Gradient-index Medium Research Articles

Propagation characteristics of non-diffracting Lommel beams in a gradient-index medium

We investigate the propagation of the Lommel beams in a gradient-index medium. Based on the ABCD transfer matrix and Collins diffraction integral formula, the analytical expressions of the complex amplitude of the Lommel beams through the gradient-index medium are derived. The intensity distribution and the Poynting vector of the Lommel beams are studied analytically and numerically. The results show that the Lommel beam is focused at the singularities and the corresponding periodic beam profile is induced by the symmetrical refractive index pattern during the beam propagating in the gradient-index medium. The effects of the asymmetry factor and the topological charge on the propagation characteristics of the Lommel beams in the gradient-index medium are discussed in detail. Especially, the propagation trajectories of the Lommel beams become visible as the topological charge increases and the symmetry of the Lommel beam’s transverse intensity pattern can be controlled by varying the asymmetry factor. It is expected that the results obtained in this paper may be useful for potential applications of optical control and trapping.

Propagation properties of Gaussian vortex beams through the gradient-index medium

Based on the extended Huygens–Fresnel principle, the analytical expression for Gaussian vortex beams propagating through a gradient-index medium is obtained. The propagation trajectory, the intensity distributions, and the phase distributions of Gaussian vortex beams propagating in the gradient-index medium are demonstrated, and the effects of gradient-index parameter β and topological charge m on these propagation properties are analysed in detail. It is found that the propagation trajectory and intensity distributions change on periodically with the increasing of propagation distance, and the periodicities are determined by the gradient-index parameters. As the increasing of the gradient-index parameters, the periodic variations on the propagation trajectory and intensity distributions are slowed down. Interestingly, the phase always circulates about the central null in the medium, but the rotation direction will change. The phase distributions also change on periodically in the medium, but the periodicities are different when the topological charge is odd or even. Furthermore, increasing the topological charge will increase the size of the beam spot and the circular dark hollow in the medium.

Symplectic ray tracing based on Hamiltonian optics in gradient-index media.

A method based on Hamiltonian optics for ray tracing through gradient-index (GRIN) media is proposed. The ray equation that describes light-ray paths can be written in the form of the Hamiltonian equations. Although the Hamiltonian equations can be numerically calculated using a finite-difference explicit method, deviations from the exact equations are generally inevitable at subsequent time steps. An optical Hamiltonian can be constructed of two independent terms, i.e., one term dependent on position and the other term dependent on momentum. The symplectic integrator is applicable to such a separable optical Hamiltonian system and makes the optical Hamiltonian equations form invariant at each time step of numerical calculations. Accuracies of light-ray paths calculated using the first-order symplectic ray tracing in GRIN lenses approximate those calculated on the basis of the fourth-order Runge-Kutta algorithm, which shows the promising potential of the symplectic-ray-tracing method.

Enhancing optical transmission of multilayer composites using interfacial nanostructures

We demonstrate the suppression of light reflections at solid-solid interfaces in multilayer thin and thick films using interfacial nanostructures. The embedded nanostructures have subwavelength features and function as a gradient-index medium to eliminate Fresnel losses induced by refractive index mismatch between dissimilar materials. Suppressing the interfacial reflection can reduce interference effects in thin films, and the transmittance measurement of a polymer on a silica substrate demonstrates a two-fold decrease in interference fringe contrast. A thick multilayer composite consisting of three fused silica and two polymer layers has also been fabricated and demonstrates the enhancement of optical transmission up to 30% at high incident angles. The effects of the interfacial structure geometry are examined by theoretical models based on rigorous coupled-wave analysis methods. The experimental results agree well with simulation models, which predicts that further improvements can be achieved using the optimized tapered profile. This work indicates that interfacial nanostructures can improve the broadband and wide-angle response of multilayers and can find applications in thin-film optics, optoelectronic devices, and composite windows.

Three-dimensional localized Airy-Gaussian wave packets in a gradient-index medium

By solving the Schrödinger equation in spatial and temporal coordinates, the exact solution of the chirped Airy-Gaussian (CAiG) wave packets in a gradient-index medium is obtained. Based on that, we conduct a theoretical analysis about the properties of CAiG wave packets and discover that the chirp factor, the distribution factor, the initial velocity, as well as the propagation distance have an effect on the wave packets in both the propagation process and the spatiotemporal profiles. Among these, the Gaussian distribution factor, which is first added to the initial pulses, also alters the peculiarities of the Airy-Gaussian beams in the temporal domain and spatial domain. In addition, the investigation about the radiation force illuminates certain evolution properties of the CAiG wave packets well.

