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Nanostructure arrays in free-space: optical properties and applications

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Reports on Progress in Physics Rep. Prog. Phys. 77 (2014) 126402 (33pp)

doi:10.1088/0034-4885/77/12/126402

Review Article

Nanostructure arrays in free-space: optical properties and applications Stéphane Collin Laboratoire de Photonique et de Nanostructures (LPN-CNRS), Route de Nozay, 91460 Marcoussis, France E-mail: [email protected] Received 22 May 2014, revised 3 August 2014 Accepted for publication 4 August 2014 Published 26 November 2014

Invited by Masud Mansuripur Abstract

Dielectric and metallic gratings have been studied for more than a century. Nevertheless, novel optical phenomena and fabrication techniques have emerged recently and have opened new perspectives for applications in the visible and infrared domains. Here, we review the design rules and the resonant mechanisms that can lead to very efficient light–matter interactions in sub-wavelength nanostructure arrays. We emphasize the role of symmetries and free-space coupling of resonant structures. We present the different scenarios for perfect optical absorption, transmission or reflection of plane waves in resonant nanostructures. We discuss the fabrication issues, experimental achievements and emerging applications of resonant nanostructure arrays. Keywords: nanophotonics, arrays, nanostructures, plasmonics, perfect absorption, photonic crystals, high-contrast-gratings (Some figures may appear in colour only in the online journal)

1. Introduction

waves many decades later by Fano in 1941 [6] and Hessel and Oliner in 1965 [7]. They distinguished Rayleigh anomalies (passing off of a diffracted order) from resonant phenomena. Rigorous electromagnetic theories of gratings were first developed in the 1970s [8]. This period was also marked by the fabrication of holographic gratings by laser sources, the development of phenomenological theories and the achievement of total absorption of light by a metallic grating [9]. In the 1980s and 1990s, research on resonant gratings was progressively extended to a much wider variety of structures and wavelengths. This was made possible thanks to increasing computing capabilities, novel numerical methods and the development of micrometer and nanometer scale fabrication techniques. During this period, the emergence of photonic crystals (PC) [10, 11] and guided-mode resonance (GMR) structures [12] drove research on dielectric nanostructure arrays. In 1998, nearly a century after Wood’s anomaly, Ebbesen et al reported extraordinary optical transmission (EOT) observed through sub-wavelength hole arrays in metal films [13].

In 1902, Wood published his famous article entitled ‘On a remarkable case of uneven distribution of light in a diffraction grating spectrum’ in the Philosophical Magazine [1]. While diffraction gratings were already well known, this work was certainly one of the very first observations of a resonance phenomenon in nanostructure arrays. Wood studied diffraction gratings made of grooves on a flat metal surface and reported abrupt changes in the reflection spectra of white light sources. The position of the bright and dark bands of these ‘singular anomalies’ changed with the angle of incidence and their spectral width was estimated to be 2–3 nm. Wood concluded that this ‘problem is one of the most interesting that I have ever met with’. He published later experimental works in 1912 and 1935 [2, 3]. The role of diffraction effects was first highlighted by Rayleigh in 1907 [4, 5], but Wood’s anomalies were properly interpreted as a resonant phenomenon associated with the excitation of surface 0034-4885/14/126402+33$33.00

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© 2014 IOP Publishing Ltd  Printed in the UK

Review Article

Rep. Prog. Phys. 77 (2014) 126402

This discovery raised considerable attention in the nanophotonics community and beyond. More than fifteen years later and despite this long history, the interest of the scientific community toward resonant metallic and dielectric nanostructure arrays is still unfailing and the flow of discoveries and applications is still extremely impressive. The scope of this article is to review the optical properties of resonant nanostructure arrays coupled to free space, in the visible and infrared wavelength ranges. We will focus on the recent advances in light absorption in arrays fabricated on a mirror and on the transmission and reflection properties of free-standing structures. In the far field, sub-wavelength nanostructure arrays behave as a homogeneous layer (no diffraction). The geometry and refractive index of their unit cell can induce a wide variety of optical phenomena. In the metamaterial (MM) framework, nanostructure arrays would be described as artificial materials whose electromagnetic properties are defined by the shape of building blocks much smaller than the wavelength and expressed by effective permittivities and permeabilities. Resonant MM slabs would be characterized by their resonant unit cells. Here, in contrast with the MM approach, we consider the whole slab as a resonant structure. With this approach, we will show that the temporal coupled-mode theory (TCMT) provides a fruitful tool to enhance free-space couplings and light–matter interactions. Efficient numerical methods are now available for the calculation of the optical response of periodical structures [14], such as rigorous coupled-wave analysis (RCWA), the finite element method (FEM) and the finite difference time domain (FDTD). They will not be reviewed or discussed in this article, but we warmly recommend to any reader the insightful review by Barnes on the comparison of numerical calculations and experiments, in particular in plasmonics [15]. In this review, section  2 will be devoted to the basics of optical resonances in sub-wavelength nanostructure arrays. In section 3, the TCMT will be explained. Based on simple considerations, the design rules for the achievement of perfect optical absorption, transmission or reflection in resonant nanostructure arrays will be discussed. The role of symmetries, radiative coupling and internal (absorption) losses will be highlighted. It will be shown that this formalism provides guidelines for engineering the free-space coupling of nanostructure arrays, with important consequences for applications. In section  4, resonant nanocavity arrays fabricated on a mirror will be reviewed. These structures can lead to perfect optical absorption in nanometer scale layers, with applications in optical detection (visible and infrared), biosensing and coherent thermal emission. We will present perspectives for multi-resonant structures and broadband absorption, with applications to solar photovoltaics. In section 5, the optical properties of free-standing metal nanostructure arrays will be described. The role of propagating and localized surface plasmons (LSP) in metal films with one-dimensional (1D) or two-dimensional (2D) nanohole arrays will be detailed. The conditions and limits for the achievement of nearly perfect optical transmission through nanoapertures in metal films will be discussed. Applications in infrared imaging and optical filters will be presented.

Figure 1.  Generic nanostructured membranes: a slab made of any material (metal, semiconductor, dielectric) is surrounded by a vacuum and drilled by (a) 1D slits (lamellar grating) or (b) 2D holes.

In section 6, the optical properties of free-standing dielectric nanostructure arrays will be described. Depending on the size of the nanostructures, different resonant mechanisms can occur in free-standing PC slabs, high-index sub-wavelength gratings (or high-index gratings, HCG) and nanorod arrays. They result in a wide variety of applications: optical filters, broadband mirrors for vertical-cavity surface-emitting lasers (VCSEL), and optomechanics. 2.  Resonances in nanostructure arrays We consider resonant nanostructure arrays and gratings illuminated by free-space plane waves. Semi-infinite metallic and dielectric membranes (slabs surrounded by a vacuum), periodically nanostructured in one or two dimensions, are depicted in figure  1. In the 1D case, the structure is infinite in y and infinitely periodic in x, the electromagnetic solutions are independent of y. In 2D, the structure is infinitely periodic in x and y. The electromagnetic field above and below a 2D array is described as the sum of propagating and evanescent plane waves (Rayleigh expansion), whose in-plane wavevectors are quantified: 

⎛ ⎛ 2π ⎞ 2π ⎞ k(pm, q) = ⎜kx 0 + m ⎟ ux + ⎜k y0 + q ⎟ uy ⎝ dx ⎠ dy ⎠ ⎝

(1)

where kx0ux + ky0uy is the in-plane wavevector of an incident plane wave, (dx, dy) are the periods of the array along the directions given by the unit vectors (ux, uy) and (m, q) are integers. The basics of the electromagnetic theory of gratings and ­periodic nanostructures can be found in [8, 16]. In this section, we provide an overview of the wide range of optical phenomena that can occur in such a nanostructured membrane. 2.1.  Dielectric nanostructure arrays: an overview

Let us first consider a simple dielectric membrane of subwavelength thickness, refractive index n, drilled with a 2

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Rep. Prog. Phys. 77 (2014) 126402

Figure 2.  1D free-standing dielectric gratings at different scales: (a) diffraction grating, (b) resonant (sub-wavelength) grating and

(c) effective medium slab.

Figure 3.  Free-standing sub-wavelength dielectric gratings (λ ≃ d) with various filling fractions and resonant mechanisms.

periodical arrangement of 1D rectangular grooves. The optical mechanisms in this structure depend firstly on the ratio of the period d to the wavelength λ (­figure 2). They also depend on the polarization, due to the 1D structuration. Transverse electric (TE) (respectively transverse magnetic (TM)) polarization is defined by the direction of the electric (respectively magnetic) field along the groove direction. In the short wavelength range (λ  ≪  d), this transmission grating acts as a diffraction grating. Such free-standing diffraction gratings are commercially available for x-ray or ultraviolet spectroscopy, for instance. In the long wavelength limit (λ ≫  d), the drilled membrane behaves like a homogeneous layer with an effective refractive index, in accordance with the effective medium theory [17–19]. In the case of slightly sub-wavelength gratings (λ  ⩾  d), different regimes can be depicted, depending on the filling fraction f (figure 3). For f  →  1, sub-wavelength dielectric gratings exhibit GMR effects that can be described by a perturbative model of waveguide modes in the slab weakly coupled to free-space. These structures were extensively studied in the 1990s [12, 20, 21]. For f ∼ 0.5, the structure is often referred as a PC slab whose optical properties are basically described by interference effects involving forward and backward guided waves [22]. Below the light cone, PC can be designed to exhibit photonic band gaps (PBG) for guided waves in the slab. Above the light cone, guided modes can interact with freespace radiation in complex ways [23]. As a recent example, it has been shown that HCG with f ∼ 0.5 also exhibit resonant features that lead to high reflectivity over a broad spectral range [24, 25]. For f → 0, the structure can no longer be described as a guided-mode structure: it is made of nanorods with a diameter much smaller than the wavelength. Its optical response can be described by a simple model based on multiple scattering [26].

Depending on the filling fraction, the resonance can be attributed to either a collective mechanism involving a large number of unit cells connected via guided waves or free-space interactions, or a resonance localized in one or a few unit cells. Similarly, metal-based nanostructure arrays exhibit both collective and localized resonances. Nevertheless, the smallest size of dielectric structures supporting localized resonances is roughly ∼λ/2n. Introducing metallic nanostructures further widens the family of resonances, enabling new kinds of (surface) guided waves and localized resonances at the deep subwavelength scale. 2.2.  Plasmonics in metal nanostructures

Light–matter interactions with metals are mainly driven by the free electrons close to the metal surface. In the visible and infrared wavelength ranges, the complex permittivity of metals can be described, as a first approximation, by the Drude model: 

ϵm = 1 −

ωp2 ω + jωγ 2

,

(2)

where ωp is the plasma frequency and γ the damping constant. The real part of ϵm is negative in a wide range of frequencies above ωp. The two main optical properties of noble metals result from their permittivity: their opacity (skin depth δ ≃ c/ωp ≃ 25 nm for noble metals) and the existence of electromagnetic surface waves called surface plasmon polaritons (SPP) [27, 28]. These surface waves originate from a resonant interaction between photons and free electrons in the metal. 2.2.1. SPP propagating at an interface.  We now consider

a plane interface between two semi-infinite media: a metal (ϵm) and a dielectric (ϵd). We are looking for a homogeneous solution of Maxwell’s equation at the interface, i.e. a surface wave propagating along the interface, that decays into the 3

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Figure 4.  Propagating (SPP) and LSP resonances in model systems: (a) SPP at a planar metal interface, (b) LSP in a metallic particle, (c) LSP in the gap between two metal particles and (d) LSP in a MIM resonator.

