Wirbeltiere sind in der Lage, die Brechkraft ihres optischen Systems dynamisch so anzupassen, dass Objekte aus verschiedenen Entfernungen fokussiert auf der Netzhaut abgebildet werden. Diese Fähigkeit bezeichnet man als Akkommodation. Je nach Lebensraum und visuellem Anspruch haben sich evolutionär sehr unterschiedliche Akkommodationsmechanismen entwickelt. Die größten Unterschiede ergeben sich aufgrund der verschiedenen Umgebungsmedien zwischen terrestrischen, aquatischen und amphibisch lebenden Wirbeltieren. Die grundlegende Funktionsweise der Mechanismen ist qualitativ bekannt, jedoch liegen detaillierte Untersuchungen, wie sich die optischen Komponenten und die Abbildungsqualität während der Akkommodation verändern, nur zum menschlichen Auge vor. In der vorliegenden Arbeit wurde mittels der Methode des Optikdesigns, ein naturnahes Augenmodell eines aquatischen (Goldfisch) und amphibisch lebenden Wirbeltieres (Kappensäger) modelliert und simuliert, welches die quantitative Untersuchung des jeweiligen Akkommodationsmechanismus ermöglicht. Durch die Modelle konnten neue Erkenntnisse zum Akkommodationsvermögen aquatischer und amphibisch lebender Wirbeltiere gewonnen werden. Die bereits bestehenden Kenntnisse zur Akkommodation und Abbildungsqualität konnten auf Plausibilität geprüft werden. Weiterhin erlauben die Augenmodelle einen detaillierten Einblick, wie sich die geometrisch optischen Parameter sowie die theoretische Abbildungsqualität während der Akkommodation verändern. Zudem wird sichtbar, welchen Einfluss die einzelnen Parameter haben und wie sie voneinander abhängen.:Abkürzungsverzeichnis III 1 Einführung . . . . . . . . . 1 1.1 Motivation . . . . . . . . . 2 1.2 Grundlagen zur Akkommodation . . . . . . . . . 4 1.3 Anatomische und physiologische Besonderheiten der Akkommodation bei verschiedenen Wirbeltieraugen . . . . . . . . . 7 1.4 Optikdesign . . . . . . . . . 13 1.5 Aufgaben- und Zielstellung . . . . . . . . . 17 2 Publikationsmanuskripte . . . . . . . . . 19 2.1 Comprehensive optical design model of the goldfish eye and quantitative simulation of the consequences on the accommodation mechanism . . . . . . . . . 19 2.2 Amphibious vision - Optical design model of the hooded merganser eye . . . . . . . . . 27 3 Zusammenfassung . . . . . . . . . 39 Literaturverzeichnis . . . . . . . . . 50 Abbildungsverzeichnis . . . . . . . . . 51 Tabellenverzeichnis . . . . . . . . . 53 4 Anhang . . . . . . . . . 55 4.1 Daten zum Liou-Brennan-Augenmodell . . . . . . . . . 55 Selbständigkeitserklärung . . . . . . . . . 61 Publikationen . . . . . . . . . 63 Danksagung . . . . . . . . . 65

The dissertation presents theoretical and experimental studies on the physical origin of the signal in photothermal microscopy of single particles. This noninvasive optical far field microscopy scheme allows the imaging and detection of single absorbing nanoparticles. Based on a heat-induced pertur- bation in the refractive index in the embedding medium of the nanoscopic absorber, a corresponding probe beam modification is measured and quantified. The method is well established and has been applied since its first demonstration in 2002 to the imaging and characterization of various absorbing particle species, such as quantum dots, single molecules and nanoparticles of different shapes. The extensive theoretical developments presented in this thesis provide the first quantitative assess- ment of the signal and at the same time enlarge its phenomenology and thereby its potential. On the basis of several approximation schemes to the Maxwell equations, which fundamentally gov- ern the interaction of light with inhomogeneities, several complementing models are devised which describe the photothermal signal both qualitatively and quantitatively. In succession an interdepen- dent and self-consistent set of theoretical descriptions is given and allows important experimental consequences to be drawn. In consequence, the photothermal signal is shown to correspond to the action of a nanoscopic (thermal) lens, represented by the spherically symmetric refractive index pro- file n(r) which accompanies the thermal expansion of the absorber’s environment. The achieved quantification allows the direct measurement of absorption cross-sections of nanoparticles. Further, a qualitatively new phenomenology of the signal is unraveled and experimentally demonstrated. The separate roles of the probing and the heating beams in photothermal microscopy is dismantled and the influence of their relative alignment shown to allow for a controlled adjustment of the effective detection volume. For the first time, both positive and negative signals