Arbitrary Refractive Index Profile Research Articles

Tackling the inverse problem in ellipsometry: analytic expressions for supported coatings with nonuniform refractive index profiles in the thin film and weak contrast limits

Ellipsometry is a powerful tool for the evaluation of the refractive index profile of a film or coating supported on a solid substrate. A well-acknowledged problem, however, is the inverse problem: Given a set of data, under what conditions can a refractive index profile be determined unambiguously? To this end, a series expansion of the Abeles matrix method has been applied to an arbitrary refractive index profile to determine analytic expressions of the ellipsometric ratio ρ. Two types of expressions are found: The thin film limit in which the film thickness L is much less than the wavelength of the incident light beam ( L≪λ ) and the weak contrast limit in which the refractive index of the coating is very near the refractive index of the supporting substrate. In the thin film limit, the first two terms in the series expansion are relatively straight-forward, and they depend on two types of integrals involving the difference between the dielectric profile of the coating and the dielectric constant of the substrate. While higher order terms are possible, they are quite convoluted and do not assist in the inverse problem. In the weak contrast limit, however, the series expansion of ρ depends on the moments of the difference between the dielectric profile of the coating and the dielectric constant of the substrate, allowing an analytic expression that applies to coatings that are even much larger than the incident light beam. The expressions associated with both limits are verified through comparison to the numerical evaluation of ρ with the Abeles matrix method. The results demonstrate that through judicious selection of the substrate refractive index and incident wavelength, conditions can be created that permit critical insights into the inverse problem for either thin coatings or for coatings that are very near the refractive index of the substrate.

An Ab Initio, Fully Coherent, Semi-Analytical Model of Surface-Roughness-Induced Scattering

Integrated optics and silicon photonics is a rapidly maturing technology and is progressing in telecom, computation, and sensing. Surface-roughness-induced scattering is the primary source of optical loss in most photonic integrated circuits, and as such, ultimately limits the performance of their applications. However, a closed-form description for arbitrary refractive index profiles remains lacking, even though such a one is essential to enable an objective-oriented design of the waveguide platform. We present an ab initio, fully coherent, analytical model based on the volume current method that uses the surface roughness' autocorrelation function and the unperturbed mode's electromagnetic fields to predict the loss coefficient in closed form. An improved expression for the perturbation facilitates the application also to high-index-contrast systems. Hence, it is flexible concerning wavelength, materials, fabrication process, geometry and mode. Consequently, our model may be seamlessly integrated into electromagnetic simulation software suites, once the surface roughness is known for the utilized fabrication process. To verify our model, we compare the calculated scattering losses to measured propagation losses and established models for a wide range of waveguide systems in literature. We find that the previously neglected correlation along the waveguide height significantly impacts the scattering, which necessitates the holistic statistical analysis of the surface roughness. We believe these comprehensive prediction capabilities to be a useful tool for the optimization of silicon photonics design and fabrication, especially for low-confinement applications like sensors.

Modified theory of coupled waves and a numerical method for the analysis of Bragg mirrors with an arbitrary refractive index profil

An analytical and effective numerical approach to the analysis of the optical characteristics of Bragg mirrors with an arbitrary refractive index profile is developed. Analytical expressions obtained using the modified theory of coupled waves for the coefficients of reflection / transmission of light from through the BM with high accuracy predict its main characteristics at both low and high contrast of the refractive index. The hybrid numerical method includes the numerical calculation of the transfer matrix for one period of the knowledge base and its analytical multiplication for an arbitrary number of periods. The developed methods are applied to the analysis of the properties of practically important BMs made on the basis of sprayed pairs of Ta2O5/SiO2 dielectrics, Group III nitrides and arsenides. The characteristics of the knowledge base with smooth (gradient) interfaces are considered in detail. Keywords: vertically emitting lasers, reflection coefficient, reflection bandwidth.

General method based on WKB approximation and digital holography to characterize planar waveguides

Future optical technologies are basically depending on and employing different types of optical waveguides. The fabrication of waveguides with different properties represents an expanding direction in both the industrial and scientific fields. In the current paper, we focus on studying the polymeric planar waveguides fabricated by the method of UV–excimer–laser lithography. We propose a general technique based on the WKB approximation and digital holographic interferometry to characterize slab waveguides with arbitrary refractive index profile. The necessary theoretical model is derived and the characterization method given in detail. Two samples are fabricated from polymethylmethacrylate (PMMA) polymeric material using different parameters for the used UV-laser. Then a phase-shifting digital holographic technique with the help of a Mach–Zehnder-like interferometer, is used to characterize the samples and measure their modified refractive index profiles. The resulting profiles are then fed to the proposed theoretical model and a full TE mode analysis is obtained. These results prove the ability of the proposed technique to analyze planar waveguides having any complicated refractive index profile.

