We propose, analyze and optimize a two-dimensional conical photonic crystal geometry to enhance light extraction from a high refractive index material, such as an inorganic scintillator. The conical geometry suppresses Fresnel reflections at an optical interface due to adiabatic impedance matching from a gradient index effect. The periodic array of cone structures with a pitch larger than the wavelength of light diffracts light into higher-order modes with different propagating angles, enabling certain photons to overcome total internal reflection (TIR). The numerical simulation shows simultaneous light yield gains relative to a flat surface both below and above the critical angle and how key parameters affect the light extraction efficiency. Our optimized design provides a 46% gain in light yield when the conical photonic crystals are coated on an LSO (cerium-doped lutetium oxyorthosilicate) scintillator.
© 2015 Optical Society of America
Efficient light extraction is crucial for many photonic devices where light is generated from the inside of high refractive index materials, such as scintillators. Often used for medical imaging or cosmic radiation detection, scintillators generate visible light after absorbing high-energy electromagnetic waves such as X-ray or gamma radiation [1, 2]. Scintillator materials are generally isotropic, so the visible light they generate is emitted at all possible angles, whereas the optoelectronic transducer occupies only one facet as shown in Fig. 1(a). When the generated visible light is coupled with an interface between a high index material and an outer environment such as air, light transmission is often limited by Fresnel reflection due to the difference in refractive indices between two materials. Light extraction efficiency is further limited by total internal reflection (TIR), a limitation that becomes more severe for high refractive index materials, because of the low critical angle. Since light generation in a scintillator material can be assumed to be isotropic, all light whose incidence angle is larger than the critical angle should be reflected back and trapped in the material [3, 4].
Recently, there has been growing interest in functional micro- and nano-structures that can enhance the light extraction efficiency of materials such as scintillators and LEDs. Anti-reflective moth-eye nanostructures of subwavelength period can increase the light transmission by suppressing the Fresnel reflection [5–8]. However, their sub-wavelength scale is not appropriate for the scintillator application since only the zeroth-order light is allowed to propagate through the nanostructured interface, and thus TIR still occurs. Instead, diffraction gratings, sometimes referred to as “photonic crystals”, have also been used for enhancing the extraction efficiency. To overcome TIR, these patterns are redirecting some of the light to diffraction orders, propagating beyond the critical angle [1, 3, 4, 9–12]. Typically, these structures consist of square lattices of cylindrical holes with spacing larger than the wavelength. However, in these cases the transmission under the critical angle must be partly sacrificed, not only because of Fresnel reflection, but also because the zeroth-order must now be divided into several diffracted orders (some of which still do not successfully overcome the TIR problem.) Although extensive research has been carried out on enhancing light extraction efficiency [13–23], none of the referenced works shows sufficient advantages both for under the critical angle region and beyond the critical angle region.
Here we propose a conical diffraction grating/photonic crystal as a highly efficient light extraction layer shown in Fig. 1(b), as an attempt to balance more successfully Fresnel reflection and TIR. Essentially, we combine two concepts: the conical shape is meant to reduce Fresnel reflection for the zeroth-order of light at a broad range of angles of incidence, whereas the diffractive structure is meant to redirect at least a portion of the light that is incident beyond the critical angle. However, the two goals are in conflict in the sense that the first function (reducing Fresnel reflection) requires a subwavelength pitch, whereas the second function (diffracting light that would otherwise be TIR’ed) requires a diffractive structure with period bigger than wavelength. Fortunately, if the period is near the wavelength, a mix of both effects can be observed [24–26] and we can hope to balance them effectively to maximize light extraction.
In order to analyze and optimize the proposed structures, we considered several approaches. The analytical approach, based on Fourier optics, of course fails for periods shorter than approximately two wavelengths, so it is not suitable for this problem. The most reliable method is, of course, FDTD (see Appendix) but it is not easily utilized for optimization because of the computational effort required .
