Gradient-index Alvarez lenses: publisher's note.

This publisher's note contains a correction to Appl. Opt.62, 3485 (2023)APOPAI0003-693510.1364/AO.487089.

Gradient-index Alvarez lenses.

Gradient-index Alvarez lenses (GALs), a new, to the best of our knowledge, type of freeform optical component, are surveyed in this work for their unique properties in generating variable optical power. GALs display similar behavior to conventional surface Alvarez lenses (SALs) by means of a freeform refractive index distribution that has only recently been achievable in fabrication. A first-order framework is described for GALs including analytical expressions for their refractive index distribution and power variation. A useful feature of Alvarez lenses for introducing bias power is also detailed and is helpful for both GALs and SALs. The performance of GALs is studied, and the value of three-dimensional higher-order refractive index terms is demonstrated in an optimized design. Last, a fabricated GAL is demonstrated along with power measurements agreeing closely with the developed first-order theory.

Calculation of zonal power and astigmatism of a freeform gradient index lens with freeform surfaces.

Freeform gradient index (F-GRIN) lenses have been recently shown to enable compact optical design. However, aberration theory is only fully developed for rotationally symmetric distributions with a well-defined optical axis. The F-GRIN has no well-defined optical axis, and rays are continuously perturbed along their trajectory. Optical performance can be understood without abstracting optical function to numerical evaluation. The present work derives freeform power and astigmatism along an axis through a zone of an F-GRIN lens with freeform surfaces. Zonal power and astigmatism can be assessed without tracing any rays, capturing mixed contributions of the F-GRIN and freeform surface. Theory is compared with a commercial design software numerical raytrace evaluation. The comparison shows that the raytrace-free (RTF) calculation represents all raytrace contributions within a margin of error. In one example, it is demonstrated that linear terms of index and surface alone in an F-GRIN corrector can correct the astigmatism of a tilted spherical mirror. Accounting for the induced effects of the spherical mirror, RTF calculation provides the amount of astigmatism correction of the optimized F-GRIN corrector.

Multi-material freeform gradient-index spectrometers.

With the advent of 3D printing optical elements, new techniques for manufacturing gradient-index (GRIN) optics have been realized that blend multiple materials in the deposition process. A method to achieve spectral splitting using multi-material GRIN optics is presented. The GRIN is additionally used to generate optical power, allowing for planar entrance and exit surfaces. It is shown that this simultaneous focusing and spectral splitting is not feasible with a two-material GRIN. A comparative design study is then conducted using three and four-material GRIN. A four-material design with optimized materials is also presented to showcase the potential for this new design form.

Achromatization of multi-material gradient-index singlets.

Recent advancements in additive manufacturing have enabled new methods of fabricating gradient-index (GRIN) optics by blending multiple materials in the deposition process. A design study highlighting the advantages of multi-material GRIN optics is presented. It is shown that additional materials in the GRIN allow for higher orders of color correction. A new multi-material refractive index representation, which constrains the GRIN to real materials, is also presented.

Freeform gradient index generalized Coddington's equations.

Coddington's equations and their generalized forms are useful for lens design and analysis of optical performance. Generalized Coddington's equations (GCE) exist in literature for analysis of decentered systems and freeform surfaces, but not for gradient index (GRIN) lenses. In this work, GCE are presented for the analysis of freeform GRIN lenses with freeform surfaces. Examples are shown where the presented theory converges on Coddington's equations and known paraxial GRIN behavior. The method also correctly shows known afocal behavior proximate to azimuthally directed rays in a cylindrical GRIN. The latter case is one of analytically validated local freeform behavior.

Replacing optical surfaces with gradient index functions which preserve ray trajectory.

Laws of reflection and refraction between homogeneous media and gradient index (GRIN) ray behavior are both derived from Fermat's principle. Design methods for GRIN can be difficult to analytically develop. This Letter proposes a foundation for complete replacement of refracting and total internally reflecting optical interfaces in existing designs with GRIN distribution. The proposed method can aid in incorporating GRIN into existing optical designs. Refraction in GRIN is specified to match the ray striking and leaving the optical interface in both position and angle. This result is shown for a collection of similar GRIN functions. One GRIN function is analyzed over a full space of attainable ray bend angles. A local arbitrarily oriented planar interface is replaced with GRIN distribution, and ray behavior is maintained.

Polychromatic annular folded lenses using freeform gradient-index optics.

