Plane Wave Spectrum Research Articles

Design and fabrication of crossed gratings with multiple zero-reference marks for planar encoders

We design and fabricate a new type of crossed gratings with multiple zero-reference marks for planar encoders. The crossed grating together with an indicator can output multiple zero-reference positions along both the x and y directions for 2D displacement measurement. The zero-reference marks are designed on basis of the plane-wave angular spectrum analysis including the diffraction effect. A crossed grating with multiple zero-reference marks was fabricated by holographic exposure and contact ultraviolet exposure. The spacing error of the crossed grating was 0.03λ (λ = 632.8 nm) and the average diffraction efficiency of the crossed grating’s four principal nonzero symmetrical orders reached 14%. The measurement repeatability of the zero-reference position was 15.1 nm, while the zero-reference marks only reduced the average diffraction efficiency of the crossed grating by 0.55%.

Scattering by a Dielectric Sphere Buried in a Half-Space With a Slightly Rough Interface

Analytical expressions for the scattering coefficients of a dielectric sphere buried under a rough interface are presented. The proposed method combines the small perturbation method (SPM) and the Mie solution by using the expansion of plane waves in terms of vector spherical functions (VSFs) and vice versa. First, using SPM, the zeroth- and the first-order perturbative scattered fields of a rough interface for illuminations from above and below are derived. Using these solutions, the field transmitted to the lower half-space is determined as a spectrum of down-going plane waves. The scattered fields from the sphere are then calculated using the vector Mie solution. Subsequently, the VSFs are expanded in terms of up-going plane waves. These plane waves illuminate the interface, and using SPM, the scattered fields in the upper and lower regions are determined as infinite summations of plane waves. The reflected plane waves are once again scattered by the sphere and the scenario repeats. By inspecting the form of the fields resulting from the few first interactions of the sphere and the rough interface, a recursive form is obtained for the scattered fields. This recursive form is then used to rewrite the system of equations in a form containing all interactions in a single-step formulation. Accordingly, the zeroth- and the first-order closed-form scattered fields are obtained. The derived expressions are analytically and numerically validated. Finally, the numerical results for the case of the rough interface with sinusoidal profile are presented and briefly discussed.

Processing an acoustic microscope’s spatiotemporal signal to determine the parameters of an isotropic layer

This paper presents a method for measuring the thickness and velocities of body waves and the density of an isotropic layer by a pulse scanning acoustic microscope. The method is based on recording the microscope signal as a function of the displacement magnitude of the focused ultrasonic transducer along its axis perpendicular to the sample surface and on the decomposition of the recorded 2D spatiotemporal signal into the spectrum of plane pulse waves. The velocities of the longitudinal and transverse waves and the layer’s thickness are calculated from the relative delays of the components of the spectrum of plane waves reflected from the surfaces of the layer and the density is computed by the amplitudes of these components. An experimental investigation of a test sample in the form of a glass plate carried out in the 50-MHz range shows that the error in measuring the thickness and velocities of body waves does not exceed 1% and the density measurement error does not exceed 10%.

STT Analysis of the Time-Dependent Reflected Field From a Moving Dielectric–Magnetic Planar Discontinuity

This paper is concerned with obtaining closed-form exact solution for the 2-D canonical problem of the reflected field from a moving planar discontinuity of a dielectric–magnetic medium that is excited by an impulsive line current. The spectral theory of transient (STT) that originally deals with nondispersive frequency-domain plane-wave spectra is used in this paper for addressing a time-variant scattering problem. The scattering from a moving object yields dispersive and anisotropic PW spectra. Nevertheless, we demonstrate that the STT is capable of dealing with these types of spectra and obtain the desired exact time-dependent solutions. The unique wave phenomena that are associated with the medium and scatterer velocity are explored.

Coherent field propagation between tilted planes.

