Hyperbolic Secant Refractive Index Profile Research Articles

3D-printed gradient-index phononic crystal lens for transcranial focused ultrasound

Transcranial focused ultrasound (tFUS) shows great promise as a noninvasive tool to treat neurological conditions such as essential tremor. The existing clinical phased array systems are mostly intended for ultrasound delivery to the center of the brain (as in thalamotomy for essential tremor), in addition to being complex and expensive. To seek an alternative focusing approach especially for the brain periphery, we explore a 3D-printed gradient-index (GRIN) lens as a simple and an orders of magnitude more cost-effective approach. The lens is constructed using a phononic crystal (PC) architecture with varying lattice geometry and hence refractive index distribution. Specifically, the lens uses an axisymmetric hyperbolic secant refractive index profile to focus ultrasonic waves generated by a 1 MHz single-element ultrasonic transducer. Finite element simulations are performed to design and analyze the GRIN-PC lens, and to explore the effects of various parameters such as the distance from the skull and the incidence angle. The numerical results are validated experimentally for a 3D-printed lens by scanning the 3D pressure field generated through a temporal bone. This cost-effective approach to tFUS can open new possibilities to 3D print lenses based on patient computed tomography scans for various applications from tissue ablation to neurostimulation.

Broadband achromatic flexural wave Mikaelian lens for high resolution focusing

Conformal transformation method (CTM) has been extensively applied to control propagation of electromagnetic waves and acoustics waves due to the form-invariant property of Maxwell equations and acoustic equations. However, CTM’s application in elastic waves is rarely reported due to the governing equation of elastic waves do not have form invariant property. In this paper, through igniting evanescent waves at the interface of conformally mapped Mikaelian lens with hyperbolic secant refractive index profile, CTM is successfully used to achieve highly efficient (above 75%) broadband (30−80 kHz) achromatic high-resolution flexural wave focusing in thin plate with full width at half maximum (FWHM) around 0.2λ. The proposed Mikaelian lens is designed by linking refractive index with the thickness in plates. Simulated results agree well with theoretical prediction. This high performance for flexural wave focusing could be used for energy harvesting and medical imaging.

Mechanically Tuned Gradient-Index Phononic Crystal Lens

A two-dimensional gradient-index phononic crystal composed of aluminum scatterers in air matrix for frequency ofλ=6.1a is designed and simulated. We gradually change the radiuses to make a hyperbolic secant refractive index profile. Then the path of waves inside the structure is shown by analytical solutions. In this work, filling factor is mechanically increased which results in a shift in the focal point whereas the spot size remains almost constant. In order to enable the structure to be mechanically tuned, the conservators have been utilized and changed the circular cross section of scatterers slightly. However, due to this fact that the results of circular and non-circular structures do not show any differences, we used the circular scatterers for calculating the refractive indices. However, the amount of focal pressure can be slightly changed. The calculation of band structure as well as the simulation of acoustic waves’ propagation inside the structure has been accomplished using the finite element method.

MCVD-based GRIN-axicon for the generation of scalable Bessel-Gauss beams.

In this Letter, we introduce a graded-index (GRIN)-lens combination named GRIN-axicon, which is a versatile component capable of generating high-quality scalable Bessel-Gauss beams. To the best of our knowledge, the GRIN-axicon is the only optical component that can be introduced in both larger-scale laboratory setups and miniaturized all-fiber optical setups, while having an easy control of the dimensioning of the generated focal line. We show that a GRIN lens with a hyperbolic secant refractive index profile with a sharp central dip and no ripples generates a Bessel-Gauss beam with a high-intensity central lobe when coupled to a simple lens. Such fabrication characteristics are very suitable for the modified chemical vapor deposition (MCVD) process and enable easy manufacturing of an adaptable component that can fit in any optical setup.

