Angular Spectrum Research Articles

Adaptive layer-based computer-generated holograms.

We propose a novel, to the best of our knowledge, and fast adaptive layer-based (ALB) method for generating a computer-generated hologram (CGH) with accurate depth information. A complex three-dimensional (3D) object is adaptively divided into layers along the depth direction according to its own non-uniformly distributed depth coordinates, which reduces the depth error caused by the conventional layer-based method. Each adaptive layer generates a single-layer hologram using the angular spectrum method for diffraction, and the final hologram of a complex three-dimensional object is obtained by superimposing all the adaptive layer holograms. A hologram derived with the proposed method is referred to as an adaptive layer-based hologram (ALBH). Our demonstration shows that the desired reconstruction can be achieved with 52 adaptive layers in 8.7 s, whereas the conventional method requires 397 layers in 74.9 s.

Dark ages, a window on the dark sector. Hunting for ultra-light axions

Measurements of 21 cm intensity mapping (IM) during the dark ages can potentially provide us with an unprecedented window on high redshifts and small scales. One of the main advantages this can bring involves the possibility to probe the nature of dark matter. Tests of dark matter models with the large-scale structure of the Universe are limited by non-linearities and astrophysical effects, which are not present for IM measurements during the dark ages. In this paper we focus on constraining the model in which dark matter is comprised, totally or in part, by ultra-light axion-like particles around the 10-18– 10-22 eV mass scale. For this model, the angular power spectrum of 21 cm brightness temperature fluctuations will exhibit a small-scale suppression. However, this effect is intertwined with the imprint of baryon-dark matter relative velocity at recombination, causing at the same time an enhancement at large-scales, which is affected by the mass and abundance of axion dark matter. In this work we forecast how future radio arrays will be able to constrain ultra-light axion mass through both these effects on the angular power spectrum.

Strong diffraction of nonlinear surface acoustic waves in crystals

Nonlinear distortion and shock formation in planar surface acoustic waves in anisotropic crystals have been modeled without [Hamilton et al., JASA (1999)] and with [Cormack et al., JASA 2022] piezoelectricity. Weak diffraction has been included in the paraxial approximation for nonlinear surface wave beams in isotropic solids [Shull et al., JASA (1995)]. Here, a procedure for including strong diffraction, i.e., without the paraxial approximation, in the model for nonlinear surface waves in crystals is presented, which incorporates the angular spectrum approach described by Kharusi and Farnell (JASA 1970). Anisotropy is defined by expressing the phase speed of a plane wave, and therefore, the magnitude of the corresponding wavenumber, as a function of the direction of propagation. Next, an explicit expression relating the two wavenumber components in the planar surface along which the wave propagates must be obtained as a function of direction, requiring iterative solution of a transcendental relation. The beam is then propagated incrementally away from the source, advancing the angular spectrum in k-space and including nonlinear interactions in the spatial domain to characterize the combined effects of diffraction and nonlinearity. Preliminary results are presented for converging nonlinear surface waves in crystals. [Work supported by IR&D at ARL:UT.]

Full-polarization angular spectrum modeling of scattered light modulation

The exact physical modeling for scattered light modulation is critical in phototherapy, biomedical imaging, and free-space optical communications. In particular, the angular spectrum modeling of scattered light has attracted considerable attention, but the existing angular spectrum models neglect the polarization of photons, degrading their performance. Here, we propose a full-polarization angular spectrum model (fpASM) to take the polarization into account. This model involves a combination of the optical field changes and free-space angular spectrum diffraction, and enables an investigation of the influence of polarization-related factors on the performance of scattered light modulation. By establishing the relationship between various model parameters and macroscopic scattering properties, our model can effectively characterize various depolarization conditions. As a demonstration, we apply the model in the time-reversal data transmission and anti-scattering light focusing. Our method allows the analysis of various depolarization scattering events and benefits applications related to scattered light modulation.

Small-scale Anisotropies of Cosmic Rays from Turbulent Flow

Within the classical convection–diffusion approximation, we show that the angular distribution of cosmic rays (CRs) in a highly turbulent flow may exhibit significant small-scale anisotropies. The CR intensity angular power spectrum C ℓ is then a direct reflection of interstellar turbulence, from which one expects C ℓ ∝ ℓ −γ−1 for ℓ ≫ 1, where γ is the power-law turbulence spectral index. Observations by IceCube and HAWC at TeV energies can be explained approximately with the Kolmogorov law γ = 5/3 with a convection velocity dispersion of 20 km s−1 on the scale of 10 pc.

