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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 24522–24535
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Efficient optimization method for the light extraction from periodically modulated LEDs using reciprocity

O. T. A. Janssen, A. J. H. Wachters, and H. P. Urbach  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 24522-24535 (2010)
http://dx.doi.org/10.1364/OE.18.024522


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Abstract

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

1. Introduction

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 [1

1. T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84(6), 855–857 (2004). [CrossRef]

], but there is also much interest in periodic structures i.e. two-dimensional gratings (photonic crystals) [2

2. M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75(8), 1036–1038 (1999). [CrossRef]

4

4. Y. J. Lee, S. H. Kim, G. H. Kim, Y. H. Lee, S. H. Cho, Y. W. Song, Y. C. Kim, and Y. R. Do, “Far-field radiation of photonic crystal organic light-emitting diode,” Opt. Express 13(15), 5864–5870 (2005). [CrossRef] [PubMed]

,6

6. A. L. Fehrembach, S. Enoch, and A. Sentenac, “Highly directive light sources using two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 79(26), 4280–4282 (2001). [CrossRef]

,7

7. A. David, H. Benisty, and C. Weisbuch, “Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs,” J. Disp. Technol. 3(2), 133–148 (2007). [CrossRef]

]. 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.

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 , ŷ, and 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 [17

17. E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006). [CrossRef]

].

Fig. 1 (left) A typical photonic crystal LED geometry. (right) Schematic of one periodic cell with pitch Λ of a periodic LED. Two different dipole sources p𝒪 and p are shown, one in the active layer 𝒪 and the other in the half-space above interface Γ.

To calculate the far-field emission of an LED, we first consider the radiation in the direction of the unit vector 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 = –p𝒪δ(r rp𝒪) and radiates the field denoted by Ep𝒪(r). Let rp = rp 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:
Ep𝒪(k^rp)p=Ek(rp𝒪)p𝒪.
(2)
If rp is sufficiently large, the field that is emitted by p and that is incident on the LED can be approximated by the field of a spherical wave with electric field vector:
Ekinc(r)=k24πɛ0reikreik(k^r)(k^×p)×k^,
(3)
which for large distances r from the source is incident as a plane wave. The wave vector of this incident plane wave is kinc=kk^=k0ɛ2k^=k, where k0 = ω/c is the wave number in vacuum and ɛ2 is the relative permittivity in the half-space above the LED. The vector kinc points, of course, in the direction opposite to the direction of emission in which the field radiated by dipole p𝒪 is observed. Without restricting the generality we may assume that p is perpendicular to kinc. Then the incident field given by Eq. (3) can be simplified to
Ekinc(r)=k24πɛ0reikreikincrp.
(4)
The plane of incidence of this plane wave is through kinc and (see 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, 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:
ω2μ0ɛ(r)Ekν(r)××Ekν(r)=0,Ekν(r)Ekinc,ν(r)satisfiestheo.r.c,
(5)
with ν = S, P. Here o.r.c. stands for the “outgoing radiation conditions” that apply to the scattered fields, i.e. to the total fields minus the incident fields. For a periodically varying surface Γ as shown in Fig. 1, the fields Ekν(r) 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 EkS(r) and EkP(r) have been computed numerically, the Ŝ and component of the field radiated by dipole p𝒪 at rp𝒪 in the direction follows from reciprocity principle in Eq. (1):
limrpEp𝒪(rpk^)ν^=Ekν(rP𝒪)p𝒪.
(6)
Using that Ŝ and are orthonormal, the total radiated intensity at r → ∞ in the direction of due to dipole p𝒪 is proportional to:
limrp|Ep𝒪(rpk^)|2=limrpν=S,P|Ep𝒪(rpk^)ν^|2=ν=S,P|Ekν(rp𝒪)p𝒪|2.
(7)
The time-averaged intensity in the direction of due to incoherent dipoles at rp𝒪 with random orientation of the dipole vector p𝒪, is obtained by integrating Eq. (7) over all directions of p𝒪. By substituting p𝒪 = p𝒪 (cos ϕ sin θ + sin ϕ sin θŷ + cos θ) into Eq. (7) and averaging over the unit sphere of directions for p𝒪, we obtain for the intensity emitted in the direction per solid angle due to randomly isotropically oriented dipoles at rp𝒪 with all the same strength p𝒪:
I(k,rp𝒪)=12ɛ0ɛ2μ014π02π0πν=S,P|Ekν(rp𝒪)p𝒪|2sinθdθdϕ=p𝒪26ɛ0ɛ2μ0ν=S,P|Ekν(rp𝒪)|2.
(8)
This result is valid for isotropically oriented dipoles only. For LEDs using multiple quantum well structures it is known, however, that the dipoles are mainly oriented in the plane of the active layer [18

