Kerker Conditions upon Lossless, Absorption, and Optical Gain Regimes

The directionality and polarization of light show peculiar properties when the scattering by a dielectric sphere can be described exclusively by electric and magnetic dipolar modes. Particularly, when these modes oscillate in phase with equal amplitude, at the so-called first Kerker condition, the zero optical backscattering condition emerges for nondissipating spheres. However, the role of absorption and optical gain in the first Kerker condition remains unexplored. In this work, we demonstrate that either absorption or optical gain precludes the first Kerker condition and, hence, the absence of backscattered radiation light, regardless of the particle’s size, incident wavelength, and incoming polarization. Finally, we derive the necessary prerequisites of the second Kerker condition of the zero forward light scattering, finding that optical gain is a compulsory requirement.

Figure 1

(a) Real (dash-dotted red) and imaginary part (dashed-red) of the refractive index contrast ($m$) vs the incident wavelength ($\lambda$) for a Ge sphere. Maximum of the expected value of the EM helicity in solid blue, $⟨\mathrm{\Lambda }{⟩}_{\max }$, for a Ge sphere vs $\lambda$ under plane wave illumination with $\sigma =+1$. (b) Color map of $⟨\mathrm{\Lambda }⟩$ vs $\lambda$ and particle’s size ($R$) for a Ge sphere under plane wave illumination with $\sigma =+1$. The visible range is encompassed by a dashed rectangle. As mentioned in the text, in this region helicity conservation is never fulfilled.

Figure 2

Normalized scattering patterns by Ge spheres calculated from Eq. (12) in the telecom spectral regime (a) [$\lambda =2100\mathrm{nm}$ and $R=223\mathrm{nm}$] and visible spectral range (b) [$\lambda =632\mathrm{nm}$ and $R=48\mathrm{nm}$] and (c) [$\lambda =575\mathrm{nm}$ and $R=35\mathrm{nm}$]. The $g$ parameter is given by $g=0.5$, 0.3, 0.1, respectively, in the dipolar regime. The color map and the 3D spatial distribution represent the normalized far field differential scattering cross section as a function of the scattering angle.

Figure 3

$g$ parameter vs the imaginary part of the refractive index contrast $\Im \{m\}$ and the $x=ka$ size parameter under well-defined EM helicity ($\sigma =+1$) plane wave illumination. In this range, the optical response is purely of a dipolar nature. The first and second Kerker conditions are depicted by black and white circles, respectively.