Free and Guided Optical Beams

Table of Contents

Preface

List of Participants

Basic theoretical elements enabling the numerical calculation of the diffraction integrals are introduced. The related practical procedures are also explained in some detail and several application cases corresponding to significant problems in the field of optics are addressed.

Although the theory of partial polarization of uniform electromagnetic beams is well-developed, it cannot be applied to random three-dimensional (3D) waves such as optical near fields. A general formulation of partial coherence and polarization in arbitrary 3D electromagnetic fields must be based on the 3 × 3 cross-spectral density tensors. After the fundamentals of electromagnetic coherence theory, we recall some recent researches on the spatial-coherence and spectral changes brought on by surface modes and evanescent waves. Partial polarization of arbitrary electromagnetic fields containing three orthogonal components is then addressed in terms of the 3 × 3 coherence matrix, and the variation of the 3D degree of polarization in near fields generated by semi-infinite thermal sources is analyzed. Finally, the similarities and differences of the conventional 2D and the new 3D formulations of partial polarization are briefly discussed.

The differential equation governing, within the paraxial regime, the propagation of an optical field through a medium with variable refractive index is formally identical to the time-dependent Schroedinger equation for a particle in two dimensions. In particular, the wave function of the particle plays the role of the transverse disturbance of the optical field, the potential acts as the modulation of the refractive index, and time evolution is replaced by the propagation along the longitudinal coordinate. An interesting consequence of such analogy is that well known propagation features of light in some media can afford a model for the time evolution of quantum wavefunctions of a particle within the corresponding potential. Finding the stationary states of a given potential, for instance, turns out to be equivalent to finding the propagation modes in a suitably chosen guiding structure.

The simplest case is that of an optical beam propagating inside a homogeneous medium and it corresponds to a free particle on a plane evolving in time. Well, the time evolution of the particle wavefunction can be predicted, starting from its expression at the starting time, using a propagator that can be immediately derived from the Fresnel integral. Yet, it is not difficult to conceive, by suitably superposing eigenstates of the energy of the free particle, a wavefunction whose squared modulus does not change with time, in perfect analogy with an optical nondiffracting Bessel beam.

A slightly less trivial case is that of the propagation inside an optical fibre having parabolic refractive index. It is known that the modes of such a fibre are Hermite-Gaussian modes, whose width is determined by the parameter governing the refractive index variation. The quantum analog is a two-dimensional harmonic oscillator, whose energy eigenstates are, of course, the same Hermite-Gaussian functions, and this is, actually, even better known. But the optical analogy allows us also to predict behaviors of the quantum particle that are not completely trivial. It is the case, for instance, of a fundamental Gaussian beam that is injected into the fiber off axis. If the width of the Gausian is chosen as that of the propagation modes of the fiber, the beam will propagate inside the fibre without changing its width, but its center will oscillate around the axis of the fiber. This is due to the “lens-like” effect of a fiber with parabolic refractive index. This corresponds to a coherent state of the harmonic oscillator. Due to the same effect, if we inject off-axis a Gaussian beam with a different spot size, then the beam not only oscillates around the axis, but also changes its width, passing periodically from smaller to larger values of the spot size. This corresponds to a squeezed state of the harmonic oscillator.

Further significant cases can be considered, such as those pertinent to optical waveguides, that correspond to potential wells for the quantum particle. Even in such cases, classical results concerning the propagation of the optical fields can help to solve analogous quantum problems, as far as the determination of the stationary states and of the energy eigenvalues, and the time evolution of the system is concerned.

Finally, starting from the above analogy, a correspondence between partially coherent beams and mixed quantum states can be established. It stems from the equivalence between the cross-spectral density function, defined within the second-order coherence theory, and the density matrix, used to describe quantum mixed states. Results obtained in the field of the partial coherence optics, such as those pertinent to the well-known Gaussian Schell-model sources, can be easily transposed into the quantum realm.

Note from Publisher: This article contains the abstract only.

Micro-optics includes a family of optical components and systems that are fabricated with the aid of modern micromachining. Elements utilizing either refractive or diffractive surfaces are now found in applications ranging from laser beam shaping in laser material processing to optical interconnects in telecom applications. We introduce the micro-optics family members and present system concepts.

Bose-Einstein condensates (BECs) realized with dilute alkali atoms can be considered as a source of coherent matter waves, what is sometimes termed atom laser. We will review the experimental methods used to create and manipulate BECs. We will also illustrate some simple atom-optical elements that can be created. We will review the propagation of coherent matter waves in periodic potentials showing how it is possible to control matter wave dispersion and create gap solitons.

No abstract received.

The main concepts and results in laser beam characterization and their propagation through ABCD-type optical systems are reviewed by using the method of second-order moments. A better understanding of laser beams and optical systems is obtained.