Модифицированная теория связанных волн и численный метод для анализа брэгговских зеркал с произвольным профилем показателя преломления

An analytical and effective numerical approach to the analysis of the optical characteristics of Bragg mirrors with an arbitrary refractive index profile is developed. Analytical expressions obtained using the modified theory of coupled waves for the coefficients of reflection / transmission of light from / through the BM with high accuracy predict its main characteristics at both low and high contrast of the refractive index. The hybrid numerical method includes the numerical calculation of the transfer matrix for one period of the knowledge base and its analytical multiplication for an arbitrary number of periods. The developed methods are applied to the analysis of the properties of practically important BMs made on the basis of sprayed pairs of Ta2O5 / SiO2 dielectrics, Group III nitrides and arsenides. The characteristics of the knowledge base with smooth (gradient) interfaces are considered in detail.

Elliptical fibre dielectric waveguides: a transverse transmission line analysis

The authors present a new method for determining the optical properties of arbitrary refractive index elliptical core fibres. The method applies the elliptic cylindrical coordinate system to Maxwell's equations and appropriately, using Fourier Transforms, operated on an elliptical core optical fibre waveguide, derives a flexible algorithm, which can be used to determine the optical properties of elliptical fibre waveguides. By dividing the fibre into a series of uniform consecutive thin dielectric layers transversely across the elliptical cross section with suitable transforming functions representing voltage and current themselves based on the electromagnetic field components, leads to transverse transmission line equations. The resonance frequencies of the cascaded matrices determine the mode propagation constants. The analysis leads to a numerical algorithm for calculating the exact modes of propagation constants. The method is analytically exact and computationally accurate, leading to algorithmic implementation, without making use of Mathieu functions which are solutions for step index waveguide. The advantage of this work is also the fact that it can be applied to arbitrary refractive index profiles. They present results of modal b ¯ − V diagrams for various fibre ellipticities and also apply this method to determine the birefringence of graded index profile fibres and mode cut-off frequencies.

Development of large diameter nanostructured GRIN microlenses enhanced with temperature-controlled diffusion.

Nanostructured GRIN components are optical elements which can have an arbitrary refractive index profile while retaining flat-parallel entry and exit facets. A method of their fabrication requires assembly of large quantities of glass rods in order to satisfy subwavelength requirement of the effective medium theory. In this paper, we present a development of gradient index microlenses using a combination of methods: nanostructurization of the preform and controlled diffusion process during lens drawing on a fiber drawing tower. Adding a diffusion process allows us to overcome limits of the effective medium theory related to maximum size of nanorods in the lens structure. We show that nanorods are dissolved during the fiber drawing process in high temperature and glass components are locally quasi-uniformly distributed. To demonstrate feasibility of the proposed approach, we have developed and experimentally verified the performance of a nGRIN microlens with a diameter of 115 µm composed of 115 rods on the diagonal, and length of 200 µm devoted to work for the wavelength over 658 nm.

Gradient-index phononic crystals for highly dense flexural energy harvesting

Gradient-index (GRIN) refers to a system where the refractive index changes spatially within a specific region. GRIN phononic crystals are capable of not only amplifying the magnitude of wave energies but also controlling the directional nature of the wave propagation, thus offering substantial benefits with regard to energy harvesting (EH) improvements. Here, we propose a systematic design method for GRIN phononic crystals which combine the two-dimensional Reissner–Mindlin plate model and a genetic algorithm for optimization. This design process allows us to design a GRIN phononic crystal with any arbitrary refractive index profile or complex shape of the unit cells. The experimentally verified focusing capability of the GRIN phononic crystals led to the realization of piezoelectric energy harvesting with a maximum areal power density value of up to 240.4 mW/m2, considerably outperforming the existing non-GRIN-based EH systems without direction controllability.

Rayleigh backscattered radiation produced by an arbitrary incident mode in multimode optical fibers.

The recently developed diffraction technique of analytical investigation of the Rayleigh backscattering produced by an incident fundamental mode in a multimode optical fiber with an arbitrary refractive index profile is generalized to admit an arbitrary incident mode, either radial or azimuthal. The relative powers of all backscattered modes are determined with explicit formulas via the properly normalized transverse distributions of the incident and backscattered mode fields within the fiber cross section. The regularities conditioned by azimuthal indices are expressed via a universal set of coefficients, and dependences on the radial mode indices are estimated numerically. Excitation coefficients are shown to be symmetrical for any pair of incident and backscattered modes.

