Forbidden Directions Research Articles

Hyperbolic polaritonic crystals with configurable low-symmetry Bloch modes

Photonic crystals (PhCs) are a kind of artificial structures that can mold the flow of light at will. Polaritonic crystals (PoCs) made from polaritonic media offer a promising route to controlling nano-light at the subwavelength scale. Conventional bulk PhCs and recent van der Waals PoCs mainly show highly symmetric excitation of Bloch modes that closely rely on lattice orders. Here, we experimentally demonstrate a type of hyperbolic PoCs with configurable and low-symmetry deep-subwavelength Bloch modes that are robust against lattice rearrangement in certain directions. This is achieved by periodically perforating a natural crystal α-MoO3 that hosts in-plane hyperbolic phonon polaritons. The mode excitation and symmetry are controlled by the momentum matching between reciprocal lattice vectors and hyperbolic dispersions. We show that the Bloch modes and Bragg resonances of hyperbolic PoCs can be tuned through lattice scales and orientations while exhibiting robust properties immune to lattice rearrangement in the hyperbolic forbidden directions. Our findings provide insights into the physics of hyperbolic PoCs and expand the categories of PhCs, with potential applications in waveguiding, energy transfer, biosensing and quantum nano-optics.

Attitude maneuvers avoiding forbidden directions

Many space missions require the execution of large-angle attitude slews during which stringent pointing constraints must be satisfied. For example, the pointing direction of a space telescope must be kept away from directions to bright objects, maintaining a prescribed safety margin. In this paper we propose an open-loop attitude control algorithm which determines a rest-to-rest maneuver between prescribed attitudes while ensuring that any of an arbitrary number of body-fixed directions of light-sensitive instruments stays clear of any of an arbitrary number of space-fixed directions. The approach is based on an application of a version of Pontryagin’s Maximum Principle tailor-made for optimal control problems on Lie groups, and the pointing constraints are ensured by a judicious choice of the cost functional. The existence of up to three first integrals of the resulting system equations is established, depending on the number of light-sensitive and forbidden directions. These first integrals can be exploited in the numerical implementation of the attitude control algorithm, as is shown in the case of one light-sensitive and several forbidden directions. The results of the test cases presented confirm the applicability of the proposed algorithm.

Optimal attitude maneuvers in the presence of prohibited directions

AbstractWe consider the problem of realigning a space telescope from one observation target to the next in the presence of prohibited directions. More specifically, we want to steer the space telescope – modelled as a gyrostat – from an initial attitude at rest to a final attitude at rest within a fixed time interval. During the motion, the line of sight of the telescope must be kept away from forbidden directions towards bright objects like the sun, moon or earth due to power or thermal requirements. The kinematics of the spacecraft motion are governed by a differential equation on the rotation group SO(3). Treating the angular velocities as control variables, this equation takes the form of a controlled dynamical system. To ensure reorientation maneuvers satisfying these pointing constraints, we introduce a cost functional penalizing proximity of the line of sight of the telescope to any of the forbidden directions. Furthermore, we include penalty terms which provide a smooth motion of the satellite and ensure the execution of a rest‐to‐rest maneuver. The chosen cost functional is minimized over all possible trajectories of the controlled dynamical system between the prescribed initial and target attitudes, which leads to an optimal control problem on SO(3), which is solved by applying a version of Pontryagin's Maximum Principle tailor‐made for optimal control problems on Lie groups. Parametrizing SO(3) in terms of Cardan angles, the solution is formulated as a boundary‐value problem on Euclidean space and hence can be solved numerically by conventional methods. The existence of two first integrals is established and exploited to reduce the computational effort. The applicability of this approach is shown in concrete examples.

Enabling propagation of anisotropic polaritons along forbidden directions via a topological transition.

Polaritons with directional in-plane propagation and ultralow losses in van der Waals (vdW) crystals promise unprecedented manipulation of light at the nanoscale. However, these polaritons present a crucial limitation: their directional propagation is intrinsically determined by the crystal structure of the host material, imposing forbidden directions of propagation. Here, we demonstrate that directional polaritons (in-plane hyperbolic phonon polaritons) in a vdW crystal (α-phase molybdenum trioxide) can be directed along forbidden directions by inducing an optical topological transition, which emerges when the slab is placed on a substrate with a given negative permittivity (4H-silicon carbide). By visualizing the transition in real space, we observe exotic polaritonic states between mutually orthogonal hyperbolic regimes, which unveil the topological origin of the transition: a gap opening in the dispersion. This work provides insights into optical topological transitions in vdW crystals, which introduce a route to direct light at the nanoscale.

