Nonlinear Optical Response of a Plasmonic Nanoantenna to Circularly Polarized Light: Rotation of Multipolar Charge Density and Near-Field Spin Angular Momentum Inversion

Nonlinear Optical Response of a Plasmonic Nanoantenna to Circularly Polarized Light: Rotation of Multipolar Charge Density and Near-Field Spin Angular Momentum Inversion Published as part of the ACS Photonics virtual special issue “Frontiers and Applications of Plasmonics and Nanophotonics”. Marina Quijada, Antton Babaze, Javier Aizpurua,*and Andrei G. Borisov* Cite This: ACS Photonics 2023, 10, 3963−3975 Read Online ACCESS Metrics & More Article Recommendations * sı Supporting Information ABSTRACT: The spin and orbital angular momentum carried by electromagnetic pulses open new perspectives to control nonlinear processes in light−matter interactions, with a wealth of potential applications. In this work, we use time-dependent density functional theory (TDDFT) to study the nonlinear optical response of a free-electron plasmonic nanowire to an intense, circularly polarized electromagnetic pulse. In contrast to the wellstudied case of the linear polarization, we find that the nth harmonic optical response to circularly polarized light is determined by the multipole moment of order nof the induced nonlinear charge density that rotates around the nanowire axis at the fundamental frequency. As a consequence, the frequency conversion in the far field is suppressed, whereas electric near fields at all harmonic frequencies are induced in the proximity of the nanowire surface. These near fields are circularly polarized with handedness opposite to that of the incident pulse, thus producing an inversion of the spin angular momentum. An analytical approach based on general symmetry constraints nicely explains our numerical findings and allows for generalization of the TDDFT results. This work thus offers new insights into nonlinear optical processes in nanoscale plasmonic nanostructures that allow for the manipulation of the angular momentum of light at harmonic frequencies. KEYWORDS: nonlinear optics, light polarization, circularly polarized light, high-harmonic generation, plasmonic nanostructure, time-dependent density functional theory ■INTRODUCTION Modern technologies enable the design and nanofabrication of photonic devices for manipulation of optical fields on spatial scales much smaller than the wavelength of light. 1 In particular, the resonant coupling of photons with collective electronic excitations in metals and two-dimensional (2D) materials, i.e., plasmons, can be used to engineer strongly enhanced near fields confined to the atomic scale. 2−5 Near-field enhancement boosts the nonlinear optical response so that plasmonic systems find practical applications not only in the linear 5−7 but also in the nonlinear 8−10 regime. Nonlinear metrology, 11 nonlinear sensing, 12,13 ultrafast spectroscopy, 14−16 and nonlinear integrated photonic circuits operation 17−19 exemplify various fields that take advantage of the nonlinear optical response of plasmonic systems. The recent interest in the use of structured light 20 has dynamized research on plasmonic metasurfaces exploiting spincontrolled nonlinear optical processes to obtain beam shaping through manipulation of orbital angular momentum (OAM) and spin angular momentum (SAM) of light. 21−30 Along with gas-phase techniques, 31−34 the use of nonlinear metasurfaces 35 for the generation of vacuum ultraviolet (VUV) and extreme ultraviolet (XUV) coherent light that carries angular momentum opens exciting perspectives in ultrafast spectroscopies and time-resolved experiments to probe chiral systems. The development of devices for on-chip control of nonlinear fields requires knowledge of the nonlinear optical response of individual plasmonic nanoparticles and plasmonic molecules, which are the building blocks of such nonlinear devices. To this end, the hydrodynamic description adopted to address the Received: June 9, 2023 Published: October 24, 2023 Article pubs.acs.org/journal/apchd5 © 2023 The Authors. Published by American Chemical Society 3963 https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 This article is licensed under CC-BY 4.0 Downloaded via UNIV DEL PAIS VASCO on January 9, 2024 at 19:03:17 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. nonlinear response of conduction electrons, and first applied to characterize the second-harmonic generation from metals and metal surfaces, 36−41 has been further developed recently. Efficient numerical approaches to address the nonlinearity of plasmonic nanoparticles have been thus proposed. 9,42−48 In this context, the situation where the fundamental wave is linearly polarized has been studied both theoretically and experimentally, providing a deep understanding about the main processes that control the second-order 45−47,49−57 and the third-order 42,58−63 response of plasmonic nanoantennas and subnanometric plasmonic gaps prone to sustain optically assisted tunneling. 64−66 However, with the exception of chiral systems 9,67−70 (where one is naturally interested in the nonlinear activity triggered by SAM-carrying incident fields), the case of a circularly polarized fundamental wave interacting with typical plasmonic nanoantennas 71 has received less attention for nonlinear plasmonic applications. In this work, we address the nonlinear optical response of a plasmonic nanostructure to a SAM-carrying incident field. We use time-dependent density functional theory (TDDFT) to study the dynamics of conduction electrons triggered by an intense electromagnetic pulse in a free-electron cylindrical nanowire. The electric field of the pulse is circularly polarized in the transversal plane of the nanowire (see Figure 1). As a reference, we also perform calculations for linearly polarized fundamental field as studied in detail in previous works. 