Graded-index Waveguide Research Articles

New soliton configurations for two different models related to the nonlinear Schrödinger equation through a graded-index waveguide

The present paper studies two various models with two different types: the nonlinear Schrödinger equation with power-law nonlinearity and the (3 + 1)-dimensional nonlinear Schrödinger equation. We perform the modified (G′G)-expansion method to find some exact solutions and to construct various types of solitary wave phenomena for each equation. The received aspects contribute to the firm mathematical foundation and might be essential to the soliton waves. As a result, we obtain all the solutions from Wazwaz [Math. Comput. Modell. 43, 178–184 (2016)] and also obtain some new soliton solutions.

Controlling the Chain Orientation and Crystal Form of Poly(9,9-dioctylfluorene) Films for Low-Threshold Light-Pumped Lasers

Neat thin films of semiconducting polymers are attractive as efficient gain media toward optically pumped lasers. However, the optical loss and out-coupling of isotropic polymer thin films are far from being rationally regulated from the perspective of chain orientation and crystal form. Herein, we accomplished a simultaneous control of both chain orientation and crystal form in large-area highly ordered poly(9,9-dioctylfluorene) (PFO) thin films through epitaxial crystallization. The well-arranged PFO lamellae naturally shape a low-loss, graded-index waveguide to provide spatially distributed optical feedback. Moreover, much more horizontally oriented dipoles significantly enhance the light out-coupling efficiency. Thus, the only 65 nm-thickness oriented PFO films demonstrate excellent amplified spontaneous emission with a low excitation threshold (7.1 μJ/cm²) and a narrow full width at half-maximum (2.2 nm) measured in the perpendicular direction of PFO chains. This strategy opens an effective pathway to prepare high-performance polymer thin-film lasers and electrically pumped polymer lasers.

Two-Color Localized Optical Vortices in a Nonlinear Waveguide with Zero Dispersion of the Group Velocity at the Second Harmonics

Using the method of averaged Lagrangian and diffraction approximation, an approximate analytical solution is obtained and analyzed, which describes the propagation of two-color vortex light bullets in a quadratic-nonlinear graded-index waveguide.

Mode Structure and the Envelope of a High-Power Pulse in an Asymmetric Graded-Index Waveguide Layer

Mode Structure and the Envelope of a High-Power Pulse in an Asymmetric Graded-Index Waveguide Layer

Electromagnetic waves in graded-index planar waveguides

The propagation of guided TE and TM modes through graded-index planar waveguides is reported in this paper. Both a real-valued and complex-valued position-dependent refractive index are supposed for a film layer. It is possible to set various refractive index profiles based on five distribution parameters. For the position-dependent refractive index, the governing equations are transformed to Heun’s differential equation, an exact local solution expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The calculation of guided modes requires evaluation of the general solution in the interval containing two regular singular points. For this purpose, the generalized Heun function is introduced and employed in general solutions to the governing equations. The applicability of the generalized solutions is demonstrated by the calculation of guided modes for both the real-valued and complex-valued refractive index.

Snake—Modulation Dynamics of an Optical—Terahertz Soliton in a Graded-Index Waveguide

A nonlinear stage of the effect of snake and modulation instabilities on the dynamics of an optical—terahertz soliton in a quadratically nonlinear graded-index waveguide has been studied theoretically. It has been shown that both instabilities are of fundamental importance and cannot be separated from each other. If the carrier frequency of the optical component is in the region of an anomalous dispersion of the group velocity, these instabilities are developed in a blow-up regime, leading to the self-focusing of the soliton. In the case of the normal dispersion of the group velocity, the mutual effect of the waveguide and snake—modulation dynamics results in the formation of a stable spatiotemporal soliton.