Deblurring the early Universe: reconstruction of primordial power spectrum from Planck CMB using image analysis techniques

While the simplest inflationary models predict the primordial perturbations to be near scale-invariant, the primordial power spectrum (PPS) can exhibit oscillatory features in many physically well-motivated models. We search for hints of such features via free-form reconstructions of the PPS based on Planck 2018 CMB temperature and polarization anisotropies. In order to robustly invert the oscillatory integrals and handle noisy unbinned data, we draw inspiration from image analysis techniques. In previous works, the Richardson-Lucy deconvolution algorithm for deblurring images has been modified for reconstructing PPS from the CMB temperature angular power spectrum. We extensively develop the methodology by including CMB polarization and introducing two new regularization techniques, also inspired by image analysis and adapted for our cosmological context. Regularization is essential for improving the fit to the temperature and polarization channels (TT, TE and EE) simultaneously without sacrificing one for another. The reconstructions we obtain are consistent with previous findings from temperature-only analyses. We evaluate the statistical significance of the oscillatory features in our reconstructions using mock data and find the observations to be consistent with having a featureless PPS. The machinery developed here will be a complimentary tool in the search for features with upcoming CMB surveys. Our methodology also shows competitive performance in image deconvolution tasks, which have various applications from microscopy to medical imaging.

Collimated hot electron generation from sub-wavelength grating target irradiated by a femtosecond laser pulse of relativistic intensity

We investigate the production of hot electrons from the interaction of relativistically intense (I>1018 W/cm2) ultrashort (25 fs) laser pulses with sub-wavelength grating target. We measure the hot electron angular distribution and energy spectra for grating target and compare them with those from a planar mirror target. We observe that hot electrons are emitted in a collimated beam along the specular direction of the grating target. From the measurements, we see fast electron temperature and flux for grating are higher than those for mirror due to a stronger coupling with the laser. We performed numerical simulations, which are in good agreement with experimental results, and offer insights into the acceleration mechanism by resulting electric and magnetic fields. Such collimated fast electron beams have a wide range of applications in applied and fundamental science.

Enabling support for Octave within the FOCUS software package

FOCUS, the “Fast Object-Oriented C+ + Simulator” (https://www.egr.msu.edu/∼fultras-web/), is a free software package that enables rapid calculations of continuous-wave and transient pressure fields generated by single transducers and phased arrays. FOCUS achieves small errors in relatively short computation times through linear memory-efficient calculations with the fast nearfield method. This capability extends to quick and accurate biomedical imaging simulations within FOCUS. Moreover, FOCUS supports some nonlinear pressure calculations, such as the continuous-wave and transient Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations for circular and spherical transducers. Additionally, the Angular Spectrum Approach (ASA), a frequency-domain solution to linear computational methods, ideal for large volumetric field computations, is also included within the FOCUS package. Initial success with MATLAB motivates the creation of an Octave version that replicates the core functionalities of the original FOCUS package. GNU Octave is similar to MATLAB, but unlike MATLAB, Octave is a free software. Building and compiling FOCUS in Octave required a few minor alterations due to discrepancies in syntax and function calls. The use of Octave-compiled MEX files simplified the conversion process while introducing addition memory overhead compared to rewriting C+ + code for Oct-file structuring. The performance and capabilities of FOCUS in Octave are demonstrated and discussed.

Terahertz super-radiance from picosecond electron bunches moving through a micro-undulator

We suggest using emission from the photoinjector-formed electron bunches moving through micro-undulators for generation of powerful super-radiant pulses in the terahertz/far infrared frequency range. Within the time-domain quasi-optical approach, we demonstrate the spatial coherence of emission with narrow angular spectrum from the electron bunches with transverse size limited by the Fresnel parameter NF ∼ 1, when the diffraction effects together with slippage provide synchronization of radiation from the entire volume of the extended bunch. For picosecond-duration, 5 MeV, 250 pC electron bunches, and a micro-undulator with a period of 3 mm, the peak power of 15 THz SR pulses can be about of 20 MW.

