Phase Front Curvature Research Articles

Pulsating behavior of an optical beam induced by initial phase-front curvature in strongly nonlocal media

An exact Gaussian solution with initial phase-front curvature for the Snyder-Mitchell linear model is presented. It is found that the propagation of the beam in strongly nonlocal media is deeply affected by the initial phase-front curvature, which can cause pulsating behavior of the Gaussian beam and lead to the occurrence of a periodic interference pattern during the interaction of two Gaussian beams. Also, we show that these properties are still valid for the nonlocal nonlinear Schr\odinger equation provided that the characteristic response length of the nonlocal media is much broader than the width of the optical beam. It is useful for further understanding the properties of optical beams in strongly nonlocal media and may be applied to precision measurements.

Injection, trapping, and acceleration of electrons in a three-dimensional nonlinear laser wakefield

It is demonstrated that the accelerating and focusing phases of the nonlinear three-dimensional axisymmetric laser wake can almost entirely overlap starting from a certain distance behind the laser pulse in homogeneous plasma. Such field structure results from the curvature of phase fronts due to the radially inhomogeneous relativistic shift of plasma frequency. Consequently, the number of trapped low-energy electrons can be much greater than that predicted by the linear wake theory. This effect is favorable for quasimonoenergetic acceleration of a considerable charge (several hundreds of pC) to about 1GeV per electron in the plasma wakefield driven by an ultrashort (∼30fs) weakly focused (r0∼100μm) petawatt laser pulse.

Competition of pattern forming instabilities due to phase front curvature in an optical system

In this paper we analyze experimentally and theoretically the competition between two pattern forming instabilities in a single mirror feedback scheme with sodium vapor as the nonlinear medium. Two types of structures with different transverse wave numbers are observed experimentally, if the spatial phase modulation of the light field is varied. This phenomenon results from the combination of a nonlinear self-lensing effect on the one hand and of the externally controlled phase front curvature of the light field on the other. A linear stability analysis yields two instabilities whose length scales match quite well the experimental findings. Further analysis reveals the mechanism of length-scale selection in this system and demonstrates the possibly crucial role of phase front curvature in optical pattern formation.

Theory for Gaussian beam diffraction in 2D inhomogeneous medium, based on the eikonal form of complex geometrical optics

A simple and effective method to describe Gaussian beams propagation and diffraction in arbitrary smoothly inhomogeneous 2D medium has been developed based on the eikonal form of complex geometrical optics. The method assumes the eikonal equation can be solved in paraxial approximation in curvilinear frame of references, connected with the central ray. The Riccati-type ordinary differential equation is derived for complex parameter characterizing the Gaussian beam width and phase front curvature. The same parameter was proved to define both the modulus and the argument of the complex amplitude. As a result, the problem of the Gaussian beam diffraction in inhomogeneous media has been reduced to the solution of the ordinary differential equation of the first order, which can be readily calculated numerically for arbitrary profile of dielectric permittivity.

Paraxial theory of slow self-focusing.

We present a theory of slow self-focusing that is paraxial in nature, gives the field including the phase and eikonal explicitly, while it also agrees with the results of variational and moments theories. After presenting the features of the theory, particularly its similarity to the central force problem, we go on to reformulate the theory for an absorbing medium. We find that the laser beam focuses to a constant beamwidth with a small phase-front curvature depending on the extent of absorption. The theory is applicable to a whole range of saturating nonlinearities although it specializes to two plasma cases, the ponderomotive force based and the relativistic electron quiver based nonlinearities, for definitive results.

Laser wakefield structure in a plasma column created in capillary tubes

The structure of the wakefield is studied in a plasma column, created by a monomode laser pulse propagating in a capillary tube, filled with gas affected by tunneling ionization. Linear analytical considerations as well as self-consistent numerical simulations show that in the central bulk part of a plasma column where the laser intensity exceeds the ionization threshold, the wakefield structure is similar to that of an infinite homogeneous plasma. Near the wall of the capillary tube, where the laser intensity decreases below the ionization threshold and where the plasma density falls to zero, the curvature of the plasma wave phase front increases with the distance from the laser pulse, resulting in small-scale radial electric field which may undergo phase mixing.

Chirp control in the direct space-to-time pulse shaper

The dispersion properties of the direct space-to-time pulse shaper are investigated for the first time to our knowledge. We demonstrate that phase-front curvature of the input spatial profile leads to a chirp in the output temporal waveform, which one can compensate for by varying the separation between the pulse-shaping lens and slit. Furthermore, the output intensity profile remains invariant as the chirp is manipulated. These properties are fundamentally different than in the well-known Fourier-transform pulse shaper.

