### Nonlinear, tunable, and active optical metasurface with liquid film Download： 587次

Metamaterials are composite man-made or natural materials that possess emergent optical properties, which stem from specific spatial arrangements of the constituent subwavelength basic units^{1}^{,}^{2} and lead to numerous effects, such as formation of bandgap in dielectric photonic crystals,^{3}^{,}^{4} suppression of plasmon frequency in metallic metamaterials,^{5} control over the radiation dynamics of embedded active materials^{6}^{,}^{7} (see also Ref. 8 and references therein), wave front control using thin elements,^{9} polarization and phase control in both transmission^{10} and reflection^{11} modes, and control of light properties in low-loss and high-index dielectric resonant Mie nanoparticles.^{12} Dynamical tuning of optical metamaterial properties is particularly appealing as it allows one to study new regimes and effects of light–matter interaction, holds promise for future metamaterial-based devices with functionalities achieved by structuring matter on the subwavelength scale,^{13} and may be also of interest as platforms to simulate many body quantum effects, which are challenging to realize in real quantum systems.^{14} In particular, numerous studies, partly fueled by seminal advances in condensed matter physics, such as the discovery of graphene,^{15} have revealed analogies between propagation of light in photonic crystals to dynamics of relativistic Dirac fermions near Dirac points in crystals and dynamics of electrons in topological insulators. In the case of two-dimensional (2-D) photonic structures, Dirac cones have attracted significant interest because of the existence of robust surface states due to the breaking of parity and time-reversal symmetry,^{16} and intriguing transport properties such as pseudodiffusive transmittance,^{17} persistence of the Klein effect,^{18} and breakdown of conical diffraction due to symmetry breaking of hexagonal symmetry^{19} or due to nonlinear interactions.^{20}^{,}^{21} Furthermore, 2-D periodic structures play an important role in realizing lasing effects in so-called distributed feedback (DFB) structures, which provide continuous coherent backscattering from the periodic structures without mirrors; these were originally proposed in one-dimensional (1-D)^{22}^{,}^{23} and later were extended to 2-D systems enabling lasing amplification of waveguide modes in photonic crystals^{24} and of surface plasmon polariton (SPP) oscillations in metal structures,^{25} and more recently also in thin polymer membranes.^{26} Since metasurfaces in general and 2-D periodic structures in particular are conventionally constructed using solid metals and high index dielectrics, their tuning properties are constrained by physical properties, such as carrier^{27} and material density, which can be manipulated by external electrical, magnetic, acoustic, and temperature fields,^{28} and also by mechanical stretching of the elastic DFB structure, which affects the resonant lasing frequencies.^{29}

Liquids on the other hand provide an attractive platform to introduce significant changes to the optical properties of metamaterials and metasurfaces, due to the liquids’ capability to induce relatively large dynamical changes of their dielectric function, which stems from their adaptive property to fill microchannels of desired shape,^{30} compliance under external stimuli, and ability to sustain changes of their physicochemical properties. Prominent examples include light-induced collective orientation of liquid crystal molecules,^{31} pressure-induced control of lasing frequency in a 1-D array of droplets,^{32} magnetic induced ferrofluid-based hyperbolic metamaterial,^{33} and chemical composition-induced changes enabling lasing frequency tuning in 1-D systems^{34} and in 2-D photonic crystals.^{35} Furthermore, recent studies introduced a novel interaction between SPPs and a thin liquid dielectric (TLD) film due to geometrical changes of the gas–fluid (or fluid–fluid) interface facilitated by the thermocapillary (TC) effect: theoretical study of self-induced focusing and defocusing effects of propagating SPPs due to nonlocal interaction^{36} (where refractive index changes extend beyond the regions of maximal optical intensity) as well as formation of an optical liquid lattice of a fixed square symmetry, and experimental demonstration of TC-assisted optical tuning of surface plasmon resonance coupling angle.^{37} While these two studies demonstrated a significant coupling between the topography of the gas–fluid interface and propagation of SPP modes, these studies did not leverage surface tension nor surface optical modes to study bandstructure and lasing modes tuning due to optically induced changes of the liquid lattice symmetry.

