2017 Poly Web Optical Artificial Materials.pdf

Outline (course 3A Ph. Lalanne – 22H) LIGHT INTERACTION with NANOSTRUCTURES Actualisé Sept 2017

Coordinateur Philippe Lalanne Intervenants : Philippe Lalanne, Kevin Vynck, Etienne Brasselet (LOMA) Objectifs. L’électromagnétisme est au cœur de tous les dispositifs et phénomènes de la photonique. L’objectif de ce cours est d’apprendre des notions avancées d’électromagnétisme pour comprendre les fondements de la photonique moderne et de ces applications. Le cours comprend 12 séances de 2 heures 1) 18/09 13h15 (PhL) 2) 19/09 13h15 (PhL) 3) 20/09 10h15 (PhL) 4) 25/09 13h15 (PhL) 5) 26/09 15h30 (Kevin V) 6) 27/09 10h15 (Kevin V) 7) 02/10 13h15 (Kevin V) 8) 03/10 10h15 (Kevin V) 9) 04/10 10h15 (Kevin V) 10) 09/10 13h15 (Etienne B) 11) 11/10 10h15 (PhL) 12) 16/10 13h15 (PhL) 13) ??/?? ??h?? Exam Séance 1 (Philippe) Introduction 1/ Introduction to optical nanostructures Historical overview of nanostructures (= overview of the course): Bragg mirrors, diffraction gratings, photonic crystals, effective medium, metamaterials, nanoantennas, disordered media Séance 2 & 3 (Philippe) II Light propagation in periodic media 1/Thin film stack: effective index 2×2 matrix approach Bloch-wave approach Bands and gaps 2/ Periodic thin-film stacks with small periods Static limit of 1D periodic structures Effective index

TD: Taylor expansion for small periods 3/ The photonic bandgap Bandgap opening Penetration depth Multi-dimension Bragg mirrors 4/ Bloch modes in photonic structures with finite lengths 5/ Photonic crystal -waveguide (slow light) -cavity References: A. Yariv and P. Yeh, Optical waves in crystals, J. Wiley and Sons eds., New York, 1984 OR P. Yeh, Optical waves in layered media, J. Wiley and Sons eds., New York, 1988.

Séance 4 (Philippe) Subwavelength diffractive optics 1/ Homogeneisation of subwavelength gratings Grating equation Single mode approximation Evanescent Bloch modes 2/ Form birefringence 3/ Wire grid polarizer Homogeneisation of metal slits (skin depth) Surface plasmons and Wood anomaly 4/ Antireflection coats with artificial dielectrics 5/ Diffractive optical elements Echelette diffractive optical elements TD: Wavelength-dependence of the diffraction efficiency of échelette Blazed-binary diffractive optical elements References: R. Halir et al., Optical Properties-Subwavelength Scale taken from the Waveguide sub-wavelength structures: a review of principles and applications, Laser Photonics Rev. (2014) / DOI 10.1002/lpor.201400083

Séance 5 (Kévin) 1/ TD: grating equation and zeroth-order gratings (1H) Grating equation in conical mounts 2/ TD: Anomalous refraction in photonic crystals (group velocity, iso-frequency curves, ...) Séance 6 (Kévin) Resonance mode of cavity Mode lifetime Poynting theorem ( complex and Q definition) TD: Fabry-Perot model of a resonance mode -vg of the bouncing mode -penetration depth Séance 7 (Kévin) Green tensor formalism Séance 8 (Kévin) Diffusion by small particles 1/General concepts Scattering, absorption, extinction cross-sections Optical theorem 2/Examples of cross-sections for spherical particles 3/Subwavelength particles Dipolar approximation, electrical polarisability Cross sections Séance 9 (Kévin) Light transport in disordered media 1/ Introduction to radiative transfer

Why can we neglect wave interference upon average: intuitive picture with speckles Derivation of the RTE Typical length scales (absorption, scattering, transport mean free path) Example of applications (atmospheric transport, spectroscopy of tissues) 2/ Beyond the RTE: Localization Reference: E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007)