are demonstrated to occur and to be the characteristic signature of the lens-like action on the probe beam. The detection of the probe beam’s modification is also shown to sensitively depend on the aperture used in the detection chan- nel, and a signal optimization is shown to be feasible. Also, a generalization of the detectable signal via the use of a quadrant photodiode is achieved. Specifically, measuring the far field beam deflec- tion the result of the beam passing the lens off-center manifests in a laterally split detection volume. Hereby, finally each classical photothermal spectroscopic techniques has been shown to possess its microscopic counterpart. Central to the understanding of this generalized and new phenomenology is a scalar wave-optical model which draws an analogy between the scattering of a massive particle wave-packet by a Coulomb potential and the deflection of a focused beam by a photonic potential connected with the thermal lens. The significance of the findings is demonstrated by its methodological implications on photother- mal correlation spectroscopy in which the diffusion dynamics of absorbing colloidal particles can be studied. The unique split focal detection volumes are shown to allow the sensitive measurement of a deterministic velocity field. Finally, the method is supplemented by a newly introduced sta- tistical analysis method which is capable of characterizing samples containing a heterogeneous size distribution.

Diese Dissertation demonstriert die Nutzung von einzelnen optisch geheizten Goldnanopartikeln als Sensoren f ¨ur die Untersuchung von W¨armetransport und als Intensit¨atsmodulator f ¨ur Licht. Die beschriebenen Experimente basieren auf der photothermischen Mikroskopie, die die selektive Abbildung and Untersuchung von einzelnen absorbierenden Objekten erm¨oglicht. Goldnanopartikel werden optisch angeregt. Die Relaxation erfolgt durch nichtstrahlende Prozesse, die zu einer lokalen Erh¨ohung der Temperatur f ¨uhren. Die Erw¨armung f ¨uhrt zu einer Verringerung der Brechzahl, die als thermische Linse wirkt und dadurch die Ausbreitung eines zweiten nicht absorbierten Lichtstrahls vera¨ndert. Da die thermische A¨ nderung der Brechzahl sehr gering ist, wird das photothermische Signal durch das moduliertes Detektionsverfahren verst¨arkt. Der Heizlaserstrahl wird intensit¨atsmoduliert und erzeugt dadurch eine geringe Modulation der Strahlbreite des Detektionslaserstahls. Damit ver¨andert sich die Leistung, die durch eine Blende transmittiert wird. Diese Modulationsamplitude and Phaseverz¨ogerung werden mittels eines phasenempfindlichen Gleichrichters detektiert. Amplitude und Phase h¨angen von Modulationsfrequenz und thermischer Diffusivit¨at ab. Die frequenzaufgel¨oste Messung der beiden Gr¨oßen und deren Modellierung mittels einer verallgemeinerten Lorenz–Mie Theorie erm¨oglicht die Messung von der thermischen Diffusivit¨at des Mediums, das das Goldnanopartikel umgibt. In der zweiten Variante wird die Ausbreitung der W¨arme beobachtet. Ein Nanopartikel wird optisch geheizt und die ausgedehnte thermische Linse wird mit Hilfe der Ablenkung eines zweiten Laserstrahls vermessen. Das Ablenkungssignal wird mittels eines strahlenoptischen Models berechnet, um die thermische Diffusivit ¨at des Materials zu bestimmen, das das Nanopartikel umgibt. In einem weiteren Experiment wird das große Potential von optisch geheizten Nanopartikeln verdeutlicht. Einzelne Goldnanopartikel werden in eine d¨unne nematische Fl¨ussigkristallschicht eingebettet, deren Dicke darauf abgestimmt ist, dass die Schicht eine l/2-Platte darstellt. Die Goldnanopartikel werden optisch geheizt und steuern damit den Phasen¨ubergang von der nematischen zur isotropen Phase. Damit wird die Transmission eines zweiten Laserstrahls im Polarisationskontrast ge¨andert. Mit Hilfe dieser Anordnung kann die Intensit¨at eines Lichtstrahls um bis zu 100% moduliert werden. / This dissertation demonstrates the use of individual optically heated gold nanoparticles as sensors for investigations of heat transport and intensity modulation of light. The experiments employ the photothermal effect, which allows the selective detection and investigation of individual absorbers. The photothermal contrast is based on absorbing particles that are optically excited and relax via nonradiative processes. The absorbers act as nanosources of heat. The local temperature elevation leads to a local refractive index change due to thermal expansion which then acts as a lens. This thermal lens alters the propagation of a second non-absorbed beam of light. As the refractive index change with temperature is minuscule, the transmission