Adiabatic extraction models for beam dynamics

Il presente lavoro prende le mosse da un problema di dinamica dei fasci relativo a un metodo di separazione di un fascio di particelle recentemente messo in funzione nell’sincrotrone PS al CERN. In questo sistema, variando adiabaticamente i parametri di un campo magnetico, nello spazio delle fasi si creano diverse isole di stabilità (risonanze) in cui le particelle vengono catturate. Dopo una parte introduttiva in cui si ricava, a partire dalle equazioni di Maxwell, l’hamiltoniana di una particella sottoposta ai campi magnetici che si usano negli acceleratori, e una presentazione generale della teoria dell’invarianza adiabatica, si procede analizzando la dinamica di tali sistemi. Inizialmente si prende in considerazione l’hamiltoniana mediata sulle variabili veloci, considerando perturbazioni (kick) dei termini dipolare e quadrupolare. In ognuno dei due casi, si arriva a determinare la probabilità che una particella sia catturata nella risonanza. Successivamente, attraverso un approccio perturbativo, utilizzando le variabili di azione ed angolo, si calcola la forza della risonanza 4:1 per un kick quadrupolare.

Shaping transverse beam distributions by means of adiabatic crossing of resonances

Non-linear effects are responsible for peculiar phenomena in charged particles dynamics in circular accelerators. Recently, they have been used to propose novel beam manipulations where one can modify the transverse beam distribution in a controlled way, to fulfil the constraints posed by new applications. One example is the resonant beam splitting used at CERN for the Multi-Turn Extraction (MTE), to transfer proton beams from PS to SPS. The theoretical description of these effects relies on the formulation of the particle's dynamics in terms of Hamiltonian systems and symplectic maps, and on the theory of adiabatic invariance and resonant separatrix crossing. Close to resonance, new stable regions and new separatrices appear in the phase space. As non-linear effects do not preserve the Courant-Snyder invariant, it is possible for a particle to cross a separatrix, changing the value of its adiabatic invariant. This process opens the path to new beam manipulations. This thesis deals with various possible effects that can be used to shape the transverse beam dynamics, using 2D and 4D models of particles' motion. We show the possibility of splitting a beam using a resonant external exciter, or combining its action with MTE-like tune modulation close to resonance. Non-linear effects can also be used to cool a beam acting on its transverse beam distribution. We discuss the case of an annular beam distribution, showing that emittance can be reduced modulating amplitude and frequency of a resonant oscillating dipole. We then consider 4D models where, close to resonance, motion in the two transverse planes is coupled. This is exploited to operate on the transverse emittances with a 2D resonance crossing. Depending on the resonance, the result is an emittance exchange between the two planes, or an emittance sharing. These phenomena are described and understood in terms of adiabatic invariance theory.