18. M. Yamanishi and I. Suemune, “Comment on polarization dependent momentum matrix elements in quantum well lasers,” Jpn. J. Appl. Phys. 23(Part 2, No. 1), L35–L36 (1984). [CrossRef]

]. In that case θ = π/2 and averaging should be done over 0 ≤ ϕ ≤ 2π, yielding:
I˜(k,rp𝒪)=12ɛ0ɛ2μ012π02πν=S,P|Ekν(rp𝒪)p𝒪|2dϕ=p𝒪24ɛ0ɛ2μ0ν=S,P(|Ek,xν(rp𝒪)|2+|Ek,yν(rp𝒪)|2).
(9)
To be specific and for simplicity, we shall use 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 per solid angle of randomly oriented incoherent dipoles with fixed length p𝒪2=6μ0/ɛ0ɛ2 in the entire active layer 𝒪 is found by integrating rp𝒪 over the active layer:
I(k^)=𝒪ν=S,P|Ekν(rp𝒪)|2d3rp𝒪.
(10)
Writing = cos ϕ sin θ + sin ϕ sin θ + cos θ, the total radiation emitted inside the cone 0 ≤ θθC, with solid angle Ω = 2π (1 cos θC), is given by
I(Ω)=I(k^)dΩ,
(11)
where dΩ = r2 sin θ dθ dϕ. To compute the integral numerically, I() has to be calculated for sufficiently many θ, ϕ. Each angle requires solving two quasi-periodic rigorous scattering problems.

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 [19

19. M. Cui, H. P. Urbach, and D. K. G. de Boer, “Optimization of light extraction from OLEDs,” Opt. Express 15(8), 4398–4409 (2007). [CrossRef] [PubMed]

]. 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.

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 [20

20. D. G. Luenberger, Optimization by Vector Space Methods (Wiley-Interscience, 1967).

] which in the direction h is given by
δI(Γ;h)=limξ01ξ[I(Γ+ξh)I(Γ)]=2Re[𝒪δE(Γ;h)E(Γ)*dxdz],
(15)
where δE(Γ; h) is the Gateaux derivative of the electric field E(Γ) in the direction of h, and * denotes the complex conjugate. To obtain δE(Γ;h) we consider two electromagnetic boundary value problems. The first is that of an incident plane wave Einc on an LED with surface Γ as described by Eq. (5) and the second is that of the same incident plane wave on an LED with surface Γ+ξh:
ω2μ0ɛ(Γ)E(Γ)××E(Γ)0,E(Γ)Eincsatisfiestheo.r.c,
(16)
ω2μ0ɛ(Γ+ξh)E(Γ+ξh)××E(Γ+ξh)=0,E(Γ+ξh)Eincsatisfiestheo.r.c,
(17)
with in both cases quasi-periodic boundary conditions on the sides (x = 0, x = Λ) of the cell. By subtracting Eq. (17) from Eq. (16) we get
ω2μ0ɛ(Γ)[E(Γ)E(Γ+ξh)]××[E(Γ)E(Γ+ξh)]=ω2μ0[ɛ(Γ+ξh)ɛ(Γ)]E(Γ+ξh),E(Γ)E(Γ+ξh)satisfiestheo.r.c.
(18)
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
ɛ(Γ+ξh)(x,z)=ɛ(Γ)(x,zξh(x))forallx,z.
(19)
Hence, in the limit ξ → 0 the permittivity difference becomes, according to the definition of the derivative, proportional to its partial derivative with respect to the z coordinate:
limξ0[ɛ(Γ+ξh)(x,z)ɛ(Γ)(x,z)]=limξ0[ɛ(Γ)(x,zξh(x))ɛ(Γ)(x,z)]=ɛ(Γ)z(x,z)h(x)𝟙SbΓ(x,z),
(20)
where 𝟙SbΓ(x,z) is the function that is 1 in the region SbΓ and 0 otherwise. Hence, Eq. (18) becomes in the limit ξ → 0:
ω2μ0ɛ(Γ)δE(Γ;h)××δE(Γ;h)=ω2μ0ɛ(Γ)zhE(Γ)𝟙SbΓ.δE(Γ;h)satisfiestheo.r.c.
(21)
We conclude that δE(Γ; h) is the electric field radiated by the current density defined by
JΓ(x,z)=iωɛ(Γ)z(x,z)h(x)E(Γ)(x,z)𝟙SbΓ(x,z),
(22)
in the configuration with surface Γ. Note that JΓ depends linearly on the perturbation h of the interface and is nonzero only inside the strip SbΓ that surrounds the curve z = Γ(x).

