Secant Profile Research Articles

Structurally embedded gradient index lens for guided wave amplification in polymers

Guided wave inspection is a commonly used technique in structural health monitoring, yet there are challenges associated with ultrasonic inspection of polymeric structures due to high material damping resulting in wave attenuation. In this work, we present a structurally embedded gradient-index phononic crystal (GRIN-PC) lens for elastic wave amplification in polyvinyl chloride (PVC) structures to overcome the attenuation problem in long-range guided wave inspection. The GRIN-PC lens is made up of cylindrical brass stubs of varying heights or diameters embedded in the PVC base structure, providing a hyperbolic secant profile of refractive index distribution transverse to the wave propagation direction. We first demonstrate flexural wave focusing in a PVC plate by implementing the embedded lens design via numerical simulations and experiments. Then, the embedded GRIN-PC lens concept is implemented to PVC pipe for multi-mode wave focusing using a single lens design which is studied through numerical simulations. The proposed GRIN-PC lens design reduces the amplitude decay due to material damping by amplifying the wave energy at the focal regions of the lens. The embedded lens concept developed for polymers can also benefit other applications requiring the transmission of wave energy in structures with high material damping.

Conformal gradient-index phononic crystal lens for ultrasonic wave focusing in pipe-like structures

We explore a conformal gradient-index phononic crystal lens integrated within a pipe to amplify guided wave modes toward improved ultrasonic inspection of pipelines. The proposed conformal lens is composed of an array of cylindrical steel stubs attached to the outer surface of a steel pipe, which are tailored according to the hyperbolic secant profile of refractive index in the circumferential direction of the pipe. Hence, the ultrasonic guided wave energy is focused in the axial direction of the pipe and amplified at the focal point of the lens. Refractive indices are calculated using dispersion curves obtained from the finite element simulations of the stubbed unit cells, and the curved lens is designed for the second longitudinal wave mode of the pipe, which is commonly used in guided wave testing. The proposed lens design is implemented on a steel pipe, which is typically used in the distribution networks utilized in cities, and simultaneous focusing of longitudinal wave modes in a broad frequency range is verified through both numerical models and experimental measurements.

Multi-branch subwavelength focusing of the lowest-order antisymmetric Lamb mode in a gradient-index phononic crystal

We demonstrate a metasurface design of a multi-branch dispersion that can converge plane wave into a focal region at sub-wavelength scale. The design is based on overlapping the periodic array of steel pillars, whose role is to obtain non trivial multi-branch dispersion for the lowest-order antisymmetric Lamb (A0) mode, upon the periodic array of air holes in an aluminum plate. The radii of air holes are gradually modulated along the transverse direction to impose the hyperbolic secant profile for the refractive index. The air-coupled method and laser ultrasonic technique are applied to excite the flexural wave and to map the out-of-plane displacement within the gradient-index (GRIN) lens. The multi-branch focusing is observed within two different frequency ranges and its experimental character shows excellent agreement with the numerical results.

Phononic crystal lens with an asymmetric scatterer

A novel 2D gradient index phononic crystal (GRIN PC) lens was designed by rotating scatterers that theoretically differs from the traditional GRIN PC lens by altering fill fraction, lattice constant or material parameters. Three-petal epoxy cylinders were designed as scatterers and embedded into a homogenous steel matrix. Rotation of the three-petal cylinder results in an asymmetry about x axis, and there is a change in the first dispersion band which is related to group velocity and the refractive index. The equivalent refraction index along the y direction was modulated to satisfy the distribution of a hyperbolic secant profile using a specific rotation angle of the scatterer. The focus of the acoustic wave in the PC lens with a rotating three-petal scatterer was investigated using a numerical method, and the results showed that the focus length agreed with the trajectories derived from the analytical solution. Results also showed that the range of the operation reduced frequency is 0.18. The influence of the operation frequency on the focusing efficiency and focal length was also discussed. The work provides a convenient method for fabricating a phononic crystal lens and makes the lens suitable for relevant applications.

Broadband focusing of underwater sound using a transparent pentamode lens.

