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Cooperative Optical Wavefront Engineering with Atomic Arrays Nanophotonics 2021; 10(7): 1901–1909 Research article Kyle E. Ballantine and Janne Ruostekoski* Cooperative optical wavefront engineering with atomic arrays https://doi.org/10.1515/nanoph-2021-0059 such metamaterials, known as metasurfaces, can impart Received February 9, 2021; accepted March 26, 2021; an abrupt phase shift on transmitted or reflected light, published online April 15, 2021 allowing for unconventional beam shaping over subwave- length distances [2, 3]. An important example of a metasur- Abstract: Natural materials typically interact weakly with face is the Huygens’ surface, based on Huygens’ principle, the magnetic component of light which greatly limits their that every point acts as an ideal source of forward prop- applications. This has led to the development of artificial agating waves [4, 5]. By engineering crossed electric and metamaterials and metasurfaces. However, natural atoms, magnetic dipoles, a physical implementation of Huygens’ where only electric dipole transitions are relevant at opti- fictitious sources can be realized, providing full transmis- cal frequencies, can cooperatively respond to light to form sion with the arbitrary 2 phase allowing extreme control collective excitations with strong magnetic, as well as elec- π and manipulation of light [6–10]. tric, interactions together with corresponding electric and The use of artificial metamaterials for these applica- magnetic mirror reflection properties. By combining the electric and magnetic collective degrees of freedom, we tions is due to the restriction that most natural materials show that ultrathin planar arrays of atoms can be utilized interact weakly with magnetic fields at optical frequencies, as atomic lenses to focus light to subwavelength spots at to the extent that the magnetic response can be consid- the diffraction limit, to steer light at different angles allow- ered negligible. However, arrays of atoms with a dominant ing for optical sorting, and as converters between different electric dipole transition can have collective excitations angular momentum states. The method is based on coher- which interact strongly with the magnetic light component ently superposing induced electric and magnetic dipoles [11, 12]. As the ability to use the electric dipole response to engineer a quantum nanophotonic Huygens’ surface of of regular ultrathin 2D planar arrays of atoms to control atoms, giving full 2π phase control over the transmission, and manipulate light has already been explored in many with close to zero reflection. contexts [13–26], this opens the way to combine both the electric and magnetic degrees of freedom. Subwavelength Keywords: beam focusing; beam steering; cooperative arrays of atoms can operate at a single-photon quantum optical response; Huygens’ surface; metasurfaces; quan- level [27–32] and are increasingly experimentally achiev- tum optics. able [33–36]. Indeed, recent experiments have already investigated the cooperative response of a single layer of atoms and found characteristic spectral narrowing below 1 Introduction the fundamental quantum limit for a single atom [33]. Here we illustrate how an atomic Huygens’ surface[11] can be used for novel beam-shaping and optical manipu- The quest for artificial materials with a strong magnetic, lation applications. Collective coherent, uniform electric- as well as electric, response at optical frequencies, has dipole and magnetic-dipole excitations of atomic arrays fueled the rich and rapidly expanding field of metamate- have different fundamental ectionrefl properties, corre- rials [1]. Thin, effectively two-dimensional (2D), layers of sponding to electric and magnetic mirrors, for which in the latter case the standard π phase shift of the reflected beam *Corresponding author: Janne Ruostekoski, Department of Physics, is absent. We show how superpositions of such collective Lancaster University, Lancaster LA1 4YB, UK, excitations can be used to create a nearly reflection-less E-mail: [email protected]. Huygens’ surface to engineer the wavefront of the light, https://orcid.org/0000-0001-7197-0942 Kyle E. Ballantine, Department of Physics, Lancaster Univer- so that the atomic array acts as an ultrathin flat lens, sity, Lancaster LA1 4YB, UK, E-mail: [email protected]. an optical sorter, or a converter to different orbital angu- https://orcid.org/0000-0002-2419-9961 lar momentum (OAM) states of light. The simulations of Open Access. ©2021 Kyle E. Ballantine and Janne Ruostekoski, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 1902 | K. E. Ballantine and J. Ruostekoski: Cooperative optical wavefront engineering the atomic array produce diffraction-limited focusing of be