Quality Factor of Inhomogeneous Plane Waves

We first compare two apparently dissimilar expressions for the quality factor Q of inhomogeneous plane waves in isotropic viscoelastic media, where Q is defined as the result of an energy-balance relation. In this case, it is the ratio of twice the strain energy to the dissipated energy. Both expressions give the same Q for P-waves and Q for S-waves is independent of the inhomogeneity angle γ (isotropic media). Then, we consider the more general balance equation, which holds for anisotropic viscoelastic media, where anomalous behaviors are observed when γ exceeds some critical value (forbidden directions of propagation). This problem has already been solved analytically for SH-waves. Here, we consider the qP–qS case, which requires a numerical solution of the dispersion equation to obtain the wavenumber and attenuation factor, i.e., the real and imaginary parts of the wave vector, respectively. The forbidden directions appear when the phase velocity approaches a zero value. Generally, the phase velocity of homogeneous waves (γ = 0) exceeds that of inhomogeneous waves, while the latter show stronger attenuation (lower quality factor). In the vicinity of the forbidden directions, the opposite behavior may occur.

Dynamic Nonreciprocity in Loss-Compensated Piezophononic Media

In controlling sound, it is not trivial to engineer a device that can strongly break the reciprocity (symmetry under time reversal) of acoustic wave propagation. Despite significant progress, it is still difficult to obtain a large contrast ratio between the allowed (passing) and forbidden propagation directions, over a broad frequency range, without distortion or attenuation of the signal in the passing direction. The authors propose an approach, based on the acousto-electric effect in piezoelectric semiconductors, that could achieve all of these goals for acoustic waves in elastic solids, to the benefit of noise control, energy harvesting, and transducer technology in general.

Straight-line motion of classical point particles in a three-dimensional lattice

By means of the usual definition of inner product and of the Gauss condition on the sum of squares of three integers in number theory, it can be seen that there exist specific directions in continuous space in which a classical point particle moving in a three-dimensional lattice cannot propagate. When representing all directions for which propagation is possible as points on a unitary sphere, the forbidden directions appear as vacancies on this sphere. By means of a stereographic projection of the allowed direction, it is argued that propagation is not allowed for specific sets of points on the stereographic plane. The present work can be considered as an interdisciplinary lecture for advanced high-school students or to first-year college students.

Forbidden directions for the fracture of thin anisotropic sheets: an analogy with the Wulff plot.

It is often postulated that quasistatic cracks propagate along the direction allowing fracture for the lowest load. Nevertheless, this statement is debated, in particular for anisotropic materials. We performed tearing experiments in anisotropic brittle thin sheets that validate this principle in the case of weak anisotropy. We also predict the existence of forbidden directions and facets in strongly anisotropic materials, through an analogy with the description of equilibrium shapes in crystals. However, we observe cracks that do not necessarily follow the easiest direction but can select a harder direction, which is only locally more advantageous than neighboring paths. These results challenge the traditional description of fracture propagation, and we suggest a modified, less restrictive criterion compatible with our experimental observations.