(figure 4(c)). A similar phenomenon is observed in metal– insulator–metal (MIM) nanocavities made of two flat metal nanoparticles separated by a thin dielectric spacer, see figure  4(d). The MIM cavity can be modeled as a truncated planar metallic waveguide. Its fundamental TM mode has no cut-off frequency, it is made of two SPP waves propagating at the metal–insulator interfaces. In the case of a very thin insulator, the two SPP modes are strongly coupled and the effective index of the TM mode increases [34–36]. In a classical approximation, it can be written as n eff = nd 1 + 2δ / t where nd is the refractive index of the insulator layer, t its thickness and δ the metal skin depth [35]. In the limit of extremely thin gaps between metal surfaces, quantum and non-local effects may become predominant, they are the subject of intense research [37–40]. This plasmonic mode goes back and forth between the ends of the cavity, resulting in a Fabry–Perot-type resonance [34, 41, 42]. The resonance wavelength can easily be tuned with the cavity width w [43, 44], according to the simple formula w = mλ/(2neff) where the phase shift due to reflection at the edges of the cavity is neglected and m is the order of the Fabry–Perot mode [346]. This MIM structure has also been a building block for the conception of MM. In this framework, they are described as magnetic atoms which induce an effective magnetic permeability different from 1 (or even negative) at visible or infrared frequencies [45]. They can even result in negative refractive indices [46–49]. LSP resonances have been widely studied as single objects in the frame of near-field optics and more recently nanoantennas [32, 50, 51]. They also offer numerous degrees of freedom for the control of light–matter interactions in ensembles of tiny structures.

two media exponentially with the distance from the interface (­figure 4(a)). The solution is found easily by introducing the tangential (electric and magnetic) fields across the interface in each medium (i = m, d), Eti and Hti and the surface impedance Zi defined as Eti = Zi ni × Hti [29], where ni is an unit vector normal to the interface (nm = − uz and nd = uz). The dispersion relation of a surface wave is obtained by writing the continuity of the tangential fields across the interface: 

Z m + Zd = 0.

(3)

A solution, the SPP mode, exists only if the magnetic field is parallel to the interface (TM polarization, Hz = 0 and Zi = kzi/ ωϵ0ϵi with ω = 2 π c/λ). In a more familiar form, the in-plane wavevector of the SPP mode can be written as 

kSPP = k 0

1 1 ϵd

+

1 ϵm

.

(4)

The surface nature of this electromagnetic mode is reflected by a real part of kSPP larger than the wavevector of light in free-space (in the dielectric medium): SPP cannot be excited directly by a plane wave. Excitation of SPP is usually achieved by evanescent coupling, or with a periodical structure that adds an additional momentum to match the SPP wavevector. In the latter case, the dispersion relation is folded in the first Brillouin zone, in the light cone. It is worth mentioning that polar crystals exhibit similar electromagnetic properties to metals in various infrared wavelength ranges. The real part of their permittivity is negative due to phonon excitations. The electromagnetic problem can be treated in the same way and leads to surface phonon polaritons (SPhP). The wavelength of SPP can be much smaller than the wavelength of light in free space, enabling optical confinement beyond the diffraction limit and localized resonances in plasmonic resonators much smaller than λ/2n.

2.3.  Optical modes and dispersion diagrams: localized and collective resonances

2.2.2.  Localized surface plasmons.  Surface plasmon modes

can also occur in metal nanoparticles, allowing a strongly localized electromagnetic field in three dimensions [30]. In the case of spherical particles (figure 4(b)), optical modes can be described analytically by the Mie theory [31]. In more complex cases, LSP modes are usually determined by solving Maxwell’s equations numerically. Their solution depends on the size and shape of the metal nanoparticles [32, 33]. When two particles are very close to each other, strong optical confinement can be achieved inside the gap

The optical response of any semi-infinite nanostructure array can be analyzed through the S-matrix formalism. This formalism is closely related to the reflection, transmission or diffraction intensity measured in far-field spectroscopic experiments, i.e. the response of the optical system to excitation by a collimated light beam. In the vicinity of a resonance, it is characterized by its optical mode and dispersion relation.

4

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In the following, we are interested in 2D infinite periodic structures (xOy plane) illuminated by a plane wave (plane of incidence xOz, angle of incidence θ). Every cell of the periodic structure is excited with the same amplitude and a phase relation given by the in-plane wavevector kp = kx = ω/c sin θ. In this case, the eigenmode is defined by a real wavevector and a complex frequency:  Figure 5.  Schematic description of a two-port optical system made

2.3.1. The S-matrix of an optical system.  We consider a system whose optical response can be described by the general scattering matrix (or S-matrix) formalism:

(OUT) = [S ] (IN)

2.3.3.  Lorentzian response of a resonant mode.  The amplitude of the mode in the frequency domain is a Lorentzian function: α0 . a (ω ) =  (9) (ω − ω 0 ) − j γ

(5)

where the vector of the out-going wave amplitudes (OUT) is expressed as a function of the amplitudes of the incoming waves (IN) (see figure 5). In the case of a simple dielectric slab of a homogeneous material illuminated by a plane wave, the S-matrix is a 2 × 2 matrix containing the reflection and transmission coefficients. In the case of a 2D periodic structure, the electromagnetic field above and below the slab can be expressed as the superposition of plane and evanescent waves (Rayleigh expansion) whose amplitudes are linked by the S-matrix.

This resonant mode is characterized by its frequency ω0 and full width at half maximum (FWHM) 2γ. The quality factor of the resonance is defined as 2π times the ratio of the timeaveraged energy stored in the system to the energy loss per cycle [29]: Stored energy Q = ω0 = ω0 /(2γ ). Power loss 2.3.4. Dispersion diagrams.  Dispersion diagrams repre-

sent the location of the optical modes in the (ω, k) plane, where only the real part of ω is considered. The imaginary part can be represented in a separate graph [54]. The angular dispersion of resonances is a direct consequence of the inplane spatial coherence of the resonant mode and depends strongly on the nature of the resonant mechanism. The dispersion relations of localized and collective resonances are illustrated in figure 6.

2.3.2.  Eigenmodes of a resonant system and dispersion relation. 

In case of a resonant structure, an eigenmode is defined as a state of the system such that the amplitude of the optical response goes to infinity when the excitation amplitude is constant (or equivalently, a solution exists without external excitation). It is a solution of Maxwell’s equations with proper boundary conditions. With the S-matrix formalism, this problem is formulated as: 

[S −1] (OUT) = 0.

2.3.5. Localized resonances.  Localized resonances are nearly independent of the in-plane wavevector: each unit cell is made of resonant nanostructures (high-index-contrast dielectric nanostructures, localized plasmonic resonances,...). Light propagation along the slab and the phase difference between unit cells play a minor role. As a consequence, localized resonances appear as flat features in dispersion diagrams.

(6)

It is called the homogeneous problem [8]. The solutions are the zeros of det[S−1] and are also called leaky modes. These eigenmodes of the resonant system are defined by their excitation condition (photon energy, in-plane momentum, polarization). The ensemble of solutions is described by the dispersion relation 

f (ω, k ) = 0

(8)

where the losses are expressed by the decay time τ = 1/γ. The complex frequency ω is a pole of the S-matrix [52] and can be calculated by various numerical methods [53].

of a semi-infinite resonant slab. The S-matrix links the input and output plane waves.



a (x, t ) = a 0 e j(ω0t − kpx )e−t / τ

2.3.6. Collective resonances.  When collective resonances

are excited in nanostructure arrays, the phase difference between the unit cells plays an important role in the resonant mechanism. It is controlled by the in-plane wavevector kg of guided waves propagating along the slab. This is the case for GMR in nanostructured dielectric slabs (dielectric waveguides), or in metallic films with periodical corrugations (SPP waves, kg ≃ kSPP). In multiple coherent scattering phenomena, optical interactions between nanostructures is achieved through free-space waves (kg ≃ k0). Collective resonances are characterized by a strong angular dispersion (kp-dependence of the dispersion curves). In the case of SPP resonances on metallic surfaces with weak corrugations, the spatial coherence of the mode is given by the propagation length of surface waves along the interface.

(7)

where the frequency ω = ω0 + jγ and the wavevector k = kp + jκ have complex values a priori. The imaginary parts of ω and k represent the losses of the resonant mode. For the sake of simplicity, we consider single-mode resonators with low losses (the width of the resonance is much smaller than the frequency, Δ ω  ≪  ω0). The solutions can be restricted to more simple cases, according to the excitation mechanism. In the case of a guided mode propagating in the x-direction and excited by a monochromatic wave at the position x = 0, the amplitude of the mode can be written in the space–time domain with a real frequency: a(x, t) = a0ej(ω0t − kpx)e−x/δ where δ = 1/κ is the decay length of the mode in the propagating direction. 5

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propagation modes coupled to each other and to deduce a few elementary properties from conservation of energy and reciprocity. It has been developed and extensively applied in the framework of optical waveguides [57–59]. This approach considers the coupling of modes in space, each guided mode consisting in both a forward and a backward wave. The coupled-mode theory has been further extended to the temporal domain by studying the coupling of modes in time, e.g. wave coupling to optical resonators [60–62]. It applies to numerous physical systems involving propagating waves and resonant modes. During the past ten π/d 0 kx kx years, it has been used to analyze the free-space coupling of optical resonances, such as plane wave coupling with a semi-infinite Figure 6.  Generic dispersion curves. (a) A planar structure slab [63]. The approach used in the following for resonant modes surrounded by a vacuum: propagating mode with an effective index coupled to plane waves was primarily developed by Haus [60] neff ≃ 1.2 (guided wave, SPP,...) in red and localized mode in blue. and extended by Fan et al [63]. This TCMT is a powerful tool to (b) Periodic structure (1D): the dispersion curve of the collective (propagating) mode is folded in the first Brillouin zone, with analyze resonant nanostructure arrays and to design efficient phobandgaps at kx = 0 and kx = π/d. Optical modes in the yellow region tonic devices coupled to several input and output plane waves. (light cone) are coupled to free space. We consider a resonant single-mode optical system coupled to N input/output channels and monochromatic input waves Guided waves are excited by a diffracted wave (order (m, q)) (time-dependance ejωt). The width of the resonance is assumed when their in-plane wavevectors are equal. This condition proto be much smaller than the resonance frequency and the kpvides a first-order approximation of the dispersion relation: dependence of the mode is not taken into account (kp is fixed). (m, q) The temporal variation of the mode amplitude is written as a | k | ≃ k .  (10) p g function of energy dissipation (decay rate γ) and input wave If kg  =  neffk0 with a constant neff, the dispersion relation is a amplitudes (In) (equation (11)). The output wave amplitude (Om) straight line, its slope is given by the effective index neff (red in channel m is the sum of the direct coupling of input waves lines in figure 6). and the radiative dissipation of the system (equation (12)). The amplitudes are defined such that ∣ a ∣2 is the energy in the resona2.3.7. Experimental determination of dispersion diagrams.  tor and ∣ In ∣2 (∣ On ∣2) is the input (output) power in channel n: Angular spectroscopy provides an experimental mean to probe the dispersion properties of optical modes [55]. SpecN da ular reflection (R(λ, θ)) or transmission (T(λ, θ)) spectra  = (jω0 − γ ) a + ∑ αni . In (11) dt n=1 are measured for various incidence angles θ of an incident collimated beam with a small increment δθ. With a simple N coordinate transformation, they are translated into reflection O = ∑ Cnm . In + αmoa (12) m (R(ω, k)) or transmission (T(ω, k)) dispersion diagrams that  n=1 are usually plotted in a color scale. The dispersion curves appear as bright or dark bands. The resonance width is related where [Cnm] is the S-matrix for the direct pathways between to the total decay rate γ of optical modes (temporal coher- the incoming and outgoing waves, and αni (αno ) are the input ence, τ) and intensity variations reflect both their free-space (output) coupling coefficients of the mode. The reciproccoupling and internal losses, as discussed in the next section. ity theorem can be used to demonstrate that αni = αno = αn. Using the conservation of energy and γn =∣αni∣2 /2, the total ω

(a)

light cone

ω

(b)

N

radiative decay rate of the resonant mode is γr = ∑n = 1 γn. The non-radiative decay rate (internal dissipation) is γnr = γ − γr. Note that with this notation, the decay rate of the energy stored in the system is 2γ (lifetime: τ/2). The mode amplitude is given by:

3.  Nanostructure arrays: free-space coupling The dispersion relation provides important information on the intrinsic properties of nanostructure arrays: the resonant frequencies of optical modes and their temporal and spatial coherence which gives insight into their localized or collective nature. The free-space coupling and internal losses of resonant systems are of primary importance for practical applications, since they are directly related to the efficiency of devices conceived for light harvesting, filtering, or emission. The TCMT provides a very useful framework to address this issue.

