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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 25 — Dec. 3, 2012
  • pp: 27530–27541
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Light extraction by directional sources within optically dense media

James R. Nagel  »View Author Affiliations


Optics Express, Vol. 20, Issue 25, pp. 27530-27541 (2012)
http://dx.doi.org/10.1364/OE.20.027530


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Abstract

Light extraction efficiency (LEE) from a light-emitting diode is commonly referenced against an isotropic radiator within a dense dielectric medium. However, this description is not necessarily accurate for photonic devices with directional source elements. We therefore derive exact solutions for the LEE of a directive radiating source next to a planar dielectric boundary, accounting for any Fresnel reflections at the interface. These results can be used to validate numerical simulations and to quantify the baseline LEE for different source models. Four variations are explored, including the isotropic radiator, parallel and perpendicular orientations of the Hertzian dipole, and Lambertian scattering. Due to index matching, Fresnel reflections are generally negligible for materials with large escape cones, but reduce LEE by 20 % or more when critical angle is below 25°.

© 2012 OSA

1. Introduction

The problem of light extraction from an optically dense medium is one of the great challenges in developing efficient light emitting diodes (LEDs). Due to the high indices of refraction found in many typical semiconductor materials, the escape cone at a planar interface with the surrounding medium is also very narrow. This places heavy limits on efficiency since any photons that fail to exit the device will eventually be reabsorbed and converted into waste heat.

Two of the more commmon remedies for light extraction include random surface roughening [1

1. I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30 % external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63(16), 2174–2176 (1993). [CrossRef]

, 2

2. 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]

] and photonic crystal texturing [3

3. 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]

, 4

4. C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs - designing light extraction,” Laser & Photon. Rev. 3(2), 262–286 (2009). [CrossRef]

]. Other techniques may incorporate surface plasmon resonances [5

5. J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quant. Electron. 36(10), 1131–1144 (2000). [CrossRef]

] or embedded arrays of nanoparticles [6

6. T. Yamasaki, K. Sumioka, and T. Tsutsui, “Organic light-emitting device with an ordered monolayer of silica microspheres as a scattering medium,” Appl. Phys. Lett. 76(10), 1243–1245 (2000). [CrossRef]

]. In general, the ultimate goal of such methods is to simply disrupt the planar symmetry of the interfaces and scatter light within the substrate. This has the tendency of giving light more opportunity to find an escape cone as well as maintaining a uniform distribution of excited modes within the substrate and epitaxial layers.

From a theoretical perspective, the performance of a given light extraction method is often referenced against an isotropic radiator adjacent to an antireflective boundary [4

4. C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs - designing light extraction,” Laser & Photon. Rev. 3(2), 262–286 (2009). [CrossRef]

, 7

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

]. While such a model is very simple mathematically, it is only valid for perfectly incoherent sources with equal polarization in all directions. This is not necessarily a justified assumption since certain LED designs can have a preferred polarization of the dipole moments that radiate photons. For example, LEDs based on quantum wells have been shown to radiate with a polarization ratio of 7:1 [8

8. M. F. Schubert, S. Chhajed, J. K. Kim, and E. F. Schubert, “Polarization of light emission by 460 nm GaInN/GaN light-emitting diodes grown on (0001) oriented sapphire substrates,” Appl. Phys. Lett. 91(5), 051117 (2007). [CrossRef]

]. Even some bulk alloys have also been shown to exhibit varying degrees of polarization as a function of alloy composition [9

9. K. B. Nam, J. Li, M. L. Nakarmi, J. Y. Lin, and H. X. Jiang, “Unique optical properties of AlGaN alloys and related ultraviolet emitters,” Appl. Phys. Lett. 84(25), 5264–5266 (2004). [CrossRef]

].