Gradient-index dark hole based on conformal mapping with etendue conservation.

An omnidirectional light collector consisting of an axisymmetric spatially gradient refractive index medium can almost perfectly absorb light rays, regardless of where they come from. Based on the conformal mapping with complex gradient-index medium, the omnidirectional light collector, which is here called a dark hole, is able to be designed with an exponential function. The dark hole, however, has a reflective boundary where the Fresnel reflection occurs, which might lessen the absorption efficiency. To design a dark hole with consideration of the Fresnel reflection loss, a method to estimate its absorptance is necessary. Therefore, a formula to calculate the absorptance of the dark hole is derived based on the Lagrangian optics with the etendue conservation. Absorptances calculated using the formula agree well with those calculated using the Mie scattering theory in refractive index small-difference limit, which validates the formula. Absorptance of a dark hole with a silicon core and another dark hole with a complex gradient-index intermediate medium are calculated using the formula to be more than 98.8%. A micro-size dark hole is also shown to efficiently collect light rays with an absorptance of more than 95% using FDTD (finite-difference time-domain) simulation.

Propagation of cosine complex variable function Airy–Gaussian beams through the gradient-index medium

We introduce a new class of cosine complex variable function Airy–Gaussian (Cos-CVFAiG) beams and investigate their propagation through the gradient-index medium. Deriving the analytical expression, we show the intensity and phase distributions, the propagation trajectory and the Poynting vector. We carry out the numerical experiment to verify the rationality of the model and find that the light intensity distribution is parallel to the y axis in the transverse plane. With the distribution factor increasing, the axial intensity maximum and the number of the side lobes increase as well. Additionally, the distribution factor can change the bend extent of the ballistic trajectory and the intensity distribution region in the x–z plane. We also compare the influences of the gradient refractive index coefficient with the distribution factor on the propagation properties.

Propagation of a Bessel-Gaussian beam in a gradient-index medium.

Based on the ABCD matrix method and Collins diffraction integral formula, analytical expression for Bessel-Gaussian beam propagation in a gradient-index medium is derived. The propagation trajectory, intensity, and phase distributions of the zeroth-order, second-order, and superposition cases are numerically investigated. The effect of beam waist radius w0 on the properties of beam propagation in a gradient-index medium is discussed in detail. The result shows that the beam is focused at z/L=N/2 (N=0,1,2,…) and propagates periodically in the medium. Evolution of the vortical structure of the superposed Bessel-Gaussian beam is investigated, showing that the superposed beam forms new singularities, and the rotation of the beam occurs mainly near the singularities.

Propagation properties of Airy Ince–Gaussian wave packets in gradient-index media

We study localized Airy Ince–Gaussian (AiIG) wave packets in gradient-index (GRIN) media by solving the spatiotemporal evolution equation analytically. The generation and manipulation of AiIG wave packets in GRIN media are affected by the direction of the self-accelerating, elliptical, refractive index distribution parameter, initial velocity and mode number. In addition, energy flow, angular momentum and radiation force, which change periodically, are also thoroughly discussed. The Airy helical Ince–Gaussian wave packets exhibit multiple vortices and carry orbital angular momentum, which have potential applications in optical tweezers and optical communications.

Adaptability analysis of radiative transport diffusion approximation in planar-graded-index media

To study the applicability of diffusion approximation method, the influence of the optical thickness, scattering albedo, and wall emissivity of the examined medium on the accuracy of the diffusion approximation is analyzed. The accuracy is determined by comparison with results obtained by using the Monte Carlo method. By calculating the dimensionless radiation heat flux distributions at the bottom of the medium for various parameter combinations, the influence of each optical parameter is analyzed. In addition, the adaptability of the diffusion approximation method to the case of a gradient-index medium is examined. The results show that in the homogeneous-refractive-index medium and the gradient-index medium, the accuracy of diffusion approximation method is improved by an increase in the optical thickness or the scattering albedo and by a decrease in the wall emissivity. Moreover, the maximum relative errors are primarily distributed at the medium boundary and center. In the case of increased optical thickness or decreased wall emissivity, the error of the diffusion approximation method becomes stable and remains at a small value.

