Solid Harmonics Research Articles

The three-dimensional harmonic oscillator and solid harmonics in Bargmann space

The three-dimensional harmonic oscillator is solved in Bargmann space. The treatment is pedagogically more transparent than the standard ones, at the price of introducing the Bargmann transform in the context of the one-dimensional oscillator. The standard solid harmonics are similarly derived with minimal technical effort, amounting to a complete self-contained exposition suitable for introductory courses in quantum mechanics or mathematical methods of physics. It provides an early exposure to wavelets, with important contemporary applications in signal analysis and quantum optics.

Rayleigh Scattering from a Sphere Located Near a Planar Rigid Boundary

Rayleigh scattering from a spherical object located near a planar rigid boundary at distances smaller than the wavelength is calculated. Low frequency analysis reduces a scattering problem to a sequence of potential problems. An analytical solution based on expansion in spherical solid harmonics and the use of addition theorem is presented. Analytical perturbation approach is validated by comparison with numerical calculations. The velocity of the center of the particle and the scattering amplitude are determined. In the lowest order in wavenumber, the scattering amplitude is expressed in terms of the monopole and dipole components. In contrast to the behavior of a bubble, under the same conditions, dipole oscillations of the particle in the direction normal to the boundary are not excited and the monopole component of the scattering amplitude does not depend on the location of the particle relative to the boundary.

Analytical method for reconstructing the stress on a spherical particle from its surface deformation

The mechanical forces that cells experience from the tissue surrounding them are crucial for their behavior and development. Experimental studies of such mechanical forces require a method for measuring them. A widely used approach in this context is bead deformation analysis, where spherical particles are embedded into the tissue. The deformation of the particles then allows to reconstruct the mechanical stress acting on them. Existing approaches for this reconstruction are either very time-consuming or not sufficiently general. In this article, we present an analytical approach to this problem based on an expansion in solid spherical harmonics that allows us to find the complete stress tensor describing the stress acting on the tissue. Our approach is based on the linear theory of elasticity and uses an ansatz specifically designed for deformed spherical bodies. We clarify the conditions under which this ansatz can be used, making our results useful also for other contexts in which this ansatz is employed. Our method can be applied to arbitrary radial particle deformations and requires a very low computational effort. The usefulness of the method is demonstrated by an application to experimental data.

Defects of the nearest-neighbor tight-binding model in the study of solid harmonics

Defects of the nearest-neighbor tight-binding model in the study of solid harmonics

Elasticity of spheres with buckled surfaces.

The buckling instabilities of core-shell systems, comprising an interior elastic sphere, attached to an exterior shell, have been proposed to underlie myriad biological morphologies. To fully discuss such systems, however, it is important to properly understand the elasticity of the spherical core. Here, by exploiting well-known properties of the solid harmonics, we present a simple, direct method for solving the linear elastic problem of spheres and spherical voids with surface deformations, described by a real spherical harmonic. We calculate the corresponding bulk elastic energies, providing closed-form expressions for any values of the spherical harmonic degree (l), Poisson ratio, and shear modulus. We find that the elastic energies are independent of the spherical harmonic index (m). Using these results, we revisit the buckling instability experienced by a core-shell system comprising an elastic sphere, attached within a membrane of fixed area, that occurs when the area of the membrane sufficiently exceeds the area of the unstrained sphere [C. Fogle et al., Phys. Rev. E 88, 052404 (2013)1539-375510.1103/PhysRevE.88.052404]. We determine the phase diagram of the core-shell sphere's shape, specifying what value of l is realized as a function of the area mismatch and the core-shell elasticity. We also determine the shape phase diagram for a spherical void bounded by a fixed-area membrane.

Slater Tipi Orbitaller Bazında Bir- ve İki-Merkezli Coulomb İntegrallerinin Özellikleri

In this study, two-electron one- and two-center Coulomb integrals with the same and different screening parameters are investigated numerically in the real Slater type orbital (STO) basis using Fourier transform method. In momentum space firstly, for atomic, i.e. one-center, Coulomb integrals are calculated, and analytical expressions are obtained in terms of binomial coefficients. Then, the solutions of the two-center Coulomb integrals are made with the modified Bessel function of second kind and the results are expressed in terms of binomial and Gaunt coefficients, irregular solid harmonics, and finite sum of STOs. A computer program is written in the MATHEMATICA language to determine the accuracy of the analytical expressions that are highly suitable for programming. The numerical results obtained from the program are given in the tables, and it is shown that the results agree with the literature.

