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 subwavelength 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 subwavelength 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 𝜎 ,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, 𝜎 𝜎 𝜎 𝜎 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 = 𝜎⟩ transition. Such a geometry, illus- field, where  ( j) then represents the probability amplitude trated in Figure 1, could be realized as a bilayer optical 𝜎 of the excitation on level 𝜎 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 𝛿 + i𝜐 ,where𝛿 is the collective line shift and 𝜐 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 𝛾 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 b𝜎+3 j−1 = 𝜎 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 ≲𝜆, 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 coupling 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 effective 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, approximate a tion shown in Figure 1(c),they component of polarization circular loop of a continuous azimuthal electric polariza- is anti-symmetric in x. For the uniform, coherent, excitation density. In this collective eigenmode of Figure 1(c), tion of a large lattice with all dipoles oscillating in phase, the electric dipoles at each point due to the electronic the scattered field in the forward and backward direc- transitions therefore leads to an equivalent circulating (−) tions depends only on this in-plane component, and Es = current around the center producing a magnetic dipole (+) −Es .Hence,whenthemagnetic dipolelatticedisplaysfull moment, analogous to a classical continuous distribution reflection it is with r = 1, and an antinode at the plane of of oscillating charges on a ring. This subwavelength cur- the mirror. This distinction means that an emitter placed in rent loop is close to indistinguishable in the far-field from the near-field will interfere constructively with its image, a fundamental magnetic dipole [11, 12]. rather than destructively, as for an electric mirror [53]. Strong light-mediated interactions between different Figure 2(a) shows the magnitude and phase of the unit cells then lead to collective radiative excitations of the reflection amplitude for an incident Gaussian beam, as the ( j) ( j) whole lattice that synthesize effective electric and mag- detuning Δ=Δ휎 =Ω−휔휎 of the laser frequency from ( j) netic dipole arrays. In particular, the lattice in Figure 1(a) the identical atomic resonance 휔휎 of each level 휎 on atom has two collective excitation eigenmodes of interest. One is j is tuned through the resonance of the collective magnetic a uniform repetition of coherent, in-phase, electric dipole excitation. While the magnetic dipole mode is very subra- moments oriented along the y directiononeachunitcell, diant for large lattices, it can be well excited by an incident and has a collective linewidth comparable with a single plane wave or Gaussian because the exp(ikx) rapid phase atom [46]. The other is a similarly coherent, uniform, exci- variance in the x direction has an overlap with the anti- tation of magnetic dipole moments oriented along the z symmetric polarization amplitudes of the mode. We find direction, which has a significantly narrower linewidth for the numerical value r ≈ 0.99 exp(0.1i)onresonance,close large lattices. to the expected value r = 1. This full reflection is a result of the collective nature of the excitation, and depends 2.2 Magnetic mirror on the uniform, in-phase, coherent nature of the magnetic dipole oscillations, as well as the symmetry of the Full reflection from an array of dipoles can occur when individual dipole radiation. The occupation L of the collec- (+)  (+) (−) Es =− ,whereEs (Es ) is the scattered field in the tive excitation eigenmode corresponding most closely to forward (backward) direction, leading to destructive inter- this ideal uniform magnetic dipole excitation is shown in ference in the transmitted light and a standing wave in Figure 2(b), along with the occupation of all other modes. the backward direction. As seen in Figure 1(b), the elec- Although the magnetic mode is dominant, it is the small tric dipole collective excitation eigenmode has polariza- contribution from other modes, with different symmetry, tion amplitude that is symmetric around x = 0, and so which leads to a minor deviation from r = 1. (−) (+) Es = Es . Defining the complex reflection and transmis- In Figure 3, the spatial dependence of the intensity sion amplitudes, of the total (incident plus scattered) light is shown at the [ ] (−) ̂ ⋅  (+) ê ⋅ E ey + Es y s , , (2) r = ̂ ⋅  t = ̂ ⋅  ey ey