The method that we did use is based on Rigorous Coupled Wave Analysis (RCWA) for a 2D cross-section of the proposed structures, i.e. triangular for diffraction analysis and a 3D cone shape for optimization process. We settled on this compromise because it gives reasonably reliable results for the range of geometrical and refractive index parameters considered (checked with 3D FDTD in Appendix for selected cases), and it is also reasonably time-effective. The methodology for using RCWA for this problem, as well as a preliminary analysis of the effects of pitch on the ability of the structure to retain light emitted at different angles are presented in Section 2. With these insights, we proceed to fully optimize the structures with respect to all geometrical parameters in Section 3. Concluding remarks are in Section 4.
2. Analysis of light extraction efficiency using RCWA
For the flat surface as shown in Fig. 2(a), if the emission angle (θe) is larger than the critical angle determined by Snell’s law θc = sin−1(n2/n1), all light is internally reflected inside the material.
The relationship from Snell’s law can also be explained by the conservation of the tangential momentum (k||) parallel to the coupling interface, i.e. phase-matching condition. In the phase-matching diagram in the wave number space shown in Fig. 2(c), only light with k|| = k0nscsinθe smaller than nairk0 = k0nscsinθc can radiate into the ambient medium for a flat surface, where nsc and nair are the refractive indices of the scintillator and air, respectively. Therefore light with an angle larger than the critical angle (θc) cannot escape the scintillator through the interface.
If the in-plain component of the wave vector of emitted light is coupled with a reciprocal lattice vector or grating vector G, and satisfies the phase matching relation |kgsin(θe) ± mG| < k0, where m denotes the order of the diffraction, light can be extracted into the air as shown in Fig. 2(b). The reciprocal lattice vector G of the photonic crystal can be represented as a function of the periodicity of cones (G = 2π/P). The light extraction through diffraction by the photonic crystal is thus explained by Bragg’s diffraction law , which is described as:
White dotted lines depicted in Fig. 3 define the boundaries plotted from Eq. (2) in θe-P domain for the first three orders of diffracted light. Consequently, the dotted lines in Fig. 3 represent the boundaries confining the area where each diffraction order is effectively generated.
The diffraction efficiency is also calculated using RCWA as depicted in Fig. 3(a) for TE polarization and Fig. 3(b) for TM polarization, respectively. The color contours confirm the relationship between the transmission of diffracted light and the main parameters such as the periodicity and the emission angle. The transmission of each diffraction mode was calculated for various periodicities (0 μm ≤ P ≤ 3 μm) and emission angles (0° ≤ θe ≤ 90°) with fixed height H = 0.42 μm.
The relationship between each diffraction order calculated from Eq. (2) and the light transmission becomes obvious in each diffraction mode’s transmission calculated numerically as shown in Fig. 3. The areas of the positive 1st, 2nd, and 3rd modes and negative 1st, 2nd, and 3rd modes, as shown in Figs. 3(a) and 3(b), are defined by following the lines of the diffraction modes, respectively. For example, the area above the white dotted line in Fig. 3 is the area where the positive diffraction orders can be transmitted, and the area confined by dotted lines in Fig. 3 indicates the area where the negative diffraction orders can be transmitted. The 0th order is dominated by the critical angle (θc = 33.3°) between the scintillator (nsc = 1.82) and air (nair = 1).
Further, the total light transmission was also calculated using RCWA for different periodicities (0 μm ≤ P ≤ 3 μm) and emission angles (0° ≤ θe ≤ 90°) with the same height H = 0.42 μm. The results are shown in Fig. 4. The total light transmission is exactly the same as the summation of all the orders from 0th to higher orders shown in Figs. 3(a) and 3(b), and hence we can analyze the diffraction efficiency and its contribution to the complicated total transmission shown in Fig. 4.
The vertical border at an angle of 33.3° from the critical angle separates the high transmission area on the left side and relatively low transmission area on the right. Less light is transmitted beyond the critical angle when the pitch is smaller since little light can be extracted due to TIR. In particular, when the pitch is smaller than half of the wavelength, there is no light extraction after the critical angle, as also shown in Fig. 7(a) in the Appendix.
