## Light extraction by directional sources within optically dense media |

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

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]

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. 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]

## 2. Light extraction efficiency defined

**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.

*z*=

*a*: 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.

*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 Solving for the Jacobian and converting the coordinate system of integration then leads to It is worth pointing out that in this form,

*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.

*G*of the source element using [13] 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

*ε*

_{1}>

*ε*

_{2}[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]

*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 The significance of Eq. (12) is that LEE may now be computed as a spherical surface integral of the directive gain pattern

_{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

*G*= 1 over all

*θ*and

*ϕ*while 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 This is the expression for the total solid angle subtended by a cone with apex angle

*θ*and naturally evaluates to

_{c}**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 where

*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.

*z*-oriented (perpendicular) Hertzian dipole, the gain pattern is given as Similarly, for an

*x*-oriented (parallel) Hertzian dipole, the gain becomes Assuming perfect transmittance through the escape cone in accordance with Eq. (13), the LEE for each case evaluates to It is also instructive to note that 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

*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 We next define the polarization angle

*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 The total transmittance can now be explicitly written as In this context,

_{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*.

*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 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 Note that a singularity occurs if

**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.

**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 Taking the cross product with

**ẑ**therefore gives which, in spherical unit vectors, is simply

**n̂**= −

**.**

*ϕ̂***p̂**=

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

**ê**

_{⊥}= −

**so that comparison with Eq. (29) indicates Since this is identically zero, we may conclude that all fields radiated by a perpendicularly-oriented Hertzian dipole will be TM-polarized.**

*θ*̂**p̂**=

**x̂**. For this case, radiated electric fields take on the form Solving for cos

*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

*ε*

_{r}_{2}= 1.

## 5. Light extraction by a Lambertian source

*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.

*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 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.

_{c}## 6. Discussion

*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]

*θ*= 39°. Some visible-light LEDs are made out GaN and have indices of

_{c}*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).

*θ*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

_{c}*θ*= 25°.

_{c}*θ*→ 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

_{c}*θ*.

_{c}*θ*= 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.

_{c}*θ*→ 0.

_{c}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]

*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]

## 7. Summary

## Acknowledgments

## 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. |

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. |

3. | A. David, H. Benisty, and C. Weisbuch, “Optimization of light-diffracting photonic-crystals for high extraction efficiency LEDs,” J. Disp. Technol. |

4. | C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs - designing light extraction,” Laser & Photon. Rev. |

5. | J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quant. Electron. |

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. |

7. | E. F. Schubert, |

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. |

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. |

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. |

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. |

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

13. | D. H. Staelin, A. W. Morgenthaler, and J. A. Kong, |

14. | J. D. Jackson, |

15. | F. T. Ulaby, |

16. | J. A. Kong, |

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 |

18. | M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. |

**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

- 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]
- 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]
- 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]
- 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]
- J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quant. Electron.36(10), 1131–1144 (2000). [CrossRef]
- 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]
- E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- M. Cui, P. Urbach, and D. K. G. de Boer, “Optimization of light extraction from OLEDs,” Opt. Express15(8), 4398–4409 (2007). [CrossRef] [PubMed]
- D. H. Staelin, A. W. Morgenthaler, and J. A. Kong, Electromagnetic Waves (Prentice Hall, 1998).
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
- F. T. Ulaby, Fundamentals of Applied Electromagnetics (Prentice Hall, 2007).
- J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2000).
- 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. A23(5), 1207–1213 (2006). [CrossRef]
- 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]

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