**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 Weisbuch^{3, 4}
^{1} Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA USA
^{2} 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
∗ 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 by (*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 PhCs^{1}. The thick free-standing *m*-plane GaN substrate below the PhCs approaches a semi-inﬁnite 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 n_{PhC} ∼ 2.03 for *ﬀ* = 35%, we calculated 9 guided modes with eﬀective indices *n*_{eﬀective} (where ) between n_{GaN} ∼ 2.41 and n_{PhC}. 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 P_{x} 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 and a parallel wavevector component are diﬀracted along the *a*-direction if . Its parallel diﬀracted component of the wavevector (green vector) is (Fig. S1a). The external angle θ is related to the internal angle * *as . Combining these expressions, the in-plane angle α can be written as
(1)
and the dispersion relation of these modes is
(2)
**Fig. S1: Diﬀraction of modes propagating oﬀ-axis. a.** Three dimension schematics of the diﬀraction of oﬀ-axis modes into the m-a plane. The original in-plane wavevector diﬀracted by resulting in. ** b. **Angle resolved measurement along the *m-a* plane for both and polarizations, normalized by the lineshape and intensity of emission from the QWs. Calculated dispersion relation for modes with *n*_{eﬀective} = 1.73 (dashed-blue) and *n*_{eﬀective} = 1.78 (dashed-green) matches the experimental data.
Figure S1b shows the angle resolved measurement in the m-a plane for both and 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 and ), which in the present case occurs at large angles (∼ 40^{o}). Therefore the crossing between the modes and the replicas is better seen for the polarization as the intensity of the 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 polarization is weak at angles close to 0^{o} only for modes excited by the P_{x} dipole (λ ∼ 465 nm). polarized modes excited by the P_{z} dipole with λ ∼ 430 nm behave similarly to polarized modes excited by the P_{x} 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 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 = 90^{o} the polarizer is parallel to the *m*-direction.
**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:
Combining this expression to the Fresnel’s law and to its derivative yields
Therefore the intensity of light per solid angle measured in air () corresponds approximately to .
**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. |