Spatial resolution enhancement in holographic imaging via angular spectrum expansion

Digital holography numerically restores three-dimensional image information using optically captured diffractive waves. The required bandwidth is larger than that of hologram pixel at a closer distance in the Fresnel diffraction regime, which results in the formation of aliased replica patterns in digital hologram. From the analysis of sampling phenomenon, the replica functions are revealed to be the components of higher angular spectra of hologram. A undersampled hologram consists of the moire patterns formed by the modulation of original function by complex exponential function. There is a one-to-one correspondence between the replicas in both real and Fourier spaces. The acquisition methods of high-resolution images over a wide field view are proposed on the basis of the expansion process of angular spectrum by using replicas. Only the captured hologram with low numerical aperature or low resolution restores a high-resolution image when using an optimization algorithm. Numerical simulations and optical experiments are performed to investigate the proposed scheme.

Stochastic diffusion within expanding space–time

The paper examines stochastic diffusion within an expanding space–time framework motivated by cosmological applications. Contrary to other results in the literature, for the considered general stochastic model, the expansion of space–time leads to a class of stochastic equations with non-constant coefficients that evolve with the expansion factor. The Cauchy problem with random initial conditions is posed and investigated. The exact solution to a stochastic diffusion equation on the expanding sphere is derived. Various probabilistic properties of the solution are studied, including its dependence structure, evolution of the angular power spectrum and local properties of the solution and its approximations by finite truncations. The paper also characterizes the extremal behaviour of the random solution by establishing upper bounds on the probabilities of large deviations. Numerical studies are carried out to illustrate the obtained theoretical results.

Turbulence on open string worldsheets under non-integrable boundary conditions

We demonstrate the turbulent dynamics of the Nambu-Goto open string in the AdS3 spacetime. While the motion of a classical closed string in AdS is known to be integrable, the integrability of an open string motion depends on the boundary conditions at the string endpoints. We numerically solve the equations of motion of the open string under the boundary conditions where the endpoints are i) fixed to a finite radial coordinate in AdS, and ii) free. For i), we find turbulence on the string, that shows a cascade in the energy and angular momentum spectra. This result indicates the non-integrability of the open string with this type of boundary conditions. For ii), we find no turbulence. This is consistent with the integrability of the open string with the free boundary conditions.

Metasurface for Engineering Superimposed Ince-Gaussian Beams.

Ince-Gaussian beams (IGBs) are the third complete family of exact and orthogonal solutions of the paraxial wave equation and have been applied in many fields ranging from particle trapping to quantum optics. IGBs play a very important role in optics as they represent the exact and continuous transition modes connecting Laguerre-Gaussian and Hermite-Gaussian beams. The method currently in use suffers from the high cost, complexity, and large volume of the optical system. The superposition of IGBs can generate complicated structured beams with multiple phase and polarization singularities. A metasurface approach is proposed to realizing various superpositions of IGBs without relying on a complicated optical setup. By superimposing IGBs with even and odd modes, multiple phase, and polarization singularities are observed in the resultant beams. The phase and polarization singularities are modulated by setting the initial phase in the design and controlling the incident linear polarization. The compactness of the developed metasurface devices and the unique properties of the generated beams have the potential to impact many practical applications such as particle manipulation, orbital angular momentum spectrum manipulation, and optical communications.

Analytical calculation of beam profile and orbital angular momentum spectrum of Laguerre Gaussian beams reflected from a graphene plasmonic structure.

In this paper, Laguerre Gaussian (LG) beams with different topological charges are used for excitation of surface plasmon polaritons (SPPs) through a graphene layer inserted in the Otto-configuration. By utilizing the angular spectrum representation (ASR) and Lorenz-gauge vector potential, an explicit analytical expression is derived for the electromagnetic fields of the reflected beam. At the optimal excitation state of graphene SPPs, the reflected beam exhibits a distinctive field profile characterized by two identical crescent-shaped lobes separated by a vertical strip with null intensity. Furthermore, in the absence of external magnetic field, the orbital angular momentum (OAM) spectrum of the reflected beam at the optimal excitation of SPPs reveals the annihilation of central OAM mode and the generation of two equal OAM sidebands, regardless of the incident OAM topological charge. Furthermore, the phase distributions of electric field of the reflected beam confirm the existence of OAM sidebands in the vicinity of optimal SPPs excitation. As the system is taken away from the optimal excitation of SPPs by introduction of an external magnetic field or increasing the chemical potential or increasing the incident angle, both central and sideband modes appear in the OAM spectrum of the reflected beam. In this case, when the topological charge of the incident wave increases, the weight of central OAM mode decreases while the weight of sidebands increases. In contrast, in the presence of external magnetic field, at the optimal excitation of SPPs, both central OAM and sidebands modes exist in the reflected beam such that the weight of central modes increases with the external magnetic field. This effect is also confirmed by plotting the phase distributions of the reflected beam at different external magnetic fields and for different incident topological charges. Therefore, the manipulation of graphene plasmons characteristics leads to the control of OAM sideband generation.