43,47,72−75 Without any a priori assumptions, our TDDFT results reveal that the optical response at the nth harmonic of the circularly polarized fundamental wave is determined by the multipole moment of order nof the induced nonlinear charge density that rotates around the nanowire axis at the fundamental frequency. In particular, the induced near field is circularly polarized at all harmonics of the fundamental frequency and reveals an SAM inversion. Moreover, the frequency conversion in the far field is suppressed for circularly polarized incident pulses. We further demonstrate that these results are a direct consequence of the symmetry of the system as can be fully described and understood within an analytical approach. Therefore, our findings are qualitatively robust and provide a new paradigm for the design of nonlinear nanoscale optical devices. Unless otherwise stated, atomic units (a.u.) are used throughout the paper. ■METHODS The details on the method including the modeling of plasmonic nanoparticles and the real-time TDDFT calculations of the electron dynamics can be found in prior works. 76,77 Thus, only the aspects specific to this study will be described here. We consider a plasmonic nanowire represented as a freeelectron metal cylinder of radius Rc, infinite along the z-axis (see Figure 1). The nanowire is described using the stabilized jellium model 78 characterized by the Wigner−Seitz radius of gold (and silver), rs= 3.02 a0(a0= 0.0529 nm is the Bohr radius), and a work function of 5.49 eV. The radius of the nanowire is set to Rc= 66.4 a0(≈3.5 nm), which is a good compromise between the feasibility of the TDDFT calculations and a sufficiently small value of the surface Landau damping 79,80 so that well-resolved localized plasmon resonances can be observed in the linear optical response. The free-electron model is well suited to quantitatively address the linear and nonlinear optical response of nanoparticles formed by prototype metals such as alkali metals and aluminum. For noble metals, as far as the fundamental frequency is below the onset of interband transitions involving localized d-electrons, the symmetry-protected aspects of the nonlinear optical response can be nicely understood within the framework of the free-electron model, considering that the nonlinear currents are created by the quasi-free conductionband electrons. 9,37,38,44,47,51 The contribution of d-electrons to the dynamical screening of the fundamental and harmonic fields can be treated in a model way; 37,38,81−84 however, a fully quantitative assessment of linear and nonlinear properties of noble metal nanoparticles would require further studies. The nonlinear optical response of the nanowire is triggered by a Gaussian pulse of electric field E(t) left-handed circularly polarized (SAM of 1) in the (x,y)-plane t E t t t tE e e( ) cos( ( )) sin( ( )) e x y t t t 0 0 0 0 p 2 = [ + ] i k j j j j y { z z z z (1) where e x(e y) stands for the unit length vector along the x- (y-) axis, E0is the field amplitude, Ωis the fundamental frequency, tpis the duration of the pulse, and t0is the delay time. We also perform reference calculations for a linearly polarized fundamental field given by t E t tE e( ) cos( ( )) e x t t t 0 0 0 p 2 = i k j j j j y { z z z z (2) In this work, we use Ω= 1.5 eV, t5 570 a.u. p 2 = = (≈ 13.7 fs), and E0= 0.064 ×10−2−1.07 ×10−2a.u. corresponding to an average power of the circularly polarized pulse of 3.6 ×1010 −1×1013 W/cm2, and twice smaller average power of the linearly polarized pulse. To analyze the nonlinear optical response of the nanowire to left-handed circularly polarized (SAM = 1) illumination, it is convenient to rewrite eq 1 in the form Figure 1. Sketch of the studied system. (a) Plasmonic nanowire of radius Rc= 66.4 a0(≈3.5 nm) infinite along the z-axis. The external field is indicated with a blue arrow. It is circularly polarized in the (x,y)‑plane and rotates anticlockwise (left-handed) with a fundamental frequency Ω. The red dot located at the x-axis at a distance dfrom the surface of the nanowire indicates the position at which we calculate the induced near field. (b) Cross-section of the nanowire in the (x,y)-plane, definition of the cylindrical coordinates used in the paper. ACS Photonics pubs.acs.org/journal/apchd5 Article https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 3964 tEi i Ei i e E e e e e e e e e ( ) 2( )e ( )e e 2( )e e ( )e e x y i t t x y i t t t t t r i i t t r i i t t t t t 0 ( ) ( ) 0 ( ) ( ) 0 0 0 p 2 0 0 0 p 2 = { + + } · = { + + } i k j j j j y { z z z z i k j j j j y { z z z z (3) where the second expression is given in cylindrical (r,φ) coordinates, and e rand e φare cylindrical unit vectors (see Figure 1b). The fundamental field given by eq 1 or eq 3 can be obtained, for instance, using p-polarized optical pulses where one pulse (with carrier-envelope phase equals to zero) is propagating along the y-axis, and another pulse (with carrier-envelope phase equals to 2 ) is propagating along the x-axis. Furthermore, the present model applies to a situation in which a circularly polarized laser pulse impinges on a plasmonic nanowire along its symmetry axis. In this case, the height hof the nanowire has to be small compared to the wavelength so that there is no effect of plasmon propagation along the z-axis. Moreover, hhas to be significantly larger than the Fermi wavelength of electrons and large enough for the top and bottom surface effects to be neglected. 