The incoherent emission of periodically structured Light Emitting Diodes (LEDs) can be computed at relatively low computational cost by applying the reciprocity method. We show that by another application of the reciprocity principle, the structure of the LED can be optimized to obtain a high emission. We demonstrate the method by optimizing one-dimensional grating structures. The optimized structures have twice the extraction efficiency of an optimized flat structure.
© 2010 Optical Society of America
There has always been much interest in coupling light in or out of optical gratings. In the last decade, light-emitting diodes (LEDs) and solar cells have attracted much attention. Whereas for LEDs it is desirable that light is efficiently extracted out of the device, in photovoltaics light from the exterior should be absorbed inside the device. For both applications, highly efficient designs are required to fulfill their promise as a sustainable light and energy source for the near future.
The extraction efficiency of new LEDs is often enhanced by patterning the semiconductor-air interface. Good results have been obtained by random surface roughening , but there is also much interest in periodic structures i.e. two-dimensional gratings (photonic crystals) [2–4,6,7]. In unpatterned LEDs, the light is mostly trapped by total internal reflection and it radiates primarily from the sides of the device. In photonic crystal LEDs, the mode structure of the photonic crystal helps to prevent that light is radiated laterally. However, it does not guarantee that the light radiates into the direction that is desired. This paper focuses on describing an efficient method for optimizing the radiated intensity of LEDs into a desired cone.
The complexity of current designs requires accurate, rigorous, large-scale, three-dimensional (3D) electromagnetic simulations which need a lot of memory and extensive computations. In this paper, we first describe a very efficient method, based on the reciprocity principle, to calculate the radiation in a single direction from a complete photonic crystal LED for any number of incoherent dipoles at any position in the LED. It requires solving only two small quasi-periodic scattering problems on a single cell of the periodic structure. The calculation of the entire radiation pattern of an LED is thus split up into many small computational problems, one for each direction of emission and polarization. Each of the simulations can be done on a standard PC, but in practice we use a supercomputer or cluster to run many angles in parallel. Next to computing the far field radiation, we apply the reciprocity to derive a useful expression of the derivative of the radiated intensity with respect to the grating surface. With this expression the radiated intensity in a certain cone is maximized. Our approach is related to the inverse problem of reconstructing shapes of scattering objects in scatterometry. In this field, an initially unknown permittivity distribution or object shape is reconstructed using only limited field or intensity information, such as far field scattering information. Several of such methods have been developed for gratings  and for more general object shapes . Our formulation is specifically derived for structures which contain incoherent sources such as LEDs.
In classical electrodynamics, the electric field E radiated by a time-harmonic current density J is derived. The relationship between the field and the source remains unchanged if one interchanges the position of the source and that of the observer. This so-called Lorentz reciprocity principle can be written as10], while in  an overview is given of the application of the reciprocity theorem in optics. The theorem is valid for absorbing and anisotropic media, but in general not for nonlinear and active media. In , it is suggested that the reciprocity principle can be used to calculate the radiation of a single dipole in a planar dielectric stack. In the field of near-field imaging , reciprocity is specifically used for extended incoherent sources. For photonic crystal LEDs, the theorem is applied in combination with a rigorous modal simulation method in  and also in  to design a highly directive light source. The efficient calculation of the derivative of the radiated intensity using the reciprocity principle is based on the adjoint method, described in . It is used for inverse scattering problems in for instance [15, 16].
In this paper, we first describe in Section 2 the efficient use of the reciprocity principle for computing the emission pattern of incoherent sources in periodic structures and apply this to LEDs. Although interesting in itself, we moreover need this derivation in the iterative optimization of the topography of the LED, which is the topic of Section 3. In Section 4 we discuss LED topographies consisting of one-dimensional gratings that have been optimized by our method.
