Near field excitation of optical confined modes

Resumo

This thesis, which is framed in the field of nanophotonics, focuses on how nanopatterned structures can affect the interaction between light and matter to produce new phenomena that have potential application to new improved opticd technology. Due to the advances produced from this research, nowadays it is possible to fabricate and characterize nanostructures with a very high degree of accuracy, on a large production scale, and with moderate costs, making it a valuable ally in the design of novel devices that are more efficient and versatile. The field of optics can profit from such developments because the optical properties are strongly dependent on the size of the structures, and many phenomena only appear when the features are similar or smaller than the wavelength of the light. For the visible and near-infrared parts of the spectrum, where the typical wavelengths are in the order of hundreds of nanometers, an accurate control over the nanoscale is required. Optical devices such as biosensors, photo-voltaic cells, and light emission diodes (LEDs) are already engineered based on the possibilities and benefits of nanofabrication, but the potential of nanotechnology in the industry is still far from being completely explored. These advances represent the outcome of a scientific synergy where fields such as metamaterials, plasmonics, electron-beam lithography, colloidal synthesis, and sensing go hand by hand with the result of a highly productive research with great potential and currently emerging commercial applications. Today, nanophotonics is a fundamental area for smart materials, energy, and information industries, while it is increasingly finding applications in an even broader range of technologies. In this thesis, we found that the presence of ordered sets of nanoparticles can lead to new resonances that support optical guided modes. We extensively explored the properties of these resonances and their interaction with different structures. We showed that the characterization of such modes has a relevant impact in the accurate optical description of the structures, which should have a positive impact in the understanding and engineering of new optical devices that include particle arrays (PAs), both by improving the simulations and by taking direct advantage of their specific properties. We also studied the interaction of guided modes with electron beams, and how this interaction could excite light through these modes. Furthermore, this technique of electron-induced light production is able to produce quantum states of light such as single photons, with high rates of photon yield, which could be advantageous for the rising field of quantum computing. Scope of the work The effect of order in arrangements of different photonic structures, such as nanoparticles and nanoholes, has been extensively reported, and diffraction-driven phenomena such as enhanced transmission [1], where a screen drilled with small holes can show a transmission coefficient larger than the ratio of hole-to-screen areas, have been the subject of intensive work due to their physical relevance and potential applications. Most of these studies, either analytical or numerical, as well as experimental, focus on the radiative part of the dispersion diagram (i.e., inside the light cone, k < k_0) since it is the most accessible region for external sources of light. In the first part of this thesis we revealed that for a 2-dimensional ordered array of small scatterers, the region of bound states (i.e., outside the light cone) also presents remarkable characteristics. Under the dipolar approximation, the self-consistent solution of the Maxwell equations in an array of nanoparticles shows divergences in the reciprocal-space representation of the Green function at the position of the Wood anomalies. This allows us to formulate an uncoupled system of equations to obtain a solution to the self-sustained electromagnetic field. The dispersion relation of this solution is to the right of the light line in a frequency-wave vector diagram and therefore corresponds to a confined mode that propagates freely along the directions within the plane of the array while is unable to do so along the normal direction, in which the resulting fields are evanescent. We obtained an analytical expression for this dispersion relation, that depends on the lattice parameter of the array, a, as well as in the polarizabilities of the particles both electric and magnetic, ¿, and the wave vector of the light, k_0. This analytical results were checked against converged rigorous solutions of the Maxwell equations using the Korringa-Kohn-Rostoka layer method with a multipolar description of the response of the particles [2, 3]. Both calculations show a remarkable agreement, proving the validity of the model and the actual existence of the confined modes. The existence of these resonances has an important role in the description of the near-field scattered by the array. The confined modes have associated evanescent tails that can produce a field enhancement even at the proximity of the array, but when the mode is close to the light line, so that it is poorly confined, this enhancement can happen also at relatively large distances from the structure. Continuing with the analytical study of the modes, we looked into the influence of the aspect ratio (AR) of the particles. The analytical description can be approached for photonic crystal (PC) particles, showing that transverse electric (TE) and TM mode can be tuned separately. Interestingly, the confinement remains finite even for vanishing AR (infinitely thin disks). The fabrication and design specifications often require that a substrate is placed in close proximity of the photonic device. In the case of a PA, the impact of a substrate in the confined resonance is dramatic. For very small distances, the constructive interference that leads to the resonance is strongly suppressed by the presence of a dielectric interface, thus preventing the very existence of the modes in this asymmetric environments. For larger separations, we proposed a Fabry-Perot model that also showed excellent agreement relative to the self-standing PAs when comparing the aforementioned numerical methods. The position of the resonance matches that of the array in the symmetric environment for distances above a few wavelengths. As the PA and the substrate become closer, the dispersion relation of the mode is pushed towards the light line, thus decreasing the degree of confinement of the mode. Once the distance reaches a certain critical value, the mode is no longer placed at the bound region