More light transmission is observed upon increasing pitch, even beyond the critical angle due to the diffraction effect. The optical transmission distribution is divided by additional colored boundaries located across different diagonal directions all along the graph. It is noted that the lines drawn by Eq. (2) are perfectly matched with the boundary lines, shown in Figs. 4(a) and 4(b), depicted from numerical calculation, which clearly proves that the complicated distribution of light transmission through conical photonic crystals are mainly governed by different diffraction orders. The total number of diffraction modes and their kinds generated with certain pitch and angle determine the amount of light extracted through the conical photonic crystal surface. Even though the complex transmission distribution makes it difficult to select an optimal pitch for the best possible light extraction efficiency over all the emission angles, we know where the boundaries originate and what the relationship is between key parameters such as the pitch and the angle. We can also analyze which diffraction orders contribute to the total transmission and what parameter we have to choose to increase diffraction efficiency, which will eventually contribute to light extraction efficiency.
In addition, the refractive index of the light-extracting-layer is an important factor determining the light extraction efficiency. Generally speaking, the higher the index, the better transmission as the light transmission is largely reliant on the diffraction effect, which can be enhanced by a higher index contrast between the nanostructure material and environment. While some gain is expected even with a low refractive index material, for example n = 1.5, the gain becomes even higher when we increase the refractive index to 2.3, as shown in Fig. 5. However, the practical refractive index of the light-extracting-layer is limited by the actual material we can utilize. For example, if the structure is coated in a form of an imprinted film made of polymer, the practical limit of the layer’s refractive index is at most nmax = 1.67 at the wavelength of 420 nm, assuming the absorption coefficient of the polymer is almost zero . Nevertheless, if a higher index polymer with minimal absorption can be developed, such as a TiO2 mixed polymer , or if an inorganic material such as Si3N4 can be patterned into conical photonic crystal easily, a higher gain is expected.
While the pitch of the conical photonic crystal is the most critical parameter to determine the diffraction behavior of the conical photonic crystal, the height can also affect the diffraction efficiency and hence affect the total transmission of light. If the height is too low, the surface is close to a flat surface and hence the anti-reflectivity or diffraction effect is small. There should also be an upper limit for the height, as it negatively affects the diffraction efficiency of the surface if it is too high. Further, there are additional potential problems related to the height of cones such as a loss depending on the material absorption and a difficulty in fabricating the structures. In fact, a height of approximately 2λ is close to the optimal value as shown in the optimization results in Fig. 6.
Assuming that we utilize the material with a refractive index of 1.67, which is the practical limit of the polymer that we can utilize , we calculated the optimal pitch and height of the conical photonic crystal on an LSO (cerium-doped lutetium oxyorthosilicate, Lu2SiO5: Ce3+) scintillator (nLSO = 1.82 and λemission = 420 nm) when coupled with air and index matching liquid (noil = 1.5).
In order to optimize the light extraction efficiency, we have to consider isotropic light generation in the scintillator , which means that the generated light incident on the coupling surface has all incident and azimuthal angles from 0° to 90°. All the polarization components (TE and TM) should also be included in the simulation. We first calculated and averaged out the optical transmission for all the azimuthal angles and polarization components, and then plotted the averaged value of transmission along different emission angles (0° ≤ θe ≤ 90°) for a specific pitch and height of the conical photonic crystal. In addition, we need to compute the probability for the number of photons incident on the coupling surfaces as a function of incidence angles, which can be affected by the geometry of the scintillator, refractive indices of materials and wrapping conditions . The probability factor can be calculated using Monte-Carlo simulation , and then used for weighting each component of incidence angles when calculating the total number of photons extracted from each geometry.