Role of the inhomogeneity angle in anisotropic attenuation analysis

The inhomogeneity angle (the angle between the real and imaginary parts of the wave vector) is seldom taken into account in estimating attenuation coefficients from seismic data. Wave propagation through the subsurface, however, can result in relatively large inhomogeneity angles [Formula: see text], especially for models with significant attenuation contrasts across layer boundaries. Here we study the influence of the angle [Formula: see text] on phase and group attenuation in arbitrarily anisotropic media using the first-order perturbation theory verified by exact numerical modeling. Application of the spectral-ratio method to transmitted or reflected waves yields the normalized group attenuation coefficient [Formula: see text], which is responsible for amplitude decay along seismic rays. Our analytic solutions show that for a wide range of inhomogeneity angles, the coefficient [Formula: see text] is close to the normalized phase attenuation coefficient [Formula: see text] computed for [Formula: see text] [Formula: see text]. The coefficient[Formula: see text] can be inverted directly for the attenuation-anisotropy parameters, so no knowledge of the inhomogeneity angle is required for attenuation analysis of seismic data. This conclusion remains valid even for uncommonly high attenuation with the quality factor [Formula: see text] less than 10 and strong velocity and attenuation anisotropy. However, the relationship between group and phase attenuation coefficients becomes more complicated for relatively large inhomogeneity angles approaching so-called ‘‘forbidden directions.’’ We also demonstrate that the velocity function remains practically independent of attenuation for a wide range of small and moderate angles [Formula: see text]. In principle, estimation of the attenuation-anisotropy parameters from the coefficient [Formula: see text] requires computation of the phase angle, which depends on the anisotropic velocity field. For moderately anisotropic models, however, the difference between the phase and group directions should not significantly distort the results of attenuation analysis.

Plane waves in viscoelastic anisotropic media-II. Numerical examples

SUMMARY Properties of homogeneous and inhomogeneous plane waves propagating in unbounded viscoelastic anisotropic media in an arbitrarily specified direction are studied numerically. The analytical expressions and algebraic equations of the sixth degree derived in the companion paper (Paper I, this issue) are used for this purpose. Studied characteristics are phase velocity, amplitude decay along the propagation direction, attenuation, attenuation angle and polarization vector. Simple analytical expressions are used to describe the behaviour of SH plane waves propagating in a plane of symmetry of a viscoelastic anisotropic medium. Characteristics of P and S plane waves are determined by solving a complex-valued algebraic equation of the sixth degree. Combined effects of attenuation and anisotropy are illustrated on several models of viscoelastic anisotropic media. The explanation of the so-called forbidden directions known from the directional specification of the slowness vector is given. Non-physical results related to forbidden directions do not arise in the algorithm used.

Directed diffraction without negative refraction

Using the FDTD method, we investigate the electromagnetic propagation in two-dimensional photonic crystals, formed by parallel air cylinders in a dielectric medium. The corresponding frequency band structure is computed using the standard plane-wave expansion method. It is shown that within partial bandgaps, waves tend to bend away from the forbidden directions. This phenomenon perhaps need not be explained in terms of negative refraction or `superlensing' behavior, contrast to what has been conjectured.

Guiding optical flows by photonic crystal slabs made of dielectric cylinders.

We investigate the electromagnetic propagation in two-dimensional photonic crystals, formed by parallel dielectric cylinders embedded in a uniform medium. The frequency band structure is computed using the standard plane-wave expansion method, while the propagation and scattering of the electromagnetic waves are calculated by the multiple scattering theory. It is shown that within partial band gaps, the waves tend to bend away from the forbidden directions. Such a property may render novel applications in manipulating optical flows. In addition, the relevance with the imaging by flat photonic crystal slabs will also be discussed.

Dynamics of a faceted nematic-smectic-B front in thin-sample directional solidification.

We present an experimental study of the directional-solidification patterns of a nematic-smectic-B front. The chosen system is C4H9-(C6H10)2CN (in short, CCH4) in 12 microm-thick samples, and in the planar configuration (director parallel to the plane of the sample). The nematic-smectic-B interface presents a facet in one direction-the direction parallel to the smectic layers--and is otherwise rough and devoid of forbidden directions. We measure the Mullins-Sekerka instability threshold and establish the morphology diagram of the system as a function of the solidification rate V and the angle straight theta(0) between the facet and the isotherms. We focus on the phenomena occurring immediately above the instability threshold when straight theta(0) is neither very small nor close to 90 degrees. Under these conditions, we observe drifting shallow cells and a type of solitary wave, called "faceton," which consists essentially of an isolated macroscopic facet traveling laterally at such a velocity that its growth rate with respect to the liquid is small. Facetons may propagate either in a stationary or an oscillatory way. The detailed study of their dynamics casts light on the microscopic growth mechanisms of the facets in this system.