The goal of this paper is to analyze the problem of light extraction by a directional point source. Our approach to solving the problem will be treated as one of electromagnetic radiation by a small dipole element adjacent to a planar boundary. The basic physics of this problem have been well analyzed in terms of the field propagation and total radiated power [10

10. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. I. Total radiated power,” J. Opt. Soc. Am. 67(12), 1607–1615 (1977). [CrossRef]

, 11

11. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67(12), 1615–1619 (1977). [CrossRef]

], but such work did not investigate light extraction efficiency directly. Further analysis of extraction efficiency was also explored for the case of organic LEDs by accounting for directivity of the dipole [12

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

], but the majority of these results were numerical in nature and focused only on two-dimensional models.

This paper begins with a mathematical definition of light extraction efficiency (LEE) from a planar interface in terms of the electromagnetic Ponyting’s vector. We then convert the system into spherical coordinates to derive a compact expression for LEE in terms of directive gain of the source element and the transmittance through the boundary. This allows for direct comparison of several radiative models, including the isotropic radiator, the Herzian dipole, and a perfect Lambertian surface. We also account for Fresnel reflections (FR) by the planar interface and compare the results against perfectly antireflective (AR) models.

2. Light extraction efficiency defined

The problem of light extraction from an LED is essentially that of a small, radiating current density J placed at the origin with a nearby planar dielectric interface as shown in Fig. 1. The source region (Region 1) is defined by a permittivity ε1 and permeability μ1, while the target region (Region 2) is defined by the parameters ε2 and μ2. Because most semiconductor materials are nonmagnetic, we can generally assume that μ1 = μ2 = μ0, where μ0 is the permeability of free space. We shall also assume that both regions are perfectly lossless, such that ε1 and ε2 are both real values rather than complex. Finally, we shall assume that the source region is more optically dense than the target region, meaning that ε1 > ε2. This means light rays emitted by the source will not be able to exit into Region 2 unless they fall within the escape cone of the material interface.

Fig. 1 A current density J is radiating light from within a dense dielectric medium. Light that falls within the escape cone defined by θc has the potential to escape the LED. The inset shows the surface of integration Ω for evaluating Prad. The exact geometry for Ω is arbitrary, just so long as it encloses the current element defined by J.

We now define the light extraction efficiency (LEE) as the fraction of total radiated power that crosses the planar boundary at z = a:
LEE=1PradS(x,y,a)z^dxdy.
(4)
Taken together, Equations (3) and (4) are the most generic expressions for determining LEE and would generally be applied in the context of numerical simulation. This is useful for arbitrary configurations like roughened surface textures and photonic crystals, which do not necessarily have simple, closed-form solutions.

If we strictly limit ourselves to the case of a flat, planar interface, it is possible to derive a very convenient expression for LEE by converting the integration into spherical coordinates (r,θ,ϕ). Because the plane of integration is fixed at z = a, the position coordinates x and y can be expressed along the interface as
x=atanθcosϕ,
(5)
y=atanθsinϕ.
(6)
Solving for the Jacobian and converting the coordinate system of integration then leads to
LEE=1Prad02π0π/2S(θ,ϕ)z^(a2sec2θtanθ)dθdϕ.
(7)
It is worth pointing out that in this form, r = asecθ is a dependent variable of θ and therefore not a direct part of the integration. We shall need to make use of this fact later.

The final step in our conversion is to define the directive gain pattern G of the source element using [13

13. D. H. Staelin, A. W. Morgenthaler, and J. A. Kong, Electromagnetic Waves (Prentice Hall, 1998).

]
G(θ,ϕ)=Si(θ,ϕ)Prad/4πr2.
(11)
Note that use of G effectively ignores the effects of any evanescant waves in the near-field of the source. This is perfectly acceptable, since no evanescant wave can transfer real power into Region 2 when ε1 > ε2 [10

10. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. I. Total radiated power,” J. Opt. Soc. Am. 67(12), 1607–1615 (1977). [CrossRef]

]. However, this is only true if the boundary at z = a is perfectly flat. When this is not the case (as with roughened surface textures), then we would need to assume that the source element is a proper distance away from the boundary such that any surface features are well-within the far-field zone of the arriving wave fronts. Substitution for Si thus leads to
LEE=14π02π0π/2T(θ,ϕ)G(θ,ϕ)sinθdθdϕ.
(12)
The significance of Eq. (12) is that LEE may now be computed as a spherical surface integral of the directive gain pattern G weighed against the transmittance T at the boundary. This allows for exact computation of LEE for a wide range of simplified models. We can also see that LEE is entirely independent of Prad as well as the distance a to the interface. So while Prad itself certainly does vary with a, the fraction of Prad that escapes into Region 2 remains constant.

