Lens-like Medium Research Articles

Shadows of structured beams in lenslike media.

The self-healing phenomenon of structured light beams has been comprehensively investigated for its important role in various applications including optical tweezing, superresolution imaging, and optical communication. However, for different structured beams, there are different explanations for the self-healing effect, and a unified theory has not yet been formed. Here we report both theoretically and experimentally a study of the self-healing effect of structured beams in lenslike media, this is, inhomogeneous lenslike media with a quadratic gradient index. By observing the appearance of a number of shadows of obstructed structured wave fields it has been demonstrated that their self-healing in inhomogeneous media are the result of superposition of fundamental traveling waves. We have found that self-healing of structured beams occurs in this medium and, interestingly enough, that the shadows created in the process present sinusoidal propagating characteristics as determined by the geometrical ray theory in lenslike media. This work provides what we believe to be a new inhomogenous environment to explain the self-healing effect and is expected to deepen understanding of the physical mechanism.

Equivalent optical model for lenslike gain medium in continuously single-end-pumped solid-state laser

In order to conveniently and precisely estimate the spot size of laser modes in continuously single-end-pumped solid-state lasers, a ray matrix for the thermal-induced lenslike medium with exponential heat distribution along the axial direction is, for the first time to our best knowledge, derived under assumptions of only radial heat flow and uniform pump spot size. An equivalent optical model is also developed and especially modified for a lenslike gain medium in a continuously single-end-pumped solid-state laser. The result is verified by discussing how it fits the practical cases much better than the previous simplified model and by numerical simulation in a practice problem. This work is especially important in cases with a relatively long gain medium, which has been used more and more frequently in recent years.

Gaussian beams in inhomogeneous media: A review

The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich’s method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich’s method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich’s method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names. The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.

Diffraction of a Gaussian beam in a three-dimensional smoothly inhomogeneous medium: an eikonal-based complex geometrical-optics approach

We present an ab initio account of the paraxial complex geometrical optics (CGO) in application to scalar Gaussian beam propagation and diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with quadratic expansion of the complex eikonal and reduces the wave problem to the solution of ordinary differential equations of the Riccati type. This substantially simplifies the description of Gaussian beam diffraction as compared with full-wave or parabolic (quasi-optics) equations. For a Gaussian beam propagating in a homogeneous medium or along the symmetry axis in a lenslike medium, the CGO equations possess analytical solutions; otherwise, they can be readily solved numerically. As a nontrivial example we consider Gaussian beam propagation and diffraction along a helical ray in an axially symmetric waveguide medium. It is shown that the major axis of the beam's elliptical cross section grows unboundedly; it is oriented predominantly in the azimuthal (binormal) direction and does not obey the parallel-transport law.

Propagation of Ince-Gaussian beams in a thermal lens medium

The propagation of Ince-Gaussian beams in a thermal lens medium is studied in this paper. Based on the ABCD matrix for Gaussian beams passing through a thermal lens medium, distinct expressions for the beam transverse intensity distributions and the longitudinal phase shift are deduced and discussed. Similar to Laguerre and Hermite-Gaussian beams, Ince-Gaussian beams, which constitute the third complete family of exact and orthogonal solutions of the paraxial wave equation, can also be used in other inhomogeneous media such as lenslike media and saturated absorption media.

Recurring beams in hollow metal waveguides: paraxial approximation.

For optical and near-optical applications in electromagnetics, the directed propagation of waves in free space and in lenslike media is often in the Cartesian form of Gaussian or more general Hermite-sinusoidal-Gaussian beams. It has been shown that recurring (rather than continuing) forms of such beams are possible in the paraxial approximation for certain hollow metal waveguides, in which multiple reflections from the waveguide walls may occur. Limitations on this recurrence behavior implicit in use of the paraxial approximation are considered here, and estimates are obtained for the maximum propagation distance before the onset of significant distortion of the recurring beams.

Beam propagation in a linear or nonlinear lens-like medium usingABCD ray matrices: the method of moments

We derive exact expressions for the evolution of the second order moment of the intensity distribution of an arbitrary beam propagating in a nonlinear Kerr medium with a quadratic index profile. The results can be recast in terms of theABCD matrix formalism after introducing a generalized complex radius of curvature,Q(z). Various definitions of the beam quality factor are introduced. Numerical simulations reveal the interest of this approach.

Propagation of Gaussian beams in dielectric waveguides

The propagation of Gaussian beams in both a homogeneous medium and a lenslike medium is treated. The correct Helmholz equation derived by the author is used to obtain the analytic solutions, taking into account the diffraction effect. Based on this, the beamwidth and the form of the wavefront are considered. The beamwidth will increase monotonically in the homogeneous medium and will be sinusoidal in the lenslike medium; that is, in the lenslike medium, the width will alternately increase and decrease. >

Transformation of a Gaussian Beam into a Uniform Beam by a Lenslike Medium with Gain or Loss

The propagation of Gaussian beams through lenslike media with gain or loss is analysed in order to discuss the transformation of these beams into uniform beams. The results obtained are applied to a lenslike medium with gain increasing along the radial direction. © 1991 Taylor & Francis Ltd.

