Light distribution close to focus in biaxially birefringent media

The effect of focusing into a biaxially birefringent medium on the light distribution in the focal region of a high-NA optical system is investigated with the Debye approach to vectorial diffraction theory. Attention is limited to media with small birefringence. The electric field in the focal region is the sum of the field of the two polarization eigenmodes of the biaxially birefringent medium. Both modes are generally astigmatically aberrated, are defocused with respect to each other, and have a polarization field that is nonuniform over the pupil. The diffraction integrals are calculated numerically on the basis of an expansion of the field close to focus in terms of partial waves.     

Asymptotic behavior of the spatial frequency response of an optical system with defocus and spherical aberration

Asymptotic expressions are derived for the two-dimensional incoherent optical transfer function (OTF) of an optical system with defocus and spherical aberration. The two-dimensional stationary phase method is used to evaluate the aberrated OTF at large and moderately large defocus and spherical aberration. For small aberrations, the OTF is approximated by a power series in the aberration coefficients. An accurate approximation (in elementary functions) to the OTF is obtained for a defocused optical system with a circular pupil. We experimentally demonstrate the validity of the OTF approximations in sharp-focus image restoration from two defocused images. A digital focusing method is presented.     

Dynamically shifting the focal spot with tunable pupil filters

Abstract

It is popularly believed that when the focal spot is shifted far away from the optical axis of optical system with the pupil filter, it will be significantly aberrated. Therefore, the pupil filter is usually only used to adjust the distribution of the focal spot at the optical axis of optical system. In this paper, the special combination of the phase pupil filter shifting the focal spot and the high-pass filter eliminating the aberration of the focal spot is designed so that the focal spot can be shifted dynamically at the focal plane. By utilizing the Debye vector diffractive theory, the shifting behaviours of the focal spot are investigated in detail. The proof-of-principle experiment demonstrates the validation of the method for dynamically controlling the position of focal spot. This work is helpful to realize dynamically shifting the focal spot with the pupil filter and may find valuable applications in particle trapping, microscopes, optical engineering, and so on.     

Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus.

The effects of birefringence on the light distribution in the focal region of a high-NA optical system are investigated with use of the Debye approach to vector diffraction theory. The attention is limited to uniaxially birefringent media with symmetry axis along the optical axis of the imaging system. The radially (p) and tangentially (s) polarized fields in the exit pupil produce spots in the focal region that are defocused with respect to each other. For small birefringence values the relative defocus causes a distortion and broadening of the spot; for larger values the two spots separate completely. As a corollary to the theory it is shown that there is a tangential tornadolike flow of energy in the focal region when the polarization in the entrance pupil is elliptical.     

光学四象限分光器参数的优化设计

以几何光学的矩阵变换为基础，通过光线追迹的方法，对四象限分光的多透镜成像复杂光学系统进行仿真建模。由所得到的基本参数的关系曲线，优化确定系统的放大镜放大系数为20，分光镜离轴距离为3mm，系统出瞳直径1mm。对复杂光学系统参数的优化提供了新的思路和方法。     

Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system

The judgment of the imaging quality of an optical system can be carried out by examining its through-focus intensity distribution. It has been shown in a previous paper that a scalar-wave analysis of the imaging process according to the extended Nijboer-Zernike theory allows the retrieval of the complex pupil function of the imaging system, including aberrations as well as transmission variations. However, the applicability of the scalar analysis is limited to systems with a numerical aperture (NA) value of the order of 0.60 or less; beyond these values polarization effects become significant. In this scalar retrieval method, the complex pupil function is represented by means of the coefficients of its expansion in a series involving the Zernike polynomials. This representation is highly efficient, in terms of number and magnitude of the required coefficients, and lends itself quite well to matching procedures in the focal region. This distinguishes the method from the retrieval schemes in the literature, which are normally not based on Zernike-type expansions, and rather rely on point-by-point matching procedures. In a previous paper [J. Opt. Soc. Am. A 20, 2281 (2003)] we have incorporated the extended Nijboer-Zernike approach into the Ignatowsky-Richards/Wolf formalism for the vectorial treatment of optical systems with high NA. In the present paper we further develop this approach by defining an appropriate set of functions that describe the energy density distribution in the focal region. Using this more refined analysis, we establish the set of equations that allow the retrieval of aberrations and birefringence from the intensity point-spread function in the focal volume for high-NA systems. It is shown that one needs four analyses of the intensity distribution in the image volume with different states of polarization in the entrance pupil. Only in this way will it be possible to retrieve the "vectorial" pupil function that includes the effects of birefringence induced by the imaging system. A first numerical test example is presented that illustrates the importance of using the vectorial approach and the correct NA value in the aberration retrieval scheme.     

