Optical Magnetism and Huygens’ Surfaces in Arrays of Atoms Induced by Cooperative Responses

By utilizing strong optical resonant interactions in arrays of atoms with electric dipole transitions, we show how to synthesize collective optical responses that correspond to those formed by arrays of magnetic dipoles and other multipoles. Optically active magnetism with the strength comparable with that of electric dipole transitions is achieved in collective excitation eigenmodes of the array. By controlling the atomic level shifts, an array of spectrally overlapping, crossed electric and magnetic dipoles can be excited, providing a physical realization of a nearly reflectionless quantum Huygens’ surface with the full $2\pi$ phase control of the transmitted light that allows for extreme wavefront engineering even at a single photon level. We illustrate this by creating a superposition of two different orbital angular momentum states of light from an ordinary input state that has no orbital angular momentum.

Figure 1

(a) Array with square unit cells of lattice constant $d$ and unit cell of size $a$ to generate an array of magnetic dipoles pointing normal to the plane. (b) Bilayer array forming a Huygens’ surface with crossed electric ($\mathbf{d}$) and magnetic ($\mathbf{m}$) dipoles.

Figure 2

(a) Occupations of collective excitation eigenmodes (ordered by linewidth ${\upsilon }_i$) in a steady state for a $20\times 20(\times 4)$ array ($d=0.5\lambda$, $a=0.15\lambda$), shown in Fig. 1, illuminated by a Gaussian profile ${\mathcal{E}}_0(y,z)$ with $1/e^2$ radius $7\lambda$, when detunings are optimized to target the collective eigenmode (red square) which is closest to a uniform repetition of the unit-cell electric dipole mode. (b) As in (a), but with a target magnetic dipole eigenmode closest to a uniform distribution of magnetic dipoles in the $x$ direction (red square). Insets (a),(b): the resulting atomic dipoles on a central unit cell.

Figure 3

(a) Transmission $T$ (left axis) and phase (right axis) of the incident and scattered light in the forward direction as a function of the laser frequency for plane-wave illumination of a $32\times 32$ lattice ($d=0.8\lambda$, $a=0.15\lambda$) showing the full $2\pi$ range with transmission everywhere exceeding 90%. (b) Decomposition of the corresponding radiative excitation into (curves from top to bottom) electric and magnetic dipole, and electric quadrupole contributions. The relative level shifts of all levels are held fixed, and identical on each unit cell.

Figure 4

(a) Transmitted intensity and (b) phase of the total electric field, calculated from each dipole at a distance $5\lambda$ for a $20\times 20$ lattice ($d=0.8\lambda$, $a=0.15\lambda$) shown in Fig. 1. An incident Gaussian beam with waist $w_0=5\lambda$ is transformed into an equal superposition of states with orbital angular momentum 0 and $\hslash$ per photon. The intensity varies with angle due to interference between the two components.