Diffraction of Gaussian beams by periodic and aperiodic rulings

The diffraction of Gaussian beams by periodic and aperiodic rulings is considered. The theory of diffraction is based on the Rayleigh-Sommerfeld integral equation with Dirichlet conditions. The transmitted power and the normally diffracted energy are analyzed as a function of the beam radius. Two methods to determine the Gaussian beam radius by means of periodic and aperiodic lamellar gratings are proposed. One is based on the maximum and the minimum transmitted power, and the other one considers the normally diffracted energy. Small and large Gaussian beam radii can be treated with these two methods.     

Bowtie limited diffraction beams for low-sidelobe and large depthof field imaging

Limited diffraction beams such as Bessel beams and X waves have a large depth of field and thus could have many applications. However, these beams have higher sidelobes as compared to conventional focused beams in their focal planes. In this paper, a new class of limited diffraction beams is developed. These beams are termed bowtie limited diffraction beams because they have bowtie shapes in a plane perpendicular to the beam axis. To obtain pulse-echo images of low sidelobes and a large depth of field, a bowtie limited diffraction beam is used in transmission and its 90° rotated response (around the beam axis) is used in reception. Unlike the summation-subtraction method developed previously, this method does not reduce image frame rate or dynamic range of signals and is not motion sensitive. The theory of the bowtie limited diffraction beams is developed. Computer simulation of the theoretical beams under practical conditions, such as finite aperture, finite bandwidth, and causal excitation, is performed with the Rayleigh-Sommerfeld diffraction formula. The simulated beams are very close to those predicted analytically over a large depth of field     

A new approach to obtain limited diffraction beams

Limited diffraction beams could have applications in medical imaging, tissue characterization, and nondestructive evaluation, as well as other wave related areas such as electromagnetics and optics. In this paper, we develop a novel approach that can convert any diffracting solution of the isotropic-homogeneous wave equation to a limited diffraction solution. As an example, this approach was applied to an n-dimensional wavelet solution that we generalized from the three-dimensional solution obtained by Kaiser et al. (1992). This example establishes a relationship between localized limited diffraction beams and the wavelet theory. The resulting limited diffraction beam was compared with those discovered previously     

A beam theory for thin-walled composite beams

A beam theory is presented that is formulated in terms of the in-plane elastic properties of the panels of the cross-section of a thin-walled composite beam. Shear deformation is accounted for by using a suitable form of the Timoshenko beam theory together with a modified form of the shear coefficient. The theory gives both the bending deflection and the shear deflection of a beam loaded by an applied transverse load. Numerical and graphical results obtained from a computer code show the effects of using different composite material systems and lay-ups in the panels of typical beams.     

Designing limited diffraction beams

Theoretically, limited diffraction beams can only be produced with an infinite aperture. In practice, they can be closely approximated with a finite aperture over a large depth of field. Because of this property, these beams could have applications in medical imaging, tissue characterization, Doppler velocity estimation, and nondestructive evaluation (NDE) of materials, as well as other physics-related areas such as electromagnetics and optics. In this paper, a new method is developed to design limited diffraction beams of desired beam shapes within a finite aperture of interest. It uses previously discovered limited diffraction beams such as Bessel beams and X waves as basis functions, and constructs new beams with linear superpositions of the bases. To construct a new beam of a desired shape, coefficients of the basis functions in the linear superposition are chosen so that the difference between the new beam and a desired beam is minimized under the criterion of least-squares error within the aperture. This procedure is implemented by digitizing both the basis beams and desired beams in the aperture and solving a system of linear equations from its normal equation. The method is applied to several desired beams that are limited diffraction beams known previously. Results show that the designed beams and the desired beams are virtually identical. If the desired beams are not solutions to the wave equation, the designed beams are new limited diffraction beams that are similar in shapes to the desired beams. This suggests that the method may be a powerful and practical tool for developing new limited diffraction beams of desired properties.     

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.     

Further study on the refined theory of rectangle deep beams

Based on the refined theory for narrow rectangular deep beams, two different displacement boundary conditions of the fixed end of a cantilever beam are used to study the deformation of the beam. One is the conventional simplified displacement boundary condition, and the other is a new boundary condition determined by the least squares method. Three load cases are investigated, which are a transverse shear force at the free end of the beam, a uniformly distributed load at the top surface, and a linearly distributed load at the top surface, respectively. Solutions are given for both the refined theory and the Timoshenko beam theory and are compared with the known solutions from the elastic theory and results by the finite element method. It is shown that the solutions of the refined theory coincide with those of the elastic theory; the solutions from the Timoshenko theory by using the two different displacement boundary conditions are the same; the refined theory by using the new boundary condition provides better results than using the conventional boundary condition and also better than those of the Timoshenko beam theory.     