of the dispersion diagram and the resonance disappears, recovering the previous limit of small separation. The robustness of the modes against the asymmetry of the environment was also tested for the index contrast between the matrix of the array and the substrate, obtaining again the two limiting cases of symmetric environment when the ratio is 1 and the suppression of the modes for large index mismatch. We found that the contrast needed to suppress the resonance is fairly small, ~ 10% in the particular instance investigated in this work. In the particular case where the index in the substrate is larger than in the array envrionment, the described effects are affected by the leakage of the mode into the substrate medium. This induces a smooth transition between the configurations that support and do not support the mode, in contrast to the sharp transitions when the substrate is optically less dense. The impact of these modes is hardly visible in a far-field characterization due to the fact that the involved light couples poorly to them. Nevertheless this coupling can be increased by several factors that might appear as requirements of the design or be introduced with the purpose of taking advantage of the properties of the guided modes. We studied the coupling to the modes by two different methods: the presence of a high index prism that can induce a refractive coupling; and the intrinsic capability of the array to act as a diffraction grating. The refractive case can be approximated with the same Fabry-Perot model used to describe the presence of a substrate. We performed calculations in order to design a device that would efficiently couple light plane waves of designated wavelength. Using the resulting designed parameters, arrays of gold nanodisks with different parameters were fabricated by means of imprint lithography over a SiO2 substrate and covered with a capping layer of the same material to produce a separation from the array to the interface, thus preventing the suppression of the resonance. We placed the sample in a rotation stage and employed an optically dense F2 glass prism to couple a polarized white laser source. The external light impinged the interface from the prism side producing both reflected and transmitted beams. When the incidence angle was larger than the critical angle for total internal reflection, the transmitted field consisted in evanescent tails rather than propagating waves. When the parallel momentum of this evanescent fields matched that of the confined modes, there was a maximum coupling to guided light. In contrast to the all-dielectric geometries studied beforehand, the metallic nanodisks are highly lossy and the energy dissipates rapidly, thus decreasing the available power for the reflected beam. The specularly reflected light that we collected displayed a dip in intensity that had a very high degree of agreement with respect to our simulations in angular and energy position, as well as spectral shape and depth, thus proving that the predicted existence and characteristics of the modes are reproduced experimentally. We were also able to determine characteristic values of the mode such as the dispersion relation, the penetration length in the direction normal to the plane of the array, and the propagation distance. The largest mismatch between theory and experiment happened for the latter, due to its intrinsic sensitivity to fabrication imperfections in the lattice and the particles. The confined modes can also couple to propagating light trough momentum transfer provided by the lattice to the diffracted beams. Due to the small confinement characteristic of our modes, the relevance of this coupling is small for light momenta well below the diffraction edge, k = 2¿/a, but we found that its effects are important for larger energies. In order to account for this effects, we modified the analytical model to include the <0, 0> and <0, ±1> diffracted beams, which are the ones expected to play a significant role in the final results. We checked the validity of this model against rigorous numerical solutions of Maxwell¿s equations for an array of gold nanospheres above a water/glass interface. Our three-beam model proved to faithfully reproduce the characteristics of the system, specially in the regions around the diffraction edges, where the simpler model has divergent behavior. This demonstrates that, in order to accurately reproduce the far-field response of a PA, for certain conditions of the relation between the wavelength and the lattice parameter, we must include the effect of the confined modes. As the confined modes are strongly affected by the asymmetry of the environment, their influence in the far-field can be tuned by the presence of a substrate. We performed a systematic study of how the distance to the interface and the index contrast can affect the performance of the array. This study should help in the design and simulation of future devices including PAs. The Fabry-Perot model can also be used to describe the interaction between the guided modes of the PAs and other neighboring waveguides, both when the latter are similar or dissimilar. We analyzed the interaction of the described modes and those of a dielectric slab waveguide and a metal-dielectric interface supporting surface-plasmon polaritons (SPPs). In each case, the behavior showed to be similar: for large separation distances we recover the two modes associated with each individual guide, which are degenerate when the two structures are symmetric; when this distance is reduced, there is hybridization of both resonances that resembles that of a dimer, producing two solutions that are shifted in momentum in the direction of increasing splitting. Thus, the interaction produces an effective ¿repulsion¿ between both modes that becomes stronger for smaller distances. In the case of symmetric waveguides, there are two branches emerging from the degenerate resonance that correspond to symmetric and antisymmetric solutions for the hybrid system. For dielectric-embedded structures, such as PAs and the dielectric slabs, the interaction drives an increasing separation between the two hybrid solutions when the distance is decreased, until the less confined (to the left) eventually reaches the light line. At this cutoff point, this guided mode disappears, while the more confined mode (to the right) continues to increase its degree of confinement indefinitely. The situation is different for metal-insulator-metal (MIM) structures. Since light does not propagate freely within a metal due to its negative dielectric constant, the leakage to the environment that prevented the existence of the confined modes