Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams

On the basis of the truncated second-order moments method in the cylindrical coordinate systems and the expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions, an approximate method used to calculate the generalized beam propagation factor (M2 factor) is proposed. The approximate analytical expressions of the generalized M2 factor for rotationally symmetric hard-edged diffracted flattened Gaussian beams defined by Gori [Opt. Commun. 107, 335 (1994)] and Li [Opt. Lett. 27, 1007 (2002)] are derived, respectively; we show that it depends on the beam order N and the beam truncation parameter delta. Some typical numerical examples are given to illustrate its applications that we compare by using the obtained analytical method and the numerical integration method.     

Generalized M2 factor of a partially coherent beam propagating through a circular hard-edged aperture

The second-order intensity moments and the beam-propagation M2 factor of partially coherent beams that propagate through a circular-symmetry hard-edged aperture are in the cylindrical coordinate system. AJo-correlated Schell-model beam with a Gaussian intensity distribution is an example. The analytical expression for the generalized M2 factor is derived. The numerical calculation results are analyzed.     

Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures

The second-order intensity moments and beam-propagation factor (M2 factor) of partially coherent beams have been generalized to include the case of hard-edged diffraction. A laser beam with amplitude modulation and phase fluctuation and a Gaussian Schell-model beam are taken as two typical examples of partially coherent beams. Analytical expressions for the generalized M2 factor are derived.     

Generalized M* factor of truncated partially coherent controllable dark-hollow beams

On the basis of the fact that a hard-edged aperture function can be expanded into an approximate sum of complex Gaussian functions with finite numbers and the method of truncated second-order moments, the generalized beam propagation factor of truncated partially coherent controllable dark-hollow beams is derived. Some typical numerical simulations are given to illustrate the relations of the generalized beam propagation factor to four parameters: beam parameter ε, beam order N, truncation parameter F and coherence parameter T.     

Generalized M2 factor of hard-edged diffracted flattened Gaussian beams

On the basis of generalized truncated second-order moments, a closed-form expression for the generalized M2 factor of hard-edge diffracted flattened Gaussian beams is derived that is determined by the beam order and the truncation parameter. Special cases are discussed. Moreover, it is shown that the M2 factor of truncated plane waves is equal to 4sqaure root3/3, independent of the aperture width.     

Complex Gaussian functions expansion method applied to truncated Gaussian beams

An analytical expression for the beam propagation factor (M ² factor) of truncated Gaussian beams was derived by using the complex Gaussian functions expansion method. The reasonability of the approximation of complex Gaussian functions expansion method is studied, and a comparison of this method with the generalised truncated second-order moments method and the asymptotic analysis method is also made. In general, an easy analytical expression for the M ² factor of truncated laser beams can be derived by using the complex Gaussian functions expansion method. The M ² factor obtained by using the complex Gaussian functions expansion method is more consistent with that in practice than that obtained by using two other methods. The analytical results obtained by using the complex Gaussian functions expansion method can be reduced to that for the non-truncated case when the truncation parameter is sufficiently large. Therefore, the complex Gaussian functions expansion method is a suitable approximation method for studying the M ² factor of truncated laser beams.     

Far-field intensity distribution and M2 factor of hollow Gaussian beams

The far-field intensity distribution of hollow Gaussian beams was investigated based on scalar diffraction theory. An analytical expression of the M2 factor of the beams was derived on the basis of the second-order moments. Moreover, numerical examples to illustrate our analytical results are given.     

Aliasing errors due to quadratic nonlinearities on triangular spectral /hp element discretisations

In this paper, consideration is given to how aliasing errors, introduced when evaluating nonlinear products, inexactly affect the solution of Galerkin spectral/hp element polynomial discretisations on triangles. A theoretical discussion is presented of how aliasing errors are introduced by a collocation projection onto a set of quadrature points insufficient for exact integration, and consider interpolation projections to geometrically symmetric ollocation points. The discussion is corroborated by numerica examples that elucidate the key features. The study is first motivated with a review of aliasing errors introduced in one-dimensional spectral-element methods (these results extend naturally to tensor-product quadrilaterals and hexahedra.) Within triangular domains two commonly used expansions are a hierarchical, or modal, expansion based on a rotationally non-symmetric collapsed-coordinate system, and a Lagrange expansion based on a set of rotationally symmetric nodal points. Whilst both expansions span the same polynomial space, the construction of the two bases numerically motivates a different set of collocation points for use in the collocation projection of a nonlinear product. The purpose of this paper is to compare these two collocation projections. The analysis and results show that aliasing errors produced using a collocation projection on the rotationally non-symmetric, collapsed-coordinate system are significantly smaller than those for a collocation projection using the rotationally symmetric nodal points. In the case of the collapsed coordinate projection, if the Gaussian quadrature order employed is less than half the polynomial order of the integrand, then it is possible for the aliasing error to modify the constant mode of the expansion and therefore affect the conservation property of the approximation. However, the use of a collocation projection onto a polynomial expansion associated with a set of rotationally symmetric nodal points within the triangle is always observed to be non-conservative. Nevertheless, the rotationally symmetric collocation will maintain the overall symmetry of the triangular region, which is not typically the case when a collapsed coordinate quadrature projection is used.     