76 The dynamics of the electron density within the nanowire in response to the optical excitation is obtained from real-space, real-time TDDFT calculations within the Kohn−Sham (KS) scheme. 85,86 Since the system is invariant with respect to a translation along the z-coordinate, the time-dependent electron density is sought in the form n(r,t) = ∑jχj|ψj(r,t)|2, where ψj(r,t) are the KS orbitals of the noninteracting electron system, and r= (x,y) is the 2D position vector. The sum runs over the occupied KS orbitals with ground-state energies jF ( F is the Fermi energy), and the statistical factors jj 2 2 F = account for spin degeneracy as well as for the electron motion along the z-axis. The orbitals ψj(r,t) evolve in time according to the 2D timedependent KS equations, where the nonretarded approximation is used consistent with the small relevant dimensions of the system itt T V n t V n t V V t t r r r r r r ( , ) ( , ) ( , ) ( ) ( , ) ( , ) j j H xc st = [ + [ ] + [ ] + + ] (4) The initial conditions ψj(r,t= 0) correspond to the KS orbitals of the ground-state system. In eq 4,Tis the kinetic-energy operator, VH[n(r,t)] is the Hartree potential, Vxc[n(r,t)] is the exchange−correlation potential, and Vst(r) is the stabilization potential. The time evolution of the electron density, n(r,t), introduces a time dependence to the Hartree and exchange− correlation potentials. The kernel formulated by Gunnarsson and Lundqvist 87 within the adiabatic local-density approximation (ALDA) 86 is used in this work to compute Vxc[n(r,t)]. Finally, V(r,t) = r·E(t) is the potential of the optical field. From the time evolution of the electron density, we obtain all the time-dependent quantities of interest such as the induced charge density t n t nr r r( , ) ( , ) ( ) 0 = [ ] (5) and the induced electric near field t V tE r r( , ) ( , ) ind ind = (6) where n0(r) is the electron density of the ground state of the system, and Vind(r,t) = −{VH[n(r,t)]−VH[n0(r)]} is the induced potential. We also calculate the multipole moments Qm(t) of the charge density induced in the nanowire per unit length in z Q t m r Rt mr r( ) 1d e ( , ), 0 m m im2 c = | | | | i k j j j j j y { z z z z z (7) This expression is given in cylindrical (r,φ) coordinates. From the charge neutrality of the system, the monopole moment is zero, Q0(t) = 0. The induced dipole moment per unit length can be found from t Q t Q t i Q t Q tp e e( ) ( ( ) ( ) ( ) ( ) ) R x y 21 1 1 1 c = [ + ] + [ ] . For the sake of compactness, we use the term “multipole moments” below and understand that the corresponding quantities are calculated per unit length along the z-coordinate. Any frequency-resolved magnitude ( ) reported in this work is obtained from the time-dependent result t( ) using the time-to-frequency Fourier transform. In what follows, without loss of generality, we consider positive frequencies, ω> 0, and ( )e i t as the time dependence of the spectral components. Along with the time-to-frequency Fourier transform of Qm(t) given by eq 7, the multipole moments Qm(ω) at harmonics ω=nΩof the fundamental frequency can also be obtained from the corresponding spectral components of the induced charge density as Qm r R mr r r Rr r r( ) 1d e ( , ) 1d ( , ) m m im m m 2 c c = | | = | | | | | | i k j j j j j y { z z z z z i k j j j j j y { z z z z z (8) The coefficients δϱm(r,ω) are defined in the angular eimφbasis using a representation of the induced charge density rr( , ) 1 2( , )e m m im = (9) The multipole moments Qm(ω) given by eq 8 can be used to express the induced potential V r Q R r ( , , ) ( ) e m m m imind c = | | i k j j jy { z z z (10) and the near field r Q m R ri Qm R ri E e e e e ( , , ) ( ) e ( ) e mm m mr im mm m mx y i m ind c 1 c 1 ( 1) =| | [ ] =| | [ ] | | | |+ | | | |+ + (11) induced at harmonic frequencies ω=nΩof the fundamental field. ACS Photonics pubs.acs.org/journal/apchd5 Article https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 3965 ■RESULTS AND DISCUSSION Linear Response. Prior to the discussion of the nonlinear optical response, we analyze the linear optical response and plasmon resonances of the nanowire using TDDFT. Consistent with the symmetry of the system, the plasmon modes studied here correspond to the cylinder plasmons 88−90 in the limit of zero wavevector along the z-axis. These plasmon modes are localized multipolar plasmons and can be identified with the multipole order massociated with the eimφ dependence of the plasmon-induced electron density, potential, and near field. The ±mplasmon modes are degenerate in frequency. Therefore, below we consider only positive values of m. The localized multipolar plasmons evolve in the (x,y)-plane and can be understood as surface plasmons sustained along the circumference of the nanowire, which leads to a quantized wavenumber qm m Rc = . 77,91 The excitation of localized multipolar plasmons in the nanowire is revealed by a resonant profile in the frequency dependence of the multipolar polarizabilities αm(ω) calculated with TDDFT as shown in Figure 2a (see SI for the definition of αm(ω)). The analysis of the resonances yields the frequencies ωmand lifetimes τm= 1/Γmof the underlying plasmon modes. Here, Γmis the full width at half maximum of the resonance. The resonant profile of Im{αm(ω)} is often perturbed by additional features related to the decay of plasmons into electron−hole pair excitations. 