Bunch frequency multiplication by RF injection into an isochronous ring

La collaboration CLIC (Compact LInear Collider, collisionneur linéaire compact) étudie la possibilité de réaliser un collisionneur électron-positon linéaire à haute énergie (3 TeV dans le centre de masse) et haute luminosité (1034 cm-2s-1), pour la recherche en physique des particules. Le projet CLIC se fonde sur l'utilisation de cavités accélératrices à haute fréquence (30 GHz). La puissance nécessaire à ces cavités est fournie par un faisceau d'électrons de basse énergie et de haute intensité, appelé faisceau de puissance, circulant parallèlement à l'accélérateur linéaire principal (procédé appelé « Accélération à Double Faisceau »). Dans ce schéma, un des principaux défis est la réalisation du faisceau de puissance, qui est d'abord généré dans un complexe accélérateur à basse fréquence, puis transformé pour obtenir une structure temporelle à haute fréquence nécessaire à l'alimentation des cavités accélératrices de l'accélérateur linéaire principal. La structure temporelle à haute fréquence des paquets d'électrons est obtenue par le procédé de multiplication de fréquence, dont la manipulation principale consiste à faire circuler le faisceau d'électrons dans un anneau isochrone en utilisant des déflecteurs radio-fréquence (déflecteurs RF) pour injecter et combiner les paquets d'électrons. Cependant, ce type de manipulation n'a jamais été réalisé auparavant et la première phase de la troisième installation de test pour CLIC (CLIC Test Facility 3 ou CTF3) a pour but la démonstration à faible charge du procédé de multiplication de fréquence par injection RF dans un anneau isochrone. Cette expérience, qui a été réalisée avec succès au CERN au cours de l'année 2002 en utilisant une version modifiée du pré-injecteur du grand collisionneur électron-positon LEP (Large Electron Positron), est le sujet central de ce rapport. L'expérience de combinaison des paquets d'électrons consiste à accélérer cinq impulsions dont les paquets d'électrons sont espacés de 10 cm, puis à les combiner dans un anneau isochrone pour obtenir une seule impulsion dont les paquets d'électrons sont espacés de 2 cm, multipliant ainsi la fréquence des paquets d'électrons, ainsi que la charge par impulsion, par cinq. Cette combinaison est réalisée au moyen de structures RF résonnantes sur un mode déflecteur, qui créent dans l'anneau une déformation locale et dépendante du temps de l'orbite du faisceau. Ce mécanisme impose plusieurs contraintes de dynamique de faisceau comme l'isochronicité, ainsi que des tolérances spécifiques sur les paquets d'électrons, qui sont définies dans ce rapport. Les études pour la conception de la Phase Préliminaire du CTF3 sont détaillées, en particulier le nouveau procédé d'injection avec les déflecteurs RF. Les tests de haute puissance réalisés sur ces cavités déflectrices avant leur installation dans l'anneau sont également décrits. L'activité de mise en fonctionnement de l'expérience est présentée en comparant les mesures faites avec le faisceau aux simulations et calculs théoriques. Finalement, les expériences de multiplication de fréquence des paquets d'électrons sont décrites et analysées. On montre qu'une très bonne efficacité de combinaison est possible après optimisation des paramètres de l'injection et des déflecteurs RF. En plus de l'expérience acquise sur l'utilisation de ces déflecteurs, des conclusions importantes pour les futures activités CTF3 et CLIC sont tirées de cette première démonstration de la multiplication de fréquence des paquets d'électrons par injection RF dans un anneau isochrone.<br/><br/>The Compact LInear Collider (CLIC) collaboration studies the possibility of building a multi-TeV (3 TeV centre-of-mass), high-luminosity (1034 cm-2s-1) electron-positron collider for particle physics. The CLIC scheme is based on high-frequency (30 GHz) linear accelerators powered by a low-energy, high-intensity drive beam running parallel to the main linear accelerators (Two-Beam Acceleration concept). One of the main challenges to realize this scheme is to generate the drive beam in a low-frequency accelerator and to achieve the required high-frequency bunch structure needed for the final acceleration. In order to provide bunch frequency multiplication, the main manipulation consists in sending the beam through an isochronous combiner ring using radio-frequency (RF) deflectors to inject and combine electron bunches. However, such a scheme has never been used before, and the first stage of the CLIC Test Facility 3 (CTF3) project aims at a low-charge demonstration of the bunch frequency multiplication by RF injection into an isochronous ring. This proof-of-principle experiment, which was successfully performed at CERN in 2002 using a modified version of the LEP (Large Electron Positron) pre-injector complex, is the central subject of this report. The bunch combination experiment consists in accelerating in a linear accelerator five pulses in which the electron bunches are spaced by 10 cm, and combining them in an isochronous ring to obtain one pulse in which the electron bunches are spaced by 2 cm, thus achieving a bunch frequency multiplication of a factor five, and increasing the charge per pulse by a factor five. The combination is done by means of RF deflecting cavities that create a time-dependent bump inside the ring, thus allowing the interleaving of the bunches of the five pulses. This process imposes several beam dynamics constraints, such as isochronicity, and specific tolerances on the electron bunches that are defined in this report. The design studies of the CTF3 Preliminary Phase are detailed, with emphasis on the novel injection process using RF deflectors. The high power tests performed on the RF deflectors prior to their installation in the ring are also reported. The commissioning activity is presented by comparing beam measurements to model simulations and theoretical expectations. Eventually, the bunch frequency multiplication experiments are described and analysed. It is shown that the process of bunch frequency multiplication is feasible with a very good efficiency after a careful optimisation of the injection and RF deflector parameters. In addition to the experience acquired in the operation of these RF deflectors, important conclusions for future CTF3 and CLIC activities are drawn from this first demonstration of the bunch frequency multiplication by RF injection into an isochronous ring.<br/><br/>La collaboration CLIC (Compact LInear Collider, collisionneur linéaire compact) étudie la possibilité de réaliser un collisionneur électron-positon linéaire à haute énergie (3 TeV) pour la recherche en physique des particules. Le projet CLIC se fonde sur l'utilisation de cavités accélératrices à haute fréquence (30 GHz). La puissance nécessaire à ces cavités est fournie par un faisceau d'électrons de basse énergie et de haut courant, appelé faisceau de puissance, circulant parallèlement à l'accélérateur linéaire principal (procédé appelé « Accélération à Double Faisceau »). Dans ce schéma, un des principaux défis est la réalisation du faisceau de puissance, qui est d'abord généré dans un complexe accélérateur à basse fréquence, puis transformé pour obtenir une structure temporelle à haute fréquence nécessaire à l'alimentation des cavités accélératrices de l'accélérateur linéaire principal. La structure temporelle à haute fréquence des paquets d'électrons est obtenue par le procédé de multiplication de fréquence, dont la manipulation principale consiste à faire circuler le faisceau d'électrons dans un anneau isochrone en utilisant des déflecteurs radio-fréquence (déflecteurs RF) pour injecter et combiner les paquets d'électrons. Cependant, ce type de manipulation n'a jamais été réalisé auparavant et la première phase de la troisième installation de test pour CLIC (CLIC Test Facility 3 ou CTF3) a pour but la démonstration à faible charge du procédé de multiplication de fréquence par injection RF dans un anneau isochrone. L'expérience consiste à accélérer cinq impulsions, puis à les combiner dans un anneau isochrone pour obtenir une seule impulsion dans laquelle la fréquence des paquets d'électrons et le courant sont multipliés par cinq. Cette combinaison est réalisée au moyen de structures déflectrices RF qui créent dans l'anneau une déformation locale et dépendante du temps de la trajectoire du faisceau. Les résultats de cette expérience, qui a été réalisée avec succès au CERN au cours de l?année 2002 en utilisant une version modifiée du pré-injecteur du grand collisionneur électron-positon LEP (Large Electron Positon), sont présentés en détail.

