Um estudo do efeito da composição dos vidros teluretos sobre os índices de refração linear e não linear

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Modelo teórico de sensores ópticos baseados em fibras com grade de Bragg

Ao se aplicar uma tração exterior a uma fibra óptica, seu índice de refração pode modificar devido ao efeito foto-elástico. Quando a fibra contém uma grade de Bragg inscrita em seu núcleo, além do índice de refração seu período espacial também modifica, permitindo sua aplicação como sensores ópticos de temperatura e/ou deformação de alta sensitvidade. Do ponto de vista teórico, até o presente a operação de sensores baseados em fibras ópticas com grade de Brag (FGB) é explicada pelo efeito foto-elástico, através de componentes adequadas do tensor foto-elástico de fibras de sílica dopadas com germânio (germano-silicatadas). Em praticamente toda a literatura previamente estudada, apenas este tipo de fibra óptica tem suas propriedades optomecânicas bem definidas, o que pode representar uma restrição. Nesta dissertação apresentamos uma formulação teórica alternativa para o estudo das propriedades de um sensor de deformação longitudinal baseado em fibra óptica com grade de Bragg (FGB) que não depende do conhecimento prévio das componentes do tensor foto-elástico, e que leva a resultados que dependem apenas da razão de Poisson da fibra, uma característica mecânica facilmente mensurável, e de seu índice de refração efetivo. Os resultados obtidos com este modelo apresentam ótima concordância com aqueles obtidos pelo modelo conhecido da literatura, além de permitir estimar o comportamento de outros parâmetros relacionados ao espectro de refletividade da FGB.

Simulative Investigation on the Electronic, Vibrational and Optical Properties of the Ge2Sb2Te5 Chalcogenide

Chalcogenides are chemical compounds with at least one of the following three chemical elements: Sulfur (S), Selenium (Sn), and Tellurium (Te). As opposed to other materials, chalcogenide atomic arrangement can quickly and reversibly inter-change between crystalline, amorphous and liquid phases. Therefore they are also called phase change materials. As a results, chalcogenide thermal, optical, structural, electronic, electrical properties change pronouncedly and significantly with the phase they are in, leading to a host of different applications in different areas. The noticeable optical reflectivity difference between crystalline and amorphous phases has allowed optical storage devices to be made. Their very high thermal conductivity and heat fusion provided remarkable benefits in the frame of thermal energy storage for heating and cooling in residential and commercial buildings. The outstanding resistivity difference between crystalline and amorphous phases led to a significant improvement of solid state storage devices from the power consumption to the re-writability to say nothing of the shrinkability. This work focuses on a better understanding from a simulative stand point of the electronic, vibrational and optical properties for the crystalline phases (hexagonal and faced-centered cubic). The electronic properties are calculated implementing the density functional theory combined with pseudo-potentials, plane waves and the local density approximation. The phonon properties are computed using the density functional perturbation theory. The phonon dispersion and spectrum are calculated using the density functional perturbation theory. As it relates to the optical constants, the real part dielectric function is calculated through the Drude-Lorentz expression. The imaginary part results from the real part through the Kramers-Kronig transformation. The refractive index, the extinctive and absorption coefficients are analytically calculated from the dielectric function. The transmission and reflection coefficients are calculated using the Fresnel equations. All calculated optical constants compare well the experimental ones.