Test optics error removal

Wave-front or surface errors may be divided into rotationally symmetric and nonrotationally symmetric terms. It is shown that if either the test part or the reference surface in an interferometric test is rotated to N equally spaced positions about the optical axis and the resulting wave fronts are averaged, then errors in the rotated member with angular orders that are not integer multiples of the number of positions will be removed. Thus if the test piece is rotated to N equally spaced positions and the data rotated back to a common orientation in software, all nonrotationally symmetric errors of the interferometer except those of angular order kNθ are completely removed. It is also shown how this method may be applied in an absolute test, giving both rotationally symmetric and nonsymmetric components of the surface. A general proof is given that assumes only that the surface or wave-front information can be described by some arbitrary set of orthognal polynomials in a radial coordinate r and terms in sin θ and cos θ. A simulation, using Zernike polynomials, is also presented.     

M2-factor of truncated partially coherent beams propagating through atmospheric turbulence

A method of studying the M2-factor of truncated partially coherent beams both in free space and in turbulence is proposed, i.e., the method of the window function being expanded into a finite sum of complex-valued Gaussian functions. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, the analytical formula of the M2-factor of truncated GSM beams propagating through atmospheric turbulence is derived. It is shown that the M2-factor decreases as the truncation parameter δ and the coherence parameter α increase. However, the M2-factor in turbulence is more sensitive to δ than that in free space. On the other hand, the M2-factor of truncated partially coherent beams with smaller δ is more affected by turbulence. In addition, the effect of turbulence on the M2-factor of truncated GSM beams is less sensitive to the coherence parameter α than that of nontruncated GSM beams.     

Air-Coupled Transmission Coefficient Reconstruction Using a 3-D Complex-Transducer-Point Voltage Model

Results are presented on the reconstruction of the transmission coefficient in air-coupled ultrasound experiments where the excitation is either a Gaussian sheet beam or a rotationally symmetric Gaussian beam. The effects of nonideal limitations in the method are also reported. The angular spectrum of the transmitted signal is obtained by performing a spatial Fourier transform on data acquired in a coordinate scan of the receiving transducer, an operation that yields, in the 2-D case, exactly the plane-wave transmission coefficient. This equality is shown using an analysis based on the complex transducer point technique. The transmission function reconstructed from a 3-D rotationally symmetric Gaussian beam differs only slightly, in isotropic media, from the plane-wave transmission coefficient. In addition, the influence of a finite coordinate scan and finite step size are also studied. As a demonstration, elastic constants of isotropic and anisotropic materials are obtained from reconstructed transmission functions by inverting the experimental data. The reconstructed results are compared with independent measurements.     

Weak forms for modelling of rotationally symmetric,multilayered structures,including anisotropic poro‐elastic media

A weak form of the anisotropic Biot's equation represented in a cylindrical coordinate system using a spatial Fourier expansion in the circumferential direction is presented. The original three dimensional Cartesian anisotropic weak formulation is rewritten in an arbitrary orthogonal curvilinear basis. Introducing a cylindrical coordinate system and expanding the circumferential wave propagation in terms of orthogonal harmonic functions, the original, geometrically rotationally symmetric three dimensional boundary value problem, is decomposed into independent two‐dimensional problems, one for each harmonic function. Using a minimum number of dependent variables, pore pressure and frame displacement, a computationally efficient procedure for vibro‐acoustic finite element modelling of rotationally symmetric three‐dimensional multilayered structures including anisotropic porous elastic materials is thus obtained. By numerical simulations, this method is compared with, and the correctness is verified against, a full three‐dimensional Cartesian coordinate system finite element model. Copyright © 2012 John Wiley & Sons, Ltd.     

Weak formulation of Biot's equations in cylindrical coordinates with harmonic expansion in the circumferential direction

A weak symmetric form of Biot's equation in cylindrical coordinates with a spatial Fourier expansion in the circumferential direction is presented. The solid phase displacement and the pore pressure are used as the dependent variables. The original three‐dimensional boundary value problem is here, due to the orthogonality of the harmonic functions and the rotationally symmetric geometry, decomposed into independent two‐dimensional problems, one for each harmonic function. This formulation provides a computationally efficient procedure for vibroacoustic finite element modelling of rotationally symmetric three‐dimensional multilayered structures including porous elastic materials. By numerical simulations, this method is compared with, and verified against, full three‐dimensional Cartesian coordinate system finite element models. Copyright © 2009 John Wiley & Sons, Ltd.     

Beam-quality optimization of partially polarized fields

For light fields propagating through rotationally symmetric first-order optical systems, the possibility of improvement of the beam-propagation factor is shown to arise when the vectorial behavior is taken into account. For partially polarized beams, we find the optimized value of the beam-quality parameter that can be attained by using this kind of system. This value is given in terms of the beam qualities associated with the transverse polarization components of the vector field. On the basis of the so-called intensity-moment formalism, the general conditions that should be fulfilled at some plane to reach such an optimized value are determined. A procedure to experimentally get the optimization conditions is also proposed.     

