Identifying regions of minimal backscattering by a relativistically moving sphere

The far-field backscattering amplitude of an electric field from a relativistically moving sphere is analyzed. Contrary to prior research, we do so by expressing the fields in the helicity basis and we highlight here its advantages when compared to the commonly considered parity basis. With the purpose of exploring specific scattering phenomena considering relativistic effects, we identify conditions that minimize the backscattered field, leading to a relativistic formulation of the first Kerker condition. The requirements to be satisfied by the sphere are expressed in terms of Mie angles, which constitute an effective parametrization of any possible optical response a sphere might have. By considering different speeds of the sphere and angles of incidence, we are able to identify multiple combinations of Mie angles up to octupolar order via gradient-based optimization that satisfy our relativistic Kerker condition, that is, where the backscattered energy is at most 0.1% of the average scattered energy. Our results can be extended to involve multiple particles forming a metasurface, potentially having direct implications on the design of light sails as considered by the Breakthrough Starshot Initiative.

Figure 1

Pictorial representation of the sphere in (a) the beam's frame $S^{\parallel }$ and the rotated frame $S$ containing an external observer (represented by the eye) and (b) the sphere's inertial reference frame $S^{\prime }$. In $S$ the sphere is seen to be moving with velocity $v\widehat{\mathbf{z}}$. The wave vector ${\widehat{\mathbf{k}}}_{\mathrm{i}}$ is incident on the sphere with angle ${\mathrm{\Theta }}_{\mathrm{i}}$. In (a) the direction of backscattering ${\widehat{\mathbf{k}}}_{\mathrm{BS}}$ (denoted by the red arrow) is in the opposite direction to ${\widehat{\mathbf{k}}}_{\mathrm{i}}$ and it is this direction that is considered when formulating a relativistic Kerker condition. In (b) the sphere is stationary ($v=0$) in $S^{\prime }$, while the Lorentz-boosted wave vector of the incident field is given by ${\widehat{\mathbf{k}}}_{\mathrm{i}}^{\prime }$ and is incident with angle ${\mathrm{\Theta }}_{\mathrm{i}}^{\prime }\ne {\mathrm{\Theta }}_{\mathrm{i}}$ when ${\mathrm{\Theta }}_{\mathrm{i}}\notin \{0,\pi \}$ and ${\mathrm{\Theta }}_{\mathrm{i}}^{\prime }={\mathrm{\Theta }}_{\mathrm{i}}$ when ${\mathrm{\Theta }}_{\mathrm{i}}\in \{0,\pi \}$. Moreover, the direction of backscattering ${\widehat{\mathbf{k}}}_{\mathrm{BS}}^{\prime }$ in this frame is in general not opposite to ${\widehat{\mathbf{k}}}_{\mathrm{i}}$ or ${\widehat{\mathbf{k}}}_{\mathrm{i}}^{\prime }$. (a) Frames $S^{\parallel }$ and $S$ and (b) Frame $S^{\prime }$.

Figure 2

(a) Amplitude in the form of a transverse cross section of the incident beam given by $|{\mathbf{E}}_{\mathrm{i}}^{\parallel }\left(\mathbf{r}\right)|$ with arbitrary wavelength $L$ and waist $w_0=10L$. The overlaying red arrows depict the left-handed circular polarization of the beam (${\lambda }_{\mathrm{i}}=+1$) scaled with $|{\mathbf{E}}_{\mathrm{i}}^{\parallel }\left(\mathbf{r}\right)|$. (b) Schematic for the finite interaction between the sphere and the incident beam.

Figure 3

(a) Doppler shift ${\mathrm{\Theta }}_{\mathrm{i}}^{\prime }$ of ${\mathrm{\Theta }}_{\mathrm{i}}$ as a function of ${\mathrm{\Theta }}_{\mathrm{i}}$ and $\beta$ determined using Eq. (14). This demonstrates that the direction of the incident wave as perceived in $S^{\prime }$ is different from that in $S$. (b) Normalized Doppler-shifted incident frequency ${\omega }_{\mathrm{i}}^{\prime }/\gamma {\omega }_{\mathrm{i}}$ as a function of ${\mathrm{\Theta }}_{\mathrm{i}}$ and $\beta$, where ${\omega }_{\mathrm{i}}^{\prime }$ is the Doppler-shifted incident frequency determined using Eq. (15). When ${\mathrm{\Theta }}_{\mathrm{i}}=0$ and $\beta \to 1$, the object is moving away from the field source, thus causing the incident frequency in $S^{\prime }$ to decrease (redshift). When ${\mathrm{\Theta }}_{\mathrm{i}}=\pi$ and $\beta \to 1$, the object is moving towards the field source, thus causing the incident frequency in $S^{\prime }$ to increase (blueshift). When ${\mathrm{\Theta }}_{\mathrm{i}}=\pi /2$, ${\omega }_{\mathrm{i}}^{\prime }/\gamma {\omega }_{\mathrm{i}}=1$ for all $\beta$. This is due to ${\widehat{\mathbf{k}}}_{\mathrm{i}}$ and $\mathbf{v}$ being perpendicular to each other. The factor $1/\gamma$ is necessary to eliminate the exaggeration of ${\omega }_{\mathrm{i}}^{\prime }$ when $\beta \to 1$ and ${\mathrm{\Theta }}_{\mathrm{i}}\to \pi$. Without this factor, the other frequency values would appear too suppressed.