Spectral shift of solid high-order harmonics from different channels in a combined laser field

We investigate theoretically the spectral shift of the high-order harmonic generation (HHG) in ZnO driven by a combined laser field by solving the two-band semiconductor Bloch equations (SBEs) in the velocity gauge. The combined laser field is synthesized by a fundamental laser pulse and its seventh-frequency laser pulse. When the seventh-frequency laser pulse is added to the rising or falling parts of the fundamental laser field, we find that the spectral blueshift or redshift appears, which is due to the unequal contribution of the rising and falling parts in the fundamental laser field to the harmonics. By analyzing the time-dependent conduction band population in k space, we found that, in addition to the tunneling ionization channel, there is also the resonant electron injection channels which is induced by the seventh-frequency laser pulse. The harmonics generated by the different channels show the spectral redshift or the spectral blueshift, respectively. Through analyzing the k -integrated transient conduction band population of the electrons from different channels, we found that if there is a certain delay in the process of the electron excitation, it will lead to the delay in the harmonic emission, which results in the spectral redshift of the harmonics.

Expansion of multicenter Coulomb integrals in terms of two-center integrals

Three- and four-center Coulomb integrals in the solid spherical harmonic Gaussian basis are solved by expansion in terms of two-center integrals. The two-electron Gaussian product rule, coupled with the addition theorem for solid spherical harmonics, reduces four-center Coulomb integrals into a linear combination of two-center Coulomb integrals and one-center overlap integrals. With this approach, three- and four-center Coulomb integrals can be reduced to the same form of two-center integrals. Resulting two-center Coulomb integrals can be further simplified into a simpler form, which can be related to the Boys function. Multi-center Coulomb integrals are solved hierarchically: simple two-center Coulomb integrals are used for calculation of more complicated two-center Coulomb integrals, which are used in the calculation of multicenter integrals.

Spherical harmonic based noise rejection and neuronal sampling with multi-axis OPMs

In this study we explore the interference rejection and spatial sampling properties of multi-axis Optically Pumped Magnetometer (OPM) data. We use both vector spherical harmonics and eigenspectra to quantify how well an array can separate neuronal signal from environmental interference while adequately sampling the entire cortex. We found that triaxial OPMs have superb noise rejection properties allowing for very high orders of interference (L=6) to be accounted for while minimally affecting the neural space (2dB attenuation for a 60-sensor triaxial system). We show that at least 11th order (143 spatial degrees of freedom) irregular solid harmonics or 95 eigenvectors of the lead field are needed to model the neural space for OPM data (regardless of number of axes measured). This can be adequately sampled with 75–100 equidistant triaxial sensors (225–300 channels) or 200 equidistant radial channels. In other words, ordering the same number of channels in triaxial (rather than purely radial) configuration may give significant advantages not only in terms of external noise rejection but also by minimizing cost, weight and cross-talk.

Generation, manipulation, and application of high-order harmonics in solids

The generation of high-order harmonics based on the interaction between ultrafast intense laser and matter provides a platform for studying the light-matter interaction in the non-perturbative region. It is also the main route to generating desktop extreme ultraviolet light source and attosecond pulse. The non-perturbative solid high-order harmonic involves the core content of ultrafast strong field physics, condensed matter physics, materials science, information science and other fields. Since it was first experimentally observed in 2011, it has rapidly become the research frontier of strong field physics and attosecond science. This review summarizes the research progress and important applications of solid high-order harmonics from the perspective of an experimentalist. Firstly, distinct characteristics are shown for solid high-order harmonic by comparing the dependence of harmonic yield and cut-off energy on driving laser parameters with gas high-order harmonic. Then, the progress of manipulation and application are highlighted for solid high-order harmonic, including the precise control of harmonic yield, polarization, space-time distribution through the design of target structure or laser field, as well as the application of solid high-order harmonic spectroscopy in the fields of material structure characterization and ultrafast electron dynamics. Finally, the future is prospected for the study of solid high-order harmonics.