Untersuchungen zur Strahldynamik am Mainzer Mikrotron

Diese Arbeit untersucht die longitudinale und transversaleStrahldynamik am Mainzer Mikrotron MAMI. Die gemessenen Abbildungseigenschaften werden mit den Design-Rechnungen verglichen. Dadurch konnte die Strahlqualitaet von MAMI B und das Design der neuen HDSM Mikrotronstufe verbessert werden. Es wurde eine Stoerungsrechnung formuliert, um die 6-DAbbildungsmatrix entlang der Beschleunigungsstrecke zu berechnen. Ausgehend von der linearisierten Hamilton Funktion wird die Transfermatrix M in eine unendliche Summe ueber Matrizen M(n) zerlegt, die jeweils eine n-fache Wechselwirkung des Strahls mit dem Quadrupolanteil des Fuehrungsfeldes darstellen. Dank des tieferen Einblicks in die Auswirkung von Feldfehlern konnte damit das Mikrotron-Modell leicht an die gemessenen Transfermatrizen angepasst werden. Ferner wurde die Identifizierung und Korrektur anti-symmetrischer Feldfehler in den Mikrotron-Dipolen untersucht. Es wurde ein Messverfahren entwickelt, um kleine Feldkomponenten in der Bahnebene von der Groessenordnung 10E-3 zu erkennen. Das vorgeschlagene Verfahren wurde mit Hilfe des Simulationsprogramms TOSCA ausgetestet. Schliesslich wurde die Stabilitaet der Longitudinaloptik verbessert. Dadurch konnte eine hochpraezise Energiestabi-lisierung verwirklicht werden. Bei 855 MeV Strahlenergie wird eine Stabilitaet von etwa 10E-6 erreicht.

