Parabolic Refractive Index Profile Research Articles

Waveguide fabrication with integrated coupling optic

We present the fabrication of a waveguide with integrated injection optics using a laser photoinscription technique in the bulk of a chalcogenide glass. The injection optic is a gradient index lens with a parabolic refractive index profile. Its focusing characteristics depend on the amplitude of the index gradient and the thickness of the lens. These two parameters can be used to optimize light injection when the lens is written in front of a waveguide. We show experimentally that the coupling efficiency can be superior to that obtained by conventional injection with an external lens. Numerical simulations show that this coupling efficiency can be as high as 85%. The simultaneous inscription of a waveguide and its injection lens result in a monolithic component that is compact, lightweight and insensitive to mechanical or thermal perturbations. Performance has been evaluated in the mid-infrared range, but these results can be extended to the visible.

Spatial Photo-Patterning of Nematic Liquid Crystal Pretilt and its Application in Fabricating Flat Gradient-Index Lenses.

Liquid crystals offer a dynamic platform for developing advanced photonics and soft actuation systems due to their unique and facile tunability and reconfigurability. Achieving precise spatial patterning of the liquid crystal alignment is critical to developing electro-optical devices, programmable origami, directed colloidal assembly, and controlling active matter. Here, a simple method is demonstrated to achieve continuous 3D control of the directions of liquid crystal mesogens using a two-step photo-exposure process. In the first step, polarized light sets the orientation in the plane of confining substrates; the second step uses unpolarized light of a prescribed dose to set the out-of-plane orientation. The method enables smoothly varying orientational patterns with sub-micrometer precision. As a demonstration, the setup is used to create gradient-index lenses with parabolic refractive index profiles that remain stable without external electric fields. The lenses' focal length and sensitivity to light polarization are characterized through experimental and numerical methods. The findings pave the way for developing next-generation photonic devices and actuated materials, with potential applications in molecular self-assembly, re-configurable optics, and responsivematter.

Six-wave interaction with double wavefront reversal in multimode waveguides with Kerr and thermal nonlinearities

Spatial selectivity of six-wave radiation converters, which perform double wavefront conjugation of a signal wave in long multimode waveguides with both Kerr and thermal nonlinearities, is studied. Waveguides with infinitely conductive surfaces, with a parabolic refractive index profile, were used. It is shown that the spatial structure of the first pump wave does not affect the quality of doubled wavefront conjugation in a waveguide with Kerr nonlinearity, but only slightly affects the quality of doubled wavefront conjugation in a waveguide with thermal nonlinearity. A decrease in the radius of the second Gaussian pump wave on the back face of the waveguide leads to an improvement in the quality of the doubled wavefront reversal both in the case of six-wave interaction in the Kerr and thermal nonlinearities. In a parabolic waveguide, when the zero mode of the waveguide is excited by pump waves at a constant frequency of the second pump wave, an increase in the frequency of the first pump wave worsens the quality of the double wavefront conjugation.

Dynamics of three-dimensional spatiotemporal solitons in multimode waveguides

In this work, we present a detailed study of the dynamics and stability of fundamental spatiotemporal solitons emerging in multimode waveguides with a parabolic transverse profile of the linear refractive index. Pulsed beam propagation in these structures can be described by using a Gross–Pitaevskii equation with a two-dimensional parabolic spatial potential. Our investigations compare variational approaches, based on the Ritz optimization method, with extensive numerical simulations. We found that, with a Kerr self-focusing nonlinearity, spatiotemporal solitons are stable for low pulse energies, where our analytical results find a perfect agreement with the numerical simulations. However, with progressively increasing energies, solitons eventually undergo wave collapse: this occurs below the theoretical limit, which is predicted within the variational approach. In a self-defocusing scenario, a similar trend is found, where the good agreement persists for low energies. For large soliton energies, however, complex spatiotemporal dynamics emerge.

