Focal swift for a Gaussian beam: an experimental study

The beam radius along a focused, unapertured Gaussian beam was measured and used to calculate the dependence of the geometrical Fresnel number on the effective Fresnel number of the beam as it emerged from the focusing lens. The resulting data clearly demonstrate a focal shift away from the focal plane given by geometrical optics. The data agree very well with a theory due to Carter. This theory indicates that a significant focal shift occurs in these beams if the focusing lens is placed within the near field of the focal plane.     

Optimum truncation of a Gaussian beam for propagation through atmospheric turbulence

The mean on-axis far-field (or focal-plane) irradiance of a Gaussian beam that is truncated by a circular aperture in the presence of atmospheric turbulence is considered. In the absence of turbulence, an accurate analytic approximation for the irradiance distribution that is valid within the main central lobe of the beam is presented. Based on this approximation, the mean on-axis far-field irradiance and the corresponding turbulence Strehl ratio for the truncated Gaussian beam are then obtained. By maximization of the on-axis irradiance, the optimum ratio of the beam diameter to the aperture diameter in the presence of turbulence is obtained, and the results for the corresponding maximum on-axis irradiance as a function of the strength of turbulence are presented. In particular, for D/r(0) > 1, where D is the aperture diameter and r(0) is Fried's coherence length, optimum truncation of a Gaussian beam and uniform illumination of a circular aperture (where the same total power isuniformly distributed over the aperture) result in the same on-axis irradiance in the presence of uncompensated turbulence.     

Polarization distribution in a Gaussian beam reflected from a resonant medium

It is shown that the distribution of parameters of light polarization in the cross section of a Gaussian beam reflected from a resonant medium is substantially inhomogeneous. This inhomogeneity is especially pronounced in the wavelength range of light absorption by metals.     

Beam scintillations for ground-to-space propagation. Part 2: Gaussian beam scintillation

On the basis of the analytic techniques presented in the first of these two companion papers [J. Opt. Soc. Am. A27, 2169 (2010)] we present the complete asymptotic analysis of the axial beam scintillation index for coherent Gaussian beams on the ground-to-space propagation paths. The ratio of turbulence layer thickness to overall propagation path length contributes an additional small parameter to the analysis. We show that it is possible to use three dimensionless parameters to describe the problem and that the general arrangement of the asymptotic regions established in our earlier work [Waves Random Media 4, 243 (1994)]) is preserved. We find that on a slant propagation path, collimated beams can experience the unusual double-scattering-dominated scintillation found originally for focused beams.     

Causality in propagation of a pulse in a nonlinear dispersive medium

We investigate the causal propagation of the pulse through dispersive media by very precise numerical solution of the coupled Maxwell–Bloch equations without any approximations about the strength of the input field. We study full nonlinear behaviour of the pulse propagation through solid state media like ruby and alexandrite. We have demonstrated that the information carried by the discontinuity, i.e. front of the pulse, moves inside the media with velocity c even though the peak of the pulse can travel either with sub-luminal or with super-luminal velocity. Our numerical demonstration is subject to the condition that the background refractive index of the medium is unity. We extend the argument of Levi-Civita to prove that the discontinuity would travel with velocity c even in a nonlinear medium.     

Photobleaching of a solid photochromic medium by a Gaussian laser beam

Equations that describe the photobleaching of photochromic layers illuminated by Gaussian laser beams are given. The photochromic samples are made of thionine, triethanolamine, and polyvinyl alcohol and follow simple kinetics law. Spatial absorbance and time-dependent power transmittance are well described by the developed equations. The model is used to calculate the Gaussian dimension of helium-neon laser beams from kinetics data.     

Analysis of beam propagation in thick nonlinear media

We present a theoretical analysis of beam propagation in thick nonlinear media by using the Gaussian decomposition method and considering a thick medium as a stack of thin media. Simple analytic solutions of Z-scan characteristics and optical limiting with thick nonlinear media are obtained. Comparisons of these results with those obtained by use of a distributed-lens model and Gaussian-Laguerre mode decomposition are made. Good agreement is obtained with a distributed-lens model.     

Radiation force on a nonlinear microsphere by a tightly focused Gaussian beam

We determine the characteristics of the radiation force that is exerted on a nonresonant nonlinear (Kerr-effect) rigid microsphere by a strongly focused Gaussian beam when diffraction and interference effects are significant (sphere radius a < or = illumination wavelength lambda). The average force is calculated from the surface integral of the energy-momentum tensor consisting of incident, scattered, and internal electromagnetic field vectors, which are expressed as multipole spherical-wave expansions. The refractive index of a Kerr microsphere is proportional to the internal field intensity, which is computed iteratively by the Rytov approximation (residual error of solution, 10(-30). The expansion coefficients for the field vectors are calculated from the approximated index value. Compared with that obtained in a dielectric (linear) microsphere in the same illumination conditions, we find that the force magnitude on the Kerr microsphere is larger and increases more rapidly with both a and the numerical aperture of the focusing objective. It also increases nonlinearly with the beam power unlike that of a linear sphere. The Kerr nonlinearity also leads to possible reversals of the force direction. The proposed technique is applicable to other types of weak optical nonlinearity.     

Modeling the interferometric radius measurement using Gaussian beam propagation

We model the interferometric radius measurement using Gaussian beam propagation to identify biases in the measurement due to using a simple geometric ray-trace model instead of the more complex Gaussian model. The radius measurement is based on using an interferometer to identify the test part's position when it is at two null locations, and the distance between the positions is an estimate of the part's radius. The null condition is observed when there is no difference in curvature between the reflected reference and the test wavefronts, and a Gaussian model will provide a first-order estimate of curvature changes due to wave propagation and therefore changes to the radius measurement. We show that the geometric ray assumption leads to radius biases (errors) that are a strong function of the test part radius and increase as the radius of the part decreases. We tested for a bias for both microscaled (<1 mm) and macroscaled parts. The bias is of the order of parts in 10(5) for micro-optics with radii a small fraction of a millimeter and much smaller for macroscaled optics. The amount of bias depends on the interferometer configuration (numerical aperture, etc.), the nominal radius of the test part, and the distances in the interferometer.     