respectively, gives r =−1whent = 0.Whileherewecon- sider a collective mode of effective dipoles formed on each unit cell, this result is well-known for uniform arrays of coherently oscillating electric dipoles [47–50], including equivalent single layers of atoms [17, 18, 23]. The r =−1 condition leads to a node of the standing wave at the array, equivalent to reflection from a perfect electrical conductor. Figure 2: Collective resonance of a magnetic mirror. (a) Magnitude Indeed, reflection from the electric-dipole collective exci- (left axis) and phase (right axis) of reflection coefficient r showing tation of a single-layer atomic array with subwavelength r ≈ 1 close to magnetic resonance. (b) Occupation L of magnetic dipole mode and L′ of all other modes, normalized such that spacing has now been observed in experiments [33]. An ∑ j Lj = 1 at the center of the resonance. Gaussian beam with alternative condition, with r = 1, occurs for reflection from  푤 . 휆 (0) = 1, beam waist radius 0 = 6 4 ,incidenton25× 25 × 4 . 휆 . 휆 magnetic dipole excitations in metal [51], dielectric [52–54] lattice with dy = dz = 0 7 , ax = ay = 0 15 . 1904 | K. E. Ballantine and J. Ruostekoski: Cooperative optical wavefront engineering

Figure 3: Magnetic and electric mirror formed by a bilayer of atoms. Figure 4: Principle of Huygens’ surface of atoms. Possible complex 2 2 (+) Spatial variation of |E| ∕|(0)| in the z = 0 plane, resulting from values for the forward scattered field Es and the total transmitted  (+)  interference between incoming Gaussian beam and reflected light field E = + Es due to an incident field for (a) a uniform electric from (a) magnetic mirror, when the incident light is resonant with dipole excitation and (b) crossed electric and magnetic dipoles at the collective magnetic dipole mode, showing antinodes at integer each site. The circles show range of possible values traced out as and half-integer distances from the lattice corresponding to r ≈ 1, the detuning is varied. (c) Numerical calculation of the scattered and (b) on resonance with the electric dipole mode with r ≈−1. field from plane wave ( = 1) incident on an atomic Huygens’ surface Parameters as in Figure 2. as the detuning ⟨Δ⟩ of the laser from the average single-atom resonance is varied. The phase of the scattered light (dashed line, right axis) ranges from π∕2to3π∕2 as a function of the laser frequency of the collective magnetic mode, and at the col- frequency, while the magnitude (solid line, left axis) can be almost lective electric dipole mode resonance. As well as almost double the input field. (d) Magnitude (left axis) and phase (right axis) of total forward-propagating field showing high transmission complete reflection, a clear 휆∕4 shift in the positions of the over the full 2π range. (c) and (d) are for 20 × 20 × 4 lattice with peaks of the resulting standing wave is evident. . 휆 . 휆 . 휆 dy = dz = 0 8 , ax = 0 12 and ay = 0 11 .

3 Huygens’ surface laser frequency, and the circles illustrate the range of possible values these fields can take as the detuning varies. As resonance is approached, the scattered field grows orig- We superpose the collective electric-dipole and magnetic- inally at a phase 2 to the incident field, reaching a dipole excitations to form a nearly reflection-less Huygens’ π∕ maximum amplitude when E(+) =−, and then falling off surface that controls the phase of the transmitted light s to engineer its wavefront. A Huygens’ surface is a physical as the phase approaches 3π∕2, tracing out the blue circle.  implementationofHuygens’principle[4],whichstatesthat The total transmitted field E meanwhile starts equal to every point in a propagating wave acts as a source of further far from resonance, then decreases to zero where the phase  forward-propagating waves. While electric and magnetic shifts from π∕2to−π∕2, before returning to ,tracingout < < dipoles both scatter light forwards and backwards with theorangecircle.Theresultis−π∕2 arg(t) π∕2, and | | ≤ | | equal amplitude, a crossed electric and magnetic dipole t 1, with t = 0onresonance. can lead to destructive interference in the backward direc- For the simultaneous excitation of the uniform collec- tion and constructive interference in the forward direc- tive modes corresponding to crossed magnetic and electric dipoles, illustrated in Figure 4(b),wewriteE(±) = E(±) + tion, providing a physical realization of Huygens’ fictitious s s,d (±) where (±) ( (±) ) is the contribution from the electric sources [5, 55]. Es,m Es,d Es,m (+) The principle of how simultaneous excitation of both (magnetic) dipoles. Again, away from resonance, Es = 0 modes can lead to full