Parabolic Index Profile Research Articles

Six-wave interaction in multimode waveguides with Kerr nonlinearity with allowance for the Gaussian structure of pump waves

The quality of wavefront conjugation in the case of six-wave interaction in two-dimensional multimode waveguides with Kerr nonlinearity is analyzed under the condition that one of the pump waves excites a zero waveguide mode and the amplitude distribution of the other pump wave at the waveguide end facet varies according to the Gauss law. It is shown that in a waveguide with infinitely conducting walls, the half-width of the modulus of the point spread function of a six-wave radiation converter is completely determined by the transverse size of the waveguide and weakly depends on the width of the Gaussian pump wave. In a parabolic-index profile waveguide, a decrease in the width of the Gaussian pump wave at the waveguide ends commonly leads to a monotonic decrease in the half-width of the point spread function modulus.

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.

Focusing of partially coherent light by a graded-index lens.

We use coherence theory to study how the focusing of an optical beam by a graded-index (GRIN) lens is affected when the incoming beam is only partially coherent. The Gaussian-Schell model is used to show that the intensity of a partially coherent beam exhibits self-imaging and evolves in a periodic fashion in a GRIN medium with a parabolic index profile. Spatial coherence of the beam affects a single parameter that governs how much the beam is compressed at the focal point. Our results show that the focal spot size depends on the fraction of the beam's diameter over which coherence persists. Focusing ceases to occur, and the beam may even expand at the focal point of a GRIN lens, when this fraction is below 10%.

Nonparaxial corrections for short cavities and fibers

This paper describes optical propagation beyond the paraxial limit in rotational symmetric short Fabry-P\'erot cavities and long fibers with parabolic-index profiles. Frequency shifts due to paraxial and nonparaxial effects, the so-called spectral fine structure and vector corrections, are calculated with perturbation theory and expressed in a single dimensionless expansion parameter. The results obtained for short cavities and fibers have similar functional forms but are surprisingly different. These similarities and differences are pointed out and discussed, including their consequences for mode multiplexing in fibers.

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.

Arrayed waveguide lens for beam steering.

Integrated planar lenses are critical components for analog optical information processing that enable a wide range of applications including beam steering. Conventional planar lenses require gradient index control which makes their on-chip realization challenging. Here, we introduce a new approach for beam steering by designing an array of coupled waveguides with segmented tails that allow for simultaneously achieving planar lensing and off-chip radiation. The proposed arrayed waveguide lens is built on engineering the evanescent coupling between adjacent channels to realize a photonic lattice with an equi-distant ladder of propagation constants that emulates the continuous parabolic index profile. Through coupled-mode analysis and full-wave numerical simulations, we show that selective excitation of waveguide channels enables beam steering with large field-of-views of ∼60°. The proposed arrayed waveguide lens can serve as a compact component in integrated photonic circuits for applications in imaging, sensing, and metrology.

Spatial single mode laser source interaction with measured pulse based parabolic index multimode fiber

Abstract The study clarified spatial single mode laser interaction with measured pulse based parabolic index multimode fiber. Peak power level margin, maximum/minimum signal amplitude margin after parabolic index multimode fiber are measured with core radius of 25 µm, cladding thickness of 10 µm, refractive index peak of 1.4142, length of 300 m, and refractive index step of 1%. Maximum signal power margin against spectral frequency after PIN light detector based parabolic fiber properties is tested under the same operating parameters. The signal power amplitude/power within parabolic index multimode fiber is also measured based on variations of fiber lengths and relative refractive index step. The study implies the multimode graded index fibers with parabolic or near parabolic index profile cores have transmission bandwidths than other multimode fibers.