The results of the optimization processes are shown in Fig. 6. When an LSO scintillator is coupled with air, the geometry with a periodicity 1 μm and a height 0.7 μm shows the best possible extraction efficiency, with a gain of approximately 46% compared to that of a flat scintillator surface. If the scintillator is coupled with an index matching liquid with the index of noil = 1.5, the possible gain is estimated as 12% when the pitch and the height of the structure is approximately 2.5 μm and 0.7 μm, respectively. The air coupled case exhibits higher gain than the index matching liquid case due to a higher index contrast between the light extracting material and ambient. It is noted that if multiple reflections from a reflector shown in Fig. 1(a) are incorporated in the calculation, the maximum gain from the optimization process becomes higher . The optimal geometry ranges over a certain window rather than a single point, which is beneficial when fabricating the structure because the window will allow a certain amount of fabrication tolerance.
We can use the same optimization process for various types of light generating materials with different refractive indices, wavelengths, and coupling environments. For examples, related applications such as LEDs or OLEDs can utilize the same concept for enhancing light extraction through the proposed design and optimization process. The higher index contrast in the case of LEDs has the potential to generate even more gain than a scintillator case discussed in this paper.
We demonstrate that the proposed conical photonic crystal can serve as an efficient light extracting layer by keeping both anti-reflection and diffraction effects. For the optimized geometry of the conical photonic crystals, around 46% more light can be extracted compared to a flat LSO scintillator coupled with air. Further improvement is expected with a higher refractive index polymer such as TiO2 mixed materials . The design described here is potentially applicable to a wide range of light emitting materials. By using the same concept of design and the optimization process, the conical photonic crystals can play a part in increasing the light extraction efficiency not only for scintillators, but also for LEDs and OLEDs.
Appendix 3D FDTD method for analysis of light extraction in near-wavelength periodic cone arrays
We used 3D FDTD to compare light extraction from an inorganic scintillator for different pitches of conical structures superimposed on the exit surface. A commercial-grade simulator based on the FDTD method (Lumerical Solutions, Inc., FDTD Solutions 8.0) was used to perform the calculations. We first compared conical photonic crystal structures with different periodicity as shown in Fig. 7(a). The common conditions for these simulations are transverse electric (TE) polarized irradiation, illuminated at different incident angles from 0° to 90°.
In Fig. 7(a), the top part represents conventional anti-reflective moth-eye or nanocone structures with a subwavelength pitch of P/λ=0.33, which exhibit excellent transmissivity when the incidence angle is below the critical angle (θc ≅ 33.3°). The gradient in the index profile of nanocones minimizes the Fresnel reflection because of adiabatic impedance matching between air and the scintillator substrate. However, no light propagating beyond the critical angle can be extracted, indicating that the sub-wavelength nanostructure cannot overcome TIR. However, as the periodicity gets larger, light gains are observed beyond and below the critical angle. The light extraction beyond the critical angle is due to the lateral periodicity of the conical photonic crystal generating a diffraction effect. The light transmission under the critical angle is also higher than on the flat surface shown in Fig. 7(b), because the reflection is suppressed due to the gradient index effect from the tapered geometry [5, 6] of the proposed design.
A comparison among conventional subwavelength nanocones, conical photonic crystals and a flat scintillator surface is shown in Fig. 7(b). The conical photonic crystal shows higher transmission compared to a flat scintillator surface due to the gradient index effect from the tapered shape of the design, while there also is some light extraction beyond the critical angle generated by diffraction from the periodicity of the conical photonic crystals. In Fig. 7(b) the results calculated using 3D FDTD are compared with the values calculated using RCWA, which gives reasonably reliable match with each other for the range of geometrical parameters considered.
In the numerical simulations used in this paper including RCWA and FDTD, the mesh sizes were less than 5nm, which is far below the convergence range for the conical photonic crystal. Total seven diffraction orders (+3, +2, +1, 0, −1, −2 and −3) shown in Fig. 3 were considered in all RCWA calculations. By summing all the efficiencies of seven diffraction orders, we have calculated the light transmission through conical photonic crystals for different pitches and angles. In 3D FDTD, simulation space is confined to one unit of a cone structure and then periodic boundary condition was applied.
This work was supported by the U.S. Department of Homeland Security, Domestic Nuclear Detection Office, under the competitively awarded contract HSHQDC-13-C-B0040. This support does not constitute an express or implied endorsement on the part of the government.
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