transmission, with arbitrary phase, and t = 1. As the resonance is approached, however, the is illustrated in Figure 4.InFigure 4(a),theindividual two contributions lead to double the amplitude, and on response of either of the collective excitation eigenmodes resonance E(+) E(+) ,givingt 1, after which the s,m = s,d =− =− (electric or magnetic) to an incident field  is shown, where scattered field decreases again to zero. In between the (+) the arrows display the forward-scattered field Es and the phase of E varies over the full 2π phase range while the sym- total transmitted field  (+) in the complex plane for metry condition E(−) E(−) ensures r 0 as expected [5]. E = + Es s,d =− s,m = some particular detuning from resonance of the incident The destructive interference of the reflected fields, which K. E. Ballantine and J. Ruostekoski: Cooperative optical wavefront engineering | 1905

is a pre-condition for a Huygens’ surface leading to full E(+) E(+) ,andE(−) E(−) .Theresultiscloseto s,d ∼ s,m ∼− s,d ∼− s,m transmission, is known as the Kerker effect [56, 57]. full transmission with a π phase shift. As the occupation The numerically calculated scattered light from the of the electric dipole mode is a the direct result of coupling atomic array in Figure 4(c) shows that indeed the scat- from the magnetic mode, the occupation of both modes tered field approaches twice the magnitude of the incident falls away on either side of the magnetic resonance, and field. While the phase of the scattered field is constrained the total scattered field decreases to zero. between π∕2and3π∕2, as is the case for a single uniform excitation, this increased magnitude means that the total field in Figure 4(d) covers a full 2π phase range, while the 4 Beam focusing transmission |t| = |E|∕|| remains close to one. We find from a multipole decomposition that at the center of the We show how an ultrathin flat atomic array, acting as resonance the normalized electric dipole moment from a a Huygens’ surface, can achieve tight diffraction-limited single unit cell is 0.55, while the magnetic dipole moment focusing in a propagation distance of a few wavelengths. is 0.44, with the remaining contribution from quadrupole Thetransmittedphaseprofileischosentobeequivalentto or higher moments. alens,namely[60, 61] To simultaneously excite the electric and magnetic ( )( √ ) 2휋 collective excitation eigenmodes, we take the level shifts 휙 = f − f 2 + 휌2 , (3) ( j) 휆 Δ휎 to vary independently between atomic levels within √ a unit cell, while keeping them identical between each where f is the focal length of the lens and 휌 = y2 + z2, unit cell. The relative level shifts, along with the unit cell but because of the abrupt phase change at the array, this size and lattice constants, are numerically optimized to is achieved in a vastly smaller propagation distance than maximize transmission, and here in the studied examples a solid lens. Here, the phase profile is implemented by vary between ±4훾. The level shifts could be achieved by ac adding a slowly varying spatially dependent shift to all Stark shifts [58] of standing waves, with similar periodicity levels, in the range ±훾, such that the locally transmitted as the unit cells of the lattice, such that the pattern is phase varies as predicted by Figure 4(d).Infreespace,the repeated on each cell, while for a slow variation across the spot size is limited by the Abbe–Rayleigh diffraction limit, array also microwave or magnetic fields can be suitable. 휆∕ (2sin휃) , where 휃 is the maximum angle of incoming The result of the numerical optimization we use here rays, such that tan 휃 = L∕(2f ) for an array of side length L. can be understood as inducing a coupling between the superradiant collective mode of uniform electric dipoles on each unit cell, with 휐 = 1.3훾,andthesubradiantcol- lectivemodeofuniformmagneticdipoleswith휐 = 0.1훾. For a single atom, although the incident light directly only drives the atomic dipoles along the y direction, relative level shifts of the m =±1 states lead to coupling between x and y polarization amplitudes. Then, over the entire lattice, repeated patterns of varying level shifts can be used to engineer an effective coupling between collective modes ofthearray,allowingmodestobeoccupiedeveniftheyare not directly driven [11, 17, 46, 59].The maximum magnitude of the scattered field in Figure 4(c) occurs when the incident light is on resonance with the magnetic dipole mode, which as a result is highly occupied with L = 0.88. The resulting scattered field from the magnetic dipoles again (+) Figure 5: Beam focusing by ultrathin atomic lens at a focal length approximately cancels the incident field, with E , ∼−, s m 휆 | |2 ||2  as in the magnetic mirror case shown in Figure 3. However, f = 10 .