79,92 We therefore define ωmnot as the frequency at which Im{αm(ω)} is maximum but as its mean frequency. The dipolar plasmon mode (m= 1) at ω1= 6.30 eV is slightly redshifted from the classical nonretarded surface plasmon frequency ωs=ωp/ 2 = 6.35 eV (ωp= 8.98 eV is the bulk plasma frequency) because of the spill-out of the induced electron density. 81,93,94 Note that the plasmon frequencies of the nanowire considered here are significantly higher than those of actual noble metal nanostructures, since the dynamical screening associated with d-band electrons is not considered in our jellium model. 81,93 However, the differences in the nonlinear response of a plasmonic system induced by a SAM-carrying fundamental field or by a linearly polarized field stem from robust symmetry properties. Thus, the influence of bound electrons on the field screening and on the plasmon resonances does not alter the qualitative findings reported here. Interestingly, while the classical nonretarded theory predicts ωm=ωsirrespective of m, the frequencies of the plasmon modes calculated with TDDFT depend on m. As shown in Figure 2b, ωmfirst redshifts with increasing mup to m= 3 and then monotonically blueshifts for larger m⩾4. This finding can be understood considering the picture of localized multipolar plasmons as surface plasmons with quantized wavenumber qm m Rc = propagating along the circumference of the cross-section of the nanowire in the (x,y)-plane. 77,91 The dispersion of ωmwith mcan be then associated with the wellknown dispersion relationship of the surface plasmon frequency on a planar free-electron metal surface as a function of the wavenumber qparallel to the surface. 82 Finally, Figure 2c nicely illustrates the multipolar surface character of the m= 1−5 plasmon modes. The induced charge density is mainly located at the metal−vacuum interface and features the characteristic eimφangular dependence. Nonlinear Response. We start the discussion of the nonlinear optical response of the nanowire with the analysis of the induced dipole moment p, which determines the efficiency of the frequency conversion in the far field. Figure 3 Figure 2. Linear optical response of a metallic nanowire calculated with TDDFT. (a) Imaginary part of the multipolar polarizabilities of order mper unit length, Im{αm(ω)}. Results are shown as a function of frequency. Each spectrum is vertically offset for clarity. (b) Frequency ωmof the localized multipolar plasmon resonance sustained by the metallic nanowire as a function of the multipole order m. (c) The induced charge density δϱ given by the coherent superposition of the ±mmultipolar plasmon modes, for different mas labeled above each interpolated image. Results are shown in the (x,y)-plane, and the system is translationally invariant with respect to the z-axis. The red (blue) color corresponds to positive (negative) values of δϱ as indicated with a color bar. ACS Photonics pubs.acs.org/journal/apchd5 Article https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 3966 summarizes our results obtained for linearly (panel a) and circularly (panel b) polarized fundamental field. The fundamental frequency Ω= 1.5 eV used here is far from the plasmon resonances of the system so that no plasmon ringing is produced in the response. The time dependence of the induced dipole moment, p(t), shown in the inset of Figure 3a,b follows the time dependence of the fundamental electric field E(t). For circularly polarized illumination, this results in a π/2 phase shift between the px(t)- and py(t)- components of the induced dipole, as shown in the inset of panel b. We use offresonance incident field on purpose to avoid strong energy deposition into the nanowire that would lead to an efficient electron excitation and eventually to an electron emission with charging of the nanowire. We have explicitly checked that with the present choice of the pump pulse the calculated quantities show the expected scaling with the nonlinearity order nand with the amplitude of the fundamental electric field. With this off-resonance condition, however, we can only observe the resonant enhancement of the nonlinear optical response 46,53,54,60,74,75,95−99 due to the resonance between the harmonic frequency nΩand the multipolar plasmon frequency ωmand not due to the resonance between the fundamental frequency Ωand, e.g., the dipolar plasmon frequency ω1of the nanowire. The frequency spectra of the induced dipole shown in Figure 3 reveal that the far-field emission at harmonic frequencies ω=nΩ(n> 1) is only efficient for a linearly polarized fundamental field. Consistently with the symmetry of the nanowire, the TDDFT results in Figure 3a show that a nonlinear dipole is induced at odd harmonics, allowing the emission into the far field for linearly polarized illumination. 