Investigation of photonic band gaps with special emphasis on hyperuniform structures

Abstract

A toolbox of considerable size was collected within the course of this work that enables the study of photonic meta materials. It is now possible to successfully simulate, fabricate and moreover characterise meta materials with a photonic band gap. This is of great interest for applications, where waveguides are one possible object of interest, as well as fundamental theoretical investigations, namely identify the properties a pattern needs to posses to form such a photonic band gap, for example. Not only periodic structures were investigated, but also the dataset of an amorphous point pattern was employed, provided by our collaborators Marian Florescu (Princeton University), Paul Chaikin (New York University) and Paul Steinhardt (Princeton University). The pattern posses the property of hyperuniformity, its long range density fluctuations vanish, and according to most recent theories, hyperuniformity is one criterion that enables a band gap. Built upon the experience from preliminary work of Ropers and Knappe, samples from this data set were fabricated by means of the direct laser writing (DLW) technology. Despite a low refractive index contrast and a filling fraction much less than the recommended value given by our collaborators, surprisingly, reduced transmittance was found. Close to lambda=1550nm wavelength, important for modern telecommunications, an ever so small dip in transmission was found. This peculiar finding needed thorough testing, as even the performed finite-difference time-domain (FDTD) simulations only indicate the weakest change in transmittance, if at all. The simulations done with the Meep software package show that substantially reduced transmittance emerges only for higher refractive index contrasts. Having examined this behaviour further with a systematic parameter sweep, other interesting questions arise, regarding the exact behaviour of a wave in such a medium. The field distribution for waves with frequencies at the band edges could be examined in further research, in order to stress a band gap origin of the reduced transmittance. For waveguides in photonic band gap structures, the penetration depth can give a limit on how close two waveguides can be built before they interact and couple into each other, making further simulations uttermost worthwhile. Point sources inside the structure might be simulated and the flow through concentric flux spheres calculated. By comparing the findings to structures obtained from e.g. Poisson patterns would allow to determine the influence of the structure factor, of interest for the fundamental theory. By writing the Pointpatterntool, a software was established that allows for convenient handling of point pattern data sets and generation of basic statistics, and further research is pioneered. By comparing results with the well known woodpile structure, the authenticity of the measurement of the hyperuniform sample was further assured. The periodicity of the woodpile structure furthermore permits utilisation of Bloch's theorem, allowing to a different kind of simulation: the plane wave expansion method, implemented in the MPB software package. With that, the complete three dimensional dispersion relation was calculated successfully, for a high and low refractive index contrast and furthermore the density of states (DOS) was deduced. A genuine band gap was found for high refractive index contrasts, in compliance with literature. For the low refractive index, interesting properties regarding the coupling of the light into the structure were deduced. A cross check with the FDTD simulation indeed reveals reduced transmittance, for either case. With the woodpile structure, the fabrication process by means of DLW was greatly improved. It became evident that the final fabricated structures properties depend on a very delicate interplay of the parameters. Moreover, the parameters are not independent of each other, but interact. For example, the line width changes dependent on the very structure itself, for otherwise the same settings. This influences the filling fraction as a fundamental parameter that determines the formation of a band gap. For every structure that one wants to fabricate, the influence of the parameters need to be tested thoroughly. Only then reliable statements e.g. on the filling fraction can be done and a well defined sample can be fabricated. As the simulations predict a more pronounced gap for higher refractive indices, material conversion e.g. with the introduced sol-gel processes together with the calcination of the keratin structure is promising to enhance any present band gap. The need for non-destructive methods arose, and a scattering experiment was set up. It is possible to collect diffraction patterns of a sample, revealing information on characteristic lengths in the sample. The decoupling of the diffracted light into air was regarded, and the measurement of the characteristic length verified by imaging the samples with the SEM. Thus, with the scattering experiment a valuable method was employed that allows to measure the sample quality without destroying the sample. As the filling fraction is such a crucial quantity, further methods should be developed. One could exploit the change in effective refractive index by infiltration with refractive index liquids for the woodpile structures. With that, the optical path length would change slightly, which is then measurable with an interferometer setup. This should be done with large wavelengths with respect to the structural variation, which would imply far infra red wavelengths. These are less prone to scattering but behave like in a homogeneous medium with effective refractive index, which was stressed by the simulations. With the reconditioned spectroscopy setup transmittance spectra of the samples can now be measured reliably, as proven on the example of the woodpile structure. The measured dip corresponds to the simulated transmittance and dispersion relation, within the range of uncertainty, further stressing the authenticity of the measured reduced transmittance of the hyperuniform structure. By replacing the photo diode in the spectrometer experiment with a camera, and the thermal light source with a tunable laser source, the spectrometer and scattering experiment can be combined. A frequency resolved diffraction pattern can then be measured, revealing more properties of the sample. Furthermore, a spatially resolved transmission spectrum can be taken. This may not only allow for conclusions to a band gap, but also on the light propagation behaviour near a band edge. The structural colour found in nature, for example in the green medullary cells of the feather barbs of Agapornis roseicollis, indicates that a reduced transmittance is very well possible for such low refractive indices. Further examination is consequential, and a thin slice reconstruction with the focused ion beam (FIB) was performed in order to extract the keratin volume information. A dedicated extraction method was contrived, that allows to extract the point pattern of such a keratin network. This extraction method was beforehand tested on artificially generated thin slices to delimit its capabilities. The minimal needed resolution is close to the maximally achievable resolution of the FIB facility, regarding the thickness of a slice. Despite great challenges in the proper settings of the SEM and FIB device, a preliminary data set could be harvested and the extraction method was applied as an experiment. The results proves to be promising and the findings justify further research: instead of possessing a tetrahedral network topology like the hyperuniform data set, a trihedral topology is indicated, contradicting theories in literature. Especially with contrast enhancement, achieved by the sol-gel process, more reliable data can be expected to verify or falsify these very surprising findings. Summarising, by testing the accuracy of the spectrometer setup, cross checking results with a periodic structure, considering existent band stop structures from nature and monitoring the band gap evolution with change in the refractive index contrast with simulations, strong evidence is given that the reduced transmittance measured for the hyperuniform sample truly stems from a band gap origin. This is the first time reported that an amorphous structure, fabricated purely by means of DLW, possesses a measurable band stop. Furthermore, with the collected methods the path for future research is cleared.