(a) E ∕ from plane wave = 1 incident on lattice at 휆 . x = 0. (b) Focal spot along the y axis at x = 10 , z = 0 and (c) along the coupling also leads to a small occupation, L = 0 08, of the x axis at y, z = 0. Focal spot has full-width-at-half-maximum theoff-resonanceelectricdipolemode.Thesmalleroccu- (FWHM) 0.77휆 in the plane and 4.0휆 along the x axis. Lattice pation of this mode is compensated by the larger linewidth, parameters as in Figure 4. Atomic positions are illustrated by black leading to a similar magnitude of scattered field with dots. 1906 | K. E. Ballantine and J. Ruostekoski: Cooperative optical wavefront engineering

at an angle of 15◦ to the x axis. Beam steering from atoms could be used to sort light for different subsequent stages, e.g., as part of a modular quantum information archi- tecture. In contrast to fabricated plasmonic or dielectric systems, the detuning gradient can in principle be varied in situ, changing the steering angle and allowing for the dynamic sorting and redirection of different light beams. Ultrathin atomic lattices thus offer a major advantage for beam steering, combined with focusing and other beam shaping, allowing multiple independent steps with sepa- ration on the order of wavelengths, all without losses or Figure 6: Beam steering by 15◦ by using an atomic Huygens’ surface. | | fabrication inconsistencies. Real part of electric field Re(Ey )∕ (0) for incident Gaussian beam being steered by atomic lattice at x = 0. Incident beam has (0) = 1 푤 . 휆 and beam waist radius 0 = 6 4 .20× 20 × 4 lattice with . 휆 . 휆 . 휆 . 휆 dy = 0 82 , dz = 0 65 , ax = 0 11 and ay = 0 1 . 6 Orbital angular momentum An example is shown in Figure 5,forf = 10휆.Theresulting spot size has a subwavelength FWHM of 0.78휆, approxi- Metasurfaces can be used to generate vortex beams with mately equal to the diffraction limit in this case. Focusing OAM [65, 66]. The angular momentum of paraxial beams close to the diffraction limit can be achieved for a range of light can be separated into spin angular momentum, of focal lengths, down to f ≈ 휆. More complex phase vari- depending on its polarization and OAM, depending on the ations could, in principle, be used to correct aberrations spatial variation of the field. Figure 7(a)–(d) shows the and further limit the focal spot size as has been achieved creation of vortex beams with a phase winding exp(il휙), in plasmonic and dielectric metalenses [62]. achieved by an additional angle-dependent level shift, for integer l = 1, −2, corresponding to a quantized OAM lℏ per photon. The incoming light, with no angular momentum, 5 Beam steering hasbeenimparted withquantized OAM,withacorrespond- ing torque on the atomic array. The integer quantization We use an atomic array to produce beam steering, a of OAM provides a larger alphabet for quantum logic resource in controlling the flow of light, by redirecting an than typical two-state bases, a useful resource for quan- incoming beam in different directions. This is achieved tum computation [67]. Particularly important for quantum by adding an additional level shift gradient, which varies applications is the ability to form coherent superpositions linearly across the lattice, such that the phase of the trans- of different OAM values. This can be achieved by matching mitted light also varies linearly [6, 63, 64]. To steer the beam the phase to that of the desired transmitted beam, with an by an angle 휃 away from the normal requires the phase pro- example shown in Figure 7(e) and (f). While in all cases file 휙 = 훼z where 훼 = 2π sin 휃∕휆. For discrete dipoles with the incoming light has an intensity maximum at the cen- spacing dz in the z direction, this phase profile is unique ter, destructive interference leads to a vortex beam with | 휃| < 휆 modulo 2π if sin ( ∕dz) − 1. An example is shown in zero intensity at the beam center, and along lines of phase Figure 6 where an incident Gaussian beam is transmitted discontinuities. Despite the change in intensity along the

| | | | Figure 7: Converting an optical vortex-free beam to beams with OAM. E ∕ (0) (left of each pair) and arg(Ey ) (right) for several OAM vortex beams. (a) and (b) l = 1, (c) and (d) l =−2, (e) and (f) superposition of l = 1andl =−1, in the yz plane at x = 6휆. Gaussian beam with (0) = 1 푤 . 휆 and beam waist radius 0 = 6 4 is incident on 20 × 20 × 4 lattice with parameters as in Figure 4. K. E. Ballantine and J. Ruostekoski: Cooperative optical wavefront engineering | 1907

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