Controlling electromagnetic fields with graded photonic crystals in metamaterial regime

Fabrications of twisted moiré photonic crystal and random moiré photonic crystal and their potential applications in light extraction

Twisted moiré photonic crystal is an optical analog of twisted graphene or twisted transition metal dichalcogenide bilayers. In this paper, we report the fabrication of twisted moiré photonic crystals and randomized moiré photonic crystals and their use in enhanced extraction of light in light-emitting diodes (LEDs). Fractional diffraction orders from randomized moiré photonic crystals are more uniform than those from moiré photonic crystals. Extraction efficiencies of 76.5%, 77.8% and 79.5% into glass substrate are predicted in simulations of LED patterned with twisted moiré photonic crystals, defect-containing photonic crystals and random moiré photonic crystals, respectively, at 584 nm. Extraction efficiencies of optically pumped LEDs with 2D perovskite (BA)2(MA)n-1PbnI3n+1ofn= 3 and (5-(2'-pyridyl)-tetrazolato)(3-CF3-5-(2'-pyridyl)pyrazolato) platinum(II) (PtD) have been measured.

Dynamic Beam Steering and Focusing Graphene Metasurface Mirror Based on Fermi Energy Control

Beam steering technology is crucial for radio frequency and infrared telecommunication signal processing. Microelectromechanical systems (MEMS) are typically used for beam steering in infrared optics-based fields but have slow operational speeds. An alternative solution is to use tunable metasurfaces. Since graphene has gate-tunable optical properties, it is widely used in electrically tunable optical devices due to ultrathin physical thickness. We propose a tunable metasurface structure using graphene in a metal gap structure that can exhibit a fast-operating speed through bias control. The proposed structure can change beam steering and can focus immediately by controlling the Fermi energy distribution on the metasurface, thus overcoming the limitations of MEMS. The operation is numerically demonstrated through finite element method simulations.

Directivity enhancement of a cylindrical wire antenna by a graded index dielectric shell designed using strictly conformal transformation optics

A transformation-optical method is presented to enhance the directivity of a cylindrical wire antenna by using an all-dielectric graded index medium. The strictly conformal mapping between two doubly connected virtual and physical domains is established numerically. Multiple directive beams are produced, providing directive emission. The state-of-the-art optical path rescaling method is employed to mitigate the superluminal regions. The resulting transformation medium is all-dielectric and nondispersive, which can provide broadband functionality and facilitate the realization of the device using available fabrication technologies. The realization of the device is demonstrated by dielectric perforation based on the effective medium theory. The device’s functionality is verified by carrying out both ray-tracing and full-wave simulations using finite-element-based software COMSOL Multiphysics.

Transformation cavities with a narrow refractive index profile

Recently, gradient index cavities, or so-called transformation cavities, designed by conformal transformation optics, have been studied to support resonant modes with both high Q-factors and emission directionality. We propose a new, to the best of our knowledge, design method for transformation cavities to realize a narrower width of the refractive index profile, a great advantage in experimental implementations, without losing the benefits of conformal mapping. We study resonant modes with both high Q-factors and directional emission in newly designed transformation cavities, where the refractive index profile is 50% narrower than in previously proposed transformation cavities. By varying a system parameter with a fixed maximal value of the refractive index profile inside the cavity, the width of the refractive index profile narrows, the Q-factors become higher, and the near and far field patterns maintain their properties, namely, conformal whispering gallery modes and bidirectional emission, respectively.

Metamaterial-based ultrashort multimode waveguide taper with low intermodal crosstalk

We propose and theoretically demonstrate an ultrashort multimode waveguide taper based on the all-dielectric metamaterial. Attributed to the gradient index distribution of the metamaterial, the spot sizes of the four lowest-order transverse magnetic (TM) modes can be expanded in a short distance of 6 μm with negligible mode conversions. Numerical results prove that the insertion losses of the taper are lower than 1 dB, 1.12 dB, 1.26 dB and 1.66 dB for the TM₀ - TM₃ modes, respectively, and the intermodal crosstalk values are below -15 dB for the four modes, both in the wavelength range of 1.5 μm - 1.6 μm. To the best of our knowledge, this is the first multimode waveguide taper that has low intermodal crosstalk of < -15 dB over a 100-nm bandwidth.