On-chip arbitrary-mode spot size conversion

Abstract Manipulating on-chip optical modes via components in analogy with free-space devices provides intuitional light control, and this concept has been adopted to implement single-lens–assisted spot size conversion using integrated device. However, the reported schemes have been demonstrated only for fundamental mode, while high-order or irregular modes are preferred in specific applications. The 4-f system is widely used in Fourier optics for optical information processing. Under the inspiration of the 4-f system and the beam expander in bulk optics, a spot size converter (SSC) with two metamaterial-based graded-index waveguides is proposed and demonstrated. The proposed device is capable of widening an arbitrary mode while preserving its profile shape. Compared with conventional SSC using adiabatic taper, the footprint can be reduced by 91.5% under a same intermode crosstalk. Experimentally, an expansion ratio of five is demonstrated for regular modes. Furthermore, for an irregular mode, the functionality is numerically verified without structure modification. This work offers a universal solution to on-chip spot size conversion and may broaden the on-chip application prospects of Fourier optics.

Two-color ‘dancing’ light bullets in the graded-index waveguide

The propagation of a spatiotemporal soliton in a focusing quadratic-nonlinear waveguide is theoretically investigated. This soliton is a bound state of localized bunches of light energy at the fundamental frequency and at its second harmonics. It is shown that with anisotropic spatial distribution of the refractive index in the cross section of the waveguide, the soliton trajectory can be a spatial Lissajous figure. Such spatiotemporal solitons are called two-color ‘dancing’ light bullets. Unlike dancing light bullets in a waveguide with Kerr nonlinearity, the stability of the dancing bullets discussed here does not require additional restrictions on their apertures, powers, and temporal durations. In addition, a two-color dancing light bullet can be formed both under anomalous and normal dispersion of group velocity. It also distinguishes a two-color dancing light bullet in a quadratic nonlinear waveguide from a dancing light bullet in a Kerr nonlinear waveguide.

Dark–dark and bright–dark solitons for a (2+1)-dimensional variable-coefficient nonlinear Schrödinger system in a graded-index waveguide

In this letter, we study a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient nonlinear Schrödinger system, which describes an optical beam inside the two-dimensional graded-index waveguide with polarization effects. Through the Kadomtsev–Petviashvili hierarchy reduction, the [Formula: see text] dark–dark soliton and [Formula: see text] bright-dark soliton solutions in terms of the Gramian are obtained, where [Formula: see text] is a positive integer. We analyze the interaction and propagation of the dark–dark solitons graphically. With the different values of the diffraction coefficient [Formula: see text], periodic-, cubic- and parabolic-shaped dark–dark solitons are derived. With the different values of the gain/loss coefficient [Formula: see text], periodic- and arctangent-profile background waves are obtained. Moreover, we discuss the effects from the dimensionless beam width [Formula: see text], [Formula: see text] and [Formula: see text] on the solitons and background waves: Shapes of the solitons are affected by [Formula: see text] and [Formula: see text], while profiles of the background waves are affected by [Formula: see text] and [Formula: see text].

Configurational entropy of optical bright similariton in tapered graded-index waveguide

Configurational Entropy (CE) consists of a family of entropic measures of information used to describe the shape complexity of spatially-localized functions with respect to a set of parameters. We obtain the Differential Configurational Entropy (DCE) for similariton waves traveling in tapered graded-index optical waveguides modeled by a generalized nonlinear Schrödinger equation. It is found that for similariton's widths lying within a certain range, DCE attains minimum saturation values as the nonlinear wave evolves along the effective propagation variable ζ(t). In particular, saturation is achieved earlier for lower values of the width, which we show correspond to global minima of the DCE. Such low entropic values lead to minimum dispersion of momentum modes as the similariton waves propagate along tapered graded-index waveguides, and should be of importance in guiding their design.