The evolution of the main parameters (spot size, radius of curvature, far-field divergence and/or beam quality factor) of laser radiation pulses propagating in stable and unstable cavities with variable reflectivity mirrors is investigated both theoretically and experimentally as a function of the number of round trips. It is shown that a theoretical analysis, formerly developed to study the growth of coherence of a Schell-Gauss model beam propagating through a periodic sequence of Gaussian apertures, provides a satisfactory description of the laser radiation temporal evolution.

The study of transient states is very important in practice because high-gain, short-pulse lasers as excimers or copper-vapor lasers, generate pulses making only a few transits inside the resonator. In this situation, beam parameters seldom reach a steady state hence, an analysis yielding the beam parameters at each round-trip is needed for a better characterization of the laser beam quality and can be of great interest in many application fields.

The experimental techniques used to analyze the temporal evolution of the main parameters of laser radiation pulses few ten of nanoseconds long are at first presented. Then, the propagation of laser radiation pulses in plane-parallel cavities with a variable reflectivity mirror is analyzed. Mirrors with Gaussian, super-Gaussian and step-reflectivity profile are tested in order to investigate the effects of the mirror reflectivity profile on the temporal evolution of the laser radiation parameters. Gaussian reflectivity mirrors of different spot size E are also considered to investigate the effect of E on the radiation beam-quality-factor evolution.

An experimental study of coherence evolution versus the Fresnel number of Gaussian cavities is also presented. To this end, plane parallel cavities of different length and with a Gaussian reflectivity mirror have been applied to a XeCl excimer laser (λ=308 nm) and the results on the resonator-length effects on the coherence growth of the laser radiation are presented. It is also shown that in Gaussian cavities of equal length and effective Fresnel number, the properties of the intracavity radiation are dependent on the Gaussian mirror distance from the active medium, since the onset of laser action.

Finally, experimental and numerical results on the beam quality and coherence growth of the laser radiation in unstable cavities equipped with a variable reflectivity mirror as output coupler are presented. Variable reflectivity unstable cavities are generally applied to pulsed laser to get a faster coherence growth of the oscillating radiation.

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No abstract received.

The Gaussian beam is a fundamental concept in optical beam propagation, but is based upon assumptions of paraxial optics. There are several ways of generalizing the concept of che Gaussian beam to the non-paraxial case, but some of these have limitations if a solution is required that is valid for a complete three-dimensional space. A solution that satisfies che requirements is based on the complex source-sink representation. This can be applied in a full vectorial treatment, in which a lineary polarized (LPO1) mode is generated from transverse and orthogonal electric and magnetic dipoles. The individuai TM and TE modes are represented by electric or magnetic dipoles alone. All three of these beam modes reduce to the ordinary TEMOO mode in the paraxial approximation. Axial dipoles result in radial or azimuthal TM or TE modes. Higher order multipoles generate higher order beams that can be expressed in terms of surns over che Laguerre-Gauss beams of paraxial theory. Another important beam is the Bessel beam, recognized as the fundamental solution of the wave equation in cylindrical coordinates. This too can be generalized to a non-paraxial vectorial form.

Ultra-short pulses can be modeled based on these types of beam. Again there are several ways of constructing solutions corresponding to different assumptions for the spatial distribution of the spectral components. A particulary useful type is the isodiffracting pulse, corresponding to the field of a mode-locked laser.

Beams and pulses can be treated by methods based on Fourier or phase space representations, generalized to the non-paraxial case. It can also be shown that there are relationships between these different representations.

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The behaviour of partially coherent Gaussian beams under free propagation and under passage through lenslike media or first order systems (FOS) is governed by the smplectic group of real linear canonical transformations repesented by the ray-transfer matrix of the system under consideration. And this, coupled with the Wigner representation of the cross-spectral density of the beam, reduces the propagation problem to one of 2×2 or 4×4 matrices depending on whether the beam under consideration is rotationally symmetric or asymmetric about its axis, thus often resulting in complete answers to interesting questions, without too much effort. These considerations hold good not only for Gaussian beams, but also for non-Gaussian partially coherent beams as long as one’s interest is restricted (as often happens to be the case) to the first and second moments of the beam. The combined power offered by the Wigner representation on the one hand and the waveoptic representation of canonical transformations on the other will be briefly explored in this sequence of three talks. No prior knowledge or experience with group structures is reqired to follow the arguments and to appreciate the results.

Lecture 1: The first obvious question in respect of partially coherent Gaussian beams is this: Given a hermitian Gaussian two-point function, how to test if it is 'nonnegative' so that it qualifies to be a physical cross-spectral density in some transvese plane? This nontrivial issue gets resolved relatively easily by the 'combined power' referred to above. The evolution of the beam parameters as the beam propagates is obtained as a bye-product. In particular, the invariants or quality parameters associated with a beam present themselves with no additional effort. And so are generalizations of the ABCD-law as well as the presence of the twist phase.

