Four ‐ dimensional relativistic scattering of electromagnetic waves from an arbitrary collection of moving lossy dielectric spheres

Four ‐ dimensional (4D) relativistic scattering of electromagnetic waves from an arbitrary collection of uniformly translational moving lossy dielectric spheres is discussed. Two reference frames, four 4D coordinate systems and Lorentz transformation are used to obtain the scattered electromagnetic fields. The direct scattering of the spheres and their interactions are considered with a novel approach. The introduced method is straightforward and the analytical relations for the fields are achieved. To check the validity of the proposed method, different examples for both stationary and moving scatterers are investigated. The effects of key parameters such as the size, material, velocity, number, position of the spheres and also the frequency of the incident wave are discussed. The derived scattered fields are valid for low, medium and high velocities but according to practical applications low and moderate velocities are highlighted in numerical results.


| INTRODUCTION
A very interesting subject in electromagnetics is the scattering of electromagnetic (EM) waves from moving objects which has been investigated by researchers over the last century. From the standpoint of applications, for low and moderate speed cases, it can be used to calculate the attenuation and transmission of EM waves for rainy, snowy and dusty mediums, which is very important in meteorology, satellite communications, environmental issues, radar applications and remotely sensed data. Furthermore, for the profile of highspeed objects, it has applications in the understanding of scattering by relativistically moving interstellar dust grains [1], moving plasma columns [2][3][4] and mass flows in pneumatic pipes [5].
The important properties of objects such as shape, material, and velocity could be obtained by processing the scattered fields, which is known as inverse scattering. Furthermore, with post-processing, other significant practical characteristics of a collection of objects such as scattering, extinction and absorption cross-sections could be derived. The main challenge in random and multiple scattering is obtaining the scattered fields from a collection of moving scatterers. In this case, 4 four-dimensional (4D) coordinate systems for the rest and moving frames and Special Theory of Relativity (STR) should be considered which causes mathematical difficulties.
Electromagnetic scattering of a translational moving body with STR, which was first introduced by Einstein in 1905 [6], has been used for a moving perfectly reflecting mirror [7,8]. By applying STR, EM scattered waves for different moving shapes such as dielectric medium [9,10] cylinder [11][12][13][14][15], conducting sphere [16], electrically small chiral sphere [17], small particle [18], perfectly conducting flat plate [19], rough surface [20], arbitrary obstacle [21][22][23][24][25][26][27], wedge [28,29] and half-plane [30][31][32] have been derived. Also, scattering characteristics (scattering cross-section, extinction and absorption) for a uniformly moving object [33] and a moving concentrically layered sphere [34] are discussed. The backscattered signal by a uniformly moving sphere considering the incident wave to be a pulsed plane wave, is investigated [35]. All foregoing works discuss only one moving object whereas practical applications mostly deal with a collection of many moving random objects. In that case, not only investigation on the relativistic translational motion of the individual object is required, the mutual interactions of the moving objects also have significant effects, which make the solution more complicated.
Time-domain scattered fields from an arbitrary collection of uniformly translational moving lossy-dielectric spheres are calculated in the far-field region. The size, material, velocity, number, position of the spheres and the frequency of the incident wave can be selected arbitrarily, which makes studying of effective parameters possible. By considering the intrinsic inaccuracies of using numerical techniques, such as the finite-difference-time-domain method (FDTD) and Lorentz precise integration time-domain method (Lorentz-PITD), for a moving object [36][37][38][39][40][41][42], here the STR and Mie theory [43][44][45][46] are employed. Also, other kinds of motions for an individual object such as rotational [47][48][49][50][51] and vibrational [52][53][54] have been investigated, but, here the translational motion is of interest.

| FORMULATION OF THE PROBLEM
A collection of uniformly translational moving spheres of radius a, a complex refractive index of n ¼ n 0 þjn″ and a constant velocity of ν → ¼ vẑ moving along the z-direction is considered. Four 4D coordinate systems (three dimensions are associated with the position and one with the time) are considered regarding the rest and moving frames which are denoted with K and K 0 , respectively, as shown in Figure 1. In the rest frame, the spheres appear to be moving and in the moving frame, the spheres seem to be stationary. To synchronize times, K and K 0 are considered to coincide at time t ¼ t 0 ¼ 0. Each frame has two spatial coordinate systems. To characterize and solve the whole problem, Cartesian (x,y,z) and spherical ðr; θ; φÞ coordinate systems are considered for the rest frame (K) and similarly, the prime forms (x 0 ,y 0 ,z 0 ) and ðr 0 ; θ 0 ; φ 0 Þ are chosen to indicate the quantities in the moving frame (K 0 ). Considering that all spheres are moving in the rest frame, it would be more appropriate to transform the problem to the moving frame. In other words, the problem is solved in K 0 and then the resulting scattered fields transformed back into K. The incident wave and the observation point are given in the K frame. According to symmetry characteristics of spheres, without losing generality, an incident plane wave is considered to propagate in the negative x^direction 1 and has a polarization in the y direction which can be expressed in the rest frame by: where η is the intrinsic impedance of the free space and k is the wave number in the rest frame.