Séance 10 (Etienne) Optomechanics 1/Light momenta and mechanical principles : linear and angular (spin and orbital) 2/Optomechanics : momentum balance under light-matter interaction 3/Applications : optical trapping and manipulation of microscale entities 4/Structured light-matter interaction for advanced optical manipulation Séance 11 (Philippe) Plasmonic 1/Drude model 2/Interaction of light with metals (hot electrons, phonon relaxation) 3/Surface plasmon polaritons of flat interface and MIM 4/Metal nanoparticles 5/Launching SPP with isolated slits and slit arrays 6/Field scattered by a single tiny slit on the metal surface 7/Anomaly de Wood Séance 12 (Philippe) Metamaterials 1/ Historical perspective 2/ Veselago’s flat lens 3/ Negative index materials at microwave frequencies 4/ Effective parameters 5/ Negative index materials at optical frequencies 6/ The perfect lens 7/ Super-resolution with metamaterials References: L. Solymar and E. Shamoniva, Waves in metamaterials, (Oxford University Press, Oxford, 2009).

Light interaction with nanostructures Philippe Lalanne (CNRS-LP2N) Kevin Vynck (CNRS-LP2N) Etienne Brasselet (CNRS-LOMA) Cours en relation étroite avec les TPs de simulations physiques

On est objectivement dans le siècle des nanos On se doit de se demander si les propriétés optiques des nanostructures sont différentes de celles des macro-objets On se doit de se demander ce qu’on peut faire en optique avec les nanostructures

Young’s slit experiment

Thomas Young (1773-1829)

La vraie manip d’ Young Manip des Franges d’Young: preuve (belle et simple) de la nature ondulatoire de la lumière 1801 feuille aluminium

bougie

écran

Fentes Young en éclairant une seule fente 2007 feuille aluminium

bougie

écran

Fentes Young en éclairant une seule fente

Si on n’utilise les mêmes fentes que Young (largeur de 1 mm), on n’observe aucune frange d’interférence 100 nm

Kuzmin et al., Opt. Lett. 32, 445 (2007). S. Ravets et al., JOSA B 26, B28 (2009).

Fentes Young en éclairant une seule fente

- Exactement le même interfrange - Juste le contraste est diminué par un facteur 2

2007 feuille aluminium

bougie

écran 100 nm

Subwavelength domain

Resonant domain

Scalar domain

Ray optics

a/l 1

10

n Merging optics and electronics requires nanoscale optics

1 µm

50 nm

Subwavelength domain

Resonant domain

Scalar domain

Ray optics

a/l 1

Morpho butterflies

10

écailles

strie

Metamaterial: ultra-flat graded-index lens

n

1 µm

Photonic crystal: hollow core waveguides

1 µm

photonic crystal: slow light

intensity enhancement ( 1/vg)

slow light

space compression ( 1/vg) •more pulses are stored per unit length (unique route for miniaturization) •intensity enhancement (light-matter interaction processes are made easier)

photonic crystal: Single molecule detection with nanocavity

Response

~ Re

 ~   Im 

Frequency

500 nm

ratio

transmission

Plasmonics : extraordinary transmission

l (nm)

air

metal 200 nm

métal

Plasmonics : extraordinary transmission (Images de la physiques 2010)

superlentille

89 nm linewidth Plasmonics : superlens (Science April 2005)

Single photon source with semiconductor nanowires

Emission control : nanowires (Nature Photonics March 2010)

Surface plasmons : guiding light to the nanoscale

E

k

Negative refractive index at optical frequency

p = 860 nm ~ l/2

Refractive index

Optical negative refraction

from Snell’s law

l (µm) J. Valentine et al., Nature (London) 455, 376 (2008)

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

An introduction to optical Bloch modes

Contents 1.1. Thin film stack: effective index ........................................................................................ 2 1.2. Periodic thin-film stacks with small periods....................................................................... 7 1.2.1. Static limit of 1D periodic structures .......................................................................... 9 1.2.2 Taylor expansion for small periods.............................................................................11 1.3 The photonic gap ...........................................................................................................14 1.3.1. Bandgap opening ....................................................................................................16 1.3.2 Penetration depth .....................................................................................................22 1.3.3 Structural slow light .................................................................................................24 1.3.4 Multi-dimension Bragg mirrors .................................................................................26 1.4 Bloch modes of real photonic structures with finite lengths .................................................28 References..........................................................................................................................31

The theory of the electronic transport in periodic potentials has lead to the concept of energy levels and band structures in semiconductors. This requires the use of periodic wave functions, called Bloch functions (after the Swiss physicist Félix Bloch who established the quantum theory of solids during his thesis in 1928), and to solve Schrödinger equations. The result of this analysis is that the energy levels are grouped in bands, separated by energy band gaps. The analysis of electromagnetic fields in periodic media shares the same concepts, such as Bloch modes, Brillouin zones and forbidden bands.