N



a (ω ) =

∑ αnIn n=1

j (ω − ω0 ) + (γr + γnr )

,

(13)

and the non-radiative loss power is: 2γnr ∣ a ∣2. The coupling constants of the direct pathways [Cnm] are also related to αn. In the following, we restrict our discussion to particular cases that will be used in this review for the design of various optical elements.

3.1.  The TCMT

The coupled-mode theory was first introduced by Pierce in 1954 [56]. Its purpose is to consider general linear systems with 6

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T (ω ) =

4γ1γ2 . (ω − ω0 )2 + (γr + γnr )2

(15)

The transmission maximum is T(ω0) = 4 γ1 γ2/(γr + γnr)2. Total transmission occurs if there is no loss (γnr  =  0) and if the coupling strengths of the two radiative channels are equal (figure  8(b)). It is important to emphasize that a symmetric system is not required: perfect transmission can occur in a lossless, asymmetric system, provided γ1 = γ2. A lossless, symmetric Fabry–Perot resonator is a simple illustration of perfect transmission at the resonant frequency. Furthermore, this analysis provides a quantitative value for the transmission efficiency in the case of asymmetric mirrors or internal losses. In the case of a symmetric but lossy system, the transmission maximum is given by:

Figure 7.  A schematic of a resonant slab and coupling coefficients: direct reflection (C11) and transmission (C12), radiative (γ1 and γ2) and non-radiative (γnr) loss channels.

3.1.1.  Example: a resonant slab.  We now consider a resonant slab made of a periodic structure, surrounded by a vacuum. The input/output radiations are plane waves propagating in the two half-spaces separated by the slab. If all but the zero-order diffracted waves are evanescent (e.g. subwavelength period at normal incidence), the photonic slab can be modeled as an optical resonant system excited by an incident plane wave (I1) in the upper half-space and coupled to three loss channels (decay rates γ1, γ2 and γnr), as depicted in figure 7:



3.4.  Fano resonances, perfect reflection and transmission in dielectric nanostructure arrays

We now consider a lossless symmetric structure with a direct pathway for the transmission/reflection. In case of lossless systems, the time-reversal symmetry can be used to deduce a relation between the [C] matrix and the coupling coefficients [63]. With C21  =  C12  =  jt and C11 = C22 = r, the transmission intensity can be written:

Using (13), the TCMT provides the absorption intensity in the slab: A (ω ) =

4γ1γnr . (ω − ω0 )2 + (γr + γnr )2

(16)

This result has been used to determine experimentally the radiative and non-radiative losses of real optical structures [64].

• radiative losses via out-going plane waves in the upper (incident) half-space (reflection amplitude: O1/I1), • radiative losses via out-going plane waves in the lower half-space (transmission amplitude: O2/I1), • non-radiative losses (absorption) in the case of lossy materials (we do not consider the case of the gain medium).



⎛ γr ⎞2 T (ω0 ) = ⎜ ⎟ . ⎝ γr + γnr ⎠

(14)

The TCMT provides a simple and powerful tool to analyze resonant slabs in different configurations, as depicted in figure 8.



t (ω − ω0 ) ± rγr T (ω ) = j (ω − ω0 ) + γr

2

(17)

where t and r are real positive numbers, r2  +  t2  =  1 and the ± sign depends on the symmetry of the optical mode with respect to the mirror plane (α2 = ± α1). In the vicinity of the resonance, the transmission intensity is fully described by three parameters: t, ω0 and γr. With a direct transmission pathway (t ≠ 0), the transmission intensity takes the form of a Fano resonance:

3.2.  Perfect optical absorption in nanocavity arrays

From equation (14), the absorption maximum is: A(ω0) = 4 γ1 γnr/(γr + γnr)2. In the case of a symmetric system (γ1  =  γ2) with no direct transmission (C12  =  0), it can be shown that the absorption maximum is 50% when γ1 = γnr/2. If a perfect mirror is placed behind the slab (γ2  =  0, γr = γ1), perfect optical absorption can be achieved for γr = γnr (­figure  8(a)). This is called the critical coupling condition. The latter case will be used to create perfect absorbers with nanoscale resonator arrays, see section 4.



ω − ω0z T (ω) = t ω − ω0p

2

,

(18)

leading to an asymmetric lineshape with a transmission minimum close to a transmission maximum: T has a complex pole ω0p and a real zero ω0z. The transmission intensity goes from 0 to 100% 4 (figure 8(c)). It is worth mentioning that the zero of T is either on the long-wavelength side or on the

3.3.  Perfect transmission through metallic nanostructure arrays

We now assume no direct transmission (C12 = 0). The transmission amplitude is α2a and the transmission intensity is written as

4

  This property can also be found with a phenomenological approach based on the time-reversal symmetry and the geometrical symmetries of the ­system [65]. 7

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Figure 8.  Illustration of the TCMT applied to various resonant nanostructure arrays (a) deposited on a mirror (perfect absorption in the critical coupling), or (b) and (c) surrounded by a vacuum. Single-mode structures exhibit resonant absorption (a) or transmission (b) spectra with a Lorentzian lineshape, or with a Fano lineshape if a direct transmission pathway exists (c).

short-wavelength side of the pole, depending on both the mode symmetry and the magnitude of the direct transmission coefficient t/r. 4.  Absorption in metallic nanocavity arrays In this section, we consider resonant sub-wavelength arrays of nanostructures deposited on a mirror, or drilled in a bulk metal such as gold or silver. There is only one radiative and one nonradiative loss channel, as a result of both the back mirror (no transmission) and the sub-wavelength period (no diffraction). The critical coupling condition is reached when their decay rates are balanced (γr = γnr), leading to total absorption of light at the resonance frequency. It can be schematically explained as follows. Reflected light can be described as the sum of (i) direct reflection and (ii) light radiated by the resonant mode. If these two waves have the same amplitudes with opposite phases, they cancel out, resulting in perfect absorption in the structure. When the critical coupling condition is not fulfilled, the absorption efficiency at resonant wavelength is given by: 

Amax =

4γrγnr . (γr + γnr )2

Figure 9.  Shallow metallic gratings with sinusoidal corrugation. Absorption losses depend weakly on the grating height (red). Radiative losses (grating coupling to free space) increase with the height (blue) and reach non-radiative losses (critical coupling, perfect absorption).

1976 [67, 68] and evidenced experimentally by Hutley and Maystre [9]. The sinusoidal gratings were fabricated by interference lithography and gold coating. The primary resonant phenomenon arises from the excitation of SPP, similarly to their first observation by Wood in 1902 [1]. This has been reviewed recently [69]. For an angle θ0  =  6.6°, which corresponds to the SPP resonance k p(−1) ≃ kSPP at the excitation wavelength λ  =  647  nm (period d  =  555.5  nm), TM-polarized light is totally absorbed for a height h = 37 nm of the groove profile. Maystre and co-workers have studied the reflectivity efficiency R(θ, h) as a function of the incidence angle θ and the grating height h. A phenomenological approach based on the pole and zero of R(θ0, h) has been used to explain total light absorption. It can also be analyzed in light of the TCMT, as illustrated in figure  9. For shallow profiles (h  →  0), the absorption losses of the surface plasmon resonance (SPR) are independent of the height h (red curve). In contrast, the radiative coupling is canceled for h = 0 and increases monotonically with the grating height (blue curve in figure 9), until

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Electromagnetic absorbers were first developed in the frame of stealth radar applications. The Salisbury screen, patented in 1952, is an early example of resonant structures developed for this purpose. It was made of a slab of lossy materials placed at a quarter-wave distance of a ground (metallic) plane [66]. This concept based on destructive interference was extended to periodic structures (frequency selective surfaces) [66]. In the visible range, perfect resonant absorption of light was first discovered in shallow metallic gratings in the 1970s.

(

4.1.  Shallow metallic gratings

Total absorption of light by a shallow sinusoidal metallic grating was first predicted theoretically by Maystre et al in 8

)

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Figure 10.  Absorption in metal grooves and voids. (a) The fabrication process of high aspect ratio 1D metal grooves by metal deposition

on a mold. (b) Hexagonal arrays of metal voids fabricated by electrochemical deposition of gold through a template of self-assembled latex spheres. (c) Absorption spectra for a various thicknesses t. Total light absorption is demonstrated for a thickness equal to 1.08 times the void diameter. (a) Reproduced with permission from [82]. Copyright 2008, AIP Publishing LLC. (b) Reproduced with permission from [83]. Copyright 2006 American Physical Society. (c) Reproduced with permission from [84]. Copyright 2008 Nature Publishing Group.

the critical coupling condition is reached. This primary work on 1D gratings has been recently extended to total absorption of unpolarized light by crossed gratings by Popov et al [70]. Total absorption at normal incidence has also been studied numerically in 1D sub-wavelength metallic gratings with double-period grooves by Tan et al [71]. A long-period grating provides additional degrees of freedom in order to engineer the radiative coupling of the resonant mode and reach total absorption. These results are remarkable: these periodical structures are essentially metallic mirrors which become perfect absorbers at the resonance frequency. Total resonant absorption has been obtained recently with a wide variety of nanostructured metals using various resonant mechanisms.

independent of the angle of incidence. Nanoscale groove widths lead to high effective indices for the waveguide mode and strong electric field enhancements [80, 81]. In high aspect ratio metallic grooves, high-order Fabry–Perot resonances can be obtained [79]. Total absorption in high aspect ratio 1D nanocavities has been experimentally demonstrated in the visible [85] and midinfrared [82, 86] wavelength ranges. In [82], narrow (100 nm) and deep (1000  nm) rectangular grooves in gold have been fabricated by inversion from an inverted sacrificial master mold made of Si (see figure 10(a)). Alternatively, the mold can be made of patterned resist layers or GaAs wafers [85, 86]. Omni-directionality of optical absorption in deep grooves has been demonstrated in [86]. Planar metallic slabs containing a 2D lattice of spherical nanocavities have also been proposed to act as resonant light absorbers [87, 88]. Samples were fabricated by electrochemical deposition of gold over a monolayer of latex spheres deposited on a gold substrate [83, 84]. The latex spheres are self-assembled in a compact hexagonal lattice. They are removed by chemical etching to leave empty spherical voids buried in gold with various thicknesses, from widely open truncated voids to fully encapsulated cavities, see figure 10(b). The thickness impacts both the resonant mechanism (localized void plasmons versus SPP surface modes) and the radiative coupling [83, 89–91], leading to omnidirectional perfect absorption [84] (figure 10(c)). Inspired by the properties of grooves and voids in metals, various nanostructured metal surfaces have been proposed and studied, such as pyramidal grooves [92] and crossed slits in metal [93, 94]. Introducing more sophisticated morphologies has also allowed the extension of the spectral bandwidth of the optical absorption [95], for instance with pyramidal or convex grooves [96–98].

4.2.  Deep metal grooves and voids

A 1D metallic grating with rectangular grooves, also called a lamellar grating, is a canonical structure that exhibits both propagating and localized plasmonic resonances [7]. It can also be described as a metallic membrane with rectangular slits, deposited on a metallic mirror. First, similarly to the case of plasmonic resonances in sinusoidal gratings, surface plasmon waves can be excited resonantly on the top metallic surface [72]. Following a perturbative approach, propagating surface waves (SPP) are excited when the wavevector of the (evanescent) first-order diffracted waves is close to the SPP wavevector on flat surfaces: k p(±1) ≃ k spp for a given incidence angle. Second, incident plane waves also couple directly to the fundamental TM waveguide mode of the groove, which is also plasmonic in nature [35]. When the groove is deep enough, this mode resonates between the upper end of the groove and the bottom mirror, leading to Fabry–Perot resonances [73–78]. The latter mechanism is 9

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Figure 11.  Fabrication of MIM nanocavity arrays by soft-UV nanoimprint lithography (period: 400 nm, width: 200 nm). (a) Deposition of

the metallic and dielectric layers on a glass substrate. (b) A soft nanoimprint in UV-curable resist; a thin PMMA layer is first spin-coated to facilitate the lift-off. (c) Pattern transfer: reactive ion etching of the residual layer (CHF3/O2) and of the PMMA (O2). (d) Metallic deposition and lift-off. (e) The master mold made from Si wafers by electron beam lithography and reactive ion etching and the Si replica. (f) A flexible h-PDMS/PDMS stamp is fabricated from a Si master mold and used in the nanoimprint lithography.