Figure 4

The (a) ${\lambda }_{\mathrm{s}}=+1$ and (b) ${\lambda }_{\mathrm{s}}=-1$ components of ${\log }_{10}\left(D_{\mathrm{BS}}\right)$ when ${\theta }_{\mathrm{ED}}={\theta }_{\mathrm{MD}}=\frac{\pi }{3}$. Note in (b) the dark diagonal line, which means that the backscattering vanishes when ${\theta }_{\mathrm{EQ}}={\theta }_{\mathrm{MQ}}$. This corresponds to the case where the sphere is a dual scatterer. Since the incident helicity is given by ${\lambda }_{\mathrm{i}}=+1$, these vanishing components are to be expected. (c) Total directivity ${\log }_{10}\left(D_{\mathrm{BS}}\right)$ when combining (a) and (b). (d) For comparison ${\log }_{10}\left(D_{\mathrm{BS}}\right)$ when ${\theta }_{\mathrm{ED}}=\frac{\pi }{9}$ and ${\theta }_{\mathrm{MD}}=-\frac{\pi }{4}$. In all cases, we have that $\beta =0.2$ and ${\mathrm{\Theta }}_{\mathrm{i}}=\frac{\pi }{4}$.

Figure 5

Directivity $D_{\mathrm{BS}}$ (in ${\log }_{10}$ scale) as a function of $\beta$ and ${\mathrm{\Theta }}_{\mathrm{i}}$ with ${\theta }_{\mathrm{ED}}=\frac{\pi }{4}$, ${\theta }_{\mathrm{EQ}}=-\frac{\pi }{9}$, ${\theta }_{\mathrm{MD}}=\frac{\pi }{3}$, and ${\theta }_{\mathrm{MQ}}=\frac{\pi }{7}$.

Figure 6

Backscattered directivity $D_{\mathrm{BS}}$ up to octupolar order as a function of the radius $R$ of a spherical SiC particle with refractive index $n_{\mathrm{SiC}}=2.59$ for a fixed angle of incidence ${\mathrm{\Theta }}_{\mathrm{i}}=\frac{\pi }{4}$ and three fixed speeds $\beta =1\times {10}^{-6}$, 0.2, and 0.5. Here $R$ is given as multiples of the wavelength $L=1\mu \mathrm{m}$ of the incident beam in $S$.

Figure 7

Variation of the optimized (a) Mie angles and (b) directivity $D_{\mathrm{BS}}$ with respect to the speed $\beta$ for a fixed angle of incidence ${\mathrm{\Theta }}_{\mathrm{i}}=\frac{\pi }{4}$. (c) and (d) analogous study but with varying ${\mathrm{\Theta }}_{\mathrm{i}}$ and fixed $\beta =0.2$.

Figure 8

Absolute value of the dipole expansion coefficients ${\mathcal{A}}_{10}$ $({\ell }^{\prime }=1,m^{\prime }=0)$ given by Eq. (27) as a function of the normalized frequency $({\omega }_{\mathrm{i}}-{\omega }^{\prime })/{\omega }_{\mathrm{i}}$ when $\beta =0.1$ (blue solid line), $\beta =0.2$ (orange dotted line), $\beta =0.5$ (yellow thick solid line), and $\beta =0.9$ (purple dashed line). In all cases, ${\mathrm{\Theta }}_{\mathrm{i}}=\frac{\pi }{4}$ and ${\lambda }_{\mathrm{i}}=+1$ which, since helicity is conserved under Lorentz boosts, means that ${\lambda }^{\prime }={\lambda }_{\mathrm{i}}=+1$. Note that the width of the peak increases with speed. This is due to the increasing effect of the Doppler shift ${\omega }^{\prime }$ of ${\omega }_{\mathrm{i}}$. Since the incident beam is monochromatic, $|{\mathcal{A}}_{10}|$ tends to a $\delta$-distribution-like peak as the speed decreases. As the speed increases, the plane-wave components of the incident beam all Doppler shift differently due to their differing angular orientations. This leads to nonzero values for $|{\mathcal{A}}_{10}|$ when ${\omega }_{\mathrm{i}}\ne {\omega }^{\prime }$.