3. Light extraction by antireflective boundaries

As a test case, consider the familiar example of an isotropic radiator adjacent to a planar boundary. For simplicity, we shall assume that the boundary is perfectly reflectionless for all angles within the escape cone and perfectly reflective for all angles outside. This is equivalent to setting G = 1 over all θ and ϕ while
T(θ,ϕ)={1,θ<θc0,otherwise.
(13)
In practice, such a model for T is actually a reasonable representation of what occurs at a flat planar boundary that has been treated with a good antireflective coating. Plugging in for LEE therefore leads to
LEE=14π02π0θcsinθdθdϕ.
(14)
This is the expression for the total solid angle subtended by a cone with apex angle θc and naturally evaluates to
LEE=1cosθc2.
(15)

Although this result is simple and intuitive, the isotropic radiator does not physically exist for any coherent distribution of source elements in J. This motivates us to solve for LEE under the assumption that J is a Hertzian dipole, which is the standard model for a point source of current density [13

13. D. H. Staelin, A. W. Morgenthaler, and J. A. Kong, Electromagnetic Waves (Prentice Hall, 1998).

, 14

14. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

]. In principle, this is equivalent to treating the source element as an infinitessimal current density with the form
J(r)=pJ0δ(rr0),
(16)
where δ(r) is the Dirac delta function and r = x + yŷ + z is the position vector in space. The position r0 is the location of the source element, while the source magnitude J0 is given in units of A·m. The polarization vector p can be broken down into two elements, denoted p and p for the perpendicular and parallel polarizations with respect to the planar boundary. For simplicity, we may set p = and p = in accordance with the system geometry. Any arbitrary polarization vector p must therefore lie somewhere between these two extremes in orientation.

For a z-oriented (perpendicular) Hertzian dipole, the gain pattern is given as
G(θ,ϕ)=32sin2θ.
(17)
Similarly, for an x-oriented (parallel) Hertzian dipole, the gain becomes
G||(θ,ϕ)=32(1sin2θcos2ϕ).
(18)
Assuming perfect transmittance through the escape cone in accordance with Eq. (13), the LEE for each case evaluates to
LEE=12+116[cos(3θc)9cosθc],
(19)
LEE||=12132[15cosθc+cos(3θc)].
(20)
It is also instructive to note that
13(LEE+2LEE||)=1cosθc2,
(21)
which shows that the isotropic case is equivalent to an incoherent average between the three orthogonal polarizations of the dipole.

4. Light extraction by reflective boundaries

Fig. 2 Intersection between the plane of incidence (POI) and the planar dielectric boundary at z = a. The POI is defined by the vectors and , with a unit normal vector .

At this point, the only unknown parameter is the polarization angle ψ. We can solve for this by treating the radiated fields as a series of rays that propagate away from the source and then pierce the dielectric boundary at various locations in θ and ϕ. If we let denote the unit normal vector to the POI and ê denote the polarization of the incident electric field, then the polarization angle necessarily satisfies
cosψ=n^e^.
(29)
To find the POI, we simply need to know the incident wave vector k and the unit normal vector to the interface. This geometry is shown in Fig. 2, which illustrates the POI as it intersects with the plane at z = a. Because the cross product between any two vectors is always normal to the plane defined by those vectors, we may define the unit normal vector to the POI as
n^=k^×z^|k^×z^|.
(30)
Note that a singularity occurs if = . However, this condition implies normal incidence to the boundary and thus no distinction between TE or TM polarizations. We may therefore default to cosψ = 1 for this case without any error.

