Plane-symmetric Optical Systems Research Articles

Third-Order Intrinsic Aberration Modification with Second-Order Accuracy Transfer of Pupil Coordinates and Extrinsic Aberration of Plane-Symmetric Optical System with a Two-Dimensional Field Light Source

A plane-symmetric optical system with a two-dimensional field light source, such as a soft X-ray and vacuum ultraviolet optical system, is widely used for synchrotron radiation source devices, microscopes and so on. Therefore, this paper studied third-order intrinsic aberration modification with second-order accuracy transfer of pupil coordinates and extrinsic aberration of these kinds of optical systems. First, we discuss the extrinsic aberration calculation method and derive the corresponding aberration expressions; in addition, we apply optical geometric relations and the polynomial fitting method to obtain a second-order accuracy transfer relationship expression of the pupil coordinates between adjacent optical surfaces and derived the modification calculation expressions of the intrinsic aberration caused by the above factor. Finally, the third-order intrinsic aberration modification with second-order accuracy transfer of pupil coordinates and extrinsic aberration expressions derived in this paper are applied to calculate ray spot diagrams of two soft X-ray and vacuum ultraviolet optical systems with meridional and sagittal field angle light sources, and these results are compared with those obtained from the ray-tracing software Shadow; this showed that the calculation accuracy of the aberration expressions is sufficient.

Determination Position and Initial Value of Aspheric Surface for Fisheye Lens Design

The aspheric surface is a commonly used method to improve the imaging quality of the fisheye lens, but it is difficult to determine the position and initial value. Based on the wave aberration theory of the plane-symmetric optical system, a method of using an aspheric surface to design a fisheye lens is proposed, which can quickly determine the appropriate aspheric surface to improve the imaging performance. First, the wave aberration of each optical surface of the fisheye lens is calculated and its aberration characteristics are analyzed. Then, a numerical evaluation function is reported based on the aberration distribution of the fisheye lens on the image plane. According to the functional relationship between the evaluation function and the aspheric coefficient, the position of the aspheric surface and the initial value of the aspheric coefficient can be calculated. Finally, the adaptive and normalized real-coded genetic algorithm is used as the evaluation function to optimize the fisheye lens using an aspheric surface. The proposed method can provide an effective solution for designing a fisheye lens using an aspheric surface.

Intrinsic aberration coefficients for plane-symmetric optical systems consisting ofspherical surfaces.

This paper presents the analytical form of the intrinsic aberration coefficients for spherical plane-symmetric optical systems expressed as a function of first-order system parameters and the paraxial chief and marginal ray angles and heights. The derived aberration coefficients are in the third and fourth groups with the multiplication of two or three vector products of pupil and field vectors.

Third-order aberration of soft X-ray optical systems with orthogonal and coplanar arrangement of the main planes of elements.

A third-order aberration analytical analysis method of soft X-ray optical systems with orthogonal and coplanar arrangement of the main planes of elements is proposed. Firstly, the transfer equations of the aperture ray and the principle ray are derived; then, based on the third-order aberration theory with the aperture-ray coordinates on the reference exit wavefront of a plane-symmetric optical system, the aberration expressions contributed by the wave aberration and defocus of this kind of optical system are studied in detail. Finally, the derived aberration calculation expressions are applied to calculate the aberration of two design examples of such types of optical systems; the images are compared with ray-tracing results obtained using the Shadow software to validate the aberration expressions. The study shows that the accuracy of the aberration expressions is satisfactory. The analytical analysis method of aberration is helpful in the design and optimization of the soft X-ray optical systems with orthogonal and coplanar arrangement of the main planes ofoptical elements.

Calculation of the wave aberration of field curvature and color aberrations of an ultrawide-angle optical system

Based on the aberration theory of plane-symmetric optical system, this paper develops a method for calculating the wave aberration of the field curvature and color aberrations of each optical surface in an ultrawide-angle optical system, which enables us to analyze, like Seidel aberration theory, the contribution and balance of these aberrations between different optical surfaces. We then apply the resultant formulas to calculate the wave aberrations of the field curvature and color aberrations of two optical lenses, and compare the results with the corresponding wave aberrations values obtained from Seidel aberration theory and Zemax software. The study shows that accuracy of the results obtained with our formulas are quite satisfactory even in the large range of field view while the accuracy of those with the Seidel aberrations is limited by the field angle. Our method can provide an effective means for aberration analysis and optimization design of an ultrawide-angle optical systems.