100,101 In contrast, for circularly polarized illumination (Figure 3b), the induced dipole is suppressed at all harmonics of the fundamental frequency so that the frequency conversion into the far field is quenched. The nonlinear electric near field calculated with TDDFT at a distance d= 1.3 nm from the surface of the nanowire is analyzed in Figure 4a,b. In contrast to the far-field emission analyzed in Figure 3, we find that all harmonics (odd and even) are present in the near field both for linear (panel a) and circular (panel b) polarizations of the fundamental field. Thus, the harmonic decomposition of the induced near field points to the presence of higher-order multipole moments (m> 1) of the induced nonlinear charge density. Furthermore, for lefthanded circularly polarized incident field (SAM = 1) the projections of the induced nonlinear near field on xand yaxes satisfy the relation Ey ind(ω) = −iEx ind(ω) (see the −π/2 phase between Ey ind(ω) and Ex ind(ω) for each harmonic frequency in the inset of Figure 4b, represented with the blue dots). This relation between xand yfield components corresponds to right-handed circular polarization (SAM = −1) of the induced nonlinear near field, which is opposite to that of the fundamental field. The inversion of the SAM between the fundamental field and the nonlinear near field holds for all harmonic frequencies encompassed here. We demonstrate below with an analytical approach that this SAM inversion stems from the symmetry of the system and that it is not specific to a given observation point but rather a general property of the near field. The excitation of nonlinear multipole moments at harmonic frequencies nΩis evidenced with the results shown in Figure 4c,d. In this figure, we show the spectral analysis of the timedependent multipole moments Qm(t) of the induced charge density δϱ(r,t) calculated with TDDFT using eq 7. The multipole moments Qm(ω), obtained from the time-tofrequency Fourier transform of Qm(t), feature well-resolved harmonic contributions at ω=nΩ. At a fixed harmonic frequency nΩ, for a linearly polarized incident field (Figure 4c) one or several multipole moments Qm(nΩ) of the nonlinear induced charge density are excited with ±mdegeneracy, and |m|=n−2j(j= 0, 1..., and 2j<n). In particular, multipole moments Q±1(ω) of order m=±1 (and thus a dipole moment p(ω) given by their linear combination) are present at odd harmonics, and absent at even harmonics, consistent with the results reported in Figure 3a. For a circularly polarized incident field (Figure 4d), a qualitatively different nonlinear response is obtained. Namely, only multipole moments Qn(nΩ) with positive m=nare excited in this situation, i.e., the nonlinear induced charge density is characterized by the multipole moment of the same order mas the harmonic order n. It is also worth noting that the fourth harmonic of the fundamental frequency Ω= 1.5 eV overlaps the localized multipolar plasmon m= 4 of the nanowire. This occurs because of the finite width, Γm, of the latter (see Figure 2a,b) Figure 3. Dipolar response of the nanowire to a linearly (a) and circularly (b) polarized fundamental field with frequency Ω= 1.5 eV. The TDDFT calculations are performed for an amplitude of the fundamental field E0= 5.2 ×10−3a.u. corresponding to an average power of the circularly polarized optical pulse of 2.4 ×1012 W/cm2. The spectra |px(ω)|2and |py(ω)|2of the scalar projections of the induced dipole moment, p(t), on the x- (px) and y- (py) directions are shown as a function of frequency measured in units of the fundamental frequency. The blue (red dotted) line is used for px(py). For the case of linear polarization, the electric field is x-polarized so that py= 0. All the quantities are calculated per unit length of the nanowire. ACS Photonics pubs.acs.org/journal/apchd5 Article https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 3967 allowing for the condition 4 m m 2 2 m m + to be fulfilled. The m-order multipolar plasmon is characterized by the eimφangular dependence of the surface charges (see Figure 2c). The component of the induced nonlinear charge density δϱm(r,ω) eimφ(see eq 9) which contributes to the multipole moment Qm(see eq 8) has the same angular dependence. This leads to a resonant enhancement of the multipole moment Q4(4Ω) for the circularly polarized fundamental field. Indeed, the amplitude of Q4(4Ω) stands off the general trend in Figure 4d that shows a decreasing sequence of |Qn(nΩ)|with increasing n. The resonance enhancement is also observed in Figure 4c for the multipole moments Q±2(4Ω) and Q±4(4Ω) excited by linearly polarized fundamental field (see also ref 101 for the discussion of the resonant enhancement of the fourthharmonic generation in a polarized spherical Al nanoparticle). Analytical Interpretation of the TDDFT Results. The main physics behind the TDDFT results can be understood using an approach 102,103 often evoked in the context of nonlinear metamaterials, 22−24,35 which is based on the symmetry of the system and Neumann’s principle for tensors. 104,105 To this end, it is convenient to use cylindrical coordinates and to introduce the basis of the anticlockwise (SAM = +1) and clockwise (SAM = −1) rotating waves defined as i ie e e e e 1 2 1 2 e x y r i 1= [ ± ] = [ ± ] ± ± (12) The fundamental field E(Ω) can be expressed in the e ±1basis as follows: EE e( ) ( ) 1 = =± (13) which is position-independent in the absence of retardation effects. For circularly polarized illumination, eq 13 results in E(Ω) = E±1(Ω)e ±1, where ±1 stands for the SAM of the fundamental field. For linearly polarized illumination E(Ω) = E+1(Ω)e +1 +E−1(Ω)e −1, where E−1(Ω) = E+1(Ω) for an xpolarized incident field and E−1(Ω) = −E+1(Ω) for a ypolarized incident field. For circular and linear polarizations, E(−Ω) = [E(Ω)]*, where [Z]*stands for the complex conjugate of a complex number Z. Using the basis of rotating waves, we introduce the nonlinear multipolar hyperpolarizabilities αm;μd 1...