The annular folded lens (AFL) is a design form offering large aperture, high-resolution imaging in a very axially compact package. The folded optic can be made monolithic for easier fabrication and alignment, yet the introduction of refractive surfaces with a dispersive optical material gives way to chromatic aberrations. AFL designs using homogeneous media are generally limited to the monochromatic regime, with polychromatic performance greatly reduced. By introducing freeform gradient-index (F-GRIN) media, monolithic AFL designs can achieve higher monochromatic performance as well as provide color correction for diffraction-limited polychromatic imaging. Monochromatic and polychromatic design methodologies are surveyed where the F-GRIN is constrained to remain feasible for fabrication.

Freeform gradient index progressive addition lens raytrace performance evaluation.

Raytrace evaluation capable of evaluating progressive addition lens (PAL) designs with freeform surface and gradient index (GRIN) contributions is presented. The method is validated on an analytically generated PAL start design and on optimized surface designs. Surface raytrace evaluations are compared with the surface-geometric evaluation commonly presented for freeform surface PAL designs. The evaluation is also tested on analytically generated freeform GRIN PAL designs with spherical and plano surfaces. The raytrace method agrees with the analytic performance and surface-geometric performance near the center of the lens and deviates at the edge of the lens, due to ray obliquity with the surfaces and aggregate contributions of surfaces and/or GRIN. These deviations are expected, as the raytrace model accounts for more physical contributions to optical performance, including pupil diameter and eye position. This raytrace method enables the evaluation of lens performance contributions other than from polished surfaces on homogeneous materials, enabling further exploration of GRIN in PAL designs.

Freeform gradient-index media: a new frontier in freeform optics.

Freeform optics enable irregular system geometries and high optical performance by leveraging rotational variance. To this point, for both imaging and illumination, freeform optics has largely been synonymous with freeform surfaces. Here a new frontier in freeform optics is surveyed in the form of freeform gradient-index (F-GRIN) media. F-GRIN leverages arbitrary three-dimensional refractive index distributions to impart unique optical influence. When transversely variant, F-GRIN behaves similarly to freeform surfaces. By introducing a longitudinal refractive index variation as well, F-GRIN optical behavior deviates from that of freeform surfaces due to the effect of volume propagation. F-GRIN is a useful design tool that offers vast degrees of freedom and serves as an important complement to freeform surfaces in the design of advanced optical systems for both imaging and illumination.

Prescribed irradiance distributions with freeform gradient-index optics.

Generating a prescribed irradiance distribution given a source distribution is an inverse problem that sits at the heart of illumination design. The growing prevalence of freeform optics has inspired several design methods for obtaining a prescribed irradiance distribution possessing no symmetry. Up to now, these methods have relied exclusively on freeform optical surfaces for generating freeform irradiances. This paper presents a design method that, for the first time, applies gradient-index (GRIN) optics to solving this inverse problem. Using a piecewise-continuous freeform gradient-index (F-GRIN) profile, a single optic with two planar surfaces can be designed to produce a far-field prescribed irradiance distribution from a point source. The design process is herein presented along with two design examples which demonstrate some of the unique properties of F-GRIN illumination optics.

Simultaneous interferometric measurement of linear coefficient of thermal expansion and temperature-dependent refractive index coefficient of optical materials.

Characterizing the thermal properties of optical materials is necessary for understanding how to design an optical system for changing environmental conditions. A method is presented for simultaneously measuring both the linear coefficient of thermal expansion and the temperature-dependent refractive index coefficient of a sample interferometrically in air. Both the design and fabrication of the interferometer is presented as well as a discussion of the results of measuring both a steel and a CaF2 sample.

Gradient-index materials based on thiol-ene networks.

Gradient-index (GRIN) optics offer spatially varied refractive indexes that can enhance current imaging technologies. Current methods to fabricate GRIN optics are highly complex and costly. Here we report a simple and efficient method that utilizes commercially available reagents to fabricate polymeric GRIN optics with significant refractive index differences (Δn = 0.04). First, two different mixtures of network precursors are layered and time allotted for molecular diffusion in the liquid state, prior to curing. The resulting, partially mixed layers are UV-cured to yield clear, glassy molecular networks with fixed refractive index gradients. The fully cured network resins exhibit smoothly varying composition and refractive index over centimeter length scales, confirmed by spectroscopy and interferometry.