An inhomogeneous acoustic metamaterial lens based on spatial variation of refractive index for broadband focusing of underwater sound is reported. The index gradient follows a modified hyperbolic secant profile designed to reduce aberration and suppress side lobes. The gradient index (GRIN) lens is comprised of transversely isotropic hexagonal microstructures with tunable quasi-static bulk modulus and mass density. In addition, the unit cells are impedance-matched to water and have in-plane shear modulus negligible compared to the effective bulk modulus. The flat GRIN lens is fabricated by cutting hexagonal centimeter scale hollow microstructures in aluminum plates, which are then stacked and sealed from the exterior water. Broadband focusing effects are observed within the homogenization regime of the lattice in both finite element simulations and underwater measurements (20-40 kHz). This design approach has potential applications in medical ultrasound imaging and underwater acoustic communications.

Experimental validation of the underwater sound focusing properties of a pentamode gradient index metamaterial lens

A gradient index (GRIN) acoustic lens is a material slab with spatially varying properties that refracts an incident acoustic wave to produce a desired transmitted field. Hyperbolic secant index profiles have widely been used to create focusing lenses with minimal aberration and good sidelobe suppression. This work reports on the underwater experimental characterization of a GRIN lens that employs a modified secant profile for improved aberration reduction when compared to the hyperbolic secant profile. The index profile is produced using a two-dimensional pentamode material created from air-filled aluminum hexagonal unit cells. The lens is fabricated from waterjet cut aluminum plates that are stacked and sealed together. The test apparatus is comprised of a custom plane-wave source, an automated horizontal planar scanner, a pentamode GRIN lens, a custom support structure, and a calibrated hydrophone and associated electronics. The source was tested for amplitude planarity across its aperture at a constant distance to confirm its usability for this system. Experimental results of lens performance over the frequency range of 10-40 kHz are compared with the isolated source and model predictions. [Work supported by ONR.]

A transparent broadband metallic lens for focusing underwater sound

Diffractive lenses usually suffer from spherical aberration. Even though it is possible to minimize aberration by design, the quality of focused sound is still limited owing to the side lobes caused by aperture diffraction. In recent years, hyperbolic secant index profiles have been widely used in gradient index (GRIN) metamaterials design to reduce aberration and suppress side lobes. Here, we introduce a 1D transformation to modify the secant profile for further aberration reduction, and present a GRIN lens comprised of hexagonal unit cells obeying the modified profile for focusing underwater sound in the near field. The transversely isotropic unit cells are designed with water-like acoustic properties using foam mechanics and optimized using a homogenization technique based on finite element method. The in-plane shear modulus of the unit cells are minimized to suppress the shear modes in the GRIN lens. Moreover, the impedances of the unit cells are tuned close to water, such that most of the energy carried by the normally incident plane wave propagate across the interface between the lens and water. This lens can focus underwater sound with high efficiency over a broadband (10 kHz—40 kHz).

Conversion of sound radiation pattern via gradient acoustic metasurface with space-coiling structure

We designed, fabricated, and experimentally demonstrated a gradient acoustic metasurface to manipulate sound radiation patterns. The gradient metasurface is constructed on the basis of a coiling-up space in a tunable interdigitated structure, which exhibits relative refractive index in a discretized classic hyperbolic secant profile. Capable of generating secondary sound sources with desired gradient phase shifts, the metasurface shows the ability of controlling sound radiation such as by cylindrical-to-plane-wave conversion, plane wave focusing, and effective tunable acoustic negative refraction. Owing to its deep-subwavelength thickness, the metasurface may reduce the size of acoustic devices and offer potential applications in imaging and scanning systems.

Analog-to-digital optical waveguide conversion at sampling periods greater than the free-space wavelength

Nyquist sampling theorem reveals the possibility of sampling the continuous refractive index profiles of optical waveguides at periods greater than the free-space wavelength, λ(o). Binary encoding of these analog waveguides is investigated using the zero-order effective medium theory, while conserving the quantization of the modal spectrum implied by their boundary conditions. Both analytical and numerical approaches are developed for this analog-to-digital (A-to-D) conversion. An example is presented for the A-to-D conversion of a graded index waveguide with a hyperbolic secant profile at a sample period of 1.3λ(o). The results are confirmed using a beam propagation method.

