Design of accelerating beams based on caustic method

<sec> With the development of laser technology, the application scope of nondiffracting beams, such as Bessel beams, Mathieu beams, cosine beams, and parabolic beams, which remain invariant along their propagation, continues to expand. During its propagation, the main lobes of these beams tend to bend towards off-axis position, which is called self-accelerating (or self-bending) property. A Bessel-like beam with self-acceleration can realize the propagation of the main lobe along a curved trajectory while maintaining the non-diffraction. Because of the above property, Bessel-like beams have been utilized in various areas such as guiding particles along arbitrarily curved trajectories, self-accelerating beams in nonlinear medium, plasma guidance, and laser-assisted guiding of electric discharges around objects. </sec><sec> In this paper, we propose a method of bending the trajectory of Bessel-like beams by using a magnetic fluid deformable mirror (MFDM) instead of traditional spatial light modulator (SLM) and Pancharatnam-Berry (PB) phase manipulation. The MFDM provides a method without pixelation, where all parameters can be rapidly modified for fine-tuning. Furthermore, compared with the conventional deformable mirror, the MFDM has the advantages of a continuous and smooth mirror surface, large shape deformation, low manufacture cost, easy extension, and large inter-actuator stroke. Therefore, it is easy for the MFDM to generate the ideal shape of an axicon. Firstly, according to geometric analysis, the asymmetrical mirror profile to produce a self-accelerating Bessel-like optical beam is proposed. The proposed mirror profile can be used to compensate for the difference in optical path length for each annular slice of the axicon. If a collimated Gaussian beam is incident on the mirror combining the axicon and the asymmetrical mirror profiles, which can obtain Bessel-like beams with arbitrarily curved trajectories. Secondly, the resultant of the self-accelerating Bessel-like beams along parabolic trajectories is validated by the simulation in MATLAB. Finally, a prototype of MFDM consisting of the dual-layer arrays of miniature electromagnetic coils, a Maxwell coil and the magnetic fluid filled in a circular container is fabricated for the experiment. The experimental results show that the Bessel-like beams propagate along the parabolic trajectories, with the MFDM used, and the accuracy of the curved trajectories is verified. The proposed method in this paper opens a new experimental way to the study of Bessel-like beam; the theoretical approach can also be generalized mathematically for other non-paraxial beam propagation. </sec>

An appropriate design of wavefront will enable light fields propagating along arbitrary trajectories thus forming accelerating beams in free space. Previous ways of designing such accelerating beams mainly rely on caustic methods, which start from diffraction integrals and only deal with two-dimensional fields. Here we introduce a new perspective to construct accelerating beams in phase space by designing the corresponding Wigner distribution function (WDF). We find such a WDF-based method is capable of providing both the initial field distribution and the angular spectrum in need by projecting the WDF into the real space and the Fourier space respectively. Moreover, this approach applies to the construction of both two- and three-dimensional fields, greatly generalizing previous caustic methods. It may therefore open up a new route to construct highly-tailored accelerating beams and facilitate applications ranging from particle manipulation and trapping to optical routing as well as material processing.