Two octave supercontinuum generation in a non-silica graded-index multimode fiber

The generation of a two-octave supercontinuum from the visible to mid-infrared (700–2800 nm) in a non-silica graded-index multimode fiber is reported. The fiber design is based on a nanostructured core comprised of two types of drawn lead-bismuth-gallate glass rods with different refractive indices. This yields an effective parabolic index profile and ten times increased nonlinearity when compared to silica fibers. Using femtosecond pulse pumping at wavelengths in both normal and anomalous dispersion regimes, a detailed study is carried out into the supercontinuum generating mechanisms and instabilities seeded by periodic self-imaging. Significantly, suitable injection conditions in the high power regime are found to result in the output beam profile showing clear signatures of beam self-cleaning from nonlinear mode mixing. Experimental observations are interpreted using spatio-temporal 3+1D numerical simulations of the generalized nonlinear Schrödinger equation, and simulated spectra are in excellent agreement with experiment over the full two-octave spectral bandwidth. Experimental comparison with the generation of supercontinuum in a silica graded-index multimode fiber shows that the enhanced nonlinear refractive index of the lead-bismuth-gallate fiber yields a spectrum with a significantly larger bandwidth. These results demonstrate a new pathway towards the generation of bright, ultrabroadband light sources in the mid-infrared.

Prediction of First Higher Order Modal Field for Graded Index Fiber in Presence of Kerr Nonlinearity

AbstractWe report evaluation of first higher order modal field for dual mode optical fiber having step and parabolic index profiles. The study is carried out both in absence as well as in presence of Kerr nonlinearity. The analysis is based on a simple iterative method involving Chebyshev formalism. Taking some typical step- and parabolic-index fibers as examples, we show that our results agree excellently with the exact results which can be obtained by applying rigorous methods. Thus, our simple formalism stands the merit of being considered as an accurate alternative to the existing cumbersome methods. The prescribed formalism provides scope for accurate estimation of different propagation parameters associated with first higher order mode in such kinds of fibers in presence of Kerr nonlinearity. The execution of formalism being user friendly, it will be beneficial to the system engineers working in the field of optical technology.

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.

Saturation and Refractive Index Geometry Effects on Localization of a Spatial Soliton in a Waveguide with Parabolic Rectangular Index Profile

In this paper, we have investigated the effects of well numbers, maximum refractive index variation Δn0, saturable nonlinearity β and soliton amplitude N on behavior of a spatial soliton propagating through a waveguide having a parabolic rectangular index profile. The aim of this work is to find the optimal values for these parameters in order to achieve the best soliton behavior. We found that the best system is one with N = 3, β = 0.4 and well number = 20. Knowing these optimal values helps experimenters to select the optimal system. In addition, to study the behavior of the propagating soliton, localization of the soliton was investigated under variations of a variety of the above-mentioned quantities. Our results show that localization of a soliton for the propagation distance z 10, the effective width (Weff) of the soliton shows an oscillatory behavior. The amplitude of Weff oscillations with increasing well number decreases, and the soliton becomes more localized. In contrast, with increase of β, Weff oscillations increase.

Inscription of Bragg gratings in nanostructured graded index single-mode fibers.

We report on efficient inscription of fiber Bragg gratings (FBGs) in a new type of single mode fiber with nanostructured core and with an effective parabolic graded index profile, using the standard phase mask method and a 248 nm pulsed laser. A nanostructured core allows to obtain high concentration of GeO2 in subwavelength glass rods and simultaneously to maintain low average germanium dopant level of silica similarly to standard single mode fibers. We showed that in a nanostructured core fiber, a factor of 3 better efficiency in gratings inscription was achieved, although the fiber has 20% lower average concentration of GeO2 with respect to SMF-28. In the nanostructured fiber we obtained a significant improvement in temperature sensitivity, while the strain sensitivity of FBG in nGRIN optical fiber is the same as in case of standard single-mode fiber (SMF-28). We have measured the strain sensitivity of 0.72 × 10-6 1/με (1.11 pm/με@1.53μm), and the temperature sensitivity is about 30% higher than for FBG in SMF-28 and equals to 10.2 × 10-6 1/K (15.6 pm/K@1.53μm).

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.

Propagation characteristics of some intricate planar waveguides of practical interest by finite difference method

Using the finite difference method (FDM), we analyse some practically interesting planar waveguide structures and develop the proper FDM algorithm to predict the relevant transverse electric and transverse magnetic modes with their propagation constants (b) over a long range of normalised frequency (V). In order to crosscheck our results from FDM, we use the step index profile, parabolic index profile and complementary error index profile as a test case to standardise our program. Then, with our program, we consider two more structures such as, diffused planar waveguide and planar directional coupler, to predict the corresponding b−V curves which are yet to be, completely, analysed by FDM.