Untersuchungen zur Strahldynamik am Harmonischen Doppelseitigen Mikrotron von MAMI-C

Das Institut für Kernphysik der Universität Mainz betreibt seit 1990 eine weltweit einzigartige Beschleunigeranlage für kern- und teilchenphysikalische Experimente – das Mainzer Mikrotron (MAMI-B). Diese Beschleunigerkaskade besteht aus drei Rennbahn-Mikrotrons (RTMs) mit Hochfrequenzlinearbeschleunigern bei 2.45 GHz, mit denen ein quasi kontinuierlicher Elektronenstrahl von bis zu 100 μA auf 855MeV beschleunigt werden kann.rnrnIm Jahr 1999 wurde die Umsetzung der letzten Ausbaustufe – ein Harmonisches Doppelseitiges Mikrotron (HDSM, MAMI-C) – mit einer Endenergie von 1.5 GeV begonnen. Die Planung erforderte einige mutige Schritte, z.B. Umlenkmagnete mit Feldgradient und ihren daraus resultierenden strahloptischen Eigenschaften, die einen großen Einfluss auf die Longitudinaldynamik des Beschleunigers haben. Dies erforderte die Einführung der „harmonischen“ Betriebsweise mit zwei Frequenzen der zwei Linearbeschleuniger.rnrnViele Maschinenparameter (wie z.B. HF-Amplituden oder -Phasen) wirken direkt auf den Beschleunigungsprozess ein, ihre physikalischen Größen sind indes nicht immer auf einfache Weise messtechnisch zugänglich. Bei einem RTM mit einer verhältnismäßig einfachen und wohldefinierten Strahldynamik ist das im Routinebetrieb unproblematisch, beim HDSM hingegen ist schon allein wegen der größeren Zahl an Parametern die Kenntnis der physikalischen Größen von deutlich größerer Bedeutung. Es gelang im Rahmen dieser Arbeit, geeignete Methoden der Strahldiagnose zu entwickeln, mit denen diese Maschinenparameter überprüft und mit den Planungsvorgaben verglichen werden können.rnrnDa die Anpassung des Maschinenmodells an eine einzelne Phasenmessung aufgrund der unvermeidlichen Messfehler nicht immer eindeutige Ergebnisse liefert, wird eine Form der Tomographie verwendet. Der longitudinale Phasenraum wird dann in Form einer Akzeptanzmessung untersucht. Anschließend kann ein erweitertes Modell an die gewonnene Datenvielfalt angepasst werden, wodurch eine größere Signifikanz der Modellparameter erreicht wird.rnrnDie Ergebnisse dieser Untersuchungen zeigen, dass sich der Beschleuniger als Gesamtsystem im Wesentlichen wie vorhergesagt verhält und eine große Zahl unterschiedlicher Konfigurationen zum Strahlbetrieb möglich sind – im Routinebetrieb wird dies jedoch vermieden und eine bewährte Konfiguration für die meisten Situationen eingesetzt. Das führt zu einer guten Reproduzierbarkeit z.B. der Endenergie oder des Spinpolarisationswinkels an den Experimentierplätzen.rnrnDie Erkenntnisse aus diesen Untersuchungen wurden teilweise automatisiert, so dass nun den Operateuren zusätzliche und hilfreiche Diagnose zur Verfügung steht, mit denen der Maschinenbetrieb noch zuverlässiger durchgeführt werden kann.

Generalizing of Neural Nets: Functional Nets of Special Type

Special generalizing for the artificial neural nets: so called RFT – FN – is under discussion in the report. Such refinement touch upon the constituent elements for the conception of artificial neural network, namely, the choice of main primary functional elements in the net, the way to connect them(topology) and the structure of the net as a whole. As to the last, the structure of the functional net proposed is determined dynamically just in the constructing the net by itself by the special recurrent procedure. The number of newly joining primary functional elements, the topology of its connecting and tuning of the primary elements is the content of the each recurrent step. The procedure is terminated under fulfilling “natural” criteria relating residuals for example. The functional proposed can be used in solving the approximation problem for the functions, represented by its observations, for classifying and clustering, pattern recognition, etc. Recurrent procedure provide for the versatile optimizing possibilities: as on the each step of the procedure and wholly: by the choice of the newly joining elements, topology, by the affine transformations if input and intermediate coordinate as well as by its nonlinear coordinate wise transformations. All considerations are essentially based, constructively and evidently represented by the means of the Generalized Inverse.

Dynamical Systems in Description of Nonlinear Recursive Regression Transformers

The task of approximation-forecasting for a function, represented by empirical data was investigated. Certain class of the functions as forecasting tools: so called RFT-transformers, – was proposed. Least Square Method and superposition are the principal composing means for the function generating. Besides, the special classes of beam dynamics with delay were introduced and investigated to get classical results regarding gradients. These results were applied to optimize the RFT-transformers. The effectiveness of the forecast was demonstrated on the empirical data from the Forex market.