Receiver phase alignment using fitted SVD derived sensitivities from routine prescans

Magnetic resonance imaging radio frequency arrays are composed of multiple receive coils that have their signals combined to form an image. Combination requires an estimate of the radio frequency coil sensitivities to align signal phases and prevent destructive interference. At lower fields this can be accomplished using a uniform physical reference coil. However, at higher fields, uniform volume coils are lacking and, when available, suffer from regions of low receive sensitivity that result in poor sensitivity estimation and combination. Several approaches exist that do not require a physical reference coil but require manual intervention, specific prescans, or must be completed post-acquisition. This makes these methods impractical for large multi-volume datasets such as those collected for novel types of functional MRI or quantitative susceptibility mapping, where magnitude and phase are important. This pilot study proposes a fitted SVD method which utilizes existing combination methods to create a phase sensitive combination method targeted at large multi-volume datasets. This method uses any multi-image prescan to calculate the relative receive sensitivities using voxel-wise singular value decomposition. These relative sensitivities are fitted to the solid harmonics using an iterative least squares fitting algorithm. Fits of the relative sensitivities are used to align the phases of the receive coils and improve combination in subsequent acquisitions during the imaging session. This method is compared against existing approaches in the human brain at 7 Tesla by examining the combined data for the presence of singularities and changes in phase signal-to-noise ratio. Two additional applications of the method are also explored, using the fitted SVD method in an asymmetrical coil and in a case with subject motion. The fitted SVD method produces singularity-free images and recovers between 95–100% of the phase signal-to-noise ratio depending on the prescan data resolution. Using solid harmonic fitting to interpolate singular value decomposition derived receive sensitivities from existing prescans allows the fitted SVD method to be used on all acquisitions within a session without increasing exam duration. Our fitted SVD method is able to combine imaging datasets accurately without supervision during online reconstruction.

High-harmonic generation in metallic titanium nitride

High-harmonic generation is a cornerstone of nonlinear optics. It has been demonstrated in dielectrics, semiconductors, semi-metals, plasmas, and gases, but, until now, not in metals. Here we report high harmonics of 800-nm-wavelength light irradiating metallic titanium nitride film. Titanium nitride is a refractory metal known for its high melting temperature and large laser damage threshold. We show that it can withstand few-cycle light pulses with peak intensities as high as 13 TW/cm2, enabling high-harmonics generation up to photon energies of 11 eV. We measure the emitted vacuum ultraviolet radiation as a function of the crystal orientation with respect to the laser polarization and show that it is consistent with the anisotropic conduction band structure of titanium nitride. The generation of high harmonics from metals opens a link between solid and plasma harmonics. In addition, titanium nitride is a promising material for refractory plasmonic devices and could enable compact vacuum ultraviolet frequency combs.

Mapping gradient nonlinearity and miscalibration using diffusion-weighted MR images of a uniform isotropic phantom.

PurposeTo use diffusion measurements to map the spatial dependence of the magnetic field produced by the gradient coils of an MRI scanner with sufficient accuracy to correct errors in quantitative diffusion MRI (DMRI) caused by gradient nonlinearity and gradient amplifier miscalibration.Theory and MethodsThe field produced by the gradient coils is expanded in regular solid harmonics. The expansion coefficients are found by fitting a model to a minimum set of diffusion‐weighted images of an isotropic diffusion phantom. The accuracy of the resulting gradient coil field maps is evaluated by using them to compute corrected b‐matrices that are then used to process a multi‐shell diffusion tensor imaging (DTI) dataset with 32 diffusion directions per shell.ResultsThe method substantially reduces both the spatial inhomogeneity of the computed mean diffusivities (MD) and the computed values of the fractional anisotropy (FA), as well as virtually eliminating any artifactual directional bias in the tensor field secondary to gradient nonlinearity. When a small scaling miscalibration was purposely introduced in the x, y, and z, the method accurately detected the amount of miscalibration on each gradient axis.ConclusionThe method presented detects and corrects the effects of gradient nonlinearity and gradient gain miscalibration using a simple isotropic diffusion phantom. The correction would improve the accuracy of DMRI measurements in the brain and other organs for both DTI and higher order diffusion analysis. In particular, it would allow calibration of MRI systems, improving data harmony in multicenter studies.

Ellipsoidal inhomogeneity with anisotropic incoherent interface. Multipole series solution and application to micromechanics

The ellipsoidal inhomogeneity with transversely isotropic incoherent interface is considered in the conductivity and elasticity context. The general imperfect interface is modeled by the first order approximation of thin transversely isotropic interphase layer. This model expands the Gurtin et al. (1998) theory of curved deformable interfaces in solids with a nanometer-scale microstructure to the incoherent interfaces between the dissimilar elastic materials and provides a certain insight into the interface moduli. The rigorous analytical solution to the conductivity and elasticity problems has been obtained by the multipole expansion method in terms of ellipsoidal solid harmonics. An accurate fulfillment of the imperfect interface conditions reduces the model boundary value problem to the linear algebraic system for multipole strengths. These results apply equally to the inhomogeneities with anisotropic interphases and nano level incoherent interfaces. The obtained solutions are valid for the non-uniform far loading and are readily incorporated in the many-particle (finite cluster or representative unit cell) model of heterogeneous solid with anisotropic incoherent interface. The tensors of effective conductivity and elastic stiffness of composite with ellipsoidal inhomogeneities are evaluated using Maxwell homogenization scheme. The obtained accurate numerical data indicate that the effective properties of composite may vary widely due to shape of inhomogeneities and the interface anisotropy ratio. Taking the incoherency of interface into account may increase reliability of predicting the behavior of nanostructured solids.