Efficient coupling of a laser to a waveguide using a taper designed by conformal mapping

Maximising the optical power collected in a waveguide from the diffracting field of a semiconductor laser is desirable in optical fibre communication systems. However, the spot size and phase front curvature of the laser field usually makes a poor overlap with the mode of the receiving waveguide. Various proposals have been made to improve this coupling. This paper presents the design of a tapered waveguide section, having the correct geometry and refractive index profile, to efficiently capture and transform the rapidly diffracting light from a semiconductor laser to a planar wavefront in a straight waveguide. Experimentally, such an approach requires the refining of available techniques (UV exposure, ion implantation or diffusion) to obtain the required grading of the refractive index profile within the tapered input section of the receiving waveguide.

Influence of the phase front parameters on the path of a beam of rays

The geometric optics approximation is used to derive an equation describing the path of a beam of rays in an inhomogeneous but locally isotropic medium. The predicted additional twisting, proportional to the average phase front curvature, is confirmed. The effect disappears in the case of a rectilinear path.

Modeling of thermal lensing and higher order ring mode oscillation in end-pumped C-W Nd:YAG lasers

The influence of thermal lensing and mode matching on the oscillating mode in the end-pumped Nd:YAG laser is investigated. Thermal lensing was measured by probing the output beam divergence of a dynamically stable resonator. The magnitude of lensing expected due to thermal dispersion was confirmed in a distinct range of adequate mode matching. Residual phase front curvature and superimposed higher order ring mode oscillation of the active resonator were found to limit precise quantization. Measurements on arrangements which typically exhibit higher order ring mode oscillation were performed, and compared to a refined model. The refined model is based on a modified numerical calculation of the eigenstate of the electrical field in the active resonator.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonlinear dynamics of beam parameters and intensity in a unidirectional ring laser with a homogeneously broadened line

A model of the dynamics of a single-longitudinal-mode unidirectional ring laser with a homogeneously broadened medium is proposed. The model includes the transverse variation of the field and of the material parameters and is based on a treatment of the round-trip propagation of a beam in a nonlinear medium rather than on an expansion in terms of the empty-cavity modes. In particular, the case of adiabatically eliminated polarization and inversion is studied. A three-dimensional point mapping is obtained and analyzed. It is shown that this model describes stationary regimes with mode deformation and spatiotemporal (quasi-periodic or periodic) oscillations of both the beam parameters (beam radius and phase-front curvature) and the intensity on the axis.

Computation of body-wave seismograms in absorbing 2-D media using the Gaussian beam method: comparison with exact methods

SUMMARY The accuracy and applicability of the Gaussian beam method for the computation of synthetic seismograms in absorbing 2-D laterally inhomogeneous media is discussed in this paper. A computer program was developed for the computation of P-SV- and SH-waves along horizontal and vertical seismic profiles in absorbing 2-D inhomogeneous media with first-order discontinuities in density and velocities. Subdividing the model into triangles with linear density and velocity laws not only allows flexible modelling of complicated structures but also results in fast kinematic and dynamic ray tracing. With this program it is possible to compute phases which are specified by single or multiple reflections, refractions and/or conversions at the different discontinuities in the medium. The width and phase-front curvature of each beam is controlled by its complex beam parameter e. Along with several e-options proposed by other authors, a newly developed option to select the beam parameter was tested for a large set of models. The new option correlates the width of each beam to the size of the triangles the beam has passed through and thus to the structure of the medium. Slowly varying media represented by large triangles will give broad Gaussian beams whereas complicated and rapidly changing media which have to be described by many small triangles will produce narrow Gaussian beams. For the computation of reference seismograms the reflectivity method, the frequency-domain finite-difference method and the finite-difference method were used. The conclusion from these tests is, that the newly developed e-option is the optimum choice, whereas the other options sometimes fail and do not give accurate seismograms for critical incidence and at caustics.

Theoretical study of FEL active guiding in the small signal regime

We use a self-consistent small signal FEL field equation to study active guiding. We show that self-similar mode propagation is possible only in the exponential regime. These modes must be either growing or decaying; the purely oscillatory self-similar mode is not confined. By a Gaussian approximation, we investigate the eigenvalue structure of the fundamental mode analytically, and derive a cubic equation for the complex growth rate, the laser mode size, and the radius of the phase front curvature.