changes of the detection are tiny as well. The photothermal signal is amplified by the use of a modulated detection scheme which enables the methods high sensitivity and provides a time scale for the measurement of thermal transport. The heating laser beam is intensity-modulated and thereby produces a small modulation of the beam waist of the detection laser beam and thus the transmitted power through an aperture. This modulation amplitude and phase are detected by a lock-in amplifier. Amplitude and phase depend on the modulation frequency and the thermal diffusivity of the material surrounding the nanoparticle. The frequency-resolved measurement of the two observables and their modeling using a generalized Lorenz–Mie theory allows the measurement of thermal diffusivities. In the second variant, the spread of heat into space is observed. A nanoparticle is optically heated, and the extended thermal lens is characterized by the deflection of a second laser beam. The deflection signal is modeled using ray optics to determine the thermal diffusivity of the material surrounding the nanoparticle. In a further experiment, the great potential of optically heated nanoparticles is demonstrated. Individual gold nanoparticles are embedded in a thin nematic liquid-crystal layer acting as a half-wave plate. The gold particles are optically heated. They control the transmission of a detection laser set up in polarization contrast. The intensity of the detection beam is modulated by up to 100%.

We present a HiLo microscope with an electrically tunable lens for high-contrast three-dimensional image acquisition. HiLo microscopy combines widefield and speckled illumination images to create optically sectioned images. Additionally, the depth-of-field is not fixed, but can be adjusted between widefield and confocal-like axial resolution. We incorporate an electrically tunable lens in the HiLo microscope for axial scanning, to obtain three-dimensional data without the need of moving neither the sample nor the objective. The used adaptive lens consists of a transparent polydimethylsiloxane (PDMS) membrane into which an annular piezo bending actuator is embedded. A transparent fluid is filled between the membrane and the glass substrate. When actuated, the piezo generates a pressure in the lens which deflects the membrane and thus changes the refractive power. This technique enables a large tuning range of the refractive power between 1/f = (-24 . . . 25) 1/m. As the NA of the adaptive lens is only about 0.05, a fixed high-NA lens is included in the setup to provide high resolution. In this contribution, the scan properties and capabilities of the tunable lens in the HiLo microscope are analyzed. Eventually, exemplary measurements are presented and discussed.

No description available.

The dissertation presents theoretical and experimental studies on the physical origin of the signal in photothermal microscopy of single particles. This noninvasive optical far field microscopy scheme allows the imaging and detection of single absorbing nanoparticles. Based on a heat-induced pertur- bation in the refractive index in the embedding medium of the nanoscopic absorber, a corresponding probe beam modification is measured and quantified. The method is well established and has been applied since its first demonstration in 2002 to the imaging and characterization of various absorbing particle species, such as quantum dots, single molecules and nanoparticles of different shapes. The extensive theoretical developments presented in this thesis provide the first quantitative assess- ment of the signal and at the same time enlarge its phenomenology and thereby its potential. On the basis of several approximation schemes to the Maxwell equations, which fundamentally gov- ern the interaction of light with inhomogeneities, several complementing models are devised which describe the photothermal signal both qualitatively and quantitatively. In succession an interdepen- dent and self-consistent set of theoretical descriptions is given and allows important experimental consequences to be drawn. In consequence, the photothermal signal is shown to correspond to the action of a nanoscopic (thermal) lens, represented by the spherically symmetric refractive index pro- file n(r) which accompanies the thermal expansion of the absorber’s environment. The achieved quantification allows the direct measurement of absorption cross-sections of nanoparticles. Further, a qualitatively new phenomenology of the signal is unraveled and experimentally demonstrated. The separate roles of the probing and the heating beams in photothermal microscopy is dismantled