Theoretical and experimental studies on tightly focused vector vortex beams

The high-NA focusing properties of vector vortex beams are studied theoretically and experimentally. The vector vortex beams are generated by space-variant segmented subwavelength metallic gratings first. Then the mathematical expressions for the focused fields are derived based on the vector diffraction theory, and some numerical simulations are presented that show that the focused fields are not dark at the center and the focusing spot size of vector vortex beams with high topological charges approaches the diffraction limitation at high NA. Finally, to verify the theoretical analysis, the tightly focused fields are measured based on a confocal microscopy system when the NA of the objective lens is 0.90. The research results confirm the potential of vector vortex beams in some applications, such as optical trapping, laser printing, lithography, and material processing.     

Imaging of generalized Bessel-Gauss beams

Abstract

Different kinds of Bessel-Gauss beam have been recently introduced. By considering the effect of a lens on the field, we analyse how these different sets of fields transform into one another and illustrate how a superposition model of Gaussian beams allows this transformation to be clearly interpreted.     

Scalar modified Bessel-Gauss beams and waves

For modified Bessel-Gauss beams, the modulating function for the Gaussian, instead of a Bessel function of real argument, is a Bessel function of imaginary argument. The modified Bessel-Gauss beams and their full wave generalizations are treated with particular attention to the spreading properties on propagation for the azimuthal mode numbers m=0 and m=1. The spreading on propagation of the peak and the null in the radiation pattern obtained in the propagation direction for m=0 and m=1, respectively, is substantially less for the modified Bessel-Gauss waves than that for the corresponding Bessel-Gauss waves. The total power transported by the waves is determined and compared with that of the corresponding paraxial beam to assess the quality of the paraxial beam approximation for the wave. The powers in the Bessel-Gauss wave and the modified Bessel-Gauss wave are finite in contrast to that in the Bessel wave. With respect to both the spreading properties and the quality of the paraxial beam approximation, the modified Bessel-Gauss beam is an improvement over the Bessel-Gauss beam.     

Diffraction field of tightly focused laser beams and its influence to laser accelerator

In this paper, we have studied the dynamic characteristics of relativistic electron injected into stationary intense vacuum laser fields. We found the dynamic trajectories can basically be classified as three categories, namely Inelastic Scattering (IS), Capture and Acceleration Scenario (CAS) and Penetrate into Axial Region and Move (PARM) trajectory. The physical mechanism as to the three kinds of electrons have been examined. In particular, the PARM trajectory which we presented in this paper is different from the CAS and IS trajectory which we had already found in our previous work. We will show the PARM stems from the strong diffraction effect of a tightly focused laser field. In addition, the initial condition for the three kinds of electrons to emerge were detailed investigated. It has been found that there are four factors which chiefly decide the appearance of the three kinds of dynamics trajectories, namely the laser beam width w₀ and intensity a₀, the electrons incident angle θ and initial transversal momentum p_ti. The implication of the PARM electrons to the planned vacuum laser accelerators is illustrated.     

Electromagnetic modified Bessel-Gauss beams and waves

The transverse magnetic (TM) modified Bessel-Gauss beams and their full-wave generalizations are treated. Attention is paid to the spreading properties on propagation of the null in the radiation intensity pattern for the azimuthal mode numbers m=0 and 1. The rate of spreading of the null in the propagation direction is significantly less for the TM modified Bessel-Gauss waves than those for the corresponding TM Bessel-Gauss waves. The total power transported by the waves is determined and compared with that of the corresponding paraxial beam to estimate the quality of the paraxial beam approximation of the wave. The dependence of the quality of the paraxial beam approximation on the azimuthal mode number, the beam shape parameter, and the ratio of the beam waist to the wavelength has a regular pattern for the TM Bessel-Gauss wave and not for the TM modified Bessel-Gauss wave.     

Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams

We show that the elegant Laguerre-Gauss light beams of high radial order n are asymptotically equal to Bessel-Gauss light beams. The Bessel-Gauss beam equivalent to each elegant Laguerre-Gauss beam is found and shown to have almost identical propagation factors M2. In the limit n-->infinity, elegant Laguerre-Gauss beams can be identified with Durnin's Bessel beam. Our results suggest a new experimental procedure for generating light beams with nondiffractinglike properties directly from the output of a stable resonator.     

Effective Fresnel number and the focal shifts of focused partially coherent beams

We define the effective Fresnel number of the cylindrical lens illuminated by a plane wave or Schell-model beams. On the basis of the concept of the effective Fresnel number, the focusing properties of the cylindrical lens illuminated by the Schell-model beam are investigated in a simple way. It is shown that the relative focal shift can be evaluated by an analytical formulation, which is expressed as a function of the effective Fresnel number. To evaluate our approach, we make the comparison between the results obtained by our method and the numerical calculation based on the diffraction integral. The results indicate that we can simply and exactly evaluate the focal shifts with our method.     

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.     

Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals

A simple correspondence between the paraxial propagation formulas along the optical axis of a uniaxial crystal and inside an isotropic medium is found in the case of beams with linearly polarized circularly symmetric boundary distributions. The electric fields of the ordinary and the extraordinary beams are related to the corresponding expressions in a medium with refractive index n(o) and n(e)2/n(o), where n(o) and n(e) are the ordinary and the extraordinary refractive indexes, respectively. Closed-form expressions for Laguerre-Gauss and Bessel-Gauss beams propagating through an anisotropic crystal are given.     

Electromagnetic field for a tightly focused beam incident upon ordinary and layered plane surfaces

A solution procedure is developed for the determination of the electromagnetic field that results from the interaction of a tightly focused beam with a plane surface with and without a layer. The effects of angle of incidence, relative index of refraction, polarization, layer thickness, and incident beam profile on the resulting electromagnetic field distribution are demonstrated.     

Eigenfields and output beams of an unstable Bessel-Gauss resonator

We evaluate the eigenfields of an unstable Bessel-Gauss resonator (UBGR) by use of the transfer-matrix method in which the transverse profiles and their corresponding losses of the UBGR are considered as the eigenvectors and eigenvalues of a transfer matrix so that the dominant mode fields and their losses of the UBGR can be readily extracted in terms of the matrix eigenvalue algorithm. Moreover, based on the eigenfields across two mirrors that resulted from the transfer-matrix method, we simulate the field distributions in the cavity and the propagation of output beams by means of the angular spectrum method. The computation results show that the UBGR easily produces a fundamental Bessel-Gauss mode of good quality, and the output beams retain the original Bessel-Gauss distribution during propagation.     

受光阑限制贝塞耳-高斯光束的衍射特性

对贝塞耳-高斯光束经不同几何构形光阑的衍射作了比较研究。结果表明,在选取合适的光阑参数情况下,锯齿光阑能减弱所关心区域(F=20)轴上光强的衍射调制,而超高斯光阑能抑制大部分菲涅耳区(F>3.7)的衍射调制。锯齿光阑和超高斯光阑都有较大的填充因子,使用合适的超高斯光阑可得到有较大填充因子(如fSuper-Gaussian=0.25)又更平滑的横向光强分布。数值计算说明,为有效抑制光强的衍射调制,适当选取超高斯光阑的阶数n和半宽ws(如n=4,ws=0.4mm)是必要的。     

Bessel-Gauss beams as rigorous solutions of the Helmholtz equation

The study of the nonparaxial propagation of optical beams has received considerable attention. In particular, the so-called complex-source/sink model can be used to describe strongly focused beams near the beam waist, but this method has not yet been applied to the Bessel-Gauss (BG) beam. In this paper, the complex-source/sink solution for the nonparaxial BG beam is expressed as a superposition of nonparaxial elegant Laguerre-Gaussian beams. This provides a direct way to write the explicit expression for a tightly focused BG beam that is an exact solution of the Helmholtz equation. It reduces correctly to the paraxial BG beam, the nonparaxial Gaussian beam, and the Bessel beam in the appropriate limits. The analytical expression can be used to calculate the field of a BG beam near its waist, and it may be useful in investigating the features of BG beams under tight focusing conditions.     

