Gradient Index Lens Research Articles

Additional Error of Optical Path Measurement Caused by Radial GRIN Lens With Mispositioned Sample

Radial gradient index (GRIN) lens is widely used in optical path measurement based on white light interferometry (WLI), yet the additional error caused by the GRIN lens when the sample under test (SUT) is not at the ideal position has rarely been discussed. To deal with this problem, a ray-tracing model is established to reveal the influence of angular tilt and axial offset of SUT when GRIN lens is applied. Simulations based on WLI are done using the parameters of a real GRIN lens. The results show that the maximum additional optical path change is as larger as 0.99μm within the range of ±0.1° angular tilt and ±2000μm axial offset of SUT, which can be regarded as additional error if the influence of GRIN lens is ignored. Experimental results of optical path change and insertion loss at different tilt angles of SUT simultaneously agree well with one set of simulation.

Tunable spatial fractional derivatives with graphene-based transmit arrays.

The optical implementation of mathematical spatial operators is a critical step toward achieving practical high-speed, low-energy analog optical processors. In recent years, it has been shown that using fractional derivatives in many engineering and science applications leads to more accurate results. In the case of optical spatial mathematical operators, the derivatives of the first and second orders have been investigated. But no research has been performed on fractional derivatives. On the other hand, in previous studies, each structure is dedicated to a single integer order derivative. This paper proposes a tunable structure made of graphene arrays on silica to implement fractional derivative orders smaller than two, as well as first and second orders. The approach used for derivatives implementation is based on the Fourier transform with two graded index lenses positioned at the structure's sides and three stacked periodic graphene-based transmit arrays in middle. The distance between the graded index lenses and the nearest graphene array is different for the derivatives of order smaller than one and between one and two. In fact, to implement all derivatives, we need two devices with the same structure having a slight difference in parameters. Simulation results based on the finite element method closely match the desired values. Given the tunability of the transmission coefficient of the proposed structure in the approximate amplitude range of [0,1] and phase range of [-180, 180], on top of the acceptable implementation of the derivative operator, this structure allows obtaining other spatial multi-purpose operators, which are a prelude to achieving analog optical processors and even improving the optical studies performed in image processing.

Enabling High Precision Gradient Index Control in Subsurface Multiphoton Lithography.

Multiphoton lithography inside a mesoporous host can create optical components with continuously tunable refractive indices in three-dimensional (3D) space. However, the process is very sensitive at exposure doses near the photoresist threshold, leading previous work to reliably achieve only a fraction of the available refractive index range for a given material system. Here, we present a method for greatly enhancing the uniformity of the subsurface micro-optics, increasing the reliable index range from 0.12 (in prior work) to 0.37 and decreasing the standard deviation (SD) at threshold from 0.13 to 0.0021. Three modifications to the previous method enable higher uniformity in all three spatial dimensions: (1) calibrating the planar write field of mirror galvanometers using a spatially varying optical transmission function which corrects for large-scale optical aberrations; (2) periodically relocating the piezoelectrically driven stage, termed piezo-galvo dithering, to reduce small-scale errors in writing; and (3) enforcing a constant time between each lateral cross section to reduce variation across all writing depths. With this new method, accurate fabrication of optics of any index between n = 1.20 and 1.57 (SD < 0.012 across the full range) was achieved inside a volume of porous silica. We demonstrate the importance of this increased accuracy and precision by fabricating and characterizing calibrated two-dimensional (2D) line gratings and flat gradient index lenses with significantly better performance than the corresponding control devices. As a visual representation, the University of Illinois logo made with 2D line gratings shows significant improvement in its color uniformity across its width.

Gain Enhancement of Millimeter Wave Antenna by Ultra-thin Radial Phase Gradient Metasurface for 5G Applications

In this article, gain augmentation of a microstrip antenna (MSA) operating at 28 GHz has been enunciated in a cost-effective manner using a phase-graded index lens. Different metamaterial unit cells with transmissive characteristics have been designed and placed in an array to create a radial phase graded metalens with locally varying refractive index and spatially varying surface impedance. Transformation of spherical wavefronts into planar wavefronts results in beam collimation which helps in obtaining a directive beam in the 5G frequency band. The ratio of focal length to the diameter (f/d) of the lens is 0.38. The gain of MSA is enhanced by 4.4 dB with radiation efficiency above 95% due to the focusing effect of the lens placed vertically above the antenna. The proposed planar lens is compact in size and easy to fabricate when compared with the conventional 3-D lens reported in existing works. The microstrip patch integrated lens prototype exhibits a gain of 12.4 dBi operating at 28 GHz. The performance of the lens has been analyzed theoretically and evaluated further by conducting experiments on the fabricated prototype. Measured results validate the full–wave simulation results carried out using finite element method.