Another approach consists in the design of multi-resonant structures by introducing several nanostructures per period. With several identical deep grooves per period, Barbara and co-workers have evidenced new types of coupled modes localized in one or several grooves and which can also couple to propagating surface plasmons [99, 100].

more specifically studied in 1D metal stripes [104–106] and 2D disc patches [106]. The experimental demonstration of perfect resonant absorption in MIM nanocavity arrays was achieved at λ ≃ 1.5  µm in 2010 [107, 108] and omni-directionality was subsequently demonstrated by angle-resolved spectroscopy in 2011 [42, 109, 110]. Since then, MIM nanocavity arrays have been the basis of numerous studies (see references in [111]), motivated by several applications that will be described in the following sections. MIM nanocavity arrays have been fabricated by nanoimprint lithography on large surfaces [42], as described in figure  11. First, a single Si master mold is fabricated by electron-beam lithography. Second, it is replicated to create other Si master molds, or it is used for the fabrication of PDMS stamps, figures 11(e) and (f). Then, the PDMS stamp is used in a simple and low-cost process. Nanoimprint lithography enables the replication of sub-20 nm features [112] with industrially relevant processes that could be exploited in various practical applications [113]. Figure 12 shows angular spectroscopy experiments of 1D MIM arrays. On the one hand, nearly perfect optical absorption (>98%) is demonstrated at any incidence angle for the

4.3.  MIM nanocavity arrays

The MIM nanocavity arrays are an important class of resonant absorbers. They are composed of 2D arrays of metallic patches with square or disc shapes, deposited on a metallic mirror with a thin dielectric spacer. As explained in section  2.2, the fundamental resonance mode of the MIM nanocavity originates from Fabry–Perot-type resonances of a strongly confined plasmonic mode. Perfect absorption at a chosen wavelength is obtained as follows: the back mirror prevents light transmission, the resonance wavelength is mainly determined by the cavity width and the radiative coupling by the filling fraction. The first studies were solely focused on the electromagnetic coupling between a metal nanoparticle array and a metallic surface [101–103]. Resonant absorption was

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Figure 12.  Absorption in 1D MIM nanocavity arrays. (a) A schematic of the cross-section of a MIM nanocavity. (b) Absorption spectra of

1D MIM resonators measured under TM polarization (magnetic field parallel to the wires) at different angles of incidence (period: 400 nm, w = 200 nm, tM = tD = 20 nm). (c, d) Simulations showing the magnetic field intensity below the metal wires for the fundamental (m = 1, θ = 0°) and second-order mode (m = 2, θ = 30°), under plane wave illumination. (e, f) Determination of the radiative and non-radiative losses γr and γnr at several angles for m = 1 (f) and m = 2 (e); the critical coupling condition for the second-order mode is fulfilled for angles of incidence between 45° and 50°. Reproduced from [42]. Copyright 2011 American Chemical Society.

fundamental resonance (m = 1, λ1 = 1280 nm). On the other hand, the absorption efficiency is strongly angular-dependent for the second-order resonance (m  =  2, λ2  =  740  nm). This effect can be explained by the symmetry of the secondorder mode. Due to the mirror symmetry of the nanocavity in the x-direction, the resonant modes have either an even (m  =  1) or an odd (m  =  2) amplitude profile (figures 12(c) and (d)). At normal incidence, the incident plane wave has an even amplitude profile and cannot couple to the secondorder mode: the radiative decay rate vanishes at θ = 0° for m = 2. To sum up, both resonance wavelengths λ1 and λ2 are independent of the incidence angle, as expected for localized resonances, but the free-space coupling (γr) of the secondorder resonance does depend on the incidence angle for symmetry reasons. Using equation  (14) with γ1 = γr, absorption resonances can be fitted by a Lorentzian function and the radiative and non-radiative decay rates can be determined from absorption experiments for each angle of incidence. The results are shown in fi ­ gures  12(e) and (f ). They illustrate the role of the radiative and non-radiative decay rates in the achievement of perfect absorption. The critical coupling condition γr = γnr is fulfilled by tuning the incidence angle at θ ≃ 45−50° for m = 2. It was recently demonstrated that etching the insulator between the MIM cavities provides another convenient way to tune the radiative

and non-radiative decay rates and reach the critical coupling condition [114]. Decreasing the size of the MIM nanocavities is an important goal for various applications requiring high field enhancement and low volume for light–matter interactions. It can be achieved with thinner gaps between the two metal surfaces, which also induces (i) an increase of the effective index of the plasmonic mode and (ii) a decrease of the cavity width for a given resonant wavelength. Such light confinement in MIM-based absorbers has been achieved recently by use of ultrathin dielectric spacers based on nanoscalethick polymer or atomic layer deposition deposited dielectric layers [115, 116]. Using silver nanocubes separated from a gold mirror with very thin organic layers, Moreau et al have demonstrated resonant absorption at λ = 700 nm with 74  nm wide nanocubes and 6  nm thick spacers (see figure  13). The nanocubes are not regularly spaced, but interestingly the disorder seems to have little effect on the absorption efficiency. MIM absorbers have been investigated by numerous groups during the past few years, see for instance references in [111] and [117]. Better control of the spectral bandwidth of MIM resonators is an important issue for several applications, such as photovoltaics and thermophotovoltaics. A first way to broaden the absorption bandwidth is to use crossed trapezoidal arrays as the top layer of the MIM structure [118]. This strategy has

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Figure 13.  Silver nanocubes synthesized chemically, before (a) and after (b) deposition on a gold mirror. The very thin spacing between

the cubes and the mirror is precisely controlled by an organic layer, as depicted in (d). (c) The modulus of the magnetic field (red, high values; blue, low values) for the fundamental MIM mode. (e) The position of the resonance as a function of the spacer thickness. Inset: corresponding experimental reflectance spectra for the different spacer thicknesses (θ = 25°). Reproduced with permission from [115]. Copyright 2012 Nature Publishing Group.

led to resonant light absorption over the 400–700  nm wavelength range, with an average measured absorption of 71%. Another approach is to combine several MIM resonators with different sizes in the unit cell of the 1D or 2D periodical structure [119–125]. This results in multi-resonant absorption spectra, see figure  14. Each resonance wavelength can be tuned separately and at a given resonance wavelength, most of the absorption is routed toward a single MIM element [121]. The small size of nanocavities is a key element for the combination of several resonators in a sub-wavelength unit cell. MM have also been a fruitful approach for the design and fabrication of absorbers. MM perfect absorbers are reviewed in detail in [117]. The basic design principle is to tune the effective permittivity ϵeff and permeability μeff in order to achieve ϵeff = μeff. This ensures that the MM impedance equals the free-space impedance, preventing optical reflection at the top interface. This concept was first proposed in 2002 [126] and demonstrated in the microwave regime in 2008 [127]. It was rapidly extended to shorter wavelengths (from THz to near-infrared), wide angle and polarization insensitive operation [128–133]. The geometry has moved towards MM deposited on a ground plate (metallic mirror), leading to geometries very similar to the MIM absorbers described above [108, 134]. The similarities of the approaches have been discussed together with analytical models [135, 136]. The extension of these approaches to nanopillar arrays made of stacks of multiple metal/dielectric layers results in high absorption efficiency over a broad spectral range [137–139]. The principles used in periodic structures with grooves or MIM nanocavities can be extended to other geometries exhibiting localized resonances. For instance, ensembles of coreshell spherical nanospheres deposited on metallic mirrors have been proposed to act as resonant absorbers [140, 141].

Figure 14.  Multi-resonant absorption in MIM nanocavities.

Experimental absorptivity of the dual-band structure: each unit cell consists of two crosses of different sizes, resulting in two absorption peaks at 6.2 µm and 8.3 µm. Reproduced from [120]. Copyright 2010 American Physical Society.

according to Kirchhoff’s law, the emissivity of an object equals its absorptivity [142], so that modifying the spectral and directional properties of the absorbance of a body results in an analogous modification of its emissivity once used as thermal light-emitting source. Perfect absorbers are also blackbody emitters. This principle was used to demonstrate coherent thermal emission by Greffet et al in 2002 [72]. They fabricated a 1D SiC grating exhibiting nearly perfect absorption due to the resonant excitation of SPhP. Reciprocally, directional emission of infrared light was evidenced (λ ≃ 10  µm), see figures  15(a) and (b). This directionality is related to the coherence of thermal currents along the grating surface. It originates in the coherent properties of surface waves (either SPP or SPhP). This work has inspired numerous studies during the past ten years. Coherent thermal emitters made of tungsten have also been demonstrated [143], with a directivity comparable to the directivity of a CO2 laser. They are based on the excitation of SPP on 1D gratings. The directional and polarization properties can be further controlled with 2D gratings [144,

4.4.  Application: coherent thermal emission of light

Thermal emitters, such as blackbodies, are usually considered as incoherent and broadband light sources. However, 12

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Figure 15.  Coherent thermal emission. (a, b) Directional emissivity (λ = 11.36 µm) from a 1D SiC grating (period 0.55λ, depth λ/40).

(a) An AFM image. (b) Polar plot of the emissivity for TM polarization (red: experiments, green: theoretical calculation). (c, d) Spectrally selective emissivity from the 2D MIM array depicted in (c). Dashed curves: perfect blackbody emission. (a) and (b) Reproduced with permission from [120]. Copyright 2011 American Physical Society. (c) and (d) Reproduced with permisison from [147]. Copyright 2008, AIP Publishing LLC.

LSP resonances and the weak optical cross-section of metal nanoparticles. Nanocavity-based perfect absorbers constitute an excellent platform for the conception of novel biosensors: (i) strong optical confinement in nanoscale cavities ensures a better overlap of the optical mode with grafted molecules, (ii) every incoming photon interacts with the cavity mode via perfect absorption and (iii) the FOM can be enhanced with relatively good quality factors. These design principles have been applied to 2D metal nanopyramidal gratings [156] and MIM nanocavity arrays [42, 107, 157], as illustrated in figure  16. The bulk refractive index sensitivity is of the order of S ≃ 300−400 nm/RIU (refractive index unit). Recently, mushroom-shaped plasmonic resonators have demonstrated excellent performances [158, 159], with FOM reaching ≃ 100. The various geometries offer different confinement factors, FOM and practical usability. Recent achievements are listed in the supplementary information of reference [159].