Constructing Metastructures with Broadband Electromagnetic Functionality

Electromagnetic metastructures stand for the artificial structures with a characteristic size smaller than the wavelength, which may efficiently manipulate the states of light. However, their applications are often restricted by the bandwidth of the electromagnetic response of the metastructures. It is therefore essential to reassert the principles in constructing broadband electromagnetic metastructures. Herein, after summarizing the conventional approaches for achieving broadband electromagnetic functionality, some recent developments in realizing broadband electromagnetic response by dispersion compensation, nonresonant effects, and several trade-off approaches are reviewed, followed by some perspectives for the future development of broadband metamaterials. It is anticipated that broadband metastructures will have even more substantial applications in optoelectronics, energy harvesting, and information technology.

Silicon nitride waveguide devices based on gradient-index lenses implemented by subwavelength silicon grating metamaterials

The rapid development of photonic integrated circuits demands the design of efficient and compact waveguide devices such as waveguide tapers and crossings. Some components in the silicon nitride (SiN) waveguide platform are superior to their counterparts in the silicon waveguide platform. Designing a compact SiN waveguide taper and crossing is crucial to reduce the size of SiN photonic components. In this paper, we utilize the focusing property of the Luneburg lens to design an SiN taper connecting a 10-µm-wide waveguide to a 1-µm-wide waveguide. Three-dimensional full-wave simulations indicate that the designed 13-µm-long taper has an average transmission efficiency of 92% in the wavelength range of 1500-1600 nm. We also present an in-plane SiN waveguide crossing based on the imaging property of the square Maxwell's fisheye lens designed with quasi-conformal transformation optics. The designed waveguide crossing occupies a compact footprint of 5.65µm×5.65µm, while its average insertion loss is 0.46 dB in the bandwidth of 1500-1600 nm. To the best our knowledge, the designed SiN waveguide taper and crossing have the smallest footprints to date.

Optimization of conformal whispering gallery modes in limaçon-shaped transformation cavities

Directional light emission from high-Q resonant modes without significant Q-spoiling has been a long standing issue in deformed dielectric cavities. In limaçon-shaped gradient index dielectric cavities recently proposed by exploiting conformal transformation optics, the variation of Q-factors and emission directionality of resonant modes was traced in their system parameter space. For these cavities, their boundary shapes and refractive index profiles are determined in each case by a chosen conformal mapping which is taken as a coordinate transformation. Through the numerical exploration, we found that bidirectionality factors of generic high-Q resonant modes are not directly proportional to their Q-factors. The optimal system parameters for the coexistence of strong bidirectionality and a high Q-factor was obtained for anisotropic whispering gallery modes supported by total internal reflection.

Multimode waveguide crossing based on a square Maxwell's fisheye lens

Mode-division multiplexing (MDM) is an emerging large-capacity data communication technology utilizing orthogonal guiding modes as independent data streams. One of the challenges of multimode waveguide routing in MDM systems is decreasing the mode leakage of waveguide crossings. In this article, a square Maxwell's fish-eye lens as a waveguide crossing medium based on quasiconformal transformation optics is designed and implemented on a silicon-on-insulator platform. Two approaches were taken to realize the designed lens: graded photonic crystal and varying the thickness of the silicon slab waveguide. Three-dimensional numerical simulations show that the designed multimode waveguide crossing has an ultrawide bandwidth from 1260 to 1675 nm with a compact footprint of only 3.77×3.77  μm². For the first three transverse electric modes (TE0, TE1, and TE2), the designed waveguide crossing exhibits an average insertion loss of 0.24, 0.55, and 0.45 dB; a crosstalk of less than -72, -61, and -27  dB; and a maximum return loss of 54, 53, and 30 dB, respectively. The designed waveguide crossing supports low-distortion pulse transmission with a high fidelity factor of 0.9857. Furthermore, the proposed method can be expanded to design waveguide crossings with an even higher number of supporting modes by increasing the size of the lens.

Sharp bend in two-dimensional optical waveguide based on gradient refractive index structure

In this work, we propose the design of a sharp bend in a two-dimensional optical waveguide which has super-ellipse curve boundaries and a gradient refractive index structure in its core. Numerical simulations are presented to show the efficient light propagation in the waveguide bend, as well as the efficient light coupling between the proposed waveguide bend and a straight waveguide, for TE0 and TM0 modes. The proposed design strategy is also useful for designing other compact optical and photonic components.

