k_z^{\left(1\right)}=\sqrt{k_0^2{n}_1^2-k_x^2}{k}_z^{\left(2\right)}=\sqrt{k_0^2{n}_2^2-k_x^2}$ for a wave that is incident on the interface from the side of layer 2. We can see that, the far field electric field consists of two factors: The first factor is the emission by the source in a homogeneous polymer. The second factor is due to the transmission at the polymer-air surface and to multiple reflections in the polymer layer.

In order to optimize the thickness of the polymer layer, the metal-source distance was fixed at 100nm and the total emitted energy as a function of the thickness was calculated.

The result is shown in Fig. 8. We can see that because of the total reflection on the polymer-air surface, the radiated energy into the air is less than the case in homogeneous polymer space shown in Fig. 5. At a thickness of 280nm, the emitted energy reaches its maximum. It is interesting that this function is not smooth: At d=330nm and d=560nm there are discontinuities in the derivative. These correspond to thicknesses at which the number of waves that are guided by the polymer layer increases.

 

Fig. 8. Total radiated energy as function of the thickness of the polymer layer for a dipole in the one-layer system shown in Fig.7-a). The distance of the dipole to the metal is 100nm.

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We can in principle extend this method to more layers, such as glass, an electron transport layer or a hole transport layer, but the calculation will be rather complex. In reality, the glass layer is very thick (several hundreds of micrometers for example), therefore the two-layer system can be modeled by a one-layer system.

4. The effect of a small disturbance

Up to now, we have analyzed the emitted energy by a current in a configuration with flat interfaces. As follows from the numbers in the upper-left of the intensity patterns, the totally radiated energy varies a lot with the configurations, although the current is always the same. To explain the difference in emitted energy, we choose a closed surface ∂Ω around our current source. The surface is chosen so small that the region inside the surface can be considered as homogeneous. We integrate the normal component of the Poynting vector over the surface. By using Gauss’s law, we can change this integral into the integral over the volume. When absorption is negligible we find,

We can see that, the energy emitted by the current is determined by the local field at the current and the current strength. During our simulation, the current is always unity, but a photonic structure can change the local electric field and therefore the total radiated energy.

4.1 Diffraction in homogeneous space: Born approximation

As a possible way to enhance the extraction efficiency, we consider a periodic pattern in the ITO, as indicated in Fig. 1(b). Consider now a row of 2M+1 scatterers located above a line source G as shown in Fig. 9(a). The background is polymer with n₀=1.58 and we first assume that there is no metal plate. The scatterers are made of ITO regions with refractive index n_s=1.8.

 

Fig. 9. (a). A current source with a row of small scatterers. (b). Numerical solution of the radiated intensity generated by a current source and an infinitely long row of periodic scatterers. (c). Rayleigh anomaly.

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Each scatterer is supposed to be infinitely long along the y-direction and its surface area on the (x,z)-plane is denoted by A₀. The distance between two nearby scatterers is p=400nm.

If the scatterers are so small that they will not influence each other, we can use the Born approximation [7]. The total emitted field can be written as the sum of the field emitted by the source without scatterers, and the fields scattered by the ITO. The ITO regions can be considered as secondary sources with unknown source strength. The Born approximation contains an approximate formula for the strength of the secondary sources in terms of the field emitted by the primary source at the position of the secondary sources, the jump of the refractive index and the surface area of the scatterers. We thus obtain for the far field:

Figure 9(b) shows the intensity in the far field as M→∞. These patterns have been computed numerically using a FEM program. We verified that the Born approximation predicts similar patterns as the rigorous numerical model, provided that the ITO regions are sufficiently small. The sharp peaks at θ=±0.13 and θ=±0.74 are caused by the Rayleigh anomaly when one of the diffracted orders is at grazing angle as shown in Fig. 9(c). By using the Grating Formula, k_x^(m)=k_x⁽⁰⁾+2πm/p m=1,2,3. It follows that, when the zero order is at grazing angle, the first order shows a peak at sinθ=0.13 and the second order shows a peak at sinθ=0.74.

4.2 Diffraction with a perfect metal plane

 

Fig. 10. (a). A periodic diffractive structure consisting of small ITO regions inside a polymer half space on top of a perfectly conducting metal plane. b) The far field intensity patterns with small scatterers for the metal-source distances of z₀=50nm, l00nm, 150nm and 200nm.