AMICO galaxy clusters in KiDS-DR3: Cosmological constraints from the angular power spectrum and correlation function

We study the tomographic clustering properties of the photometric cluster catalogue derived from the third data release of the Kilo Degree Survey (KiDS), focusing on the angular correlation function and its spherical harmonic counterpart: the angular power spectrum. We measured the angular correlation function and power spectrum from a sample of 5162 clusters, with an intrinsic richness of λ* ≥ 15, in the photometric redshift range of z ∈ [0.1, 0.6]. We compared our measurements with theoretical models, within the framework of the Λ cold dark matter cosmology. We performed a Markov chain Monte Carlo (MCMC) analysis to constrain the cosmological parameters, Ωm and σ8, as well as the structure growth parameter, S8 ≡ σ8√Ωm/0.3. We adopted Gaussian priors on the parameters of the mass-richness relation, based on the posterior distributions derived from a previous joint analysis of cluster counts and weak-lensing mass measurements carried out on the basis of the same catalogue. From the angular correlation function, we obtained Ωm = 0.32−0.04+0.05, σ8 = 0.77−0.09+0.13, and S8 = 0.80−0.06+0.08, which are in agreement, within 1σ, with the 3D clustering result based on the same cluster sample and with existing complementary studies on other data sets. For the angular power spectrum, we checked the validity of the Poissonian shot noise approximation, also considering the mode-mode coupling induced by the mask. We derived statistically consistent results, in particular, Ωm = 0.24−0.04+0.05 and S8 = 0.93−0.12+0.11; while the constraint on σ8 alone is weaker with respect to the one provided by the angular correlation function, σ8 = 1.01−0.17+0.25. Our results show that the 2D clustering from photometric cluster surveys can provide competitive cosmological constraints with respect to the full 3D clustering statistics. We also demonstrate that they can be successfully applied to ongoing and forthcoming spectrometric and photometric surveys.

Flat-sky angular power spectra revisited

We revisit the flat-sky approximation for evaluating the angular power spectra of projected random fields by retaining information about the correlations along the line of sight. For the case of projections with broad, overlapping radial window functions, these line-of-sight correlations are suppressed and are ignored in the commonly adopted Limber approximation. However, retaining the correlations is important for narrow window functions or unequal-time spectra but introduces significant computational difficulties due to the highly oscillatory nature of the integrands involved. We deal with the integral over line-of-sight wave-modes in the flat-sky approximation analytically, using the FFTlog expansion of the 3D power spectrum. This results in an efficient computational method, which is a substantial improvement compared to any full-sky approaches. We apply our results to galaxy clustering (with and without redshift-space distortions), CMB lensing and galaxy lensing observables in a flat ΛCDM universe. In the case of galaxy clustering, we find excellent agreement with the full-sky results on large (percent-level agreement) and intermediate or small (subpercent agreement) scales, dramatically out-performing the Limber approximation for both wide and narrow window functions, and in equal- and unequal-time cases. In the cases of lensing, we show on the full-sky that the angular power spectrum of the lensing convergence can be very well approximated by projecting the 3D Laplacian (rather than the correct angular Laplacian) of the gravitational potential, even on large scales. Combining this approximation with our flat-sky techniques provides an efficient and accurate evaluation of the CMB lensing angular power spectrum on all scales. We further analyse the clustering and lensing angular power spectra by isolating the projection effects due to the observable- and survey-specific window functions, separating them from the effects due to integration along the line of sight and unequal-time mixing in the 3D power spectrum. All of the angular power spectrum results presented in this paper are obtained using a Python code implementation, which we make publicly available.