Stabilization of spatiotemporal dissipative solitons in multimode fiber lasers by external phase modulation

In this work, we introduce a method for the stabilization of spatiotemporal (ST) solitons. These solitons correspond to light bullets in multimode optical fiber lasers, energy-scalable waveguide oscillators and amplifiers, localized coherent patterns in Bose–Einstein condensates, etc. We show that a three-dimensional confinement potential, formed by a spatial transverse (radial) parabolic graded refractive index and dissipation profile, in combination with quadratic temporal phase modulation, may permit the generation of stable ST dissipative solitons. This corresponds to combining phase mode-locking with the distributed Kerr-lens mode-locking. Our study of the soliton characteristics and stability is based on analytical and numerical solutions of the generalized dissipative Gross–Pitaevskii equation. This approach could lead to higher energy (or condensate mass) harvesting in coherent spatio-temporal beam structures formed in multimode fiber lasers, waveguide oscillators, and weakly-dissipative Bose–Einstein condensates.

Spatial beam self-cleaning and supercontinuum generation with Yb-doped multimode graded-index fiber taper based on accelerating self-imaging and dissipative landscape.

We experimentally demonstrate spatial beam self-cleaning and supercontinuum generation in a tapered Ytterbium-doped multimode optical fiber with parabolic core refractive index profile when 1064 nm pulsed beams propagate from wider (122 µm) into smaller (37 µm) diameter. In the passive mode, increasing the input beam peak power above 20 kW leads to a bell-shaped output beam profile. In the active configuration, gain from the pump laser diode permits to combine beam self-cleaning with supercontinuum generation between 520-2600 nm. By taper cut-back, we observed that the dissipative landscape, i.e., a non-monotonic variation of the average beam power along the MMF, leads to modal transitions of self-cleaned beams along the taper length.

Numerical Studies on Large-Mode Area Fibers With Nanostructured Core for Fiber Lasers

We present numerical studies on bend-induced effective single-mode large-mode area fibers with various refractive index profiles: step index, parabolic, and triangular and hyperbolic for laser applications. We report the fiber with effective mode area as large as 1530 μm2 for 110-μm all-solid core diameter using triangular refractive index profile, in single-mode regime. We also report the fiber with parabolic refractive index profile, 70-μm core diameter with effective single-mode area of 1170 μm2. Development of novel active very large nanostructured core fiber with dedicated gradient index profile is discussed. Owning to nanostructurization, the extremely large core in the all-solid fiber can be obtained and used in a laser system to diminish undesirable nonlinear effects and increase the output power level with diffraction-limited beam quality. The nanostructurization method allows us to break the limits of classical manufacturing methods and opens new opportunities in precise shaping of the refractive index distribution.

Robustness of Light-Transport Processes to Bending Deformations in Graded-Index Multimode Waveguides.

Light transport through a multimode optical waveguide undergoes changes when subjected to bending deformations. We show that optical waveguides with a perfectly parabolic refractive index profile are almost immune to bending, conserving the structure of propagation-invariant modes. Moreover, we show that changes to the transmission matrix of parabolic-index fibers due to bending can be expressed with only two free parameters, regardless of how complex a particular deformation is. We provide detailed analysis of experimentally measured transmission matrices of a commercially available graded-index fiber as well as a gradient-index rod lens featuring a very faithful parabolic refractive index profile. Although parabolic-index fibers with a sufficiently precise refractive index profile are not within our reach, we show that imaging performance with standard commercially available graded-index fibers is significantly less influenced by bending deformations than step-index types under the same conditions. Our work thus predicts that the availability of ultraprecise parabolic-index fibers will make endoscopic applications with flexible probes feasible and free from extremely elaborate computational challenges.

Integrating Free-Form Nanostructured GRIN Microlenses with Single-Mode Fibers for Optofluidic Systems

We present both a theoretical and an experimental study of a novel compact lensed fiber system utilizing a nanostructured GRIN lens. The lens can be integrated with an optical fiber, which ensures a unique and efficient focusing in any high index medium, such as a liquid. We use the effective medium approach to design lenses with arbitrary refractive index. To fabricate lenses, we utilize a discrete array of nano-sized rods made of two types of glasses, and apply a standard stack-and-draw fiber drawing technology. The fabricated nanostructured GRIN lenses have a parabolic refractive index profile with a diameter of a standard fiber, very short working distances (55 µm in the air) and a high numerical aperture (NA = 0.16). As a proof-of-concept of the new micro-lensed fiber system, we demonstrate an experiment on optical trapping of micrometer-sized glass beads. We also show that our method is compatible with optical fiber technology and allows for any shape of the refractive index distribution in 2D. Thanks to that a new functionality could be achieved by replacing the GRIN lens with an axicon lens, vortex type elements, micro-lenses arrays or diffraction elements.