Inkjet-printed freeform gradient refractive index Alvarez-Lohmann optics for vision correction in respiratory masks

Inkjet-print additive manufactured freeform gradient refractive index representations of an Alvarez-Lohmann lens were developed and optimized for use in vision-correction inserts for respiratory masks. A series of planar 30-mm by 20-mm Alvarez-Lohmann lens pair were printed and characterized. Select lens pairs were mounted with printed mechanical lens mounts and translation guides optimized for vision correction. Metrology showed accurate reproduction of the cubic refractive phase functions. Tests matching simulated performance showed a large diopter adjustment over a large eyebox and a wide gaze angle. The printed optics were environmentally tested and then mounted in a respiratory mask and demonstrated with glove-accessible control electronics. The performance demonstrates the benefits of using printed three-dimensional freeform gradient index optics for implementing complex refractive optical functions for challenging applications including artificial and virtual reality systems.

Detailed derivation of the generalized Snell-Descartes laws from Fermat's principle.

Beginning with Fermat's principle, we provide a detailed derivation of the generalized laws of refraction and reflection for a geometry realizing a metasurface. We first solve the Euler-Lagrange equations for a light ray propagating across the metasurface. The ray-path equation is found analytically, and the results are supported by numerical calculations. We get generalized laws of refraction and reflection that have three main features: (i)They are relevant in gradient-index optics and in geometrical optics; (ii)A collection of rays emerges from the metasurface as a result of multiple reflections inside the metasurface; and (iii)The laws, although derived from Fermat's principle, differ from previously published results.

Terahertz spoof surface plasmon polariton gradient index lens

Plasmonics takes advantage of surface plasmon polaritons (SPPs), which have the characteristics of strong subwavelength field confinement and enhancement. Plasmonic devices that confine light in regions with dimensions that are smaller than the wavelength have important application prospects in on-chip integrated circuits. Nonetheless, at terahertz frequencies, the weak confinement of SPPs and immature near-field technology hinder the development of related devices. Here, a highly confined SPP mode (spoof SPP) is implemented by structuring metal surfaces. Gradient index lenses with variable indices are constructed by fine-tuning the size of the pillars, and the characteristics of several different lenses are examined and explored by scanning near-field terahertz microscopy. Thus, terahertz waves are regulated and more possibilities are provided for further miniaturization and compactness of THz devices.

Beamforming Phased-Array-Fed Lenses With &gt;0.5λ-Spaced Elements

We propose a phased-array-fed lens (PAFL) antenna that is capable of beamforming like a phased array but with array elements spaced beyond <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.5\lambda $ </tex-math></inline-formula> . The PAFL produces high-quality scanned beams using only five active feeds. This architecture represents dramatic cost and power savings over conventional phased arrays while providing many of the features. We present an optimal beamforming method to achieve the maximum gain at any angle using a subset of feeds and a multiobjective optimizer using particle swarm optimization for more granular pattern control. The method is applied to several simulated state-of-the-art lens antennas with good performance confirming the generality of the method. The theory is demonstrated with a prototype PAFL comprising a 4 inch aperture gradient-index (GRIN) lens antenna, an 8-element <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.725\lambda $ </tex-math></inline-formula> -spaced linear patch array operating at 29 GHz, and a commercial Ka-band SATCOM beamformer integrated circuit (IC). The prototype achieves the maximum gain at all angles and improves the scan loss by 4 dB at ±50°.

Inverse design of high-NA metalens for maskless lithography

Abstract We demonstrate an axisymmetric inverse-designed metalens to improve the performance of zone-plate-array lithography (ZPAL), one of the maskless lithography approaches, that offer a new paradigm for nanoscale research and industry. First, we derive a computational upper bound for a unit-cell-based axisymmetric metalens. Then, we demonstrate a fabrication-compatible inverse-designed metalens with 85.50% transmission normalized focusing efficiency at 0.6 numerical aperture at 405 nm wavelength; a higher efficiency than a theoretical gradient index lens design (79.98%). We also demonstrate experimental validation for our axisymmetric inverse-designed metalens via electron beam lithography. Metalens-based maskless lithography may open a new way of achieving low-cost, large-area nanofabrication.

Spatial coupling efficiency of collimators based on gradient-index lens with an angle polish

As high coupling efficiency and return loss are crucial in fiber-optic transmission systems, they have attracted widespread attention. This study proposes a ray-transfer matrix-based mathematical analysis method and experimentally demonstrates a collimator based on a gradient-index lens with an angle polish. The propagation characteristics and coupling mechanisms of the collimators are introduced. A beam-steering technology based on a wedge prism and flat glass is proposed to improve the coupling efficiency of the Gaussian beam using collimators. The proposed method is validated via simulation and experiment. The results are significant in free-space optical communication, optical signal processing, and optical fiber connectors.