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 such that δI(Γ; ) 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 density
J𝒪=E(Γ)*𝟙𝒪.
(23)
This current is only nonzero in the active region 𝒪. Let the field EJ𝒪 (Γ) be the electric field that is radiated by this current J𝒪 in the configuration with curve Γ. Hence EJ𝒪(Γ) satisfies:
ω2μ0ɛ(Γ)EJ𝒪(Γ)××EJ𝒪(Γ)=iωμ0E(Γ)*𝟙𝒪,EJ𝒪(Γ)satisfiestheo.r.c.
(24)
We now use Eq. (23) and the reciprocity principle to rewrite Eq. (15) as follows:
δI(Γ;h)=2Re[𝒪δE(Γ;h)J𝒪dxdz]=2Re[JΓEJ𝒪(Γ)dxdz]=2Im[ωSbΓɛ(Γ)zhE(Γ)EJ𝒪(Γ)dxdz]
(25)
where the reciprocity principle is used in the second step. From 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. 0Λh(x)2dx=1, then the perturbation of given norm for which δI(Γ; h) is maximum is given by the for which the integral at the right of Eq. (25) is maximum. Since h is only a function of x, is given by
h˜(x)=CIm[Γ(x)bΓ(x)+bɛ(Γ)z(x,z)E(Γ)(x,z)EJ𝒪(x,z)dz],
(26)
where C is such that 0Λh˜(x)2dx=1.

4. Results

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 [21

21. X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. 24(3), 866–881 (2007). [CrossRef]

].

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 [22

22. J. J. Wierer, A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photonics 3(3), 163–169 (2009). [CrossRef]

]. 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 = kmaxkfwhm) up to 105.

Fig. 2 Angle dependent intensity at the wavelength of 450nm of isotropically radiating dipoles in a geometry with 700nm thick GaN, an active layer 100nm above the metal contact, and a grating with a pitch of 250nm and a depth of 5nm. Values are normalized by the intensity of the flat geometry Iflat0 at kx = 0. The solid and dashed vertical lines depict, respectively, the location of the S and P waveguide modes of the flat geometry. The insets show the corresponding radiation patterns. (a) The metal contact is considered a perfect metallic conductor. (b) The metal contact is considered to be silver (Ag) with complex permittivity.

Using tabulated permittivity data of silver [23

23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

], 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 Δfwhm = 10−6, the angles corresponding to 0 ≤ x ≤ 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(x) 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 [24

24. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]

].

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.

Fig. 3 Gateaux derivative at the surface Γ as defined for Fig. 2. The x-axis is cut as to emphasize the derivative close to the resonances. (a) Gateaux derivative with respect to the average layer thickness t. (b) Gateaux derivative with respect to the grating depth d.

Fig. 4 Radiated intensity of an LED with a block grating and pitch Λ = 450nm, normalized to the maximum intensity obtainable with an unpatterned geometry. The indicated paths are optimization paths from arbitrary starting points leading to several local maxima.

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.

Fig. 5 Plot of the radiated intensity in a cone with 60° opening angle (x < 0.5) resulting from a local incoherent dipole as a function of its position within the grating. The plot on the right is the radiated intensity of incoherent dipoles in a thin flat active layer at the corresponding z-coordinate. (a) After optimizing with surface 2 parameters. (b) After optimizing a surface parameterized by 10 parameters.

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.

Fig. 6 Radiated intensity of a grating with t = 700nm and d = 300nm as a function of the normalized pitch Λ and angle x. Red shades indicate contributions from p-polarized emission, while blue shades indicate the contribution by s-polarized emission. The plot on the right gives the total radiated intensity in a cone with a 60° opening angle (x < 0.5).

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.

5. Conclusions

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.