Wave localization in two dimensional parabolic periodic refractive index profiles: a 4th order Runge–Kutta study

In this work, we have studied the wave localization in a two dimensional parabolic periodic refractive index profile. Our calculations have been performed by developing a 4th order Runge–Kutta method. Effects of different parameters in refractive index profile and incident wave shape on the wave intensity and shape in the future are shown. Effects of the mentioned parameters on localization degree and total momentum of the system are also investigated. We find different parameters change intervals within which the excitation disperses. Thus no bound state is possible. Finally, we show when a bound state is present.

TE modes of UV-laser generated waveguides in a planar polymer chip of parabolic refractive index profile

The UV-laser lithographic method is used for the preparation of Polymeric integrated-optical waveguides in a planar polymer chip. The waveguide samples are irradiated by an excimer laser of wavelength 248 nm with different doses and with the same fluencies. The refractive index depth profile for the waveguides, in the first zone is found to have a parabolic shape and Gaussian shape in the second one that can be determined by Mach–Zehnder interferometer. Both the mode field distribution and the effective mode indices for the first zone only are determined by making use of the theoretical mode and the experimental data. It is found that the model field distribution is strongly dependent on the refractive indices for each zone.

Fast and accurate modeling of nonlinear pulse propagation in graded-index multimode fibers.

We develop a model for the description of nonlinear pulse propagation in multimode optical fibers with a parabolic refractive index profile. It consists of a 1+1D generalized nonlinear Schrödinger equation with a periodic nonlinear coefficient, which can be solved in an extremely fast and efficient way. The model is able to quantitatively reproduce recently observed phenomena like geometric parametric instability and broadband dispersive wave emission. We envisage that our equation will represent a valuable tool for the study of spatiotemporal nonlinear dynamics in the growing field of multimode fiber optics.

Variational method to spatial soliton propagation in a waveguide with periodic parabolic index profile

In this paper, we study spatial soliton propagation in a waveguide with periodic parabolic refractive index profile. Wave equation in the present of the refractive index profile includes diffraction, self-focusing (SF) and self-defocusing (SDF). To solve the wave equation, we use variational method and finally discuss the effect of self-defocusing and maximum potential depth on soliton behaviour.

Generation of soliton-like wave packets and wave packets with linear phase modulation in gaining optical wave-guides with saturable nonlinearity

We have considered the dynamics of soliton-like pulses and stable frequency-modulated self-similar pulses in an active medium with saturable nonlinearity and a parabolic refractive-index profile. We show that, based on gaining gradient fibers with a parabolic distribution of the refractive index and saturable nonlinearity, it is possible to create complexes that ensure generation of laser pulses with a high (above 1 TW) peak power.

Technological aspects of manufacturing silica optical fibers with large central defect of graded refractive index profile for fiber optic sensors based on few-mode effects

The paper deals with results of the study on the main technological aspects relating to a full production cycle of silica multimode graded-index fibers with the refractive index profile having central defect in the form of a large dip. Preform synthesis conditions for implementation of the mentioned defect via MCVD method have been analyzed and optimized. We have carried out research of the effect of geometrical irregularities, induced by drawing optical fibers under the manual control of the outer diameter stability, on attenuation coefficient of the graded-index 50/125 μm optical fibers with a large dip in the center of the refractive index profile. It is shown that variations of the outer diameter within the limits ± 3.5 μm lead to an increase of attenuation by 2–5 dB/km at the wavelength λ = 1310 μm as compared to the optical fibers fabricated under the automatic maintenance of the outer diameter stability. It has been determined that in the latter case fibers with the parabolic refractive index profile, corresponding to numerical aperture of 0.20, and the dip depth equal to 0.0115 demonstrate the attenuation about 5 dB/km in the second and third optical fiber transmission windows. Applying the Weibull distribution, a statistical evaluation of mechanical properties of the optical fibers drawn at various temperatures has been carried out. Based upon measurements, tensile strength of the fibers was estimated to be 5.07–5.49 GPa, that is comparable with the strength properties of silica telecom fibers. The manufactured multimode fibers are attractive candidates for developing sensing elements of registering external influences in systems of fiber-optic sensor networks based on few-mode effects.