变焦鱼眼镜头系统设计

The zoom fish-eye lens had the characteristics of much larger field-of-view angle, much larger relative aperture, and much larger anti-far ratio. In the work described in this paper, the initial structure of a fish-eye lens with fixed focal length was firstly designed using the theory of a plane symmetric optical system. And then the initial structure components of the fish-eye lens were divided into two groups, the former-group and the rear-group, and then the entire zoom fish-eye lens system was optimized using Gaussian optical theory. Finally, a zoom fish-eye lens system with good imaging quality was obtained. This fish-eye lens system had a field-of-view angle of 180° with an 8 mm short focal length and a field-of-view angle of 90° with a 16 mm long focal length. Its relative aperture was 1/3.5. The design results show that the modulation transfer function of this zoom lens system at different focal lengths is no less than 0.45 when the spatial frequency is 50 lp/mm. This zoom fish-eye photographic objective lens had higher imaging quality than other zoom fish-eye lenses.

Method of calculating the aberrations of soft X-ray and vacuum ultraviolet optical systems.

Based on the the third-order aberration theory of plane-symmetric optical systems, this paper studies the effect on aberrations of the second-order accuracy of aperture-ray coordinates and the extrinsic aberrations of this kind of optical system; their calculation expressions are derived. The resultant aberration expressions are then applied to calculate the aberrations of two design examples of soft X-ray and vacuum ultraviolet (XUV) optical systems; images are compared with ray-tracing results using SHADOW software to validate the aberration expressions. The study shows that the accuracy of the aberration expressions is satisfactory.

Chromatic aberration of plane-symmetric optical systems.

This paper presents a calculation method of chromatic aberration of such kinds of optical systems based on the aberration theory of plane-symmetric optical systems, and the analytic expressions of the axial chromatic aberration and chromatic change in the magnification at an image plane of any position are derived. The resultant expressions are used to calculate the chromatic aberration of two single-lens optical systems with a large incidence angle, and the calculation results are compared to the ones obtained from a ray-tracing program to validate the chromatic aberration expressions. The study shows that the calculation accuracy of the chromatic aberration expressions is satisfactory. The analytical analysis of chromatic aberration is helpful in the imaging analysis and design of plane-symmetric optical systems.

Sixth-order wave aberration theory of ultrawide-angle optical systems.

In this paper, we develop sixth-order wave aberration theory of ultrawide-angle optical systems like fisheye lenses. Based on the concept and approach to develop wave aberration theory of plane-symmetric optical systems, we first derive the sixth-order intrinsic wave aberrations and the fifth-order ray aberrations; second, we present a method to calculate the pupil aberration of such kind of optical systems to develop the extrinsic aberrations; third, the relation of aperture-ray coordinates between adjacent optical surfaces is fitted with the second-order polynomial to improve the calculation accuracy of the wave aberrations of a fisheye lens with a large acceptance aperture. Finally, the resultant aberration expressions are applied to calculate the aberrations of two design examples of fisheye lenses; the calculation results are compared with the ray-tracing ones with Zemax software to validate the aberration expressions.

Aberrations of soft x-ray and vacuum ultraviolet optical systems with orthogonal arrangement of elements

The wave aberration of plane-symmetric optical systems is expressed with the aperture-ray coordinates on the reference exit wavefront in the paper; the defocus aberration caused by the meridional field of source is analyzed in detail. Based on the expressions of the wave aberration and the defocus, the aberrations of the soft x-ray and ultraviolet (XUV) optical systems with an orthogonal arrangement of elements are studied as a whole. The resultant aberration expressions are used to calculate the aberrations of two design examples; the images are compared to the ray-tracing results with Shadow to validate the aberration expressions. The study shows that the accuracy of the aberration expressions is satisfactory. The analytical analysis of aberrations is helpful in the design and optimization of the XUV optical systems with an orthogonal arrangement of elements.

Two-mirror telescope design with third-order coma insensitive to decenter misalignment

Misalignments always occur in real optical systems. These misalignments do not generate new aberration forms, but they change the aberration field dependence. Two-mirror telescopes have been used in several applications. We analyze a two-mirror telescope configuration that has negligible sensitivity to decenter misalignments. By applying the wave aberration theory for plane-symmetric optical systems it is shown that the asphericity in the secondary mirror, if properly chosen, can compensate for any decenter perturbation allowing third-order coma unchanged across the field of view. For any two-mirror system it is possible to find a configuration in which decenter misalignments do not generate field-uniform coma.

