GRIN Media Research Articles

Cauchy–Riemann beams in GRIN media

We investigate the propagation characteristics of Cauchy–Riemann beams in gradient-index media. Our study reveals two key findings: first, the preservation of their form during propagation; and second, the surprising feasibility of obtaining the two-dimensional fractional Fourier transform for any arbitrary entire function. We elucidate an explicit and straightforward expression for this transform. Additionally, our results contribute to a deeper understanding of the behavior of Cauchy–Riemann beams in inhomogeneous media, offering insights that may have implications for various applications in optics and wave propagation.

The Propagation Properties of a Lorentz–Gauss Vortex Beam in a Gradient-Index Medium

Based on the Huygens–Fresnel integral and ABCD matrix, the propagation equation for the Lorentz–Gauss vortex beam (LGVB) in a gradient-index medium (GRIN) is rederived. The evolution of the intensity and phase distributions of an LGVB through a GRIN medium are numerically calculated as a function of the gradient-index parameter with changes in the incident beam parameters. The results showed that the propagation path and intensity distributions changed periodically with increasing propagation distance. In contrast, phase distributions change at multiples of π/β or 2π/β, depending on whether the M values are odd or even, respectively. At the same time, the parameters of the gradient index determine the periodic values of the Lorentz–Gauss vortex beams during propagation, and as β increased, the period of evolution decreased. The Lorentz–Gauss vortex beam propagating through the gradient index will develop from a square beam to a Gaussian vortex beam more quickly with an increase of w0x=w0y. In addition, the topological charge affects the size of the dark spot at the center of the beam and the size of the beam, causing the phase distributions to change periodically in the medium. This study is beneficial for laser optics and optical communications.

Bohm approach to the Gouy phase shift

By adapting the Madelung–Bohm formalism to paraxial wave propagation we show, by using Ermakov–Lewis techniques, that the Gouy phase is related to the form of the phase chosen in order to produce a Gaussian function as a propagated field. For this, we introduce a quantum mechanical invariant, that it is explicitly time dependent. We finally show that the effective Bohm index of refraction generates a GRIN medium that produces the focusing needed for the Gouy phase.

Crystalline lens paradoxes revisited: significance of age-related restructuring of the GRIN.

The accommodating volume-constant age-dependent optical (AVOCADO) model of the crystalline lens is used to explore the age-related changes in ocular power and spherical aberration. The additional parameter m in the GRIN lens model allows decoupling of the axial and radial GRIN profiles, and is used to stabilise the age-related change in ocular power. Data for age-related changes in ocular geometry and lens parameter P in the axial GRIN profile were taken from published experimental data. In our age-dependent eye model, the ocular refractive power shows behaviour similar to the previously unexplained "lens paradox". Furthermore, ocular spherical aberration agrees with the data average, in contrast to the proposed "spherical aberration paradox". The additional flexibility afforded by parameter m, which controls the ratio of the axial and radial GRIN profile exponents, has allowed us to study the restructuring of the lens GRIN medium with age, resulting in a new interpretation of the origin of the power and spherical aberration paradoxes. Our findings also contradict the conceptual idea that the ageing eye is similar to the accommodating eye.

Generalized revival and splitting of an arbitrary optical field in GRIN media.

Assuming a non-paraxial propagation operator, we study the propagation of an electromagnetic field with an arbitrary initial condition in a quadratic GRIN medium. We show analytically that at certain specific periodic distances, the propagated field is given by the fractional Fourier transform of a superposition of the initial field and of a reflected version of it. We also prove that for particular wavelengths, there is a revival and a splitting of the initial field. We apply this results, first to an initial field given by a Bessel function and show that it splits into two generalized Bessel functions, and second, to an Airy function. In both cases our results are compared with the numerical ones.

Revival and splitting of a Gaussian beam in gradient index media.

The short range revival of an arbitrary monochromatic optical field, which propagates in a quadratic GRIN rod, is a well-known effect that is established assuming the first-order approximation of the propagation operator. We discuss the revival and multiple splitting of an off-axis Gaussian beam propagating to relatively long distances in a quadratic GRIN medium. These effects are obtained assuming the second-order approximation of the propagation operator in this medium.

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.

Ray tracing through the crystalline lens using the decomposition method

The problem of tracing light rays through an inhomogeneous graded index medium is solved using the decomposition method. This is a powerful method for solving nonlinear and/or stochastic differential equations. We apply this method to arrive at the solution for ray tracing through the human crystalline eye lens having a radially varying refractive index distribution. This method offers a simpler method to solve the ray equation in GRIN media.

실안의 수정체 굴절률 분포를 갖는 정밀모형안 설계

젊은 한국인 정시안에 대한 임상 자료를 가지고 구면수차 이외에 비축수차인 비점수차, 상면만곡과 왜곡수차들을 포함하는 실안의 광학적 성능과 유사한 정밀모형안을 설계하였다. 또한 수정체가 갖는 Gradient index 형태의 매질을 광학적으로 조사하였고, GRIN 매질을 수정체에 도입하여 정밀모형안을 설계하였다. In this study, clinical data for emmetopia in young Koreans was taken in order to design the finite schematic eye, which had optical properties of real eyes including spherical aberration, astigmatism, field curvature and distortion. Furthermore, the crystalline lens with GRIN medium was optically analyzed, and the finite schematic eye with the GRIN crystalline lens was designed.

New Bose Operator Realization of SU(2) Algebra and Its Application in Optical Propagation**The project supported by National Natural Science Foundation of China under Grant No. 10175057

We introduce a new Bose operator realization of SU(2) Lie algebra and applied it to optical propagation in GRIN media or some lens-setup.

Gaussian beam propagation and imaging in hyperbolic secant profile media

The propagation of Gaussian beams in GRIN media characterized by the hyperbolic secant (HS) index profile is studied. It is shown that the profile can contract, expand or maintain the width of a beam incident with the beam waist at the input end of the medium. We derive an accurate analytical expression for the mode width of the fundamental mode of a HS medium. The imaging characteristics of the medium are also studied and compared to selfoc media. It is shown that, in the appropriate limit, the expressions for the HS profile reduce to those of a selfoc medium.