Self-frequency shift effect for chirped self-similar solitons in a tapered graded-indexed waveguide

This work studies the transmission dynamics of self-similar optical beams through an inhomogeneous tapered graded-index waveguide under diffraction and nonlinearity management as well as self-frequency shift arising from stimulated Raman scattering (SRS). A similarity transformation connecting with the constant-coefficient higher-order nonlinear Schrö dinger’s equation (HNLSE) is employed to construct new self-similar solitons for wave field. We demonstrate that the self-frequency shift term alone turns the self-similar pulses to be nonlinear chirped. The results indicate that the nonlinear chirp related to the self-similar localized waves is directly proportional to the intensity of the optical beam. The necessary and sufficient conditions on tapered graded-index nonlinear waveguide parameters for such nonlinearly chirped self-similar localized waves exist in this model are suggested. The dynamical behaviors of those chirped optical similaritons are also investigated in a periodically distributed waveguide as well as a diffraction decreasing waveguide.

Low-Loss Multimode Glass Waveguides With Beam-Expanded Fiber Connectors Enabling On-Board Optical Links

Ion-exchange glass waveguides are a very promising technology with a balanced trade-off between optical loss, integration density, and fabrication costs. Graded-index multimode waveguides show low propagation loss of only 0.035 dB/cm at wavelength of 1310 nm in glass, compatible with large scale manufacturing. For fiber cable connectivity, expanded beam connectors were used to terminate the waveguides by passive pick-and-place assembly, resulting in a coupling loss of ∼1 dB per interface. Design and fabrication of glass waveguides are studied, and 2-m-long spiral waveguides along with 12 × 12 optical shuffle waveguide layouts made on a 9′′ × 12′′ area glass sheet are demonstrated experimentally.

Laser-Driven Phase Segregation and Tailoring of Compositionally Graded Microstructures in Si–Ge Nanoscale Thin Films

The ability to manipulate the composition of semiconductor alloys on demand and at nanometer-scale resolutions is a powerful tool that could be exploited to tune key properties such as the electronic band gap, mobility, and refractive index. However, existing methods to modify the composition involve altering the stoichiometry by temporal or spatial modulation of the process parameters during material growth, limiting the scalability and flexibility for device fabrication. Here, we report a laser processing method for localized tailoring of the composition in amorphous silicon-germanium (a-SiGe) nanoscale thin films on silicon substrates, postdeposition, by controlling phase segregation through the scan speed of the laser-induced molten zone. Laser-driven phase segregation at speeds adjustable from 0.1 to 100 mm s-1 allows access to previously unexplored solidification dynamics. The steady-state spatial distribution of the alloy constituents can be tuned directly by setting the laser scan speed constant to achieve indefinitely long Si1-xGex microstructures, exhibiting the full range of compositions (0 < x < 1). To illustrate the potential, we demonstrate a photodetection application by exploiting the laser-written polycrystalline SiGe microstripes, showing tunability of the optical absorption edge over a wavelength range of 200 nm. Our method can be applied to pseudobinary alloys of ternary semiconductors, metals, ceramics, and organic crystals, which have phase diagrams similar to those of SiGe alloys. This study opens a route for direct laser writing of novel devices made of alloy microstructures with tunable composition profiles, including graded-index waveguides and metasurfaces, multispectral photodetectors, full-spectrum solar cells, and lateral heterostructures.

Modeling multivariate dependencies for manufacturing single-mode I/O structures of integrated MMI-based splitters in glass sheets

For the realization of integrated optical interconnect systems, optical waveguide components are needed. A promising approach to manufacturing graded-index waveguide components is their embedding in thin glass sheets by ion-exchange processes. Conventionally, the modeling of integrated waveguide components manufactured by thermal diffusion processes involves the numerical solution of differential equations. In this work, a semi-analytical approach for the modeling of the concentration profile is extended by the Gauss–Newton optimization algorithm. This is done with the specific purpose of accurately modeling single-mode input and output waveguide structures of multimode-interference-based (MMI-based) optical splitters. The constraints imposed by the multivariate manufacturing process for the realization of these MMI-based components are elaborated. Corresponding correlations between these constraints and the according modeling of the devices are established.