Electrochemical Fabrication of Flat, Polymer‐Embedded Porous Silicon 1D Gradient Refractive Index Microlens Arrays

Gradient refractive index (GRIN) optics has attracted considerable interest due to the ability to decouple optical performance from optical element shape. However, despite the utility of GRIN optical components, it remains challenging to fabricate arbitrary GRIN profiles and the available refractive index contrast remains small, particularly at the microscale. Here, using mathematical transformations that the authors developed, the electrochemical waveform required to electrochemically etch arrays of bulk Si microstructures into 1D porous Si (PSi) GRIN microlens arrays (MLAs) is determined. This waveform is then used to form high refractive index contrast MLAs containing precisely‐defined, arbitrary refractive index profiles. The MLAs are then embedded in a transparent optical polymer, mechanically detached from the host Si substrate, and planarized via simple polishing. Cylindrical microlenses and 1D axicons are demonstrated and characterized, and the optical behavior is found to be in agreement with theory. These MLAs could find applications in displays, photodetectors, and optical microscopy.

Influence of a variable Rayleigh scattering-loss coefficient on the light backscattering in multimode optical fibers.

The recently developed diffraction technique of analytical investigation of the Rayleigh backscattering produced by an incident fundamental mode in a multimode optical fiber with an arbitrary refractive index profile is supplemented by taking into account the Rayleigh scattering-loss coefficient, which could be variable within the fiber cross section. The relative changes in various radial and azimuthal modes' excitation levels, due to some typical radial dependences of this coefficient, are computed for the quadratic- and step-index fibers. It is stated that the excitation efficiency could either rise or decay for different modes. The effect of the variable Rayleigh scattering-loss coefficient is shown to be more noticeable in the fibers with a quadratic refractive index profile, whereas it is negligible in actual multimode step-index fibers.

Design of arbitrary modal electric field in planar waveguides

The purpose of this study is to propose a method for designing planar waveguides of arbitrary modal electric field distribution. Normally, the authors know or specify a waveguide and they solve for the mode propagation constant which allows them to plot the electric field. A novel and efficient algorithm is proposed here for determining arbitrary refractive index profile planar waveguides with a given desired mode field distribution. This inverse problem is generally solved using complex and variable refractive index profile planar waveguides. The authors present a method for determining the required refractive index profile for both real and imaginary parts. This inverse problem is solved via modelling the waveguide transversely as a transmission line and work inversely from the electric field. They offer example reconstructions of refractive index profiles which are useful for high-power lasers, mode matching between planar and circular waveguides and complex conjugate electric fields. The method also has applications in designing optical fibres for sensors, where one may require specific shape electric field from a waveguide, and also in refractive index profiling instrumentation.

Numerical Investigation of Microbending Loss in Optical Fibres

Microbending plays a key role in the bend loss of optical fibres. To numerically investigate a microbending induced loss, an analytical model for microbending in optical fibres with arbitrary refractive index profiles is presented. In this model, random perturbations of the fibre core along the fibre axis are described by an analytical function whose power spectral density is derived from an exponential autocorrelation function. Using the model together with the beam propagation method, microbending loss is investigated for several different types of optical fibre, which include the traditional single-mode/multimode fibres (SMF/MMF) as used in existing optical networks, and typical few-mode/ring-core fibres (FMF/RCF) with the potential for future ultra high-speed optical networks. The validity of the proposed model is demonstrated by comparing the microbending loss of a SMF computed using coupled mode theory and the results. Simulation results show that, under the condition of launching only the fundamental mode into the optical fibres, the SMF and RCF supporting only one radial mode have nearly equal microbending loss, whilst the FMF and MMF have relatively low microbending loss. In addition, the microbending loss of the RCF is shown to be dependent on the ring core thickness.

Mode-dependent characteristics of Rayleigh backscattering in weakly-coupled few-mode fiber

We theoretically investigate the mode-dependent characteristics of Rayleigh backscattering arising in weakly-coupled few-mode fibers (FMFs). Based on the theory of Lorentz reciprocity and surface wave excitation, we derive a general analytical equation of excitation efficiency and power distribution of Rayleigh backscattering light among backward propagation modes under the condition of impulse response. Thus, we are able to characterize the Rayleigh backscattering of weakly-coupled FMF with arbitrary refractive index profile. As for the weakly-coupled FMF with a step-index profile, the power distribution ratio of individual modes in the Rayleigh backscattering light is mainly determined by the forward propagation mode. In particular, the backscattering mode with the same profile as the forward propagation one has the highest excitation efficiency. The FMF parameters have influence on the total backscattering power, but little effect on the mode power distribution ratio.

