Comment on ‘Two-state Description of Total Polarization by Means of Singular Eigenvalue Problems’

We derive a formulation that can be used to determine the electromagnetic field distribution in the focal region of a wide-angle spherical mirror illuminated by an obliquely incident and linearly polarized plane monochromatic wave. The integrals representing the diffracted fields derived in part 1 [1] for the two cases of polarization of the incident field are first suitably expanded in series form. The integrals with respect to the azimuth 0 are then evaluated analytically with the help of some new integration formulae developed in this paper. The diffracted field components for the two cases of incident polarization are finally expressed in forms suitable for numerical computation. Our formulae can be used to determine the three-dimensional field distribution in the caustic region of the mirror for any obliquity of the incident beam. Simpler expressions for the field components for a few special cases of interest are also derived. In particular, it is shown that the series solution obtained for the case of oblique incidence reduces to the simpler results found by previous authors for the case of normal incidence.     

Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures

We show that the contribution of the electric field components into the focal region can be controlled using binary phase structures. We discuss differently polarized incident waves, for each case suggesting easily implemented binary phase distributions that ensure a maximum contribution of a definite electric field component on the optical axis. A decrease in the size of the central focal spot produced by a high numerical aperture (NA) focusing system comes as the result of the spatial redistribution of the contribution of different electric field components into the focal region. Using a polarization conversion matrix of a high NA lens and the numerical simulation of the focusing system in Debye's approximation, we demonstrate benefits of using asymmetric to polar angle ? binary phase distributions (such as arg[cos ?] or arg[sin 2?]) for generating a subwavelength focal spot in separate electric field components. Additional binary structure variations with respect to the azimuthal angle also make possible controlling the longitudinal distribution of light. In particular, the contribution of the transverse components in the focal plane can be reduced by the use of a simple axicon-like structure that serves to enhance the NA of the lens central part, redirecting the energy from focal plane. As compared with the superimposition of a narrow annular aperture, this approach is more energy efficient, and as compared with the Toraldo filters, it is easier to control when applied to three-dimensional focal shaping.     

Use of confocal microscopes in conoscopy and ellipsometry. 1. Electromagnetic theory

An optical system consisting of two objective lenses in a confocal arrangement is examined. It is shown that a simple algebraic relation exists between the electric field in the back focal plane of the first objective lens, which focuses the incident light, and the Fourier transform of the electric field in the focal plane of the same lens. The relation holds for high angles. If a thin object is placed in the focal plane it is possible to write the electric field by use of a Fourier transform relation at the exit aperture of the second lens. The theory is generalized for objects that are positioned at oblique angles with respect to the optical axis of the system. This configuration is clearly identical to the setup of a spatially resolving ellipsometer.     

Focal shifts in focused nonuniformly polarized beams.

We present a simple formula to evaluate the relative focal shift in a circular-aperture lens system illuminated by a nonuniformly polarized (NUP) light wave. Specifically, it is shown that the relative focal shift is determined by the effective Fresnel number. The effective Fresnel number is equal to the product of the Fresnel number of the lens aperture and the parameter sigma, which describes the uniformity of the polarization distribution of the NUP beam across the lens aperture. Some examples are given to illustrate the use of this approach. The influence of the polarization distribution of the incident NUP light wave on the polarization distribution in the axial points of the focused field is also presented.     

Linear focusing by a plane grating with curved grooves

We study the field diffracted by a plane grating with curved (parabolic) grooves. We will demonstrate that when a monochromatic plane wave is incident on a grating with parabolic grooves the diffracted field has a focal line whose position depends on the curvature radius of the parabolas and the incidence angle of the light onto the grating. The effect described has potential applications in grating-based devices for focusing light without requiring any additional optics.     

Experimental study on polarization lens formed by asymmetrical metallic hole array

A polarization bifocal lens based on the polarization effect caused by asymmetrical hole arrays had been designed, fabricated, and characterized experimentally. By considering the fact that the skin depth of an infrared electromagnetic field inside metal is much shorter than the incident wavelength, a polarization bifocal lens composed of high deep-width ratio metallic holes was realized by using a gold-coated silicon structure to replace the one directly formed on a thick metal film. An infrared optical experiment setup is built based on the secondary imagery method for characterizing the focal length of the designed bifocal lens. The measured focal lengths of the fabricated bifocal lens coincide well with the designed values, which proves the validity for realizing the polarization elements with the proposed structure and the feasibility of the fabrication process.     

Transformation of the V-dipole by an astigmatic lens

The transformation of the V-dipole by an astigmatic lens is studied analytically and numerically. It is found that after passing through the lens, the V-dipole splits into four C-points which make up two C-dipoles with same singularity index and opposite handedness. By a suitable choice of the astigmatic coefficient, off-axis distance, or propagation distance, the creation and annihilation of more than two C-dipoles may take place, as well as C-points may occur at the focal plane of the astigmatic lens. As a special case, only two C-dipoles annihilate or recreate and no polarization singularities appear at the focal plane in the presence of an ideal lens.     

Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects

Transmission of an arbitrarily polarized plane wave by an arbitrarily oriented spheroid in the short-wavelength limit is considered in the context of ray theory. The transmitted electric field is added to the diffracted plus reflected ray-theory electric field that was previously derived to obtain an approximation to the far-zone scattered intensity in the forward hemisphere. Two different types of cross-polarization effects are found. These are (a) a rotation of the polarization state of the transmitted rays from when they are referenced with respect to their entrance into the spheroid to when they are referenced with respect to their exit from it and (b) a rotation of the polarization state of the transmitted rays when they are referenced with respect to the polarization state of the diffracted plus reflected rays.     

Optical Diffraction in Close Proximity to Plane Apertures. I. Boundary-Value Solutions for Circular Apertures and Slits

In this paper the classical Rayleigh-Sommerfeld and Kirchhoff boundary-value diffraction integrals are solved in closed form for circular apertures and slits illuminated by normally incident plane waves. The mathematical expressions obtained involve no simplifying approximations and are free of singularities, except in the aperture plane itself. Their use for numerical computations was straightforward and provided new insight into the nature of diffraction in the near zone where the Fresnel approximation does not apply. The Rayleigh-Sommerfeld integrals were found to be very similar to each other, so that polarization effects appear to be negligibly small. On the other hand, they differ substantially at sub-wavelength differences from the aperture plane and do not correctly describe the diffracted field as an analytical continuation of the incident geometrical field.     

Possibility of an optical focal shift with polarization masks

The polarization phase shift (PPS) has emerged as an important analytical tool in optical metrology. The present study utilizes the concept of controlling the polarization phase in applications such as focal shift and automatic focusing. When elliptically polarized light, in general, is incident upon a circularly symmetric polarization mask consisting of circular and annular zones with each zone having a unique linear polarizability, the polarization-phase difference introduced between the polarization-masked zones is also circularly symmetric. With the mask at the lens aperture, the polarization phase introduced is multiplicative with the lens function and is shown to result in a shift of the Gaussian focus plane. Because the polarization phase can be controlled by variation of the polarization parameters, the effective focal length of the imaging system can be varied within a small range. A study of the point-spread functions at the shifted focal planes has shown that the quality of the focal patch in these planes is comparable with that produced by a diffraction-limited imaging system at Gaussian focus. The shift of focus can be achieved by control of the polarization of the input beam. It is anticipated that this technique may find application in areas for which dynamic focusing within a small range is required.     

Programmable birefringent lenses with a liquid-crystal display

We study the properties of a novel birefringent lens constructed by combination of a regular glass lens and a programmable diffractive lens addressed to a liquid-crystal display (LCD). The LCD affects only the vertical polarization state. Consequently the birefringent lens produces two images of an input object with different locations and magnifications for the two orthogonal polarization states. Using a properly oriented analyzer polarizer produces interference fringes. We then show how the imaging system acts as a common-path polarization interferometer for wave-front analysis of objects in the input plane. Finally, we subtract the two images to produce an edge-enhanced version of the input image. All these effects can be controlled because we can program lenses with different focal lengths onto the LCD.     

Self-pumped phase-conjugate interferometer with a photorefractive iron-doped lithium-niobate crystal

A single object wave is amplitude divided by a beam splitter into two waves of equal intensity that are made to interfere at the back surface of an iron-doped lithium-niobate crystal so that the normal to the back surface is the angular bisector of the input waves. The interference results in the formation of a phase grating (Bragg grating) in the volume of the crystal. These waves are diffracted at the Bragg grating on both the front focal plane and the back focal plane of the crystal. The wave diffracted in the back focal plane from the Bragg grating and counterpropagating to the incident wave is observed to be the phase conjugate of the input object wave. The wave diffracted in the front focal plane of the Bragg grating is incorporated into the design of an interferometer to measure a specific in-plane displacement of the object wave. It is theoretically evaluated and experimentally demonstrated that interferometers such as those that incorporate conjugate-wave pairs are highly sensitive.     

Vector diffraction analysis of optical disk readout

The optical disk readout signals from ROM disks are presented by use of a rigorous three-dimensional vector diffraction method. The optical disk is modeled as a crossed metal grating without restriction on the form of the information marks, and the permittivity of the metal is taken into account. The diffracted field from the disk is obtained by means of decomposing the focused incident beam into a spectrum of plane waves and then calculating the diffracted plane waves for each respective incident component. The readout signal is obtained by integration of the energy-flux density of the diffracted field according to the detection scheme of the optical disk system. A typical digital versatile disk (DVD) system is applied with this theory, and the result is far from that of scalar diffraction theory.     