and the influence of their relative alignment shown to allow for a controlled adjustment of the effective detection volume. For the first time, both positive and negative signals are demonstrated to occur and to be the characteristic signature of the lens-like action on the probe beam. The detection of the probe beam’s modification is also shown to sensitively depend on the aperture used in the detection chan- nel, and a signal optimization is shown to be feasible. Also, a generalization of the detectable signal via the use of a quadrant photodiode is achieved. Specifically, measuring the far field beam deflec- tion the result of the beam passing the lens off-center manifests in a laterally split detection volume. Hereby, finally each classical photothermal spectroscopic techniques has been shown to possess its microscopic counterpart. Central to the understanding of this generalized and new phenomenology is a scalar wave-optical model which draws an analogy between the scattering of a massive particle wave-packet by a Coulomb potential and the deflection of a focused beam by a photonic potential connected with the thermal lens. The significance of the findings is demonstrated by its methodological implications on photother- mal correlation spectroscopy in which the diffusion dynamics of absorbing colloidal particles can be studied. The unique split focal detection volumes are shown to allow the sensitive measurement of a deterministic velocity field. Finally, the method is supplemented by a newly introduced sta- tistical analysis method which is capable of characterizing samples containing a heterogeneous size distribution.:Contents Bibliographic description Abbreviations 1 Introduction 2 Theoretical Background 2.1 The current literature on the subject of the photothermal signal 2.2 Thermal conduction, and the temperature field around heated nanoparticles 2.3 The linear thermo-refractive response and the thermal lens 2.4 MAXWELL equations and approximation schemes 2.4.1 The MAXWELL equations 2.4.2 HELMHOLTZ equations 2.4.3 Paraxial HELMHOLTZ equation for the field components 2.4.4 Geometrical optics and the eikonal ansatz 2.5 Diffraction and the optical resolution limit in far field microscopy 2.5.1 Transmission scanning microscopy 2.5.2 Point spread functions and aberrations 2.5.3 Scalar diffraction approximation for weakly focused beams 2.5.4 Vectorial diffraction for highly focused electromagnetic fields 2.5.5 Theoretical description of transmission signals 2.6 Elastic scattering of light 2.6.1 Overview of optical elastic scattering theory 2.6.2 The integral equation of potential scattering and the BORN approximation 2.6.3 The generalized LORENZ-MIE theory 2.6.4 The electromagnetic fields 2.6.5 Description of the incident field: beam shape coefficients 2.6.6 Multilayered scatterers 2.6.7 POYNTING vector and field decomposition 2.6.8 Energy balance & total cross-sections 2.6.9 Optical theorem & the extinction paradox 2.6.10 Small particle scattering: the RAYLEIGH-limit 2.7 Optical properties of gold nanoparticles & Surface plasmon resonances 2.7.1 Dielectric function of gold 2.7.2 Total cross-sections of plasmonic nanoparticles properties of gold nanoparticles & Surface plasmon resonances 2.8 (Hot) BROWNian motion, diffusion and their statistical analysis 2.8.1 (Hot) BROWNian motion 2.8.2 Diffusion and correlation analysis 2.8.3 Methods regarding the signal statistics of diffusing tracer particles 2.9 RUTHERFORD scattering of charged particles 2.9.1 Classical RUTHERFORD scattering 2.9.2 Quantum mechanical COULOMB scattering 3 Experimental Setup 3.1 Sample preparation 3.2 Photothermal microscopy setup 4 Photothermal Imaging: Results and Discussion 4.1 MAXWELL equations: Exact treatment of the PT signal 4.1.1 Angularly resolved powers: Fractional cross-sections 4.1.2 Incident power and background normalization 4.1.3 Fractional scattering and extinction cross-sections (off-axis) 4.1.4 Fractional scattering and extinction cross-sections (on-axis) 4.1.5 Small particle approximation(on-axis) 4.1.6 General properties of transmission scans 4.1.7 The thermal lens n(r) in the MIE-scattering framework 4.1.8 The photothermal signal F in the MIE scattering framework 4.2 Geometrical optics: Photonic RUTHERFORD scattering (ray optics) 4.2.1 FERMAT’s principle for a thermal lens medium 4.2.2 Gaussian beam transformation by a thermal lens 4.2.3 Experiments using weakly focused, i.e. nearly Gaussian beams 4.3 HELMHOLTZ equation: Photonic RUTHERFORD scattering (wave optics) 4.3.1 Plane-wave scattering 4.3.2 Focused beam scattering 4.3.3 Connection to the far field 4.3.4 Photothermal Rutherford scattering