Simultaneous direct holographic fabrication of photonic cavity and graded photonic lattice with dual periodicity, dual basis, and dual symmetry

For the first time, to the authors' best knowledge, this paper demonstrates the digital, holographic fabrication of graded, super-basis photonic lattices with dual periodicity, dual basis, and dual symmetry. Pixel-by-pixel phase engineering of the laser beam generates the highest resolution in a programmable spatial light modulator (SLM) for the direct imaging of graded photonic super-lattices. This technique grants flexibility in designing 2-D lattices with size-graded features, differing periodicities, and differing symmetries, as well as lattices having simultaneously two periodicities and two symmetries in high resolutions. By tuning the diffraction efficiency ratio from the SLM, photonic cavities can also be generated in the graded super-lattice simultaneously through a one-exposure process. A high quality factor of over 1.56 × 10⁶ for a cavity mode in the graded photonic lattice with a large super-cell is predicted by simulations.

Optimized invisibility cloaks from the Logarithm conformal mapping

Invisibility cloaks designed from the coordinate transformation method have attracted increasing interest recently. Conformal transformation optics scheme leads to cloaks that possess isotopic media, thus provides a prospective way to facilitate easier realization. Reducing the maximum value of the refractive index required by the cloaks is very important in practical imple- mentation. This letter studies on how the parameters in the logarithm conformal mapping control the cloaking effect. The optimized invisibility cloaks are designed. The maximum values of the refractive index required from the first kind and the second kind of logarithm conformal mappings are reduced to 9.779 and 12.936, respectively.

On-chip sub-terahertz surface plasmon polariton transmission lines with mode converter in CMOS

An on-chip low-loss and high conversion efficiency plasmonic waveguide converter is demonstrated at sub-THz in CMOS. By introducing a subwavelength periodic corrugated structure onto the transmission line (T-line) implemented by a top-layer metal, surface plasmon polaritons (SPP) are established to propagate signals with strongly localized surface-wave. To match both impedance and momentum of other on-chip components with TEM-wave propagation, a mode converter structure featured by a smooth bridge between the Ground coplanar waveguide (GCPW) with 50 Ω impedance and SPP T-line is proposed. To further reduce area, the converter is ultimately simplified to a gradual increment of groove with smooth gradient. The proposed SPP T-lines with the converter is designed and fabricated in the standard 65 nm CMOS process. Both near-field simulation and measurement results show excellent conversion efficiency from quasi-TEM to SPP modes in a broadband frequency range. The converter achieves wideband impedance matching (<-9 dB) with excellent transmission efficiency (averagely -1.9 dB) from 110 GHz-325 GHz. The demonstrated compact and wideband SPP T-lines with mode converter have shown great potentials to replace existing waveguides as future on-chip THz interconnects. To the best of the author's knowledge, this is the first time to demonstrate the (sub)-THz surface mode conversion on-chip in CMOS technology.

Manipulating Water Wave Propagation via Gradient Index Media

It is challenging to realise the perfect manipulation of water waves within a broad range of frequencies. By extending conformal transformation principles to water waves, their propagation can be controlled via gradually varying water depths, permitting the realisation of a desired refractive index profile for linear water surface waves. Wave bending, directional wave emission and wave focusing are analysed experimentally with accompanying simulations. The results demonstrate desired wave manipulations within a broad range of frequencies, confirming the accuracy and effectiveness of conformal transformation for water waves.

Spatially variant periodic structures in electromagnetics

Spatial transforms are a popular technique for designing periodic structures that are macroscopically inhomogeneous. The structures are often required to be anisotropic, provide a magnetic response, and to have extreme values for the constitutive parameters in Maxwell's equations. Metamaterials and photonic crystals are capable of providing these, although sometimes only approximately. The problem still remains about how to generate the geometry of the final lattice when it is functionally graded, or spatially varied. This paper describes a simple numerical technique to spatially vary any periodic structure while minimizing deformations to the unit cells that would weaken or destroy the electromagnetic properties. New developments in this algorithm are disclosed that increase efficiency, improve the quality of the lattices and provide the ability to design aplanatic metasurfaces. The ability to spatially vary a lattice in this manner enables new design paradigms that are not possible using spatial transforms, three of which are discussed here. First, spatially variant self-collimating photonic crystals are shown to flow unguided waves around very tight bends using ordinary materials with low refractive index. Second, multi-mode waveguides in spatially variant band gap materials are shown to guide waves around bends without mixing power between the modes. Third, spatially variant anisotropic materials are shown to sculpt the near-field around electric components. This can be used to improve electromagnetic compatibility between components in close proximity.