Controlling the three-dimensional shape of a focal spot with a three-annular-zones-polarizing filter

A linear polarization through a high numerical aperture yields an elliptical spot on the focal plane due to depolarization, which is explained by the vector diffraction theory. Depolarization results from orthogonal polarizations are generated after diffraction by a high numerical aperture lens. In this paper, it has been shown that the elliptical focal shape can be suppressed by adding perpendicular polarization to the incident beam. By a specially designed three-annular-zones-polarizing filter which modifies the linear polarization properly, the ellipticity and volume size of the focal spot are reduced from 0.395 to 0.10 and 0.405λ³ to 0.381λ³, respectively. Furthermore, the filter elongates the focal spot longitudinally by 50.9% and divides the focus into two spots longitudinally separated by 0.754λ.     

Numerical calculation of a converging vector electromagnetic wave diffracted by an aperture using Borgnis potentials. II. Application to the study of focal shift

Focal shift of the converging spherical wavefront light diffracted by a circular aperture is numerically studied with the method of calculating the vector diffractive field by using Borgnis potentials given in Part I [J. Opt. Soc. Am. A23, 872 (2006)]. The quantitative dependence of the focal shift on the geometric parameters is discussed. The focal shift is mainly determined by the Fresnel number (N(f)) on the geometric focusing plane of the converging light, and an empirical formula between the fractional focal shift and the Fresnel number is deduced for N(f)<2. The focal shift of the same geometry is also studied on the basis of the scalar Rayleigh theory of diffraction, and its comparison with and difference from the result of our method are presented.     

用于毫米波焦面阵成像系统的扩展半球介质透镜

毫米波成像是近年来毫米波领域的一个研究热点,而焦面阵成像因其具有实时成像的优点更加受到重视。中分析可用为面阵成像的扩展半球介质透镜。这种焦面阵成像结构将集成天线阵贴附在透镜背面接收透镜聚焦的功率,消除了集成天线工作在毫米波频段时存在的表面波对天线性的影响,具有尺寸紧凑、损耗小的特点。采用Stratton-Chu公式和射线追迹分析了电磁波入射到扩展半球透镜上时在其背面的场分布,即透镜的焦区场分布,以获得透镜用于焦面阵成像时的性能。为验证分析方法的正确性,对平面波垂直入射和会聚高斯束入射两种情形进行了实验验证,实验结果与理论分析吻合较好。该透镜天线还可用来消除常规集成毫米波系统中抛物面天线与集成前端之间的过渡,以降低损耗,改善系统性能,也可用于与准光系统的连接或耦合。该结果将对上述应用提供理论指导。     

Multipole theory for tight focusing of polarized light, including radially polarized and other special cases

A multipole expansion, based on spherical harmonics, provides an efficient method for calculating the field in the focal region of a lens for radially polarized illumination, or other illumination polarization and phase distributions, including vortex beams. The multipole approach also has the benefit of providing a simple measure of the purity of the longitudinal field mode. The method is also convenient for calculation of fields scattered by particles and calculation of optical trapping forces.     

Simple focal-length measurement technique with a circular Dammann grating

A novel technique for focal-length measurements with a circular Dammann grating is presented. In the back focal plane of the lens under test, a one-order circular Dammann grating with limited aperture will produce double-humped radial rings. The separation between the two lobes varies with the displacement of the observed plane from the focal plane of the lens. By searching for the position at which the separation is minimal, the focal point of the lens can be located and hence the back focal length can be determined. Experimental results demonstrated that this method is efficient and can be used effectively for a quick check of focal length.     

Field control of magneto-optic gratings

The Faraday or Kerr rotation of the plane of polarization of light incident on adjacent magnetic domains has opposite sense. Thus light transmitted through a film containing stripe domains or reflected from it is diffracted, as by a grating. The period of such a grating can be controlled by the application of a uniform magnetic field. We report on experimental and theoretical work that explores the range of grating field control under quasistatic conditions, using real films, specifically     

Measurement of longitudinal displacement using lateral shearing cyclic path optical configuration setup and phase shifting interferometry

We present a technique for the measurement of longitudinal displacement using a lateral shearing cyclic path optical configuration (CPOC) setup and phase shifting interferometry. In the technique, a plane mirror mounted on a linear translation stage, placed slightly away from the focal plane of a lens, introduces a longitudinal focal shift to the incident focusing beam. The resulting spherical wavefront emerging from the lens is sheared into two orthogonally polarized beams using the CPOC setup. By applying polarization phase shifting interferometry (PPSI), the longitudinal focal shift of the beam focus is calculated by determining the slope of the optical path difference variation between the sheared beams. Similarly, the additional focal shift introduced due to longitudinal translation of the mirror, by an unknown amount, is determined using PPSI. Half of the difference between the two longitudinal focal shifts measured gives the longitudinal displacement of the mirror. The technique can be used for an extended range of distance measurement. The novelty of the technique is the introduction of CPOC for the distance measurement. The advantages of the technique compared to other related methods are discussed.