High pressure surface plasmon spectroscopy

Der Einfluß von Druck auf die Eigenschaften dünner dielektrischer Filme wurde mit Hilfe von Oberflächenplasmonen-Spektroskopie untersucht. Die Arbeit kann aus der Perspektive der Materialcharakterisierung und aus apparativer Sicht betrachtet werden, da z.B. eine neue Hochdruckzelle konstruiert wurde, die kombinierte Oberflächenplasmonen-Elektrochemie Messungen erlaubt. SiO2-Solgel Filme wurden optimiert und auf ihre Widerstandsfähigkeit in Bufferlösungen und ihre Oberflächeneigenschaften hin untersucht. Eine Anwendung lag in der Charakterisierung von thermoresponsiven Acrylsäureisopropylamid Hydrogelen, die einen Volumenphasenübergang aufwiesen, dessen Eigenschaften durch Druck und die Beschränktheit des Films auf die Oberfläche beeinflußt wurden.Die Charakterisierung von DNA Hybridisierungsreaktionen an Oberflächen wurde mit einer Fluoreszenz-erweiterten Hochdruckapparatur durchgeführt. Zunächst wurde die Stabilität der zugrundeliegenden Bindematrix sichergestellt. Bei DNA Einzelsträngen führten Temperatur und Druck zu jeweils verringertem bzw. erhöhtem Signal. Entropie und Änderungen der Lösungsmitteleigenschaften wurden für die Signaländerungen verantwortlich gemacht. Die Eigenschaften der Doppelhelix wurden im Langmuir-Bild beschrieben. Der Brechungsindex von Kohlendioxid wurde bis zu 200 MPa gemessen und mit vorhandenen Gleichungen verglichen. Weiterhin wurde das Schwellverhalten von PMMA in scCO2/MMA-Mischungen untersucht. Mit Druck und MMA-Konzentration nimmt das Polymer vermehrt Kohlendioxid auf. Dadurch schwillt es an und sein Brechungsindex nimmt ab. Berechnungen unter Annahme einer idealen Mixtur ergaben gute qualitative Übereinstimmung mit den Meßwerten.Abschließend wurde eine neue Elektrochemie-Hochdruckzelle vorgestellt, die an Kaliumhexacyanoferrat(III)-(II) getestet wurde. Die Elektropolymerisation von Thiophen optisch untersucht.

Polymer properties measured by micromechanical cantilever technique

Surface stress changes induced by specific adsorption of molecules were investigated using a micromechanical cantilever sensor (MCS) device. 16 MCS are grouped within four separate wells. Each well can be addressed independently by different liquid enabling functionalization of MCS separately by flowing different solutions through each well and performing sensing and reference experiments simultaneously. In addition, each well contains a fixed reference mirror, which allows measuring the absolute bending of MCS. The effect of the flow rate on the MCS bending change was found to be dependent on the absolute bending value of MCS. In addition, the signal from the reference mirror can be used to follow refractive index changes upon mixing different solutions. Finite element simulation of solution exchange in wells was compared with experiment results. Both revealed that one solution can be exchanged by another one after a total volume of 200 µl has flown through. Using MCS, the adsorption of thiolated deoxyribonucleic acid (DNA) molecules and 6-mercapto-1-hexanol (MCH) on gold surfaces, and the DNA hybridization were performed. The nanomechanical response is in agreement with data reported by Fritz et al.1 Thus, the multiwell device is readily applicable for sensing of multiple chemical and biological recognition events in a single step. In this context controlled release and uptake of drugs are currently widely discussed. As a model system, we have used polystyrene (PS) spheres with diameters in the order of µm. The swelling behavior of individual PS spheres in toluene vapor was studied via mass loading by means of micromechanical cantilever sensors. For 4–8% cross-linked PS a mass increase of 180% in saturated toluene vapor was measured. In addition, the diameter change in saturated toluene vapor was measured and the corresponding volume increase of 200% was calculated. The mass of the swollen PS sphere decreases with increasing exposure time to ultraviolet (UV) light. The swelling response is significantly different between the first and the second exposure to toluene vapor. This is attributed to the formation of a cross-linked shell at the surface of the PS spheres. Shape persistent parts were observed for locally UV irradiated PS spheres. These PS spheres were found to be fluorescent and cracks occur after exposure in toluene liquid. The diffusion time of dye molecules in PS spheres increases with increasing chemical cross-linking density. This concept of locally dissolving non cross-linked PS from the sphere was applied to fabricate donut structures on surfaces. Arrays of PS spheres were fabricated using spin coating. The donut structure was produced simply after liquid solvent rinsing. The complete cross-linking of PS spheres was found after long exposure time to UV. We found that stabilizers play a major role in the formation of the donut nanostructures.

