Aplanatic Optical System Research Articles

Aplanatism in stigmatic optical systems.

The minimization of spherical and coma aberrations in optical imaging systems is currently accomplished through the use of corrective aspheric optical surfaces. In this work, we develop a new, to the best of our knowledge, theory for the design of rigorously aplanatic optical systems, considering as a starting point the rigorous stigmatism theory of optical systems composed of Cartesian surfaces. The main characteristic of these surfaces is their, a priori, zero spherical aberration. In a general parametric formulation for systems made up of a set of these surfaces, the Abbe sine condition is adapted to simultaneously obtain the stigmatism and aplanatism conditions. Thus, we achieved the design of optical systems that in theory are immune to both coma and spherical aberrations.

Analytic aplanatic singlet lens: setting and design for three-point objects and images in the meridional plane

We present the setting and design of an aplanatic optical system for three-point objects and images. The system is composed of a singlet aspheric lens and its entrance pupil. The aspheric lens and the entrance pupil are such that the three-point images and objects lay on the same plane. The image at the center is free of spherical aberration, and the other off-axis images are free of coma.

Manipulating the magnetic energy density and energy flux by cylindrically symmetric state of polarization

Using the Richards and Wolf formulas for an arbitrary cylindrical vector (CV) beam, we obtain explicit expressions for all components of the electric and magnetic field strength vectors near the focus, as well as expression for the energy flux in an aplanatic optical system. Based on such analytical models, it reveals that the intensity pattern of the magnetic field at the focus can be tailored by appropriately adjusting the initial phase, peak-centered, doughnut, and flat-topped magnetic fields can be achieved using this method. For the energy flows, in contrast, they are almost the same for arbitrary CV beams, which exhibit hollow shaped patterns for both the transverse and longitudinal components. Unlike the longitudinal component, however, the hollow shaped pattern of the transverse component is separated into two regions, arriving from the null transverse energy flow in the focal plane. Besides, the directions of the transverse energy flow are reversed between these two regions, which are directed inwardly and outwardly along the radial direction, respectively.

Pancharatnam-Berry phase shaping for control of the transverse enhancement of focusing.

We show that elongating a tightly focused field in the direction perpendicular to the optical axis is possible. We demonstrate our approach by specially shaping the Pancharatnam-Berry (PB) phase. Moreover, the analytical formulae required to calculate the strength vectors and energy flux of the three-dimensional electromagnetic fields near the focus of an aplanatic optical system are derived using the Richards and Wolf vectorial diffraction methods. Calculations reveal that the transverse enhancement is controllable and depend on the phase index in the PB phase, thereby giving rise to a focus with tunable length and subwavelength width in the focal plane.

Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy.

Using the Richards-Wolf formulas for an arbitrary circularly polarized optical vortex with an integer topological charge m, we obtain explicit expressions for all components of the electric and magnetic field strength vectors near the focus, as well as expressions for the intensity (energy density) and for the energy flux (components of the Poynting vector) in the focal plane of an aplanatic optical system. For m=2, from the obtained expressions it follows that the energy flux near the optical axis propagates in the reversed direction, rotating along a spiral around the optical axis. On the optical axis itself, the reversed flux is maximal and decays rapidly with the distance from the axis. For m=3, in contrast, the reversed energy flux in the focal plane is minimal (zero) on the optical axis and increases (until the first ring of the light intensity) as a squared distance from the axis.

Backward flow of energy for an optical vortex with arbitrary integer topological charge

We analyze the sharp focusing of an arbitrary optical vortex with the integer topological charge m and circular polarization in an aplanatic optical system. Explicit formulas to describe all projections of the electric and magnetic fields near the focal spot are derived. Expressions for the near-focus intensity (energy density) and energy flow (projections of the Pointing vector) are also derived. The expressions derived suggest that for a left-hand circularly polarized optical vortex with m > 2, the on-axis backward flow is equal to zero, growing in the absolute value as a power 2(m – 2) of the radial coordinate. These relations also show that upon the negative propagation, the energy flow rotates around the optical axis.

