The Lunenburg lens (also known as Lüneburg lens) is an advanced optical structure with a specific and unique refractive index distribution. It operates fundamentally differently from standard lenses, and its properties allow it to focus or deflect electromagnetic waves—primarily in the radio and microwave frequency ranges—in ways that traditional lenses cannot achieve. Let’s dive deeper into its design, principles, and applications.

1. Design and Structure

The Lüneburg lens is a spherical lens with a specific refractive index profile that decreases radially from the outer surface towards the center. In a traditional lens, the refractive index typically increases or decreases as a function of the distance from the center of the lens. In contrast, the Lüneburg lens has the following characteristics:

• Spherical Geometry: It is shaped like a sphere, where the refractive index changes as a function of the radial distance from the center.

• Refractive Index Distribution: The key feature is the radial variation of the refractive index, which is described mathematically as:

  

  
  

2. Principle of Operation

The Lüneburg lens operates on the principle of controlling the speed and direction of electromagnetic waves (such as light, microwaves, or radio waves) by varying the refractive index across the lens. This variation in refractive index affects how waves propagate through the material.

Wave Propagation:

• Electromagnetic Wave Focusing: Due to the way the refractive index decreases from the edges toward the center, the waves bend towards the center as they travel through the lens. The lens focuses parallel waves to a point at the opposite side of the sphere, making it a perfect focusing lens for certain frequencies of electromagnetic waves.

• Ray Optics and Refraction: The refraction at each point on the surface of the lens depends on the local refractive index, directing the rays of light or waves toward a common focal point. Unlike regular lenses where curvature alone dictates refraction, the Lüneburg lens controls refraction via both geometry (spherical shape) and the varying refractive index.

Focusing Characteristics:

• The spherical symmetry of the lens leads to its ability to focus incoming waves from any direction to a single focal point. This is in stark contrast to traditional lenses that require specific curvature and material properties for focused light.

• Ideal for Wavefront Shaping: The Lüneburg lens can focus incoming plane waves into a single point, which is ideal for applications where wavefronts need to be manipulated with high precision.

3. Mathematical Explanation

The refractive index profile is based on a spherical coordinate system, and the mathematical derivation behind it is rooted in ray optics and Snell's law (the law of refraction). The formulation 

comes from solving the wave equation under boundary conditions that would focus incoming plane waves to a single focal point on the other side of the sphere.

This refractive index profile ensures that:

• The wave’s direction is controlled as it travels through the lens,
• The lens can focus waves from an arbitrary direction,
• The behavior is consistent with the principles of geometrical optics but differs due to the unusual index distribution.

4. Applications of the Lüneburg Lens

While Lüneburg lenses were initially a theoretical development, they have found practical applications, particularly in fields that rely on microwave and radio frequency (RF) technology. Some key applications include:

A. Antenna and Radar Systems

• Microwave and Radio Frequency: In radar and communication systems, the Lüneburg lens is used to focus and direct electromagnetic waves (such as microwaves or radio signals) more efficiently than conventional systems.

• Radar Antennas: The Lüneburg lens can be used in radar systems to create beamforming mechanisms, where electromagnetic waves are focused or steered. This is especially important for directing signals without mechanical moving parts.

• Wide-Angle Radar: In radar applications, the Lüneburg lens can help in creating wide-angle beams or highly focused beams. It is especially useful in multi-beam systems, where the goal is to focus energy across a broad area without distortion.

• Antenna Design: Lüneburg lenses are sometimes incorporated into antenna designs to enhance the focusing of waves from an antenna, improving the overall efficiency of the transmission and reception of signals.

B. Optical Applications (Theoretical and Experimental)

While most common applications of the Lüneburg lens are in radio and microwave frequencies, there have been theoretical proposals and experimental efforts to adapt the principles of the lens to optical wavelengths. The challenge is the manufacturing of materials with the required refractive index profiles at these smaller wavelengths.

• Photonics: In the field of photonics, researchers have looked at the potential of Lüneburg lenses for manipulating light at optical wavelengths (e.g., in laser systems or optical communication systems). However, the material and precision challenges are significant.

• Plasmonics and Metamaterials: Some experimental research explores the possibility of creating Lüneburg-like lenses using metamaterials that are engineered to have specific refractive properties at optical frequencies.

C. Signal Processing and Wave Manipulation

• The Lüneburg lens’s ability to control the direction and focusing of electromagnetic waves makes it useful in advanced signal processing systems, including waveguide designs and beam-steering technologies.

• Optical and Microwave Imaging: The lens may be applied in systems requiring wavefront shaping, where the goal is to create images or direct waves with high precision.

5. Advantages and Challenges

Advantages:

• Compactness: The Lüneburg lens is a single, spherical piece, making it a compact solution compared to traditional optical systems with complex arrangements of lenses and mirrors.
• Wide-Angle Focusing: It can focus waves from all directions, offering an advantage in systems that need a wide field of view or flexible focusing.

Challenges:

• Manufacturing Complexity: Creating a material with the precise radial refractive index profile, especially at optical wavelengths, is technically challenging.
• Material Limitations: For optical wavelengths, suitable materials with the required refractive properties are not readily available.

6. Conclusion

The Lüneburg lens is a fascinating and powerful optical device that differs from conventional lenses in its spherical design and radially varying refractive index. It has found success primarily in microwave and radio frequency applications, especially in radar and antenna systems. While challenges remain in applying it to optical frequencies, the Lüneburg lens remains a promising concept for future developments in optical and electromagnetic wave manipulation.

References__________________________________________________________________________

Li Y, Zhu Q. Luneburg lens with extended flat focal surface for electronic scan applications. Opt Express. 2016 Apr 4;24(7):7201-11. doi: 10.1364/OE.24.007201.