In this work, we theoretically demonstrate that SPPs or slab waveguide (WG) modes propagating in a TLD film, which is thinner than the penetration depth of the corresponding surface optical mode in the direction normal to the film’s surface, lead to a nonlocal and nonlinear response of the corresponding dielectric function due to optically driven surface tension effects. In particular, we take advantage of the surface tension dependence on local physicochemical conditions^{38} to show that the optically induced TC^{39} or solutocapillary (SC) effects,^{40} which stem from temperature or chemical concentration gradients, respectively, lead to self-induced changes of the dielectric function. In particular, since both surface tension effects are accompanied by flows and deformation of the TLD film, the latter lead to self-induced changes of the dielectric function that are coupled back to the propagation conditions of the surface optical mode. Importantly, the fact that both SPP and WG modes can propagate within thin dielectric films, which are thinner than the optical penetration depth of the corresponding surface optical mode into the domain outside the TLD film, facilitates a significant coupling between these surface optical modes and changes of liquid’s topography. In particular, the SPPs and slab WG modes are both guaranteed to propagate in arbitrarily thin liquid films; the former due to the fundamental capability of metal–dielectric interfaces to support SPPs,^{41} whereas the latter can be described by the transverse resonance condition of the fundamental mode.^{42} Employing this interaction for the case of interfering surface optical waves leads to a self-induced optical liquid lattice of tunable symmetry and bandstructure, which can be tuned by changing the relative propagation directions and the amplitudes of the interfering SPP or WG modes. Furthermore, applying bandstructure tuning in the case, the TLD film or the dielectric substrate admits gain properties leads to configurable DFB mechanism with gain and/or index modulation that can control the threshold condition and the corresponding lasing frequency. In particular, dielectric substrate with gain and TLD film without gain leads to index modulation, whereas the complementary case of TLD film with gain supports both index and gain modulation, as shortly discussed below.

^{43} or by stabilizing liquid film with surfactants leading to an optically thin and stable liquid sheet with a pair of parallel gas–fluid interfaces (see also Ref. 44 for a recent experiments of optical guiding in soap films).

#### Fig. 1. Schematic presentation of TLD film deformation forming optical liquid lattices (blue) due to surface tension effects triggered by interference of surface optical modes (red). (a) 2-D plasmonic liquid lattice formed by interference of SPPs. (b), (c) Suspended and supported photonic liquid lattice, respectively, formed by interference of photonic slab WG modes. Gain can be introduced into the suspended structure (c) either to the liquid or to the dielectric supporting membrane. The lateral dimensions of the liquid slots, which are bounded by solid dielectric walls (not shown) are ${d}_{y}$ and ${d}_{z}$ . (d)–(f) The corresponding 1-D optical liquid lattices in a liquid slot of length ${d}_{z}$ induced by pairs of (d) counterpropagating SPPs or (e) and (f) slab WG modes.

This paper is structured as follows. First, we describe the nonlinear self-induced interaction between surface optical modes and the TLD film driven by either the TC or SC effects and derive the underlying complex nonlocal Ginzburg–Landau equation that governs the dynamics of the corresponding envelope function. We then solve the TLD film equation and describe tuning of the optical liquid lattices and of the corresponding symmetries due to changes of the propagation directions and amplitudes of the surface optical modes. As an illustrative example, we first consider the simpler 1-D case and analyze the threshold change and lasing frequency tuning, due to self-induced and nonself-induced amplitude modulation of the formed liquid lattice. Afterward, we present numerical simulation results of bandstructure tuning due to the breaking of 2-D hexagonal and square symmetries and demonstrate formation of Dirac points in liquid lattices with a broken hexagonal symmetry. Finally, we demonstrate that symmetry changes of 2-D optical liquid lattices leads to tuning of the corresponding lasing frequencies and the corresponding emission directions.

## 1 Results

## 1.1 Light-Induced Interaction between Surface Optical Modes and a TLD Film

The set of coupled governing equations that describes light–fluid interaction due to TLD film thickness changes includes Maxwell equations, Navier–Stokes equations for an incompressible Newtonian fluid, balance condition between viscous and surface tension stresses on the gas–fluid (or fluid–fluid) interface, and heat/mass-transport equations depending on the specific light-induced mechanism, which triggers local changes of the surface tension. We employ heat-transport and mass-transport equations, relevant for the TC effect and SC effect, respectively, which constitute the coupling mechanism between light propagation and dynamics of TLD film. Specifically, light-induced changes of the surface tension lead to deformation of the TLD film, ^{36}^{45} and optical power of ^{46} which are beyond the scope of this work.