μd n (n)as Q n E E( ) ( )... ( ) mm n 0 ,..., ; ... ( ) n n n 1 1 1 = (14) where Qm(ω=nΩ) is the mth order multipolar moment of the charge density induced in the nanowire per unit length (eq 8), and indexes μ1,...,μncan take values of ±1 independently. Here, we consider the smallest possible nonlinear order in the field and neglect the contribution of the higher-order terms such as ∝Eμd 1(Ω)...Eμd n(Ω)Eμd n+1(Ω)Eμd n+2(−Ω). Let us perform an anticlockwise rotation of the xand yaxes by an angle of βaround the nanowire axis. In cylindrical coordinates, this rotation is Figure 4. Spectral analysis of nonlinear near fields (a, b) and multipole moments (c, d) induced by the linearly (a, c) and circularly (b, d) polarized field with a fundamental frequency Ω= 1.5 eV. Results are shown as a function of the frequency measured in units of the fundamental frequency Ω. The TDDFT calculations are performed for an amplitude of the fundamental field E0= 5.2 ×10−3a.u. corresponding to an average power of the circularly polarized optical pulse of 2.4 ×1012 W/cm2. (a, b) Frequency-resolved xand ycomponents of the induced near field calculated at the xaxis at a distance d= 1.3 nm from the surface of the nanowire (see geometry in Figure 1). For the color code, see the inset of panel (a). The inset of panel b shows the phase between Exand Eycomponents of the near field calculated at harmonic frequencies ω=nΩ(blue dots). The red dot indicates the corresponding phase for the fundamental field. (c, d) Frequency-resolved multipole moments |Qm(ω)|of the induced charge density. The color code corresponds to the multipole order mas explained in the insets. For the x-polarized fundamental field the results obtained for ±m are degenerate. For a fundamental field with SAM = 1, the Qm(ω) are zero for nonpositive m≤0. ACS Photonics pubs.acs.org/journal/apchd5 Article https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 3968 expressed as φ=φ′+β, where the variable with an apostrophe refers to the rotated coordinate system. Consequently, the vector components are transformed as Eμ ′(ω) = e+iμβEμ(ω), the multipolar moments are transformed as Qm ′(ω) = e+imβQm(ω), and the multipolar hyperpolarizabilities are transformed as e m n i m m n ; ... ( ) ( ... ) ; ... ( ) n n n1 1 1 { } = (15) For a nanowire with axial symmetry, the Neumann’s principle 104,105 implies that a rotation by any angle must preserve the form of the multipolar hyperpolarizability tensor. The following selection rule is then obtained m... n1 = + + (16) The hyperpolarizabilities αm;μd 1...μd n (n)are nonzero only if eq 16 is fulfilled. Consider first a linearly polarized fundamental field, which contains both components E±1(Ω) equal in absolute value in the basis of e ±1. The condition given by eq 16 can be fulfilled by several combinations of (m;μ1, ..., μn) with m=± |n−2j| (j= 0, 1, ..., and 2j<n). Consequently, only the corresponding Qm(nΩ) multipole moments are excited. For example, for the second harmonic n= 2, we have (±2; ±1, ±1) leading to the formation of a quadrupole moment. As another example, for the third harmonic n= 3, we have (±3; ±1, ±1, ±1), leading to an octupole moment, and (±1; ±1, ±1, ∓1), leading to a dipole moment. Notice that the nonlinear dipole moment (m=±1), and thus the frequency conversion into the far field, is possible only for odd harmonics consistent with the symmetry of the system, as shown in the TDDFT results in Figure 3a. Consider now a left-handed circularly polarized fundamental field (SAM = 1). The only nonvanishing component in the e ±1 basis is E+1(ω). Therefore, μ1=μ2= ... = μn= 1, and μ1+ ... + μn=n. It thus follows from eq 16 that, in this situation, the nonlinear response at the nth harmonic of the fundamental frequency exclusively allows for the formation of an n-order multipole moment Qn(nΩ) of the induced charge density δϱ(r,nΩ). For n≥2, the nonlinear dipole moment is zero so that the frequency conversion into the far field is suppressed, as shown in the TDDFT results in Figure 3b. The induced nonlinear near field can be obtained from eq 11, and it is given by ( ) n n Q n iE r e e( , ) ( )e R rn i n x y ind ( 1) n n c 1 = [ ] + + . That is, for the circularly polarized fundamental field with SAM = +1, the induced field is circularly polarized with SAM = −1. The SAM inversion is obtained in the near field irrespective of the position and for all harmonics. The general consequences of the symmetry of the system deduced above explain the TDDFT results reported in Figure 3and in Figure 4 when changing from linear polarization of the fundamental field to circular polarization, namely: (i) suppression of the frequency conversion into the far field, (ii) SAM inversion in the nonlinear near field for all harmonics of the fundamental frequency, and (iii) exclusive formation of Qn(nΩ) multipole moment of the nonlinear charge density at nΩharmonic frequency for circularly polarized illumination. Rotating Nonlinear Charge Density. To gain deeper insight into the nonlinear response of the system and to validate the theoretical analysis presented above, we show in Figure 5 the maps of the charge density Re{δϱ(r,nΩ)} induced at the fundamental frequency Ω(n= 1; linear response), and at higher harmonic frequencies nΩ(n= 2, 3, 4; nonlinear response). Re{Z} stands for the real part of the complex number Z. The TDDFT results are shown as a function of xand y-coordinates in the transversal plane of the nanowire for linear (upper row of panels) and circular (lower row of panels) polarizations of the fundamental field. The dipolar character of the induced density characterizing the linear response (n= 1) can be clearly seen for both Figure 5. Maps of the nonlinear charge density Re{δϱ(r,nΩ)}, n= 1, 2, 3, 4, induced in a cylindrical nanowire at a fundamental frequency Ω= 1.5 eV, and at higher harmonics. Re{Z} stands for the real part of the