Electronic Focus Steering Capabilities of a Diagnostic-Type Linear Ultrasound Array Designed for High Power Therapy and Its Visualization

The focus steering capabilities of a 1 MHz linear phased array transducer (64 rectangular elements, 14.8 × 51.2 mm aperture) intended for drug delivery applications in abdominal organs were assessed and compared with its design-stage computer model. Acoustic fields generated by the transducer and predicted by the models of an ideal array with uniformly vibrating elements and either a plane or a cylindrically focused surface were simulated using the Rayleigh integral and angular spectrum methods. The boundary conditions for the transducer were reconstructed from acoustic holography measurements performed for selected focusing configurations of the array and also synthesized from holography data measured for each of its individual elements. It was shown that the transducer field with electronic focus steering can be accurately synthesized based on the holography data of its elements, which significantly simplified acoustic field characterization. Variability of the power and directivity patterns of the array elements were analyzed. A twofold smaller range of electronic steering in the transverse direction for the transducer compared to its computer model is discussed.

Cosmological parameters derived from the final Planck data release (PR4)

We present cosmological parameter constraints using maps from the last Planck data release (PR4). In particular, we detail an upgraded version of the cosmic microwave background likelihood, HiLLiPoP, that is based on angular power spectra and relies on a physical modeling of the foreground residuals in the spectral domain. This new version of the likelihood retains a larger sky fraction (up to 75%) and uses an extended multipole range. Using this likelihood, along with low-ℓ measurements from LoLLiPoP, we derived constraints on ΛCDM parameters that are in good agreement with previous Planck 2018 results, but with smaller uncertainties by 10% to 20%. We demonstrate that the foregrounds can be accurately described in the spectral domain, with a negligible impact on ΛCDM parameters. We also derived constraints on single-parameter extensions to ΛCDM, including AL, ΩK, Neff, and ∑mν. Noteworthy results from this updated analysis include a lensing amplitude value of AL = 1.039 ± 0.052, which is more closely aligned with theoretical expectations within the ΛCDM framework. Additionally, our curvature measurement, ΩK = −0.012 ± 0.010, is now fully consistent with a flat universe and our measurement of S8 is closer to the measurements derived from large-scale structure surveys (at the 1.5σ level). We also added constraints from PR4 lensing, making this combination the most tightly constrained data set currently available from Planck. Additionally, we explored the addition of baryon acoustic oscillation data, which tightens the limits on some particular extensions to the standard cosmology.

FURION: modeling of FEL pulses propagation in dispersive soft X-ray beamline systems.

Modern X-ray free-electron lasers (XFELs) can generate pulses with durations ranging from femtoseconds to attoseconds. The numerical evaluation of ultra-short XFEL pulses through beamline systems is a critical process of beamline system design. However, the bandwidth of such ultra-short XFEL pulses is often non-negligible, and the propagation cannot be simply approximated using the central wavelength, especially in dispersive beamline systems. We developed a numerical model which is called Fourier optics based Ultrashort x-Ray pulse propagatION tool (FURION). This model can not only be used to simulate dispersive beamline systems but also to evaluate non-dispersive beamline systems. The FURION model utilizes Fresnel integral and angular spectrum integral to perform ultra-short XFEL pulse propagation in free space. We also present the method for XFEL pulse propagation through different types of dispersive gratings, which are commonly used in soft X-ray beamline systems. By using FURION, a start-to-end simulation of the FEL-1 beamline system at Shenzhen superconducting soft X-ray free electron laser (S3FEL) is carried out. This model can also be used to evaluate gratings-based spectrometers, beam splitters, pulse compressors, and pulse stretchers. This work provides valuable insights into the start-to-end simulation of X-ray beamline systems.

Needle of longitudinally polarized light using the circular Airy beam.

An optical needle is created using a radially polarized circular Airy beam with a conical angle, stemmed from the auto-focusing property of light beams. The utilization of the angular spectrum representation serves to illustrate the field distributions of the optical needle, and an explicit formula is provided to describe the angular spectrum of the light beam. The findings suggest that the optical needle exhibits a long depth of focus and well uniformity, and the full width at half maximum of the transverse field distribution is approximately 0.38 λ beyond the diffraction limit. The uniformity of the optical needle can be tailored by adjusting the width of the primary ring, the decay parameter, and the conical angle. Additionally, the depth of focus of the optical needle significantly improves as the radius of the primary ring increases while still maintaining well uniformity. It may find applications in high-resolution optical imaging and optical manipulation.