Fast and Simple Method for Evaluation of Polarization Correction to Propagation Constant of Arbitrary Order Guided Modes in Optical Fibers with Arbitrary Refractive Index Profile

This work presents fast and simple method for evaluation of polarization correction to scalar propagation constant of arbitrary order guided modes propagating over weakly guiding optical fibers. Proposed solution is based on earlier on developed modified Gaussian approximation extended for analysis of weakly guiding optical fibers with arbitrary refractive index profile in the core region bounded by single solid outer cladding. Some results are presented that illustrate the decreasing of computational error during the estimation of propagation constant when polarization corrections are taken into account. Analytical expressions for the first and second derivatives of polarization correction are derived and presented.

Dispersion characteristics of complex refractive-index planar slab optical waveguide by using finite element method

In this paper, finite element method (FEM) mode analyses of planar slab optical waveguide having complicated refractive index profile are presented. We try to estimate the dispersion graph, mode cut-off condition, group delay and waveguide dispersion for the case of α-power and chirped-type refractive index profile. In order to obtain the more accurate result, we have derived the higher-order polynomial, which establishes the suitable relationship between b and V for different profile of optical waveguide. On the basis of the derived polynomials, the waveguide dispersion is analyzed for different type of refractive index profile waveguide. Our study shows that the waveguide dispersion can be substantially reduced when we deployed the optical waveguide having linearly chirped-type refractive index profile. Earlier too, the arbitrary refractive index profile has been analyzed but to the best of our knowledge chirped-type refractive index profile has not been analyzed till date for the case of planar slab optical waveguide.

Graphics processing unit–accelerated finite-difference time-domain method for characterization of photonic crystal fibers

In the work, we have presented the technique based on the graphics processing unit accelerated finite-difference time-domain (FDTD) method for characterization of a single-mode photonic crystal fiber (PCF) with an arbitrary refractive index profile. In contrast to other numerical methods, the FDTD allows studying the mode propagation along the fiber. Particularly, we have focused attention on the method details that allowed us to reduce dramatically the computation time. It has been demonstrated that the accuracy of dispersion obtained by the FDTD method is comparable to the one provided by the finite elements method while possessing lower computation time. The method has been used to determine the fundamental mode cut-off of all-normal dispersion PCF and to find fiber losses beyond this wavelength.

Accurate Modeling and Design of Graded-Index Fiber Probes for Optical Coherence Tomography Using the Beam Propagation Method

Fiber-optic probes for sensing and biomedical imaging applications such as optical coherence tomography (OCT) frequently employ sections of graded-index (GRIN) fiber to re-focus the diverging light from the delivery fiber. Such GRIN fiber microlenses often possess aberrations that cause significant distortions of the focused output beam. Current design methods based on ABCD matrix transformations of Gaussian beams cannot model such effects and are therefore inadequate for the analysis and design of high-performance probes that require diffraction-limited output beams. We demonstrate use of the beam propagation method (BPM) to analyze beam distortion in GRIN-lensed fibers resulting from index profiles that exhibit a deviation from the ideal parabolic shape or artifacts such as ripples or a central dip. Furthermore, we demonstrate the power of this method for exploring novel probe designs that incorporate GRIN phase masks to generate wavefront-shaped output beams with extended depth-of-focus (DOF). We present results using our method that are in good agreement with experimental data. The BPM enables accurate simulation of fiber probes using non-ideal or custom-engineered GRIN fibers with arbitrary refractive index profiles, which is important in the design of high-performance fiber-based micro-imaging systems for biomedical applications.

Dielectric waveguides of arbitrary cross sectional shape

Numerical guided mode solutions of arbitrary cross sectional shaped waveguides are obtained using a finite difference (FD) technique. The standard FD scheme is appropriately modified to capture all discontinuities, due to the change of the refractive index, across the waveguides’ interfaces taking into account the shape of each interface at the same time. The method is applied to the vector Helmholtz equation formulated to describe the electric or magnetic fields in the waveguide (one field is retrieved from the other through Maxwell’s equations). Computational cost is kept to a minimum by exploiting sparse matrix algebra. The waveguides under study have arbitrary cross sectional shape and arbitrary refractive index profile.

Radiation forces on a microsphere in an arbitrary refractive index profile

The force equations describing the radiation forces on a microsphere in an arbitrary refractive index profile are derived here by using the photon-stream method in a ray-optics regime. A loosely focused Gaussian beam was employed as the radiating illumination beam. The radiation forces on a spherical microsphere were calculated in a time-varying refractive index profile. The refractive index profile of the surrounding medium was evaluated according to the concentration distribution obtained from the diffusion equation. The scattering and gradient forces on a microsphere were calculated for different refractive indices (1.22, 1.33, 1.43, and 1.59), and the radiation forces on a perfectly reflecting microsphere were calculated. The results were compared with previous results to validate the derivation.