Propagating electromagnetic light fields between nonparallel planes is of special importance, e.g., within the design of novel computer-generated holograms or the simulation of optical systems. In contrast to the extensively discussed evaluation between parallel planes, the diffraction-based propagation of light onto a tilted plane is more burdensome, since discrete fast Fourier transforms cannot be applied directly. In this work, we propose a quasi-fast algorithm (O(N3 log N)) that deals with this problem. Based on a proper decomposition into three rotations, the vectorial field distribution is calculated on a tilted plane using the spectrum of plane waves. The algorithm works on equidistant grids, so neither nonuniform Fourier transforms nor an explicit complex interpolation is necessary. The proposed algorithm is discussed in detail and applied to several examples of practical interest.

Simulation framework for compressive sensing‐based through‐wall detection of moving targets

Detection of moving targets in cluttered environments is important in numerous applications, such as disaster search-and-rescue, law enforcement, urban warfare and so on. One of the popular techniques employed for this purpose is the use of stepped-frequency continuous-wave radar due to its low cost and reduced complexity. However, the stepped-frequency radar suffers from long data acquisition time. This study focuses on the detection of moving targets in cluttered medium; such as behind walls, using stepped-frequency radar enhanced by compressive sensing (CS). The application of CS enables the reconstruction of the target space using fewer frequencies, which decreases the acquisition time. This study presents simulation results that demonstrate the applicability of CS to stepped-frequency radar measurements in accurately locating and tracking targets behind walls. The plane-wave spectrum approach is employed in conjunction with through-wall transmission coefficients for calculating the scattered fields of targets behind the wall. The received voltage at the antenna terminals is then computed by adopting the concept of complex antenna factor frequently used in Electromagnetic Compatibility (EMC) community, which fully takes into account the antenna effects, enabling an accurate simulation.

A Propagating Plane-Wave-Based Near-Field Transmission Equation for Antenna Gain Determination from Irregular Measurement Samples

In order to accurately determine the far-field gain of an antenna under test from near-field measurements, a general transmission equation between the ports of a transmitting and a receiving antenna is derived. The formulation is based on a propagating plane-wave expansion of the electric field and allows one to use a derivation based on far-field concepts separately for all entries of the plane-wave spectrum. It is shown that the equation is a generalization of the Friis transmission equation to the near-field case and the equation is compared with spherical multipole-based transmission equations for single and multiple antenna systems. The new transmission equation is integrated into the Fast Irregular Antenna Field Transformation Algorithm, which now allows one to determine the antenna gain pattern not only from canonical measurement surfaces as planes and spheres but also from highly irregular sample distributions. Examples using near-field data from both simulations and measurements are given to prove the validity of the formulation and to illustrate the capabilities of the implementation. In particular, it is also shown that accurate gain determination is possible from incomplete near-field measurements and that appropriate probe correction is important.

Angular scattering of an Airy beam light sheet by a concentric sphere

Angular distribution of scattered light by a particle reveals information of its topology and refractive index, which has been proved to be an effective method to identify and characterize the particle. In this paper, numerical study on angular scattering of a polarized one-dimensional Airy beam light sheet by concentric spheres is presented. Plane-wave spectrum method is adopted to simplify the problem into classic Mie scattering problem, which significantly improves computational efficiency. The effects of size parameters, refractive index and position of concentric spheres and Airy beam radius on the angular distribution of scattered light are investigated. This paper provides theoretical results for light scattering measurement where Airy beam light sheet is employed for reducing effects of multiple scattering.

Average scattered field from a random PEC cylinder buried below a slightly rough surface

Scattering from a perfect electric conducting cylinder with random radius buried below a half space dielectric homogenous interface is studied. The cylindrical wave scattered by cylinder is expanded in terms of plane wave spectrum. Small perturbation method is used to study the interaction of each plane wave with the interface. The zeroth order term yields solution for a flat interface, whereas scattering from a rough surface is given by first-order term. Results are obtained for both TM and TE polarizations. Analytical expressions of the average scattered field are obtained and verified using numerical evaluation. Different scattering scenarios are simulated by varying the distribution of the radius. It is observed that average scattering cross section of an ensemble with normal/uniform distribution is almost equal to that of a cylinder with mean radius.