6. Appendix

We consider first the limit b → 0 of the integrals in Eqs. (28) and (29) for the tangential component. Since this component is continuous across the interface we can take the limit b → 0 to get
0Λɛ0ɛ2h(x)E||(Γ)(x,Γ(x))EJ𝒪,||(x,Γ(x))dx0Λɛ0ɛ1h(x)E||(Γ)(x,Γ(x))EJ𝒪,||(x,Γ(x))dx=0Λɛ0(ɛ2ɛ1)h(x)E||(Γ)(x,Γ(x).EJ𝒪,||(x,Γ(x))dx.
(32)
The perpendicular component of the electric field is discontinuous across the interface, but the displacement field D= ɛE is continuous. We therefore have
E(x,Γ(x)+0)E(x,Γ(x)0)=(1ɛ2ɛ1)E(x,Γ(x)+0).
(33)
Using this, the sum of the integrals in Eqs. (28) and (29) become for the perpendicular component in the limit b → 0:
0Λɛ0ɛ2(1ɛ2ɛ1)h(x)E(Γ)(x,Γ(x)+0)EJ𝒪,(x,Γ(x)+0)dx.
(34)
One can alternatively also express the limit b → 0 of integrals in 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, E||(Γ)z(x,z) and EJ𝒪,||z(x,z) 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. Since
limb0E(Γ)z=[E(Γ+0)E(Γ0)]δΓ,
(35)
we can write Eq. (30) in the limit b → 0, using Eq. (33), as:
0Λh(x)ɛ0ɛ2(1ɛ2ɛ1)E(Γ)(x,Γ(x)+0)EJ𝒪,(x,Γ(x)+0)dx.
(36)
The integral in 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
δI(Γ;h)=2ωɛ0Im0Λh(x)[(ɛ2ɛ1)E||(Γ)(x,Γ(x))Ej𝒪,||(x,Γ(x))ɛ2(1ɛ2ɛ1)E(Γ)(x,Γ(x)+0)EJ𝒪,(x,Γ(x)+0)]dx.
(37)
Hence for a given perturbation h of the curve Γ, the Gateaux derivative δI(Γ; h) of the radiated intensity in a given direction is obtained by computing the fields E(Γ) of a unit amplitude incident plane wave with wave vector k, either S- or P-polarized, and the field EJ𝒪 radiated by the current density J𝒪 inside the active region 𝒪. Hence, only two boundary value problems (per polarization) have to be solved on the unit cell to compute δI(Γ;h).

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 (x) is now given by
h˜(x)=CIm[(ɛ2ɛ1)E||(Γ)(x,Γ(x))EJ𝒪,||(x,Γ(x))ɛ2(1ɛ2ɛ1)E(Γ)(x,Γ(x)+0)EJ𝒪,(x,Γ(x)+0)],
(38)
where C is such that 0Λh˜(x)2dx=1.

Acknowledgments

This work is supported by the Dutch Technology Foundation (STW) with project code STW-10301.

References and links

1.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84(6), 855–857 (2004). [CrossRef]

2.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75(8), 1036–1038 (1999). [CrossRef]

3.

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, “FDTD Simulation for Light Extraction in a GaN-Based LED,” J. Korean Phys. Soc. 49, 877–880 (2006).

4.

Y. J. Lee, S. H. Kim, G. H. Kim, Y. H. Lee, S. H. Cho, Y. W. Song, Y. C. Kim, and Y. R. Do, “Far-field radiation of photonic crystal organic light-emitting diode,” Opt. Express 13(15), 5864–5870 (2005). [CrossRef] [PubMed]

5.

H. Rigneault, F. Lemarchand, and A. Sentenac, “Dipole radiation into grating structures,” J. Opt. Soc. Am. A 17(6), 1048–1058 (2000). [CrossRef]

6.

A. L. Fehrembach, S. Enoch, and A. Sentenac, “Highly directive light sources using two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 79(26), 4280–4282 (2001). [CrossRef]

7.

A. David, H. Benisty, and C. Weisbuch, “Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs,” J. Disp. Technol. 3(2), 133–148 (2007). [CrossRef]

8.

A. Roger and D. Maystre, “Inverse scattering method in electromagnetic optics- Application to diffraction gratings,” J. Opt. Soc. Am. 70(12), 1483–1495 (1980). [CrossRef]

9.

S. J. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. 106(5), 2653–2660 (1999). [CrossRef]

10.

L. D. Landau and E. M. LifshitzElectrodynamics of Continuous Media (Pergamom Press, 1960).

11.

R. J. Potton, “Reciprocity in Optics,” Rep. Prog. Phys. 67(5), 717–754 (2004). [CrossRef]

12.

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56(3), 133–237 (1997). [CrossRef]

13.

C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B 36(9), 4990–5000 (1987). [CrossRef]

14.

A. Roger, “Reciprocity theorem applied to the computation of functional derivatives of the scattering matrix,” Electromagnetics 2(1), 69–83 (1982). [CrossRef]

15.

S. Bonnard, P. Vincent, and M. Saillard, “Cross-borehole inverse scattering using a boundary finite-element method,” Inverse Probl. 14(3), 521–534 (1998). [CrossRef]

16.

A. Litman and K. Belkebir, “Two-dimensional inverse profiling problem using phaseless data,” J. Opt. Soc. Am. 23(11), 2737–2746 (2006). [CrossRef]

17.

E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006). [CrossRef]

18.