The chapter provides an introduction to Bloch modes in periodic structures formed by composites of homogeneous isotropic media. The objective is to provide a comprehensive 1

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

description. We start with the very important textbook example of one-dimensional photonic crystals, which are nothing else than periodic thin film stacks. This simple system illustrates most of the physical features of the more complex two- and three-dimensional photonic crystal systems.

The formalism is provided in Section II.1 and is followed by an analysis of the asymptotic case for which the period of the stack is much smaller than the wavelength. In this limit, the stack can be homogenized and is equivalent to a uniaxial metamaterial. Section II.3 is concerned by the Bloch modes in the bandgap. The Bloch mode is evanescent and is responsible for the reflectivity of the stack. The latter is called a Bragg mirror.

1.1. Thin film stack: effective index Let us start with the simplest structure, a periodic stack composed of thin uniform layers with relative permittivities 1 et 2, see Fig. 1.1, which are assumed to be independent of the wavelength. Material dispersion is therefore neglected.

Figure 2-1: Periodic thin-film stack composed of alternate uniform layers of permittivity 1 and 2. Parameter f is the fillfactor. We assume that the material are not magnetic, µ = µ0 everywhere. We are looking for the electromagnetic modes (i.e. the solutions of the Maxwell’s equations without source) that may propagate into the periodic thin-film stack. Because the structure is invariant in the y and z directions, the y and z dependence of the fields can be written exp(ikxy+ikyz), the exp(it) being omitted in the following. The propagation constants kx and ky are arbitrary a priori. Since the continuity conditions are satisfied at the interfaces for any x and y, note that kx and ky are the same in every layer: they are invariants. As such, we often say that the parallel wavevector components, kx and ky in the present case, are conserved. This is nothing else than Snell’s law. For the sake of simplicity, hereafter, we take kx = ky = 0.

2

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

The solution to our problem is fully analytical and can even be solved with closed-from expressions, see for example the remarkable textbook [Yeh88], but the calculations are rather long, even in this simple bi-layer 1D case. The approach used hereafter is much less analytical, but is applicable to the more general cases of structures that are periodic in two or three dimensions. Because kx = ky = 0, without loss of generality, the modes can be written Ex(z) = e(z) exp(jt),

(1.1a)

Hy(z) = h(z) exp(jt).

(1.1b)

In addition, the Bloch theorem (also called the Floquet theorem because of the works of the French mathematician Gaston Floquet) that is easily demonstrated if one assumes that the solution is unique, stipulates that the solution is pseudo-periodic; the electromagnetic fields e(z) and h(z) can be written as the product of a periodic function by exp(jkzz), with kz a complex number in general. Expending in Fourier series the periodic contribution, one obtains h(z) = mUm exp[j(kz+mK)z],

(1.2a)

e(z) = mSm exp[j(kz+mK)z],

(1.2b)

where K=2/a is the reciprocal-vector modulus. We will define the effective index of the Bloch mode by kz = k0 neff avec k0 = /c,

(1.3)

c being the speed of light in a vacuum. Because of the periodicity, kz is defined modulo K and the definition of Eq. (1.3) rises up some important issues that will be discussed later. Using Eqs. (1.2a) and (1.2b) into Maxwell curl equations (in the absence of charge and current densities the divergence equations of Maxwell’s equations are satisfied since the magnetic and electric fields are curls), jµ0h(x) = e/x,

(1.4a)

j0(x)e(x) = h/x,

(1.4b)

and by introducing the Fourier series (z) = p ˆ p exp[jpKz], we obtain 3

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

jµ0mUm exp[j(kz+mK)z] = m j(kz+mK) Sm exp[j(kz+mK)z],

(1.5a)

j0p ˆ p exp[jpKz] qSq exp[j(kz+qK)z] = m j(kz+mK) Um exp[j(kz+mK)z].

(1.5b)

Identifying each term of the Fourier series, and using the change of variable p+q=m in Eq. (1.5b), we obtain two sets of equations satisfied for any relative integer m µ0Um = (kz+mK) Sm,

(1.6a)

0q ˆm q Sq = (kz+mK) Um.