complex number Z. The nonlinear charge density is shown as a function of xand y-coordinates in the transverse plane of the nanowire. Results are normalized independently for each panel such that the variation is contained within the [−1, + 1] interval. The color scale is defined with color bars at the right of the figure. The upper (lower) row of panels show results obtained with linear (circular) polarization of the fundamental field. The TDDFT calculations were performed for an amplitude of the fundamental field E0= 1.1 ×10−2a.u. corresponding to an average power of the circularly polarized pulse 1013 W/cm2. ACS Photonics pubs.acs.org/journal/apchd5 Article https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 3969 polarizations. Obviously, for circularly polarized fundamental field, the induced dipole must rotate following the direction given by the electric field vector. In the absence of retardation effects, for linearly polarized illumination, the nonlinear response of the free-electron homogeneous nanowire should be driven by the surface polarization at even harmonics, and predominantly by the bulk polarization at odd harmonics 9,38,39,44,46,51 (see discussion in the SI). We indeed observe this trend for the second (n= 2) and third (n= 3) harmonics. However, for the fourth harmonic (n= 4), the bulk contribution to δϱ(r, 4Ω) is also clearly visible in addition to the nonlinear charges generated at the surface. Similarly, for circularly polarized illumination, the nonlinear polarization of a homogeneous nanowire is possible only at the surface owing to the symmetry break (see the discussion in the SI). This is consistent with the results obtained for n= 2. However, for higher harmonics, δϱ(r,nΩ) has a strong bulk component. We attribute this effect to the break of the homogeneous approximation because of the Friedel oscillations of the ground-state charge density. As we show in the SI, Friedel oscillations persist in the bulk of the nanowire because of its relatively small radius (see also discussion of the currents induced inside a plasmonic nanoparticle in ref 71). The most important information stems, however, from the symmetry of the charge density maps. For linearly polarized fundamental field (upper panels in Figure 5), Re{δϱ(r,nΩ)} is symmetric with respect to the x-axis, and from the symmetry with respect to the y-axis it follows that the dipole moment is formed only at odd harmonics. In sheer contrast, for circularly polarized fundamental field (lower panels in Figure 5), the induced charge density Re{δϱ(r,nΩ)} at harmonic frequency ω=nΩpossesses a well-defined axial symmetry of order nwith respect to the nanowire axis z. Using the many-body response theory, we show in the SI that because of the axial symmetry of the nanowire, only the term with m=nis nonzero in eq 9. The nonlinear charge density induced at harmonic frequency can be thus expressed as n r nr( , ) ( , )e n in 1 2 = so that only Qn(nΩ) multipolar moment can be excited in full agreement with the conclusions derived from Neumann’s principle. It is however important to realize that the inverse is not true, i.e., the selection rule Qm≠n(nΩ) = 0 can also be satisfied if ∫r|m|+1drδϱm(r,nΩ) = 0 (see eq 8) and does not necessarily require that δϱm≠n(r,nΩ) = 0. The einφangular dependence of the nonlinear charge density at harmonic nof the fundamental frequency obtained for circularly polarized illumination has appealing consequences for the dynamics of the system. Indeed, in this situation, the time evolution of the nonlinear charge density of harmonic nis given by Figure 6. Time evolution of the multipolar moments Qm (n)(t) for linearly and circularly polarized incident fields (left and right column of panels, respectively). Here, mstands for the order of the multipole, and nstands for the harmonic order. The Qm (n)(t) are obtained using the frequencyresolved maps of the induced charge density δϱ(r,nΩ) (see Figure 5). Results are shown as a function of time for the linear response (n= 1) as well as for n= 2, 3, 4 harmonics of the fundamental frequency Ω= 1.5 eV. In the case of circular polarization, we indicate with dashed lines the time dependence of the corresponding multipolar moments calculated in a frame rotating around the z-axis anticlockwise with angular frequency Ω. For further details, see the text of the paper. ACS Photonics pubs.acs.org/journal/apchd5 Article https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 3970 t n r n r r( , ) Re ( , )e Re 1 2( , )e n in t n in t ( ) ( ) { } = { } = (17) Regardless of the harmonic order n, the time evolution of the nonlinear induced charge density δϱ(n)(r,t) can be seen as a rotation around the z-axis of the rigid multipolar charge distribution calculated for t= 0, r nRe ( , )e n 1 2 in { } . The angular frequency of this rotation equals the fundamental frequency Ω, and the direction of the rotation is the same as that of the circularly polarized fundamental field. Consequently, the oscillation frequency of the nonlinear multipole moments, near fields, and other physical quantities at the nth harmonic frequency stem from the symmetry of this rotating charge distribution, where the same spatial profile is retrieved in a fixed reference frame ntimes per fundamental period. For linearly x-polarized fundamental field, the induced nonlinear charge density can be expressed as t r n mr( , ) Re 1 2e ( , ) cos( ) n m in t m ( ) { } = (18) where m=n−2j(j= 0, 1, ..., and 2j<n). Equation 18 thus reflects the superposition of several charge distributions with angular dependencies