Diffusive models and chaos indicators for non-linear betatron motion in circular hadron accelerators

Understanding the complex dynamics of beam-halo formation and evolution in circular particle accelerators is crucial for the design of current and future rings, particularly those utilizing superconducting magnets such as the CERN Large Hadron Collider (LHC), its luminosity upgrade HL-LHC, and the proposed Future Circular Hadron Collider (FCC-hh). A recent diffusive framework, which describes the evolution of the beam distribution by means of a Fokker-Planck equation, with diffusion coefficient derived from the Nekhoroshev theorem, has been proposed to describe the long-term behaviour of beam dynamics and particle losses. In this thesis, we discuss the theoretical foundations of this framework, and propose the implementation of an original measurement protocol based on collimator scans in view of measuring the Nekhoroshev-like diffusive coefficient by means of beam loss data. The available LHC collimator scan data, unfortunately collected without the proposed measurement protocol, have been successfully analysed using the proposed framework. This approach is also applied to datasets from detailed measurements of the impact on the beam losses of so-called long-range beam-beam compensators also at the LHC. Furthermore, dynamic indicators have been studied as a tool for exploring the phase-space properties of realistic accelerator lattices in single-particle tracking simulations. By first examining the classification performance of known and new indicators in detecting the chaotic character of initial conditions for a modulated Hénon map and then applying this knowledge to study the properties of realistic accelerator lattices, we tried to identify a connection between the presence of chaotic regions in the phase space and Nekhoroshev-like diffusive behaviour, providing new tools to the accelerator physics community.

Influence of nonlinear stiffness on the dynamics of a slender elastic beam under torsional oscillations

This study focuses on analysing the effects of nonlinear torsional stiffness on the dynam-ics of a slender elastic beam under torsional oscillations, which can be subject to helical buckling.The helical buckling of an elastic beam confined in a cylinder is relevant to many applications. Someexamples include oil drilling, medical cateters and even the conformation and functioning of DNAmolecules. A recent study showed that the formation of the helical configuration is a result of onlythe torsional load, confirming that there is a different path to helical buckling which is not related tothe sinusoidal buckling, stressing the importance of the geometrical behaviour of the beam. A lowdimensional model of an elastic beam under torsional oscillations is used to analyse its dynamical be-haviour with different stiffness characteristics, which are present before and after the helical buckling.Hardening and softening characteristics are present, as the effects of torsion and bending are coupled.With the use of numerical algorithms applied to nonlinear dynamics, such as bifurcation diagramsand basins of attraction, it is shown that the nonlinear stiffness can shift the bifurcations and inducechanges in the stability of the desirable and undesirable solutions. Therefore, the proper modellingof these stiffness nonlinearities seems to be important for a better understanding of the dynamicalbehaviour of such beams.

Relative Dynamics and Control of an Ion Beam Shepherd Satellite

The ion beam shepherd (IBS) is a recently proposed concept for modifying the orbit and/or attitude of a generic orbiting body in a contactless manner, which makes it a candidate technology for active space debris removal. In this paper we deal with the problem of controlling the relative position of a shepherd satellite coorbiting at small separation distance with a target debris. After deriving the orbit relative motion equations including the effect of the ion beam perturbation we study the system stability and propose different control strategies.

A simple torsion-free nonlinear beam element for multibody dynamics

We propose in this work a very simple torsion-free beam element capable of capturing geometrical nonlinearities. The simple formulation is objective and unconditionally con- vergent for geometrically nonlinear models with large displacements, in the traditional sense that guarantees more precise numerical solutions for finer discretizations. The formulation does not employ rotational degrees of freedom, can be applied to two and three-dimensional problems, and it is computationally very efficient.