Active beam steering and afocal zooming by nematic liquid crystal-infiltrated graded index photonic structures

This study presents active beam steering and afocal zooming of light by incorporating liquid crystals (LCs) with graded index photonic crystals (GRIN PCs). The GRIN PC structures are composed of low refractive index polymer annular rods with holes of gradually varying radii. To actively manipulate incident light, the annular rods are infiltrated with nematic LCs. By applying an external voltage to the infiltrated LCs, the effective index profile of the low-index GRIN PC structure is modulated without introducing any mechanical movement. The incident beam deflection and corresponding focal distance modulation are tuned only by controlling the applied bias voltage. In the present work, the hyperbolic secant refractive index profile is chosen to design GRIN PC structures. To design a GRIN PC structure with annular PCs, the Maxwell–Garnett effective medium approximation is employed. We analytically express the relation between infiltrated LCs and the gradient parameter to show the physical background of the tuning ability of the proposed devices. Beam steering and afocal zooming devices are analytically investigated via geometrical optics, and numerically realized with the help of a finite-difference time-domain method. A beam deflection with an angle change of Δθout = 44° and a light magnification with maximum ×2.15 are obtained within operating frequencies of a/λ = [0.10–0.15] and a/λ = [0.15–0.25], respectively, where ‘a’ is the lattice constant and λ is the incident wavelength. The corresponding operating frequency bandwidths are calculated as 40% and 50% for the beam steering and afocal zooming applications, respectively. LCs are inexpensive materials and work under low voltage/power conditions. This feature can be used for designing an electro–optic GRIN PC device that has the potential for use in a wide variety of optical applications.

Conformally Mapped Multifunctional Acoustic Metamaterial Lens for Spectral Sound Guiding and Talbot Effect.

We demonstrate a conformally mapped multifunctional acoustic metamaterial Mikaelian lens. Mikaelian lens is a gradient medium with a hyperbolic secant refractive index profile that can realize functions like beam self-focusing. Unlike the conventional design approaches, with a conformal transformation method, only isotropic material parameters with gradient refractive index profiles are required for the construction of such lens. To realize desired gradient index distribution, we carefully design a new type of cross-channel-shaped acoustic metamaterial, whose refractive index can be effectively modulated by simply changing the slit opening size. The distinct capabilities of the metamaterial Mikaelian lens in manipulating acoustic waves are experimentally verified with the fabricated sample. Simultaneous sound guiding and Talbot effects, which normally require respective geometrical and wave acoustic approximations, are observed in simulations and experiments. Furthermore, those effects of shaping acoustic wave propagations were validated within a relatively wide frequency range. Our study reveals how the conformal transformation method can help to bridge the ray acoustics with wave acoustics. It offers opportunities to the development of novel multifunctional acoustic devices for various applications, such as sound and particle manipulations.

A tolerance analysis on design parameters of parabolic and hyperbolic secant active GRIN materials for laser beam shaping purposes

The present paper considers two gain GRIN media, characterized by a complex parabolic and hyperbolic secant refractive index profile, for the design of uniform beam shaper systems. A general condition for beam shaping is obtained from the equation describing the evolution of the half-width of a plane Gaussian beam in the GRIN media. The simulation of the irradiance evolution of an input plane Gaussian beam—operating at 575 nm and beam waist radius of 0.45 mm—in each material is shown, in order to examine the beam shaping quality in terms of thickness of the active GRIN media and input beam wavelength.

Design of a Wavelength Selective Medium by Graded Index Photonic Crystals

In this paper, a novel technique for spatial wavelength division using graded index photonic crystals (GRIN PCs) is proposed. The designed GRIN PC structure with hyperbolic-secant refractive index profile at a fixed frequency generates spatial shifts in the longitudinal direction with respect to different incident wavelengths so that distinct path trajectories are followed by propagating beams through the graded medium. Due to gradual variation of the refractive index profile (incremental decrease), an incident beam with an appropriately selected wavelength undergoes continuing refraction and finally, total internal reflection (TIR) occurs at the turning point. Due to the wavelength-dependency characteristic of GRIN PCs, TIR takes place at different positions for different wavelengths and thus, light wave emanates from the GRIN medium at different locations with different exit angles. It is revealed that the designed structure can work in three distinct frequency regimes and is able to separate a certain number of incident beams with different wavelengths, viz. from three to seven different wavelengths. This condition enables the deployment of the GRIN PC configuration for the design of wavelength selective media.