Registering functional defects into periodic holographic structures

In this paper, we present two methods for registering desired defect lattices within background periodic lattices through spatial light-modulator-based holographic lithography. In the first method, the diffraction efficiency from the engineered phase pattern was used to locally modify the fill fraction of polymerized materials in holographic structures, and, at the same time, we achieved the lattice matching between modified and background regions. In the second method, we registered spatially variant lattices for a 90 deg bend within the background periodic lattices through two steps of phase engineering of the laser beam.

Transformation optics with Fabry-Pérot resonances

Transformation optics is a powerful tool to design various novel devices, such as invisibility cloak. Fantastic effects from this technique are usually accompanied with singular mappings, resulting in challenging implementations and narrow bands of working frequencies. Here in this article, Fabry-Pérot resonances in materials of extreme anisotropy are used to design various transformation optical devices that are not only easy to realize but also work well for a set of resonant frequencies (multiple frequencies). As an example, a prototype of a cylindrical concentrator is fabricated for microwaves.

Logarithm conformal mapping brings the cloaking effect

Over the past years, invisibility cloaks have been extensively discussed since transformation optics emerges. Generally, the electromagnetic parameters of invisibility cloaks are complicated tensors, yet difficult to realize. As a special method of transformation optics, conformal mapping helps us design invisibility cloak with isotropic materials of a refractive index distribution. However, for all proposed isotropic cloaks, the refractive index range is at such a breadth that challenges current experimental fabrication. In this work, we propose two new kinds of logarithm conformal mappings for invisible device designs. For one of the mappings, the refractive index distribution of conformal cloak varies from 0 to 9.839, which is more feasible for future implementation. Numerical simulations by using finite element method are performed to confirm the theoretical analysis.

Tight control of light beams in photonic crystals with spatially-variant lattice orientation

Spatially-variant photonic crystals (SVPCs), in which the orientation of the unit cell changes as a function of position, are shown to be capable of abruptly controlling light beams using just low index materials and can be made to have high polarization selectivity. Multi-photon direct laser writing in the photo-polymer SU-8 was used to fabricate three-dimensional SVPCs that direct the flow of light around a 90 degree bend. The lattice spacing and fill factor were maintained nearly constant throughout the structure. The SVPCs were characterized at a wavelength of 2.94 μm by scanning the faces with optical fibers and the results were compared to electromagnetic simulations. The lattices were shown to direct infrared light of one polarization through sharp bends while the other polarization propagated straight through the SVPC. This work introduces a new scheme for controlling light that should be useful for integrated photonics.

Holographic fabrication of functionally graded photonic lattices through spatially specified phase patterns

In this paper, we present a method for the mathematically formulated phase engineering of interfering laser beams through a spatial light modulator for a holographic fabrication of graded photonic lattices. The desired phases can be programmed at specific locations by assigning gray levels in cellular structures. The method is demonstrated by embedding single-lattice structures or missing lattices in dual-lattice periodic photonic structures. The demonstrated method can be potentially combined with the coordinate transformation technique in transformation optics for the fabrication of graded photonic devices.

Subwavelength micropillar array terahertz lasers

We report on micropillar-based terahertz lasers with active pillars that are much smaller than the emission wavelength. These micropillar array lasers correspond to scaled-down band-edge photonic crystal lasers forming an active photonic metamaterial. In contrast to photonic crystal lasers which use significantly larger pillar structures, lasing emission is not observed close to high-symmetry points in the photonic band diagram, but in the effective medium regime. We measure stimulated emission at 4 THz for micropillar array lasers with pillar diameters of 5 µm. Our results not only demonstrate the integration of active subwavelength optics in a terahertz laser, but are also an important step towards the realization of nanowire-based terahertz lasers.

Mode-expansion theory for inhomogeneous meta-surfaces

Modeling meta-surfaces as thin metamaterial layers with continuously varying bulk parameters, we employed a rigorous mode-expansion theory to study the scattering properties of such systems. We found that a meta-surface with a linear reflection-phase profile could redirect an impinging light to a non-specular channel with nearly 100% efficiency, and a meta-surface with a parabolic reflection-phase profile could focus incident plane wave to a point image. Under certain approximations, our theory reduces to the local response model (LRM) established for such problems previously, but our full theory has overcome the energy non-conservation problems suffered by the LRM. Microwave experiments were performed on realistic samples to verify the key theoretical predictions, which match well with full-wave simulations.