Off-axis complex-argument polynomial-Gaussian beams in optical systems

Off-axis light-beam propagation in optical systems representable by complex ABCD matrices is considered in detail. The results are presented in terms of modal parameter transformations similar to Kogelnik’s ABCD law. The modes emphasized here can be expressed in terms of polynomial-Gaussian functions of complex argument. These modes may be both astigmatic and asymmetric and can describe the propagation of off-axis beams of light in lenslike media with spatial gain or loss variation.

Focal Shift in Lenslike Media with Gain or Loss

Abstract Focal shift in lenslike systems having loss or gain is evaluated. The results obtained show a shift in the geometrical focus with respect to the lossless case. The best focus is determined for uniform plane and Gaussian illumination.

Path-integral derivation of the complex ABCD Huygens integral

The Huygens integral, or Green’s function, for paraxial beam propagation in a generalized lenslike medium is derived as a Feynman path integral involving a summation over real rays. This formulation provides a physically intuitive understanding of the structure of the Green’s function. The path integral is evaluated by introducing in a consistent manner the notion of complex rays, and the familiar result for the Huygens integral in terms of the complex ABCD matrix of the medium is obtained.

Gaussian light beam in a lens-like medium with a nonlinear complex susceptibility

An analytic solution is obtained of the equation for a Gaussian light beam propagating in an inhomogeneous medium with a nonlinear complex susceptibility. It is shown that an allowance for nonlinear absorption alters greatly the condition for the waveguide propagation regime. The limits of validity of the parabolic approximation for the wave equation are identified. It is shown that self-focusing is possible even in the case when the real part of the nonlinear susceptibility is negative.

Representation of Gaussian beams by complex rays

This paper shows that a fundamental Gaussian beam propagating in a lenslike medium with cylindrical symmetry can be generated by the rotation about its axis of a skew ray which obeys the laws of geometrical optics. A complex representation: X(z) = ξ(z) + jη(z), where ξ(z) and η(z) are the projections of the skew ray on two perpendicular meridional planes, is discussed. It is found that the beam radius is equal to the modulus of X(z) and the on-axis phase to the phase of X(z). Using this representation, we derive a general expression for the on-axis phase shift ΔΦ experienced by a beam with an input complex beam parameter q through an optical system whose ray matrix is [ABCD]:ΔΦ=phase of(A+B/q). When the beam is matched to the optical system (output q = q), ΔΦ can be written cos–1(A + D)/2. This representation also provides a useful beam tracing method which is demonstrated and a simple interpretation for the known representation of Gaussian modes by ray packets.

Factorization of the transfer matrix for symmetrical optical systems

In the paraxial approximation a symmetrical optical system may be represented by a 2 × 2 matrix. It has been the custom to describe each optical element by a transfer matrix representing propagation between the principal planes or through an interface for thin elements. If the focal-plane representation is used instead, any focusing element or combination of elements is represented by the same antidiagonal matrix whose nonzero elements are the focal lengths: The matrix represents propagation between the focal planes. For propagation between any two arbitrary planes, the system transfer matrix can be decomposed into the product of two upper triangular matrices and an antidiagonal matrix. This decomposition yields the above-mentioned focal-plane matrix, and the two upper triangular matrices represent propagation between the input and the output planes and the focal planes. Because the matrix decomposition directly yields the parameters of interest, the analysis and the synthesis of optical systems are simpler to carry out. Examples are given for lenses, diopters, mirrors, periodic sequences, resonators, lenslike media, and phase-conjugate mirror systems.

Directivity of radiation leaving a misaligned resonator with a lenslike medium

Directivity of radiation leaving a misaligned resonator with a lenslike medium

Crosstalk characteristics of lens-like Ti-LiNbO3 intersecting channel waveguides

Crosstalk characteristics of intersecting channel waveguides are presented. Full diffusion times of various Ti film thicknesses are measured first. The concentration distributions in Ti-LiNbO3 crosschannel waveguides are then calculated and compared with the measured distribution. The lens-like medium is the crossover region of crosstalk between Δn and 2Δn crosschannel waveguide.

Erratum: Misaligned optical resonator with a lens-like medium [Sov. J. Quantum Electron. 10, 672–677 (June 1980)

Erratum: Misaligned optical resonator with a lens-like medium [Sov. J. Quantum Electron. 10, 672–677 (June 1980)

Comparison of Vector and Scalar Modes in a Lightguide with a Hyperbolic Secant Index Distribution

To develop a sound base on which to design efficient lightguides, it is necessary to understand the physics of the propagating modes of this transmission medium. In this paper, we examine the properties of the scalar modes in an infinite lens-like medium and the vector modes in a self-focusing fiber with an infinite homogeneous cladding for a hyperbolic secant index distribution. We find that only the meridional rays are equalized and that this only occurs when there is no material dispersion.

Misaligned optical resonator with a lens-like medium

A description is given of a method of calculating optical resonators based on solving a parabolic equation. By applying this method to unstable resonators with lens-like media, it is possible to describe the mode structure with misaligned mirrors (in addition to finding the known properties). Relationships between the spatial characteristics of the laser beam and the misalignment parameters of the mirrors and inhomogeneity of the medium are determined.