Theoretical analysis of a compensator of the secondary longitudinal chromatic aberration.

A theoretical analysis of an afocal compensator of the secondary longitudinal chromatic aberration is provided. Such a compensator is able to compensate for chromatic aberration of an achromatic optical system and make an apochromatic correction of the system. The parameters of the proposed chromatic aberration compensator are described and the principle is presented in examples.

Calculation of the wave aberration of an axial bundle of rays.

A differential equation is derived for the calculation of the wave aberration of an axial bundle of rays affected by spherical aberration. The solution of the derived equation is analyzed, and the influence of the radius of the reference sphere on the value of the wave aberration is presented. The approximate formula for wave aberration is obtained, which allows the splitting of the wave aberration into a part relative to an infinitely distant reference sphere and a part showing the dependence of the wave aberration on the radius of the reference sphere. Furthermore, the case of third-order and fifth-order spherical aberration is investigated, and the formulas for the calculation of the Strehl definition for the case of small aberrations are derived. The accuracy of the approximated formulas for the wave aberration is analyzed in an example.

Algebraic solution for aplanatic cemented doublets with aspherical surfaces of second degree.

An algebraic method of an initial design of the aplanatic cemented doublet with a unit focal length is described. Based on the third-order aberration theory, algebraic equations are derived for calculating the design parameters of an aplanatic thin cemented doublet with spherical surfaces and aspherical surfaces of the second degree. These equations make it easy to determine whether there exists a real solution of the doublet design for required parameters. The results of the calculations of selected cemented doublet solutions are presented in the examples. The obtained parameters can be used as the starting solution for the final optimization of doublet designs.

Method for algebraic calculation of initial design parameters of a two-element telecentric f-theta lens.

A method for the algebraic calculation of paraxial design parameters of a telecentric f-theta lens is described. Moreover, a third-order aberration analysis is performed for a telecentric f-theta lens composed of two optical elements. The method can be used to calculate the starting optical design of telecentric f-theta lenses, which can be used, for example, in laser material processing, optical imaging systems, and optical metrology. The initial design parameters then can be used for further optimization in optical design software.

Method of initial design of a two-element double-sided telecentric optical system.

An algebraic method for finding fundamental parameters of a starting design of a two-element double-sided telecentric lens is introduced in this work. The telecentric lens is formed by two objectives composed of an afocal meniscus lens followed by a cemented doublet. It is used the third-order aberration theory to find the fundamental parameters of the starting configuration of a given optical system. The method gives results which make possible to obtain a good initial design of such a telecentric lens for further optimization using optical design software. The proposed method is presented on an example of finding initial design parameters of the double-sided telecentric lens.

Algebraic method for finding the initial design parameters of a two-element optical system with an afocal meniscus lens.

This work introduces an algebraic method of the calculation of an initial design of a two-element optical system composed of a thick afocal meniscus lens and a thin lens element, which can be formed, for example, by a cemented doublet. The third-order aberration theory is used to find the fundamental parameters of the starting configuration of a given optical system. The method gives results that make it possible to obtain a good starting design for further optimization of such imaging systems using optical design software. The proposed method is presented on two examples of finding the initial design parameters of the objective with an afocal meniscus lens and a cemented doublet.

Stigmatic optical system with corrected third-order spherical aberration for an arbitrary position of the object.

Our paper is focused on a problem of analysis and design of a stigmatic optical system that has corrected third-order spherical aberration for an arbitrary position of the object. The relations for Seidel aberration coefficients of the system for an object at infinity that must be satisfied to ensure that the third-order spherical aberration does not depend on the position of the object are given. The method for obtaining design parameters of the initial optical system that can serve as a good starting point for further refinement using numerical optimization methods is proposed. Based on the use of modified formulas for third-order aberration coefficients, this method enables one to decide if the individual members of the optical system can be simple lenses or if these should be more complex elements (cemented doublets, triplets, etc.). As a final result, one obtains the design parameters of the above-mentioned optical system (radii of curvature, optical materials, axial separations between individual elements). The analysis is performed for a thin-lens representation of the system. The transition to the thick-lens optical system then can be done by mathematical methods of numerical optimization using commercially available optical design software. The proposed method is shown on a practical example of calculation of parameters of such an optical system.

Method of calculation of initial design parameters of microscope objective lenses with a long working distance.

We describe an analytic method of the initial design of microscope objectives with a long working distance. The optical system of a microscope objective is composed of two thin optical elements and a frontal optical element from several thick lenses. The paper describes formulas that make it possible to calculate fundamental parameters of the objective, namely Seidel aberration coefficients, radii of curvature, refractive indices of individual lenses of the microscope objective, etc. The proposed method of calculation of the initial design parameters of the microscope objective, which is based on the theory of aberrations of the third order and chromatic aberrations of the first order, enables us to obtain a good starting design for further optimization using optical design software. The method of the design is presented in an example of the microscope objective with a long working distance.

General method for replacing a thin lens with a thick lens with the same value of Seidel aberration coefficient of spherical aberration or coma.

The paper deals with the problem of replacing a thin lens with a thick lens having similar third-order aberration properties as the thin lens. The equations that make possible the calculation of parameters of the thick lens, which has the same value of focal length (or transverse magnification) and one of the Seidel aberration coefficients (either Seidel aberration coefficient of spherical aberration or Seidel aberration coefficient of coma) as the thin lens for given position of the entrance pupil and object, are derived. The application of the proposed method for calculation of parameters of the thick lens is shown in examples.

Calculation parameters of an objective consisting of three simple lenses.