Quasi-diffraction-free beams.

A diffraction-free beam is obtained by the superposing of plane waves whose wave vectors make an angle with the propagation axis. These plane waves are realized with point sources that are distributed uniformly around a circle and an infinitely large aperture lens. After the field passes through the lens it has nondiffracting properties and is described by the zero-order Bessel function. Relaxing these conditions makes the beam diffraction free within only a limited region. The beam generated from such a geometry is referred to as a quasi-diffraction-free beam. The effects of the width of the annular source on the beam spread are discussed and compared with those for a Gaussian beam. Approximate expressions for quasi-diffraction-free beams are also obtained.     

Generation of nondiffracting quasi-circular polarization beams using an amplitude modulated phase hologram

We propose an approach to the generation of nondiffracting quasi-circularly polarized beams by a highly focusing azimuthally polarized beam using an amplitude modulated spiral phase hologram. Numerical verifications are implemented in the calculation of the electromagnetic fields and Poynting vector field near the focus based on the vector diffraction theory, and the polarization of the wavefront near the focal plane is analyzed in detail by calculating the Stokes polarization parameters. It is found that the electric field, magnetic field, and Poynting vector field can simultaneously be uniform and nondiverging over a relatively long axial range of ~7.23λ. In the transverse plane, the ellipticity and azimuthal angle of the local polarization ellipse varies from point to point. No polarization singularity and phase singularity are found at the beam center, which makes the bright spot possible.     

Interferometric phase reconstruction by nonuniform shifting of the reference beam

A method for phase measurement in common-path interferometers, believed to be novel, is presented. We use the property of phase reconstruction algorithms, such as the Carré and Hariharan algorithms, that do not require uniform phase across the reference beam. Only the ratio of the phase steps must be the same at each pixel. We show phase measurement and reconstruction in a common-path interferometer by shifting either the tilt or the focus of the reference wave front. We present a theoretical explanation of phase measurement using this property. We also present results from a proof-of-principle experiment using a scatterplate interferometer, in conjunction with the tilt phase-shifting technique, to measure the reflected phase of a test optical element. Furthermore, we present a computer simulation to demonstrate the mathematical validity of this measurement technique using defocus shifting, rather than tilt shifting, in the reference wave front.     

Diffraction of optical communication Gaussian beams by volume gratings: comparison of simulations and experimental results

The diffraction effects induced by a thick holographic grating on the propagation of a finite Gaussian beam are theoretically analyzed by means of the coupled-wave theory and the beam propagation method. Distortion of the transmitted and diffracted beams is simulated as a function of the grating parameters. Theoretical results are verified by experimentation realized by use of LiNbO3 volume gratings read out by a 1550-nm Gaussian beam, typical of optical fiber communications. This analysis can be implemented as a useful tool to aid with the design of volume grating-based devices employed in optical communications.     

A methodology for identification of damage in beams

In this work, a parameter based on correlation of signals using mathematical morphology and other two parameters based on energy of a vibration signal using wavelet transform and bispectrum theory are used for approximate location of damage in a steel clamped-free beam. The experimental data are obtained through accelerometers placed along the sample. The system is excited using impact hammer. A preliminary location of the damage is found using the position of the accelerometer that has a higher energy change when it is compared to the signal of the system with and without damage. The mathematical models are obtained using 2D elasticity theory and the finite element method. The numerical and experimental data are approximated using the particle swarm optimization method, and this way, it is possible to adjust the location and severity of the damage. The procedure is also applied to a free–free steel beam with two damages and a sandwich beam.     

Airy pulsed beams

The Airy beam (AiB) has attracted a lot of attention recently because of its intriguing features; the most distinctive ones are the propagation along curved trajectories in free space and the weak diffraction. We have previously shown that the AiB is, in fact, a caustic of the rays that radiate from the tail of the Airy function aperture distribution. Here we derive a class of ultra wideband Airy pulsed beams (AiPBs), which are the extension of the AiB into the time domain. We introduce a frequency scaling of the initial aperture field that renders the ray skeleton of the field, including the caustic, frequency independent, thus ensuring that all the frequency components propagate along the same curved trajectory and that the AiPB does not disperse. The resulting AiPB preserves the intriguing features of the time-harmonic AiB discussed above. An exact closed-form solution for the AiPB is derived using the spectral theory of transients. We also derive wavefront approximations for the field in the time window around the pulse arrival, which are valid uniformly in the vicinity of the caustic. These approximations are based on the so-called uniform geometrical optics, which is extended here to the time domain.     