Two dimensional Green's function for planar grounded dielectric layer in non-integer dimensional space

Two dimensional Green's function for grounded dielectric half space geometry in non-integer dimensional space has been derived. It has been assumed that both half spaces are of non-integer dimension. Radiated fields are written in terms of unknown plane wave spectrum. Unknown spectrum functions are determined using the boundary conditions. Special cases have also been addressed considering one or both the spaces are of integer dimensional space. The solution for both the spaces are of integer dimension is used to make a comparison with previously reported results. Results are reported for different values of physical and geometrical parameters.

Scattering of one-dimensional Airy beam light sheet with finite energy by a sphere.

A one-dimensional Airy beam light sheet has the advantage of nondiffraction and self-reconstruction after encountering particles. The angular scattering of an Airy beam by a homogeneous sphere is numerically studied in this paper using Mie theory and a plane-wave spectrum method. The effects of sphere radius, refractive index, beam width, and sphere position on the angular distribution of scattering light are simulated. Correlation coefficients are employed to investigate whether Airy beam scattering can be approximated to be plane-wave scattering for computational efficiency. The results can be applied in light scattering measurement that uses an Airy beam aiming at reducing multiple scattering.

Enhancing ultraviolet spontaneous emission with a designed quantum vacuum.

We determine how to alter the properties of the quantum vacuum at ultraviolet wavelengths to simultaneously enhance the spontaneous transition rates and the far field detection rate of quantum emitters. We find the response of several complex nanostructures in the 200 - 400 nm range, where many organic molecules have fluorescent responses, using an analytic decomposition of the electromagnetic response in terms of continuous spectra of plane waves and discrete sets of modes. Coupling a nanorod with an aluminum substrate gives decay rates up to 2.7 × 103 times larger than the decay rate in vacuum and enhancements of 824 for the far field emission into the entire upper semi-space and of 2.04 × 103 for emission within a cone with a 60° semi-angle. This effect is due to both an enhancement of the field at the emitter's position and a reshaping of the radiation patterns near mode resonances and cannot be obtained by replacing the aluminum substrate with a second nanoparticle or with a fused silica substrate. These large decay rates and far field enhancement factors will be very useful in the detection of fluorescence signals, as these resonances can be shifted by changing the dimensions of the nanorod. Moreover, these nanostructures have potential for nano-lasing because the Q factors of these resonances can reach 107.9, higher than the Q factors observed in nano-lasers.

Polarisers in the focal domain: Theoretical model and experimental validation

Polarisers are one of the most widely used devices in optical set-ups. They are commonly used with paraxial beams that propagate in the normal direction of the polariser plane. Nevertheless, the conventional projection character of these devices may change when the beam impinges a polariser with a certain angle of incidence. This effect is more noticeable if polarisers are used in optical systems with a high numerical aperture, because multiple angles of incidence have to be taken into account. Moreover, the non-transverse character of highly focused beams makes the problem more complex and strictly speaking, the Malus’ law does not apply. In this paper we develop a theoretical framework to explain how ideal polarisers affect the behavior of highly focused fields. In this model, the polarisers are considered as birefringent plates, and the vector behaviour of focused fields is described using the plane-wave angular spectrum approach. Experiments involving focused fields were conducted to verify the theoretical model and a satisfactory agreement between theoretical and experimental results was found.

Improved intercostal HIFU ablation using a phased array transducer based on Fermat's spiral and Voronoi tessellation: A numerical evaluation