A broadband and omnidirectional electromagnetic wave concentrator with gradient woodpile structure

We present the first realized three-dimensional (3D) practical implementation of the so called "optical black hole" in microwave frequencies, an electromagnetic (EM) concentrator. The 3D EM wave concentrator was designed with non-resonant gradient index (GRIN) 3D woodpile photonic crystals (PCs) structure in metamaterial regime, and fabricated by Stereolithography (SL) process. Omnidirectional EM wave capture and absorbing ability of the device in a broad bandwidth (12GHz-15GHz) were validated by full-wave simulation and experiments. Such devices may have applications in microwave energy harvesting and radiation detector.

High-efficiency beam bending using graded photonic crystals

We explore beam-bending properties of graded index (GRIN) waveguide with hyperbolic secant profile. The transmission efficiency and bandwidth features are extracted for GRIN photonic crystal (PC) media composed of dielectric rods. Light guiding performance of the GRIN PC medium is analyzed for 90° and 180° waveguide bends. The finite-difference time-domain method is deployed to investigate the performance of the designed GRIN waveguides. By the help of proposed photonic configuration, bending of light is achieved with a high efficiency within a broad bandwidth, which promotes the use of GRIN PC structures for efficient light-bending purposes.

Efficient mode converter design using asymmetric graded index photonic structures

Asymmetric graded-index (GRIN) inhomogeneous medium is proposed to enable even-to-odd mode conversion. An exponential refractive index profile is implemented to induce an asymmetric refractive index distribution. A two-dimensional photonic crystal structure composed of constant dielectric radii with rectangular unit cells is deployed. Efficient mode transformation in a compact structure may promote the manipulation of light for the creation of other types of higher-order modes in asymmetric GRIN structures.

High-efficiency broadband anomalous reflection by gradient meta-surfaces

We combine theory and experiment to demonstrate that a carefully designed gradient meta-surface supports high-efficiency anomalous reflections for near-infrared light following the generalized Snell's law, and the reflected wave becomes a bounded surface wave as the incident angle exceeds a critical value. Compared to previously fabricated gradient meta-surfaces in infrared regime, our samples work in a shorter wavelength regime with a broad bandwidth (750-900 nm), exhibit a much higher conversion efficiency (∼80%) to the anomalous reflection mode at normal incidence, and keep light polarization unchanged after the anomalous reflection. Finite-difference-time-domain (FDTD) simulations are in excellent agreement with experiments. Our findings may lead to many interesting applications, such as antireflection coating, polarization and spectral beam splitters, high-efficiency light absorbers, and surface plasmon couplers.

Modified Maxwell fish-eye approach for efficient coupler design by graded photonic crystals

We present a novel design of two dimensional graded index medium that provides coupling of light with high coupling efficiency between two planar dielectric waveguides of different widths (15.46 μm vs. 2.21 μm). Poor light coupling performance of butt-coupler can be mitigated by implementing tapered coupler at the expense of long coupler section. In order to reduce coupling losses, a new coupling device approach based on graded index (GRIN) concept is proposed. The refractive index distribution is in the form of modified version of the Maxwell fish-eye lens. The inhomogeneous refractive index distribution is approximated by photonic crystals (PCs) such that the positions of each PC rods are appropriately arranged. Strong electric field focusing ability of the designed GRIN PC medium provides relatively high coupling efficiency that is around 90%. Spectral region corresponding to coupling efficiency over 75% has a bandwidth of Δω = 18.56% (284 nm). Finally, we discuss the durability of the proposed coupler against the lateral displacement and angular misalignment of output waveguides.

Recent advances in transformation optics

Within the past a few years, transformation optics has emerged as a new research area, since it provides a general methodology and design tool for manipulating electromagnetic waves in a prescribed manner. Using transformation optics, researchers have demonstrated a host of striking phenomena and devices; many of which were only thought possible in science fiction. In this paper, we review the most recent advances in transformation optics. We focus on the theory, design, fabrication and characterization of transformation devices such as the carpet cloak, "Janus" lens and plasmonic cloak at optical frequencies, which allow routing light at the nanoscale. We also provide an outlook of the challenges and future directions in this fascinating area of transformation optics.