(1.6b)

There is two important ways to look for a solution to the system of equations. With solution 1, the wave frequency  is fixed and one calculates the dispersion relation kz() or neff(). Note that since the frequency is fixed, it is straightforward to take into account material dispersion in this approach. This will not be true for the second solution, for which values are calculated from fixed kzvalues. Equations (1.6a) et (1.6b) lead to an eigenproblem that can be written in a matrix format µ0   S  kz K  U

 K  E  0

S U  0 ,  

(1.7)

where I est the identity matrix, E is a Toeplitz matrix formed by the Fourier coefficients of (z), Em,n = ˆm n , K is a diagonal matrix with Km,m = mK, S et U are column vectors formed by the coefficients Um et Sm. Equation (1.7) is a dispersion relation, which links the mode momentum kz (or effective index neff) to the mode frequency . The kz’s are obtained as the eigenvalues and the Bloch modes are the associated eigenstate. Alternatively, it is also possible to fix kz and to calculate . Because of the pseudoperiodicity (Floquet’s theorem of Eqs. (1.2a) and (1.2b)), it is sufficient to fix kz in the first Brillouin zone from –K/2 to K/2, at least as long as we are concerned with real values. With this approach that is classically used in solid-state physics to calculate the band diagram of semiconductors, one gets  0  k z I  K

E 1k z I  K  S  S        0. 0  U c U

(1.8)

4

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

A key property of the eigenproblem in Eq. (1.8) is that if (x) is real (no Ohmic loss), the eigenvalues of the matrix (kzI+K)E1(kzI+K) are real for real values of kz. This can be seen by

  and

first considering that for real (x), ˆp  ˆ p

thus E = E, where  denotes the

transpose-conjugate operation. E and therefore E1 are Hermitian matrices. Noting that (kzI+K) = (kzI+K), it follows that (kzI+K)E1(kzI+K) is Hermitian too. Illustrative results calculated with a computer using classical eigenroutine to solve Eqs. (1.7) and (1.8) are shown in Fig. 1.2. In practice the matrices of infinite dimension are truncated for the sake of implementation, and one should guaranty that enough Fourier harmonics are retained in the calculation to achieve high accuracy. High accuracy is already achieved for dielectric materials for m values varying from M = -5 to M = 5 (2M+1 = 11 Fourier harmonics are retained) in Eqs. (1.6a) and (1.6b). Note that 2(2M+1) eigenvalues are thus calculated, but they are all degenerate. As shown in [Yeh88], for given kx and ky, in every layer, the Bloch mode is composed of two planes waves that are counter-propagating, and the total number of Bloch modes is only two at a given frequency, one with a positive propagation constant kz and the other one with an opposite value –kz (not shown in Fig. 1.2). The theoretical approach we take, although general and efficient, is not describing the Bloch modes as two sets of two counter-propagating plane waves. We insist, the interested reader should look at the approach adopted in [Yeh88]. In what follows we try to understand the properties and the physical meaning of these curves.

5

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Figure 1.2: Dispersion curves, neff() (top) et kz() (bottom) obtained by solving Eqs. (1.7) et (1.8), respectively, for the set of parameters: 1 = (n1)2 = 1, 2 = (n2)2 = (2.3)2, a = 1 et f = 0.5. Both solutions represent solutions of Maxwell’s equations for the same structures shown in the inset. Two different zones appears: BANDGAPS: in some specific energy frequencies, no mode is found in the (kz) representation (bottom), or equivalently, a complex propagation constant is obtained in the kz() representation (top), although the materials are lossless ((z) is real). ALLOWED BANDS : Outside the bandgaps, the periodic stack supports truly-propagative modes with a real propagation constant (kz is real). Let us emphasize that the kz (or neff) strongly depends on w, whereas the included materials are assumed dispersion less. In the following, we first study the properties of the thin film stack in the allowed band, especially in the first band at small energies. We will introduce the “homogenized” (effective) material, and we will show that by controlling the geometry, it is possible to vary the effective properties at will. In a second step, we will study the properties of Bloch modes in the gap 6

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

and will show that the imaginary part of the propagation constant is due to the evanescent character of the mode. The computational method, previously introduced for 1D periodic structures, is based on Fourier expansion techniques and can be generalized for 2D and 3D structures. For fullyperiodic structures, such as those shown in the top of Fig. 1.3, the generalized method is known as the plane-wave method [Ho90,Zha90]. This method is popular among the photonic crystal community as a method of solving for the band structure (dispersion relation) of specific photonic crystal geometries. In general, one fixes the reciprocal wavevector k and calculate , but it is also possible to fix  and two components of k (for instance kx and ky) and to calculate the other component. [Lal98]. In practice, the structures have necessarily a finite dimension (see the bottom panel (b) in Fig. 1.3), and this adds to the complexity of the problem. Related methods using Fourier expansion techniques are available, such as the rigorous-coupled-wave analysis.

