Reliable manufacturing and high optical performance of plano-convex aspheric lenses necessitate an improved optical design characterized by an optimal refractive index of the lens material and the prevention of the slope surface shape oscillations, which provide an increased angular field of view and low sensitivity to misalignments. To formulate the criteria for the optimal refractive index, we propose an analytical representation of the optimal aspheric surface based on the Fermat's principle in the form of parametric functions of coordinates, allowing the surface to be approximated by the elliptical conic section expanded with high-order aspheric deformation terms starting from the 6th power. There are determined the ranges of optimal refractive indices for materials of plano-convex aspheric lenses: n=1.605…1.665 for an aplanatic design with the Abbe sine condition fulfilled, ensuring low lens sensitivity of the lens to angular misalignment; n=1.735…1.835 for sharp imaging of elongated longitudinal objects in accordance with the Herschel's condition.

A high-flux, high-resolution Offner imaging spectrometer with a compact configuration is proposed and demonstrated. By introducing a custom-designed lens group between the entrance slit and the primary concave mirror, the effective slit is optically transformed into a virtual slit with extended divergence and path length. This design fundamentally alleviates astigmatism and enables a larger numerical aperture (NA = 0.3), while preserving the intrinsic aberration-free benefits of the Offner configuration. Theoretical modeling and optical analysis were conducted to derive aplanatic conditions and optimize focal power distribution. A three-lens assembly, including a toroidal surface, was employed to decouple meridional and sagittal focal lengths and correct residual aberrations. The optimized system covers the 400-1000 nm spectral range with a measured spectral resolution of 1.35 nm and maintains an MTF>0.7 at 28 lp/mm across the full field of view. Compared to a conventional Offner spectrometer of equal resolution, the proposed design improves optical throughput by a factor of 2.25 and reduces system volume by 14.5%. Laboratory calibration using a mercury-argon lamp confirmed its high-resolution capabilities, and field experiments demonstrated excellent spectral and spatial imaging performance. The improved Offner spectrometer offers a compact, aberration-free, and high-resolution solution for applications requiring high-precision spectral imaging.

We propose a facile method for enhancing on-axis energy backflow relative to forward energy flow in the focused fields. By employing higher-order Laguerre-Gaussian (LG) complex amplitude profiles for left-handed circularly polarized vortex beams, we numerically demonstrate the proposed method. It is shown that for incident LG beams with a beam size parameter β4σ⩾1, the ratio of the peak absolute values of the Poynting vector for backward and forward energy flow increases with the radial index p of LG beam and the numerical aperture (NA) of the aplanatic lens. More remarkably, the area of on-axis energy backflow is simultaneously found to be more elongated for larger values of p. Besides, we further show the relative enhancement of on-axis energy backflow in the focal fields of second-order radially polarized LG beams with higher values of p. A similar enhancement effect is actually applicable to other light beams. These results hold potential applications in optical manipulation and sorting.

This manuscript introduces a method to design a freeform reflective optical system with the particularity of being aplanatic and having several stigmatic pairs. To achieve this goal the mathematical expressions in [Appl. Opt.63, 8068 (2024)APOPAI0003-693510.1364/AO.540494; Appl. Opt.62, 5260 (2023)APOPAI0003-693510.1364/AO.489061] are merged into a single set of expressions that leads to a system of differential equations that can be solved. The paper shows an illustrative example that explains how the method works. The resulting design is as expected by the theory proposed.

A detailed analysis of the determination of the starting design parameters of a thin air-spaced aplanatic doublet composed of variable focal length lenses is performed using the third-order aberration theory. For such a doublet, it is possible to vary its focal length without changing the relative position of the doublet lenses, which is not possible in conventional doublets made from optical glass. The method is presented in several examples. It provides a good starting point for further optimization of such an optical design.

We propose a method to obtain the tangential surface susceptibility of a single nonuniform and spatially dispersive metasurface achieving the same optical work as a complex optical system based on the generalized sheet transition conditions (GSTCs). To do so, a windowed Fourier transform is applied to ensure the realization of a design ray mapping condition and introduces the necessity of a locally periodic approximation. Then, we emulated and compared the optical performances of an analytically designed aplanatic singlet to the computed single metasurface. The ray mapping condition of the reference design composed of two local metasurfaces is conserved and the on-axis optical performances of the single aplanatic metasurface are the same as the reference singlet. The off-axis performances are slightly different from the reference but expected from different optical systems.

A fundamentally new, to the best of our knowledge, class of linear (2D) dual-mirror aplanats tailored to tubular absorbers is developed for the types of solar concentrators used for thermal power. It is shown that prior investigation establishing this concept possesses unrecognized additional designs, as well as having missed high-performance configurations. It is shown that our line-focus solar concentrators can attain intercept factors exceeding 0.9 at concentration values as high as 55, with practical mirror contours and assemblies. Designed expressly for tubular absorbers, they represent improvements upon previous aplanatic concentrators that were tailored to flat one-sided absorbers but applied to tubular absorbers (as well as to conventional parabolic troughs).

To break through the limitations of the classical sine condition, a multi-field cosine condition (MCC) is proposed in this paper. Building on this condition, a design method for wide field-of-view (FOV) freeform aplanatic systems is also proposed. This design method ensures that the optical system meets the MCC by correcting the mapping relationships of rays that originate from different apertures on both the object and image sides of the optical system. Using this method, we have successfully designed a chromatic-aberration-free and unobstructed freeform off-axis reflective microscope objective that is characterized by a wide FOV, an extensive spectral working range, and an extended working distance. The objective, which comprises only three freeform mirrors, has a numerical aperture (NA) of 0.4, magnification of 20×, an FOV of 2 mm × 2 mm, and an extended working distance of up to 60 mm. The objective demonstrates imaging quality close to the diffraction limit across the visible band to the short-wave infrared band (0.4-3 µm). When compared with a transmission-type wide-field microscope objective with identical NA and magnification properties, the reflective microscope objective nearly triples the FOV.