A theoretical analysis of properties of an objective composed of three thin lenses is described in this paper. The formulas for the calculation of parameters of the three-lens objective are derived and the procedure for calculation of the shape of individual lenses of the objective is described using the third-order aberration theory. The application of the described analysis is presented in an example of the design of the three-lens objective.

Hybrid liquid-membrane lenses made by one fixed and one or two active optical components.

The paper presents a methodology of calculation of the inner structure of two- and three-component hybrid liquid-membrane lenses with variable focal length that have corrected spherical aberration and coma. Specifically, the formulas for calculation of initial-design inner parameters (radii of curvatures of individual surfaces, axial thickness, and refractive indices of a material of the lens) of a thin-lens system are derived for a hybrid two-component system (doublet) made by one glass and one liquid-membrane lens, and a hybrid three-component lens (triplet) made by one glass lens and two liquid-membrane lenses, which both have variable focal length and corrected spherical aberration and coma for an object at infinity. As optimization during the optical design process requires the starting point be very close to the optimal solution, the presented approach can be successfully used for its calculation, as it is based on fundamental proven formulas of optical aberrations.

Design of zoom systems composed of lenses with variable focal length.

This paper develops a methodology for a design of zoom-systems, which are composed of thin optical components with a variable focal length (e.g., membrane lenses). The proposed procedure allows us to design not only the outer parameters of the system (focal lengths and separation of lenses), but the inner structure of individual components of the system can be calculated as well (radii of curvature, thicknesses, and refractive indices) i.e., the starting values of the mentioned parameters can be calculated and used for the next optimization.

Double curvature membrane lens.

The paper presents a theoretical analysis of properties of a specific liquid membrane lens composed of two axially symmetric membranes of different thicknesses and double curvature. These membranes enclose a space where an optical liquid is filled. Mechanical and optical properties of the lens are then changed by varying the volume of the liquid. The paper presents new formulas for calculation of membrane deflections, radii of curvatures of the membranes, and axial geometry, which offer to minimize the third-order spherical aberration of the lens for an object at infinity. The presented theory is examined on specific examples.

Diffraction by perfect and imperfect amplitude grating.

The paper presents theoretical formulas for calculation of diffraction by perfect infinite and finite amplitude gratings with Fresnel and Fraunhofer approximations. Further, general formulas for diffraction by an imperfect diffraction grating are derived where edges of the grating are described with general harmonic functions. Such a formalism provides enough power to accurately characterize imperfections of diffraction gratings, and it serves as a simple tool for a solution to a diffraction problem.

Calculation of a lens system with one or two aspherical surfaces having corrected spherical aberration.

The paper presents a detailed theoretical analysis of characteristics of a rotationally symmetric lens system with one or two aspherical surfaces having corrected spherical aberration and reduced coma aberration for a given position of the object and the image. Formulas for surface shape optimization are derived, and the procedure for calculating the aspherical system is shown. The presented formulas are verified with examples of ray tracing.

Replacing a thin lens by a thick lens.

This paper deals with the problem of replacing a thin lens by a thick lens with approximately the same properties as the thin lens. Equations enabling the calculation of the parameters of a thick lens that has the same focal length and the same value for one of the Seidel aberration coefficients (either the Seidel aberration coefficient of spherical aberration or the Seidel aberration coefficient of coma) as the thin lens are derived. A comparison of the proposed method for calculating the parameters of the thick lens with existing methods is given in examples. Further, the problem of replacing a thick lens made of optical glass with a given refractive index by another thick lens with a different refractive index but with the same focal length and the same Seidel aberration coefficient of spherical aberration is investigated.

Fundamental design parameters of two-component optical systems: theoretical analysis.

The paper presents a detailed theoretical analysis of two-component optical systems of Petzval objective, tele-objectives, reverse tele-objectives, and objectives of anallactic type. This type of optical system is popular in practice, especially in the field of photographic technologies and surveying devices (theodolites, levelling devices, etc.), where anallactic telescopes with inner focusing are used. The paper presents methods of designing of fundamental parameters of the objective, i.e., focal distances of the objective's components and their mutual distance, and radii of curvatures of individual surfaces if the components are cemented doublets. Further, a detailed analysis of aberration properties of those optical systems is presented.

Point spread function of an optical system with defocus and spherical aberration-analytical formulas.

This work describes a procedure for an analytical calculation of the point spread function (PSF) of an optical system affected by defocus and spherical aberration. Explicit formulas are derived for the approximate calculation of the PSF of an optical system with spherical aberration up to the ninth order. Application of the derived formulas is performed on an example of optical systems with spherical aberration up to the third order.

Paraxial focal length calculation using Strehl definition measurement.

This work describes a method for determination of a paraxial focal length of optical systems using measurements of the Strehl definition. The influence of spherical aberration on the value of the measured effective focal length of an optical system is analyzed, and a method is proposed for an elimination of this influence and determination of the paraxial focus position and the paraxial focal length of a lens from the measurement of the effective focal length and the Strehl definition.

Method of paraxial design of rifle scope with four-element image-erecting zoom system and high zoom factor.

An analysis and a description of methods for a paraxial design of rifle scopes with a four-element image-erecting zoom system is performed. It is described a method of the calculation of basic design parameters of the rifle scope with the four-element image-erecting zoom system, where two inner elements are movable for ensuring zooming of the rifle scope. The method is demonstrated on two examples of the rifle scope design with the zoom factor 10×. Moreover, the possibility to design the rifle scope with the fixed position of the exit pupil using the four-element image-erecting zoom system with three movable elements is analyzed. The method of the initial paraxial design is demonstrated on two examples of the calculation of the basic rifle scope design parameters. The calculated parameters from proposed methods can be used in the aberration analysis and the optimization of the rifle scope parameters in optical design software.