| Electromagnetic scattering from a moving sphere:
At time t ¼ 0, the coordinates of the ith moving sphere is represented by (x i0 , y i0 , z i0 ). Moreover, τ is the time when the ith sphere scatters a spherical wavefront and r i is the instantaneous distance between the ith moving sphere and the observation point (x p , y p , z p ), as shown in Figure 2. So the position of this sphere in the rest frame can be expressed by (x i , y i , z i ,t)¼(x i0 , y i0 , z i0 þ vτ,τ) and the angle θ i is defined as: In orther to associate K and K 0 using the Lorentz transformation [7,8] F I G U R E 1 Two reference frames for relative motion 1 Throughout this article e À jωt used to transform to the time-harmonic fields. RADPOUR ET AL. -181 where β ¼ v/c and c is the speed of light in free space. The components of the spherical coordinate system in K 0 can be written as [7,8]: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The incident plane wave is transformed into the moving frame [7,16]: By applying Mie theory to the incident field 2 for the farfield region (k 0 r 0 ≫1Þ in the moving frame, the direct scattered field can be derived as E 0 ds θ ¼ 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi and δ 0 is the angle between the incident direction (k i´) and the scattering direction (k s´) . The scattering amplitude matrix coefficients for Mie theory [45] can be stated as ½a n π n ðcos δÞ þ b n τ n ðcos δÞ� where π n ðcos δÞ ¼ À P 1 n ðcos δÞ sin δ τ n ðcos δÞ ¼ À dP 1 n ðcos δÞ dδ ð12bÞ a n ¼ k 2 p a 2 j n ðk p aÞ h kaj n ðkaÞ i 0 À k 2 a 2 j n ðkaÞ h k p aj n ðk p aÞ i 0 k 2 p a 2 j n ðk p aÞ where a is the radius of the sphere, a n , b n are the Mie scattering coefficients, j n , h n are the spherical Bessel and Hankel functions of the first kind, respectively, P n is the associated Legendre function and prime is the notation for derivation.
To transform back to the stationary frame, the following relations are used [7].
Hence, the time-harmonic expressions 3 for E θ ds and E φ ds can be written according to Equations (10), (11), and (13) as F I G U R E 2 Scattering configuration and angles for a sphere moving with velocity v along the z-direction To represent the phase factor ðe jk 0 ðr 0 i À ct 0 Þ Þ of the obtained scattered fields in an unprimed form and to relate it with τ it should be noted that By applying Equations (16) and (9) to (14) and (15), the time-harmonic direct scattered fields in the rest frame could be expressed by: It is important to state that although these direct scattered fields components are functions of τ, they represent the scattered fields in the observation point at time t ¼ τþr i /c.