2. The emission from an LED
The LEDs described in this paper consist of a metal contact layer that is covered by one or more layers of semiconductor material (Fig. 1) defined in 3D space with unit vectors x̂, ŷ, and ẑ as defined in the figure. Here and in the following, â on a bold face character denotes a vector of unit length. To improve the emission efficiency, the interface Γ between the topmost semiconductor and the surrounding medium is in general not flat. We will specifically discuss the class of interfaces that are periodic in both horizontal directions x and y, though this periodicity is not a requirement for the computational method that we use and describe. Within the semiconductor region, there is a thin region 𝒪 that is the active layer. In this layer, incoherent dipole sources are generated from multiple quantum well structures .
To calculate the far-field emission of an LED, we first consider the radiation in the direction of the unit vector k̂ of a single dipole with dipole moment p𝒪, oscillating at frequency ω, and situated at point rp𝒪 in the active layer. The corresponding current source density is: Jp𝒪 = –iωp𝒪δ(r – rp𝒪) and radiates the field denoted by Ep𝒪(r). Let rp = k̂rp be a point in the half-space above the LED and let there be a time harmonic dipole at rp with frequency ω and dipole moment p. The electric field radiated by the latter dipole will be denoted by Ek. The two sources and radiated fields satisfy the reciprocity principle:Eq. (3) can be simplified to Fig. 1). It suffices to consider two linear independent polarization stats of the incident plane wave. We choose for p the unit vectors ν̂ = Ŝ, P̂ corresponding to the S- and P-polarization respectively.
The fields at the location of dipole p𝒪 in the active layer due to an incident S- and P-polarized plane wave is found by solving the two separate simulations the two boundary value problems defined by:Fig. 1, the fields are quasi-periodic, so that only a single periodic cell has to be considered with appropriate quasi-periodic boundary conditions. Note that this method does not require periodicity of the domain, but for periodic domains only one cell has to be considered.
When the solutions and have been computed numerically, the Ŝ and P̂ component of the field radiated by dipole p𝒪 at rp𝒪 in the direction k̂ follows from reciprocity principle in Eq. (1):Eq. (7) over all directions of p𝒪. By substituting p𝒪 = p𝒪 (cos ϕ sin θx̂ + sin ϕ sin θŷ + cos θẑ) into Eq. (7) and averaging over the unit sphere of directions for p𝒪, we obtain for the intensity emitted in the direction k̂ per solid angle due to randomly isotropically oriented dipoles at rp𝒪 with all the same strength p𝒪: 18]. In that case θ = π/2 and averaging should be done over 0 ≤ ϕ ≤ 2π, yielding: Eq. (8) for the radiation pattern. In the radiation cones considered in this paper, dipoles oriented along the z-axis have a minimal influence on the radiation efficiency. Hence, we assume that the dipoles are isotropically oriented. In case this is not true, only minor modification of the derivations need to be carried out. The total intensity radiated in the direction of k̂ per solid angle of randomly oriented incoherent dipoles with fixed length in the entire active layer 𝒪 is found by integrating rp𝒪 over the active layer:
3. Optimization of the LED surface
For the radiation pattern and the total radiated energy of an LED, the shape and position of the surface Γ is essential. The main goal is to find a suitable Γ for the application under consideration. For simplicity, we restrict ourselves in this paper to a two-dimensional LED-like geometry, i.e. the interface Γ is translational invariant with respect to y and hence is specified by a periodic curve z = Γ(x) = Γ(x + Λ), with Λ the period. The radiating sources are in this two-dimensional situation current wires parallel to the y-axis. In addition, all quantities are per unit length in the y-direction. The full three-dimensional case can be tackled in the same manner as the two-dimensional case.
The assumption of radiation current lines implies that all dipoles along these wires are coherent. It can be shown that in a homogeneous medium when the dipole moments are in the y-direction, i.e. the current is parallel to the wire, the radiation pattern of these coherent dipoles is identical to that of incoherent dipoles . For dipoles oriented in other directions than the y-direction, there is a difference between the radiation pattern of the coherent and incoherent case. However, this difference is not large in a two-dimensional configuration.