A major complication for abdominal High Intensity Focused Ultrasound (HIFU) applications is the obstruction of the acoustic beam path by the thoracic cage, which absorbs and reflects the ultrasonic energy leading to undesired overheating of healthy tissues in the pre-focal area. Prior work has investigated the determination of optimized transducer apodization laws, which allow for a reduced rib exposure whilst (partially) restoring focal point intensity through power compensation. Although such methods provide an excellent means of reducing rib exposure, they generally increase the local energy density in the pre-focal area, which similarly can lead to undesired overheating. Therefore, this numerical study aimed at evaluating whether a novel transducer design could provide improvement for intercostal HIFU applications, in particular with respect to the pre-focal area. A combination of acoustic and thermal simulations was used to evaluate 2 mono-element transducers, 2 clinical phased array transducers, and 4 novel transducers based on Fermat's Spiral (FS), two of which were Voronoi-tessellated (VTFS). Binary apodizations were determined for the phased array transducers using a collision detection algorithm. A tissue geometry was modeled to represent an intercostal HIFU sonication in the liver at 30 and 50 mm behind the ribs, including subsequent layers of gel pad, skin, subcutaneous fat, muscle, and liver tissue. Acoustic simulations were then conducted using propagation of the angular spectrum of plane waves (ASPW). The results of these simulations were used to evaluate pre-focal intensity levels. Subsequently, a finite difference scheme based on the Pennes bioheat equation was used for thermal simulations. The results of these simulations were used to calculate both the energy density in the pre-focal skin, fat, and muscle layers, as well as the energy exposure of the ribs. The acoustic simulations showed that for a sonication in a single point without beamsteering, comparing the best performing clinical phased array in this study to an equivalent VTFS transducer, the maximum intensity in the focal point was increased from 19.0 to 27.0 W/mm2 for the sonication 30 mm behind the ribs, while the rib area exposed to ≥20 J/cm2 was reduced from 0.88 to 0.14 cm2 . For the sonication 50 mm behind the ribs, the maximum focal point intensity was increased from 13.4 to 21.5 W/mm2 , while the rib area exposed to ≥40 J/cm2 was lowered from 2.71 to 0.01 cm2 . The thermal simulations showed that for a circular sonication cell of 4 mm diameter in the transversal plane, sonication times for sonications 30/50 mm behind the ribs were reduced from 13.9 to 8.38 s/38.2 to 17.4 s, respectively. Energy density levels in the skin for these sonications were decreased from 5.28 to 2.22/9.45 to 3.78 J/mm2 . VTFS transducers are expected to provide improvement for intercostal HIFU applications compared to currently available clinical transducers, as they reduce both the energy density in the pre-focal zone and the energy exposure of the ribs. These characteristics allow for increasing either the re-sonication rate or the treatment volume per sonication.

Analysis of the influence of diattenuation on optical imaging system by using the theory of vector plane wave spectrum

In most of the researches of polarization aberration, the influence of diattenuation is not large enough to affect imaging quality evidently. However, the modulation transfer function decreases when optical elements with complex planar dielectric structures and low transmittance, such as beam-splitter and optical modulator, are introduced into an imaging system. In this paper, a vector optical model in Descartes coordinate system is proposed based on the concept of vector plane wave spectrum (VPWS). The results of calculation show that the VPWS model is consistent with Debye model. Compared with Debye vector diffraction integral, the VPWS method is more suitable to the description of the PA introduced by planar optical device with opaque mask, such as larger surface quantum-confined-stark-effect electro-absorption modulator, which is used to modulate the light collected by optical antenna of time-of-flight (TOF) depth system or modulating-retroreflector free-space-optical communication system. In order to simplify the calculation and obtain the conclusion of the change in imaging quality directly, the formula of optical transfer function is decomposed into three parts (TE component, TM component and the correlation of them) instead of polynomial expansion of pupil function. The influences of diattenuation on MTF is studied globally and locally in a range of cut-off frequency of optical imaging system (2NA/ ). Allowance of diattenuation is analysed by numerical calculation, and a mathematical expression is derived. The result shows that the change of diattenuation can be neglected when the spatial frequency v is less than 0.2NA/, and the range of allowance decreases with the increase of spatial frequency. According to numerical calculation shown in Fig.7 and the derived formulas (15) and (16), the ratios of reflection/transmission coefficient of s-light and p-light D should range respectively from 0.63 to 1.6(0.2NA/ v 0.8NA/) and from 0.9 to 1.11(v0.8NA/ ) when the MTF is required to be not less than 90% of the value in ideal diffraction-limited system. The range of allowance becomes larger gradually with the increase of angle n between the normal of optical interface n and the optical axis of imaging system z. If a polarization beam splitter is considered, D,n sin-1 NA should be greater than 1-3.

