## Efficient optimization method for the light extraction from periodically modulated LEDs using reciprocity |

Optics Express, Vol. 18, Issue 24, pp. 24522-24535 (2010)

http://dx.doi.org/10.1364/OE.18.024522

Acrobat PDF (1365 KB)

### 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

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]

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. 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]

**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 as where all quantities are complex field amplitudes. A concise derivation of the reciprocity principle can be found in [10], while in [11

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

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

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]

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]

## 2. The emission from an LED

**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 [17

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

**k̂**of a single dipole with dipole moment

**p**

*, oscillating at frequency*

_{𝒪}*ω*, and situated at point

**r**

_{p𝒪}in the active layer. The corresponding current source density is:

**J**

_{p𝒪}= –

*iω*

**p**

*(*

_{𝒪}δ**r**

*–*

**r**

_{p𝒪}) and radiates the field denoted by

**E**

_{p𝒪}(

**r**). Let

**r**=

_{p}**k̂**

*r*

**be a point in the half-space above the LED and let there be a time harmonic dipole at**

_{p}**r**with frequency

_{p}*ω*and dipole moment

**p**. The electric field radiated by the latter dipole will be denoted by

**E**. The two sources and radiated fields satisfy the reciprocity principle: If

_{k}*r*

**is sufficiently large, the field that is emitted by**

_{p}**p**and that is incident on the LED can be approximated by the field of a spherical wave with electric field vector: which for large distances

*r*from the source is incident as a plane wave. The wave vector of this incident plane wave is

*k*

_{0}=

*ω/c*is the wave number in vacuum and

*ɛ*

_{2}is the relative permittivity in the half-space above the LED. The vector

**k**

^{inc}points, of course, in the direction opposite to the direction of emission

**k̂**in which the field radiated by dipole

**p**

*is observed. Without restricting the generality we may assume that*

_{𝒪}**p**is perpendicular to

**k**

^{inc}. Then the incident field given by Eq. (3) can be simplified to The plane of incidence of this plane wave is through

**k**

^{inc}and

**ẑ**(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**,

**P̂**corresponding to the S- and P-polarization respectively.

**p**

*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: with*

_{𝒪}*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

*not*require periodicity of the domain, but for periodic domains only one cell has to be considered.

**Ŝ**and

**P̂**component of the field radiated by dipole

**p**

*at*

_{𝒪}**r**

_{p𝒪}in the direction

**k̂**follows from reciprocity principle in Eq. (1): Using that

**Ŝ**and

**P̂**are orthonormal, the total radiated intensity at

*r*→ ∞ in the direction of

**k̂**due to dipole

**p**

*is proportional to: The time-averaged intensity in the direction of*

_{𝒪}**k̂**due to incoherent dipoles at

**r**

_{p𝒪}with random orientation of the dipole vector

**p**

*, is obtained by integrating Eq. (7) over all directions of*

_{𝒪}**p**

*. By substituting*

_{𝒪}**p**

*=*

_{𝒪}*p*(cos

_{𝒪}*ϕ*sin

*θ*

**x̂**+ sin

*ϕ*sin

*θ*

**ŷ**+ cos

*θ*

**ẑ**) into Eq. (7) and averaging over the unit sphere of directions for

**p**

*, we obtain for the intensity emitted in the direction*

_{𝒪}**k̂**per solid angle due to randomly isotropically oriented dipoles at

**r**

_{p𝒪}with all the same strength

*p*: 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

_{𝒪}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]

