Arbitrary Focal Length Research Articles

Design and fabrication of a focus-tunable liquid cylindrical lens based on electrowetting.

In this study, a focus-tunable liquid cylindrical lens based on electrowetting was designed and fabricated. The cylindrical cavity usually used in common electrowetting zoom spherical lenses was replaced by a 20 mm × 10 mm × 8 mm cuboid cavity, in which the interface of two liquids formed a toroid owing to the electrowetting effect. The proposed liquid cylindrical lens can serve as either a converging or diverging lens with the response time under 110 ms by changing the supplied voltage. The zoom lens we fabricated worked stably under 0-110 V voltage for a long time, guaranteeing that the focal length of the liquid cylindrical lens can range within (-∞, -148.36 mm) ∪ (697.21 mm, +∞). By combining the liquid lens that we designed with a simple fixed cylindrical lens, a cylindrical lens system with an arbitrary focal length suitable for various tasks in beam manipulation can be realized.

A General Calibration Method for Dual-PTZ Cameras Based on Feedback Parameters

With the increasing application of dual-PTZ (Pan-Tilt-Zoom) cameras in intelligent unmanned systems, research regarding their calibration methods is becoming more and more important. The intrinsic and extrinsic parameters of dual-PTZ cameras continuously change during rotation and zoom, resulting in difficulties in obtaining precise calibration. Here, we propose a general calibration method for dual-PTZ cameras with variable focal length and posture under the following conditions: the optical center of the camera does not coincide with the horizontal and pitch rotation axes, and the horizontal and pitch rotation axes are not perpendicular to each other. We establish a relationship between the intrinsic and extrinsic parameters and the feedback parameters (pan, tilt, zoom value) of dual-PTZ cameras by fitting and calculating previous calibration results acquired at specific angles and zoom values using Zhang’s calibration method. Subsequently, we derive the intrinsic and extrinsic parameter calculation formula at arbitrary focal length and posture based on the camera’s feedback parameters. The experimental results show that intrinsic and extrinsic parameters computed using the proposed method can better meet precision requirements compared with the ground truth calibrated using Zhang’s method. The average focal length error is less than 4%, the cosine similarity of the rotation matrix between the left and right cameras is more than 99.8%, the translation vector error is less than 1%, and the recalculated Euler angle errors are less than 1 degree. Our work can quickly and accurately obtain intrinsic and extrinsic parameters during the use of the dual-PTZ camera.

Realization of general first-order optical systems using nine thin cylindrical lenses of arbitrary focal length and four units of free propagation distance.

General first-order optical systems-represented by a four-dimensional real symplectic group-can be realized using thin lenses and free propagation transformations. It is shown that such systems can be realized using four units of free propagation transformation and nine thin rotated cylindrical lenses (or equivalently, four thin rotated astigmatic lenses and a thin rotated cylindrical lens). If these nine thin lenses placed in five transverse planes can be realized using spatial light modulators (SLMs), then our gadget realizes any general first-order optical system using five SLMs. It is also outlined that any general first-order optical system with finite symplectic matrix entries can always be realized using at least any one of the identities presented here, when a particular decomposition demands thin lenses with impractical focal length.

Realization of general first-order optical systems using thin lenses of arbitrary focal length and fixed free propagation distance.

Any general first-order optical system can be represented using S∈Sp(4,R), where Sp(4,R) is the symplectic group with real entries in four dimensions. We prove that any S∈Sp(4,R) can be realized using not more than 18 thin lenses of arbitrary focal length and seven unit distance. New identities that realize S=S1⊕S2, where S1,S2∈Sp(2,R), are obtained. Also, it is proved any S of the form S1⊕S2 can be realized using a maximum of eight thin lenses of arbitrary focal length and three unit distance. Moreover, decompositions for examples such as differential magnifier, partial Fourier transform, and inverse partial Fourier transform are also provided. A "gadget" is proposed that can realize any S∈Sp(4,R) using thin lens transformations-which can be realized through the use of eight spatial light modulators (SLMs) and seven unit distance. Experimental limitation imposed by SLMs while realizing thin lens transformations is also outlined. The justification for the choice of unit distance according to the availability of thin lenses in a lab is given too.

Realization of first-order optical systems using thin convex lenses of fixed focal length.

A general axially symmetric first-order optical system is realized using free propagation and thin convex lenses of fixed focal length. It is shown that not more than five convex lenses of fixed focal length are required to realize the most general first-order optical system, with the required number of lenses depending on the situation. The free propagation distances are evaluated explicitly in each situation. The optimality of the decomposition obtained in each situation is brought out. Decompositions for some familiar subgroups of SL2(R) are also worked out. Convex or concave lenses of arbitrary focal length are realized using three or two convex lenses of fixed focal length, respectively. It is further shown that three convex lenses of arbitrary focal length are sufficient to realize the most general first-order optical system.