Consideration of the existence of surface wave (Rayleigh wave) solutions in anisotropic elastic crystals

From a previous analysis it is known that a proper surface wave may always occur in a stable linear anisotropic half space provided that a certain real symmetric 3*3 matrix B is not positive definite for all velocities less than a limiting velocity nu L, the velocity at which bulk wave solutions first appear. B is also the lagrangian factor matrix for a particular straight dislocation moving uniformly in an infinite medium which is elastically identical to the half space. By examining the behaviour of the dislocation lagrangian and the eigenvalues of B at nu L, it is proved that proper surface waves may always occur except in cetain special cases. In each special case there always exists at least one bulk wave solution which satisfies the free surface condition on the half space boundary. Thus, there exist no forbidden directions for steady state propagation satisfying the free surface condition, i.e. either a proper (attenuated) surface wave and/or an unattenuated 'bulk surface wave' solution is possible. One simple numerical integration suffices to determine if a given propagation direction admits a proper surface solution.

On Plane and Quasi-Optical Wave Propagation in Gyromagnetic Media

To promote the development and understanding of microwave magnetic devices, especially in the millimeter and submillimeter range utilizing quasi-optical techniques, a discussion of propagation and polarization of plane waves and narrow rays in gyromagnetic media in an arbitrary direction is considered. It is assumed that the medium can be described by a permeability tensor of the Polder type. The approach is structured after classical crystal optics but yields significantly different results since each of the two permitted rays is elliptically polarized. The ellipticities are derived. The phase surfaces are discussed for the lossless case. There are no optical axes but ranges of forbidden directions exist for one or both rays. D, B, and the wave vector n form an orthogonal set at all times. H is confined to the B, n plane; it gyrates along an ellipse such that the Poynting vector traces in time an elliptical cone which contains the wave vector as one mantle line. Therefore, a narrow ray can be understood to proceed along a helical path.

A search for forbidden directions of elastic surface wave propagation in anisotropic crystals

A low pass (LP) and complex band pass (CBP) reconfigurable analog baseband circuit for software-defined radio (SDR) receivers is presented. It achieves 1–15 MHz LP bandwidth, 2–8 MHz CBP bandwidth and 0–36 dB gain range with 1 dB step. Nulling-resistor Miller feed-forward (NRMFF) differential-mode compensation, passive left half-plane (LHP) zero common-mode compensation and Quasi-Floating Gate (QFG) technique are proposed to improve the high frequency performance and driving capability of the embedded fully differential operational amplifier (Op-Amp). The analog baseband circuit has been implemented in 65 nm CMOS. It achieves 15.2 dB m/27.1 dB m IB/OB-IIP3, −2 dB m IP1dB and 71 dB m IIP2 while consuming 3.6–9.1 mW from a 1.2 V power supply and 0.75 mm2 chip area.

Search for Forbidden Directions of Elastic Surface-Wave Propagation in Anisotropic Crystals

A detailed study of elastic surface-wave propagation on a variety of free surfaces on many anisotropic crystals has been made. Included in the search were cases of propagation on nonsymmetry as well as symmetry planes. To date, no combination of surface, direction of progagation, and crystal has been found for which an unattenuated surface wave satisfying the traction-free boundary conditions cannot be propagated. Solutions are shown for examples which occur within previously reported ``forbidden'' ranges and for cases in which the free surface is a nonsymmetry plane. The ``forbidden'' ranges appear now to be those in which the penetration of the surface wave becomes relatively deep. A fast and general computer program has been developed for this study to allow arbitrary choice of propagation conditions. The direction of energy flow is computed in the general case by effectively constructing the normal to an appropriate segment of the slowness surface. Solutions have been found for pseudo-surface waves in various ranges of directions. These are second surface-type waves which can propagate almost unattenuated.

I. General Theory of the Earth's Shadow Effect of Cosmic Radiation

An inquiry is made into the fundamental mode of origin of the allowed cone of cosmic radiation introduced by Lemaitre and Vallarta. Two kinds of allowed cone are distinguished. One of these, that discussed by Lemaitre and Vallarta, postulates the presence of an impenetrable earth; for it all forbidden directions are due to the earth's shadow effect. The second type of allowed cone, applicable in the first approximation to such problems as that of the sun's magnetic field, does not postulate the presence of an impenetrable earth; for it all forbidden directions are associated with bounded charged particle orbits. The general topological features of each of these kinds of allowed cone, and their relationship to one another, are described.