in the previous cases is not an issue in these structures, and the confined mode continues to move towards lower values of the parallel momentum until it reaches k = 0. The momentum shift produced by this interaction can be used to drive a power transfer between the two sides of the system. Since the phase velocity of the two hybrid modes is different, by selecting the strength and distance of the interaction we can generate a dephasing between them that leads to different values of the power in each side at the output of the system. We considered two neighboring waveguides whose interaction distance varies slowly along the propagation direction. Thus, we proposed an adiabatic model for the propagation of each solution, which provided us with analytical formulas that allow us to calculate the output power distribution for an specific input. We applied this model to two cases of cylinder-shaped waveguides for different values of the bending radii and minimum separations: two silver cylinders in vacuum and two cylindrical arrays of dielectric spheres. The results showed that this structure can be used to switch the power between two different channels for sensible values of the geometrical and material parameters. Interestingly, when the separation between both cylinders is smaller than the aforementioned cutoff distance, the left mode leaks to the environment and the output switches sharply to a fixed value, which, for symmetric structures, is 50% of power in each side. As noted above, metallic environments are not limited by leakage, but below the light line both the group and phase velocities of the mode decrease rapidly, increasing the level of losses, and producing a switching between modes similar to the one previously described. The last part of this thesis explores the capabilities of electrons in motion to produce light within specific confined modes and how the quantum character of the light produced by this interaction can be used for different applications. Due to conservation of energy and momentum, a charge that moves in vacuum cannot emit light. However, the presence of a material with a refraction index larger than one can trigger the emission under some specific conditions. A paradigmatic example of this effect is the ¿erenkov radiation [4] for charges that move faster than light in a homogeneous medium with a refraction index n > 1; in this scenario, the energy-momentum losses of the electron can lie within the light cone of the material, allowing direct coupling to radiative modes. Charges in motion can also produce light by matching energy and momentum of their evanescent fields to that of a guided mode. We have considered an electron that moves near a dielectric cylinder (i.e., an optical fiber) and used the boundary-element method (BEM) to calculate the light production within the confined modes. When an electron produces a photon, it suffers a loss of energy and momentum that match those transfered to the light. In the ¿erenkov effect, because the coupling involves a continuum of possible final states, it is impossible to determine that a single photon has been generated, because there are many possible combinations of emitted photons that produce a given amount of energy and momentum transfer. In contrast, in the case of confined modes the electron energy-momentum transfer only matches the guided mode resonance at some specific points and the loss must correspond to discrete values. This allows us to spot the creation of an specific photon state by inspection of the electron. Thus, we can consider the system as a single-photon emitter when we detect electrons whose lost of energy-momentum is only compatible with the emission of one specific photon within the confined mode. This proposed single-photon emitter has some advantages with respect to the currently available schemes for the generation of single photons: the setup only requires an electron microscope, the emission rate is high and the photons are produced within a confined mode, which provides a high collection efficiency and allows one to easily use them for different applications, such as quantum computing and cryptography. We proposed two different geometries that have distinctive characteristics. When the trajectory of the electron is perpendicular to a fiber supporting a guided mode, the emission rate is relatively small because the interaction time is limited to the passage of the electron across the width of the fiber, but we can retrieve from the electron the final energy and the momentum of the state of the photon. This configuration is also capable of producing entangled photons that propagate along opposite directions of the fiber. In an alternative configuration, we considered an electron traveling parallel to the axis of the fiber. This geometry always produces photons moving in the same direction as the electron. The main advantage of this configuration is that the interaction distance is much larger than in the previous case, which makes the emission rate to increase substantially, reaching values of up to one photon per electron under realistic condition. Conclusions In a first part of this thesis, we have demonstrated and analyzed the existence of optical guided modes in ordered arrays of nanoparticles. These modes are associated with Wood-like anomalies in the coherent interaction among the particles. We studied the specific properties of these modes and their interaction with neighboring structures, as well as their excitation by different techniques, combining analyticaland numerical methods, and testing their validity against experimental measurements. These modes proved to have a relevant role in the design and characterization of different structures that are common in several optical devices. We have also proposed a novel method for single-photon generation that presents interesting advances compared to the traditional setups, thus making them an efficient and versatile alternative to the current available methods. References [1] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff. Extraordinary optical transmission through sub-wavelength hole arrays. Nature, 391:667¿669, 1998. [2] N. Stefanou, V. Yannopapas, and A. Modinos. Heterostructures of photonic crystals: Frequency bands and transmission coefficients. Comput. Phys. Commun., 113:49¿77, 1998. [3] N. Stefanou, V. Yannopapas, and A. Modinos. Multem 2: A new version of the program for transmission and band-structure calculations of photonic crystals. Comput. Phys. Commun., 132:189¿196, 2000. [4] P. A. Cerenkov. Visible radiation produced by elecron moving in a medium with velocities exceeding that of light. Phys. Rev., 52:378¿379, 1937.