Light Extraction through Photonic Crystals

The characteristic band structures of PhCs lead to notable optical properties. It has been shown by S. Fan et. al. [26], that these features can be used to increase the light extraction from high-index materials. For this, usually PhC slabs are used, which are two-dimensional (2D) PhCs that have only finite extension in the third dimension. The band structure of these

CHAPTER 3 APPROACHES FOR IMPROVING LIGHT YIELD AND TIMING

Wave vector k ~ Wave vector k

a b

Figure 3.10: a) Photonic band structure calculated for a 2D PhC consisting of hexagonally arranged air columns (ε = 1) in a dielectric substrate (ε= 13). The TE (transverse electric) and TM (transverse magnetic) modes are equivalent to S- and P-polarization, respectively.

The horizontal axis shows |⃗k| along the outline of the so-called irreducible Brillouin zone depicted as blue triangle in the inset. b) Equivalent of (a) for a PhC slab. The horizontal axis shows the wave vector component within the slab plane. The broken symmetry along the third dimension leads to a continuum of radiative modes that are not guided within the slab (shaded area). The border of this continuum is called the light cone. Images taken from Ref. [130].

slabs differ from those of conventional PhCs (see Fig. 3.10b), as there must be distinguished between guided modes that are trapped inside the PhC and the continuum of radiative modes that leave the slab. The increase of light extraction through PhC slabs is based on various effects, such as tailored band structures [26, 139] or diffraction of guided modes into the ambient medium [140–142]. Through fabricating such a PhC slab on the exit face on a scintillator crystal (see Fig. 3.11), this effect can be transferred to PET detectors in order to avoid light trapping.

3.2.3.1 Light Diffraction through Photonic Crystals

An intuitive way to understand the effect of a PhC slab at a material interface is treating it as a biperiodic dielectric diffraction grating. The concept of PhC-enhanced light extraction studied in this work (see Fig. 3.11) considers grating structures with dimensions in the range of the wavelength (λ≈420 nm) on mm²-sized scintillator surfaces and layer thicknesses (e.g.

of the optical glue) of 100 micrometers or more. Hence, the involved diffraction effects can be treated within the Fraunhofer regime. In Fig. 3.12, the interaction of light with a material boundary is illustrated for a plain interface versus an interface equipped with a diffraction grating. In the case of light impinging with θ > θ_TIR on the interface without diffraction grating, all photons are reflected due to TIR. However, in the presence of a diffraction grating, there exist certain diffraction orders which facilitate the extraction of light even for the case θ > θ_TIR. A disadvantage of the grating is, that there are also various diffraction orders in the reflection direction, which can lead to increased reflection for θ < θTIR compared to the 46

3.2 PHOTONIC CRYSTALS

Optical glue Photosensor

Scintillator µ₁ µ₂ µ₁ µ₂

PhC Slab

Figure 3.11: Illustration of the concept to improve the light extraction from scintillators using PhC slabs. In a conventional detector with polished scintillator surfaces (left), only photons with θ₁ < θ_TIR can be extracted whereas photons with θ₂ > θ_TIR are trapped.

Inserting a PhC slab at the interface scintillator/glue (right) bears the potential to increase the light extraction.

plain interface. For a 1D grating, the angle θ_m of the m-th transmitted diffraction order is given through the grating equation

a(n₂sinθ_m−n₁sinθ) =mλ₀ , (3.33) where a is the grating pitch,n₁ and n₂ are the RIs of the first and second medium, respec-tively, and λ₀ is the vacuum wavelength. The geometry used to derive the grating equation is shown in Fig. 3.13. The diffraction patterns of 2D gratings can be obtained through a su-perposition of two 1D gratings. Eq. 3.33 also facilitates the distinction between two kinds of diffraction orders. Diffraction orders with an indexm that yield|sinθ_m|<1 are propagating orders representing EM waves that are reflected or transmitted by the grating. Diffraction orders that lead to |sinθ_m| ≥ 1 are evanescent orders. These have a complex ⃗k-component perpendicular to the grating which indicates that the intensity of the corresponding EM waves decays exponentially with the distance from the grating and cannot be detected at distances larger than a few wavelengths [143].

For a given configuration of n1, n2 and a fixed incident angle, the position and total number of all non-evanescent diffraction orders depend only on the grating pitch a and not on any other property of the grating. Nevertheless, for the efficient light extraction through a PhC grating, it is important that as many scintillation photons as possible are diffracted into an extracted order, i.e. a propagating diffraction order pointing towards the ambient medium.

The amount of light that is diffracted into a certain order is given by its diffraction efficiency, which depends on the details of the grating geometry and the RIs of the grating materials.

Through adjusting these parameters, the diffraction efficiencies can be manipulated which influences the overall transmission characteristic of the PhC. This can be used to tailor a PhC that yields increased light extraction for a given angular distribution of incident photons. The

CHAPTER 3 APPROACHES FOR IMPROVING LIGHT YIELD AND TIMING

Figure 3.12: Illustration of light extraction through a diffraction grating. Without grating at the interface and for θ₁ < θ_TIR (a), incident waves can be refracted (blue) and reflected (red), whereas forθ₂ > θ_TIR only reflection is allowed. In the presence of a diffraction grating (c,d), there exist various diffraction orders in both directions. In the case ofθ₂ > θ_TIR, these orders can lead to the extraction of light despite TIR (d).

calculation of diffraction efficiencies is usually done with numerical methods, as analytical solutions can only be found for very simple cases [143]. There is a variety of numerical methods, all of which are based on the Maxwell equations and solve for the distribution of the scattered EM field in certain spatial directions, i.e. diffraction orders. An overview of these methods is given in Ref. [143].

Once the diffraction efficiencies for a given grating are known, the obtained information is two-fold: i) the overall reflection and transmission coefficient of the PhC grating can be calculated through summing over all efficiencies; ii) the new wave vector ⃗k^′ of the photon after the diffraction process is given by the angles of the order into which the photon is diffracted. Another important aspect is the spectral behavior of PhC gratings. The grating equation indicates that the positions and efficiencies of the diffraction directly depend on the wavelength of the incident light. Since scintillator crystals have emission spectra with a width of a few hundreds of nanometers, this wavelength-dependent aspect cannot be neglected and must be taken into account when studying the effects of PhCs.