Figure 1.3: (a) Examples of periodic structures 1D, 2D et 3D whose band structure can be calculated with the plane-wave method. In every direction, the structures are either periodic or invariant. (b) Related structures (1D et 2D) with a finite thickness on a substrate.

1.2. Periodic thin-film stacks with small periods In most optical materials the atomic or molecular structure is so fine that the propagation of light within them may be characterized by their refractive indices and the materials are considered as homogeneous. When an object has structure which is larger than the 7

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

wavelength of light, its influence on the propagation of light may be described by the classical laws of diffraction, refraction and reflection. Between these two extremes is a region in which there is structure that is too fine to give rise to diffraction in the usual sense but is too coarse for the medium to be strictly considered as homogenous. In this region, often referred to as the "subwavelength domain", homogenization techniques do not strictly apply but give however a good physical understanding of the medium properties. Developments in microlithography and associated technologies now make it possible to put these principles into practice and in particular to produce "artificial media" or “metamaterials”, which operate as effective media with some remnant properties of the resonance domain. Although fully-vectorial electromagnetic computational tools nowadays exist to analyze subwavelength periodic structures, it is very important to intuitively understand how those structures behave, simply because force-brute calculation is often hiding simple phenomena and cannot be implemented to reach the level of automatic design. Homogenization theory of electromagnetic waves in composite media with subwavelength inhomogeneities is an old but still very active subject. In principle, the inhomogeneity could be randomly or regularly distributed. However, if it is random, there will in fact be a full spectrum of spatial frequencies present. Furthermore, if the distribution of spatial frequencies is such that there is a significant proportion at wavelengths which are similar to that of the light, then the medium will scatter. It is therefore preferable to produce structures which are regular and periodic. In this way it is possible to control the spatial frequencies that are present and avoid random scattering. Moreover the theory of composite materials is made easier for periodic structures because of the Bloch theorem. Initially, various effective medium approaches like the Maxwell-Garnett formula or Bruggeman’s effective medium formulae [Jac74], first developed for random and dilute distributions of inclusions in a matrix, were used to determine the dielectric constant of periodic composite materials, like those of Fig 1. It was later realized that those approaches which rely on a spatial average but which ignore the fine geometry of the inhomogeneity were inadequate even in the long-wavelength limit (a/  0), i.e. when the period is infinitely smaller than the wavelength. In this limit, very nice mathematical theorems exist, and the equivalence between 1D periodic system and homogenous materials has been demonstrated. We will admit that a 2D or 3D periodic structure is also equivalent to biaxial or uniaxial materials. Finally it should be noted that the same ideas apply without change to a number of different physical situations, so that the formulae quoted here for effective permittivities and 8

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

permeabilities also apply to electrical conductivity, thermal conductivity, diffusivity, fluid permeability and the shear matrix of anti-plane elasticity. 1.2.1. Static limit of 1D periodic structures Main result. The most important result is the genuine equivalence between 1D periodic composites and homogeneous media [Bou85]: In the long-wavelength limit a/  0), a periodic structure is mathematically equivalent to a homogeneous material. 1D periodic composites are equivalent to uniaxial materials (form birefringence) and 2D or 3D periodic composites are equivalent to biaxial materials.