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Airy beam is a self-sustained light beam that propagates in free space along a parabolic trajectory with a constant lateral acceleration. It is a diffraction-free beam with an electric field pattern described by the Airy function containing a main lobe and side lobes. Unlike Gaussian beam that expands after the focus, a practical finite-energy Airy beam keeps its pattern without spreading over a relatively long distance during its propagation. This unique feature makes the Airy beam a great potential of designing optical systems for three-dimensional (3D) imaging. In this work, we develop a new optical 3D imaging methodology using Airy beam. The proposed imaging method has an advantage of increased depth of field (DOF) that is important in biological optical imaging. In the first step, we perform a fundamental research on specific parameters that can affect the properties of the Airy beams. The finite-energy Airy beams are generated by optical Fourier transform using a lens with the incoming plane wave that is truncated by an aperture. Our simulation and experiments showed that by changing the aperture size and the focal length of the lens, the Airy beam pattern is modified. In addition, the DOF, the beam size and the main lobe energy are changed. As a result, Airy beam can be designed for specific imaging applications. Next, we consider the phase in the cubic phase pattern for the phase modulation to generate the Airy beam. This problem is related to the wavelength dependent phase modulation since the phase in cubic phase pattern is designed for a specific wavelength whereas a broadband light source is normally used in optical coherence tomography (OCT). A liquid crystal display (LCD) panel from a projector is used as the SLM. The cubic phase patterns with different gray values displayed on the computer screen provide different phase modulations in the SLM. The experimental and simulated results show that the phase modulation affects the beam shape of the Airy beam. If the maximum phase modulation is larger than 1.7pi, the Airy beam can keep its pattern and the unmodulated Gaussian beam can be neglected. Further, we design a 3D imaging system using the phase-space method based on the Wigner distribution function (WDF). We theoretically build the WDF for the Airy beam and show that the WDF of the Airy beam is shifted in the phase-space as the transverse position of incoming Airy beam changes, and the WDF is tilted as changing the axial position (along the propagation direction). A larger truncating factor can reduce the energy contribution in WDF for the side lobes and modify the shape of the main lobe close to a straight line. In this case, the WDF of the Airy beam is similar to that of the Gaussian beam. However, the DOF of the Airy beam is greatly improved. We perform experimental measurements of WDF using the basic idea of measuring both the space and spatial frequency information in a 4-f system. Our experimental results match the simulation and validate the 3D imaging using Airy beam in the phase space.

Generally, most of the photonic spin Hall effect (SHE) are investigated based on the Gaussian beam, and the research of the photonic SHE in novel beams with non‐Gaussian distribution is relatively rare. The Airy beam is one of the novel beams whose initial field distribution is asymmetric and tunable. Thus, it is necessary to explore the photonic SHE in the Airy beam. Here analytical solutions for the transverse and in‐plane shifts of the Airy beam are derived, and the peculiar photonic SHE of the Airy beam is systematically researched. The theoretical research results show that the transverse shifts of the left‐ and right‐handed circularly polarized beams are asymmetric, and the in‐plane shifts of the left‐ and right‐handed circularly beams are identical. More importantly, the variation rules of the transverse and in‐plane shifts with the incident centroid position are obtained, and the incident centroid position is regulated by the decay factor of the Airy beam. That is, the spin‐dependent transverse and in‐plane shift can be manipulated flexibly and purposefully by adjusting the beam parameter. It is believed these findings will provide more ideas for exploring the application of the photonic SHE based on novel beams.

The Wigner formalism is a convenient way of describing quantum mechanical effects through a framework of distribution functions in phase space. Currently, there are stochastic and deterministic approaches in use. In our deterministic method, the critical discretization of the diffusion term is done through the utilization of an integral formulation of the Wigner equation. This deterministic method is studied in the context of superposed quantum states as a precursor to simulations of entangled states.

A nonseparable wave-like integro-differential equation for the time evolution of the Wigner distribution function in phase space is educed from the corresponding separable kinetic equation. By employing the quantum hydrodynamical description, a non-local evolution wave equation is also derived by synthesizing the Hamilton‐Jacobi equation with that of continuity, which predicts the generation of nonlocal and quadrupole quantum phenomena in the propagation of the spatial probability density.

A plasma consists of charged particles moving relatively freely, but interacting through the electric and magnetic fields produced by each particle. It is well known that all except a very small (but nevertheless important) fraction of the detectable universe is a plasma. Most plasma physics can be expressed in terms of five fairly straightforward equations, the four Maxwell equations for electromagnetism and the equation of motion for charged particles in electromagnetic fields. Why then is plasma physics such a mathematically difficult subject and why is numerical simulation so important? The most basic reason is that any experimental plasma contains a very large number of particles, 1018–1020 in a typical fusion experiment. Moreover, particles in plasmas have a tendancy to act independently of each other. In conventional hydrodynamics, large numbers of particles can be described by a few simple numbers such as their temperature, density and mean velocity. Sometimes we can do this with a plasma but even then we have a few more numbers to include such as current and charge density, along with the electromagnetic fields. More usually, plasmas are not so simply described. Plasmas are not usually Maxwellian and we have to think in terms of distribution functions in phase space. Whenever we can, we model the non—Maxwellian nature of the plasma by transport models, e.g. for heat flow, but this is a specialised skill in itself. When one considers that conventional hydrodynamics is very complex and involves advanced numerical simulation, it is hardly surprising that simulation plays such an important role in plasma physics.