Planar microlens with front-face angle: design, fabrication, and characterization

This paper studies the effect of microlens front-face angle on the performance of an optical system consisting of a planar-graded refractive index (GRIN) lens pair facing each other separated by a free-space region. The planar silica microlens pairs are designed to facilitate low-loss optical signal propagation in the free-space region between the opposing optical waveguides. The planar lens is fabricated from a 38-μm-thick fluorine-doped silica layer on a silicon substrate. It has a parabolic refractive index profile in the vertical direction, which is achieved by controlled fluorine incorporation in the silica film to collimate the optical beam in the vertical direction. Horizontal beam collimation is achieved by incorporating a horizontal curvature at the front face of the lens defined by deep oxide etch. A generalized 3×3ABCDGH transformation matrix method has been derived to compute the coupling efficiency of such microlens pairs to take front-face angles that may be present due to fabrication variations or limitations and possible input/output optical fiber offset/tilt into considerations. Pairs of such planar GRIN lens with various free-space propagation distances between them ranging from 75 to 2500 μm and with front-face angles of 1.5 deg, 2 deg, and 4 deg have been fabricated and characterized. Beam propagation method simulations have been carried out to substantiate the theoretical and experimental results. The results indicate that the optical loss is reasonably low up to 1.5 deg of front-face angles and increases significantly with further increase in the front-face angle. Analysis shows that for a given system with specific microlens front-face angle, the optical loss can be significantly reduced by properly compensating the vertical position of the input and output fibers.

Compression of ultra-short light pulses using the graded refractive index one-dimensional photonic crystals

The one-dimensional photonic crystals (1D PCs) containing a graded refractive index layer have been theoretically utilized to compress the positively chirped ultra-short pulses of light. Two types of simple and graded index multi-layer structures consisting alternating layers of TiO2 and SiO2 with the same total thicknesses and periodicity have been investigated and compared. For the graded structure, three different refractive index distributions including linear, exponential and parabolic profiles have been considered. The results revealed that replacing one of the homogeneous layers of the unit cells in simple photonic crystal with a graded material having parabolic refractive index profile efficiently improves compression behavior of the structure. The compress factors of as much as 47% and 78% depending on the pulse's initial chirp rate obtained with parabolic profile of such the structures.

Tapered GRIN fiber microsensor.

The sensitivity of an optical fiber microsensor based on inter-modal interference can be considerably improved by tapering a short extension of the multimode fiber. In the case of Graded Index fibers with a parabolic refractive index profile, a meridional ray exhibits a sinusoidal path. When these fibers are tapered, the period of the propagated beam decrease down-taper and increase up-taper. We take advantage of this modulation -along with the enhanced overlap between the evanescent field and the external medium- to substantially increase the sensitivity of these devices by tuning the sensor's maximum sensitivity wavelength. Moreover, the extension of this device is reduced by one order of magnitude, making it more propitious for reduced space applications. Numerical and experimental results demonstrate the success and feasibility of this approach.

High temperature sensing with fiber Bragg gratings in sapphire-derived all-glass optical fibers.

A structured sapphire-derived all-glass optical fiber with an aluminum content in the core of up to 50 mol% was used for fiber Bragg grating inscription. The fiber provided a parabolic refractive index profile. Fiber Bragg gratings were inscribed by means of femtosecond-laser pulses with a wavelength of 400 nm in combination with a two-beam phase mask interferometer. Heating experiments demonstrated the stability of the gratings for temperatures up to 950°C for more than 24 h without degradation in reflectivity.

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.