Zernike representation and Strehl ratio of optical systems with variable numerical aperture

We consider optical systems with variable numerical aperture (NA) on the level of the Zernike coefficients of the correspondingly scalable pupil function. We thus present formulas for the Zernike coefficients and their first two derivatives as a function of the scaling factor ε ≤ 1, and we apply this to the Strehl ratio and its derivatives of NA-reduced optical systems. The formulas for the Zernike coefficients of NA-reduced optical systems are also useful for the forward calculation of point-spread functions and aberration retrieval within the Extended Nijboer–Zernike (ENZ) formalism for optical systems with reduced NA or systems that have a central obstruction. Thus, we retrieve a Gaussian, comatic pupil function on an annular set from the intensity point-spread function in the focal region under high-NA conditions.     

Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation

Phase retrieval on an annular pupil and an annular sector pupil by using the eigenfunction method to solve the transport of intensity equation is proposed. The analytic expressions of Laplacian eigenfunctions with the Neumann boundary condition on an annular pupil and an annular sector pupil are given. The phase can be expanded as a set of eigenfunctions on the corresponding pupil, and the coefficients of the eigenfunctions can be obtained by the integral of the eigenfunction and the intensity derivative along the optical axis. Phase retrieval by the eigenfunction method on an annular pupil and an annular sector pupil is simulated, and accurate retrieved results are obtained.     

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.     

Effect of rotation and translation on the expected benefit of an ideal method to correct the eye's higher-order aberrations

An ideal correcting method, such as a customized contact lens, laser refractive surgery, or adaptive optics, that corrects higher-order aberrations as well as defocus and astigmatism could improve vision. The benefit achieved with this ideal method will be limited by decentration. To estimate the significance of this potential limitation we studied the effect on image quality expected when an ideal correcting method translates or rotates with respect to the eye's pupil. Actual wave aberrations were obtained from ten human eyes for a 7.3-mm pupil with a Shack-Hartmann sensor. We computed the residual aberrations that appear as a result of translation or rotation of an otherwise ideal correction. The model is valid for adaptive optics, contact lenses, and phase plates, but it constitutes only a first approximation to the laser refractive surgery case where tissue removal occurs. Calculations suggest that the typical decentrations will reduce only slightly the optical benefits expected from an ideal correcting method. For typical decentrations the ideal correcting method offers a benefit in modulation 2-4 times higher (1.5-2 times in white light) than with a standard correction of defocus and astigmatism. We obtained analytical expressions that show the impact of translation and rotation on individual Zernike terms. These calculations also reveal which aberrations are most beneficial to correct. We provided practical rules to implement a selective correction depending on the amount of decentration. An experimental study was performed with an aberrated artificial eye corrected with an adaptive optics system, validating the theoretical predictions. The results in a keratoconic subject, also corrected with adaptive optics, showed that important benefits are obtained despite decentrations in highly aberrated eyes.     

Axial separation of orthogonally polarized focal field components due to a radially polarized beam

In this paper, we investigate the field distribution in the focal volume of an aberrated radially polarized beam. Using two different forms of the vectorial diffraction theory, we show that the presence of defocus in the beam displaces both the axially and the radially polarized fields parallel to the optical axis of the focusing lens, while the presence of spherical aberration primarily shifts the longitudinally polarized field only. This facilitates axial separation of the two orthogonally polarized field components, resulting in a significant boost to the ratio of the peak longitudinally polarized field to the peak laterally polarized field in the focal plane. We further show that with an appropriate combination of oppositely signed defocus and spherical aberration, the energy density in the focal volume due to the longitudinally polarized field can be caused to peak at the focal plane. The results obtained are expected to be beneficial to the applications requiring a stronger longitudinally polarized focal field relative to the laterally polarized focal field component.     