Figure 1.4: The equivalence between a periodic composite and a homogeneous medium in the static limit implies that the field scattered by the finite-size composite on the right is strictly the same as that scattered by the uniform material with the save shape (a parallelepiped on the figure) on the right. The equivalence implies that if one takes a material with a periodicity a and with an arbitrary contour (a parallelepiped in Fig. 1.4) and if one illuminates it by an incident wave with a wavelength much larger than the periodicity of the composite inside the (a/0), the electromagnetic fields scattered inside and outside the composite structure are exactly those that would be obtained if the composite would be homogeneous. For 1D composites, simple closed form expressions exist for the ordinary (no) and extraordinary (ne) effective indices of the artificial material

no  z 

1/ 2

and ne  1 z 

1 / 2

,

(1.9)

where (z) is the relative permittivity of the periodic structure, which could be either a continuous or discontinuous function of z and <.> represent spatial averaging over one

9

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

period. Thus a 1D periodic composite, periodic in the z-direction and invariant in the two others, has an effective permittivity tensor with harmonic and arithmetic averages

   eff =  0  0

  0 . 1  1  

0

0

 0

(1.10)

For the lamellar two-material composite of Fig. 1, we have no  f 2  (1  f )1 

1/ 2

ne  f 2  1  f 1 

1 / 2

and

. In general, by precisely controlling the fillfactor f, it is possible to

synthesize artificial materials with refractive indices that can be continuously monitored between (1)1/2 and (2)1/2 or between (1)1/2 and (2)1/2. Note that for metal-dielectric stacks, since 1 and 2 have opposite signs, ne may diverge for some specific frequencies leading to a “not-so-localized” plasmons lying between the plasmons of isolated particles and those on continuous metallic surfaces in their properties. It is important to realize that the form birefringence is an order of magnitude larger than the natural birefringence achieved by existing materials. For 2 = 2.32 et 1 = 1, none  0.5 for f = 0.5. This value has to be compared with the natural birefringence, n = 0.17 for the Spath crystal of Iceland (one of the crystal offering the highest n) and n = 0.01 for quartz at visible frequencies. For 2D composites that are invariant in the z-direction and periodic in the two other Cartesian axes, there is no closed-form rigorous expressions in general [Mil02], except for the zz component of the permittivity tensor, which is simply equal to <>. In this case we have

  xx  eff =  0  0 

0  yy 0

0   0 ,    

(1.11)

where xx et yy have to be determined with approximate expressions (upper and lower bounds also exists) or with fully-vectorial computations. Generally speaking, 2D structures behave as biaxial materials, except if some specific symmetry conditions are fulfilled. An interesting article can be read in [Kro02]. For 3D periodic structures, no closed-form expression exists for zz. 10

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Why harmonic and arithmetic averages? The critical dimension over which a wave significantly varies is its wavelength in the material. In the static limit a/ << 1, the wave cannot vary spatially at the scale of the uniform sections of the composite. Therefore the electric field components that are continuous at the interfaces of the stack, Ey et Ex, can be supposed to be constant, E(z)=E0, and it follows = <(z)E(z)> = <(z)E0> = <(z)>E0 = <(z)><E(z)>.

(1.12)

Defining the effective permittivity eff as an average response at the cell scale [Smi06] = eff <E>.

(1.13)

one obtains xx = yy = <(z)>, which justifies for the arithmetic mean. The z-component Dz of the displacement vector being also continuous at the interfaces, we have <Ez(z)> = = = D0<1/(z)>= <1/(z)>,

(1.14)

implying that zz = <1/(z)>1, which justifies for the harmonic mean. 1.2.2 Taylor expansion for small periods The static limit is a textbook case that is conceptually important, but which is of little importance in practice, because it is rarely reached. In order to have quantitative predictions when the period is only slightly smaller than the wavelength, it is necessary to use numerical tools to determine the material response. The problem is complicated from a theoretical point of view, and it is presently admitted that the homogenized material will present artificial magnetism and chirality, even if the constituent materials are not magnetic (µ is equal to µ0 everywhere) and are not chiral. It is much easier and important for practical applications to consider the fundamental mode of the periodic medium and to study how the propagation of the mode, the effective index, changes as one departs from the static-limit. To do that, it is convenient to admit that the effective index admits a Taylor expansion in the variable (a/) [Ryt56]. To see what’s happen to the effective index, let us again consider Eqs. (1.6a) et (1.6b), et let us introduce the scaling parameter  defined by =K/k0=/a, µ0cUm = (neff+m)Sm,

(1.15a)

11

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

µ0cq ˆm q Sq = (neff+m)Um,

(1.15b)

where we used the relation 0µ0c2 = 1. We now look for a solution of Eqs. (1.15a) and (1.15b) with a Taylor expansion in 1/ (   in the quasi-static limit)

n2eff  n(0)  n(1)  1  n(2)  2  ... , (0)

(1)

(2)

(0)

(1)

(2)

(1.16a)

Um  um  um  1  um  2  ... ,

(1.16b)

Sm  sm  sm  1  sm  2  ...