For the case of SC flows, we focus on cis–trans transformation that can be described by the simple phenomenological two-state model

Employing a perturbative expansion for the SPP and slab WG modes (see the Supplemental Material), which incorporates the effects of diffraction and gain, yields the following complex nonlocal Ginzburg–Landau (GL) equation (see the Supplemental Material for derivation): ^{48} as well as nanofocusing of SPPs in tapered plasmonic waveguides.^{49} The nonlocal integral term,

## 1.2 One-Dimensional and Two-Dimensional Optical Liquid Lattices with Tunable Symmetry

Next we show that by controlling the interference pattern of surface optical modes, it is possible to form optical liquid lattices directly from a TLD film and tune their symmetries. Consider an interference of

First, consider the simplest

#### Fig. 2. Optical liquid lattices formed from a TLD film due to interference of SPPs or WG plane wave modes with propagating direction marked with red arrows. (a) One-dimensional lattice formed by interference of two surface optical plane waves with $\theta $ -dependent periodicity given by Eq. (6). (b)–(d) Two-dimensional lattices of hexagonal symmetry and (e)–(g) rectangular symmetry, where ${\overrightarrow{V}}_{1}$ and ${\overrightarrow{V}}_{2}$ are the corresponding primitive vectors. The symmetry of the liquid lattice and can be controlled by the relative angles of the interfering beams; (b) $\theta =30\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ and (c) $\theta =45\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ are lattices with hexagonal broken/distorted symmetry, whereas (d) $\theta =60\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ is hexagonal symmetric lattice; (e) $\alpha =0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ (square symmetry), (f) $\alpha =10\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ , and (g) $\alpha =20\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ correspond to rectangular symmetry. (h), (i) Optical liquid lattice for different values of $q$ which results in (d) a phase transition from a hexagonal lattice to (h) merging of the triangular sites and (i) face-centered cubic lattice. The 2-D liquid lattices are normalized by ${\eta}_{\mathrm{max}}$ and ${\eta}_{\mathrm{min}}\equiv -{\eta}_{\mathrm{max}}$ defined in Eq. (9) and below Eq. (11).

Next, we consider the case

Expanding the deformation given by Eq. (7) up to a second-order in Taylor series around one of the maximum points [e.g., ^{51} to a face centered cubic symmetry with one primitive vector per unit cell.

A liquid lattice with a rectangular tunable symmetry can be formed by interfering four beams where the relative amplitudes satisfy

## 1.3 Self-Induced Distributed Feedback Lasing in 1-D Optical Liquid Lattices

The self-induced 1-D deformation of the TLD film given by Eq. (6) [see also ^{52}^{,}^{53} between the wave numbers

Note that, similar to other self-induced lasing mechanisms,^{54} the field intensity plays a double role; it determines both the coupling constant^{55}^{,}^{56} of the backward Bragg scattering (which provides the feedback mechanism) as well as the pump intensity necessary for lasing. Nevertheless, in our case, the amplitude of the optically induced periodic liquid lattice and consequently the coupling between the right- and the left-propagating modes is a more pronounced function of the optical intensity due to the higher difference between the typical refractive indices of liquids and gases. Notably, the relatively low optical power needed to introduce index changes induced by the TC/SC gas–liquid interface deformation [see Ref. 36, below Eq. (13) for the TC case] is expected to lead to substantially lower lasing threshold as compared to a thick liquid film or a liquid without gas–liquid interface (e.g., liquid fully occupying a microchannel), where the liquid–light interaction due to interface deformation is expected to be absent. In particular, in the absence of gas–liquid interface, index changes in liquids mostly stem from changes of polarizability, population of the electronic states (see Ref. 50 and references within), and material density; for instance, the index change due to thermo-optical effect,

In the limit of small deformation and small intensity, the corresponding coupling constant can be written as ^{54}

## 1.4 Bandstructure of WG and SPP Modes in Hexagonal and Rectangular Liquid Lattices

Consider the case of interferring plane waves of the type presented in Eq. (5), which interact via nonlocal and nonlinear integral term in Eq. (4). Representing the intensity distribution given by Eq. (5) as a linear superposition of elements in the

To get an insight into light propagation in a hexagonal crystal, one usually adopts the tight binding approximation,^{57} which describes the dynamics of light as hopping between nearest lattice sites. This approximation is applicable to cases when the potential well at each site is sufficiently deep, leading to localized intensity of the Bloch modes around the sites. For instance, one could in principle expand the potential Eq. (17) around the peak up to second-order,^{57} and then approximate it by the exponential function, to obtain closed-form expressions for the orbitals as a function of the constant ^{57} the Schrödinger equation around high symmetry points in the k-space can be represented as a pair of Dirac equations. ^{58} The latter implies the presence of a Dirac point around the corresponding high symmetry points (which usually appears around