Angular spectrum calculations for arbitrary focal length with a scaled convolution

Nyquist sampling theorem in an image calculation with angular spectrum method restricts a propagation distance and a focal length of a lens. In order to avoid these restrictions, we studied suitable expressions for the image computations depending on their conditions. Additionally, a lateral scale in an observation plane can be magnified freely by using a scaled convolution in each expression.

Parabola-doublet aplanat becomes anastigmatic when second doublet is inserted near focus.

A doublet of choice glasses may be located in the converging focal cone of the infinity-focused parabola to yield an aplanatic telescope or camera. The resulting angular field is limited by high astigmatism but is significantly larger than that of the coma-limited parabola. The spherical and chromatic aberrations are so well corrected and the coma so well balanced that the doublet may be used unaltered with a parabola of arbitrary focal length and speed with excellent results for the unvignetted rays. A second doublet nearer to the focus and designed independently of the first corrects the system's astigmatism while preserving its aplanaticism. It may also be designed for flattening the field. This arrangement may allow for greater flexibility in the placing of optical elements than does Wynne's triplet for modest-aperture systems. Equations are presented for choosing candidate glasses for the first doublet from the very limited manifold of solving glasses.

Extended fractional Fourier transforms

The concept of an extended fractional Fourier transform (FRT) is suggested. Previous FRT’s and complex FRT’s are only its subclasses. Then, through this concept and its method, we explain the physical meaning of any optical Fresnel diffraction through a lens: It is just an extended FRT; a lens-cascaded system can equivalently be simplified to a simple analyzer of the FRT; the two-independent-parameter FRT of an object illuminated with a plane wave can be readily implemented by a lens of arbitrary focal length; when cascading, the function of each lens unit and the relationship between the adjacent ones are clear and simple; and more parameters and fewer restrictions on cascading make the optical design easy.

Gradient-index Geodesic Lenses for Integrated Optics: A Uniform Theory

Abstract A new generation of planar waveguide lenses for integrated-optical circuits is proposed. They have potentially perfect focusing capabilities due to the assumed simultaneous action of the geodesic curvature of the thin film and of the gradient of its apparent refractive index. The ‘joint forces’ philosophy permits the range of the index variation, as well as the depth of the geodesic profile, to be considerably reduced or, at least, designed with a formerly inaccessible amount of flexibility, as compared with Luneburg and geodesic lenses. In the two-parts of this paper as a whole, a suitable theory of radially symmetric gradient-index geodesic lenses is formulated, the final equations of which provide a relatively simple and concise description of a family of possible full-aperture and non-full-aperture solutions with arbitrary focal length. Part I provides a detailed analysis of the full-aperture lens problem, together with two illustrative numerical examples. These two solutions are characterize...

Deformation of a Thin, Elastic Plate to a Deep Parabolic Cylinder

Equations governing the elastic deformation of thin plates through large displacements to deep parabolic cylinders are presented and solved. The solution consists of expressions for a spatially distributed surface pressure and uniform rim loads which, when applied to the plate, produce the specified, deep parabolic cylindrical shape. These forming loads are written in dimensionless form for parabolic cylinders of arbitrary focal length and arbitrary rim to rim aperture. Numerical results are presented and limiting values are discussed. The solution and results find immediate application to mechanical forming and adhesive retention of parabolic solar collector components.

Geodesic optical waveguide lens analysis*

In the geometrical approximation, ray paths over waveguide depressions (geodesic lenses) may be determimed by Fermat’s principle. In this paper an algroithm is derived, based on Fermat’s principle, to ray trace geodesic lenses with circular symmetry having an arbitrary generating curve. Ray tracing techniques are also presented for conical skirt mirror and toric edge rounding which are used to guide the rays into the geodesic lens. Furthermore, aspheric geodesic lens profiles are derived that provide perfect geometrical focusing with arbitrary focal length.

Stable and unstable cylindrical-laser-resonator theory in an angular spectrum representation

A new theory is formulated which describes the electromagnetic field inside of a laser resonator through an expansion of the field into an angular spectrum of plane waves. The resonator is composed of two identical, circular reflectors of arbitrary size, focal length, and axial separation. This theory is not limited by the usual paraxial approximations, and provides integral expressions for the spatial distribution of the field over the interior of the resonator in three dimensions. The integral expression for the transverse electric component of the field is evaluated numerically for several of the lower loss modes in confocal, spherical, and unstable resonators with linear dimensions the order of ten wavelengths. These results indicate that some of the modes in confocal and spherical resonators behave in some respects like modes that are unstable if the resonator has dimensions the order of a few wavelengths. The effect on the spatial distribution of the resonator field due to saturation of the amplifying medium is found to be slight if spatial hole burning is neglected.