*θ*=

*π*/2 and averaging should be done over 0 ≤

*ϕ*≤ 2

*π*, yielding: 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

**k̂**per solid angle of randomly oriented incoherent dipoles with fixed length

*𝒪*is found by integrating

**r**

_{p𝒪}over the active layer: Writing

**k̂**= cos

*ϕ*

**sin**

_{k̂}*θ*

**+ sin**

_{k̂}x̂*ϕ*

**sin**

_{k̂}*θ*

**+ cos**

_{k̂}x̂*θ*

**, the total radiation emitted inside the cone 0 ≤**

_{k̂}ẑ*θ*

**≤**

_{k̂}*θ*, with solid angle Ω = 2

_{C}*π*(1

*–*cos

*θ*), is given by where dΩ =

_{C}*r*

^{2}sin

*θ*

**d**

_{k̂}*θ*

**d**

_{k̂}*ϕ*

**. To compute the integral numerically, I(**

_{k̂}**k̂**) has to be calculated for sufficiently many

*θ*

**,**

_{k̂}*ϕ*

**. Each angle requires solving two quasi-periodic rigorous scattering problems.**

_{k̂}## 3. Optimization of the LED surface

*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.

*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]

*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.

*b*> 0 and define for a configuration with surface Γ the permittivity function

*ɛ*(Γ)(

*x,z*) by: The permittivity varies then smoothly in the strip

*b*→ 0 to represent a sharp transition, but until stated otherwise

*ɛ*(Γ) is smooth for fixed

*b*> 0. For a given surface Γ, the quantity that we intend to maximize is the energy emitted in a specific direction given by the unit vector

**k̂**. Obviously, we can just as well maximize the energy emitted inside the entire cone, but to keep the notation concise we choose to optimize only a single direction. Hence we maximize where

**E**

*(Γ) is the electric field due to an incident plane wave with unit amplitude and direction of incidence given by*

^{ν}*–*

**k̂**. The plane wave is either S- or P-polarized. Only one case of polarization is considered, the other can be dealt with in the same way, so that we omit in the following the superscript

*ν*from the electric field vectors due to an incident plane wave and of the incident field itself. The field

**E**(Γ) is computed by solving a single diffraction problem defined 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.

*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] which in the direction

*h*is given by where

*δ*

**E**(Γ;

*h*) is the Gateaux derivative of the electric field

**E**(Γ) in the direction of

*h*, and

***denotes the complex conjugate. To obtain

*δ*

**E**(Γ;

*h*) we consider two electromagnetic boundary value problems. The first is that of an incident plane wave

**E**

^{inc}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*: 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

*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 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: where

*ξ*→ 0: We conclude that

*δ*

**E**(Γ;

*h*) is the electric field radiated by the current density defined by in the configuration with surface Γ. Note that

**J**

_{Γ}depends linearly on the perturbation

*h*of the interface and is nonzero only inside the strip

*z*= Γ(

*x*).

*δ*

**E**(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).

*𝒪*. Let the field

**E**

_{J𝒪}(Γ) be the electric field that is radiated by this current

**J**

*in the configuration with curve Γ. Hence*

_{𝒪}**E**

_{J𝒪}(Γ) satisfies: We now use Eq. (23) and the reciprocity principle to rewrite Eq. (15) as follows: 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

**E**

_{J𝒪}. To determine these fields one scattering problem (to compute

**E**(Γ)) and one radiation problem (to compute

**E**

_{J𝒪}) 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).

*h*be normed in some way, e.g.

*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 where

*C*is such that

*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),

**E**

_{J𝒪, ||}and

**E**

_{⊥}(Γ),

**E**

_{J𝒪, ⊥}, respectively. The result of taking the limit

*b*→ 0 is then: For the details of the derivation we refer to the Appendix.

*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.

## 4. Results

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]

*Q*=

*k*

_{max}/Δ

*k*

_{fwhm}) up to 10

^{5}.

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

^{3}. These sharp resonances are potentially problematic for the reciprocity calculation method, where for each discrete

*k*value a new simulation has to be performed. To capture all the maxima, taking into account a potential Full Width Half Maximum (FWHM) of Δ

_{x}*k̂*

_{fwhm}= 10

^{−6}, the angles corresponding to 0 ≤

*k̂*≤ 1 would need to be approximated by at least 10

_{x}^{6}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

*k*, 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

_{x}*I*(

*k̂*) or where the derivative

_{x}*∂I(k̂*)

_{x}*/∂k̂*is large. New simulations are then performed until all maxima are sufficiently resolved and usually not more than 150 angles are required.