| Secondary electromagnetic scattering fields from two spheres configurations
In this section, two moving spheres are considered and the problem is to evaluate the secondary scattered fields. The secondary scattered fields are the fields scattered from a moving sphere, when illuminated by a primary scattered field from another moving sphere. Assuming that ith and jth spheres are moving along the z-direction, as represented in Figure 3, their positions in the K frame can be expressed by The incident field in Equation (1) is upon the ith sphere and this sphere scatters a field which illuminates the jth sphere, then the jth sphere scatters a field which would be calculated. Since these two spheres move with an equal velocity and have an identical direction of motion, they appear stationary to each other in the moving frame; therefore, the secondary scattered fields can be called coupling fields. It is assumed that the jth sphere is in the far-field region of the ith sphere. If the distance between the two spheres represented by d ij and the radius of the ith sphere denoted by a i then the far-field condition mathematically can be stated as According to the length-contraction property of STR, it can be written that where r j is considered as the instantaneous distance between sphere number j and the observation point (x p ,y p ,z p ) in the K frame.
The parameters (θ ij ,φ ij ) and (θ j ,φ j ) are defined as The prime forms of ðθ ij ; φ ij Þ and ðθ j ; φ j Þin moving frame can be obtained in a similar way to Equations (6) and (7).
The coupling fields would be calculated by applying Mie theory in two levels and transformations between stationary and moving frames would be employed.
The parameters S 1i , S 2i , S 1j and S 2j construct the scattering amplitude matrices that are used to calculate the primary and secondary scattered fields. The parameters S 1i , S 2i which are used for evaluation of the scattered fields from the ith sphere could be obtained with direct replacement of δ 0 i in Equations (12a) and (12b) and S 1j , S 2j are related to the jth sphere that could be achieved by replacing δ 0 ij in Equations (12a) and (12b).
The parametersk i andk s are incident and scattered field propagation directions, respectively, associated with the jth sphere. Therefore ð1 i ;2 i ;k i Þ and ð1 s ;2 s ;k s Þ are the orthonormal unit systems [45] to characterize scattering by the jth sphere which can be defined as Then the secondary scattered fields components in the moving frame can be written as Referring Equation (13) for transforming scattered fields back into the stationary frame, the time-harmonic field components can be expressed as Considering t 0 ij and t 0 j the corresponding elapsed times for d 0 ij and r 0 j , respectively; it can be written that ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Applying Equations (9), (24) and (37) to Equations (34) and (35) leads to the time-harmonic secondary scattered fields in the stationary frame It is important to notice that components of these fields represent secondary scattered fields in the observation point at The developed approach for deriving direct and coupling scattered fields can be generalized to a collection of arbitrary number of spheres by the employment of an iterative procedure to achieve the scattered fields up to the second order. This is a good approximation regarding that the spheres are in the farfield of each other, than the amplitudes of scattered fields more than second order are negligible.

| NUMERICAL RESULTS
Theoretical results achieved in the last section for a collection of both stationary and moving spheres are simulated to have a deeper physical insight into the problem. The incident field is considered to propagate in the negativex direction ðE → i ¼ e À jkxŷ Þ and the maximum value used for n in the Mie theory is set to be 100 to calculate the numerical results. The azimuth and elevation angles are angular measurements in the spherical coordinate system and have intervals of [0,2π]and [0, π], respectively.

| Fields for stationary scatterers
In this section the refractive index of the spheres is set to be n ¼ 3.2þj0.32. An individual stationary sphere with a radius of a ¼ 1 cm and size parameter of ka ¼ 10 (f ¼ 47.75 GHz) which is located at (x, y, z)¼(0, À 5m, 0) is considered. Figure 4(a) represents the three-dimensional (3D) scattered field pattern at a distance of 10 m from the origin of the coordinate system (radius r ¼ 10 m) and - Figure 4(b) illustrates the azimuth pattern for a 90 ∘ elevation angle. According to the position of the sphere, it is expected for the elevation pattern to be symmetric about the elevation angle of 90 ∘ which is confirmed by Figure 4(c).
In the following, scattered fields are calculated for two similar spheres with a ¼ 5 mm and ka ¼ 1.5 (f ¼ 14.3 GHz) which are located at (1.5 cm, 1.5 cm, 0) and (À 1.5 cm, À 1.5 cm, 0). Figure 5(a) shows the field pattern at r ¼ 10 m and the two-dimensional (2D) patterns in the azimuth and elevation planes are illustrated in Figures 5(b,c), respectively. Figure 5(b) demonstrates that the maximum level for coupling fields are approximately around the azimuth angle of 225 ∘ which is expected regarding the position of the spheres and direction of the incident field. Figure 5(c) again has the property of symmetry due to the position of spheres.
Ten similar spheres with a ¼ 5 mm and ka ¼ 3 (f ¼ 28.6 GHz) with locations shown in Table 1 are considered and the bistatic scattered field pattern at r ¼ 20 m is illustrated in Figure 6(a). Maximum amplitude in both Figures 6(a,b) occurs around azimuth angle of 180 ∘ which states the effect of the addition of direct scattered fields. Maximum deviations of the first and second-order fields, which happen at about 45 ∘ ,225 ∘ as highlighted, are due to the coupling interactions regarding the locations of spheres. These deviations show the importance of the coupling fields when the number of spheres increases. Figure 6(c) is also symmetric about the elevation angle of 90 ∘ . RADPOUR ET AL.