High-efficiency beam bending using graded photonic crystals

We explore beam-bending properties of graded index (GRIN) waveguide with hyperbolic secant profile. The transmission efficiency and bandwidth features are extracted for GRIN photonic crystal (PC) media composed of dielectric rods. Light guiding performance of the GRIN PC medium is analyzed for 90° and 180° waveguide bends. The finite-difference time-domain method is deployed to investigate the performance of the designed GRIN waveguides. By the help of proposed photonic configuration, bending of light is achieved with a high efficiency within a broad bandwidth, which promotes the use of GRIN PC structures for efficient light-bending purposes.

Propagating solitons generated by localized perturbations on the surface of deep water

We report experimental evidence of the propagating capillary solitary waves generated by a localized water surface perturbation. Solitons are formed at the air-water interface with a solid plunger and then propagate with velocities proportional to their amplitudes. The shape of the forward front of such solitons is independent of the plunger shape and is given by the hyperbolic secant profile. We find that the nonlinearity of capillary waves is responsible for a small dispersion in the capillary wave group velocity and the soliton existence. This dispersion can be further reduced by adding minute concentrations of proteins to water.

Propagation of quasisolitons in a fiber Bragg grating written in a slow saturable fiber amplifier

We show, by using numerical simulations, that quasisolitons can propagate over a long distance in a fiber Bragg grating that is written in a slow saturable fiber amplifier, such as an erbium-doped fiber amplifier. During the pulse propagation, the front end of the pulse experiences a net gain while the rear end of pulse is attenuated due to the combination of gain saturation and loss. However, the pulse profile almost does not change after propagating over a length of 5 m that is approximately 2500 times larger than the spatial pulse width. The pulse amplitude has an approximately hyperbolic secant profile. We develop a reduced model by using a multiscale analysis to study solitary-wave propagation when nonlinearity and gain are small. When gain saturation also becomes small we find analytically a new family of solitary-wave hyperbolic-secant solutions that approximately solve the reduced model. The solitary waves propagate slightly faster than Bragg solitons that propagate in fiber Bragg gratings without gain and loss.

Gradient-index phononic crystals

We designed and characterized a two-dimensional, gradient-index phononic crystal (GRIN PC) to control the propagation of acoustic waves. The GRIN PC was composed of solid cylinders arranged in a square lattice and immersed in an epoxy. The refractive index along the direction transverse to the phononic propagation was designated as a hyperbolic secant gradient distribution. This distribution was modulated by means of the density and elastic moduli of the cylinders. The effective refractive indices in each row of the GRIN PC were determined from band diagrams obtained via a plane-wave expansion method. The acoustic wave propagation was numerically investigated by a finite-difference time-domain method, and the results were compared to the analytical beam trajectories derived from the hyperbolic secant profile. These results show that the GRIN PC allows acoustic focusing over a wide range of working frequencies, making it suitable for applications such as flat acoustic lenses and couplers.

Analysis of planar waveguides using wentzel-kramers-brillouin-type methods

A presentation of four different schemes of semiclassical or Wentzel-Kramers-Brillouin-type approximations to planar waveguides is given. Attention is paid to a recent proposal for mapping a general symmetric waveguide into the hyperbolic secant or symmetric Epstein profile by exploring the dispersion relation and the fundamental mode field. As a test, these approximations are compared with the series solution of the cladded-parabolic profile.

Design and optimisation of high-brightness 980 nm laser diodes with distributed phase correction

Designs for 980 nm AlxGa1−xAs/In0.2Ga0.8As/GaAs high-power, high-brightness semiconductor lasers/amplifiers with distributed phase correction and short-cavity lengths (i.e. cavity lengths 560–1040 μm) are presented. The proposed lasers/amplifiers employ a single mode feed waveguide coupled to a power amplifier with a laterally graded effective refractive index (GRIN) profile to control the lateral mode shape and phase. The lateral index of the power amplifier has a hyperbolic secant (HYSEC) profile, which can be approximately realised by tailoring the effective refractive index of the amplifier using a series of discrete etches. The epitaxial and structure designs of the laterally discretised GRIN lasers/amplifiers are presented. Finally, a method for improving the effective refractive index discretisation is described.