Degree of polarization in tightly focused optical fields

We analyze the degree of polarization of random, statistically stationary electromagnetic fields in the focal region of a high-numerical-aperture imaging system. The Richards-Wolf theory for focusing is employed to compute the full 3 x 3 electric coherence matrix, from which the degree of polarization is obtained by using a recent definition for general three-dimensional electromagnetic waves. Significant changes in the state of partial polarization, compared with that of the incident illumination, are observed. For example, a wave consisting of two orthogonal and uncorrelated incident-electric-field components produces rings of full polarization in the focal plane. These effects are explained by considering the distribution of the spectral densities of the three electric field components as well as the correlations between them.     

Reflection and transmission of strongly focused light beams at a dielectric interface

The reflection and transmission of tightly focused azimuthally, radially and linearly polarized electromagnetic wave beams with subwavelength spot size and wavefront curvature at a dielectric interface are investigated. For a given wavefront radius, the existence of the optimal radius of beam spot corresponding to a minimal reflectance and maximal transmittance is shown. Significant lateral shift in the transmitted intensity peak is revealed for strongly focused azimuthally and radially polarized beams that are normally incident to an interface. The reflection and transmission of transverse-electric- and transverse-magnetic- polarized extremely narrow wave beams which are obliquely incident on the dielectric interface is analysed. Disappearance of the Brewster angle and total internal reflection effects for the strongly focused beams are predicted. The change in beam profile after reflection and transmission for different polarizations, incident beam spots and incidence angles are analysed.     

Comment on "Eigenfields and output beams of an unstable Bessel-Gauss resonator"

We comment on a recent paper by D. Ling et al. [Appl. Opt. 45, 4102 (2006)]. In that paper, the authors adopted the entire matrix formalism that we established in a previous work [J. Opt. Soc. Am. A 22, 1909 (2005)] for finding the eigenmodes of an unstable Bessel resonator. Nevertheless, the results are inaccurate mainly because (a) it was overlooked that light crosses through the axicon twice in a complete round trip and (b) the numerical method used to evaluate the diffraction integral equations cannot resolve the eigenvalues and eigenfields for the given resonator configuration.     

Ellipsometer measurements using focused and masked beams

While optical spectroscopic measurements using ellipsometry may be made in air and are non-destructive, the relatively large (> 2 mm) spot size has limited their use to surface regions greater than 2 mm in lateral extent. Recent developments in focusing instruments have made spot sizes on the order of 20 to 25 μm possible. The work to be presented explores the use of the 25 μm spot size to probe non-uniform nanostructured thin films. Measurements were performed on a highly non-uniform film (0 to 2 μm in thickness across 4 mm in lateral dimension) using such a 25 μm spot. Further reduction of the spot size is possible using mechanical masking with a slit. Measurements have been made to the range of a few microns in width. The practical resolution limits of beam masking may be decreased by increasing incident light intensity, improving slit alignment, and improving detection methods.     

Scattering of a tightly focused beam by an optically trapped particle

Near-forward scattering of an optically trapped 5-mum-radius polystyrene latex sphere by the trapping beam was examined both theoretically and experimentally. Since the trapping beam is tightly focused, the beam fields superpose and interfere with the scattered fields in the forward hemisphere. The observed light intensity consists of a series of concentric bright and dark fringes centered about the forward-scattering direction. Both the number of fringes and their contrast depend on the position of the trapping beam focal waist with respect to the sphere. The fringes are caused by diffraction that is due to the truncation of the tail of the trapping beam as the beam is transmitted through the sphere.