Fractional Fourier transform of truncated elliptical Gaussian beams

Based on the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical expression for an elliptical Gaussian beam (EGB) truncated by an elliptical aperture and passing through a fractional Fourier transform system is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of truncated EGBs than the usual way by using the integral formula directly, and the efficiency of numerical calculation is significantly improved.     

Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system

On the basis of expanding a hard-edged aperture function as a finite sum of complex Gaussian functions, an approximate analytical expression for the propagation of an input complex amplitude distribution passing through a general nonsymmetrical apertured double-lens system is derived. Then, the propagation result for two-dimensional flat-topped multi-Gaussian beams is given. It is shown that the apertured Lohmann's symmetrical double-lens system for fractional Fourier transform is a special case of the general apertured double-lens system. The numerical calculation, graphical illustration, and some discussions for the transformation of the two-dimensional flat-topped multi-Gaussian beam in apertured Lohmann's symmetrical double-lens systems are also presented.     

Coupling characteristics and kurtosis parameter of partially coherent beams in turbulent atmosphere

Coupling properties and kurtosis parameter (K parameter) of arbitrary beams propagating through atmospheric turbulence are investigated. A correlation factor (C⁴-factor) is introduced to describe the influence of turbulence on coupling characteristics. The general analytical expression for C⁴-factor of arbitrary beams in atmospheric turbulence is derived. It is shown that C⁴-factor of arbitrary beams in the turbulent atmosphere depends on the initial second-order moments and fourth-order moments and turbulence quantities. Taking the partially coherent anomalous elliptical hollow Gaussian (PCAEHG) beam as an example, we can obtain that C⁴-factor decreases as structure constant of the refractive index fluctuations and inner scale increase, and waist width and transverse coherence length decrease when z?>?5?km. Moreover, K parameter of PCAEHG beam in turbulent atmosphere converges to 2 when propagation distance is large enough. It indicates that the profile of PCAEHG beams in turbulent atmosphere finally evolves into fundamental Gaussian distribution.     

复宗量双曲正弦高斯光束的传输特性

对复宗量双曲正弦高斯(EshG)光束的一些基本特性进行了研究。由Collins公式出发推导出了光束通过近轴ABCD光学系统传输的场分布函数,根据二阶矩的定义和光束在近轴ABCD光学系统中的传输公式推导出了光束通过近轴ABCD光学系统光斑尺寸的解析表达式。得到了复宗量双曲正弦高斯光束在自由空间传输时束腰宽度和位置的解析表达式以及M2因子的表达式。对光强分布、光斑尺寸、束腰宽度及其位置和M2因子进行了数值计算,并对计算结果作了分析。本文所得结果具有广泛的意义,因为正弦高斯光束、双曲正弦高斯光束和复宗量正弦高斯光束均可视为其特例。     

One-dimensional description of cylindrically symmetric laser beams: application to bessel-type nondiffracting beams

We introduce a new representation of coherent laser beams that are usually described in circular cylindrical coordinates. This representation is based on the decomposition of a laser beam of a given azimuthal order into beams exhibiting Cartesian symmetry. These beams, which we call constituent waves, diffract along only one of their transverse dimensions and propagate noncollinearly with the propagation axis. A cylindrically symmetric laser beam is then considered a coherent superposition of constituent waves and is represented by an integral over an angular variable. Such a representation allows for the introduction of the propagation factor M2, defined in terms of one-dimensional root-mean-square (rms) quantities, in the treatment of two-dimensional beams. The representation naturally leads to the definition of a new rms parameter that we call the quality factor Q. It is shown that the quality factor defines in quantitative terms the nondiffracting character of a laser beam. The representation is first applied to characterize Laguerre-Gauss beams in terms of these one-dimensional rms parameters. This analysis reveals an asymptotic link between Laguerre-Gauss beams and one-dimensional Hermite-Gauss beams in the limit of high azimuthal orders. The representation is also applied to Bessel-Gauss beams and demonstrates the geometrical and one-dimensional characters of the Bessel-Gauss beams that propagate in a nondiffracting regime. By using two separate rms parameters, Q and M2, our approach gives an alternative way to describe laser beam propagation that is especially well suited to characterize Bessel-type nondiffracting beams.     

Imaging with tilted surfaces: an efficient matrix method for the generalized Scheimpflug condition and its application to rotationally symmetric triangulation

An efficient two-dimensional matrix method is presented that facilitates the design of optical systems with tilted surfaces for which the requirement or knowledge of the orientation of the image plane is necessary, i.e., for which a generalized Scheimpflug condition is needed. In more general terms, the method results in imaging properties of second-order expansion, but the method is linear. Therefore the complexity of the design process is considerably reduced. The strength of the design method is demonstrated in detail for a novel application in which a reflective optical system of several surfaces is required for rotationally symmetric triangulation.