Dispersion controlled nanocomposite gradient index lenses

The degrees of freedom afforded by nanocomposite materials and additive manufacturing allow for the precise control over the chromatic properties of gradient index (GRIN) optics. The ability to engineer nanocomposite optical materials using blends of three or more constituents makes it possible to independently specify the refractive index gradient and the dispersion of optical materials. The refractive index spectra of the primary nanocomposite feedstock are defined relative to one another using various concentrations of monomers and nanofillers. Inkjet deposition is then used to print-compose specific feedstock to form refractive index gradients with precise control over dispersion. Arrays of 4-mm-diameter spherical GRIN lenses were fabricated using different nanomaterial compositions. The ability to positively and negatively control dispersion and to obtain achromatic performance was demonstrated. Control over partial dispersion is also shown.

Sequential differential ray tracing formulation for homogeneous and gradient-index lenses using automatic differentiation

A differential ray tracing (DRT) formulation for the analysis and design of arbitrary optical systems is presented. An approach based on sequential rays is proposed to define a functional representation of the ray path, where the gradient information associated with the ray path is obtained using an automatic differentiation technique. The proposed DRT algorithm is neither constrained to a specific optical system geometry nor restricted to a set of surface types or material definition.

Compact Multifunctional Gain Enhancing GRIN Lens for Widely Separated Frequency Band Shared Aperture Antennas

AbstractGradient index (GRIN) lenses embody a powerful technology that enables control of electromagnetic wave propagation over wide frequency bands. However, GRIN lenses that achieve multifunctional (i.e., dual broadband) performance have not been realized mainly due to the limitations of conventional techniques such as Transformation Optics (TO). This paper proposes a multi‐objective inverse‐design approach for the optimization of multi‐band GRIN lenses with highly constrained aperture sizes. A candidate lens design is fabricated from a ceramic material using a new additive manufacturing approach allowing for spatially‐varying permittivities between 2–6.5 to be achieved at sub‐cm resolution. Measured results validate the candidate functionalized ceramic GRIN lens's simulated multi‐band performance and demonstrate the efficacy of the proposed inverse‐design procedure.

Graded index all-dielectric lens antenna designed by phase manipulation and optical path rescaling

A rotationally symmetric all-dielectric lens antenna is designed by defining the phase function inside the dielectric directly via a closed-form series formula. The refractive index of the lens is modified using state-of-the-art optical path rescaling to keep the refractive index within practical values. The lens optimization is done using the genetic algorithm in MATLAB, where each case is evaluated by conducting linked ray tracing and full-wave electromagnetic simulation in COMSOL. A lens prototype is shown to provide a directivity enhancement of 5.1 dB compared to the optimal conical horn antenna, an improvement of 2 dB compared to the Luneburg lens, and an improvement of 3.8 dB compared to a reference graded index lens. All the examined cases shared a similar length of 5λ for the center frequency of 10 GHz. The prototype lens antenna has a maximum refractive index of 2.1, a reflection coefficient of −22 dB, and a side-lobe level of −25 dB at the center frequency of 10 GHz. The lens’ performance is consistent in the 9–11 GHz band supported by the standard circular waveguide. The results from a prototype with a discretized refractive index simulated in CST Studio Suite are in excellent agreement with the ones with a continuous refractive index profile simulated in COMSOL.

Beam shaping for magnetoelectric transverse electromagnetic horn by using gradient refractive index lens

AbstractThis paper presents a new type of flat gradient index lens used for the beam shaping of a miniaturized magnetoelectric transverse electromagnetic (ME‐TEM) horn antenna. Novel metamaterial unit cells with variable sizes are proposed to achieve high refractive index distribution. After applying the novel meta‐lens to the modified transverse electromagnetic horn, the inherent E‐plane beam squint of the low‐profile ME‐TEM horn is corrected. The radiation patterns in E‐plane and H‐plane are focused and the peak gain at 4 GHz is improved to 17.3 dBi. The dB matching bandwidth of 0.4–4 GHz is achieved while the measured is a little poor at 0.8 GHz. The length of the total antenna is still shorter than half of the largest wavelength. The application potential of the gradient index meta‐lens in the low‐frequency band is verified.

Focusing of partially coherent light by a graded-index lens.

We use coherence theory to study how the focusing of an optical beam by a graded-index (GRIN) lens is affected when the incoming beam is only partially coherent. The Gaussian-Schell model is used to show that the intensity of a partially coherent beam exhibits self-imaging and evolves in a periodic fashion in a GRIN medium with a parabolic index profile. Spatial coherence of the beam affects a single parameter that governs how much the beam is compressed at the focal point. Our results show that the focal spot size depends on the fraction of the beam's diameter over which coherence persists. Focusing ceases to occur, and the beam may even expand at the focal point of a GRIN lens, when this fraction is below 10%.