Because the dipole is assumed to be located directly at the origin, the unit wave vector is equivalent to the unit vector . This is again due to the fact that real power can only flow along the radial direction away from the source. For convenience, it also helps to express this in spherical coordinates under rectangular unit vectors as
r^=x^sinθcosϕ+y^sinθsinϕ+z^cosθ.
(31)
Taking the cross product with therefore gives
n^=x^sinϕy^cosϕ,
(32)
which, in spherical unit vectors, is simply = −ϕ̂.

In order to solve for the polarization vector of the incident ray, we need to know the radiated electric fields from the dipole. For a perpendicular ( = ) source, the far-field electric field is given in spherical coordinates as [16

16. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2000).

]
E(r,θ,ϕ)=θ^jωμJ0sin(θ)ejkr4πr.
(33)

Note that this neglects the effects of any reactive near-fields since they contribute no real power to the target region. Clearly, the polarization vector satisfies ê = −θ̂ so that comparison with Eq. (29) indicates
cosψ=ϕ^θ^.
(34)
Since this is identically zero, we may conclude that all fields radiated by a perpendicularly-oriented Hertzian dipole will be TM-polarized.

The final case of interest is that of a parallel dipole with respect to the boundary such that = . For this case, radiated electric fields take on the form
E(r,θ,ϕ)=(cosθcosϕθ^sinϕϕ^)jωμJ0ejkr4πr.
(35)
Solving for cosψ therefore leads to
cosψ=sinϕ.
(36)
Figure 3 summarizes the calculations we have made thus far. The black curves were calculated using Equations (15), (19), and (20) and represent perfect transmission through the escape cone. The red curves account for Fresnel reflections by applying Eq. (28) and were evaluated using numerical integration. For the case of Fresnel reflection with an isotropic radiator, the field polarization was assumed to be an even mix between TE and TM components. As a convenience, the top axis also gives the corresponding index of refraction n for several critical angles, where n=εr1 and εr2 = 1.

Fig. 3 Light extraction efficiency calculated for both antireflective (AR) and Fresnel reflective (FR) surfaces. The three source conditions are (1) isotropic radiator with equal TE/TM polarization weights, (2) a parallel-oriented Hertzian dipole, and (3) a perpendicularly-oriented Hertzian dipole.

5. Light extraction by a Lambertian source

A final case worth examining is that of Lambertian scattering, defined by the directive gain pattern
G(θ,ϕ)=4cosθ(0θπ/2).
(37)
This situation is an idealized case of perfectly diffuse scattering from a roughened surface. Due to the randomized nature of light scattering, we may also assume that cosψ = 0.5 for all θ and ϕ. Such a feature has use in LED designs for randomly redistributing light after reflecting from the back contact. This helps to ensure a steady distribution of photons within the escape cone of the interface after each bounce along the top or bottom surfaces.

For the case of perfect antireflection within a finite escape cone, light extraction from a Lambertian source easily evaluates to
LEE=sin2θc.
(38)
Note that in the limit as θc → 90°, then LEE → 1 rather than 0.5. The reason for this is because the Lambertian source has no propagation in the reverse direction, thus making perfect light extraction theoretically possible. If we further account for Fresnel reflection at the surface, then the LEE integral becomes
LEE=0θc[TTE(θ)+TTM(θ)]sinθcosθdθ.
(39)
A closed-form solution for this expression is naturally unwieldy, but accurate calculation is again readily achievable through numerical integration. Figure 4 summarizes the LEE calculations for both reflective and antireflective surfaces.

Fig. 4 Light extraction efficiency for diffuse light from a Lambertian source. The black curve indicates perfect antireflection (AR) while the dashed red curve accounts for Fresnel reflections (FR) at the dielectric interface.

6. Discussion

Table 1 shows a summary of light extraction values computed for a representative range of refractive indices found in many common devices. For example, an index of n = 1.58 is a typical value for organic LEDs (OLEDs) [12

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

] and corresponds to an escape cone of θc = 39°. Some visible-light LEDs are made out GaN and have indices of n = 2.5 (θc = 25°) at a wavelength of λ = 480 nm. GaAs is common for longer wavelength LEDs and has an index of n = 3.6 (θc = 16°) at λ = 850 nm. GaSb is also seeing growth as an infrared LED with an index of n = 3.9 (θc = 15°) at a wavelength λ = 4.5 μm. The antireflective (AR) surface corresponds to a perfectly-transmitting escape cone as modeled by Eq. (13) while the Fresnel reflective (FR) surface accounts for reflections within the escape cone in accordance with Eq. (28).