Efficient calculation of Gaussian-beam seismograms for two-dimensional inhomogeneous media

This paper discusses several aspects of the calculation of theoretical seismograms for two-dimensional inhomogeneous media with the method of Gaussian beams. The most important steps of this method, kinematic and dynamic ray tracing, can be performed very efficiently, if the model cross-section is subdivided into triangles with linear velocity laws. Each Gaussian beam is characterized by a complex beam constant ∈ which determines its width and phase-front curvature. Various possibilities to choose ∈ are discussed, including cases where beam properties at the beam endpoint (and not at the beginning) are prescribed; for instance, the beam width at the endpoint can be specified. In such cases the beam constant is a function of the radiation angle at the source, and the decomposition of a cylindrical wave into beams has to take this into account by weighting the beams differently, at least in principle. The exact weight function is derived and shown to be reasonably well approximated by the weight function, corresponding to angle-independent ∈ Theoretical seismograms are presented for a laterally heterogeneous model of the crust—mantle transition which is characterized by complications in the reflection from the transition and in the refraction from below. These complications are modelled by and large with success. The seismograms, however, depend to a certain extent on the choice of the beam constant. Moreover, according to the reciprocity principle calculations with source and receiver interchanged should have the same results as calculations for the original configuration. In practice this is not so, and the difference increases with the strength of lateral heterogeneities. Hence, for a successful application of Gaussian beams the model should not vary too strongly in lateral direction.

Mean diffracted rays of an optical beam in a turbulent medium

The curvature of a mean beam phase front in a turbulent medium is shown to be higher than that of a spherical wave. A set of mean diffracted rays corresponding to the mean phase front is used for calculations of phase fluctuations of the beam under conditions of both weak and strong intensity fluctuations.

Optical parametric oscillator dependence on pump laser beam quality

Experimentation with a pulsed, angle-tuned LiNbO <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> optical parametric oscillator revealed the dependence of pump power depletion and conversion efficiency on pump laser beam divergence. Mathematical expressions, developed for these parameters, were based on a spherical-wave model. Calculated results and comparison with experimental observations indicate the degree to which phasefront curvature affects parametric oscillator performance.

Tolerances in optical mixing

Analytical expressions are derived for the power spectral density of the current in optical heterodyning when phase front curvature mismatch, tilt or axis displacement occurs between the beating light beams. The signal beam is the far field of an incoherent source of circular shape, and the local oscillator is the fundamental mode of a stabilized laser. The cases in which the signal beam either illuminates the detector coherently or has a coherence area small compared with the detector area are shown to have simple solutions. An analogous analysis is carried out for homodyning (self-beating). The dependence of the signal-to-noise ratio on the detector aperture and optimum and tolerance values for the latter are obtained for a given misalignment.

Acoustic Surface-Wave Beam Diffraction on Anisotropic Substrates

The diffraction of any distribution can be discussed in terms of its constituent Gaussian modes. Simple design formulas are developed from a two-dimensional diffraction theory in order to characterize Gaussian beam diffraction in terms of three readily determined anisotropic propagation coefficients. The relationship between these coefficients and known anisotropic propagation phenomena is explored. Beam steering, diffraction, and focusing effects are discussed in some detail. Attention is drawn to two particular effects: autocollimation and ``negative'' phase front curvature. Situations in which both occur in practice are discussed. The former should play a significant role in acoustic matched filter design. A cylindrical transducer is shown, within limits, to be an optimum structure for obtaining a single acoustic beam convergence, even in the presence of beam scattering. The parabolic diffraction theory is applied to the design of an experimental anisotropic focusing system.

Fluctuations of a focused beam wave for atmospheric turbulence probing

This paper demonstrates that a focused beam wave may be effectively used to probe the atmospheric turbulence. Unlike plane and spherical waves, the focused beam has two parameters, i.e., the beam size and the radius of curvature of phase front at the aperture which may be easily varied experimentally. General formulations for the amplitude and phase correlation functions and structure functions of a focused beam wave are given including the effects of wind velocity and time delay. The effects of the beam size and the focal length on the spectral and spatial filter functions and temporal frequency spectrum are examined in detail including the effects of separation, wind velocity, and time delay. Using these functions, procedures are shown to probe the form of the spectral density of the index of refraction, the structure constant, and the wind velocity along the path.

Analysis of Optical Resonators Involving Focusing Elements

The electromagnetic fields in optical resonators involving focusing elements are considered. The work is restricted to cases where the electromagnetic fields are small at the edges of any optical elements so that there is no significant aperturing. The paper presents the analytical solution for the fields across two reference surfaces (often the end mirrors) within the resonator, and describes graphically the fields inside and outside the resonator. The analytical solution uses Huygens’ principle, but maintains generality in that the expressions relating the geometry of the particular optical resonator are left in terms of constants, which can be simply evaluated for the particular resonator. The expressions for the fields (over the reference surfaces), resonant frequencies, and stability conditions are given. The graphical description of the fields inside and outside the resonator is in terms of two experimentally observable variables: the radius of curvature of phase fronts and the spot size. A chart is utilized which permits a simple numerical presentation of those variables as a function of position along the axis, and enables a simple calculation of the higher-order transverse mode frequencies.