## 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, 2] and photonic crystal texturing [3, 4]. Other techniques may incorporate surface plasmon resonances [5] or embedded arrays of nanoparticles [6]. 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, 7]. 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]. Even some bulk alloys have also been shown to exhibit varying degrees of polarization as a function of alloy composition [9].

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, 11], 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], 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 *ε*_{1} and permeability *μ*_{1}, while the target region (Region 2) is defined by the parameters *ε*_{2} and *μ*_{2}. Because most semiconductor materials are nonmagnetic, we can generally assume that *μ*_{1} = *μ*_{2} = *μ*_{0}, where *μ*_{0} is the permeability of free space. We shall also assume that both regions are perfectly lossless, such that *ε*_{1} and *ε*_{2} 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 *ε*_{1} > *ε*_{2}. 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.

From Snell’s law, the critical angle for a lossless dielectric interface is given by

*ε*

_{r}_{1}=

*ε*

_{1}/

*ε*

_{0}and

*ε*

_{r}_{2}=

*ε*

_{2}/

*ε*

_{0}are the dielectric constants of each region. For nonmagnetic materials, these are related to the index of refraction by $n=\sqrt{{\epsilon}_{r}}$. Because of the oscillating current element within the source region, there will necessarily be an electric field

**E**and magnetic field

**H**that fills all space. As a result, there will also be a vector field of real, time-averaged, power density flow as determined by the Poynting’s vector where ℜ{

*x*} denotes the real part of a complex value

*x*. The units on

**S**are given in W/m

^{2}, and is often called the irradiance or intensity field. The total power radiated by the source element can then by found by integrating

**S**across the surface Ω as shown by the inset in in Fig. 1. In principle, Ω can be any arbitrary closed surface, just so long as it completely encloses

**J**. Writing this out explicitly is therefore simply

*d*

**Ω**is the differential unit normal vector to the closed surface defined by Ω.

We now define the light extraction efficiency (LEE) as the fraction of total radiated power that crosses the planar boundary at *z* = *a*:

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

*r*=

*a*sec

*θ*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.

Next, we note that the Poynting’s vector **S** along the *z* = *a* plane will consist of an incident field **S*** _{i}* radiated by the source, a reflected field

**S**

*that propagates back into Region 1, and a transmitted field*

_{r}**S**

*that penetrates into Region 2. The only contribution to the integral is therefore determined by*

_{t}**S**

*, which is related to*

_{t}**S**

*by*

_{i}*T*is the transmittance between Region 1 and Region 2. Note that we are ignoring the change in directon on the

**S**

*vector due to refraction, since this has no affect on the final LEE calculation. As a result, the LEE integral now becomes*

_{t}**r̂**. We may therefore express the incident Poynting’s vector as

**S**

*=*

_{i}*S*

_{i}**r̂**, which allows for further simplification by rewriting the dot product as

**r̂**·

**ẑ**=

*a/r*= 1/sec

*θ*. Thus,

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

*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

*ε*

_{1}>

*ε*

_{2}[10]. 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

*S*thus leads to

_{i}*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

*P*as well as the distance

_{rad}*a*to the interface. So while

*P*itself certainly does vary with

_{rad}*a*, the fraction of

*P*that escapes into Region 2 remains constant.

_{rad}## 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*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

*θ*and naturally evaluates to

_{c}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, 14]. In principle, this is equivalent to treating the source element as an infinitessimal current density with the form

*δ*(

**r**) is the Dirac delta function and

**r**=

*x*

**x̂**+

*y*

**ŷ**+

*z*

**ẑ**is the position vector in space. The position

**r**

_{0}is the location of the source element, while the source magnitude

*J*

_{0}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**

_{⊥}=

**ẑ**and

**p**

_{‖}=

**x̂**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

*x*-oriented (parallel) Hertzian dipole, the gain becomes

## 4. Light extraction by reflective boundaries

Next, we shall account for nonideal values of *T* over all angles within the escape cone. The difficulty of this problem is compounded by the fact that the fields from a dipole are not going to be perfectly polarized with respect to the material interface, but instead consist of a mix between transverse electric (TE) and transverse magnetic (TM) polarizations (also called *s* and *p* polarizations, respectively). By convention, TE fields denote an electric field vector that is transverse to (i.e., perpendicular to) the plane of incidence (POI), which is depicted in Fig. 2. By the same convention, TM naturally indicates a magnetic field that is transverse to the POI. This is also equivalent to defining TM in terms of an electric field vector that is parallel to the POI. Separating the electric field into TE and TM components with respect to the plane of incidence then leads to the simple superposition

*ψ*such that Under this convention, the Poynting vector can likewise be separated into TE and TM components using where The transmitted fields into the target region may therefore be written as where

*T*and

_{TE}*T*are the transmittances with respect to TE and TM polarizations. These parameters are well-known from classical electromagnetic theory [14, 15] for a given planar boundary. Combining Equations (25) and (26) thus produces

_{TM}*T*and

_{TE}*T*will only vary as a function of

_{TM}*θ*, which represents the incidence angle along the POI. It is only the parameter

*ψ*that will vary as a function of

*ϕ*.

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 **n̂** denote the unit normal vector to the POI and **ê** denote the polarization of the incident electric field, then the polarization angle necessarily satisfies

**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

**k̂**=

**ẑ**. However, this condition implies normal incidence to the boundary and thus no distinction between TE or TM polarizations. We may therefore default to cos

*ψ*= 1 for this case without any error.

Because the dipole is assumed to be located directly at the origin, the unit wave vector **k̂** is equivalent to the unit vector **r̂**. 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

**ẑ**therefore gives

**n̂**= −

**.**

*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 (**p̂** = **ẑ**) source, the far-field electric field is given in spherical coordinates as [16]

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

The final case of interest is that of a parallel dipole with respect to the boundary such that **p̂** = **x̂**. For this case, radiated electric fields take on the form

*ψ*therefore leads to 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 $n=\sqrt{{\epsilon}_{r1}}$ and

*ε*

_{r}_{2}= 1.

## 5. Light extraction by a Lambertian source

A final case worth examining is that of Lambertian scattering, defined by the directive gain pattern

*ψ*= 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

Note that in the limit as*θ*→ 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

_{c}## 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] and corresponds to an escape cone of *θ _{c}* = 39°. Some visible-light LEDs are made out GaN and have indices of

*n*= 2.5 (

*θ*= 25°) at a wavelength of

_{c}*λ*= 480 nm. GaAs is common for longer wavelength LEDs and has an index of

*n*= 3.6 (

*θ*= 16°) at

_{c}*λ*= 850 nm. GaSb is also seeing growth as an infrared LED with an index of

*n*= 3.9 (

*θ*= 15°) at a wavelength

_{c}*λ*= 4.5

*μ*m. 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).

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 *θ _{c}* 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

*θ*= 25°.

_{c}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 *θ _{c}* → 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

*θ*.

_{c}By mathematical happenstance, the isotropic AR curve intersects with the parallel FR curve near *θ _{c}* = 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 *θ _{c}* → 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]. 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]. 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.

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