Table 1. Percent light extraction efficiencies for some common indices of refraction found in various LEDs. Comparisons are made between perfectly antireflective (AR) surfaces and Fresnel-reflective (FR) surfaces.

table-icon
View This Table

In general, Fresnel reflections are not a significant source of error as long as the escape cone is very large. Errors are only significant when θc falls below 25° and reflective losses reduce the LEE by 20 % or more. Lost power then grows larger as the escape cone is narrowed, due to the increased dielectric constast between the substrate and air. This is illustrated in Fig. 5, which shows a detailed summary of LEE values for more narrow escape cones below θc = 25°.

Fig. 5 Light extraction efficiencies for optically dense dielectrics with shallow critical angle. Black curves indicate perfect antireflection (AR) while the dashed red curves account for Fresnel reflections (FR) at the dielectric interface.

The difference between an isotropic radiator and a parallel-oriented dipole is also generally negligible for large escape cones, and the maximum error between them only reaches a factor of 1.5 as θc → 0. This is due to the peak directive gain of the parallel dipole with respect to the isotropic source being exactly 1.5. On the other hand, the perpendicular dipole extracts far less light than any other source, usually by a factor of 10 or more for small θc.

By mathematical happenstance, the isotropic AR curve intersects with the parallel FR curve near θc = 17°. For common LED materials like GaAs and GaSb, this means that LEE can be accurately approximated using Eq. (15) even though a more “correct” model is that of a parallel dipole with a reflective surface. As Table 1 shows, the two LEE values are off by a mere 0.1 % for each case.

The Lambertian source generally performs the best of all for two reasons. First, the Lambertian technically does not radiate any energy in the backwards direction, thereby raising all LEE values by a general factor of 2. Second, the Lambertian is more directive than the Hertzian dipole, even with the backward radiation accounted for. The reason is because the Hertzian dipole is omnidirectional along one of its axes while the Lambertian is not. The LEE of a Lambertian source will therefore tend towards a factor of 8/3 more than the parallel dipole as θc → 0.

Although we have limited this work to the case of a flat, planar interface, Eq. (12) may be applied to a wide variety of other geometries. For example, an etched diffraction grating is a well-studied feature for light extraction from LEDs [17

17. H. Kikuta, S. Hino, and A. Maruyama, “Estimation method for the light extraction efficiency of light-emitting elements with a rigorous grating diffraction theory,” J. Opt. Soc. Am. A 23(5), 1207–1213 (2006). [CrossRef]

]. Given a uniform plane wave striking the surface, rigorous coupled wave analysis provides a numerial tool for calculating T in the presence of arbitrary periodic structures [18

18. M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73(9), 1105–1112 (1983). [CrossRef]

]. Since any given source profile may be represented by a superposition of radiated plane waves in the far field, Eq. (12) allows us to numerically integrate over all directions and calculate LEE. Provided that the surface lies within the far-field zone of the source, this method could find useful applications for maximizing the extraction of light under highly complex structures and arbitrary gain profiles.

7. Summary

This paper provides an electromagnetic description of light extraction efficiency in terms of surface integration by the Poynting’s vector over the material interface. We then show that this definition is equivalent to the spherical integral of directive gain weighed by the transmittance function over all angles of incidence. If the transmittance is assumed to be perfect over some finite escape cone, then exact analytical expressions can be easily derived for LEE as a function of escape cone angle. Otherwise, the solutions must be calculated through numerical integration. It is found that Fresnel reflections are generally negligible for large escape cone angles, but grow more significant as the index of refraction for the source region increases.

Four source models are explored in this work, including the isotropic radiator, the parallel-and perpendicularly-oriented Hertzian dipoles, and the Lambertian scatterer. The Lambertian source has the highest LEE, due mostly to the absence of any backward radiation. The perpendicular dipole has the lowest LEE, due to the fact that most radiated power falls well outside of the escape cone, even for large values of the apex angle. The isotropic radiator is a reasonable first-order approximation for LEE, but Fresnel reflection quickly diminishes accuracy as the escape cone falls below 25°.