Lecture 2: Gori [Opt. Commun. vol. 34, 301 (1980)] was probably the first to exploit symmetry consideratons to achieve coherent-mode decomposition of a class of partially coherent beams. Reinterpretation of this approach in the light of the 'combined power' noted above, shows that this approach leads to coherent-mode decomposition of the much broader ten-parameter family of general anisotropic Gaussian Schell-model beams and, as a paticular case, to that of the twisted Gaussian beams. What may be more interesting, perhaps, is the fact that this approach leads quite simply to the first nontrivial twodimensional generalization of Factional Fourier Transform.

Lecture 3: This 'combined power' becomes also the natural setting to generalize the notion of shape-invariant propagation studied by Gori and collaborators [Opt. Commun. vo1.48, 7 (1893); Opt. Lett. vol.21, 1205 (1996)]. There are two types of shape-invariant propagatons: one in which the intensity ellipse representing the beam cross-section simply undergoes a scaling as a function of the propagation distance, retaining invariant its eccentricity as well as its orientation; and another in which only the eccentricity, and not its orientation, is maitained invariant. The 'combined power' leads to omplete characterization in respect of both types of shape-invariant propagations.

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The concept of spatially varying polarization state is examined from the viewpoint of the wave-optical engineering. The possibility for increasing the attainable diffraction efficiencies for several types of signals by using a polarization-controlling element is discussed. This type of elements are based on the simultaneous control of the phase and amplitude of the incoming field such that no energy is absorbed. The discussion is then extended to the polarization multiplexing and partial polarization. Possible research topics to be explored in near future are briefly discussed.

No abstract received.

No abstract received.

No abstract received.

This lecture is intended to provide a general introduction to quadratic nonlinear optics, along with a view on a few selected research topics of present interest. In Section 1 I will recall the quadratic effects arising from non-resonant light-matter interactions, in the absence of absorption and carriers. Specifically, I will illustrate the concept of phase matching and overview the related techniques, with emphasis on guided-wave interactions. In the last decade, novel parametric sources have been conceived and fabricated in both bulk and integrated geometries, and new phenomena and applications have been demonstrated in the domain of soliton propagation. In Section 2, on the side of novel integrated optical sources, I will focus on the recent progress towards a monolithic semiconductor OPO. In Section 3, in the framework of the phase effects associated to the cascading of χ⁽²⁾, I will deal with quadratic spatial optical solitons, i.e. free-propagating optical-field configurations where diffraction is balanced by nonlinear self-focusing.

A novel high-brilliance universal source of polarization entangled photon states is presented. It consists of a type-I non-linear crystal which operates in a ultra-stable interferometric scheme where all the photon couples emitted at a certain wavelength take part in the entanglement and can be measured. In these peculiar conditions we have obtained violation of 213 standard deviations of a Bell’s inequalities test. The brightness and the robustness of the source have been also characterized by different experiments. The possible generation of a native entangled Bessel Gauss beam by this source is also discussed.

In this work we present experimental and theoretical results for the polarization competition dynamics in the transient state of a quasi-isotropic low-pressure CO₂ laser. We show that the polarization dynamics during the switch-on is well described by means of a model including optical coherences (intrinsic anisotropy) and extrinsic linear anisotropies. Furthermore, the experiment provides a numerical assignment for the decay rate of the coherence term for a CO₂ laser. These results extend a previous work in which competition is interpreted in a phenomenological way as a parametric cross coupling between the matter polarization and electric fields.

In this paper we set forth new exact analytical Superluminal localized solutions to the wave equation for arbitrary frequencies and adjustable bandwidth. The formulation presented here is rather simple, and its results can be expressed in terms of the ordinary, so-called “X-shaped waves”. Moeover, by the present formalism we obtain the first analytical localized Superluminal approximate solutions which represent beams propagating in dispersive media. Our solutions may find application in different fields, like optics, microwaves, radio waves, and so on.

Light beams carry energy and linear momentum, but they also transport angular momentum. In order to obtain an expression for the angular momentum carried by light we will use paraxial approximation within classical electromagnetic theory. It is found that the resulting value can be divided naturally in two parts: a polarization contribution and a spatial contribution. By analogy, those two terms are also called spin part and orbital part of the angular momentum. In order to interpret this result we will consider two beam families, Laguerre-Gauss modes and Gauss Schell-model family. Several procedures used to obtain beams with angular momentum are also mentioned. Finally, different methods used to observe and measure the angular momentum transported by light beams are briefly explained.