Plasmons as sensors

“Plasmon” is a synonym for collective oscillations of the conduction electrons in a metal nanoparticle (excited by an incoming light wave), which cause strong optical responses like efficient light scattering. The scattering cross-section with respect to the light wavelength depends not only on material, size and shape of the nanoparticle, but also on the refractive index of the embedding medium. For this reason, plasmonic nanoparticles are interesting candidates for sensing applications. Here, two novel setups for rapid spectral investigations of single nanoparticles and different sensing experiments are presented.rnrnPrecisely, the novel setups are based on an optical microscope operated in darkfield modus. For the fast single particle spectroscopy (fastSPS) setup, the entrance pinhole of a coupled spectrometer is replaced by a liquid crystal device (LCD) acting as spatially addressable electronic shutter. This improvement allows the automatic and continuous investigation of several particles in parallel for the first time. The second novel setup (RotPOL) usesrna rotating wedge-shaped polarizer and encodes the full polarization information of each particle within one image, which reveals the symmetry of the particles and their plasmon modes. Both setups are used to observe nanoparticle growth in situ on a single-particle level to extract quantitative data on nanoparticle growth.rnrnUsing the fastSPS setup, I investigate the membrane coating of gold nanorods in aqueous solution and show unequivocally the subsequent detection of protein binding to the membrane. This binding process leads to a spectral shift of the particles resonance due to the higher refractive index of the protein compared to water. Hence, the nanosized addressable sensor platform allows for local analysis of protein interactions with biological membranes as a function of the lateral composition of phase separated membranes.rnrnThe sensitivity on changes in the environmental refractive index depends on the particles’ aspect ratio. On the basis of simulations and experiments, I could present the existence of an optimal aspect ratio range between 3 and 4 for gold nanorods for sensing applications. A further sensitivity increase can only be reached by chemical modifications of the gold nanorods. This can be achieved by synthesizing an additional porous gold cage around the nanorods, resulting in a plasmon sensitivity raise of up to 50 % for those “nanorattles” compared to gold nanorods with the same resonance wavelength. Another possibility isrnto coat the gold nanorods with a thin silver shell. This reduces the single particle’s resonance spectral linewidth about 30 %, which enlarges the resolution of the observable shift. rnrnThis silver coating evokes the interesting effect of reducing the ensemble plasmon linewidth by changing the relation connecting particle shape and plasmon resonance wavelength. This change, I term plasmonic focusing, leads to less variation of resonance wavelengths for the same particle size distribution, which I show experimentally and theoretically.rnrnIn a system of two coupled nanoparticles, the plasmon modes of the transversal and longitudinal axis depend on the refractive index of the environmental solution, but only the latter one is influenced by the interparticle distance. I show that monitoring both modes provides a self-calibrating system, where interparticle distance variations and changes of the environmental refractive index can be determined with high precision.

Plasmonic nanorods and nanoparticle-assemblies

Die vorliegenden Dissertation beschäftigt sich mit plasmonischen Nanopartikeln, deren Wechselwirkung mit Licht in einer Plasmonenschwingung resultiert. Suspensionen dieser Partikel zeigen kräftige Farben, da sich die Resonanzfrequenz der Plasmonenschwingung im sichtbaren Bereich des elektromagnetischen Spektrum befindet. Durch die Veränderung interner (Material, Größe, Form) oder externer Parameter (Brechungsindex der Umgebung, Abstand zu anderen plasmonischen Partikeln) lässt sich die Farbe der Partikel verändern, eine Verschiebung der Resonanzfrequenz kann beobachtet werden. Ihre Sensitivität gegenüber äußeren Bedingungen ist der Grund, weshalb plasmonische Nanopartikel als Sensoren eingesetzt werden können. Wichtig ist hierbei nicht nur, dass die Partikel eine hohe Sensitivität zeigen, sondern auch die Möglichkeit, reproduzierbar Partikel zu synthetisieren, die experimentellen Anforderungen entsprechen. In der vorliegenden Arbeit wird das Wachstum von reinen Gold- und mit Silber beschichteten Goldnanostäbchen untersucht. Des Weiteren werden plasmonische Nanopartikel als Orientierungs-, Brechungsindex- und Abstandssensoren verwendet. Die Synthese von Goldnanostäbchen erfolgt auf nasschemischen Weg, ihr anisotropes Wachstum aus isotropen Keimen wird durch zahlreiche Faktoren beeinflusst. In diesem Zusammenhang wurde ein Wachstumsmodell entwickelt, das neben dem Vorhandensein eines Stabilisators auch die Rolle von Bromid- und Silberionen herausstellt, die durch selektive Adsorption das Wachstum bestimmter Kristallflächen inhibieren. Zudem konnte gezeigt werden, dass die Potentialdifferenz zwischen Reduktions- und Oxidationsmittel klein sein muss, um ein langsames selektives Wachstum zu gewährleisten. rnDurch das Aufwachsen einer dünnen Silberschicht auf Goldnanostäbchen verbessert sich deren Qualität im Bezug auf die heterogene Linienbreite. Der “Plasmonic Focusing Effect”, die Änderung der Steigung des linearen Zusammenhangs von Plasmonenresonanz und Aspektverhältnis, konnte theoretisch berechnet und experimentell verifiziert werden. Durch die Aufnahme zeitaufgelöster Spektren und die Untersuchung des Verlaufs der Reaktion wurden sowohl Reaktionsordnung, als auch Aktivierungsenergie ermittelt. Das so gefundene kinetische Model erlaubt zudem die Vorhersage des Reaktionsprodukts zu verschiedenen Zeiten. rnEinzelne Goldnanostäbchen wurden in einer Gelmatrix bei verschiedenen Temperaturen untersucht, die Aufnahme der zeitlichen Variation der polarisationsabhängigen Streuintensität konnte genutzt werden, um den Kollaps des Gels zu charakterisieren. Neben der Verwendung einzelner plasmonischer Nanopartikel wurden auch Dimere, bestehend aus zwei Goldnanokugeln, untersucht. Nach der Kalibrierung der Resonanzfrequenz gegenüber des Abstandes der beiden Partikel durch externe Methoden (Lichtstreuung, Cryo- Elektronenmikroskopie) wurde der so gefundene exponentielle Zusammenhang verwendet, um sowohl den Brechungsindex der Umgebung, als auch den Abstand der beiden Goldnanokugeln zu bestimmen. Des Weiteren wurden Goldnanopartikeldimere benutzt, um ein als Linker verwendetes thermoresponsives Elastin-Polymer bei verschiedenen Temperaturen zu charakterisieren. Neben Aggregaten aus zwei Goldnanokugeln wurden auch so genannte “core-satellite” Strukturen synthetisiert, die um einen großen Goldnanopartikelkern viele kleine Goldnanopartikel tragen. Diese Partikel haben eine theoretisch vorhergesagte höhere Sensitivität gegenüber Brechungsindexänderungen, was in ersten Experimenten gezeigt werden konnte.