145] and spectrally selective omni-directional emitters based on localized resonances have also been demonstrated [146– 148] (figures 15(c) and (d)). 4.5.  Application: biosensing

Conventional SPR biosensors are based on the resonant excitation of propagating surface plasmons by a prism in the Kretschmann geometry. The adsorption of targeted biomolecules on a functionalized gold film changes the refractive index at the vicinity of the metal surface and results in the modification of the dispersion relation of SPP. These changes are monitored through the variation of a specific resonance angle or wavelength, or intensity change at a fixed wavelength. However, the probe size of SPR is intrinsically constrained by SPP propagation length (> several microns), and the sensitivity is limited by the small overlap of the analyte (a few nm) with the evanescent tail of SPP waves (several hundred of nm). The performance of a SPR-based sensor is most commonly characterized through the bulk sensitivity S = Δλ/Δn, where n is the refractive index of the medium in contact with the metal surface. Since it is easier to detect a given resonance shift for narrow lines, the figure of merit (FOM) = S/ FWHM (where FWHM is the FWHM of the plasmon resonance) is a more meaningful measure of the performance of the sensor. LSP resonances provide a promising alternative to further enhance the performances of biosensors [149–155], but early experiments were limited by the low FOM (∼ 2) due to broad

4.6.  Application: optical detection and photovoltaics 4.6.1. Photodetection.  Perfect absorbers can be integrated in

photodetectors to increase their performance. A photodetector is basically made of a semiconductor region that absorbs incident photons and creates electron–hole pairs and two contacts that are used to create an electric field and collect charges. Decreasing the size of photodetectors is a major goal for many applications. This is the key issue to increase the cutoff frequency of devices for optical telecommunications, or to decrease the dark current, proportional to the semiconductor 13

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Figure 16.  Perfect absorbers in biosensors. (a) An SEM image of the 2D nanocavities (square particle width w = 235 nm and pitch

400 nm). Absorption spectra under (b) TM polarization and (c) TE polarization at different angles of incidence. (d) A diagram and photograph of the biosensor based on 2D plasmonic nanocavities (pattern area 7 × 7 mm2 = 0.5 cm2) integrated in a glass–PDMS–glass fluidic chamber for the optical index sensing experiments. (e) Spectral shift of the second-order mode for different refractive index solutions: water (black line), ethanol–water solution (blue line) and pure ethanol (red line) and FOM(λ) as a function of the wavelength (green line). (f) Linear relation of the spectral shift in response to changes in refractive index resulting in the bulk sensitivity of the biosensor. Reproduced from [42]. Copyright 2011 American Chemical Society.

nanoscale absorbing semiconductor (GaAs) wire between two metallic (Ag) wires with cross-sectional dimensions of 100 nm × 40 nm. It forms a resonant nanocavity. A periodic combination of this MSM nanostructure results in a metal– semiconductor nanograting. Perfect absorption has been demonstrated for one polarization direction at λ ≃ 800 nm, with η = 9% [164]. However, the benefits of these approaches are mitigated by the complexity of the fabrication, surface recombinations and absorption losses in the metal regions. The latter metal absorption excites hot electrons that can be injected into the semiconductor over the Schottky barrier, effectively contributing to photocurrent [165, 166]. To date, the overall quantum efficiency of devices based on this process is still very low, below 1% [167, 168]. Metal absorption losses are less prohibitive in the mid-infrared range [169–171]. The dark current has been successfully reduced in quantum well infrared photodetectors (QWIP) embedded into an array of double-metal nanoantennas [163], at an operating wavelength of λ ≃ 9 µm. This achievement is shown in figures 17(e)–(g). Applications of perfect absorbers in the GHz domain have also been realized [172].

volume, which is the first limitation of mid-infrared photodetectors. However, in conventional photodetectors the thickness is limited by the absorption depth of bulk semiconductors and the lateral size is limited by the diffraction limit. Perfect absorbers based on plasmonic nanocavities bring the ability to shrink the dimensions of various types of photodetectors and to overcome the trade-off between size and efficiency [160]. MIM nanocavities are perfectly adapted to serve as a basis for the conception of photodetectors, with: (i) absorption in very small (deep sub-wavelength) volumes which can be filled with semiconductor materials, (ii) optimized efficiency through total absorption of light, (iii) spectral selectivity. The first demonstrations were carried out in 2003 with a slightly different architecture: the bottom metallic mirror was replaced by a Bragg multi-layer reflector, resulting in a resonant-cavity-enhanced sub-wavelength metal–semiconductor–metal (MSM) structure [161]. The top 1D metallic grating was made of interdigitated electrodes deposited on a 40 nm thick GaAs absorbing layer. The external quantum efficiency (EQE) η, defined as the ratio of the number of charge carriers collected by the device to the number of incident photons, reached η = 14% at resonance (λ ≃ 810 nm) [161] and was further enhanced in an optimized structure showing nearly perfect absorption and η = 40% [162], see figures 17(a)–(d). In this device, the absorption layer is twenty times thinner than the bulk GaAs absorption depth. An even more challenging architecture has been proposed [164]. The photodetection element is composed of a

4.6.2. Solar photovoltaics.  The application of resonant perfect absorbers for solar photovoltaics is in its infancy. Let us first describe the context. In conventional solar cells, light-trapping is simply achieved by use of an anti-reflection 14

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Figure 17.  Examples of photodetectors made of perfect absorbers. (a)–(d) A GaAs-based nano-MSM photodetector. (a) A schematic

and cross-section of the electric field intensity in the GaAs absorption layer (TE polarization, λ = 790 nm). Grating period: 200 nm, wire width: 100 nm, GaAs layer thickness: 40 nm. (b) An SEM image. (c, d) Reflection and absorption spectra and EQE of structure (a). (e)–(g) QWIP made of MIM cavities. (e) SEM images. (f) Simulation of the fundamental mode (displacement current). (g) Responsivity spectra for cavities with different widths s at 77 K and 1 V bias. (a)–(d) Reproduced from [162]. Copyright 2011 The Optical Society. (e)–(g) Reproduced with permission from [163]. Copyright 2014, AIP Publishing LLC.

coating on optically thick absorbers. The ability of plasmonics to reduce the absorption volume by more than one order of magnitude is key to tackling both of the main challenges of next-generation photovoltaics: cost reduction and efficiency improvement [173, 174]. Resonant absorption can compensate for the low single-pass absorption of very thin films, enabling a reduction of material consumption while maintaining good efficiency. The thickness reduction of thin-film semiconductor solar cells also leads to a reduction of carrier transit time and non-radiative recombination processes. Efficient absorption in ultra-thin layers is also key to boost the efficiency over the Shockley–Queisser limit of singlejunction devices (η > 33%), with targeted efficiencies in the 40–50% range. These efficiencies can already be achieved with multi-junction solar cells, but they involve very complex semiconductor stacks and costly fabrication processes. Alternative solutions are currently being explored. They are based on novel concepts [175, 176], in particular hot-carrier solar cells, intermediate-band solar cells and up- and down-conversion processes. These approaches may offer the possibility to absorb low-energy photons, or to convert the photon energy in excess of the band-gap. Various strategies for light-trapping in ultra-thin solar cells are under development [177–179]. They are based, for instance, on light scattering by metal nanoparticles [180–182], coupling to guided-modes [183] via nanostructured back mirrors [184, 185] or PC [186, 187], or multi-resonant structures in MIM cavities [188, 189]. In addition, plasmonic metallic structures can also play a role in collecting photogenerated charges and suppressing the transparent conductive oxide layers that are usually used as contact layers in solar cells.

However, the use of resonant absorbers for the conception of ultra-thin solar cells still faces several issues: the broad band required to harness the greatest part of the solar spectrum, the absorption losses in contact layers and the development of low cost nanofabrication processes. Nevertheless, nanostructure arrays provide new degrees of freedom for light management in solar cells and should impact the conception of the whole architecture of photovoltaic devices. 4.7.  Discussion and perspectives

This section has been restricted so far to periodic nanostructures with only one radiative channel. It is interesting to discuss, in light of the TCMT, whether or not this is the only configuration that can lead to perfect absorption at the resonant wavelength. As mentioned in section  3.2, in a free-standing resonant membrane with a plane of symmetry parallel to the slab (γ1  =  γ2), the absorption maximum is 50% if there is a single resonant mode and no direct transmission path (C12 = 0). Interestingly, 100% absorption efficiency can be achieved with two degenerate (orthogonal) modes [190]. We now assume a resonant slab with a direct transmission path (C12  ≠  0) and no symmetry (γ1  ≠  γ2). Radiative losses through channels 1 and 2 can be written, respectively, as 

O1 = C11 . I1 + α1a

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O2 = C12 . I1 + α2a .

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Similarly to the one-port critical coupling condition (γr = γnr), perfect absorption could be achieved if there is no output wave 15

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(O1 = O2 = 0): in each channel the direct path and the wave radiated by the mode cancel out (they have the same amplitude and opposite phase). The absorption maximum is written: 

Amax =

4γ1γnr . (γ1 + γ2 + γnr )2

wire grid polarizers in the long-wavelength domain, from millimeter to far-infrared wavelengths (see, for instance, [194] and references therein), with few numerical works [195]. 2D metallic meshes were developed in the 1960s, mainly in the far-infrared domain. They were used as low-pass filters, beam splitters, or reflectors in interference filters [196–198]. These 1D and 2D transmission metallic gratings were much thinner than the wavelength, with a filling fraction around 0.5. Even more importantly, their shape was not precisely controlled. As a result, their optical response did not exhibit any resonant features. The optical properties of metal films with 1D slits or 2D hole arrays have been the subject of renewed interest since the discovery of EOT by Ebbesen et al in 1998 [13]. Due to the large portion of the surface covered by metal, metal films perforated by sub-wavelength holes are mainly opaque, except in the vicinity of sharp resonances. This phenomenon has attracted the attention of numerous scientists during the past fifteen years and has been reviewed in several articles [199–201].

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It can be shown easily that the perfect absorption condition (Amax = 1) is reached only in the limit γ1 → γnr and γ2 → 0, which corresponds to the one-port configuration studied in this section. More precisely, the absorption maximum depends only on the asymmetry factor β = γ2/γ1. It is obtained for γ1 = γnr/(1 + β) and Amax = 1/(1 + β). An asymmetry of 90% (γ2  =  γ1/10) results in a theoretical absorption maximum of 90.9%. This analysis sheds light on recent numerical results of nearly perfect absorption in asymmetric structures [191, 192]. 5.  Free-standing metal nanostructure arrays In this section, we discuss the optical properties of resonant metallic nanostructure arrays. The basic structure is made of a metal film with sub-wavelength slits or holes. We are mainly interested in arrays with high metal coverage, so that the slab is mainly opaque out of resonance: in most cases, the direct transmission pathway can be neglected (C12 = 0). There is a fundamental difference between nanoslits and nanoholes. The apertures drilled in a metal film can be first viewed as independent waveguides if they are separated by a distance greater than the skin depth [35]. In this perspective, 1D slits are planar metallic waveguides with a fundamental TM mode (magnetic field parallel to the slits) at any wavelength (no cut-off frequency). On the contrary, 2D holes with small diameters do not propagate light: they behave as metallic waveguides above the cut-off wavelength of the fundamental guided mode and light transmission can only occur by tunneling through evanescent modes. The shape and size of the apertures drilled in a metal film also have a drastic impact on the resonance mechanism. Localized resonances arise from structures where each aperture behaves as an independent resonator. Alternatively, resonances can also be collective in nature, with holes coupled to each other through surface or guided waves, or through scattering events. In the following, we first present a short historical overview of free-standing metallic transmission gratings and arrays (section 5.1). Then, we describe the optical properties of metallic films drilled with nanohole arrays (section 5.2) and with nanoslits (section 5.3) and of hybrid structures made of metallo-dielectric arrays (section 5.4). We will focus more specifically on the transmission efficiency at the resonance wavelength and on the remarkable properties of free-standing structures. Applications for optical filtering and imaging are also presented (section 5.5).

5.2.  Nanohole arrays in metallic films 5.2.1.  Experimental evidence of EOT.  Ebbesen et al studied

a metal film deposited on a glass substrate and drilled by arrays of sub-wavelength holes (figure 18). They observed sharp, angle-sensitive resonances in the transmission spectra and published complete dispersion diagrams for various hole sizes and metals [202–209]. They found that, at resonance wavelength, the transmission efficiency exceeded by several orders of magnitude the expected value from Bethe’s theory of a single aperture [210] and even exceeded the surface ratio of the apertures. This phenomenon was therefore called EOT. Resonant transmission was attributed to the excitation of SPP on the surfaces of the metal film via grating coupling and subsequent tunneling through the holes [212]. As a first approximation, the dispersion relation is given by the excitation condition: |k(pm, q)| = kSPP, where k p(m, q ) is the in-plane wavevector of the (m, q) diffracted wave (equation (1)) and kspp is the surface plasmon wavevector for a flat metal surface (equation (4)). This mechanism was explored in various numerical studies [208, 213, 214] and further confirmed by complete angular spectroscopy showing that transmission maxima are associated with both reflection minima and absorption maxima [215]. These experiments brought ‘experimental evidence for transmission based on diffraction and assisted by the enhanced fields associated with surface plasmon polaritons’. However, a spectral shift was systematically observed between resonances and SPP dispersion curves. This disagreement was further confirmed by the experimental observation of the influence of the hole shape and size on the transmission resonance (figures 18(b), (c) and (e)) [211, 216, 217].