First, we suppose that the transition of the permittivity function across the interface is smooth. To be more specific, we choose b > 0 and define for a configuration with surface Γ the permittivity function ɛ(Γ)(x,z) by:Eq. (5). From the theory of reciprocity, I(Γ) is also the contribution of one polarization to the averaged intensity emitted by mutually incoherent wires in the active region, radiated in the single direction specified by k̂ as described in the previous section.
We are interested in increasing this intensity by varying the surface by a perturbation ξh(x), leading to a new surface Γ(x) → Γ(x) +ξh(x), with ξ > 0. Since z = h(x) is a function we use the Gateaux derivative  which in the direction h is given byEq. (5) and the second is that of the same incident plane wave on an LED with surface Γ+ξh: Eq. (17) from Eq. (16) we get Equation (18) is inhomogeneous with a source term that is non-zero exactly there where the permittivity function is altered by the perturbation. We shall take the limit ξ → 0 of Eq. (18). For the permittivity difference on the right-hand-side of Eq. (18), we note that, since ɛ (Γ+ξh) is the function ɛ (Γ) translated by −ξh(x) parallel to the z-axis, we have Eq. (18) becomes in the limit ξ → 0:
Hence, to compute the Gateaux derivatives δE(Γ; h) and therefore δI(Γ; h) in a particular direction h, one has to solve the radiation problem of Eq. (21). In the optimization algorithm, out of all possible h, the direction of steepest ascent should be found, i.e. the direction h̃ such that δI(Γ; h̃) is maximum compared to δI(Γ;h) for all other h of the same norm. By applying again the reciprocity principle in a manner explained below, it turns out that this optimal direction can also be found by solving only one radiation problem (per polarization).
To explain this, we first define the current densityEq. (23) and the reciprocity principle to rewrite Eq. (15) as follows: Eq. (25), the Gateaux derivative δI(Γ; h) can be determined for all perturbations h(x) at a small computational cost. In fact, to compute δI(Γ; h) for an arbitrary h one only needs to know the fields E(Γ) and EJ𝒪. To determine these fields one scattering problem (to compute E(Γ)) and one radiation problem (to compute EJ𝒪) have to be solved. Note that when both polarizations S and P are taken into account we need to solve two scattering and two radiation problems, one for every polarization. But once these four problems have been solved, the Gateaux derivative δI(Γ; h) can be computed for any perturbation h by simply computing the integral in Eq. (25).
Let h be normed in some way, e.g. , then the perturbation h̃ of given norm for which δI(Γ; h) is maximum is given by the h̃ for which the integral at the right of Eq. (25) is maximum. Since h is only a function of x, h̃ is given by
The perturbation h̃ corresponding to the direction of steepest ascent for the case of a permittivity function ɛ that is discontinuous across Γ is obtained by taking the limit b → 0 in the previous result. Taking this limit requires a careful analysis of the integral in Eq. (26) in which the components of the electric field that are parallel and perpendicular to the curve Γ have to be considered separately. These components are denoted by E||(Γ), EJ𝒪, || and E⊥(Γ), EJ𝒪, ⊥, respectively. The result of taking the limit b → 0 is then:Appendix.
Mathematical optimization means to iteratively find values for the degrees of freedom that lead to the largest figure of merit, in our case the radiated intensity. These values are found by systematically trying new values that are chosen such that every next step in the algorithm leads to a higher figure of merit. By choosing h̃(x) such as defined in Eq. (27) the surface given by Γ(x) +ξh̃(x) will lead to a higher radiated intensity, provided that the step size ξ is not too big and that Γ(x) is not already a local maximum.
To compute radiation patterns and the Gateaux derivatives when the curve Γ is optimized electromagnetic boundary value problems on a periodic computational domain have to be solved. Because the computational domain is relatively small (a single period), the problems can be tackled by almost any type of rigorous electromagnetic solvers. For this paper, calculations are performed using a implementation of the Finite Element Method .