带分划板的连续变倍双筒望远镜设计

介绍一种用两组元透镜来实现倒象和连续变倍的双筒望远镜设计。该设计在物镜焦面位置安装带刻线的分划板 ,可对目标进行瞄准和测量。特别是在系统不另设可变光栏的情况下 ,利用光学系统透镜边框作孔径光栏 ,使系统在整个变倍过程中不仅出瞳直径和出瞳距离均变化不大 ,而且系统与定倍望远镜一样具有较大的象方视场、出瞳直径和出瞳距离 ,从而使系统无论昼夜使用都具有真正的变倍作用和良好的观察效果。     

Polychromatic axial behavior of aberrated optical systems: Wigner distribution function approach

We propose a new method for the computation of the tristimuli values that correspond to the impulse response along the optical axis provided by an imaging optical system working under polychromatic illumination. We show that all the monochromatic irradiance distributions needed for this calculation can be obtained from the Wigner distribution function associated with a certain version of the pupil function of the system. The use of this single phase-space representation allows us to obtain the above merit function for aberrated systems with longitudinal chromatic aberration and primary spherical aberration. Some numerical examples are given to verify the accuracy of our proposal.     

Comparative analysis of doublets versus single-layer diffractive optical elements in eyepiece or magnifier design

We quantify the impact of eye clearance requirement on the performance of eyepieces utilizing doublets versus diffractive optical elements on aspheric substrates. In this study, the doublets were designed to be cemented on-axis elements. Specifically, four different values of eye clearance were implemented: 17, 20, 23, and 26 mm. For each value, axial and lateral color, spherical aberration, coma, astigmatism, field curvature, and distortion were compared. Each system under comparison was optimized for the same focal length, a 9 mm exit pupil, photopic wavelengths (513-608 nm), and a 40 degrees full field of view. We demonstrate that the single-layer diffractive optical element supports an eye clearance value of approximately 80% of the effective focal length, while the doublet drops below desired specifications at approximately 65% of the effective focal length.     

Phase plate to extend the depth of field of incoherent hybrid imaging systems

A hybrid imaging system combines a modified optical imaging system and a digital postprocessing step. We describe a spatial-domain method for designing a pupil phase plate to extend the depth of field of an incoherent hybrid imaging system with a rectangular aperture. We use this method to obtain a pupil phase plate to extend the depth of field, which we refer to as a logarithmic phase plate. Introducing a logarithmic phase plate at the exit pupil of a simulated diffraction-limited system and digitally processing the detector's output extend the depth of field by an order of magnitude more than the Hopkins defocus criterion. We also examine the effect of using a charge-coupled device optical detector, instead of an ideal optical detector, on the extension of the depth of field. Finally, we compare the performance of the logarithmic phase plate with that of a cubic phase plate in extending the depth of field of a hybrid imaging system with a rectangular aperture.     

Representation and focusing properties of higher-order radially polarized Laguerre–Gaussian beams

A method based on higher-order Poincaré sphere was proposed to represent the states of polarization of higher-order radially polarized Laguerre–Gaussian (LG) vectorial beams (VBs). And the focusing properties of such LG beams of different radial orders focused by a high-NA lens were discussed. By tuning the ratio of the pupil radius to the waist of the incident beams, some cage-like or needlelike electric intensity field is generated in the focal region for several specific LG VBs with high order. Modulated by diffractive optical elements, the shape of the focal field shows novel behaviors such as splitting of cage-like modes, which provides potentially a more flexible control over micro-particles.     

The Rapid Calculation of the Optical Transfer Function for On-axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials

A simple and rapid numerical quadrature is developed for the evaluation of the diffraction-based optical transfer function for on-axis systems, using a Tchebycheff polynomial expansion of the pupil function. The integration of the autocorrelation integral of the pupil function is replaced by once and for all evaluations of the cross-correlation of respective polynomials. However, the expansion coefficients themselves of the Tchebycheff series are linear sums of the sampled pupil function and thus a series of coefficients can be generated that weight the pupil function at various points to give the resultant OTF. The coefficients for a tenth-order Tchebycheff expansion are included in the paper, and a set of tables of OTF values calculated with these coefficients and 64 2 64 Gaussian quadrature for a diffraction-limited system, and one with one wavelength of primary spherical aberration.     

Focal shift and extended focal depth with tunable pupil filter

Extended focal depth and focal shift are very important in microscopy, imaging and optical storage systems, and have attracted much attention in recent years. In order to obtain the extended focal depth and focal shift, a new kind of tunable pupil filter is proposed in this article. It consists of one half-wave plate between two quarter-wave plates, and the half-wave plate is made up of two zones that can rotate with respect to each other. By analyzing the intensity distribution in the focal region of the optical system with such a device, it reveals that focal shift can be realized by rotating any zone of the half-wave plate. When the phase difference of the two zones is π, the extended focal depth and transverse superresolution can be obtained at the same time. Therefore, it may be feasible to use such a tunable pupil filter in optical systems that need focal shift and extended focal depth.     

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.