Analysis of Phase Distribution of Focused Light in High Numerical Aperture Systems

High numerical aperture focusing is becoming increasingly important for nanotechnology related applications. Rigorous, vector evaluation of the focused field, in such cases, is usually performed using the Richards-Wolf method which is based on the Debye approach. The resulting field is known to have a piecewise quasi planar phase. A corresponding result, produced by a Fresnel-Kirchhoff integral for aplanatic optical systems of medium and low numerical apertures, leads to the well known physical fact that a quadratic phase exists when the entrance pupil is not located at the front focal plane. Yet, the amplitudes produced in both ways are in a good agreement. In this work we investigated the difference, presented above, in a 2D system with the help of the Stratton-Chu diffraction integral. The amplitude obtained by the Stratton-Chu integral was quite similar to the classic results while the phase exhibited a quadratic behavior, with the quadratic coefficient depending on the numerical aperture of the optical system. For lower numerical apertures it approached the result obtained by the Fresnel-Kirchhoff integral while for higher numerical apertures it was approaching the Richards-Wolf result. A mathematical expression for the quadratic coefficient was derived and verified for various values of numerical aperture.

Tight focusing of wavefronts with piecewise quasi-constant phase

The Richards-Wolf approach to analyze tight focusing by high-numerical-aperture aplanatic optical systems was designed to treat incident waves having a planar (or negligibly curved) wavefront at the entrance pupil. In this paper we extend the approach to incorporate also wavefronts with piecewise quasi-constant phase. The evaluation of the field distribution in the vicinity of the geometrical focus is accomplished by linear superposition of the contributions from all the segments with quasi-constant phase. As an application example, a tightly focused optical field is evaluated for a -phase-step-modulated incident wavefront.

Filter performance parameters for vectorial high-aperture wave fields

Performance parameters have been presented that can be used to compare the focusing performance of different optical systems, including the effect of pupil filters. These were originally given for the paraxial case and recently extended to the high-aperture scalar regime. We generalize these parameters to the full vectorial case for an aplanatic optical system illuminated by a plane-polarized wave. The behavior of different optical systems is compared.

Mechanism of compensation of aberrations in the human eye

We studied the mechanism of compensation of aberrations within the young human eye by using experimental data and advanced ray-tracing modeling. Corneal and ocular aberrations along with the alignment properties (angle kappa, lens tilt, and decentration) were measured in eyes with different refractive errors. Predictions from individualized ray-tracing optical models were compared with the actual measurements. Ocular spherical aberration was, in general, smaller than corneal spherical aberration without relation to refractive error. However, horizontal coma compensation was found to be significantly larger for hyperopic eyes where angle kappa tended to also be larger. We propose a simple analytical model of the relationship between the corneal coma compensation effect with the field angle and corneal and crystalline shape factors. The actual shape factors corresponded approximately to the optimum shapes that automatically provide this coma compensation. We showed that the eye behaves as an aplanatic optical system, an optimized design solution rendering stable retinal image quality for different ocular geometries.

Structure of image fields in an aplanatic optical system with coherent point source illumination

Using the vector Fourier transform and stationary phase method, an integral representation of the structure of image field in an aplanatic system with polarized point source located at arbitrary position on the optical axis is obtained. Effects of wide aperture angle in object space on the structure of image field are examined numerically for the case of linearly polarized illumination. Simulation calculations show that, in a plane perpendicular to the optical axis, if the object space aperture angle is wide, the distribution of magnetic field will not be identical to that of electric field after they are rotated 90° around the optical axis with respect to each other, the contours of distribution of the electric and magnetic energy densities and the Poynting vector are approximately elliptical, and the object space aperture angle is the dominant factor that makes the distribution of Poynting vector lose circular symmetry. At the same time, the major axis of the contour of distribution of the electric energy density is perpendicular to the direction of the electric vector in the object space when using small image aperture angle, and will turn graduately to the same direction as the electric vector in the object space with the increase of image aperture angle. These conclusions are completely different from the predications of the previous theory.