Numerical simulation of fresh SCC flow in wall and beam elements using flow dynamics models

Abstract : Recently, there is a great interest to study the flow characteristics of suspensions in different environmental and industrial applications, such as snow avalanches, debris flows, hydrotransport systems, and material casting processes. Regarding rheological aspects, the majority of these suspensions, such as fresh concrete, behave mostly as non-Newtonian fluids. Concrete is the most widely used construction material in the world. Due to the limitations that exist in terms of workability and formwork filling abilities of normal concrete, a new class of concrete that is able to flow under its own weight, especially through narrow gaps in the congested areas of the formwork was developed. Accordingly, self-consolidating concrete (SCC) is a novel construction material that is gaining market acceptance in various applications. Higher fluidity characteristics of SCC enable it to be used in a number of special applications, such as densely reinforced sections. However, higher flowability of SCC makes it more sensitive to segregation of coarse particles during flow (i.e., dynamic segregation) and thereafter at rest (i.e., static segregation). Dynamic segregation can increase when SCC flows over a long distance or in the presence of obstacles. Therefore, there is always a need to establish a trade-off between the flowability, passing ability, and stability properties of SCC suspensions. This should be taken into consideration to design the casting process and the mixture proportioning of SCC. This is called “workability design” of SCC. An efficient and non-expensive workability design approach consists of the prediction and optimization of the workability of the concrete mixtures for the selected construction processes, such as transportation, pumping, casting, compaction, and finishing. Indeed, the mixture proportioning of SCC should ensure the construction quality demands, such as demanded levels of flowability, passing ability, filling ability, and stability (dynamic and static). This is necessary to develop some theoretical tools to assess under what conditions the construction quality demands are satisfied. Accordingly, this thesis is dedicated to carry out analytical and numerical simulations to predict flow performance of SCC under different casting processes, such as pumping and tremie applications, or casting using buckets. The L-Box and T-Box set-ups can evaluate flow performance properties of SCC (e.g., flowability, passing ability, filling ability, shear-induced and gravitational dynamic segregation) in casting process of wall and beam elements. The specific objective of the study consists of relating numerical results of flow simulation of SCC in L-Box and T-Box test set-ups, reported in this thesis, to the flow performance properties of SCC during casting. Accordingly, the SCC is modeled as a heterogeneous material. Furthermore, an analytical model is proposed to predict flow performance of SCC in L-Box set-up using the Dam Break Theory. On the other hand, results of the numerical simulation of SCC casting in a reinforced beam are verified by experimental free surface profiles. The results of numerical simulations of SCC casting (modeled as a single homogeneous fluid), are used to determine the critical zones corresponding to the higher risks of segregation and blocking. The effects of rheological parameters, density, particle contents, distribution of reinforcing bars, and particle-bar interactions on flow performance of SCC are evaluated using CFD simulations of SCC flow in L-Box and T-box test set-ups (modeled as a heterogeneous material). Two new approaches are proposed to classify the SCC mixtures based on filling ability and performability properties, as a contribution of flowability, passing ability, and dynamic stability of SCC.

Effects of Variations in Nonlinear Damping Coefficients on the Parametric Vibration of a Cantilever Beam with a Lumped Mass

Uncertainties in damping estimates can significantly affect the dynamic response of a given flexible structure. A common practice in linear structural dynamics is to consider a linear viscous damping model as the major energy dissipation mechanism. However, it is well known that different forms of energy dissipation can affect the structure's dynamic response. The major goal of this paper is to address the effects of the turbulent frictional damping force, also known as drag force on the dynamic behavior of a typical flexible structure composed of a slender cantilever beam carrying a lumped-mass on the tip. First, the system's analytical equation is obtained and solved by employing a perturbation technique. The solution process considers variations of the drag force coefficient and its effects on the system's response. Then, experimental results are presented to demonstrate the effects of the nonlinear quadratic damping due to the turbulent frictional force on the system's dynamic response. In particular, the effects of the quadratic damping on the frequency-response and amplitude-response curves are investigated. Numerically simulated as well as experimental results indicate that variations on the drag force coefficient significantly alter the dynamics of the structure under investigation. Copyright (c) 2008 D. G. Silva and P. S. Varoto.

Pulse train fluorescence technique for measuring triplet state dynamics

We report on a method to study the dynamics of triplet formation based on the fluorescence signal produced by a pulse train. Basically, the pulse train acts as sequential pump-probe pulses that precisely map the excited-state dynamics in the long time scale. This allows characterizing those processes that affect the population evolution of the first excited singlet state, whose decay gives rise to the fluorescence. The technique was proven to be valuable to measure parameters of triplet formation in organic molecules. Additionally, this single beam technique has the advantages of simplicity, low noise and background-free signal detection. (C) 2011 Optical Society of America