Nonparaxial diffraction-limited propagation of light in a graded-index planar medium with a hyperbolic secant refractive-index profile

Nonparaxial diffraction-limited propagation of light with amplitude distribution in hyperbolic functions through an inhomogeneous planar medium with a hyperbolic secant refractive-index profile is studied by means of the stationary phase method. The irradiance distribution at geometrical shadow, edge of shadow, and a geometrically illuminated region is analyzed for a particular case.

Diffraction-free and diffraction-limited propagation of light in graded-index planar waveguides with hyperbolic secant refractive index profile

Abstract A study of diffraction-free and diffraction-limited propagation of light with transverse amplitude in sech through a planar waveguide with hyperbolic secant refractive index profile is presented. Analytical solutions by the stationary phase method are obtained. A particular case is analysed.

Diffraction-free and diffraction-limited propagation of light in graded-index planar waveguides with hyperbolic secant refractive index profile

A study of diffraction-free and diffraction-limited propagation of light with transverse amplitude in sech through a planar waveguide with hyperbolic secant refractive index profile is presented. Analytical solutions by the stationary phase method are obtained. A particular case is analysed.

Numerical aperture for graded-index tapered planar waveguides with a hyperbolic secant refractive index profile

A study of the numerical aperture for graded-index tapered planar waveguides with a hyperbolic secant refractive index profile is presented. Results are particularized for divergent conical and convergent parabolic types of the taper function.

Off-axis Gaussian beam propagation through planar waveguides with a hyperbolic secant refractive index profile

A study of the off-axis Gaussian beam propagation through an inhomogeneous planar optical waveguide with hyperbolic secant refractive index profile is carried out using the model of Kogelnik and Li. Beam half-width evolution with refractive index parameters, wavelength and displacement of the Gaussian beam from the axis is analysed. Similarly, a comparison with beam propagation in square-law media is presented.

Collimation and focusing properties of a planar waveguide with hyperbolic secant refractive index profile: light deflector and collimator design

Light transmission through an inhomogeneous waveguide whose refractive index profile can be represented by one-dimensional hyperbolic secant function is analysed from a geometrical optics point of view. Focusing and collimation conditions are obtained and the design of a light deflector is proposed. The possibility of obtaining almost perfect collimation for an off-axis point source by polishing the exit face of the waveguide is considered.

Optical propagator in a graded-index medium with a hyperbolic secant refractive-index profile

Light transmission through a gradient-index medium, whose refractive-index profile can be represented by a one-dimensional hyperbolic secant function, is analyzed by a wave-optics approach. An asymptotic expression of the corresponding optical propagator (the Green's function) in this medium is evaluated in a such a way that nonparaxial propagation can be studied. Thus optical transformations such as focusing and collimation are obtained, and the pupil effect (diffractive effect) is shown.

Novel phase-corrected horn antenna of short length

A new design based on ray theory is presented for a phase corrected horn antenna which is shorter than one with a hyperbolic secant refractive index profile. A dielectric load is essential for realistic implementation. With load permittivity decreasing linearly towards the aperture, a smooth flared profile can be obtained. The antenna's compactness makes it useful for marine applications.

Ray theory design of hyperbolic secant horn

It is shown that ray theory can be used in designing a microwave horn with a hyperbolic secant refractive-index profile. The horn is short in length, has constant phase across the plane aperture and tapered illumination in the E-plane, resulting in a radiation pattern with low sidelobes. Experimental verification has been made on a model operating around a frequency of 10 GHz.