cos(mφ), each of them oscillating at harmonic frequency nΩ. The movies presented in the SI nicely demonstrate the striking difference in the dynamics of the nonlinear charges induced at harmonic frequencies by circularly (SAM = 1) and linearly polarized fundamental field. To further test the validity and consistency of our analysis, we use δϱ(r,nΩ) obtained from TDDFT to calculate the multipolar moments Qm(nΩ) using eq 8. The time evolution of the multipole moments at harmonic frequencies nΩof a monochromatic fundamental field is given by Qm (n)(t)≡ Re{Qm(nΩ)e−inΩt}, and it is shown in Figure 6. These results are in full agreement with the results of the Fourier analysis of the time-dependent quantities defined by eq 7 and calculated using a Gaussian envelope of the fundamental field (Figure 4c,d). Thus, while several multipole moments are present at harmonic frequency nΩfor linearly polarized illumination (left panels), for circularly polarized illumination (right panels), the only nonzero multipole moment of the nonlinear charge density at nth harmonic is Qn (n)(t). When the multipolar moments are calculated not in a fixed reference frame but in a rotating frame designed to accompany the nonlinear induced charges, their time dependence is entirely removed for the circularly polarized fundamental field (dashed lines in the lower row of panels in Figure 6). This finding confirms our conclusion on the dynamics of the induced nonlinear charge density given by the anticlockwise rotation of the rigid multipolar density distribution around the symmetry z-axis with the angular frequency Ω. ■SUMMARY AND CONCLUSIONS In summary, we used TDDFT calculations to study the nonlinear optical response of a free-electron plasmonic nanowire a few nanometers in diameter to a strong optical field circularly and linearly polarized within the plane perpendicular to the nanowire axis. We addressed the dependence of the nonlinear response on the polarization of the fundamental field and the possibility of generating SAMcarrying harmonics of the fundamental frequency. An analytical approach based on the symmetry of this homogeneous system and Neumann’s principle for the tensors 103−105 was used to elucidate the main physics behind the TDDFT results obtained without any aprioristic assumptions. In full agreement, the analytical approach and the TDDFT simulations reveal that, in this system, the nonlinear optical response to a circularly polarized fundamental field at the nth harmonic of the fundamental frequency Ωis driven by a nonlinear induced charge density with discrete n-fold symmetry with respect to the nanowire axis. This charge density rotates at the fundamental frequency Ωaround the nanowire axis, and its only nonzero multipole moment is the norder multipole Qn(nΩ). As a consequence, for a circularly polarized fundamental field: •The induced density, induced field, and induced potential at nth harmonic frequency display a cos[n(φ− Ωt) + ζ] dependence on time and azimuthal angle φin cylindrical coordinates, with z-axis along the nanowire axis (ζis some constant). •The frequency conversion into the far field is forbidden for all harmonics, in contrast to the case of linearly polarized illumination where the symmetry constraints permit the frequency conversion into the far field at odd harmonics. •All harmonic frequencies are present in the near field. •Regardless of the position, the nonlinear near field at harmonic frequencies is circularly polarized in the plane transversal to the nanowire axis, with SAM opposite to that of the fundamental field. In other words, we obtain a SAM inversion at all harmonic frequencies. One of the main take-home messages of the present work is the striking difference in the origin of the time dependence of the physical quantities at the harmonics of the fundamental frequency, depending on the polarization of the fundamental field. We demonstrated that, in the case of circular polarization, the time dependence at the nth harmonic frequency stems from the order nmultipolar symmetry of the nonlinear charge density, which rotates with angular frequency Ω(fundamental frequency) around the nanowire axis. The same spatial distribution of these charges is then retrieved in a fixed reference frame, n-times per fundamental period. In the case of linear polarization, the time dependence comes from several interfering charge distributions oscillating at the harmonic frequency nΩ. Obviously, the quantitative results reported here depend on the specific characteristics of the free-electron plasmonic nanowire used in our work. Nevertheless, it is important to note that the qualitative conclusions obtained are derived from the symmetry of the system, and thus, they can be applied to a variety of canonical plasmonic nanoantennas. For a circularly polarized fundamental field, the SAM inversion in the near field should be observed for systems with cylindrical geometry, while the axial symmetry of the plasmonic nanoobject is the only requirement for the formation of a rotating nonlinear charge distribution with a multipolar symmetry order equal to the order of the frequency harmonic. This work gets along the lines of active research devoted to the manipulation and control of light pulses carrying spin and angular momentum. In particular, our results pave the way toward the use of nanosources of circularly polarized highharmonic near fields for on-chip nonlinear applications. ACS Photonics pubs.acs.org/journal/apchd5 Article https://doi.org/10.1021/acsphotonics.3c00783 ACS Photonics 2023, 10, 3963−3975 3971