Beam characterization is the pre-requisite of any research exploiting light beams, especially in cases involving laser beams. One can rely on the beam parameters provided by the manufacturer but often they are inadequate and/or not sufficient for the experimental data analysis.

The full characterization of a laser beam can require the determination of many parameters (about ten for a generic beam); however for symmetrical beams the significant ones can reduce to only to a few. The characterization can be performed with the accuracy requested by the application and limited to the relevant parameters.

The main parameters of interest will be defined and the measurement procedures and equipment will be discussed. The ISO standards consider the following parameters mainly of interest for industrial applications:

1) Beam widths, divergence angle and beam propagation ratio.

2) Power, energy density distribution

3) Parameters for stigmatic and simple astigmatic beams

4) Parameters for general astigmatic beams

5) Geometrical laser beams classification and propagation

6) Power, energy and temporal characteristics

7) Beam positional stability

8) Beam polarization

9) Spectral characteristics

10) Shape of a laser wavefront: Phase distribution

All the above points will be briefly discussed as regards the experimental problems involved. Special attention will be given to the methods for measuring the intensity distribution and to the related instrumentation to derive the Beam propagation ratio, the Beam Quality factor M2 or the Beam Parameters Product. Examples of the parameters relevance for specific applications will be given. Depending on the spectral range, specific detectors are used: CCD cameras with detector arrays in the visible and near infrared, thermocameras with a single detector and scanning system for the medium and far IR. The major problems in data collection and processing will be discussed.

Another new and not yet fully investigated area is the characterization of laser beam by wavefront measuring instruments. One possible approach is the use of self-referencing interferometers such as the point diffraction interferometers. Alternatively wavefront gradient measuring instruments can be used such as the Hartmann-Shack sensors.

Wavefront intensity and phase joint distributions can now be measured at the same time. This can provide in addition new methods to derive the modal content. A short review of the experimental problems in this area still looking for a practical solution will be given.

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Here we present some specific application of non-destructive diagnostics in the framework of chronotomography. We designated key difficulties, bound with strongly restricted number of angular projections having a place in actual physical examinations, and have considered series of mathematical approaches (including based on Hartley transformation with as a base function), allowing to overcome them. We consider also the direct Radon transformation in time-space square in application to precision velocimetry.

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Here we present the analysis of processes happening in a volume of a monochromatic polarized radiation beam. This analysis is carried out by the means of the time-space function of the projections of a dot vector electrodynamic charge, propellented on a spiral, on two crossly perpendicular planes. All quantum properties (including of a spin presence) are explained. There is obtained, that the magnification of intensity of monochromatic polarized radiation has a limit, which depends on a radiation wavelength. The formula for the calculation of a fine structure constant is obtained.

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The paper describes the specific quasi-optical phenomena involved in the generation of the celestial and biological radiation. It examines the generation of the free space and guided biological beams in the presence of the ambient celestial primordial isotropic background. Josephson and Kerr mechanisms provide an explanation of the concept of bio-coherence as a novel possibility of electromagnetic communication.

The physical, chemical and biological processes taking place in the ocean are of primary importance for the climatic equilibrium of our planet. This explains why the more recent techniques are applied in the hydrographic studies.

In this paper, we review briefly one laser system and three satellite radiometers routinely used for the study of the phytoplankton dynamics. Their data are compared and, as a consequence, the accuracy on the measurement of relevant seawater parameters is assessed.

A variety of different semiconductor detectors from wide-band gap materials, Si, GaAs, low temperature grown GaAs, and InP were tested for fast response to X-ray pulses and for sensitivity to shallow penetrating red and blue light from LEDs. All detectors are of planar MSM structure. The detectors are intended for dosimetry of subnanosecond X-ray pulses from hot plasmas. The devices made from InP and the most defected GaAs are the fastest and mostly promising for the X-ray diagnostics. Recently we also design a new system for femtosecond timeresolved spectroscopy. In preliminary investigations we observe for about 100 fs light excitation the electrical response in the range of 10-100 ps.

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Nonlinear optical materials are expected to be active elements for optical communications and fotonics. In particular, a large number of organic compounds have been already reported as a promising materials for waveguides and devices. Several nonlinear optical polymers containing azometine side-chain groups have been obtained. In this communication we present our results of nonlinear properties of polyarylates and preparation of strip waveguides from these polymers. A patterned sequence of polymer films with different refractive index was deposited on silicon wafer in special design to produce optical switchers and modulators.

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In this paper we describe how high-energy electrons are accelerated in free space by the longitudinal electric field of a TM₀₁ laser beam. For intensities reaching 10²² W/cm², numerical calculations show that a synchronous acceleration is achieved for certain values of the phase of the laser field, leading to multi-GeV energy gains.