High-order angular response beamformer for vector sensors

A linear processing scheme for computing higher-order angular response modes of a vector sensor is described. Examples of modal response beampatterns are presented. The response modes form (in principle) a complete, orthonormal set that can be transformed into steerable sets of one or more directive beams. The linear processing scheme facilitates calibration of vector sensor measurement systems. The angular resolution that can be achieved with the new processing scheme is predicted to be (155/N_m) degrees, where N_m is the highest order of computed response mode, for the higher orders. The number of higher-order response modes appears to be limited only by the computational power available.     

Orthonormal mode sets for the two-dimensional fractional Fourier transformation

A family of orthonormal mode sets arises when Hermite-Gauss modes propagate through lossless first-order optical systems. It is shown that the modes at the output of the system are eigenfunctions for the symmetric fractional Fourier transformation if and only if the system is described by an orthosymplectic ray transformation matrix. Essentially new orthonormal mode sets can be obtained by letting helical Laguerre-Gauss modes propagate through an antisymmetric fractional Fourier transformer. The properties of these modes and their representation on the orbital Poincaré sphere are studied.     

Hypergeometric modes

A new family of paraxial laser beams that form an orthogonal basis is discussed. When propagated in uniform space, these beams preserve their structure to scale. The intensity distribution profile for such beams is similar to that for the Bessel modes, representing a set of alternating bright and dark concentric rings. The complex amplitude of these beams is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric (HyG) modes. The HyG modes are generated with a liquid-crystal microdisplay.     

Characterization of vector diffraction-free beams

It is observed that a constant unit vector denoted by I is needed to characterize a complete orthonormal set of vector diffraction-free beams. The previously found diffraction-free beams are shown to be included as special cases. The I-dependence of the longitudinal component of diffraction-free beams is also discussed.     

Design of optical coupling systems between two-dimensional quasi-stadium laser diodes and single-mode optical fibers

Optical coupling systems between a two-dimensional quasi-stadium laser diode and single-mode optical fibers using gradient-index rod lenses are designed for both stable and unstable laser resonators for the first time. A novel numerical approach using a combination of the extended Fox-Li calculation method and Gaussian beam transformations is proposed. In the case of a stable laser resonator, two kinds of beam propagation modes appear, namely the axis mode, in which an optical beam propagates along the cavity axis, and the ring mode, in which optical beams propagate along the diamond-shaped trajectory. The coupling efficiencies are found to be 54% for the axis mode and 52% for the ring mode. In contrast, an unstable laser resonator exhibits complicated modes, in which several highly directional beams are emitted from the end mirrors. The total coupling efficiency for these output beams is calculated to be 9.6%. The 3-dB tolerances for the lens pitch and alignment angles of the gradient-index rod lenses are also discussed.     

Light beams with flat-topped profiles

A necessary condition for generating optical beams with a nearly uniform profile at a certain transverse plane is proposed. This proposal leads to the introduction of a set of light beams with flat-topped profiles. This set of beams can be expressed as a finite series of lowest-order Gaussian modes (TEM(00)) with different parameters. The main features of this set of beams are investigated and compared with those predicted by existing models of flattened Gaussian beams.     

Adaptive optics for airborne platforms—Part 1: modeling

Adaptive optics systems mitigate the atmospheric turbulence-induced distortion of a propagating light wavefront. The use of adaptive optics entails the design of a feedback controller, which requires the development of a model of the plant to be controlled. In adaptive optics, the plant consists of the atmosphere through which light is traveling. Moreover, a distinct feature of the adaptive optics control application is the presence of random signals in the plant. In optics, Zernike orthonormal polynomials are commonly used as a basis set for the expansion of wavefront phase distortions. Due to the atmospheric turbulence-induced random nature of the underlying physical process, the spatial-temporal correlation functions of the Zernike polynomial phase distortion expansion coefficients must be evaluated if a proper stochastic model of the plant is to be developed and adaptive optics is to be employed. In Part 1 of this paper, these correlation functions are developed using a layered atmospheric model and calculations for the first few low-order Zernike modes are performed. Using these correlation functions, an underlying stochastic linear dynamical system, which is adequate for control design, is synthesized. This system models the plant and, in turn, provides the basis for the employment of advanced model-based control and estimation concepts in an adaptive optics system for an airborne platform application.     

