Time-domain electromagnetic scattering by a sphere in uniform translational motion

Abstract

Scattering by a uniformly translating sphere of a pulse that modulates the amplitude of a linearly polarized plane wave was formulated using the frame-hopping method involving a laboratory inertial reference frame and the sphere's comoving inertial reference frame. The incident signal was defined in the laboratory frame and transformed to the comoving frame with the Lorentz transformation, thereby altering the incident signal's spectrum, direction of propagation of the carrier plane wave, and the direction of the incident electric field, depending on the sphere's velocity. In the comoving frame, the incident signal was Fourier-transformed to the frequency domain, and the scattered field phasors were computed in all directions using the constitutive parameters of the material of the sphere at rest. The scattered signal in the comoving frame was obtained using the inverse Fourier transform. Finally, the scattered signal in the laboratory frame was obtained by inverting the original Lorentz transformation. The backscattered signal was found to depend strongly on the sphere's velocity, when the sphere's speed is an appreciable fraction of the speed of light in free space. The change in the backscattered signal compared with the backscattered signal from a stationary sphere is the greatest when the sphere's velocity is either parallel or antiparallel to the direction of propagation of the incident signal. The backscattered signal is also affected by motion transverse to the incident signal's direction of propagation; then, the backscattered signal depends on whether or not the motion is aligned with the direction of the incident electric field.

Electromagnetic pulse scattering by a spacecraft nearing light speed

“…The unit vector ̂′ indicates the functions are different in general for each scattering direction [11]. The scattered energy density in units of energy per solid angle is…”

“…We used the Lorentz transformation to convert each of the chosen scattering directions from (θʹ, ϕʹ) in Kʹ to (θ, ϕ) in K, and we added a time-domain far-zone sensor in each of the directions in XFdtd to compute the scattered electric field. Thereafter, we converted the electric field of the scattered signal to Kʹ using the Lorentz transformation, as detailed in Section 2D of Garner et al [11].…”

“…We also assumed that the object is located in the neighborhood of the origin of Kʹ at tʹ = 0, so that an interrogating beam of finite cross section could be approximated by a plane wave of infinite cross section for scattering calculations involving an object of finite dimensions. The implementation of the framehopping technique is described in detail elsewhere [11]. 3 shows the time traces of the backscattered signals from the SiC disk oriented such that ̂=̂.…”

“…Velocities in these directions result in the largest changes in the magnitude and the spectrum of the interrogating signal between Kʹ and K [11].…”

Lorentz invariance of absorption and extinction cross sections of a uniformly moving object

“…The extinction formula may also be used to compute the energy removed from a pulsed plane wave by a uniformly translating object. However, scattering is inelastic due to the twoway Doppler shift [13] except in the forward direction. Accordingly, although the extinction cross section has to be equal to the sum of the absorption and the total scattering cross sections in the co-moving inertial frame of reference [3][4][5], that equality is not guaranteed prima facie in the laboratory frame of reference which is also inertial.…”