Basis for paraxial surface-plasmon-polariton packets

We present a theoretical framework for the study of surface-plasmon polariton (SPP) packets propagating along a lossy metal-dielectric interface within the paraxial approximation. Using a rigorous formulation based on the plane-wave spectrum formalism, we introduce a set of modes that constitute a complete basis set for the solutions of Maxwell's equations for a metal-dielectric interface in the paraxial approximation. The use of this set of modes allows us to fully analyze the evolution of the transversal structure of SPP packets beyond the single plane-wave approximation. As a paradigmatic example, we analyze the case of a Gaussian SPP mode, for which, exploiting the analogy with paraxial optical beams, we introduce a set of parameters that characterize its propagation.

Reflection and transmission of Laguerre-Gaussian beams in a dielectric slab

This paper considers the reflection and transmission characteristics of a Laguerre-Gaussian (LG) beam in a dielectric slab. The fields of the reflected and transmitted beams are described based on plane-wave angular spectrum representation. Using the generalized Fresnel amplitude reflectance and transmittance, the reflected and transmitted fields in each region are expressed. With the Taylor series approximation of reflectance and transmittance, the analytical expressions of the total reflected and transmitted fields in the input and output regions are derived. The effects of the beam-waist radius and topological charge on the reflected and transmitted field intensities are simulated and discussed in detail. The centroid shifts of the reflected beam are also presented. It is concluded that the distortion of the intensity distribution including the size of the intensity contour, is influenced by the beam-waist radius and the topological charge of the incident beam. The total intensity of the slab, in particular for the case of the transmitted field, is found to be distinguishable from the case of the single interface.

Multipole expansion of circularly symmetric Bessel beams of arbitrary order for scattering calculations

A rigorous, simple and efficient approach is derived in this paper for multipole expansion of a circularly symmetric Bessel beam. Different from the existing rigorous methods which are based on the plane wave spectrum of a Bessel beam, a straight-forward integral procedure is presented in a traditional way to obtain the analytical expressions of the expansion coefficients, also called beam shape coefficients (BSCs). The convergence and correctness of the BSCs are verified numerically in detail for both on-axis and off-axis cases. The results in this paper are useful in various analytical scattering theories, such as the generalized Lorenz-Mie theory and the Null-field method, when a Bessel beam is considered.

The Active Element Patterns of a Finite Array-Space Wave and Leaky Wave Perspective

The active element patterns (AEPs) of a finite array of parallel-plate waveguides are studied. It is shown that an excited element in array environment produces a number of space and leaky wave modes (but no surface wave mode) that can be expressed in terms of branch-cut integrals and residues, respectively, in the complex wavenumber plane. The dominant space wave mode has a very sharp radiation pattern with large phase variation near its peak causing constructive and destructive interferences with the primary pattern of the excited element within a small angular region. As a result, the AEP exhibits sharp dips that are very close to the local peaks. The number and the extent of excited modes that cause interference depend on the location of the excited element in a finite array. Consequently, the shape of the AEP changes with the element location. It is found that an array generates forward and backward waves on its aperture, and for element spacing less than one wavelength, the backward wave dominates. As a result, the AEP for the left-edge element has a dip at the left-hand side of the boresight and the beam tilts toward the left. For large element spacing, multiple dips may occur on both sides. The present analysis explains several interesting characteristics of finite and infinite arrays and very closely agrees with the plane wave spectrum model.

Inductive intrinsic localized modes in a one-dimensional nonlinear electric transmission line.

The experimental properties of intrinsic localized modes (ILMs) have long been compared with theoretical dynamical lattice models that make use of nonlinear onsite and/or nearest-neighbor intersite potentials. Here it is shown for a one-dimensional lumped electrical transmission line that a nonlinear inductive component in an otherwise linear parallel capacitor lattice makes possible a new kind of ILM outside the plane wave spectrum. To simplify the analysis, the nonlinear inductive current equations are transformed to flux transmission line equations with analog onsite hard potential nonlinearities. Approximate analytic results compare favorably with those obtained from a driven damped lattice model and with eigenvalue simulations. For this mono-element lattice, ILMs above the top of the plane wave spectrum are the result. We find that the current ILM is spatially compressed relative to the corresponding flux ILM. Finally, this study makes the connection between the dynamics of mass and force constant defects in the harmonic lattice and ILMs in a strongly anharmonic lattice.