Third-order aberrations of a plane symmetric optical system

The wave aberration theory of non-symmetrical optical systems is useful for understanding the misalignment in symmetrical systems and designing of off-axis mirror systems. A theory about the third-order aberrations for sub-aperture plane symmetric optical system is developed by using the aberration theory for full-aperture axially symmetric spherical systems. This paper proves the nodal aberration theory, namely, the points in the field with zero third-order aberration will deviate from the field center except for spherical aberration. It also reveals that the nodal aberrations arise from the transformation of the aberrations in full-aperture systems. This theory can be efficiently used in non-symmetric optical system designing process.

Experimentally validated computational imaging with adaptive multiaperture folded architecture

We present experimental results of imaging and digital superresolution in a multiaperture miniature folded imaging architecture called PANOPTES. We prove the feasibility of integrating a low f-number folded imagers within a steerable multiaperture framework while maintaining a thin profile. Stringent requirements including low f-number and compact form factor, combined with the need for an ability to steer individual fields of view necessitate an off-axis design, resulting in a plane symmetric optical system. We present a detailed description of the ensuing optical design and its performance. The feasibility of this architecture is demonstrated through experiments and preliminary reconstruction results.

Geometric characteristics of aberrations of plane-symmetric optical systems

The geometric characteristics of aberrations of plane-symmetric optical systems are studied in detail with a wave-aberration theory. It is dealt with as an extension of the Seidel aberrations to realize a consistent aberration theory from axially symmetric to plane-symmetric systems. The aberration distribution is analyzed with the spot diagram of a ray and an aberration curve. Moreover, the root-mean-square value and the centroid of aberration distribution are discussed. The numerical results are obtained with the focusing optics of a toroidal mirror at grazing incidence.

Aberrations of plane-symmetric multi-element optical systems

Based on a recently developed aberration theory of plane-symmetric grating systems, its aberration coefficients are firstly extended to be applicable to the plane-symmetric refractive optical surface. To justify the linear approximation of aperture ray in a multi-element system, the ray aberrations of double-element systems with the third-order accuracy of aperture ray are derived and expressed as the form of wave aberrations. The numerical comparisons are then made between the ray tracing, the aberration expressions with the linear and with third-order accuracy of the aperture ray. Moreover, the modifications of the wave and ray aberrations are discussed in the paper when the position of the principal ray changes.

Aberration theory of plane-symmetric grating systems

Aberration theory of plane-symmetric optical systems of mirror and grating has been developed based on the wavefront aberration method. A toroidal reference wavefront surface is used to define the wavefront aberration. Based on the ray geometry, the coordinate mapping relationships of the ray between the optical element and the incident and aberrated wavefronts are derived using a polynomial-fit method; this enables the resultant coefficients of the wavefront and the transverse aberration to be kept to the fourth-order accuracy of the aperture-ray coordinates. By setting up the transfer equations of the field and aperture rays, the contribution to wavefront aberrations from each mirror and grating can be added to make the aberration calculation of multi-element systems feasible. The theory is validated by the analytic formulae of the spot diagram.

Design of Plane-Symmetric Low F-number Reflective System Based on Perturbation Aberration Model

All-reflective optical systems have theoretically no chromatic aberration and are suitable for wide spectral application, but co-axial reflective systems are difficult to design as low F-number optical systems. In this paper, the aberration of plane-symmetric optical systems with low F-number is analyzed based on the wavefront derived from optical path length. Up to third-order aberrations are classified into three categories by their characteristics. The reduction method of the dominant aberration, astigmatism-like aberration, is proposed. In the method, the Gauss image plane is modeled by means of a simplified mirror surface with third-order approximation. Because the system is plane-symmetric, symmetric cross section and asymmetric cross section are modelled differently. The design example of a three-mirror system with F/2 is shown.

Conditions for correction of linear and quadratic field-dependent aberrations in plane-symmetric optical systems.

The Abbe sine condition and the recently developed pupil astigmatism conditions provide a powerful set of relationships for describing imaging systems that are free from aberrations that have linear and quadratic dependence on field, to all orders in the pupil. We have proved both of these conditions and applied them to axisymmetric imaging systems. We now extend our approach to plane-symmetric systems. Still using Hamilton's characteristic functions, we derive the general sine conditions and the pupil astigmatism conditions that describe plane-symmetric systems that are free of all aberrations with linear and quadratic field dependence.