Bessel and conical beams and approximation with annular arrays

The Bessel beam is one of the relatively new limited-diffraction beams that have been discovered. It is compared with the conical transducer, which also gives an approximate limited-diffraction solution to the wave equation. The conical transducer's field deviates from the predicted field in the nearfield, where it is wider. Therefore, the Bessel beam is better for use in a hybrid system where a limited-diffraction beam is used for transmission and a dynamically focused beam for reception. The limited-diffraction Bessel beam of order zero can be excited on an annular transducer with equal-area division of elements and with a fixed prefocus, i.e., conventional transducers used in commercial medical imaging equipment. The element division implies that the scaling parameter must be chosen to contain the first lobe of the Bessel function in the first element. In addition, the prefocus must be such that the array is steerable to infinite depth with minor loss. Even when the Bessel beam yields a larger depth of field than that of an unfocused transducer, it has the advantage of a narrower beam. Simulated examples are shown where the approximate Bessel beam compares favorably with a spherically focused beam with a fixed focus, an unfocused beam, and a conical transducer.     

From stationary annular rings to rotating Bessel beams

In this work we use a phase-only spatial light modulator (SLM) to mimic a ring-slit aperture, containing multiple azimuthally varying phases at different radial positions. The optical Fourier transform of such an aperture is currently known and its intensity profile has been shown to rotate along its propagation axis. Here we investigate the near-field of the ring-slit aperture and show, both experimentally and theoretically, that although the near-field possesses similar attributes to its Fourier transform, its intensity profile exhibits no rotation as it propagates.     

Extraction of high-quality beams from narrow annular laser sources

We have analyzed the propagation properties of the field produced by circular arrays of coherent optical sources, obtaining the conditions for the generation of good-quality global beams. Such conditions can be obtained by the use of resonators based on the Talbot effect. This appears to be a practical method for the construction of simple, low-cost resonators for compact diffusion-cooled high-power lasers with annular format. Low-loss annular Talbot cavity configurations have also been studied.     

Modeling of CW annular arrays using limited diffraction Bessel beams

Presents a method for characterizing the linear field of any flat, continuous-wave annular array in terms of a set of known limited-diffraction Bessel beams. The technique uses Fourier-Bessel series to model the surface pressure of the transducer surface, with each term in the series giving rise to a Bessel beam with known propagation parameters. The analysis is applied numerically to two different transducers discussed previously in the literature. In both cases, a deeper understanding of the field emitted than was previously available is gained. Brief outlines for extending the technique to pulsed wave and non-annular arrays are also given     

Aberration limits for annular Gaussian beams for optical storage

Annularly apodized beams have been suggested for use in optical storage because of their potential to go beyond the conventional spot size and depth-of-focus limits. One concern for such applications is the effects of small aberrations on beams in which the energy is concentrated in a small annular ring. We present calculations and experimental results that show that annular apodization of a Gaussian beam reduces the sensitivity to defocus as well as balanced spherical and coma aberrations. The sensitivity to astigmatism is increased by a small amount.     

环状光束通过球差透镜的聚焦特性

利用偏心高斯光束相干叠加的方法,建立了柱坐标系下环状光束的新模型.从广义衍射积分理论出发,经过大量数值计算分析,详细研究了环状光束通过球差透镜的聚焦特性,定量分析了环状光束的阶数L,M、偏心高斯光束的束腰宽度w0以及球差系数C4对聚焦光场实际焦点(轴上最大光强)位置以及轴上最大光强的影响.研究结果表明,本文提出的模型可统-描述基模高斯光束和环状光束.在透镜焦距和入射光束波长一定的情况下,环状光束通过球差透镜的聚焦特性不仅与透镜的球差系数C4有关,而且还与环状光束的阶数L,M以及偏心高斯光束的束腰宽度w0有关.     

Propagation of elliptical Gaussian beams modulated by an elliptical annular aperture

An analytical propagation expression for a generalized astigmatic elliptical Gaussian beam (EGB) modulated by an elliptical annular aperture and passing through an axially nonsymmetrical optical system is obtained by the use of vector integration. The derived analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual method of using the diffraction integral directly, and the efficiency of the numerical calculation is significantly improved. Some numerical simulations are illustrated for the propagation properties of EGBs passing through a free space with an elliptical annular aperture, an elliptical screen, or an elliptical aperture. Further extensions are also pointed out.     

Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium

Log-amplitude and phase-correlation and structure functions of higher-order annular laser beams in a turbulent atmosphere are derived. A higher-order annular beam source is defined as the superposition of two different higher-order Hermite-Gaussian beams. A special case of such an excitation is the annular Gaussian beam in which two beams operate at fundamental modes of different Gaussian beam sizes, yielding a doughnut-shaped (annular) beam when the second beam is subtracted from the first beam. Our formulation utilizes Rytov approximation, which makes it applicable in the weak-turbulence regime, especially for log-amplitude fluctuations. Limiting cases of our formulations correctly match with known higher-order-mode solutions that in turn reduce to the Gaussian-beam-wave (TEM00-mode) results. Our results can be applied to determine the scintillation index and the phase fluctuations in free-space optics links under higher-order annular laser beam excitation. Except for the numerical evaluation of a specific example covering an annular Gaussian beam, the results in general are left in integral form and need to be numerically evaluated in detail to obtain quantitative results.     