(1.16c)

We then incorporate Eqs. (1.16a-c) into Eqs. (1.15a) and (1.15b), and identify terms with the same degree starting by the highest degree. Terms in O() for m  0 and in O(1) for m = 0 give us (0) 0  s(0) m  um , m  0 ,

(1.17b)

1 (0) (0) (0) (0) 0 c u(0) s0 µ0c and 0 c  ˆ0s(0) u0 , 0  n 0  n

(1.17b)

2 showing that in the limit a/  0, the zeroth-order term of the expansion of neff is given by

n (0)  ˆ0 ,

(1.18)

in agreement with Eq. (1.9). Moreover, we find that the Bloch mode is simply a plane wave (0) (0) (0) since solely u(0) 0 and s 0 differ from zero ( up  sp  0 ,  p  0): the 1D periodic composite

strictly behaves as a homogeneous material in the static limit. Let us now abandon the asymptotic limit. By identifying terms in O(1) for m  0, we obtain (1) u(0) m  0  msm ,

(1.19a)

µ0c ˆms(0) 0 = m u m ,  m  0, (1)

(1.19b)

showing that the Bloch mode is no longer a plane wave as soon as we escape the static limit (the first order terms are not null). Terms in O(1/) for m = 0 give us, (1) (0) s0  n(0) s(1) µ0c u (1) 0 , 0 = n

(1.20a) 12

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

(1) (0) µ0c ˆ 0 s(1) u0  n(0) u(1) 0 . 0 =n

(1.20b)

From which we infer using Eq. (1.15b) that n (1)  0 .

(1.21)

(2) Finally by considering terms in O(1/) for m  0 in Eq. (1.15a), we obtain µ0c u (1) m = m s m ,

which allows us to establish after injecting into Eq. (1.19b) (0) s(2) m2 ,  m  0. m  s0

(1.22)

The second order term of the expansion of the effective index is obtained by considering terms in O(1/) for m = 0 (2) (0) s0  n(0) s(2) µ0c u (2) 0 , 0 = n

(1.23a)

(0) (2) u0 , µ0cm ˆ  m s (2) =  n(2) u(0) 0 n m

(1.23b)

which allow us to obtain with Eq. (1.22), p ˆpˆ p p2  2 n (0) n (2) , or simpler 2 neff  ˆ0  

p0

ˆpˆ  p p2

a  2  Oa  4 .

(1.24)

The quadratic expansion of neff in Eq. (1.24) is valid for any periodic 1D structure (lamellar or continuous, lossy or transparent), the ˆ p ‘s being the Fourier coefficients of the periodic relative permittivity (z) ( ˆ0  z  being the mean value). An expression similar to Eq. (1.24) can be obtained for the extraordinary effective index with a similar approach and for 2D-periodic composite [Phe82,Lal96]. Please note that for transparent media ((z) is real),

ˆ p  ˆp * and neff decreases as the wavelength increase, just like most real materials unfortunately. Note however that neff varies quickly with the wavelength (like in a waveguide) and the artificial material is highly dispersive, much dispersive than natural materials. The quadratic expansion describes how the propagation constant changes as one departs from the static limit, but not the effective permittivity and one cannot simply identify (neff)2 with eff. Actually, out of the static limit, the Bloch mode is no longer a plane wave and the material cannot be replaced by a homogeneous material. 13

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Figure 1.5: As in Fig. 1.3. With the red curves, are shown results obtained with homogenization.

1.3 The photonic gap In this section, we study the gaps. In order to rapidly understand the physical meaning of the complex effective index, we first consider the reflection of a normally-incident plane wave on a thin-film stack with a finite thickness t (inset in Fig. 1.6). Figure 1.6 represents the transmission spectrum of a dielectric stack composed of six pairs with alternate layers of thicknesses equal to 0/(4n1) or 0/(4n2). The calculation is performed with the classical 22 matrix approach [Yeh88] for 0 = 1 µm, n1 = 1.6 and n2 = 2.3. This famous stack acts as a Bragg mirror and is called a quarter-wave stack for the central wavelength 0. A transmission dip appears around 0. In the absence of absorption, one minus the transmission represents the reflection, and around the central wavelength, all the incident energy is almost reflected.