Generation and propagation of Airy beams and 1-inch diameter focusing optics using three-dimensional-printed polymer optics

The additive manufacture of polymer optical elements has the promise of reducing component weight, providing new design capabilities, and enhanced performance for a wide variety of military and commercial optical systems. We review progress in the development of three-dimensional (3D)-printed gradient index (GRIN) lenses and optical phase masks. The 3D printing process uses a modified commercial inkjet printer and ultraviolet curable polymers that have specific nanoparticles added to them to modulate the index of refraction. Complex optical phase masks for the generation of Airy laser beams and polymer GRIN lenses to replace conventional glass lenses used in a telescope or riflescope are created. The generation and propagation of Airy beams using these polymer-generated optical phase masks have been investigated and analyzed through experimentation, simulations, and comparison with recent theoretical predictions. Airy beams have been generated using the conventional approach with a spatial light modulator and compared to the 3D printed optical phase masks. The maximum nondiffracting propagation distance of an aperture truncated Airy beam was experimentally measured. The results show that the maximum nondiffracting propagation distance of a laboratory generated Airy beam is proportional to x02, the Airy beam waist size squared. The size of the Gaussian envelope beam has a weaker effect on the Airy beam propagation distance. The experimental results were compared with current theoretical models. A set of 1-inch-diameter 100-mm focal length polymer GRIN lenses have been made using 3D printing. Transmission and modulation transfer function results for the lenses are reported.

Low Profile GRIN Lenses With Integrated Matching Using 3-D Printed Ceramic

In this paper, we investigate a shortened horn antenna with high gain that is enabled by a 3D-printed gradient index (GRIN) lens composed of high permittivity zirconia <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(ZrO_{2})$ </tex-math></inline-formula> . The baseline H-plane sectoral horn antenna is designed with length that is 1/3 of the optimal horn antenna and exhibits a low gain due to the high flaring rate of the horn. Increased gain is achieved by adding a flat GRIN lens at the horn aperture. High permittivity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ZrO_{2}\,\,(\varepsilon _{r} = 23)$ </tex-math></inline-formula> enables lens miniaturization; however, when interfaced with air, reflections at the air interface increase the impedance mismatch. Two different methods for mitigating the reflections are studied. One is a simple <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda /4$ </tex-math></inline-formula> matching layer that matches the bulk permittivity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ZrO_{2}$ </tex-math></inline-formula> to air. As expected, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda /4$ </tex-math></inline-formula> layers reduce the reflections in part of the 13–18 GHz band but produce high reflection in other parts. The second approach is a GRIN lens with integrated tapered matching layer to match phase and impedance simultaneously. Three tapering methods are studied (exponential, Klopfenstein, linear) for impedance matching. Analytical expressions of the minimum thickness and permittivity distribution are derived. The lens is discretized for print and three types of unit cells are proposed to create a wide range of permittivities ranging from bulk ceramic to air. A <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ZrO_{2}$ </tex-math></inline-formula> lens prototype printed with an XJet Carmel 1400 is measured and results show good agreement with simulations, including gain performance equivalent to a horn of 2.4x longer length. The measured gain and beamwidth of the lens are 5.4dB higher and 52° narrower than those of the shortened horn alone at 15 GHz, respectively.

Reflectarray Feeds Augmented With Size-Reducing GRIN Lenses for Improved Power Handling and Aperture Efficiency

The feed for reflectarrays and transmitarrays is crucial for determining the overall antenna aperture efficiency, gain, and power handling. In the past, several methods, such as altering the horn shape to achieve higher modes, have been employed to modify the feed radiation pattern to improve these key parameters. However, another approach is to use gradient index (GRIN) lenses, which offer unprecedented control over the radiation pattern and can be used to shrink the overall antenna. In this paper, we demonstrate highly effective approaches for optimizing GRIN lenses to achieve different design objectives including high gain radiation with a compact feed as well as improved aperture efficiency and power handling. Additive manufacturing is employed to fabricate the lenses, and measured results agree well with simulations.

3-D Printed All-Dielectric GRIN Lens Antenna With an Integrated Feeder

In this paper we present the design, fabrication, and experimental verification of a new type of Graded-index (GRIN) lens antenna with an integrated feeder. The continuously varying refractive index distribution is chosen appropriately to offer the rays collimation at the lens aperture. It is practically implemented by varying the material density in a host medium, thus realizing a new type of all-dielectric high gain antenna, entirely using 3D printing. This solution can find application to high gain wireless communication and measurement systems. This GRIN lens antenna is printed in one monolithic process and does not require the feeder to be placed at a focal distance, thus complying with more strict space requirements. It accepts interchangeable feeds that can cover a wide frequency range. The directivity and gain are evaluated using near-field measurements in the Ku-band. A 40% measured aperture efficiency is achieved at 14GHz. The challenges and performance limitations that come with 3D printing, as compared to the design of idealized continuous distribution GRIN lenses are discussed.