In Fig. 2 the radiated intensity from a one-dimensional grating with a shallow, 5nm deep, grating is shown. It is calculated using the reciprocity theorem as described in Section 2. The geometry is similar to that described in . It is immediately clear that the resonance peaks resulting from the patterning are sharp and several orders higher than the background intensity of the unpatterned geometry. Assuming that the metallic contact layer is perfectly conducting, the resonances have Q-factors (Q = kmax/Δkfwhm) up to 105.
Using tabulated permittivity data of silver , the Q-factors reach up to values of the order of 103. These sharp resonances are potentially problematic for the reciprocity calculation method, where for each discrete kx value a new simulation has to be performed. To capture all the maxima, taking into account a potential Full Width Half Maximum (FWHM) of Δk̂fwhm = 10−6, the angles corresponding to 0 ≤ k̂x ≤ 1 would need to be approximated by at least 106 reciprocal simulations. Since, the total radiation depends on the correct approximation of these resonances, this computational burden would render the reciprocity method unpractical. However, since simulations can be done per kx, an adaptive method can be applied. First, a relatively small number of angles (30 to 80 in our case) are simulated. Then a heuristic algorithm determines at what angles additional simulations are required. This is in general close to a local maximum of I(k̂x) or where the derivative ∂I(k̂x)/∂k̂x is large. New simulations are then performed until all maxima are sufficiently resolved and usually not more than 150 angles are required.
For this shallow grating, the locations of the resonances found correspond well to the guided modes in the GaN layer for the unpatterned geometry with a layer thickness of 698nm (Fig. 2). Although all resonances correspond to a guided mode, not all guided modes result in a radiation maximum. It is well known that for the unpatterned geometry most of the light is guided in the GaN waveguide by total internal reflection and will never couple out into the air region. Reciprocally, this means that with a plane wave we can never excite such a guided mode. For the patterned geometry, most of these modes become leaky and result in radiation. There are, however, still guided modes for this geometry that will not contribute to the LED radiation. These resonant modes can be found by a different calculation of the modes in a photonic crystal waveguide .
Figure 2 shows that even a very shallow patterning can significantly increase the radiated power over that of the unpatterned geometry and in this case almost all of the enhancement is in a cone with an opening angle of 90°. For most LED applications, it is desired that the enhancement is restricted to even smaller cones.
In Fig. 3 the Gateaux derivative δI(Γ; h) with respect to the average layer thickness t and grating depth d as indicated in Fig. 1 is shown. From the sign of the derivative it is clear that the radiated intensity of this particular shallow grating can be improved by increasing its depth d. For the average layer thickness it is also clear that the integral of the derivative over all angles is positive and we should try a larger film thickness. However, at every resonance the derivative has a negative and a positive lobe that almost cancel. This again shows that the discretization of the angles is important to obtain an accurate value for the derivative. Because the left lobe is always negative and the right lobe is always positive, the resonant peaks will shift to larger angles if we increase the average layer thickness.
Since this grating is described by only 2 parameters, namely the average layer thickness and the grating depth, it is feasible to find an optimal solution using a brute force method instead of our method based on reciprocity. In Fig. 4 we show the radiated intensity for a large range of geometries. The Fabry-Pérot maxima for the unpatterned geometry are visible in the bottom of the plot i.e. for zero grating depth, with series of weaker and stronger local maxima for deeper patterns. Using the gradient simulations described in the previous section, the 2-dimensional parameter space is also optimized using a simple steepest ascent optimization method with damped step size. Several optimization paths with random starting conditions are also shown in Fig. 4. Using the optimization method based on reciprocity, with the same computational effort, we can scan the optimization space of a surface defined by any number of parameters.
Figures 5(a) and 5(b) show two geometries at local maxima for the extraction efficiency with a surface defined by only 2 parameters (a block grating) and one that is defined by 10 parameters. It is clear from the image that the ideal location of the active layer is in both cases around z = 120nm. This position is mainly determined by the metal boundary and is very insensitive to the actual grating above it. However, for the block grating it is clear that within the patterning there are hot spots where the radiation from incoherent dipoles is much stronger than on average in the chosen active layer. The reciprocity method is thus convenient for determining the ideal position of the active layer, provided that we assume that the active layer has a similar refractive index as the surrounding semiconductor.