Definition and properties of logopoles of all degrees and orders.

Logopoles are a recently proposed class of solutions to Laplace's equation with intriguing links to both solid spheroidal and solid spherical harmonics. They share the same finite-line singularity as the former and provide a generalization of the latter as multipoles of negative order. In a previous paper [Majic and Le Ru, Phys. Rev. Res. 1, 033213 (2019)2643-156410.1103/PhysRevResearch.1.033213], we introduced and discussed the properties and applications of these new functions in the special case of axisymmetric problems (with azimuthal index m=0). This allowed us to focus on the physical properties without the added mathematical complications. Here we expand these concepts to the general case m≠0. The chosen definitions are motivated to conserve some of the most interesting properties of the m=0 case. This requires the inclusion of Legendre functions of the second kind with degree -m≤n<m (in addition to the usual n≥|m|) and we show that these are also related to the exterior spheroidal harmonics. We show that logopoles can also be defined for n≤m and discuss in particular logopoles of degree n=-m, which correspond to the potential of line segments of uniform polarization density.

Ellipsoidal inhomogeneity in elliptically orthotropic elastic solid

An unbounded elliptically orthotropic elastic solid containing a single anisotropic ellipsoidal inhomogeneity is considered. The problem statement assumes arbitrary orientation of ellipsoid with respect to the orthotropy axes and non-uniform far loading. By applying the appropriate affine transformations, the boundary value problem is reduced to that one with the isotropic constituents. The complete displacement solution to the transformed problem is obtained using the Papkovich-Neuber representation with the scalar potentials written in terms of ellipsoidal solid harmonics. By accurate fulfilling the interface conditions, the boundary value problem is reduced to the system of the linear algebraic equations for the constants entering the potentials. For the Eshelby problem (uniform remotely applied loading and perfect interface), an explicit analytical solution is written in terms of ellipsoidal harmonics. This result yields solution for the elliptical crack problem in the remarkably simple way. The numerical examples illustrate an effect of the matrix anisotropy on the stress concentration and stress intensity factors. An application of the obtained solution to the micromechanics is discussed. The effective elastic stiffness tensor of a particulate composite with elliptically orthotropic matrix is evaluated using Maxwell homogenization scheme.

Fast and accurate evaluation of geomagnetic field elements at arbitrary points in space

SUMMARY An algorithm and software are developed for fast and accurate evaluation of the elements of the geomagnetic field represented in high-degree (&amp;gt;720) solid spherical harmonics at many scattered points in the space above the surface of the Earth. The algorithm is based on representation of the geomagnetic field elements in solid ellipsoidal harmonics and application of tensor product needlets. Open source FORTRAN and MATLAB realizations of this algorithm that rely on data from the Enhanced Magnetic Models 2015, 2017 (EMM2015, EMM2017) have been developed and extensively tested. The capabilities of the software are demonstrated on the example of the north, east and down components of the geomagnetic field as well as the derived horizontal intensity, total intensity, inclination and declination. For the range from −417 m under the Earth reference ellipsoid up to 1000 km above it the FORTRAN and MATLAB versions of the software run 465 and 189 times faster than the respective FORTRAN and MATLAB versions of the software using the standard spherical harmonic series method, while the accuracy is less than 1 nT and the memory (RAM) usage is 9 GB.

Theoretical and experimental progress of high-order harmonics and attosecond pulse generation based on solid overdense plasma