Different complex wave structures described by the Hirota equation with variable coefficients in inhomogeneous optical fibers

The main concern of this paper is to investigate the Hirota equation with variable coefficients that can describe the pulse propagation in inhomogeneous fibers more realistically than its constant coefficients counterparts. A variety of complex wave solutions including new exact solutions, bright and dark soliton solutions, and similarity solutions are retrieved through $$G'/G$$ -expansion method with the aid of symbolic computation. The dynamical behaviors of the obtained solutions are illustrated using 3D- and corresponding contour plots in a graded-index waveguide.

Snakelike similaritons in tapered grating dual-core waveguide amplifiers

We present a physical model that describes bright and dark snakelike similaritons propagating in an asymmetric dual-core waveguide amplifier by considering self-focusing and self-defocusing Kerr nonlinearities. The asymmetric dual-core waveguide is composed of two adjoining, closely spaced, upper and lower cores, in which the upper one is an active tapered graded-index waveguide with a long-period grating along the medium while the lower one is a passive step-index waveguide. For different choices of tapered index profile, novel exact analytical similariton solutions undergoing a self-compression are obtained. Our results show that these optical structures can be effectively controlled by adjusting the long-period grating and amplitude of the source. Besides, the stability and interaction of two neighboring bright similaritons are discussed by means of numerical simulations. These results may simulate the possibility of experiments related to tapered grating dual-core waveguide amplifiers.

Snakelike self-similar solutions in a graded-index grating waveguide amplifier with an external source

We present a series of snakelike self-similar solutions to the nonlinear Schrodinger equation with a gain in the presence of an external source through asymmetric dual-core waveguide amplifiers with a long-period grating. The asymmetric dual-core waveguide is composed of two adjoining, closely spaced, upper and lower waveguides, in which the upper one is an active graded-index waveguide with a long-period grating along the medium, while the lower one is a passive step-index waveguide. The results show that: (1) the self-similar optical structures can be generated and effectively controlled by modulating the amplitude of the source; (2) the long-period grating can change the motion of the solutions and preserve their shapes. These results may raise the possibility of some experiments and potential applications in the graded-index grating waveguide amplifier with an external source.

One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada–Kotera equation

The main concern of the present article is to study the fifth-order variable-coefficient Sawada–Kotera (VcSK) equation which describes the motion of long waves in shallow water under the gravity. A single- and double-soliton rational solutions for this model are formally retrieved through the generalized unified method. For a single-soliton wave, the velocity, the amplitude and the shape of the wave cannot be affected by variable coefficients. There is an inelastic collision (the collision that makes change in amplitude of the soliton waves and shifts in their trajectories) between the double-soliton waves due to the time-varying field in a graded-index waveguide. It hoped that the established solutions can be used to enrich the dynamic behaviors of the VcSK equation.

Photonic potential for TM waves.

We discuss the effective photonic potential for TM waves in inhomogeneous isotropic media. The model provides an easy and intuitive comprehension of form birefringence, paving the way for a new approach on the design of graded-index optical waveguides on nanometric scales. We investigate the application to nanophotonic devices, including integrated nanoscale wave plates and slot waveguides.

Evaluation of low index contrast in lithium niobate waveguides at telecom wavelengths

For a wide range of applications in modern photonics, the low index contrast lithium niobate waveguides still offer the most established solutions to integrate a number of functions on a single optical chip. These waveguides are mostly single-mode at telecom wavelengths and knowing their index contrast (IC) in this spectral range is not straightforward. Unfortunately, the single mode behavior of the propagation at telecom wavelength makes impossible to use the M-lines technique which allows retrieving the IC only if the waveguide is multi-mode. Here, we propose and test a new hybrid (experimental and numerical) method to determine the IC at 1310 nm and 1550 nm for typical low-IC lithium niobate waveguides by fitting the experimental in-depth optical mode profiles with the Finite Element Method simulated ones. Then, by using different versions of Sellmeier equation, we reconstruct the dependence of IC on the wavelength in VIS and near-infrared (NIR) range. This hybrid method can be of use in the analysis of any surface graded-index waveguides fabricated in low-IC range.