Phase space aberrations in freeform and gradient index imaging systems

Phase space provides the natural formalism with which to formulate optical imaging problems as a system with constraints. We consider the general formulation of optical imaging problems and look at two examples. The first example is a completely asymmetric freeform prism that has titled surfaces. The second example is a gradient index medium that is exactly solvable. This gives an alternative formalism to standard Seidel aberrations and nodal aberration theory, which can be used in the design process.

Design and fabrication of integrated micro/macrostructure for 3D functional gradient systems based on additive manufacturing

Functional gradient systems have important applications in many areas. Although a 2D dielectric structure that serves as the gradient index medium for controlling electromagnetic waves is well established, it may not be suitable for application in 3D case. In this paper, we present a method to realize functional gradient systems with 3D integrated micro/macrostructure. The homogenization of the structure is studied in detail by conducting band diagram analysis. The analysis shows that the effective medium approximation is valid even when periodicity is comparable to wavelength. The condition to ensure the polarization-invariant, isotropic, and frequency-independent property is investigated. The scheme for the design and fabrication of 3D systems requiring spatial material property distribution is presented. By using the vat photopolymerization process, a large overall size of macrostructure at the system level and precise fine features of microstructure at the unit cell level are realized, thus demonstrating considerable scalability of the system for wave manipulation.

Propagation of on-axis and off-axis Bessel beams in a gradient-index medium.

Bessel beams have been increasingly used for their advantages of non-diffraction and long focal depth. In this paper, we studied the propagation of on-axis and off-axis Bessel beams in a gradient-index medium. By expressing a Bessel beam in integral form, the analytical expression of an on-axis, decentered, and tilted Bessel beam through a paraxial optical system is derived with the ABCD matrix method and Collins diffraction integral formula. Main lobe size and trajectory of the zeroth- and second-order Bessel beam are obtained, demonstrating that the Bessel beam is focused by the gradient-index medium and its main lobe trajectory is exactly the same as the corresponding geometrical ray for both the decentered and tilted Bessel beam. Effects of beam apodization are finally studied by the Fourier beam propagation method, showing that the side lobes of the Bessel beam vanish when the beam is focused inside the medium as only part of the beam enters the lens.

Propagation of a Vector Cosine-Gaussian Correlated Beam Through an Active GRIN Medium

This paper presents a convenient approach that uses the tensor method to study the propagation of a nonuniformly correlated electromagnetic Cosine-Gaussian Schell-model (ECGSM) beam through an active gradient-index (GRIN) medium. It is shown that the initial correlation structure has a significant influence on the intensity gain; an ECGSM beam ( n = 1) experiences more gain than a conventional electromagnetic Gaussian Schell-model (EGSM) beam ( n = 0). The inhomogeneous gain induced by the medium leads to a nonlinear modulation of beam parameters such as the spectral intensity, coherence, and polarization. Although the gain leads to a broadening of the beam width, the coherence gradually decreases, which yields a useful guideline for manipulating active GRIN materials to generate numerous partially coherent beams. It is found that the polarization ellipse in the center of a conventional EGSM beam gradually evolves into circular polarization. However, the state of polarization of a nonuniformly correlated ECGSM beam gradually changes into linear polarization. The ability to manipulate light beams using active GRIN materials opens an alternative avenue for the generation of complex vector beams, which promises important supports in the fields of waveguide amplifiers, beam shaping, optical sensors, and beam transformer devices.

Propagation properties of the chirped Airy beams through the gradient-index medium

Through analytical derivation and numerical analysis, the propagation properties of the chirped Airy(CAi) beams in the gradient-index medium are investigated. The intensity and the phase distributions, the propagation trajectory and the Poynting vector of the CAi beams are demonstrated to investigate the propagation properties. Owing to the special and symmetrical refractive index profile of the gradient-index medium, the CAi beams propagate periodically. The effects of the distribution factor and the chirped parameter on the propagation of the CAi beams are analyzed. As the increasing of the distribution factor, the intensity distribution of the CAi beams is more scattering. However, with the chirped parameter increasing, the focusing property of the CAi beams strengthens. The variation of the chirped parameter can change the position of the peak intensity maximum, but it cannot alter the period of the peak intensity. The variations of the initial phase and the energy of the beams in the transverse plane expedite accordingly.

