Supplementary Information
High brightness polarized light emitting diodes
Elison Matioli,1,∗ Stuart Brinkley,2 Kathryn M. Kelchner,2 Yan-Ling Hu,3
Shuji Nakamura,2, 3 Steven DenBaars,2, 3 James Speck,3 and Claude Weisbuch3, 4
1
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA USA
Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, USA
3 Materials Department, University of California, Santa Barbara, 93106 CA, USA
4 Laboratoire de Physique de la Matière Condensée, CNRS, Ecole Polytechnique, Palaiseau, France
2
∗ electronic address: elison.matioli@polytechnique.org
Guided modes in the embedded PhC structure
The embedded air-gaps form a layer of lower effective index of refraction within the LED structure, given
2
2
by nPhC  (nGaN
(1  ff )  nair
ff )1/ 2 (ff is the fill factor corresponding to the volume fraction of air in the
PhC layer). This creates a thin waveguide above the PhCs where a discrete set of guided modes is
supported, which are preferentially excited by the quantum wells (QWs), also located in the top slab
above the PhCs1. The thick free-standing m-plane GaN substrate below the PhCs approaches a semiinfinite medium, where a continuum distribution of electromagnetic solutions is supported. In a first
approximation, let us simplify this structure by a single thin cavity, corresponding to the layer above the
PhCs (Fig. 5a). Considering the PhC region as a homogeneous medium with effective index nPhC ∼ 2.03
for ff = 35%, we calculated 9 guided modes with effective indices neffective (where neffective  k|| / k 0 )
between nGaN ∼ 2.41 and nPhC.
The original polarization of the direct light and guided modes is not disturbed by the PhCs since their
different interfaces are parallel to each other, as well as to the direction of the main electric field E a. The
facets formed inside each of the 1D air gaps extend across the entire LED structure, parallel to the a
direction (Fig. 2b), conserving the initial main polarization from the Px dipole. The polarization is also
conserved for off-axis propagating modes since the PhC facets are parallel to each other and embedded
in the same optical medium (even though the inner part of the grooves is air).
Dispersion relation of off-axis modes
Off-axis modes propagating with a wavevector nGaN k0 and a parallel wavevector component
k||  neffectivek 0 are diffracted along the a-direction if k|| cos   G . Its parallel diffracted component of
the wavevector (green vector) is k||,m  k|| sin   k z tan  (Fig. S1a). The external angle θ is related to
the internal angle  as   arcsin(sin  / nGaN ) . Combining these expressions, the in-plane angle α can
be written as
 n2  n2
GaN
effective
 ( )  arcsin 

neffective

 sin    
 
tan  arcsin 

n
GaN



(1)
and the dispersion relation of these modes is
  a' neffective cos 
(2)
Fig. S1: Diffraction of modes propagating off-axis. a. Three dimension schematics of the diffraction of off-axis modes into the
k|| diffracted by G  k|| cos  resulting in k||,m . b. Angle resolved measurement
 
E || c polarizations, normalized by the lineshape and intensity of emission from the
m-a plane. The original in-plane wavevector
along the m-a plane for both
 
E || a
and
QWs. Calculated dispersion relation for modes with neffective = 1.73 (dashed-blue) and neffective = 1.78 (dashed-green) matches the
experimental data.
 
 
Figure S1b shows the angle resolved measurement in the m-a plane for both E || a and E || c
polarizations, normalized by the intensity lineshape of emission from the QWs. We observe the replicas
of the modes from the next Brillouin zone. This happens when more than one harmonic m is diffracted




within the air cone (both k  G and k  G ), which in the present case occurs at large angles (  ∼ 40o).
 
Therefore the crossing between the modes and the replicas is better seen for the E || c polarization as
 
the intensity of the E || c polarization is higher at large angles.
The observed modes and their replicas match well the modeled dispersion relation (dotted lines in Fig.
 
S1b). Moreover, notice in Fig. S1b that the intensity of the E || c polarization is weak at angles close to
 
0o only for modes excited by the Px dipole (λ ∼ 465 nm). E || c polarized modes excited by the Pz dipole
 
with λ ∼ 430 nm behave similarly to E || a polarized modes excited by the Px dipole, therefore are
stronger at small angles (These modes are only observed here due to the normalization of the data by
the lineshape of the QWs).
The intensity of these modes was also in agreement with experimental data as shown in Fig. 6. For the
 
E || a polarization, an extra cos θ term was included in equation (3) to account for the fact that the
polarizer is aligned to the direction of rotation, therefore at θ = 90o the polarizer is parallel to the mdirection.
Correction for the change in solid angle
Although the emission from the dipole inside the LED structure is isotropic in the m-c plane (Fig. 1c), the
measured radiated emission from the LED without PhC is not constant with respect to θ, as observed in
Fig. 3b. This is due to an increase in solid angle at large θ angles as the light goes from inside to outside
the LED structure, while the solid angle of light collection by the detector is constant at all angles.
Therefore the decrease of the amount of light emitted per solid angle with θ must be considered.
The ratio between solid angles inside and outside the LED is:
𝑑𝛺𝑖 2𝜋𝑠𝑖𝑛𝜃𝑖 𝑑𝜃𝑖
=
𝑑𝛺
2𝜋𝑠𝑖𝑛𝜃𝑑𝜃
Combining this expression to the Fresnel’s law 𝑛𝑖 𝑠𝑖𝑛𝜃𝑖 = 𝑠𝑖𝑛𝜃 and to its derivative 𝑛𝑖 𝑐𝑜𝑠𝜃𝑖 𝑑𝜃𝑖 =
𝑐𝑜𝑠𝜃𝑑𝜃 yields
𝑑𝛺𝑖
cos 𝜃
cos 𝜃
= 2
~ 2
𝑑𝛺
𝑛𝑖 𝑐𝑜𝑠𝜃𝑖
𝑛𝑖
Therefore the intensity of light per solid angle measured in air (𝑑𝐼/𝑑𝛺) corresponds approximately to
𝑜𝑠𝜃 𝑑𝐼/𝑑𝛺𝑖 .
References
1. Matioli E, Weisbuch C. Impact of photonic crystals on LED light extraction efficiency: approaches and
limits to vertical structure designs. J. Phys. D: Appl. Phys. 2010; 43: 354005.
2. Jackson JD. Classical Electrodynamics, Wiley, 1998.