A new scheme for the calculation of the collision integral of the nonlinear Boltzmann kinetic equation is presented, based on a piecewise constant discrete approximation of the distribution function in phase space and corresponding precalculated collision rate coefficients. The computing effort required for evaluation of the fivefold collision integral is significantly reduced by the use of precalculated rate coefficients. The method was combined with the conservative splitting method to solve the Boltzmann equation for a number of test cases to examine the performance of this method and demonstrate its efficiency. 10 refs., 6 figs.

Analytical solution for self-consistent particle equilibrium distribution in RF field with transverse focusing is found. Solution is attained in approximation of high brightness beam. Distribution function in phase space is determined as a stationary function of the energy integral. Equipartitioning for beam distribution between degrees of freedom follows directly from the choice of stationary distribution function. Analytical expressions for r-z equilibrium beam profile and maximum beam current in RF field are obtained.

The Rouse model for dilute polymer solutions undergoing homogeneous flows has been generalized to include the inertia of the beads in the equations of motion. To obtain the correct ‘‘diffusion equation’’ for the probability density distribution function in phase space, we generalize the diffusion equation derived by Murphy and Aguirre [J. Chem. Phys. 57, 2098 (1972)] from Hamilton’s equations of motion for an arbitrary number of interacting Brownian particles at equilibrium. Material functions are found, and the noninertial case is seen to be obtained as the zero mass limit in all steps of the development. In particular, the steady-state shear results are unaffected by the inclusion of inertia. It is also shown how two assumptions, ‘‘equilibration in momentum space,’’ and the neglect of acceleration, made independently by Curtiss, Bird, and Hassager in their phase-space kinetic theory, are actually the result of assuming zero mass.

A semiclassical independent electron model for ion-atom collisions

Landau theory of Fermi liquids and nuclear fluid dynamics

The entropy of an electronic system is defined in terms of the Husimi function, a nonnegative distribution function in phase space. The Husimi function is calculated by maximizing the entropy subject to the constraints that the Husimi function give a Gaussian convolution of the desity when integrating over the momentum coordinates and that its second moment with respect to momentum give a sum of Gaussian convolutions of the density and the kinetic energy density. The result is compared with the Wigner function. Equations are given for calculating the density matrix from the Husimi function. The resulting equation for the exchange energy requires a difficult numerical integration. An alternate method is used to obtain the density matrix from an approximate partially collapsed Husimi matrix that gives the maximum entropy Husimi function as its diagonal. The results are exact for the harmonic oscillator ground state. Exchange energies calculated for H and the He isoelectronic series through C+4 show slight improvements over those calculated using a maximum entropy Wigner function.

As a kind of typical self-accelerating laser beam, Airy beams have attracted much attention due to their fascinating properties and various potential applications. In this work, we carry out a full vector wave analysis of Airy beams upon reflection and refraction. A hybrid method based on the angular spectrum representation and vector potential in the Lorenz gauge is introduced to describe the vectorial structure of Airy beams upon reflection and refraction. The explicit analytical expressions for the electric and magnetic field components of arbitrarily incident Airy beams reflected and refracted at an air-medium interface are derived in detail. Local-field patterns and magnitude profiles with different parameters are displayed. The analytical formulas obtained in this work can be practically applied to explore the local dynamical characteristics, including the energy, momentum, spin, and orbital angular momentum of Airy beams upon reflection and refraction.

Significant interest has been devoted to tailoring optical fields that transversely accelerate during propagation in the form of Airy, Weber and Mathieu beams. In this work, we introduce a new type of optical field that exhibits controlled angular acceleration during propagation which is achieved by superpositions of Bessel beams with non-canonical phase functions. We demonstrate these angular accelerating fields by modulating the phase and amplitude of a supercontinuum source with the use of a phase-only spatial light modulator (SLM). We illustrate that by considering only the first diffraction order when the SLM is encoded with a blazed grating, the SLM is capable of tailoring the spatial profile of broadband sources without any wavelength dependence. By digitally simulating free-space propagation on the SLM, we compare the effects of real and digital propagation on the angular rotation rates of the resulting optical fields for various wavelengths. The development of controlled angular accelerating optical fields will be useful in areas such as particle manipulation, plasma control, material processing and non-linear optics.