Photokathoden mit internem DBR-Reflektor als Quellen hochintensiver spinpolarisierter Elektronenstrahlen

Die Erzeugung von Elektronenstrahlen hoher Intensität (I$geq$2,mA) und hoher Spinpolarisation (P$geq$85%) ist für die Experimente an den geplanten glqq Linac Ringgrqq Electron--Ion--Collidern (z.B. eRHIC am Brookhaven National Laboratory) unabdingbar, stellt aber zugleich eine enorme Herausforderung dar. Die Photoemission aus ce{GaAs}--basierten Halbleitern wie z.B. den in dieser Arbeit untersuchten GaAlAs/InGaAlAs Quanten--Übergittern zeichnet sich zwar durch eine hohe Brillanz aus, die geringe Quantenausbeute von nur ca. 1% im Bereich maximaler Polarisation erfordert jedoch hohe Laserintensitäten von mehreren Watt pro $text{cm}^{2}$, was erhebliche thermische Probleme verursacht. rnrnIn dieser Arbeit konnte zunächst gezeigt werden, dass die Lebensdauer einer Photokathode mit steigender Laserleistung bzw. Temperatur exponentiell abnimmt. Durch Einbringen eines DBR--Spiegels zwischen die aktive Zone der Photokathode und ihr Substrat wird ein Großteil des ungenutzten Laserlichts wieder aus dem Kristall herausreflektiert und trägt somit nicht zur Erwärmung bei. Gleichzeitig bildet der Spiegel zusammen mit der Grenzfläche zum Vakuum eine Resonator--Struktur aus, die die aktive Zone umschließt. Dadurch kommt es für bestimmte Wellenlängen zu konstruktiver Interferenz und die Absorption in der aktiven Zone erhöht sich. Beide Effekte konnten durch vergleichenden Messungen an Kathoden mit und ohne DBR--Spiegel nachgewiesen werden. Dabei ergibt sich eine gute Übereinstimmung mit der Vorhersage eines Modells, das auf der dielektrischen Funktion der einzelnen Halbleiterstrukturen beruht. Von besonderer praktischer Bedeutung ist, dass die DBR--Kathode für einen gegebenen Photoemissions-strom eine um einen Faktor $geq$,3{,}5 kleinere Erwärmung aufweist. Dies gilt über den gesamten Wellenlängenbereich in dem die Kathode eine hohe Strahlpolarisation (P$>$80%) produzieren kann, auch im Bereich der Resonanz.rnAus zeitaufgelösten Messungen der Ladungsverteilung und Polarisation lassen sich sowohl Rückschlüsse über die Transportmechanismen im Inneren einer Kathode als auch über die Beschaffenheit ihrer Oberfläche ziehen. Im Rahmen dieser Dissertation konnte die Messgeschwindigkeit der verwendeten Apparatur durch den Einbau eines schnelleren Detektors und durch eine Automatisierung der Messprozedur entscheidend vergrößert und die resultierende Zeitauflösung mit jetzt 1{,}2 Pikosekunden annähernd verdoppelt werden.rnrnDie mit diesen Verbesserungen erhaltenen Ergebnisse zeigen, dass sich der Transport der Elektronen in Superlattice--Strukturen stark vom Transport in den bisher untersuchten Bulk--Kristallen unterscheidet. Der Charakter der Bewegung folgt nicht dem Diffusionsmodell, sondern gibt Hinweise auf lokalisierte Zustände, die nahe der Leitungsbandunterkante liegen und Elektronen für kurze Zeit einfangen können. Dadurch hat die Impulsantwort einer Kathode neben einem schnellen Abfall des Signals auch eine größere Zeitkonstante, die selbst nach 30,ps noch ein Signal in der Größenordnung von ca. 5textperthousand der Maximalintensität erzeugt.