Possibility of excitation of azimuthal surface waves in a magnetized plasma by annular ion beams

Excitation of the azimuthal surface eigenmodes with the extraordinary polarization for the ion plasma component is shown to be possible in cylindrical waveguides with metal walls, filled partially by a cold magnetoactive neon plasma. The interaction of the oscillations with the flows of alpha particles rotating around the plasma column in a narrow layer that separates the plasma from the waveguide walls is studied. If the external magnetic field is strong enough, the resonance interaction of the beam with the waves can be realized for three minimum values of the azimuthal mode number when the wave frequency normalized by neon ion cyclotron frequency is close to the factor of five (m _Ne/m _He ≈ 5).     

Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for a Gaussian beam passing through a paraxial ABCD optical system with an annular aperture has been derived. The results could be reduced to the case of circular black screen or circular aperture. Some numerical simulations are also performed and illustrated for the propagation characteristics of a Gaussian beam through a paraxial ABCD optical system with an annular aperture, a circular black screen or a circular aperture.     

Scintillations of incoherent flat-topped Gaussian source field in turbulence

The intensity fluctuations of incoherent flat-topped Gaussian beams are evaluated when such sources are used in weakly turbulent horizontal atmospheric links. The formulation is developed for a detector having a response time much longer than the source coherence time. The flat-topped Gaussian profile is obtained by superposing many Gaussian beams, then the incoherence is introduced through delta correlation in space. The scintillation index of the incoherent flat-topped Gaussian beams is found to be smaller than the scintillation index of the corresponding incoherent Gaussian beams at the same link length, source size, and wavelength. When compared with the coherent counterparts, the intensity fluctuations of the incoherent flat-topped Gaussian beams are much smaller, yielding the same value only at the spherical wave limit, as expected. Transmitter aperture averaging is a special case of our solution.     

Saturation of Scintillations for a Light Beam in a Turbulent Medium

Assuming that the intensity of a light beam propagating along the O z axis in a turbulent medium can be written I(z) = I ₀ exp [m A(z)] where A(z) and I(z) are random fields, we discuss, for A and I, some probability distributions which could give results in agreement with experimental data, that is saturation of scintillations for large z and a log-normal distribution of I for small z; we prove that only some distributions are suitable and discuss a physical model of light propagation consistent with one of these distributions.     

Zernike annular polynomials and optical aberrations of systems with annular pupils

Zernike annular polynomials that represent orthogonal andbalanced aberrations suitable for systems with annular pupilsare described. Their numbering scheme is the same asfor Zernike circle polynomials. Expressions for standard deviationof primary and balanced primary aberrations are given.     

Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials

A general wavefront fitting procedure with Zernike annular polynomials for circular and annular pupils is proposed. For interferometric data of typical annular wavefronts with smaller and larger obscuration ratios, the results fitted with Zernike annular polynomials are compared with those of Zernike circle polynomials. Data are provided demonstrating that the annular wavefront expressed with Zernike annular polynomials is more accurate and meaningful for the decomposition of aberrations, the calculation of Seidel aberrations, and the removal of misalignments in interferometry. The primary limitations of current interferogram reduction software with Zernike circle polynomials in analyzing wavefronts of annular pupils are further illustrated, and some reasonable explanations are provided. It is suggested that the use of orthogonal basis functions on the pupils of the wavefronts analyzed is more appropriate.     

Systematic comparison of the use of annular and Zernike circle polynomials for annular wavefronts

The theory of wavefront analysis of a noncircular wavefront is given and applied for a systematic comparison of the use of annular and Zernike circle polynomials for the analysis of an annular wavefront. It is shown that, unlike the annular coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is identically the same as that with the corresponding annular polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. The interferometer setting errors of tip, tilt, and defocus from a four-circle-polynomial expansion are the same as those from the annular-polynomial expansion. However, if these errors are obtained from, say, an 11-circle-polynomial expansion, and are removed from the aberration function, wrong polishing will result by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for an annular pupil. The various results are illustrated analytically and numerically by considering an annular Seidel aberration function.     

A further comparative study of flattened gaussian beams and super-Gaussian beams

Abstract

A further comparison between flattened Gaussian beams (FGBs) and super-Gaussian beams (SGBs) is performed. It is shown that the two FGB and SGB having the same beam propagation factor (M²?factor) but different waist widths demonstrate a similar irradiance profile at the position of the equal generalized Fresnel number, while they propagate through a paraxial ABCD system. Furthermore, the focusing by a spherically aberrated lens is studied, and nearly the same propagation characteristics are achieved for the two FGB and SGB with the same M²?factor and waist width.     

Decentered Gaussian beams, ray bundles, and Bessel-Gauss beams

The concept of ray parameters for decentered Gaussian beams is developed on a formal basis. When the beam propagates through first-order optical systems, these parameters are transformed as the ray parameters of geometrical optics. It is shown how this feature helps one to understand the behavior of more sophisticated beams that can be considered as bundles of decentered Gaussian beams. In particular, the case of Bessel-Gauss beams and their recently introduced generalizations is analyzed, and simple transformation formulas are obtained.     

Zernike annular polynomials and atmospheric turbulence

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.     

Behaviour of concrete beams under torsion: NSC plain and hollow beams

A simple computation procedure is developed to predict the general behaviour of reinforced concrete beams under torsion. Both plain and hollow normal strength concrete beams are considered. Different theoretical models are used to reflect the actual behaviour of the beams in the various phases of loading. To pass from a phase to the following one, transition criteria need to be taken into consideration. Such criteria are explained. The theoretical predictions are compared with result from reported tests. Conclusions are presented. The main conclusion is that the calculation procedure described in this paper gives good predictions when compared with the actual behaviour of the plain and hollow beams.