A theoretical approach for reducing multiple monochromatic aberrations using a flat metalens doublet is proposed and verified through ray tracing simulations. The theoretical relation between the Abbe sine condition and the generalized Snell's law is revealed in the doublet system. Starting from the Abbe aplanat design, minimization conditions of astigmatism and field curvature are derived. Based on the theory, a metalens doublet is semi-analytically optimized as a compact, practical-level meta-microscope objective lens working for a target wavelength. The proposed approach also reveals how to reduce lateral chromatism for an additional wavelength. The design degree of freedom and fundamental limits of the system are both rigorously analyzed in theory and verified through ray tracing simulations. It is expected that the proposed method will provide unprecedented practical opportunities for the design of advanced compact microscopic imaging or sensing systems.

The present paper introduces a set of equations to design an aplanatic catadioptric freeform optical system. These equations form a partial differential equation system, in which a numerical solution defines the first and last surfaces of the catadioptric freeform optical system, composed of an arbitrary number of reflective/refractive surfaces with arbitrary shapes and orientations. The solution of the equation can serve as an initial setup of a more complex design that can be optimized. An illustrative example is presented to show the methodology introduced in this paper.

The exact partial differential equation to design aplanatic freeform-mirror-based optical systems is presented. The partial differential equation is not limited by the number of freeform surfaces or their orientations. The solutions of this partial differential equation can be useful as initial setups that can be optimized to meet higher criteria. One of these solutions is tested as an example of the initial setup, and the results are as expected by the theory.

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.

The optical compensation method is a promising method to compensate the image motion by driving the optical element of the system. However, the tilt motion of the optical element could induce the degradation of the image quality. Based on the Nodal Aberration Theory (NAT), this paper proposes a novel optical image motion compensation design with a secondary tilt-aplanatic two-mirror optical system (STATOS), which achieves the insensitivity on the image quality of the optical system. First, the properties of coma-free pivot point depending on the location of stop at primary mirror or secondary mirror have been analyzed respectively. Then the calculating method about construction parameters in terms of STATOS is developed. Further, the maximum tilt angle of secondary mirror could be solved using NAT under ordered specific restriction. Finally, a prototype is designed with SM-tilting result of MTF value maintain above 0.32 at the cut-off frequency of detector, where the principal distance could remain stable and the changed distortion is better than 0.03%. A test flight has been carried out to prove the desired results and feasibility of STATOS used in an aerial mapping camera.

Snapshot hyperspectral imaging is superior to scanning spectrometers due to its advantage in dimensionality, allowing longer pixel dwell time and higher data cube acquisition efficiency. Due to the trade-off between spatial and spectral resolution in snapshot spectral imaging technologies, further improvements in the performance of snapshot imaging spectrometers are limited. Therefore, we propose a cemented-curved-prism-based integral field spectrometer (CIFS), which achieves high spatial and high spectral resolution imaging with a high numerical aperture. It consists of a hemispherical lens, a cemented-curved-prism and a concave spherical mirror. The design idea of aplanatic imaging and sharing-optical-path lays the foundation for CIFS to exhibit high-resolution imaging in a compact structure. The numerical model between the parameters of optical elements and the spectral resolution of the system is established, and we analyze the system resolution influenced by the hemispherical lens and the cemented-curved-prism. Thus, the refractive index requirements of the hemispherical lens and the cemented-curved-prism for the optimal spatial and spectral resolution imaging of the system are obtained, providing guidance for the construction of CIFS. The designed CIFS achieves pupil matching with a 1.8 f-number lenslet array, sampling 268 × 76 spatial points with 403 spectral channels in the wavelength band of 400 to 760 nm. The spectral and spatial resolution are further evaluated through a simulation experiment of spectral imaging based on Zemax. It paves the way for developing integral field spectrometers exhibiting high spatial and high spectral resolution imaging with high numerical aperture.

Single-molecule localization microscopy has developed into a widely used technique to overcome the diffraction limit and enables 3D localization of single-emitters with nanometer precision. A widely used method to enable 3D encoding is to use a cylindrical lens or a phase mask to engineer the point spread function (PSF). The performance of these PSFs is often assessed by comparing the precision they achieve, ignoring accuracy. Nonetheless, accurate localization is required in many applications, such as multi-plane imaging, measuring and modelling of physical processes based on volumetric data, and 3D particle averaging. However, there are PSF model mismatches in the localization schemes due to how reference PSFs are obtained, look-up-tables are created, or spots are fitted. Currently there is little insight in how these model mismatches give rise to systematic axial localization errors, how large these errors are, and how to mitigate them. In this theoretical and simulation work we use a vector PSF model, which incorporates super-critical angle fluorescence (SAF) and the appropriate aplanatic correction factor, to analyze the errors in z-localization. We introduce theory for defining the focal plane in SAF conditions and analyze the predicted axial errors for an astigmatic PSF, double-helix PSF, and saddle-point PSF. These simulations indicate that the absolute axial biases can be as large as 140 nm, 250 nm, and 120 nm for the astigmatic, saddle-point, and double-helix PSF respectively, with relative errors of more than 50%. Finally, we discuss potential experimental methods to verify these findings and propose a workflow to mitigate these effects.