microscopy 4.3.5 Photothermal half-aperture measurements 4.4 Paraxial HELMHOLTZ equation: FRESNEL diffraction by a thermal lens 4.4.1 The diffraction integral and the phase mask for a thermal lens 4.4.2 The photothermal signal expressed via the image plane field 4.4.3 Experimental demonstration of the signal inversion 4.4.4 Connection to photothermal RUTHERFORD scattering 4.5 Plane-wave extinction & scattering by a thermal lens 4.5.1 The BORN approximation for the ideal and time-dependent thermal lens 4.5.2 The eikonal approximation for the ideal thermal lens and x>>1 4.5.3 Lessons to be learned from plane-wave scattering by thermal lenses 4.6 What is a lens? And is n(r) a lens? 5 Methodological Applications of the Results 5.1 Generalized photothermal correlation spectroscopy (incl. twin-PhoCS) 5.2 Photothermal signal distribution analysis (PhoSDA) 6 Summary and Outlook 6.1 Summary of the results 6.2 Outlook 7 Appendix 7.1 Material parameters 7.2 Calculation parameters 7.3 Interactive simulation scripts (Processing) 7.4 Vectorial scattering in the BORN-approximation 7.5 Details regarding the scattering framework 7.5.1 Connection between Gmn,TE,TM of Ref.1 and gmn,TE,TM in the GLMT 7.5.2 Off-axis BSCs including aberration (single interface) 7.5.3 Details on the incidence power Pinc 7.5.4 Details on the incidence power Pinc for arbitrary beams 7.5.5 Explicit expressions for the spherical field components of Es,i and Hs,i 7.5.6 Note on the time-dependence and the corresponding sign-conventions in M 7.5.7 Recurrence relation for Pn and tn 7.5.8 Gaussian beam shape coefficients: Off-axis 7.5.9 Multilayered Scatterer 7.5.10 POYNTING-vector and energy flow fields 7.5.11 Convergence 7.5.12 Further evaluations in the GLMT framework 7.5.13 Diffraction model: Comparison of angular PT signal pattern to the GLMT 7.6 Details on geometrical optics models 7.6.1 Geometrical optics: Exact solution r(f) for |bx|<1 7.6.2 Correspondences in photonic and partile RUTHERFORD scattering 7.6.3 On the difference in the definition of optical energy 7.6.4 Ray-opticsphotothermalsignal 7.6.5 Thick lens raytracing and the equivalent lens shape for a given aberration 7.7 Thermal lens around a wire of radius R 7.8 Twin-PhoCS: Graphic illustration of the CCF integrand Curriculum Vitae Publications Declaration Acknowledgements List of Tables List of Figures Bibliography

The main problem addressed by this dissertation is the accurate and automated determination of electron microscope imaging conditions. This enables the restoration of the object wave, which confers direct structural information about the specimen, from sets of differently aberrated images. An important member in the imaging chain is the image recording device, in many cases now a charge-coupled device (CCD) camera. Previous characterisations of these cameras often relied on the unjustified assumption that the Modulation Transfer Function (MTF) also correctly describes the spatial frequency dependent attenuation of the electron shot noise. A new theory is therefore presented that distinguishes between signal and noise transfer. This facilitates the evaluation of both properties using a detailed Monte-Carlo simulation model for the electron and photon scattering in the scintillator of the camera. Furthermore, methods for the accurate experimental determination of the signal and noise transfer functions are presented. In agreement with the Monte-Carlo simulations, experimental results for commercially available CCD cameras show that the signal transfer is significantly poorer than the noise transfer. The centrepiece of this dissertation is the development of new methods for determining the relative aberrations in a set of images and the absolute symmetric aberrations in the restored wave. Both are based on the analysis of the phase information in the Fourier domain and give each Fourier component a weight independent of its strength. This makes the method suitable even for largely crystalline samples with little amorphous contamination, where conventional methods, such as automated diffractogram fitting, usually fail. The method is then extended to also cover the antisymmetric aberrations, using combined beam tilt and focal series. The applicability of the new method is demonstrated with object wave restorations from tilt and focal series of complex inorganic block oxides and of carbon nanotubes filled with one-dimensional inorganic crystals. The latter specimens allowed for the first time a direct comparison between the phase shift in the restored object wave of a specimen with precisely known thickness and the value predicted by simulations.

No description available.