Abstract. Luneburg lens with flat focal surface has been developed to work together with planar antenna feeds for beam steering applications. According to our analysis of the conventional flattened Luneburg lens, it cannot accommodate enough feeding elements which can cover its whole scan range with half power beamwidths (HPBWs). In this paper, a novel Luneburg lens with extended flat focal surface is proposed based on the theory of Quasi-Conformal Transformation Optics (QCTO), with its beam steering features reserved. To demonstrate this design, a three-dimensional (3D) prototype of this novel extend-flattened Luneburg lens working at Ku band is fabricated based on 3D printing techniques, whose flat focal surface is attached to a 9-element microstrip antenna array to achieve different scan angles. Our measured results show that, with different antenna elements being fed, the HPBWs can cover the whole scan range.

Demetriadou A, Hao Y. Slim Luneburg lens for antenna applications. Opt Express. 2011 Oct 10;19(21):19925-34. doi: 10.1364/OE.19.019925.

Abstract. Luneburg lens is a marvellous optical lens but is extremely difficult to be applied in any practical antenna system due to its large spherical shape. In this paper, we propose a transformation that reduces the profile of the original Luneburg lens without affecting its unique properties. The new transformed slim lens is then discretized and simplified for a practical antenna application, where its properties were examined numerically. It is found that the transformed lens can be used to replace conventional antenna systems (i.e. Fabry-Perot resonant antennas) producing a high-directivity beam with low side-lobes. In addition, it provides excellent steering capabilities for wide angles, maintaining the directivity and side-lobes at high and low values respectively.

Yuan B, Liu J, Long H, Cheng Y, Liu X. Sound focusing by a broadband acoustic Luneburg lens. J Acoust Soc Am. 2022 Mar;151(3):2238. doi: 10.1121/10.0009909.

Abstract. The high-performance and aberration-free broadband acoustic lens holds promise for extensive applications, yet remains challenged. In this work, a scheme is proposed, and the experimental demonstration of a planar acoustic Luneburg lens capable of focusing broadband sound ranging from 1 to 3 kHz (relative bandwidth approaching to 100%) in an aberration-free manner is presented. Concretely, plane sound within the frequency range incident from one side can be concentrated on a same point on the opposite edge of the Luneburg lens. The demanded refractive indexes of the lens are obtained from the component space coil acoustic metamaterials, which can easily manipulate the refractive index by adjusting a structural parameter. The prototype of the proposed Luneburg lens is fabricated by three-dimensional printing technology and experimentally characterized in a two-dimensional acoustic measuring platform. The measured results are consistently in good agreement with those from the numerical simulations. Finally, the proposed Luneburg lens is employed to construct a wide-angle acoustic reflector, which can produce a strong echo propagating in the direction exactly opposite to the incident wave. These results facilitate potential possibilities for developing more acoustic functional devices capable of manipulating broadband sound.

Gunderson LC, Holmes GT. Microwave luneburg lens. Appl Opt. 1968 May 1;7(5):801-4. doi: 10.1364/AO.7.000801. 

Abstract. This article describes a two-dimensional Luneburg lens, fabricated from steps of foam glass of different refractive index as an approximation to a continuous index gradient. This is an improvement over lenses fabricated by assembly of machined parts, which unavoidably contain some air gaps, resulting in different path lengths through the lens. The foam glass lens is superior to a plastic lens since it is able to withstand higher temperatures and hence higher powers, in addition to having superior aging characteristics. Measurements have been made throughout the microwave band, and the results clearly establish the feasibility of this fabrication technique for the construction of a microwave Luneburg lens.

Wang C, Guo X, Wu X. Electrically tunable graphene plasmonic lens: from Maxwell Fisheye Lens to Luneburg Lens. Opt Express. 2023 Sep 11;31(19):31574-31586. 

Abstract. A graphene plasmonic lens with an electrically tunable focal length is proposed and numerically investigated. The design philosophy of the proposed tunable lens is based on the nonlinear relationship of surface plasmon polariton (SPP) wave index with respect to chemical potential of graphene. By controlling the gate voltage of graphene, the proposed lens can be continuously tuned from a Maxwell Fisheye lens to a Luneburg lens. A ray-tracing method is employed to find out the corresponding gate voltages for various focal lengths. Full-wave EM simulations using COMSOL show that excellent focusing performances can be achieved. This work offers a new way in exploiting active transformational plasmonic elements in the mid-infrared region.

Wang C, Guo X, Wu X. Electrically tunable virtual image Luneburg lens using graphene. Opt Express. 2024 Mar 25;32(7):12609-12619. doi: 10.1364/OE.517397.

Abstract. Virtual image lenses play essential roles in various optical devices and applications, including vision correction, photography, and scientific instruments. Here, we introduce an approach for creating virtual image Luneburg lenses (LL) on graphene. Remarkably, the graphene plasmonic lens (GPL) exhibits electrically tunable virtual focusing capabilities. The design principle of the tunability is based on the nonlinear relationship between surface plasmon polariton (SPP) wave mode index and chemical potential of graphene. By controlling the gate voltage of graphene, we can achieve continuous tuning of virtual focus. A ray-tracing technique is employed to determine the required gate voltages for various virtual focal lengths. The proposed GPL facilitates adjustable virtual focusing, promising advancements in highly adaptive and transformative nanophotonic devices.

Fuentes-Domínguez R, Yao M, Colombi A, Dryburgh P, Pieris D, Jackson-Crisp A, Colquitt D, Clare A, Smith RJ, Clark M. Design of a resonant Luneburg lens for surface acoustic waves. Ultrasonics. 2021 Mar;111:106306. doi: 10.1016/j.ultras.2020.106306. Epub 2020 Nov 24. PMID: 33290959.