Synthesis of spatially variant lattices

It is often desired to functionally grade and/or spatially vary a periodic structure like a photonic crystal or metamaterial, yet no general method for doing this has been offered in the literature. A straightforward procedure is described here that allows many properties of the lattice to be spatially varied at the same time while producing a final lattice that is still smooth and continuous. Properties include unit cell orientation, lattice spacing, fill fraction, and more. This adds many degrees of freedom to a design such as spatially varying the orientation to exploit directional phenomena. The method is not a coordinate transformation technique so it can more easily produce complicated and arbitrary spatial variance. To demonstrate, the algorithm is used to synthesize a spatially variant self-collimating photonic crystal to flow a Gaussian beam around a 90° bend. The performance of the structure was confirmed through simulation and it showed virtually no scattering around the bend that would have arisen if the lattice had defects or discontinuities.

Graded photonic quasicrystals

We introduce graded photonic quasicrystals and investigate properties of such structures on the example of a Luneburg lens based on a dodecagonal photonic quasicrystal. It is shown that the graded photonic quasicrystal lens has better focusing properties as compared with the graded photonic crystal lens in a frequency range suitable for experimental realization. The proposed graded photonic quasicrystals can be used in optical systems where compact and powerful focusing elements are required.

Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves

The arbitrary control of electromagnetic waves is a key aim of photonic research. Although, for example, the control of freely propagating waves (PWs) and surface waves (SWs) has separately become possible using transformation optics and metamaterials, a bridge linking both propagation types has not yet been found. Such a device has particular relevance given the many schemes of controlling electromagnetic waves at surfaces and interfaces, leading to trapped rainbows, lensing, beam bending, deflection, and even anomalous reflection/refraction. Here, we demonstrate theoretically and experimentally that a specific gradient-index meta-surface can convert a PW to a SW with nearly 100% efficiency. Distinct from conventional devices such as prism or grating couplers, the momentum mismatch between PW and SW is compensated by the reflection-phase gradient of the meta-surface, and a nearly perfect PW-SW conversion can happen for any incidence angle larger than a critical value. Experiments in the microwave region, including both far-field and near-field characterizations, are in excellent agreement with full-wave simulations. Our findings may pave the way for many applications, including high-efficiency surface plasmon couplers, anti-reflection surfaces, light absorbers, and so on.

Crescent shaped dielectric periodic structure for light manipulation

We present optical properties of crescent-shaped dielectric nano-rods that comprise a square lattice periodic structure named as crescent-shaped photonic crystals (CPC). The circular symmetry of individual cells of periodic dielectric structures is broken by replacing each unit cell with a reduced symmetry crescent shaped structure. The created configuration is assumed to be formed by the intersection of circular dielectric and air rods. The degree of freedom to manipulate the light propagation arises due to the rotational sensitivity of the CPC. The interesting dispersion property of designed CPC occurs due to the anisotropic nature of the iso-frequency contours that yield tilted self-collimated wave guiding. Furthermore, this feature allows focusing, routing, splitting and deflecting light beams along certain routes which are independent of the lattice symmetry directions of regular PCs. The propagation direction of light can be tuned by means of the opening angle of the crescent shape. Finally, the property of being all-dielectric structure ensures the absence of optical absorption losses that are reminiscent of employed metallic nano-particles.

Experimental demonstration of light bending at optical frequencies using a non-homogenizable graded photonic crystal

Experimental results on light bending in a non-homogenizable graded photonic crystal operating at optical wavelengths are presented in this paper. A square lattice silicon on insulator photonic crystal made of a two-dimensional chirp of the air-hole filling factor is exploited to produce the bending effect in a near bandgap frequency range. The sensitivity of light paths to wavelength tuning is also exploited to show demultiplexing capability with low insertion loss (<2dB) and low crosstalk (~-20dB). This experimental demonstration opens opportunities for light manipulation using a generalized two-dimensional chirp of photonic crystal lattice parameters. It also constitutes an alternative solution to the use of photonic metamaterials combining dielectric and metallic materials with sub-wavelength unit cells.

Design of thin-film photonic metamaterial Lüneburg lens using analytical approach

We design an all-dielectric Lüneburg lens as an adiabatic space-variant lattice explicitly accounting for finite film thickness. We describe an all-analytical approach to compensate for the finite height of subwavelength dielectric structures in the pass-band regime. This method calculates the effective refractive index of the infinite-height lattice from effective medium theory, then embeds a medium of the same effective index into a slab waveguide of finite height and uses the waveguide dispersion diagram to calculate a new effective index. The results are compared with the conventional numerical treatment - a direct band diagram calculation, using a modified three-dimensional lattice with the superstrate and substrate included in the cell geometry. We show that the analytical results are in good agreement with the numerical ones, and the performance of the thin-film Lüneburg lens is quite different than the estimates obtained assuming infinite height.