Traditional approaches to satisfying optical performance requirements within application constraints leverage the degrees of freedom (DOF) provided by multiple surface shapes distributed among lenses composed of materials with different dispersion properties. Using homogeneous-index lens materials, the radius, conic constant, or higher order aspheric surfaces provide the DOF necessary to provide optical power and address certain geometric aberrations, while using lens materials with different dispersion properties provides the DOF necessary to compensate for chromatic aberrations. Nevertheless, to achieve the DOF necessary to achieve the performance required of compact head-mounted and mobile camera imaging systems, lens stacks must be composed of a significant number of complex shaped optical elements, resulting in undesirable size, weight, and cost. Recent advancements in inkjet print manufacture of three-dimensional (3D) freeform gradient-index (GRIN) optical materials have increased available design DOF by integrating volumetric index gradients within the bulk of the optical material. The 3D index gradients achieve optical power while reducing aberrations. Moreover, when the refractive index spectra of the optical feedstock are precisely formulated, dispersion may be independently controlled within the lens. The 3D GRIN optical materials may be machined and polished with aspheric or freeform surface shapes that further increase the DOF and provide lens designer unprecedented flexibility in lens designs with reduced lens counts. The performance benefits and applications of 3D GRIN optical elements are demonstrated, showcasing their potential to reduce size, weight, and cost in optical systems.

Increased degrees of freedom afforded by 3D freeform gradient index optics

In this work, we present the fabrication of two 3D printed plano-freeform prototypes, designed in such a way that, when assembled, an Alvarez lens is formed. The freeform surface of each element was mathematically described using Zernike polynomials and verified by implementing an off-axis null-screen test. Additionally, a characterization by refraction of the assembled lens was performed. Experimental images show the suitability of additive manufacturing engineering for prototyping freeform optics by providing a practical demonstration of the Alvarez lens concept. 

Modeling, Fabrication, and Metrology of 3D Printed Alvarez Lenses Prototypes

A crucial requirement of augmented reality head‐up displays (AR‐HUDs) is continuously adjustable virtual image distance (VID), which allows adaptation to various depths in road environments and thereby avoids visual fatigue. However, usual varifocal components for near‐eye displays are unavailable because AR‐HUDs require the varifocal component's aperture to be larger than 10 cm. This study considers the Alvarez lenses, which change the optical power by in‐plane sliding two freeform lenses. Under the paraxial assumption, classic Alvarez lenses can create a quadratic wavefront profile, but the large aperture and extensive diopter variation range required by AR‐HUDs lead to significant aberrations. Thus, the classic paraxial Alvarez lens design is extended by co‐optimizing Alvarez lenses with high‐order surface profiles and a primary freeform mirror. Therefore, a novel varifocal AR‐HUD containing Alvarez lenses with apertures larger than 15 cm is proposed. The AR‐HUD generates a varifocal plane whose VID can be continuously adjusted between 2.5 and 7.5 m, and another focal plane with a fixed VID at 7.5 m. In addition, merely one display panel is used for compactness. Finally, an AR‐HUD prototype with a reduced volume of 9.8 L was built. The expected varifocal performance and qualified imaging quality were experimentally verified through the field of view, VID, and image sharpness.

A varifocal augmented reality head‐up display using Alvarez freeform lenses

Hybrid refractive-diffractive lenses combine the light efficiency of refractive lenses with the information encoding power of diffractive optical elements (DOE), showing great potential as the next generation of imaging systems. However, accurately simulating such hybrid designs is generally difficult, and in particular, there are no existing differentiable image formation models for hybrid lenses with sufficient accuracy. In this work, we propose a new hybrid ray-tracing and wave-propagation (ray-wave) model for accurate simulation of both optical aberrations and diffractive phase modulation, where the DOE is placed between the last refractive surface and the image sensor, i.e. away from the Fourier plane that is often used as a DOE position. The proposed ray-wave model is fully differentiable, enabling gradient back-propagation for end-to-end co-design of refractive-diffractive lens optimization and the image reconstruction network. We validate the accuracy of the proposed model by comparing the simulated point spread functions (PSFs) with theoretical results, as well as simulation experiments that show our model to be more accurate than solutions implemented in commercial software packages like Zemax. We demonstrate the effectiveness of the proposed model through real-world experiments and show significant improvements in both aberration correction and extended depth-of-field (EDoF) imaging. We believe the proposed model will motivate further investigation into a wide range of applications in computational imaging, computational photography, and advanced optical design. Code will be released upon publication. 

pdf/end-to-end-hybrid-refractive-diffractive-lens-design-with-6eqcrxhor5.pdf

End-to-End Hybrid Refractive-Diffractive Lens Design with Differentiable
  Ray-Wave Model

Gradient-index Alvarez lenses, a new type of variable power component, are applied to the design of a zoom visual scope. The axially fixed Alvarez lenses allow for a reduction in system Gradient-index Alvarez lenses (GALs), a recently introduced variable power component, are highlighted in this work. A zoom visual scope is designed using GALs rather than with the conventional approach of longitudinally translating, fixed power elements. The resultant design achieves a significant reduction in system length compared with a conventional zoom scope. A secondary design is also demonstrated using only commercial off-the-shelf (COTS) homogeneous elements.

Compact zoom visual scope using 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.

Gradient-index Alvarez lenses.

The geometric, intensity, and chromatic distortions that are a result of the limitations of the material and processes used to fabricate micro-optical lens arrays (MLAs) degrade the performance of light-field systems. To address these limitations, inkjet print additive manufacturing is used to fabricate planar gradient index (GRIN) lenslet arrays, in which volumetric refractive index profiles are used to embed optical functions that would otherwise require multiple homogeneous index MLA surfaces. By tailoring the optical ink feedstock refractive index spectra, independent control over dispersion is achieved, and achromatic performance is made possible. Digital manufacturing is shown to be beneficial for optimizing individual micro-optical channels in arrays wherein the shape, size, aspect ratio, focal length, and optical axis orientation of the lenslets vary as a function of the position within the optical field. Print fabrication also allows opaque inter-lens baffling and aperture stops that reduce inter-channel cross talk, improve resolution, and enhance contrast. These benefits are demonstrated in a light-field display testbed. 

Three-dimensional gradient index (GRIN) microlens arrays for light-field and holographic imaging and displays

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Gradient-index Alvarez lenses.

Three-dimensional gradient index (GRIN) microlens arrays for light-field and holographic imaging and displays