Optimized coupling of a Gaussian beam into an optical waveguide with a grating coupler: comparison of experimental and theoretical results

The coupling efficiency of grating couplers is derived for a Gaussian incident beam. Its optimum value depends on the beam waist and on the position of a light spot with respect to the coupler edge for given grating parameters. The characteristic coupling length has been experimentally determined for the grating coupler studied. Relative measurements of the coupling efficiency as a function of incident beam characteristics are in good agreement with the numerical results.     

Partially coherent flattened Gaussian beam and its paraxial propagation properties

A simple model called partially coherent flattened Gaussian beam (FGB) is proposed to describe a partially coherent beam with a flat-topped spatial profile. An explicit and analytical formula is derived for the cross-spectral density of a partially coherent FGB propagating through a paraxial ABCD optical system. The propagation factor and propagation properties of a partially coherent FGB in free space are studied in detail and found to be closely related to its coherence and beam order.     

Temperature distribution in a uniform medium heated by linear absorption of a Gaussian light beam

The linear-diffusion equation is considered for a positive half-space with heat sources represented by Gaussian functions in the transverse plane and by exponential decay along the longitudinal axis. The exact solution is presented as a single quadrature of the complementary error function (erfc). The approximate solution is suggested in the form of the product of two Gaussian functions and the hyperbolic secant function. Comparison with the exact solution shows that the error of this approximation is near 10%. The approximation may be used in different medical applications, e.g., laser angioplasty.     

Scintillation index of flat-topped Gaussian laser beam in strongly turbulent medium

In a strongly turbulent medium, the scintillation index of flat-topped Gaussian beams is derived and evaluated. In the formulation, unified solution of Rytov method is utilized. Our results correctly reduce to the existing strong turbulence scintillation index of the Gaussian beam, and naturally to spherical and plane wave scintillations. Another checkpoint of our result is the scintillation index of flat-topped Gaussian beams in weak turbulence. Regardless of the order of flatness, scintillations of flat-topped Gaussian beams in strong turbulence are found to be determined mainly by the small-scale effects. For large-sized beams in moderate and strongly turbulent medium, flatter beams exhibit smaller scintillations.     

Phase retrieval from experimental far-field intensities by use of a Gaussian beam

The noniterative phase-retrieval method by use of Gaussian filtering is applied to the reconstruction of phase objects from experimental far-field intensities. In this method, the complex amplitude of transmitted light through an object is reconstructed from three far-field intensities, which are measured with the modulation of the object by laterally shifted and unshifted Gaussian filters. In the experiment, the amplitude of a Gaussian beam illuminating objects is utilized as a Gaussian filter, and, as the phase objects, a converging lens with a small exit pupil and a plastic fiber immersed in optical adhesive are used. The experimental results show that the Gaussian beam of a laser is capable of retrieving the phases of those objects with the accuracy of the range from approximately 1/10 to 1/4 of the laser's wavelength.     

Coupled electromagnetic wave propagation in a cylindrical dielectric waveguide filled with Kerr medium: nonlinear effects

Propagation of the coupled electromagnetic wave, which is a superposition of TE and TM waves, in a dielectric circular cylindrical waveguide filled with non-linear inhomogeneous medium is studied (if the permittivity is linear, the coupled wave does not exist). Non-linear coupled TE–TM wave is characterized by two (independent) frequencies and two (coupled) propagation constants (PCs). The physical problem is reduced to a non-linear two-parameter transmission eigenvalue problem for Maxwell’s equations. The system of dispersion equations with respect to PCs is derived and solved numerically. Two types of coupled PCs and coupled guided modes are found: non-linear solutions of the first type become solutions of the corresponding linear problems as the nonlinearity coefficient tends to zero; solutions of the second type seem to be ’purely’ non-linear as they stay away from any linear solutions as coefficient of the nonlinearity tends to zero. Coupled PCs and coupled eigenmodes are calculated and plotted.     

Topological phase in a nonparaxial Gaussian beam

An analysis is made of the structure and evolution of the singularities of a nonparaxial Gaussian beam. It is shown that a Gaussian beam may be represented by a family of straight lines lying on the surface of a hyperboloid and that the wavefront of this beam is a function of a point source situated at a point on the z axis with the imaginary coordinate iz ₀. The argument of this complex function is the topological phase of the beam which characterizes the rotation of the wavefront. The singularities of a nonparaxial Gaussian beam are located in the focal plane and are annular edge dislocations. Dislocation processes near the constriction of the Gaussian beam only occur as a result of aperture diffraction. Pis’ma Zh. Tekh. Fiz. 25, 14–20 (November 26, 1999)     

Ray propagation in nonuniform random lattices

The problem of optical ray propagation in a nonuniform random half-plane lattice is considered. An external source radiates a planar monochromatic wave impinging at an angle theta on a half-plane random grid where each cell can be independently occupied with probability q(j)=1-p(j),j being the row index. The wave undergoes specular reflections on the occupied cells, and the probability of penetrating up to level k inside the lattice is analytically estimated. Numerical experiments validate the proposed approach and show improvement upon previous results that appeared in the literature. Applications are in the field of remote sensing and communications, where estimation of the penetration of electromagnetic waves in disordered media is of interest.