5.1.  A short historical perspective

5.2.2. The dispersion relation of surface modes: spoof plasmons.  In the primary description of the resonant mecha-

Free-standing 1D transmission gratings made of thin metallic wires have been fabricated and studied for more than a century [193]. They were primarily used to act as non-resonant

nism, the dispersion relation of surface waves excited on the perforated metallic surface was approximated by equations (3) 16

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Figure 18.  EOT. (a)–(c) SEM images of a gold film deposited on a glass substrate and perforated by arrays of (a) cylindrical holes and

(b, c) rectangular holes. (d) The transmission spectrum of an Ag film (thickness: 200 nm) perforated by a square array of cylindrical holes at normal incidence (period: 600 nm, diameter: 150 nm). Vertical lines indicate Rayleigh anomalies (dashed lines) and the expected

(

)

position of SPP excitation k p(m, q ) = kSPP, solid line . (e) Transmission spectra of drilled gold films (thickness: 215 nm) with different aspect ratios of the rectangular holes, with constant hole area and period 425 nm. (a) Reproduced with permission from [208]. Copyright 2001 Elsevier. (b) and (c) Reproduced with permission from [211]. Copyright 2005 American Physical Society. (d) Reproduced with permission from [202]. Copyright 1998 American Physical Society.

and (4), which hold for flat metal/dielectric interfaces. However, numerical calculations have shown that sub-wavelength holes in a perfect conductor also give rise to EOT, even though the flat perfect conductor with no holes has no surface modes. This paradox was solved by Pendry et al in 2004. They theoretically demonstrated the existence of surface modes on perfect metals perforated by holes, called spoof surface plasmons [218, 219]. Their experimental evidence followed rapidly in the microwave range [220]. It is worth noting that similar surface waves had already been observed by Ulrich and Tacke in the sub-millimeter range and published in 1973 [221]. Pendry et al also proposed an analytical model for their dispersion relation. This model has been extended to perforated metal surfaces with finite conductivity and absorption in [222]: a surface impedance model generalizes the dispersion equation of SPP modes on flat metal surfaces (equation (3)) to surface modes propagating on a wide range of nanostructured metallic surfaces. In its simplest form, it is written as 

Zd + Z m (1 − α ) + Zaα = 0,

section 2.2.1. The influence of the hole shape and size is contained in the average surface impedance of the textured metal Zm(1–α) + Zaα), which is simply driven by the propagation constant of the fundamental hole mode through Za and by the apertures’ fill factor α. The impact of the holes is interpreted as follows: opening apertures in the smooth surface induces an increase in the average impedance because of the greater penetration depth in the apertures than in the metal. It provides an explanation of the redshift of the dispersion curve for increasing hole widths observed experimentally [211, 217]. Hole arrays in metal create an effective medium with an average penetration depth. They also play the role of plasmonic crystals, due to the effect of the periodical pattern. The dispersion relation of surface modes is folded into the first Brillouin zone. It can be tuned by the shape and size of the holes [211, 216, 217], by a modification of their arrangement via rectangular arrays [55, 223], or by multi-scale patterning [224]. 5.2.3. Models for EOT.  Full numerical simulations are in

(23)

quantitative agreement with experiments [223, 225], but they do not provide insight into the microscopic resonant mechanism. Alternatively, in the case of perfect conductors and small holes, optical transmission through 2D hole arrays

where α is the apertures’ (area) fill factor and Za is the impedance of the hole mode. Zd and Zm are the surface impedance in the dielectric and metallic media, respectively, as defined in 17

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can be modeled with analytic models [226–228]. In 2008, Lalanne et al proposed a different strategy to model realistic nanohole arrays [229, 230]. It is based on a microscopic analysis of the elementary scattering events occurring among the individual hole chains of 2D arrays. This approach was motivated by experiments which had revealed that the field scattered by a hole in a metal film cannot be described only by SPP waves [231]. The microscopic model leads to analytical expressions for the transmission spectrum, in quantitative agreement with fully vectorial calculations. It also reveals that electromagnetic interactions between two rows of holes (scattering) can be decomposed in two distinct waves: an SPP and a quasi-cylindrical wave. Their respective weights depend on the metal conductivity and the separation distance between holes. This approach was validated experimentally in 2012 [232].

an original fabrication process in order to create large-area, free-standing metallic films with sub-wavelength hole arrays [236]. This procedure has been used to transfer the films on glass coverslips. The symmetry of the whole structure can be recovered by using refractive index liquids. 5.3.  Nanoslits in metallic films

Free-standing 1D metallic gratings are, at first sight, a simple variation of the well-known metallic diffraction gratings. However, the symmetry of free-standing gratings is responsible for sharp resonances with unprecedented transmission efficiency, approaching the 100% limit. 1D free-standing metallic gratings with well-controlled rectangular slits were fabricated and studied experimentally and numerically in the near-infrared domain by Lochbihler in the 1990s [237–239]. However, no sharp resonant feature was exhibited, due to wide slit widths. The optical properties of transmission metallic gratings with very narrow slits have been investigated extensively since the discovery of EOT (1998). Porto et al first showed theoretically the existence of transmission resonances for wavelengths larger than the grating period [240]. This phenomenon occurs only for TM-polarized light (magnetic field parallel to the slits). It originates from two different resonant mechanisms: (1) the excitation of coupled SPP on both surfaces of the metallic grating results in strong angular dispersion related to the delocalized nature of the resonance, (2) the coupling of incident light with waveguide resonances located in the slits is characterized by flat dispersion curves (localized resonances), independent of the incidence angle. In both cases, nearly perfect optical transmission is predicted. These phenomena have been analyzed in detail with numerical calculations based on exact and semi-analytical models [241–246] and with the calculation of complex dispersion curves including the determination of both radiative and nonradiative decay rates [54, 247, 248]. Localized resonances in the slits have been confirmed by experiments at microwave frequencies [249, 250]. In the optical domain, the first experiments were achieved with metallic gratings deposited on a substrate [251–254]. The resonances were evidenced, but the asymmetry due to the substrate was responsible for a reduction in both the transmission maximum and the resonance quality factor. The fabrication of free-standing gratings made of pure (and thus soft) noble metals like gold is unrealistic. This issue has been overcome by use of a stiff SiN-based core (figure 19). A free-standing SiN membrane was fabricated on a Si substrate, drilled by dry etching and covered by a gold coating in a multidirectional deposition process [255]. The gold layer is much thicker than the optical skin depth: the whole core-shell structure behaves as a pure gold grating made of straight bars with narrow slits. This fabrication process has been used to fabricate free-standing core-shell metallic grating on a large surface area (3 mm2), with a period d = 9.65 µm, slit width w = 2.7 µm and thickness t = 2.6 µm. Nearly perfect resonant transmission (87%) has been reported at λ ≃ 11 µm [64] (figure 20). The shapes of transmission spectra depend on the resonance mechanism. Since slits are much smaller than the wavelength, they support only one propagating waveguide mode (wavevector

5.2.4.  Limitations for EOT efficiencies.  In accordance with the TCMT (see equations  (15) and (16)), perfect (100%) optical transmission has been predicted numerically for a symmetric non-absorbing metallic film drilled with a nanohole array [212]. However, despite numerous experiments, optical transmission efficiency measurements are in the 5% to 20% range, far from the theoretical limit. These low performances strongly limit their potential applications. It was shown that they do not originate from the finite size of the sub-wavelength hole arrays [233]. The first limitation is related to the asymmetry of nanohole arrays in metal films deposited on a glass substrate. Due to the different dispersion relations of SPP at the air–metal and glass–metal interfaces, respectively, observed resonant optical transmission is usually related to the excitation of SPP on the single glass–metal interface. The asymmetry can be balanced by the use of immersion liquids with the same refractive index as glass (n  =  1.45), which do indeed boost the transmission efficiency [208]. The second limitation for efficient optical transmission is the intrinsic absorption losses in the metal. Transmission by the tunnel effect through the holes necessitates huge electric field enhancements at both interfaces of the metal film, resulting in unavoidable parasitic ohmic losses. This is the main reason why these structures betrayed their promise for application in optical filtering. 5.2.5. Fabrication of symmetric structures.  The fabrication

of suspended metallic grids or meshes is the first way to get rid of the substrate in order to achieve arrays with a plane of symmetry perpendicular to the propagation direction. Several approaches have been investigated. Suspended metallic grids have been fabricated on thin silicon nitride (SiN) membranes attached to a silicon substrate [234, 235]. However, the thin dielectric membrane used to support the metallic grid cannot be removed, so that the asymmetry is not suppressed. Ye et al deposited an additional dielectric layer on top of the metal in order to obtain a symmetric, suspended, sandwiched dielectric–metal–dielectric grid [234]. With this trick, the transmission efficiency is nearly doubled and reaches 60% at λ = 15 µm. Another way to fabricate perfectly symmetric metallic gratings or meshes consists in free-standing coreshell structures based on a stiff dielectric membrane coated by metal. It is described below (section 5.3). An alternative strategy has been proposed by the team of Odom, which developed 18

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Figure 19.  A schematic of the fabrication process of free-standing nanostructured membranes made of silicon nitride (SiN) on a silicon

wafer. The same process is used for free-standing dielectric nanorod arrays (c) and metallic core-shell gratings (d).

κ) and act as individual cavities made of planar truncated waveguides. They can also be seen as vertical MIM resonators. Hence, the zero-order transmission intensity can be described by a simple one-mode Fabry–Perot model [64, 241–243, 245]: 

T (ω, kx ) =

τ1τ2 1 − ρ2 e2iκt

2

(24)

where τi and ρ are the transmission and reflection coefficients at the interfaces (see figure 20(a)). Simple Lorentzian-shape resonances are expected for localized resonances. If surface resonances are involved, the transmission and reflection coefficients have a pole close to the Rayleigh anomaly, leading to a dispersive resonance and higher quality factor. In addition, the destructive interference between incident light and surface waves induces a zero of τ1 and τ2 and results in a Fano lineshape: T(ω) = β∣(ω − ω0)/(ω − ωp)∣2. It is worth noticing that the origin of this Fano lineshape differs from the one described in the frame of the TCMT: in the latter case, the Fano resonance comes from interferences between the direct transmission path and light outgoing from the resonator. Here, there is only one transmission path through the whole structure and the Fano lineshape comes from the zero of the in-coupling coefficient (radiative decay rate). In ­figure 20, the excellent agreement between the Fano model and experimental data was used to determine both the radiative and non-radiative decay rates for the whole angular range (0–40°) [64]. The non-radiative decay rate provides the limits for the transmission efficiency and resonance quality factor.

Figure 20.  Free-standing 1D metallic gratings. (a) A schematic

of the core-shell free-standing metallic grating. d = 9.65 µm, w = 2.7 µm and t = 2.6 µm. (b) Transmission spectra for various incidence angles: experiments (colored solid curves), calculations (gray solid curves, higher maxima) and Fano model (colored dashed curves). (c) Radiative (γr) and non-radiative (γnr) decay rates determined from experimental data by a Fano fit (red solid curves), and from eigenmode calculations (dashed curves). Inset: an SEM image of the sample. Reproduced from [64]. Copyright 2010 American Physical Society.

5.4.  Hybrid metal/dielectric nanostructure arrays

A wide variety of structures and effects can be designed on the basis of the simple metallic films drilled with 1D or 2D holes and described previously. For instance, the modification of the unit cell of transmission metallic gratings can provide additional degrees of freedom in order to make the transmission spectrum more complicated. By adding several slits per period, the transmission spectra are widened and additional modes appear [256, 257]. Also, one or more metallic nanostructure arrays can be combined with dielectric layers into more complex stacked structures. Coupling metal nanostructure arrays to a single planar dielectric waveguide leads to GMR [258, 259]. In contrast with their purely dielectric counterparts described in section  6.1, dielectric waveguides covered by metal nanostructures act as

bandpass filters. In free-standing metallic GMR structures, resonant transmission reaches 78% [260]. A variety of optical filters have been obtained [261]: polarizing filters with 1D gratings, unpolarized filters with crossed slits and selective filters with rectangular 2D slit arrays. In either case, high transmission efficiency has been demonstrated. Introducing more complicated unit cells like bi-atom structures is an additional way to simultaneously shape the spectral and the angular width of bandpass filters [262]. Metal films perforated by hole arrays exhibit EOT effects in TM polarization, due to the nature of the surface waves excited on the metal surface. Moreno et al have shown that a 19

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Figure 21.  Examples of optical filters made of nanostructure arrays. (a) Back illuminated microscope images of filters made of hole arrays

in a metal film, for various hole diameters and periods. (b) Transmission spectra (TM polarization) of single-layer and double-layer 1D metallic gratings (insets: SEM images). (c) SEM and optical images of core-shell free-standing metallic gratings with varying periods and slit widths and (d) transmission spectra. (e, f) GMR arrays based on 2D metal nanostructure arrays deposited on Si waveguides and fabricated on 200 mm Si wafers by UV lithography. (g) Corresponding transmission spectra with increasing periods. (a) Reprinted with permission from [270]. Copyright 2012 American Chemical Society. (b) Reprinted with permission from [266]. Copyright 2006 The Optical Society. (c) and (d) Reprinted with permission from [271]. Copyright 2010 AIP Publishing LLC. (g) Reprinted with permission from [259]. Copyright 2011 The Optical Society.