Bragg reflection from periodic helicoidal media with laterally graded refractive index

Light interaction with a columnar structure of InxAl1-xN where each column is a layered periodic helical medium with laterally graded refractive index is considered. It is demonstrated that such a columnar structure can be presented as a stack of layers with a gradient of the refractive index. To calculate reflectance in the proposed model, the 2 × 2 characteristic matrix method adopted for a gradient index medium was applied. The influence of the refractive indices (including absorption), parameters of the twisting, and thickness of the periodic structure on reflectance is studied. Cases of normal and oblique incident light are considered. The presented medium is a one-dimensional photonic crystal that can be utilized in many devices for light manipulation.

Propagation properties of Airy-Gaussian vortex beams through the gradient-index medium.

Propagation of Airy-Gaussian vortex (AiGV) beams through the gradient-index medium is investigated analytically and numerically with the transfer matrix method. Deriving the analytic expression of the AiGV beams based on the Huygens diffraction integral formula, we obtain the propagate path, intensity and phase distributions, and the Poynting vector of the first- and second-order AiGV beams, which propagate through the paraxial ABCD system. The ballistic trajectory is no longer conventional parabolic but trigonometric shapes in the gradient-index medium. Especially, the AiGV beams represent the singular behavior at the propagation path and the light intensity distribution. The phase distribution and the Poynting vector exhibit in reverse when the AiGV beams through the singularity. As the order increases, the main lobe of the AiGV beams is gradually overlapped by the vortex core. Further, the sidelobe weakens when the AiGV beams propagate nearly to the singularity. Additionally, the figure of the Poynting vector of the AiGV beams proves the direction of energy flow corresponding to the intensity distribution. The vortex of the second-order AiGV beams is larger, and the propagation velocity is faster than that of the first order.

Propagation of optical beams in two transverse gradient index media

We investigate the propagation of optical beams in two gradient index inhomogeneous media. The Green’s function for paraxial propagation in an inhomogeneous medium is derived as a Feynman path integral involving summation over real rays. We use a simple method based on discontinuous functions, which is similar to that used in General Relativity when studying metric discontinuities across an hypersurface, to study the propagation in two transverse refractive index gradients which are coupled in the propagation direction. We show that this method is consistent with the geometric-optics ray theory, and then with the definition of the coupled Green’s function. The propagation of Gaussian beams in two homogeneous media with different refractive indices is analyzed. We also study the propagation of Airy beams in two media with different linear transverse refractive index gradients. We demonstrate that by controlling the gradient strength of the media it is possible to reduce to zero their acceleration, yielding an Airy beam that propagates linearly.

Unified matrix approach to the description of phase-space rotators.

In optics, the rotation of a phase-space can be realized via light propagation through both an inhomogeneous medium with a radial gradient of refractive index and two special kinds of mirror-symmetrical optical systems suggested by Lohmann. Although light propagation through Lohmann's systems is described in terms of matrix optics, light propagation through the gradient-index medium is traditionally described as a solution of the wave equation. The difference in these descriptions hinders the understanding of the phase-space rotators. Fortunately, there is a matrix description of light propagation through a gradient-index medium too. A general description of the phase-space rotators is presented, which can be used to treat light propagation through both Lohmann's systems and the gradient-index medium in a unified matrix manner.

Runge–Kutta ray tracing technique for solving radiative heat transfer in a two-dimensional graded-index medium

This paper adopts the Runge–Kutta ray tracing method to obtain the ray-trajectory numerical solution in a two-dimensional gradient index medium. The emitting, absorbing and scattering processes are simulated by the Monte Carlo method. The temperature field and ray trajectory in the medium are obtained by the three methods, the Runge–Kutta ray tracing method, the ray tracing method with the cell model and the discrete curved ray tracing method with the linear refractive index cell model. Comparing the results of the three methods, it is found that the results by the Monte Carlo Runge–Kutta ray tracing method are of the highest accuracy. To improve the computational speed, the variable step-size Runge–Kutta ray tracing method is proposed, and the maximum relative error between the temperature field in the nonscattering medium by this method and the benchmark solution is less than 0.5%. The results also suggest that the Runge–Kutta ray tracing method would make the radiative transfer solution in the three-dimensional graded index media much easier.