Hypergeometric-Gaussian modes

We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric-Gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes [Opt. Lett. 32, 742 (2007)], the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some subfamilies of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre-Gaussian modes.     

The diffraction of light at ultrasonic waves in acoustic resonators

A theoretical treatment is given of the diffraction of light by hypersonic waves excited in solid prismatic samples. For a fixed frequency, a set of modes exists in such a resonator. The modes differ in the spatial distribution of the field, which corresponds to a set of partial waves propagating over an entire solid angle. The diffraction occurs only at the partial waves which satisfy the Bragg condition and the diffraction pattern has the form of a circle consisting of a large number of points. An expression is found for the angular separation of adjacent points which correspond to different modes. It is shown experimentally that for the scattered light field in the far zone the diffraction patterns are similar to Shaefer-Bergmann patterns. Explanations are given for the shape and fine structure of these patterns. The structure of the diffracted light beams in the near zone is also studied and the frequency intervals between the excitation of the various resonator modes are measured. The experimental results are in qualitative agreement with the theory.     

Mode analysis with a spatial light modulator as a correlation filter

A procedure for the real-time analysis of laser modes using a phase-only spatial light modulator is outlined. The procedure involves encoding into digital holograms by complex amplitude modulation a set of orthonormal basis functions into which the initial field is decomposed. This approach allows any function to be encoded and refreshed in real time (60 Hz). We implement a decomposition of guided modes propagating in optical fibers and show that we can successfully reconstruct the observed field with very high fidelity.     

Scattering from correlated systems

A difficulty usually encountered in formulating the problem of scattering of identical particles from correlated systems is that the customary choice of an unperturbed Hamiltonian as the target Hamiltonian plus the kinetic energy of the projectile is not symmetric under particle exchange. This choice of unperturbed Hamiltonian leads to wavefunctions which, if they are antisymmetrized, are not orthonormal. In this paper an orthonormal, antisymmetrized set of basis states is constructed. These states are then used to construct a symmetric unperturbed Hamiltonian, so that a formal scattering equation with appropriate boundary conditions can be written. An expression for a T matrix describing nucleon-nucleus scattering can then be obtained. The formalism leads to a two-potential form for the T matrix, the first term of which describes the effect of the orthogonality of the scattering state and the negative energy states.     

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler-Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.     

Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes

Quantum systems such as, for example, photons, atoms, or Bose-Einstein condensates, prepared in complex states where entanglement between distinct degrees of freedom is present, may display several intriguing features. In this Letter we introduce the concept of such complex quantum states for intense beams of light by exploiting the properties of cylindrically polarized modes. We show that already in a classical picture the spatial and polarization field variables of these modes cannot be factorized. Theoretically it is proven that by quadrature squeezing cylindrically polarized modes one generates entanglement between these two different degrees of freedom. Experimentally we demonstrate amplitude squeezing of an azimuthally polarized mode by exploiting the nonlinear Kerr effect in a specially tailored photonic crystal fiber. These results display that such novel continuous-variable entangled systems can, in principle, be realized.     

多层平面介质波导辐射模

介绍一种精确计算多层平面介质波导归一化正交辐射模的方法，在全辐射模的情况下，为了保证模的正交性，给出了准奇模和准偶模的确定方法，这此方法地计算机编程，为模传播法的建立和实际应用奠定了基础，本文最后给出几个应用的实例。     

Simple buckling and vibration analyses of beam or spring connected structures

The problem considered is the buckling or vibration of a system of n columns, or n general structures, which are suitably related to each other and are connected together by sets of springs, or inextensible beams, which are also suitably related to each other. It is shown that all the critical loads, natural frequencies and modes which are required in buckling and vibration problems can be found from n substitute systems which each consist of one column or structure and a single set of springs. The derivation of these substitute systems involves the solution of a very simple linear eigenvalue problem of order n, which has closed form solutions for several of the special cases considered. The reduction of the original system to n substitute systems has been adapted to permit the use of design procedures which avoid a complete analysis of each trial design.     