Self-induced transparency solitary waves in a doped nonlinear photonic band gap material

We derive the properties of self-induced transparency (SIT) solitary waves in a one-dimensional periodic structure doped uniformly with resonance two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. The resulting solitary waves take the form of ultrashort (picosecond) laser pulses which propagate near the band edge of the nonlinear photonic band gap (PBG) material doped with rare-earth atoms such as erbium. Solitary wave formation involves the combined effects of group velocity dispersion (GVD), nonresonant Kerr nonlinearity, and resonant interaction with dopant atoms. We derive the general Maxwell-Bloch equations for a nonlinear PBG system and then demonstrate the existence of elementary solitary wave solutions for frequencies far outside the gap where GVD effects are negligible and for frequencies near the photonic band edge where GVD effects are crucial. We find two distinct new types of propagating SIT solitary wave pulses. Far from Bragg resonance, we recapture the usual McCall-Hahn soliton with hyperbolic secant profile when the nonlinear Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ However, when the host nonresonant Kerr coefficient is nonzero, we obtain the first new type of soliton. In this case, the optical soliton envelope function deviates from the hyperbolic secant profile and pulse propagation requires nontrivial phase modulation (chirping). We derive the dependence of the solitary wave structure on the Kerr coefficient ${\ensuremath{\chi}}^{(3)},$ the resonance impurity atom density, and the detuning of the average laser frequency from the atomic transition. When the laser frequency and the atomic transition frequencies are near the photonic band edge we obtain the second type of soliton. To illustrate the second type of soliton we consider two special cases. In the first case, GVD facilitates the propagation of an unchirped SIT-gap soliton moving at a velocity fixed by the material's parameters. The soliton structure changes dramatically as the laser frequency is tuned through the atomic resonance. In the second illustrative case we set the Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ In this case, the solution is a chirped pulse which arises from the balance between GVD and the resonance interaction with the dopant atoms. Finally, we show that under certain circumstances, these solitary wave solutions may persist even in the presence of (subpicosecond) dipolar dephasing of the dopant atoms and absorption losses of the host PBG material, provided that the system is incoherently pumped. These results may be relevant to the application of PBG materials as optical devices in fiber-optic networks.

Gaussian beam propagation and imaging in hyperbolic secant profile media

The propagation of Gaussian beams in GRIN media characterized by the hyperbolic secant (HS) index profile is studied. It is shown that the profile can contract, expand or maintain the width of a beam incident with the beam waist at the input end of the medium. We derive an accurate analytical expression for the mode width of the fundamental mode of a HS medium. The imaging characteristics of the medium are also studied and compared to selfoc media. It is shown that, in the appropriate limit, the expressions for the HS profile reduce to those of a selfoc medium.

Exact envelope wave solution to nonlinear Schrödinger equation with dissipative term

In this paper, we consider the evolution of a soliton when dissipative loss exists. By means of non-perturbed method, an exact envelope wave solution of nonlinear Schrodinger equation with dissipative term is obtained. It is shown that when Γ = γ0/(1 + 2γ0 t), the solution given here still maintains the hyperbolic secant profile.

Generation of ultrahigh repetition rate soliton trains in fibre ring

A method for the generation of soliton trains at ultrahigh repetition rates is proposed. The method is based on the dual-frequency pumping of a passive fibre ring cavity. At the output port of the cavity, a train of independent pulses with hyperbolic secant profile is emitted.

An Analysis of Three-Dimensional Mountain Lee Waves in a Stratified Shear Flow

A three-dimensional, linear, hydrostatic model of stationary mountain waves in a stably stratified airstream is considered. A general solution is presented for the case of a cross-wind variation of the basic current represented by the hyperbolic secant profile; both the current and the Brunt-Väisälä frequency are independent of height. Particular solutions are presented for a variety of hill shapes. The pressure pattern and distribution of velocity at ground level are similar in many respects to the fields obtained by Smith (1980) for constant flow over a circular hill. High pressure occurs on the windward side with relatively low pressure in the lee, and the oncoming airstream tends to partially circumvent the hill resulting in a permanent streamline deflection in the lee. The flow field aloft is characterized by an upstream tilt of the phase lines, associated with outward vertical propagation of wave energy. Wave energy also is directed downstream but is totally reflected in the cross-stream direction as a consequence of the cross-stream variation of the basic current. Thus, the wave disturbance is primarily contained within a horizontal strip, at each level, that is somewhat larger than the characteristic breadth of each hill shape. This latter response is significantly different than the typical “ship wave” pattern that occurs when a constant flow passes over a hill. Some prominent features of the wave response aloft, as a function of the characteristic hill shape used to represent orography, are examined by means of an asymptotic representation of the solution.