<p indent="0mm">Since the laser was invented, harmonic generation and wavelength conversion have been important research topics in nonlinear optics regime. In 1987, McPherson A et al. used sub-picosecond KrF laser <sc>(248 nm</sc> wavelength) to interact with noble gas to obtain high-order harmonics for the first time. In 2001, the first single attosecond <sc>(10^–18 s)</sc> pulse with a pulse width of 650 as was measured experimentally by Krausz et al. by using the broadband high-order harmonic generation. Since then, high-order harmonics and attosecond pulses have been widely used in atomic and molecular physics, materials science and other fields. However, due to the low efficiency of attosecond pulses generated by the gas media, the energy of attosecond pulses is limited, which limits the detection methods of attosecond time dynamics research (currently mainly IR+XUV pumping/detection) and its applications in many fields. How to obtain high-brightness, high-energy attosecond pulses has always been the pursuit in this field since the generation of the first attosecond pulse. With the development of femtosecond laser technology, the output peak power of the femtosecond laser system has been rapidly increased in the past two decades. At present, the laser system can output the pulse with a peak power of up to 10 PW (1 PW<sc>=10¹⁵ W).</sc> Its intensity can be much higher than <sc>10¹⁸ W/cm²</sc> and its interaction with the overdense plasma can efficiently generate high-order harmonics at the level of hundreds of eV or even keV, which brings new opportunities for the development and application of high-order harmonics and attosecond pulses. Generally speaking, the interaction of high peak power femtosecond laser pulses with solid-density plasmas has unique advantages in high-luminance, high-energy high-order harmonics and attosecond pulse generation. In the ELI-ALPS plan, solid plasma high-order harmonics and attosecond pulse generation are very important parts, especially to promote their high brightness and their high photon energy. In terms of the attosecond pulse energy, the goal of their first phase is that the attosecond pulse energy reaches the μJ level, which will be advanced to the mJ level in the next phase, and the photon energy of the attosecond pulse will reach a few keV. This review gives a brief introduction to its physical mechanisms and their current experimental research progress. Three mechanisms (coherent wake emission (CWE), relativistic oscillating mirror (ROM), coherent synchrotron emission (CSE)) will be discussed. They are working under different laser intensity. CWE can emit high efficiency harmonics at the laser intensity below <sc>10¹⁸ W/cm².</sc> But when the peak laser intensity is very high, the research of the CSE mechanism shows the generation of high-luminance high-order harmonics under super-relativistic light intensity, which can be used to produce extremely high field-strength pulse of attoseconds, even tens of zeptoseconds. If the laser parameters and the plasma condition can be well controlled, the laser pulse may even generate higher harmonic radiation with field strength much higher than the pump laser field strength, which may be a possible way to study the 39th question (“What is the most powerful laser researchers can build? ”) among the 125 most challenging scientific questions published by <italic>Science</italic> on its 125th anniversary.

Local rotation invariance in 3D CNNs.

Locally Rotation Invariant (LRI) image analysis was shown to be fundamental in many applications and in particular in medical imaging where local structures of tissues occur at arbitrary rotations. LRI constituted the cornerstone of several breakthroughs in texture analysis, including Local Binary Patterns (LBP), Maximum Response 8 (MR8) and steerable filterbanks. Whereas globally rotation invariant Convolutional Neural Networks (CNN) were recently proposed, LRI was very little investigated in the context of deep learning. LRI designs allow learning filters accounting for all orientations, which enables a drastic reduction of trainable parameters and training data when compared to standard 3D CNNs. In this paper, we propose and compare several methods to obtain LRI CNNs with directional sensitivity. Two methods use orientation channels (responses to rotated kernels), either by explicitly rotating the kernels or using steerable filters. These orientation channels constitute a locally rotation equivariant representation of the data. Local pooling across orientations yields LRI image analysis. Steerable filters are used to achieve a fine and efficient sampling of 3D rotations as well as a reduction of trainable parameters and operations, thanks to a parametric representations involving solid Spherical Harmonics (SH),which are products of SH with associated learned radial profiles. Finally, we investigate a third strategy to obtain LRI based on rotational invariants calculated from responses to a learned set of solid SHs. The proposed methods are evaluated and compared to standard CNNs on 3D datasets including synthetic textured volumes composed of rotated patterns, and pulmonary nodule classification in CT. The results show the importance of LRI image analysis while resulting in a drastic reduction of trainable parameters, outperforming standard 3D CNNs trained with rotational data augmentation.

A line integral approach for the computation of the potential harmonic coefficients of a constant density polyhedron

A novel approach for the computation of the spherical harmonic coefficients of the gravity field of a constant density polyhedron is presented. The proposed method is based on the expression of these coefficients as the volume integral of solid harmonics. It is well known that the divergence theorem leads to an expression of these volume integrals as surface integrals. We show that these surface integrals can be expressed as the sum of line integrals along the edges of the polyhedron. In contrast to previous approaches, the values of the spherical harmonic coefficients at a given degree and order result directly from the computation of the line integrals. The performed numerical implementation revealed the stability of the proposed algorithm up to degree 360 for a prismatic test source.