6.  Free-standing dielectric nanostructure arrays

similar effect can be obtained in TE polarization in a slightly more complex structure [263]: a dielectric layer is added on each side of a thin metal film with 1D slits. These dielectric layers act as waveguides, enabling waves to be excited and propagate along the plane in TE polarization. Light transmission through the metal film is achieved via evanescent modes in the slits. Hence, the whole structure mimics EOT in metal films with hole arrays. Dual structures made of two metal films perforated by hole or slit arrays have been investigated by several groups [264–269]. The spacing and the lateral displacement between the two arrays can be tuned, providing additional degrees of freedom. The perforated metal films are coupled via either evanescent or propagating waves. In the particular case of dual 1D metallic gratings, not only has high-efficiency resonant transmission been found [266] (figure 21(b)), but also a perfect extinction phenomenon at wavelengths greater than the resonant transmission peak [269]. This particular feature enables the design of narrow optical filters with high transmission and very good rejection efficiency. Closely spaced metal films with hole arrays are called fishnet structures and have been widely studied in the framework of MM. They support MIM resonances localized in the spacing layer and can lead to negative-index MM [47, 49].

In this section, we consider dielectric nanostructure arrays. As a result of the use of transparent materials, a fundamental difference with their metallic counterpart is the ability to have a direct transmission pathway: they are partially transparent. Away from resonance, a nanostructure array behaves like a slab with an effective permittivity given by an average permittivity of its compounds. In the vicinity of a resonance, the direct pathway interacts with light re-emitted by the resonant structure, resulting in transmission spectra with Fano lineshapes and potentially strongly modulated efficiencies, from 0 to 1. However, the shape and size of the nanostructures have a strong impact on the resonance mechanism. 6.1.  Guided-mode resonances

Historically, dielectric slabs with periodical corrugations were first considered as waveguides coupled to free-space, resulting in GMR [276]. They have been widely studied in dielectric waveguides deposited on substrates, with weak 1D or 2D periodic modulations, as illustrated in figure 22. As discussed in section  2.3 (equation (10), figure  6), the coupling condition can be estimated by equating the in-plane wavevectors of diffracted waves (order (m, q)) and guided waves. For (m, q) ≠ (0, 0), the dispersion curve is partially in the light cone: guided waves are coupled to free space. This approach is a first approximation of the numerical calculation of the complex poles and zeros of the S-matrix of the whole structure (polology approach, phenomenological theory) [52, 65, 276– 279]. For weak corrugations, the poles and zeros provide an insightful analysis of the resonances, losses and bandgaps. GMR have been used for the conception of spectral filters [12, 20, 21]. Weak corrugations result in a narrow

5.5.  Applications: optical filters and imaging

Bandpass transmission filters are the obvious application of metal nanostructure arrays. The two spectral ranges of interest are the visible [272–274], for potential application to complementary metal-oxide-semiconductor image sensors [270, 275], and the mid-infrared (3–5µm) for imaging applications [259, 271]. Examples of recent achievements are shown in figure 21. 20

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confined within the slab. In the light cone (the grey region in figure 23(c)), the optical modes are coupled to free space. Figures 23(a) and (b) show a free-standing dielectric slab drilled with a 2D square array of holes. TE and TM guided waves vertically confined in the slab bounce back and forth between periodic interfaces, giving rise to optical modes (GMR) translated in intricate dispersion diagrams. Above the light cone, these guided-modes are leaky modes, coupled to the far field. Plane-wave excitation of GMR in PC results in Fano resonances (figure 23(b)), as expected from the TCMT (see ­equation  (18) in section  3.4). They have been analyzed theoretically by Fan et al [23, 63]. An extended review has recently been devoted to these phenomena [297]. One of the first experimental measurements was achieved on suspended AlGaAs slabs in 1997 [299], and angle-resolved measurements on silicon PC slabs were published in [298, 300] (figures 23(d) and (e)). 6.3.  High-index-contrast sub-wavelength gratings

At first glance, HCG are not different from 2D slabs. Actually, most of the time, HCG refer to deep 1D lamellar gratings that exhibit resonances that can not be thoroughly explained by in-plane guided waves coupled to free-space. This novel type of resonance appears in high-index nanostructured slabs surrounded by a low refractive index. Its most striking feature, reported in 2004, is a broadband (Δλ/λ > 30%), high reflectivity (>99%) reflection band observed at normal incidence [24, 25], see figure 24(b). These resonances have been explained by a coupled Bloch-modes model by Lalanne et al in 2006 [279] and further analyzed by several authors in [301–303]. HCG have also been extensively reviewed in [304, 305]. The singular resonances of HCG appear in the near-wavelength regime. It corresponds to the region where no diffraction occurs (λ > d) and several optical (Bloch) modes propagate in the grating region (depending on the filling fraction). In the simplest situation, two Bloch modes propagate backward and forward between the two grating interfaces with different phase velocities. At the grating boundaries, each mode is partially transmitted to free-space, partially reflected and partially coupled to the other Bloch mode. It results in a complex Fabry–Perot resonance involving two coupled Bloch modes propagating between the top and bottom slab boundaries. The resonance wavelength depends naturally on the grating thickness. It is worth noting that in this model, no direct transmission pathway is considered (C12 = 0). The radiative coupling γ2 is canceled via coupling between the two Bloch modes. This resonance mechanism is illustrated in figure  24(c). It shows the reflectivity as a function of the thickness t and wavelength. The boundaries of the near-wavelength regime are shown by white vertical lines. In the long-wavelength domain, only one Bloch mode propagates in the grating, resulting in simple fringes when the thickness is varied due to the Fabry–Perot resonances in the slab. In the near-wavelength regime, a more complex optical response arises from the interplay between the two coupled Bloch modes. Broadband reflectance is achieved for the thicknesses indicated by the horizontal arrows.

Figure 22.  GMR structures. (a, b) Schematics of planar waveguides

with a 1D grating (a) and a 2D square array of corrugation (b). (c) Theoretical and experimental reflection spectra of a GMR filter made of a 1D photoresist grating (period: 487 nm, fill factor: 0.3) and a HfO2 waveguide layer (thickness: 270 nm) on a fused-silica substrate. Reproduced with permission from [280]. Copyright 1998 The Optical Society.

linewidth due to the small radiative coupling rate γr and the absence of absorption (γnr  =  0). It can be tuned with the geometrical parameters, the profile and the symmetry of the modulations [281–283]. The angular tolerance can be controlled by introducing doubly periodic structures [284, 285] and complex bi-dimensional doubly periodic gratings can result in unpolarized, angular tolerant narrow-band filtering [286–288]. GMR structures are usually based on weak corrugations and designed to act as narrow-band reflection filters. They have been realized for various wavelength ranges, on hard and flexible substrates [288–292]. Transmission guided-mode filters have also been designed, with additional layers and increasing complexity [293]. 6.2.  Free-standing PC slabs

Dielectric slabs with stronger corrugations, such as 2D hole arrays, are more properly referred to as 2D PC slabs. In 1987, materials with periodical refractive indices were considered as the optical analogue to crystal lattices made of atoms, leading to the concept of PC [10, 11]. This perspective offers new ways to control light propagation in materials: PBG prevent light from propagating in specific directions and spectral domains. They are used as a tool to design waveguides and cavities in 2D or 3D structured materials [22, 294–296]. The dispersion diagram of a 2D PC slab is shown in figure 23(c). Below the light cone (the bright region), the optical modes are 21

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Figure 23.  2D PC slabs. (a) A schematic of a 2D PC slab made of a square lattice of air holes (period a) in a dielectric slab. (b) Transmission

spectrum at normal incidence (radius: 0.2a, thickness: 0.5a, permittivity: 12). (c) An example of a dispersion diagram for a 2D PC slab with a triangular lattice for two (TE/TM) polarizations. (d, e) A fabricated suspended silicon PC slab (thickness 340 nm, a square array of holes with a radius of 330 nm and a periodicity of 998 nm) and an experimental transmission spectrum (black) compared with FDTD simulations (red) for normally incident light. (b) Reprinted with permission from [63]. Copyright 2003 The Optical Society. (c) Reprinted with permission from [297]. Copyright 2014 Elsevier. (d) and (e) Reprinted with permission from [298]. Copyright 2004 The Optical Society.

Figure 24.  HCG. (a) A schematic. (b) An example of a spectrum showing very broadband reflectivity (Δλ/λ > 30%, R > 99%). (c) Reflectivity

as a function of both the grating thickness t/d and the wavelength (grating period d = 1.15 μm, bar width w = 460 nm, refractive index 3.17, TE polarization at normal incidence). Bright zone: perfect reflection, R = 100%. The horizontal arrows indicate regions of broadband reflectance. (b) © 2006 IEEE. Reprinted, with permission, from [24]. (c) © 2006 IEEE. Reprinted, with permission, from [279].

the linewidth by the polarizability of the particles (refractive index, shape and size) [306–308]. Each sub-wavelength rod behaves like an individual scatterer with a non-resonant dipolar response (polarizability α). The resonant mechanism can be described as follows. If the grating is illuminated by a plane wave, each rod of the array re-radiates a field proportional to α and to the incident electric field. On the other hand, the incident field on each rod is the sum of the incident plane wave (Ei) and the scattered fields from the other rods (Es), as depicted in figure  25(a). These interactions result in multiple scattering which can be modeled analytically by an effective polarizability αeff [26, 307]:

6.4.  Dielectric nanorod arrays

With much smaller bars than in the previous structures, nanorod arrays behave very differently. In contrast with their metallic counterparts, dielectric nanoparticles much smaller than the wavelength have a small scattering cross-section due to their non-resonant nature. Hence, 1D or 2D nanorod arrays with tiny diameters can not support either coupledBloch-mode resonances nor any localized resonance in the nanostructures. Nevertheless, light–matter interaction can be enhanced by coherent multiple scattering in 1D or 2D periodical ensembles of nanoparticles, offering another degree of freedom to manipulate their optical response. These geometric or lattice resonances can lead to extraordinary optical reflection from dielectric nanorod arrays with sharp resonance features, as predicted in 2006 by GÓmezMedina et al [26]. The resonance wavelength is mainly driven by the relative position of the particles (period) and



⎛1 ⎞−1 αeff = ⎜ − Gbk 2⎟ , ⎝α ⎠

(25)

where Gb is the dynamic depolarization term which depends on the geometrical parameters, i.e. the period D, the width d, 22

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Figure 25.  Dielectric nanorod arrays. (a) A schematic and (b) an SEM image of a 1D free-standing SiNx membrane made of square

rods (period D = 3 µm, width d = 500 nm). (c) Electric-field enhancement in 1D nanorod arrays as a function of the width d, at resonance (θ = 0°). Circles: full electromagnetic calculation. Line: (λ0/d)4 law, where the resonance wavelength can be approximated by λ0 ≃ 3 µm for small diameters. Inset: a map of the electric-field intensity for d = 500 nm, normalized to the incident field. The positions of the rods are shown by black squares. (d, e) Experimental (color) and theoretical (dashed) transmission (d) and reflection (e) spectra for different incidence angles. Reproduced from [309]. Copyright 2012 American Physical Society.