M. Yamanishi and I. Suemune, “Comment on polarization dependent momentum matrix elements in quantum well lasers,” Jpn. J. Appl. Phys. 23(Part 2, No. 1), L35–L36 (1984). [CrossRef]

19.

M. Cui, H. P. Urbach, and D. K. G. de Boer, “Optimization of light extraction from OLEDs,” Opt. Express 15(8), 4398–4409 (2007). [CrossRef] [PubMed]

20.

D. G. Luenberger, Optimization by Vector Space Methods (Wiley-Interscience, 1967).

21.

X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. 24(3), 866–881 (2007). [CrossRef]

22.

J. J. Wierer, A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photonics 3(3), 163–169 (2009). [CrossRef]

23.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

24.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(230.3670) Optical devices : Light-emitting diodes
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Optical Devices

History
Original Manuscript: March 31, 2010
Revised Manuscript: July 15, 2010
Manuscript Accepted: July 21, 2010
Published: November 10, 2010

Citation
O. T. A. Janssen, A. J. H. Wachters, and H. P. Urbach, "Efficient optimization method for the light extraction from periodically modulated LEDs using reciprocity," Opt. Express 18, 24522-24535 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24522


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References

  1. T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004). [CrossRef]
  2. M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999). [CrossRef]
  3. W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).
  4. Y. J. Lee, S. H. Kim, G. H. Kim, Y. H. Lee, S. H. Cho, Y. W. Song, Y. C. Kim, and Y. R. Do, "Far-field radiation of photonic crystal organic light-emitting diode," Opt. Express 13(15), 5864-5870 (2005). [CrossRef] [PubMed]
  5. H. Rigneault, F. Lemarchand, and A. Sentenac, "Dipole radiation into grating structures," J. Opt. Soc. Am. A 17(6), 1048-1058 (2000). [CrossRef]
  6. A. L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79(26), 4280-4282 (2001). [CrossRef]
  7. A. David, H. Benisty, and C. Weisbuch, "Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs," J. Disp. Technol. 3(2), 133-148 (2007). [CrossRef]
  8. A. Roger, and D. Maystre, "Inverse scattering method in electromagnetic optics- Application to diffraction gratings," J. Opt. Soc. Am. 70(12), 1483-1495 (1980). [CrossRef]
  9. S. J. Norton, "Iterative inverse scattering algorithms: Methods of computing Frechet derivatives," J. Acoust. Soc. Am. 106(5), 2653-2660 (1999). [CrossRef]
  10. L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamom Press, 1960).
  11. R. J. Potton, "Reciprocity in Optics," Rep. Prog. Phys. 67(5), 717-754 (2004). [CrossRef]
  12. J.-J. Greffet, and R. Carminati, "Image formation in near-field optics," Prog. Surf. Sci. 56(3), 133-237 (1997). [CrossRef]
  13. C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, "Dipole radiation in a multilayer geometry," Phys. Rev. B 36(9), 4990-5000 (1987). [CrossRef]
  14. A. Roger, "Reciprocity theorem applied to the computation of functional derivatives of the scattering matrix," Electromagnetics 2(1), 69-83 (1982). [CrossRef]
  15. S. Bonnard, P. Vincent, and M. Saillard, "Cross-borehole inverse scattering using a boundary finite-element method," Inverse Probl. 14(3), 521-534 (1998). [CrossRef]
  16. A. Litman, and K. Belkebir, "Two-dimensional inverse profiling problem using phaseless data," J. Opt. Soc. Am. A 23(11), 2737-2746 (2006). [CrossRef]
  17. E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006). [CrossRef]
  18. M. Yamanishi, and I. Suemune, "Comment on polarization dependent momentum matrix elements in quantum well lasers," Jpn. J. Appl. Phys. 23(Part 2, No. 1), L35-L36 (1984). [CrossRef]
  19. M. Cui, H. P. Urbach, and D. K. G. de Boer, "Optimization of light extraction from OLEDs," Opt. Express 15(8), 4398-4409 (2007). [CrossRef] [PubMed]
  20. D. G. Luenberger, Optimization by Vector Space Methods (Wiley-Interscience, 1967).
  21. X. Wei, A. J. Wachters, and H. P. Urbach, "Finite-element model for three-dimensional optical scattering problems," J. Opt. Soc. Am. A 24(3), 866-881 (2007). [CrossRef]
  22. J. J. Wierer, A. David, and M. M. Megens, "III-nitride photonic-crystal light-emitting diodes with high extraction efficiency," Nat. Photonics 3(3), 163-169 (2009). [CrossRef]
  23. P. B. Johnson, and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6(12), 4370-4379 (1972). [CrossRef]
  24. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999). [CrossRef]

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