To determine the dispersion relation for SPP or slab WG modes in the regime that is not satisfied by the tight binding approximation and captures the effect of broken hexagonal and cubic symmetries, we turn to numerical simulations. ^{59}

#### Fig. 3. Numerical simulation results presenting the bandstructure of photonic slab WG modes (i.e., frequency as a function of the corresponding directions in the k-space) formed in a suspended photonic liquid lattice of hexagonal symmetry and hexagonal broken symmetry. (a)–(c) Bandstructure of TE slab WG modes in a suspended photonic liquid lattice presented in Figs. 1(b) –1(d) due to interference formed by three plane waves at angles (a) $\theta =\pi /6$ , (b) $\pi /4$ , and (c) $\pi /3$ . (d)–(f) Bandstructure of TM slab WG modes at interference angles (d) $\theta =\pi /6$ , (e) $\pi /4$ , and (f) $\pi /3$ . Two Dirac points emerge at the angle $\theta =\pi /3$ at frequencies around 353 and 349 THz, respectively, marked at (c) and (f) by black arrows. In the simulation, the refractive index is 1.409, the mean TLD film thickness is 450 nm, and the peak-to-peak undulation amplitude is 300 nm.

## 1.5 Self-Induced Distributed Feedback Lasing and Tuning in Two-Dimensional Liquid Lattices

^{60} described by a dielectric function given by

#### Fig. 4. Numerical simulation results presenting lasing frequencies tuning in plasmonic liquid lattice and suspended photonic liquid lattice, of hexagonal and rectangular symmetries, respectively, where the angles of the interfering “writing beams” $\alpha $ and $\theta $ are described in Fig. 2 . (a) Lasing frequencies tuning of WG TE and TM modes for hexagonal symmetric photonic liquid lattice with $\theta $ values $\theta =45\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ , 48 deg, 51 deg, 54 deg, 57 deg, and 60 deg (in radians) labeled as 1 to 6 near the blue and red disks, respectively; SPP modes in rectangular symmetric plasmonic liquid lattice with $\alpha $ values $\alpha =0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ , 5 deg, 10 deg, 15 deg, and 20 deg, labeled as 1 to 5 near the red squares, respectively. (b) The location of the corresponding lasing modes in the reduced Brillouin zone in the k-space.

## 2 Discussion

In this work, we theoretically investigated optical properties of configurable liquid-made lattices, formed by interfering SPPs or slab WG modes. We leveraged the underlying complex nonlocal and nonlinear interaction described by the GL and Schrödinger equations, which capture the effect of the self-induced action of the optical mode on itself due to geometrical changes of the gas–liquid interface, to predict formation of optical liquid lattices and bandstructure tuning due to symmetry breaking of hexagonal symmetric and square symmetric lattices as well as phase transition effects between hexagonal symmetric to face centered symmetric lattices. We then applied the bandstucture tuning to demonstrate control over various properties of the lasing systems, such as gain threshold, lasing frequency, and emission direction of the corresponding lasing mode. Notably, the self-induced lasing threshold of the TLD film interacting with the surface optical mode is expected to admit much lower values as compared to a similar liquid system without gas–fluid interface and therefore has the prospect to serve as a future Lab-on-a-Film bio-sensing platform, which integrates liquid delivery with self-induced DFB lasing mechanism. Interestingly, the formation of a graphene-like liquid lattice with tunable Dirac points in lattices with hexagonal broken symmetry is substantially different from other configurable platforms introduced to date for the formation of Dirac points in optical lattices, such as schemes that incorporate cold atoms^{61} and Fermi gases.^{62} In addition, since metamaterials can be utilized as an optical computational platform, as was recently demonstrated in a solid-made and non-reconfigurable setup for solution of linear equations,^{63} the adaptive property of optical liquid lattices has the potential to allow reconfigurable computation of computationally challenging problems of systems of linear and nonlinear algebraic equations [e.g., Eq. (15)] and also has the potential to serve as an emulator of many-body quantum mechanical problems, such as electron propagation in an atomic lattice, including topological edge-state effects. We hope that our work will stimulate future experimental and theoretical studies to realize optical liquid lattices and to explore underlying nonlinear light–liquid interaction mechanisms that include also birefringence and magneto-optical effects.

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##### Article Outline

Shimon Rubin, Yeshaiahu Fainman. Nonlinear, tunable, and active optical metasurface with liquid film[J]. Advanced Photonics, 2019, 1(6): 066003.