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Next, we add a perfect metal plane below the source as shown in Fig. 10(a). By comparing Fig. 10(b) to Fig. 5(b), we can see that there are sharp peaks at sinθ=±0.13 folded on the radiation patterns obtained without scatterers. These peaks can be explained by the Rayleigh anomaly. By varying the position and the size of the ITO regions, we find that the amplitudes of the peaks strongly depend on both the relative position of the wire source and the scatterers and the size of the ITO regions, but the total emitted energy remains the same as in the case when there are no scatterers. We can explain this phenomenon by the Born approximation and the conservation of energy. The Rayleigh anomaly comes from the interference between the original source and the scatterers, and the phase differences sensitively influence the amplitude of the peaks. On the other hand, when the ITO regions are so small that they do not influence the local electric field at the source, the total radiated energy remains the same.

4.3 One-layer system with small disturbance

 

Fig. 11. When there is no guided mode inside the polymer layer, only weak peaks appear due to the Rayleigh anomaly. The source is 50nm above the metal plane and the thickness of the polymer layer is 100nm.

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Fig. 12. When the light propagates as a guided mode, there is a peak appears at its first diffraction order. The source is 100nm above the metal plane and the thickness of the polymer layer is 200nm.

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Next, we will analysis the far field pattern for a one-layer system with scatterers. As shown in Fig. 11, when the polymer layer is so thin that there is no guided mode inside this layer, we can only observe very weak peaks at the Rayleigh anomaly.When the polymer layer is so thick that there are guided modes inside the layer, there are sharp peaks appearing at the first diffractive order of the guided mode, as shown in Fig. 12.

If we further change the relative position between the source and the ITO regions, we can still obtain a similar result as in the preceeding case of a row of ITO regions inside a homogeneous half-space. Although the amplitudes of the peaks are sensitive to the relative position between the scatterers and the source, the total energy emitted is again always close to the case without scatterers.

5. A current wire inside a groove

From the previous analysis, it follows that a small perturbation of the structure can not increase the total emitted energy, but only changes the distribution of intensity over angles. We now consider a strong perturbation, namely a metal mirror with a grating structure.

 

Fig. 13. A one dimensional grating in homogeneous space

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As shown in Fig. 13, the background is homogeneous space with polymer n₀=1.58. The groove has a pitch of p=400nm and a depth of d_g=200nm. In this case, an analytical approach can not be used and therefore we have used a rigorous numerical code that is based on the Finite Element Method. Figure 14 shows both the electric near field and the intensity in the far field of three different cases corresponding to different groove widths of 100nm, 150nm and 200nm respectively. The source is at the center of the groove. When the groove width is 100nm, there are only evanescent TE waves inside the groove and therefore only little light can escape from it. When the groove width is increased to 150nm, more light comes out. When the groove width is 200nm, which is more than half of the wavelength, a propagating TE mode exists in the groove and in this case considerably more light is emitted than for a source above a flat metal surface. This phenomenon can be explained as follows: The groove confines a lot of light inside and make the local field large. In this case, the region inside the groove becomes a “golden area” for the emission.

 

Fig. 14. The near field and the radiated intensity patterns for different widths of the periodic grooves. The source is in the center of the groove. The width of the groove from left to right is 100nm, 150nm and 200nm respectively.

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6. Conclusion and prospect

In this paper, we studied the photonic structure of an OLED to increase the light extraction efficiency. We discussed the shape and the peaks in the far field intensity patterns and also analyzed the effect of a grating. At the end of the paper, we need to say that, the simulation is still far from complete. For example, our simulation is based on TE-waves which in the 2-D case mean that the dipole is polarized along the y-direction. If we have a dipole polarized along x-direction inside a grating, then the electric field has both an x and a z component and there exists always a guided TM mode inside the groove. The output coupling is then not sensitive to the groove width any more, but be primarily determined by the depth of the groove and the position of the source. We also used a perfect metal during our analytic and numerical calculation. Depending on the metal used, the conclusions can differ substantially due to absorption. Furthermore, up to now we only performed two-dimensional calculations. For in particular gratings 3-D simulation should be done because in this case the intensity radiated by a row of incoherent dipoles probably is not equivalent to that radiated by a current line (a row of coherent dipoles).