| Fields for moving scatterers
In this section, the amplitude of the scattered electric field is illustrated in a determined observation point with respect to the time. The influence of the effective parameters such as the size, material, number, velocity, position of the spheres and the frequency of the incident wave on the scattered fields of the moving spheres has been demonstrated in this section. According to the quick time varying phase of the both primary and secondary scattered fields, it is expected that the field patterns appear with a slowly varying amplitude envelop with a rapidly varying carrier.
Firstly, one sphere with a ¼ 1 mm, ka ¼ 10 (f ¼ 477 GHz), n ¼ 3.2þj0.32 and initial position in the origin of the coordinate system, moving with the velocity of v ¼ 0.5 m/s is considered. The scattered field amplitude in the observation point of (x p , y p , z p )¼(À 5 m, 0, 5 m) is illustrated in Figure 7. According to the speed of the sphere and height of the observation point, it is expected that the maximum level of field amplitude occurs at about t ¼ 10 s revealing that the forward scattering is dominant for this ka value. Since the position of the sphere is symmetric about the observation point, scattered field amplitude must be either symmetric which is in full agreement with Figure 7.
In the next step, the conditions are the same as the previous sphere except that the velocity is set to v ¼ 2 � 10 8 m/s. As can be seen in Figure 8, the amplitudes before the peak moment (t peak ) are larger than their symmetric corresponding moments (after t peak ) which is because of the effect of aberration in the propagation direction phenomenon. Also, the peak amount of the amplitude is decreased compared with the previous condition.
For two spheres scenario, a reference mode is considered and only one parameter would be changed in each following mode to have a better understanding of the intended parameter. In the reference mode, two spheres are assumed to have a ¼ 1 mm, v ¼ 1 m/sand n ¼ 3.2þj0.32. The size parameter is set to ka ¼ 10 (f ¼ 477 GHz) and coordinates of observation RADPOUR ET AL.
-187 point and primary location of spheres are given in Table 2. The maximum amplitude at t ¼ 5 s and symmetry in Figure 9 could be predicted by physical interpretation. The peak at t ¼ 5 s is about two times the peak shown in Figure 7 which indicates that the direct scattered field of each sphere has been added constructively.
This time, the radius of the spheres is changed to a ¼ 1 μm and Figure 10 shows that the amplitude reduction of the scattered field (about 1000 times) is proportional to the reduction of the radiuses. Figure 11 relates to the mode ka ¼ 5. Thus comparing this with the results of the reference mode reveals that the lower ka causes a wider beam, which is in agreement with the general fact that moving from optical through Mie and Rayleigh scattering regions makes scattering pattern more homogeneous then forward scattered pattern becomes wider.
In the next mode to represent the effect of the dielectric material, the extinction coefficient is omitted and the refractive index is set to be n ¼ 3.2, as shown in Figure 12.   Figure 13 is for a condition that spheres move with the velocity of v ¼ 20 m/s which states that the pattern has been scaled. Since the velocity of spheres is negligible when compared with the velocity of light, the electric scattered field amplitude is similar to the reference mode.
In the next situation, the observation point is approached to (x p , y p , z p )¼(À 1 m, 0, 5 m), as shown in Figure 14, which means that the closer distances result in narrower scattered beam-widths.
Next, the velocity is set to v ¼ 2 � 10 8 m/s. As depicted in Figure 15, the scattered electric field amplitude is going to be more asymmetric by increasing the velocity to the relativistic speeds due the aberration in the propagation direction phenomenon. Also, the peak amplitude decreases compared with the previous and reference mode.
In the following, scattered field amplitude for a collection of ten moving spheres with a ¼ 1 mm and v ¼ 1 m/s is calculated. The size parameter and the refractive index are set to be ka ¼ 10 (f ¼ 477 GHz) and n ¼ 3.2þj0.32, respectively. Table 3 specifies the configuration of the spheres collection and the observation point is considered to be (x p , y p , z p )¼(À 1 m, 0, 20 m). Figure 16 demonstrates the resulting scattered field amplitude.
Finally, time-domain electric scattered field amplitude for a collection of ten spheres moving in relativistic speed (β ¼ 2/3) is represented in Figure 17. The remaining parameters are the same as in the previous case. In this work, the Frame-Hopping Method (FHM) which is based on the STR is used to obtain time-domain relativistic scattered fields up to the second-order from an arbitrary collection of uniformly translational moving lossy-dielectric spheres. To gain a deeper physical insight into the problem, scattered fields for a collection of both stationary and moving spheres have been simulated. The influence of effective parameters such as the size, material, velocity, number, position of the spheres and also the frequency of the incident field on the scattered fields of a collection of moving spheres has been investigated and the obtained numerical results are in good agreement with physical concepts. Also, a wide variety of objects, such as raindrops, snowflakes and dust particles, could be approximated by spheres and the study of scattered fields from a collection of moving spheres has a substantial significance for many practical applications. The procedure applied in this work may be the basis for the study of multiple and random scattering from other collections of moving objects considering their mutual interactions.