Nonlinear modes of parametric vibrations and their applications to beams dynamics

An iterative loop combining nonlinear modes and the Rauscher method is suggested for analyzing finite degree-of-freedom nonlinear mechanical systems with parametric excitation. This method is applied to an analysis of the parametric vibration of beams.     

Free-space propagation of autofocusing Airy vortex beams with controllable intensity gradients

Vortex splitting is one of the main causes of instability in orbital angular momentum(OAM) modes transmission. Recent advances in OAM modes free-space propagation have demonstrated that abruptly autofocusing Airy vortex beams(AAVBs) can potentially mitigate the vortex splitting effect. However, different modes of vortex embedding will affect the intensity gradients of the background beams, leading to changes in the propagation characteristics of vortex beams. This study presents the unification of two common methods of coupling autofocusing Airy beams with vortices by introducing a parameter(m), which also controls the intensity gradients and focusing properties of the AAVBs. We demonstrate that vortex splitting can be effectively reduced by selecting an appropriate value of the parameter(m) according to different turbulence conditions. In this manner,the performance of OAM-based free-space optical systems can be improved.     

Generalized periodically focused beams and multiple on-axis lens imaging

A class of periodically focused propagation-invariant monochromatic beams with sharp central peak is extended to the beams, of which the patterns of intensity at discrete transverse planes regularly spaced along the propagation axis may have an arbitrary, but the same, form. It is shown that the wave field of such beams in free space may be represented as a discrete set of nonuniform phase-distorted spherical wavelets converging to the vicinities of axial points at the indicated transverse planes and possessing the property of mutual similarity. High-order azimuth angular constituents of these wavelets are shown to have phase singularities appearing in the transverse planes as dark coaxial spots with bright rims. Hybrid refractive–diffractive systems are proposed, which are suited both for the direct high efficient production of the beams considered and for the multiple imaging of arbitrary plane objects. The results of numerical simulations are given for two cases of multiple reproduction of rings with large and small radii.     

The connection of geometrical optics with the propagation of gaussian beams and the theory of optical resonators

The propagation of light near the axis of astigmatic optical systems may be described by the geometrical-optics approximation with the aid of ray-matrices. The application of the theory of diffraction to the propagation of light in such systems leads to integrals containing essentially the elements of the ray-matrices as parameters. The ABCD-law is derived by evaluating these integrals for gaussian beams. Integral equations applicable to astigmatic optical resonators, having nearly vanishing diffraction losses, are set up. They are only valid under certain conditions, which are comprehensively discussed. The eigensolutions and the eigenvalues of these integral equations are given. The spot-sizes at the resonator mirrors are derived from the eigensolutions, and the eigenvalues lead to the resonance condition. Spot-sizes and resonance condition appear as functions of the elements of the characteristic resonator matrices. The methods described here are applied to the propagation of gaussian beams through gas-lenses and to a resonator containing an internal gas-lens.     

Structural waveguide behaviour of a beam–plate system

Structures which have a constant cross-section normal to a longitudinal axis can be considered as waveguides in which vibration can propagate in the form of various waves in the longitudinal direction. The dynamic behaviour of such systems can be found by using a Fourier transform approach in terms of wavenumbers in the longitudinal direction. Analytical solutions are available for simple, infinitely long waveguides, whereas for more complex waveguides numerical approaches have been developed using finite element techniques to describe the cross-section. In the present paper an analytical Fourier transform approach is used to find the dynamic behaviour of a system consisting of two parallel beams coupled by a plate, when a point force is applied to one of the beams. Multiple waves occur in the longitudinal direction, the number of waves depending on the number of modes of the equivalent cross-section. However, the motion of the driven beam is shown to be dominated by the contribution from only one or two waves at each frequency, these having wavenumbers closest to that of the uncoupled beam. The motion of the plate is also shown to be dominated by these wavenumbers for excitation on the beam. Experimental results are obtained on beam–plate–beam systems with identical and non-identical beams, which show good agreement with the predictions. In particular, these confirm that the plate response is dominated by waves with wavenumbers in the beam direction that follow those of the excited beam.