Acknowledgments

Special thanks are given to Mark Miller for his technical discussions on this work.

References and links

1.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30 % external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63(16), 2174–2176 (1993). [CrossRef]

2.

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]

3.

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]

4.

C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs - designing light extraction,” Laser & Photon. Rev. 3(2), 262–286 (2009). [CrossRef]

5.

J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quant. Electron. 36(10), 1131–1144 (2000). [CrossRef]

6.

T. Yamasaki, K. Sumioka, and T. Tsutsui, “Organic light-emitting device with an ordered monolayer of silica microspheres as a scattering medium,” Appl. Phys. Lett. 76(10), 1243–1245 (2000). [CrossRef]

7.

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

8.

M. F. Schubert, S. Chhajed, J. K. Kim, and E. F. Schubert, “Polarization of light emission by 460 nm GaInN/GaN light-emitting diodes grown on (0001) oriented sapphire substrates,” Appl. Phys. Lett. 91(5), 051117 (2007). [CrossRef]

9.

K. B. Nam, J. Li, M. L. Nakarmi, J. Y. Lin, and H. X. Jiang, “Unique optical properties of AlGaN alloys and related ultraviolet emitters,” Appl. Phys. Lett. 84(25), 5264–5266 (2004). [CrossRef]

10.

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. I. Total radiated power,” J. Opt. Soc. Am. 67(12), 1607–1615 (1977). [CrossRef]

11.

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67(12), 1615–1619 (1977). [CrossRef]

12.

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

13.

D. H. Staelin, A. W. Morgenthaler, and J. A. Kong, Electromagnetic Waves (Prentice Hall, 1998).

14.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

15.

F. T. Ulaby, Fundamentals of Applied Electromagnetics (Prentice Hall, 2007).

16.

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2000).

17.

H. Kikuta, S. Hino, and A. Maruyama, “Estimation method for the light extraction efficiency of light-emitting elements with a rigorous grating diffraction theory,” J. Opt. Soc. Am. A 23(5), 1207–1213 (2006). [CrossRef]

18.

M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73(9), 1105–1112 (1983). [CrossRef]

OCIS Codes
(230.3670) Optical devices : Light-emitting diodes
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Optical Devices

History
Original Manuscript: October 26, 2012
Revised Manuscript: November 14, 2012
Manuscript Accepted: November 14, 2012
Published: November 27, 2012

Citation
James R. Nagel, "Light extraction by directional sources within optically dense media," Opt. Express 20, 27530-27541 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27530


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References

  1. I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30 % external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett.63(16), 2174–2176 (1993). [CrossRef]
  2. 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]
  3. 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]
  4. C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs - designing light extraction,” Laser & Photon. Rev.3(2), 262–286 (2009). [CrossRef]
  5. J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quant. Electron.36(10), 1131–1144 (2000). [CrossRef]
  6. T. Yamasaki, K. Sumioka, and T. Tsutsui, “Organic light-emitting device with an ordered monolayer of silica microspheres as a scattering medium,” Appl. Phys. Lett.76(10), 1243–1245 (2000). [CrossRef]
  7. E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006). [CrossRef]
  8. M. F. Schubert, S. Chhajed, J. K. Kim, and E. F. Schubert, “Polarization of light emission by 460 nm GaInN/GaN light-emitting diodes grown on (0001) oriented sapphire substrates,” Appl. Phys. Lett.91(5), 051117 (2007). [CrossRef]
  9. K. B. Nam, J. Li, M. L. Nakarmi, J. Y. Lin, and H. X. Jiang, “Unique optical properties of AlGaN alloys and related ultraviolet emitters,” Appl. Phys. Lett.84(25), 5264–5266 (2004). [CrossRef]
  10. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. I. Total radiated power,” J. Opt. Soc. Am.67(12), 1607–1615 (1977). [CrossRef]
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