relevant, while other multipoles, including magnetic light with very short wavelength-scale focal lengths. The dipoles, are negligible. The total electric field amplitude optical sorting is achieved by steering the beam’s propaga- is then the sum of the incident field amplitude and the tion in desired directions using the atoms with negligible scattered field from each atom acting as a point elec- reflection. tric dipole [43], and from the resulting coupled equations thelightfieldcanbesolvedexactly[39]. For subwave- length spacing, the light-mediated long-range interactions 2 Engineering optically active between the atoms are strong due to recurrent multiple scattering where a photon is scattered more than once by magnetism the same atom. The steady-state of the system in the low light-intensity limit is described by the set of polarization ( j) 2.1 Atom–light interaction amplitudes ,wherethedipolemomentofatomj is d = ∑ j ( j)ê ,and and ê are the reduced dipole matrix We consider a rectangular lattice in the yz plane, at x = 0, element and the relevant unit vector, respectively [39]. An with spacing d and d . Every lattice site consists of a unit y z identical formalism, without drive, describes the decay of a cell of four atoms displaced by ±(a ∕2)x̂ ± (a ∕2)ŷ,witha x y single-photonexcitationintheabsenceofanincidentlaser |J = 0⟩ → |J′ = 1, m = ⟩ transition. Such a geometry, illus- field, where ( j) then represents the probability amplitude trated in Figure 1, could be realized as a bilayer optical of the excitation on level of atom j [30, 44]. lattice [37, 38], with a double-well superlattice in the y The response of the array to incident light can be direction with two minima at ±(a ∕2)ŷ in every period d , y y understood in terms of the collective excitation eigen- or by optical tweezers [36]. Each atom is driven by the modes v of the radiatively coupled atoms, with eigen- coherent incident laser field = (r)eikxê (all field com- n y values + i ,where is the collective line shift and is ponents and amplitudes here and in the following refer n n n n the collective linewidth,which can differ dramatically from to the slowly varying positive frequency components with the linewidth of a single atom [45]. It is useful to define the rapid variations ∼ exp(iΩt)atthelaserfrequencyΩ the biorthogonality occupation measure of the collective filtered out), as well as by the scattered field from all other eigenmode v as [17] atoms. In the limit of a low-intensity coherent laser-drive, j theatomsbehaveasclassicallinear coupled dipoles,i.e.,as |vT b|2 a set of damped, driven, coupled, harmonic oscillators, as L = ∑ j , (1) j |vT b|2 described indetailinreferences[39–42].Theoriginofthese k k dipoles are the quantum-mechanical electronic transitions ( j) where b+3 j−1 = denote the polarization amplitudes in between atomic orbitals. Due to the selection rules, only vector form. resonant transitions corresponding to electric dipoles can To understand the collective excitations of the entire array, we first consider an isolated unit cell of four atoms. For a ≲, the scattered field from an individual unit cell can be well described by a multipole decomposition in terms of single electric and magnetic dipoles, quadrupoles, etc., at its center. Each unit cell in isolation exhibits collective eigenmodes, owing to the light-mediated cou- pling between the four atoms. The eigenmode shown in Figure 1(b) consists of all atomic dipoles oscillating in phase and pointing in one direction leading to an effec- tive electric dipole moment. While this collective mode replicates the electric dipole moment of individual atoms, radiative excitations with, e.g., magnetic properties, not Figure 1: An atomic Huygens’ surface with strong magnetic response present in individual atoms, can also be engineered by at optical frequencies. (a) The atomic array consists of a 2D square utilizing more complex collective excitation eigenmodes. lattice in the yz plane. Each site further consists of a square unit cell A second eigenmode, shown in Figure 1(c), consists of of four atoms, forming an atomic bilayer. (b) Uniform polarization on each atom leads to an effective electric dipole moment d from the an arrangement of four atoms at the corners of a rectangle, unit cell. (c) Azimuthal polarization leads to a net zero electric dipole representing a net zero electric dipole, but a perpendicular moment, but a perpendicular magnetic dipole moment m. magnetic dipole. The orientations of quantum-mechanical K. E. Ballantine and J. Ruostekoski: Cooperative optical wavefront engineering | 1903 atomic transitions in these four point-like discrete atoms, and atomic structures [12]. For the magnetic dipole excita- each of which generates an electric dipole,
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