Biological Signatures of Vaccine Responses

The set of host- and pathogen-specific molecular features of a disease comprise its “signature”. We hypothesize that biological signatures enables distinctions between vaccinated vs. infected individuals. In our research, using porcine samples, protocols were developed that could also be used to identify biological signatures of human disease. Different classes of molecular features will be tested during this project, including indicators of basic immune capacity, which are being studied at this instance. These indicators of basic immune response such as porcine cytokines and antibodies were validated using Enzyme-linked immunosorbent assay (ELISA). This is an established method that detects antigens by their interaction with a specific antibody coupled to a polystyrene substrate. Serum from naïve and vaccinated pigs was tested for the presence of cytokines. We were able to differentiate the presence of porcine IL-6 in normal porcine serum with or without added porcine IL-6 by ELISA. In addition, four different cytokines were spotted on a grating-coupled surface plasmon resonance imaging system (GCSPRI) chip and antibody specific for IL-8 was run over the chip. Only the presence of IL-8 was detected; therefore, there was no cross-reactivity in this combination of antigens and antibodies. This system uses a multiplexed sensor chip to identify components of a sample run over it. The detection is accomplished by the change in refractive index caused by the interaction between the antibody spotted on the sensor chip and the antigen present in the sample. As the multiplexed GCSPRI is developed, we will need to optimize both sensitivity and specificity, minimizing the potential for cross-reactivity between individual analytes. The next step in this project is to increase the sensitivity of detection of the analytes. Currently, we are using two different antibodies (that recognize a different part of the antigen) to amplify the signal emitted by the interaction of antibody with its cognate antigen. The development of this sensor chip would not only allow to detect FMD virus, but also to differentiate between infected and vaccinated individuals, on location. Furthermore, the diagnosis of other diseases could be done with increased accuracy, and in less time due to the microarray approach.

(Table 1) Stratigraphic position, abundance, shape, color, and refractive indices of glass shards of selected samples from DSDP Leg 60 Holes vitric muds

Precisely determined refractive indices of glass shards from 32 ash-rich, volcaniclastic sediments, mostly turbidites interbedded with nonvolcanic sediments in the Mariana Trough, range from 1.480 to 1.585 (corresponding to SiO2 ca. 75 to 49%), with most in the range 1.500 to 1.540 (SiO2 ca. 70-62%) and a second, smaller mode between ca. 1.560 and 1.585 (57 to 49% SiO2). Shards are almost exclusively colorless from 1.480 to ca. 1.530, light brown with minor colorless and green tones between 1.530 and 1.560, and dominantly brown at higher refractive indices. Tubular pumice shards are more common at higher silica percentages and non- to poorly-vesicular cuniform shards at low SiO2 values, but there is no clear correlation between shape and composition of shards. About half of the samples have bimodal shard populations with silica differences ranging up to 20 percent; unimodal layers have a range of up to about 7 percent SiO2. Of 21 samples in which one type of shard dominates, seven have the main mode in the rhyolitic composition (>69% SiO2), eight in the intermediate range (56 to 69% SiO2), and five in mafic composition (SiO2 <53%). These unusually abundant mafic shards occur mainly in site survey piston cores, SP-IA and 4E, and in Holes 454, 456, 458, and 459B. These are the sites closest to the present arc. Hole 453, containing by far the most vitric tuff turbidites, shows a gradual increase in silica content of ash layers upward to the hole from Cores 36 to 19 (about 4.6 to 3.0 Ma). A drastic decrease in ash-rich beds in the younger (Pleistocene) part of this hole was noted by the shipboard party (see site chapter, Site 453) and was interpreted by them as indicating increasing distance from the arc volcanoes as the trough opened. The increase in silica in ashes from the early to the late Pliocene at Site 453 could be interpreted in the same way and might indicate that the trough started to open in early Pliocene time.