Similarly, the ideal thickness for the active layer can be found directly from the near-field data. While for semiconductor LEDs the active layer thickness is more or less fixed by the design of the multiple quantum wells, for organic LEDs this extra information can be used to estimate the most effective light-emitting polymer layer thickness. Because the optimization method always ensures that the most radiation occurs from the chosen active layer and its dependence on the z-coordinate is mainly determined by the Fabry-Perot resonance of the incident field in the semiconductor, we chose not to explicitly optimize with respect to the thickness of the active layer.
During the optimization, the grating surface is varied causing the radiated energy to be redistributed over the available resonant modes that can be excited in the structure. In Fig. 6, the angular dependence of the radiated intensity is shown for the optimized structure in Fig. 5(a) as a function of the normalized pitch, being in essence a photonic band diagram. While very sharp resonances exist for all pitches, this does in general not lead to an enhanced extraction compared to the optimal unpatterned geometry. For this geometry, only close to Λ/λ = 0.8 a substantial enhancement occurs. Therefore, next to optimizing the grating surface, an appropriate choice for the pitch is of paramount importance.
An efficient derivative of the radiated intensity with respect to parameters such as the grating period or the angle of incidence is not practical with the method described in this paper. Often an initial guess for the period can be made from fast band structure calculations of photonic crystals. The optimization method is then used to obtain a surface that gives an optimal extraction efficiency for that chosen period. From tests, it is found that optimizing for a slightly perturbed period leads to a similar extraction efficiency for a slightly different surface. The local maxima of radiated intensity are therefore weak in a parameter space that includes both the surface and the period.
In conclusion, we have described a very efficient formalism to simulate the light extraction from grating assisted LEDs and to compute an optimal grating surface, both based on the reciprocity principle. To simulate the radiated intensity in a given direction due to the entire active layer only two small-scale boundary value problems have to be solved. Then, only two additional simulation are needed to find the Gateaux derivative with respect to the shape of the LED surface. Using two-dimensional simulations, we have shown that the method can be applied to compute radiation patterns accurately and efficiently. In addition, the data immediately show at what positions in the LED the active region should be located. The calculated Gateaux derivatives are accurate and are used to optimize the surface of the grating. We believe that by applying this method to full three-dimensional LED geometries, better solutions can be found than are currently available at much less computational costs.
As stated, Eq. (25) is valid only when ɛ(Γ) is smooth in the neighborhood of Γ. We now consider the case that the permittivity ɛ jumps across Γ, by taking the limit b → 0 in the previous result of Eq. (25). This is rather tricky, because the electric fields E(Γ) and EJ𝒪 are discontinuous across Γ. We therefore first apply a partial integration with respect to z to the integral in Eq. (25) for fixed b > 0:Eqs. (28) and (29) become for the perpendicular component in the limit b → 0: Eqs. (28) and (29) in terms of the limiting value of the electric fields from below the curve. We omit the details.
Consider now the integrals in Eqs. (30) and (31) for the tangential field components. Since E|| and EJ𝒪, || are continuous as function of z, and are possibly discontinuous across Γ but bounded. It can be shown that the bound is uniform in b → 0, hence the integrals vanish in the limit b → 0.
Next, we consider the case of the normal components. SinceEq. (30) in the limit b → 0, using Eq. (33), as: Eq. (31) for b → 0 also equals Eq. (36). Summing up, the Gateaux derivative δI(Γ; h) in Eq. (25) for a surface with a sharp material transition is given by
For a surface function Γ(x) defined by n parameters, the gradient of the intensity can be calculated using only one additional numerical simulation. Using standard finite differencing to compute the gradient from function values directly, n function evaluations are needed, i.e. 2n boundary value problems have to be solved.
Analogous to Eq. (26), the direction of steepest ascent h̃(x) is now given by
This work is supported by the Dutch Technology Foundation (STW) with project code STW-10301.
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