Design of aplanatic waveguide Fresnel lenses and aberration-free planar optical systems

A modification of the waveguide Fresnel lens design that has phase-shifting zones placed on the curved contour is suggested. It is shown that the selection of a suitable shape for this contour allows coma aberration to be eliminated. Computer simulation results are presented to demonstrate the significant improvement of the optical characteristics for curved contour lenses. The aplanatic homogeneous refracting waveguide lenses are also considered. It is found that thick aplanatic refracting lenses and the proposed Fresnel lenses have field curvature aberration of opposite sign. This fact allows planar optical systems free from low-order aberration to be designed by combining refracting and Fresnel lenses. An algorithm for producing such a system is presented. >

Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration

The structure of the diffracted electromagnetic field generated by a paraboloidal mirror having a possible central obscuration without wavefront aberrations, but allowing for defocus, is studied. Particular attention has been paid to the time average electric energy density; numerical results are displayed for various azimuthal lines. Contour plots are shown; as expected they are not rotationally symmetric. Finally, we introduce and numerically evaluate the encircled time average electric energy density. In conformity with the investigations of Yoshida and Asakura on the corresponding problem of an aplanatic optical system with central obscuration, the structure of the diffracted electromagnetic field is very dependent on the amount of obscuration.

Aplanatic optical system containing two aspheric surfaces

The design of an aplanatic optical system with two aspheric surfaces according to a method developed by Vaskas is treated in more detail. The shape of each aspheric surface is determined by a differential equation. It is shown that the ray trajectories through the optical surfaces, located between the two aspheres, are found by solving a system of N linear equations where N is the number of intermediate surfaces. An example of an aspherized aplanatic system is given.

Degrees of Freedom of Aberrated Images

The problem of degrees of freedom of the images of a coherent aplanatic optical system suffering from aberrations is investigated by means of a suitable eigenfunction technique. This analysis leads to an evaluation of the number of effective degrees of freedom in the presence of band-limited noise. This number turns out to be independent of aberrations.

Energy Flow in the Neighborhood of the Focus of a Coherent Beam*

In a previous paper [ Phys. Rev.138, B1561 ( 1965)] some new results were presented relating to the structure of the electromagnetic field near the focus of a coherent beam emerging from an aplanatic optical system. The present paper supplements the previous one by providing detailed analysis of the behavior of the Poynting vector in the focal region of such a beam. In particular, diagrams showing the flow lines and the contours of constant amplitude of the time-averaged Poynting vector are given. The energy flow is found to have vortices near certain points of the focal plane. Two diagrams showing the behavior of the flow near a typical vortex are also included.

Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam

An integral representation for the electromagnetic field in the region of focus of a coherent light beam that emerges from an aplanatic optical system has been derived by Ignatowsky (1919) and by Richards and Wolf (1959). In the present paper this representation is used to analyze the structure of the focal region in a typical case. Contours of the time-averaged electric energy density in the focal plane, in one defocused plane and in two meridional sections of the focal region of a system with angular semi-aperture 45\ifmmode^\circ\else\textdegree\fi{} are presented. The meridional diagrams refer to axial sections through a cylindrical region around the axis near focus, of length $16\ensuremath{\lambda}$ and cross-sectional diameter $10\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the wavelength of the light. It is found that the field has a strong longitudinal component at certain points of the focal plane and that longitudinal electric field strengths of the order of ${10}^{5}$ V/cm could now be attained with focused laser beams. A diagram illustrating the complete behavior of the longitudinal component in the focal plane is also given.

A Class of Aplanatic Optical Systems

An arbitrary centred optical system is followed by two aspherical mirrors which are such that the complete system is exactly aplanatic. It is shown that the equations specifying these mirrors can be reduced to a first order linear differential equation and so the solution involves two quadratures. If these integrals can be evaluated then the mirrors can be specified exactly. If the arbitrary optical system is axially stigmatic then a particular solution of the differential equation can be found by inspection and the solution then involves only one quadrature. As an example the combination of two aspherical mirrors following a parabolic reflector to form an aplanatic telescope is solved exactly.