Sharper focal spot formed by higher-order radially polarized laser beams

The intensity distributions near the focal point for radially polarized laser beams including higher-order transverse modes are calculated based on vector diffraction theory. For higher-order radially polarized mode beams as well as a fundamental mode (R-TEM01*) beam, the strong longitudinal component forms a sharper spot at the focal point under a high-NA focusing condition. In particular, double-ring-shaped radially polarized mode (R-TEM11*) beams can effectively reduce the focal spot size because of destructive interference between the inner and the outer rings with pi phase shift. Compared with an R-TEM01* beam focusing in a limit of NA=1, the full width at half-maximum values of the focal spot for an R-TEM11* beam are decreased by 13.6% for the longitudinal component and 25.8% for the total intensity.     

Elliptic Laguerre-Gaussian beams

An analytical expression for the diffraction of an elliptic Laguerre-Gaussian (LG) beam is derived and analyzed. We show that a beam with even singularity order has nonzero axial intensity for any degree of ellipticity and at any finite distance z from the initial plane, whereas at z = 0 and z = infinity the axial intensity is zero. We show that for a beam with a small degree of ellipticity and even order of singularity, two isolated intensity zeroes appear in the Fresnel zone on a straight line at an angle of 45 deg or -45 deg, depending whether the beam's spin is right or left. The theoretical conclusions are confirmed by numerical simulation and physical experiments.     

Beam wander of partially coherent Airy beams

The beam wander of a partially coherent Airy beam in a turbulent atmosphere was investigated. By using the extended Huygens–Fresnel integral, as analytical expression is derived for the second-order moment of a partially coherent Airy beam. Based on the theory proposed by Andrews, a general expression is obtained for the beam wander of a partially coherent Airy beam. With the help of the expression, various factors which impact on the beam wander are illustrated numerically. The results show that the beam wander of a partially coherent Airy beam decreases with the increase of the characteristic scale and the decrease of the coherent length or the exponent truncation factor. The value of the beam wander is a maximum when the exponent truncation factor is 0.63, no matter what the coherent lengths are. Our results provide an effective way to control the beam wander of a partially coherent Airy beam in practice.     

The refined theory of magnetoelastic rectangular beams

Summary. The problem of deducing a one-dimensional theory from a three-dimensional theory for a soft ferromagnetic elastic isotropic body is investigated. Based on the linear magnetoelasticity, the refined theory of magnetoelastic beams is presented by using the general solution for the soft ferromagnetic elastic solids and the Lure method. Based on the refined theory of magnetoelastic beams, the exact equations and solutions for the homogeneous beams are derived and the equations can be decomposed into three governing differential equations: the fourth-order equation, the transcendental equation and the magnetic equation. Moreover, the approximate equations and solutions for the beam under transverse loadings and magnetic field perturbations are derived directly from the refined beam theory. By omitting higher order terms and coupling effects, the refined beam theory can be degenerated into other well-known elastic and magnetoelastic theoretical models.     

High-speed cross-sectional imaging of valuable documents using common-path swept-source optical coherence tomography

A common-path swept-source optical coherence tomography (SS-OCT) is a promising scheme for implementing a high-speed and stable OCT system. We investigate the capability of a common-path SS-OCT system to perform the cross-sectional imaging of valuable documents translated at high speed for the check of its security feature. The influence of transport speeds, up to 2000 mm/s, on the depth resolution and the signal intensity is experimentally evaluated using a SS-OCT system equipped with a swept source at a center wavelength of 1335 nm and with a sweep repetition rate of 50 kHz. The degradation of the measured signal is in good agreement with theory.     

Focusing of radially and azimuthally polarized beams through a uniaxial crystal

We calculated and measured the difference between focal positions of radially and azimuthally polarized beams after passing through a uniaxial crystal. Calculations were carried out on the basis of the ray optics and the vector diffraction theory. The results of the calculations were in good agreement with those of the experiment. In addition, we discussed the polarization selection in a hemispherical laser cavity that was used for the generation of a radially polarized beam by use of the birefringence of a c-cut Nd:YVO4 laser crystal [Opt. Lett. 31, 2151 (2006)]. The stability range of the laser cavity length for the generation of a radially polarized beam was also in good agreement with the differences mentioned above.