The main problem addressed by this dissertation is the accurate and automated determination of electron microscope imaging conditions. This enables the restoration of the object wave, which confers direct structural information about the specimen, from sets of differently aberrated images. An important member in the imaging chain is the image recording device, in many cases now a charge-coupled device (CCD) camera. Previous characterisations of these cameras often relied on the unjustified assumption that the Modulation Transfer Function (MTF) also correctly describes the spatial frequency dependent attenuation of the electron shot noise. A new theory is therefore presented that distinguishes between signal and noise transfer. This facilitates the evaluation of both properties using a detailed Monte-Carlo simulation model for the electron and photon scattering in the scintillator of the camera. Furthermore, methods for the accurate experimental determination of the signal and noise transfer functions are presented. In agreement with the Monte-Carlo simulations, experimental results for commercially available CCD cameras show that the signal transfer is significantly poorer than the noise transfer. The centrepiece of this dissertation is the development of new methods for determining the relative aberrations in a set of images and the absolute symmetric aberrations in the restored wave. Both are based on the analysis of the phase information in the Fourier domain and give each Fourier component a weight independent of its strength. This makes the method suitable even for largely crystalline samples with little amorphous contamination, where conventional methods, such as automated diffractogram fitting, usually fail. The method is then extended to also cover the antisymmetric aberrations, using combined beam tilt and focal series. The applicability of the new method is demonstrated with object wave restorations from tilt and focal series of complex inorganic block oxides and of carbon nanotubes filled with one-dimensional inorganic crystals. The latter specimens allowed for the first time a direct comparison between the phase shift in the restored object wave of a specimen with precisely known thickness and the value predicted by simulations.

Traditional methods in optical design like ray tracing suffer from slow convergence and are not constructive, i.e., each minimal perturbation of input parameters might lead to “chaotic” changes in the output. However, so-called inverse methods can be helpful in designing optical systems of reflectors and lenses. The equations in R2 become ordinary differential equations, while in R3 the equations become partial differential equations. These equations are then used to transform source distributions into target distributions, where the distributions are arbitrary, though assumed to be positive and integrable. In this project, we derive the governing equations and solve them numerically, for the systems presented by our instructor Martijn Anthonissen [Anthonissen et al. 2021]. Additionally, we show how point sources can be derived as a special case of a interval source with di- rected source interval, i.e., with each point in the source interval there is also an associated unit direction vector which could be derived from a system of two interval sources in R2. This way, it is shown that connecting source distributions with target distributions can be classified into two instead of three categories. The resulting description of point sources as a source along an interval with directed rays could potentially be extended to three dimensions, leading to interpretations of point sources as directed sources on convex or star-shaped sets.:1 Abstract 4 2 Notation And Conventions 4 3 Introduction 5 4 ECMI Modeling Week Challenges 5 4.1 Problem 1 - Parallel to Near-Field Target 5 4.1.1 Description 5 4.1.2 Deriving The Equations 5 4.2 Problem 2 - Parallel Source To Two Targets 8 4.3 Problem 3 - Point Source To Near-Field Target 9 4.3.1 Deriving The Equations 9 4.4 Problem 4 - Point Source To Two Targets 11 5 Validation - Ray tracing 13 5.1 Splines 13 5.1.1 Piece-Wise Affine Reflectors 13 5.1.2 Piece-Wise Cubic Reflectors 14 5.2 Error Estimates For Spline Reflectors 14 5.2.1 Lemma: A Priori Feasibility Of Starting Values For Near-Field Problems 15 5.2.2 Estimates for single reflector, near-field targets 16 5.3 Ray Tracing Errors - Illumination Errors 17 5.3.1 Definition: Axioms For Errors 18 5.3.2 Extrapolated Ray Tracing Error (ERTE) 18 5.3.3 Definition: Minimal Distance Ray Tracing Error (MIRTE) 19 5.3.4 Lemma: Continuity Of The Ray Traced Reflection Projection Of Smooth Reflectors 19 5.3.5 Theorem: Convergence Of The MIRTE 20 5.3.6 Convergence Of The ERTE 21 5.3.7 Application 21 6 Numerical Implementation 21 6.1 The DOPTICS Library 21 6.2 Pseudocode Of The Implementation 21 6.2.1 Solutions Of The Problems 22 6.2.2 Ray Tracing And Ray Tracing Error 22 6.3 ERTE Implementation 25 7 Results 26 7.1 Problem 1: Results 26 7.2 Problem 2: Results 26 7.3 Problem 3: Results 27 7.4 Problem 4: Results 27 8 Generalizations In Two Dimensions 29 8.1 Directed Densities 29 8.2 Generalized, Orthogonally Emitting Sources in R2 30 8.2.1 Point Light Sources As Orthogonally Emitting Sources 30 9 Conclusion and Future Research 32 10 Group Dynamic 32 References 32