as well as the wavelength and the angle of incidence θ. The resonance is defined by the pole of αeff. Figure 25 shows experimental results obtained with freestanding arrays of 1D silicon nitride nanorods [309]. At normal incidence, the transmission spectrum reveals nearly total optical extinction (94%). This result is counterintuitive when considering (i) the low fill factor (15%) of the grating and (ii) the perfect transparency of the material (no loss at 3.2 µm). The optical response of the dielectric membrane changes in a very narrow spectral range from transparent to opaque behavior. From almost total transmission at 3 µm, the signal drops sharply at 3.2  µm to just 6%. The resonance wavelength is strongly angle-dependent. The coherent multiple scattering effect is also responsible for huge electric field enhancement in the particles (figure 25(c)). It is important to note that this formalism based on multiple interactions between nanostructures can also be used to describe various dielectric or metallic planar ensembles of particles or holes, as reviewed by Garcia de Abajo in [200]. This model becomes exact in the limit of small particles. In the case of metallic particle arrays, the polarizability of a single particle is resonant and multiple coherent scattering leads to narrowed plasmon lineshapes. This effect was found theoretically in 2004 [310, 311] and evidenced experimentally by several groups [312–315]. A free-standing monolayer of dielectric nanorods is a model structure for direct interactions in free space between

non-resonant structures. In contrast with the collective resonances described in the previous sections, interactions between nanostructures are not mediated by electromagnetic waves in matter (surface waves or guided modes in dielectric layers). It is worth noting the similarities between the coherent multiple scattering mechanism evidenced in nanorod arrays and the well-known Bragg diffraction arising in the 3D arrangement of a crystal lattice. Both phenomena are based on low crosssection scatterers. However, the constructive interference of the Bragg diffraction mechanism involves a large number of lattice planes. Here, 100% of incident photons interact with a single lattice plane of scatterers, leading to nearly perfect optical extinction. This surprising property is actually a direct consequence of the symmetry of the resonant array, as shown by the TCMT in section 3. 6.5.  Applications: mirrors in vertical cavities and VCSEL, optomechanics

The optical properties of suspended 2D PC slabs and HCG are very attractive for designing compact and efficient mirrors for filters, resonant cavities and vertically coupled opto-electronic devices. They have been primarily used for the conception and fabrication of novel VCSEL. VCSEL consist of a gain medium in a cavity generally made of two distributed Bragg reflectors (DBRs) with high reflectivity and require very thick stacks of layers. This drawback has motivated the replacement of the 23

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Figure 26.  The application of nanostructured membranes in vertical cavity-based devices. (a) A cavity composed of a bottom micro-machined

three-pair InP/air-gap Bragg reflector and a top PC membrane mirror. (b) Reflection spectra of the cavity filter (a) for different bias voltages. Inset: an SEM image of the cavity. (c) The reflectivity spectrum of a vertical Fabry–Perot cavity composed of two PC membrane mirrors. (d) A schematic of a 1550 nm VCSEL with a suspended HCG-based top mirror. (e) An SEM image of the VCSEL. (f) Light–current (solid lines) and voltage–current (dashed lines) characteristics of a HCG VCSEL and (g) the spectrum under various heat sink temperatures. (a) and (b) Reprinted with permission from [316]. Copyright 2006 The Optical Society (c) Reprinted with permission from [317]. Copyright 2007 The Optical Society. (d)–(g) Reprinted with permission from [318]. Copyright 2010 The Optical Society.

drastic reduction of the mirror mass, free-standing nanorod arrays could offer strong improvement of the optomechanical coupling. In addition, they offer additional degrees of freedom for the conception of a large-area optomechanical systems without external cavity (the membrane is the optical resonator itself). Finally, from a more fundamental point of view, electromagnetic waves can be perfectly trapped in dielectric nanostructure arrays even above the light cone [338–340]. Such bound states in the radiation continuum are modes with an infinite lifetime, even if outgoing waves are allowed: the cancelation of the free-space coupling (γr = 0) is not due to symmetry incompatibility. Leaky modes become bound states confined in the slab. They have been predicted in dual nanorod arrays [338] and in PC [339] and their first experimental observation was published in 2013 [340].

top mirror by a single nanostructured slab [316, 319–321]. The polarization of the emitted light can be controlled with 1D structuration and the resonance wavelength can be tuned by electrostatic actuation of the membrane in micro-optoelectro-mechanical systems (MOEMS) [322]. Figures 26(a) and (b) shows a vertical cavity made of a bottom Bragg reflector and a top PC membrane mirror. It can be tuned in the 1.55–1.57 µm wavelength range by applying a bias voltage of 4 V [316]. PC membranes have been a basis for the development of various microelectromechanical systems (MEMS) [323–325], high-Q cavities [326], dual-wavelength resonators [327] and mirrors with focusing capabilities [328, 329]. The total thickness of vertical Fabry–Perot cavities has also been further reduced by using two PC membrane mirrors (figure 26(c)) [317]. The development of PC membrane mirrors has led to novel, compact surface-emitting lasers [330–318] (figures 26(d)–(g)) and nanoelectromechanical tunable lasers [333]. In the case of free-standing nanorod arrays, several applications could take advantage of the very large local field enhancements without non-radiative losses: second harmonic generation [334], surface-enhanced spectroscopy, fluorescence enhancement [308], or light extraction [335]. Potential applications in optomechanics are also envisioned. In particular, an efficient optomechanical interaction has been achieved recently with unstructured membranes placed in a high-finesse cavity [336, 337]. With increased reflectivity and

7. Conclusion Resonant nanostructure arrays have a very long scientific history, starting more than a century ago. Despite their apparent simplicity, the rich variety of their optical properties has driven continuous efforts and interest in the nanophotonics community, pushed ahead by the huge development of technological and numerical capabilities during the past twenty years. 24

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[2] Wood R W 1912 Diffraction gratings with controlled groove form and abnormal distribution of intensity Phil. Mag. 23 310–7 [3] Wood R W 1935 Anomalous diffraction gratings Phys. Rev. 48 928–36 [4] Rayleigh L 1907 Note on the remarkable case of diffraction spectra described by Prof. Wood Phil. Mag. 14 60–5 [5] Rayleigh L 1907 On the dynamical theory of gratings Proc. R. Soc. Lond. A 79 399–416 [6] Fano U 1941 The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves) J. Opt. Soc. Am. 31 213–22 [7] Hessel A and Oliner A A 1965 A new theory of Wood’s anomalies on optical gratings Appl. Opt. 4 1275–97 [8] Petit R 1980 Electromagnetic Theory of Gratings (Topics in Current Physics) (New York: Springer) [9] Hutley M C and Maystre D 1976 The total absorption of light by a diffraction grating Opt. Commun. 19 431–6 [10] Yablonovitch E 1987 Inhibited spontaneous emission in solidstate physics and electronics Phys. Rev. Lett. 58 2059–62 [11] John S 1987 Strong localization of photons in certain disordered dielectric superlattices Phys. Rev. Lett. 58 2486–9 [12] Wang S S, Magnusson R, Bagby J S and Moharam M G 1990 Guided-mode resonances in planar dielectric-layer diffraction gratings J. Opt. Soc. Am. A 7 1470–4 [13] Ebbesen T W, Lezec H J, Ghaemi H F, Thio T and Wolff P A 1998 Extraordinary optical transmission through subwavelength hole arrays Nature 391 667 [14] Popov E 2012 Gratings: Theory and Numeric Application (Aix-en-Provence: Presses Universitaires de Provence) www.fresnel.fr/numerical-grating-book [15] Barnes W L 2009 Comparing experiment and theory in plasmonics J. Opt. A: Pure Appl. Opt. 11 114002 [16] Busch K, von Freymann G, Linden S, Mingaleev S F, Tkeshelashvili L and Wegener M 2007 Periodic nanostructures for photonics Phys. Rep. 444 101–202 [17] Rytov S M 1956 Electromagnetic properties of finely stratified medium Sov. Phys.—JETP 2 466–75 [18] Lalanne P and Lemercier-Lalanne D 1996 On the effective medium theory of subwavelength periodic structures J. Mod. Opt. 43 2063–86 [19] Sun Y, Edwards B, Al A and Engheta N 2012 Experimental realization of optical lumped nanocircuits at infrared wavelengths Nat. Mater. 11 208–12 [20] Wang S S and Magnusson R 1993 Theory and applications of guided-mode resonance filters Appl. Opt. 32 2606–13 [21] Magnusson R and Wang S S 1992 New principle for optical filters Appl. Phys. Lett. 61 1022–4 [22] Winn J N, Joannopoulos J D, Johnson S G and Meade R D 2008 Photonic Crystals: Molding the Flow of Light 2nd edn (Princeton, NJ: Princeton University Press) [23] Fan S and Joannopoulos J D 2002 Analysis of guided resonances in photonic crystal slabs Phys. Rev. B 65 235112 [24] Mateus C F R, Huang M C Y, Deng Y, Neureuther A R and Chang C J 2004 Ultrabroadband mirror using low-index cladded subwavelength grating IEE Photon. Technol. Lett. 16 518–20 [25] Mateus C F R, Huang M C Y, Chen L, Chang C J and Suzuki Y 2004 Broad-band mirror (1.12–1.62 µm) using a subwavelength grating IEEE Photon. Technol. Lett. 16 1676–8 [26] Gómez-Medina R, Laroche R and Sáenz J J 2006 Extraordinary optical reflection from sub-wavelength cylinder arrays Opt. Express 14 3730–7 [27] Maier S A 2007 Plasmonics: Fundamentals and Applications (New York: Springer) [28] Han Z and Bozhevolnyi S I 2013 Radiation guiding with surface plasmon polaritons Rep. Prog. Phys. 76 016402

This review article has been written with the ambition to draw a large overview of the optical properties of resonant nanostructure arrays, with special emphasis on free-space coupling. Their basic optical properties and resonant mechanisms have been described and their interactions with free space have been discussed with the help of the TCMT. We have shown that this theory provides a simple and powerful tool to analyze and enhance light–matter interactions in periodical nanostructures and can be very fruitful for the design of photonics and opto-electronics devices. Then, we have reviewed optical absorption in nanocavity arrays fabricated on bulk mirrors and the optical transmission and reflection properties of free-standing metal and dielectric nanostructure arrays. For each family of nanostructure arrays, we have given a short historical overview and we have focused our discussion on recent advances and applications. As mentioned throughout this article, some aspects of this subject have been reviewed recently in more detailed ways. It is certainly of interest to the readers to provide a selected list of review articles on the different aspects approached in this review. Many in-depth reviews on single dielectric and plasmonic nanostructures and nanoantennas have been published [30–32, 50, 51, 341], including several articles more focused on sensing and biosensing applications [149–154]. The reader can also refer to more general overviews on plasmonics [33, 199] and periodic nanostructures [16] and to reviews with emphasis on the open resonator viewpoint [342], on Fano resonances [343, 344], on light scattering in nanostructured arrays [200], or on light transmission [201, 230] and absorption [69, 117] in nanostructured metal films. The field of dielectric nanostructured arrays has been covered through general reviews on PC [294–296], on Fano resonances in 2D PC [297] and on HCG [305]. It is also worth mentioning reference books on nanooptics [345], the electromagnetic theory of gratings [8, 14], PC [22] and plasmonics [27]. Choices have to be made when reviewing such a broad scientific area. Omissions are also unavoidable. Nevertheless, I hope that this review is representative of the recent advances in the optics of resonant nanostructure arrays coupled to free space. I also hope that this review will help readers become acquainted with this field, or gain a different perspective, and that it will encourage them to participate in further developments, discoveries and applications. Acknowledgments The author acknowledges A Cattoni, A Gaucher, I Massiot, J M Castro Arias and P Lalanne for helpful comments and stimulating discussions. The author’s work was partly supported by the French ANR projects 3D-BROM, NATHISOL and ULTRACIS-M and by the French RENATECH network. References [1] Wood R W 1902 On a remarkable case of uneven distribution of light in a diffraction grating spectrum Phil. Mag. 4 396–402 25

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Nanostructure arrays in free-space: optical properties and applications.

Dielectric and metallic gratings have been studied for more than a century. Nevertheless, novel optical phenomena and fabrication techniques have emer...
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