Mineralogical and isotopic compositions of ODP Hole 107-650A

Red-brown dolomitic claystones overlay the Marsili Basin basaltic basement at ODP Site 650. Sequential leaching experiments reveal that most of the elements considered to have a hydrothermal or hydrogenous origin in a marine environment, such as Fe, Cu, Zn, Pb, Co, Ni, are present mainly in the aluminosilicate fraction of the dolomitic claystones. Their vertical distribution, content and partitioning chemistry of trace elements, and REE patterns suggest enhanced terrigenous input during dolomite formation, but no significant hydrothermal influence from the underlying basaltic basement. Positive correlations in the C and O isotopes in the dolomites reflect complex conditions during the dolomitization. The stable isotopes can be controlled in part by temperature variations during the dolomitization. Majority of the samples, however, form a trend that is steeper than expected for only temperature control on the C and O isotopes. The latter indicates possible isotopic heterogeneity in the proto-carbonate that can be related to arid climatic conditions during the formation of the basal dolomitic claystones. In addition, the dolostones stable isotopic characteristics can be influenced by diagenetic release of heavier delta18O from clay dehydration and/or alteration of siliciclastic material. Strontium and Pb isotopic data reveal that the non-carbonate fraction, the "dye" of the dolomitic claystones, is controlled by Saharan dust (75%-80%) and by material with isotopic characteristics similar to the Aeolian Arc volcanoes (20%-25%). The non-carbonate fraction of the calcareous ooze overlying the dolomitic claystones has a Sr and Pb isotopic composition identical to that of the dolomitic claystones, indicating that no change in the input sources to the sedimentary basin occurred during and after the dolomitization event. Combination of climato-tectonic factors most probably resulted in suitable conditions for dolomitization in the Marsili and the nearby Vavilov Basins. The basal dolomitic claystone sequence was formed at the initiation of the opening of the Marsili Basin (~2 Ma), which coincided with the consecutive glacial stage. The glaciation caused arid climate and enhanced evaporation that possibly contributed to the stable isotope variations in the proto-carbonate. The conductive cooling of the young lithosphere produced high heat flow in the region, causing low-temperature passive convection of pore waters in the basal calcareous sediment. We suggest that this pumping process was the major dolomitization mechanism since it is capable of driving large volumes of seawater (the source of Mg2+) through the sediment. The red-brown hue of the dolomitic claystones is terrigenous contribution of the glacially induced high eolian influx and was not hydrothermally derived from the underlying basaltic basement. The detailed geochemical investigation of the basal dolomitic sequence indicates that the dolomitization was most probably related to complex tectono-climatic conditions set by the initial opening stages of the Marsili Basin and glaciation.

Design of spherical symmetric gradient index lenses

Spherical symmetric refractive index distributions also known as Gradient Index lenses such as the Maxwell-Fish-Eye (MFE), the Luneburg or the Eaton lenses have always played an important role in Optics. The recent development of the technique called Transformation Optics has renewed the interest in these gradient index lenses. For instance, Perfect Imaging within the Wave Optics framework has recently been proved using the MFE distribution. We review here the design problem of these lenses, classify them in two groups (Luneburg moveable-limits and fixed-limits type), and establish a new design techniques for each type of problem.

On the design of spherical gradient index lenses

Classical spherical gradient index (GRIN) lenses (such as Maxwell Fish Eye lens, Eaton lens, Luneburg lens, etc.) design procedure using the Abel integral equation is reviewed and reorganized. Each lens is fully defined by a function called the angle of flight which describes the ray deflection through the lens. The radial refractive index distribution is obtained by applying a linear integral transformation to the angle of flight. The interest of this formulation is in the linearity of the integral transformation which allows us to derive new solutions from linear combinations of known lenses. Beside the review of the classical GRIN designs, we present a numerical method for GRIN lenses defined by the Abel integral equation with fixed limits, which is an ill-posed problem.