_{x}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]

*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.

*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.

*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.

*λ*= 0.8 a substantial enhancement occurs. Therefore, next to optimizing the grating surface, an appropriate choice for the pitch is of paramount importance.

## 5. Conclusions

## 6. Appendix

*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

**E**

_{J𝒪}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: Before taking the limit

*b*→ 0, we split the electric field on the curve

*z*= Γ(

*x*) into a component that is in every point of the curve tangential to the curve, and a component that is in every point of the curve perpendicular to the curve:

**E**(

*x,*Γ(

*x*)) =

**E**

_{||}(

*x,*Γ(

*x*))+

**E**

_{⊥}(

*x,*Γ(

*x*)). Note that

**E**

_{||}(

*x,*Γ(

*x*)) is continuous across the curve, but that in the limit

*b*→ 0,

**E**

_{⊥}(

*x,*Γ(

*x*)) is discontinuous. Therefore, in the latter case, we have to distinguish between limiting values taken from above and below the curve, therefore we will write

**E**(

*x*, Γ(

*x*)±0) =

**E**

_{||}(

*x*, Γ (

*x*))+

**E**

_{⊥}(

*x*, Γ(

*x*)±0).

*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

**D**

_{⊥}=

*ɛ*

**E**

_{⊥}is continuous. We therefore have Using this, the sum of the integrals in Eqs. (28) and (29) become for the perpendicular component in the limit

*b*→ 0: 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.

**E**

_{||}and

**E**

_{J𝒪, ||}are continuous as function of

*z*,

*b*→ 0, hence the integrals vanish in the limit

*b*→ 0.

*b*→ 0, using Eq. (33), as: 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

*h*of the curve Γ, the Gateaux derivative

*δI*(Γ;

*h*) of the radiated intensity in a given direction

**k̂**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

**E**

_{J𝒪}radiated by the current density

**J**

*inside the active region*

_{𝒪}*𝒪*. Hence, only two boundary value problems (per polarization) have to be solved on the unit cell to compute

*δI*(Γ;

*h*).

*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. 2

*n*boundary value problems have to be solved.

*h̃*(

*x*) is now given by where

*C*is such that

## Acknowledgments

## 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. |

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. |

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. |

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 |

5. | H. Rigneault, F. Lemarchand, and A. Sentenac, “Dipole radiation into grating structures,” J. Opt. Soc. Am. A |

6. | A. L. Fehrembach, S. Enoch, and A. Sentenac, “Highly directive light sources using two-dimensional photonic crystal slabs,” Appl. Phys. Lett. |

7. | A. David, H. Benisty, and C. Weisbuch, “Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs,” J. Disp. Technol. |

8. | A. Roger and D. Maystre, “Inverse scattering method in electromagnetic optics- Application to diffraction gratings,” J. Opt. Soc. Am. |

9. | S. J. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. |

10. | L. D. Landau and E. M. Lifshitz |

11. | R. J. Potton, “Reciprocity in Optics,” Rep. Prog. Phys. |

12. | J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. |

13. | C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B |

14. | A. Roger, “Reciprocity theorem applied to the computation of functional derivatives of the scattering matrix,” Electromagnetics |

15. | S. Bonnard, P. Vincent, and M. Saillard, “Cross-borehole inverse scattering using a boundary finite-element method,” Inverse Probl. |

16. | A. Litman and K. Belkebir, “Two-dimensional inverse profiling problem using phaseless data,” J. Opt. Soc. Am. |

17. | E. F. Schubert, |

18. | M. Yamanishi and I. Suemune, “Comment on polarization dependent momentum matrix elements in quantum well lasers,” Jpn. J. Appl. Phys. |

19. | M. Cui, H. P. Urbach, and D. K. G. de Boer, “Optimization of light extraction from OLEDs,” Opt. Express |

20. | D. G. Luenberger, |

21. | X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. |

22. | J. J. Wierer, A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photonics |

23. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

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 |

**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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