14

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Figure 1.6: Transmission spectrum of a quarter-wave thin-film stack designed for the central wavelength 0 = 1 µm. The stack is deposited on a glass substrate (refractive index n = 1.5) and is illuminated from air at normal incidence. Let us now consider the field intensity distribution in the stack for the central wavelength =0. The intensity distribution shown in Fig. 1.7 is composed of a stationary wave in the incident medium, of an exponentially damped wave in the stack, and of a weak transmitted plane-wave in the substrate. Because the field intensity at the bottom interface is very weak, the reflection at this interface is also weak, and the field in the stack is likely to be dominated by the gap Bloch-mode propagating downward in the stack. Therefore, the field envelop shown in black is proportional to exp(2Im(neff)k0z). The imaginary part of the effective index is not related to any absorption (the material are lossless), but to the evanescent character of the gap Bloch-mode.

15

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Figure 1.7: Sketch of the intensity |E|2 in the scattering by a stack of finite thickness for  = 0. The envelop of the field in the stack exponentially decreases and the penetration depth is given by 0/Im(neff). 1.3.1. Bandgap opening In this subsection, we build the band diagram in the limit of small permittivity modulations. Back to Eqs. (1.6a) and (1.6b), for any relative integer m, we have µ0Um = (kz+mK) Sm, and 0q ˆm q Sq = (kz+mK) Um. By eliminating Um, we obtain an Helmholtz-like equation in the plane wave basis (kz+mK)2 Sm + (/c)2q ˆm q Sq = 0.

(1.25)

Qualitative approach: phase matching condition. First, let us even consider the asymptotic case without any modulation, (z) = ( ˆ 0 )1/2. The composite is then homogeneous, ˆm  0 for m  0. The Bloch modes are then counter-propagative plane waves obeying the

dispersion relation /c = kz/( ˆ 0 )1/2, which are shown in Fig. 1.8a with the two black lines. As soon as the modulation is not null, even if it is infinitely small, the two plane waves becomes pseudo-periodic Bloch modes. Neglecting all the Fourier coefficients m for m  0 (they are proportional to the refractive index modulation), Eq. (1.25) becomes (kz+mK)2 Sm + (/c)2 ˆ 0 Sm = 0, whose solution is /c = (kz+mK)/( ˆ 0 )1/2.

(1.26)

Since the index modulation is very weak, one expects that the dispersion relation of the Bloch mode (shown with red lines in Fig. 1.8b) is very similar to that of the plane wave, except that the dispersion relation becomes periodic (kz+mK)=(kz), indeed.

16

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Figure 1.8: Bandgap opening. (a) Dispersion relation of the plane waves propagating in a homogeneous material with a refractive index ( ˆ 0 )1/2. (b) Dispersion relation of a Bloch mode in a periodic composite with a refractive index modulation n infinitely small compared to ( ˆ 0 )1/2. (c) Dispersion curves (kz) including the bandgaps. Note that since the relation dispersion is periodic, it is sufficient to consider it on an interval of length K, like the first Brillouin zone (–K/2 < kz < K/2), whose boundaries are shown with vertical dashed red lines.

17

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Now remains the important step of bandgap opening. The later corresponds to a backscattering phenomenon with a precise phase-matching condition. In order to intuitively understand, let us go back to Eqs. (1.4a) et (1.4b) to obtain the Helmholtz equation 2E/z2 + k020(z)E = 0. Let us insert the plane wave E= exp(jkPW z) that is propagating towards the negative z in the homogeneous material of refractive index ( ˆ 0 )1/2, and the plane wave E= exp(jkPW z) that propagates towards the positive z ( kPW  k0 ˆ0 

1/ 2

). By using the Fourier

series of (z), we obtain 2E/x2 + k02 ˆ 0 E+ k02p ˆ p exp[jpKzjkPW z] = 0,

(1.27a)

2E/x2 + k02 ˆ 0 E+ k02p ˆ p exp[jpKzjkPW z] = 0.

(1.27b)

Because ˆp  ˆ 0 for p ≠ 0, the plane waves E et E+ are almost solutions of Eqs. (1.27a) and (1.27b), and the last term can be neglected, except when a relative integer p0, such that p0K+kPW = kPW, exists. Then the previous equations become 2E/x2 + k02 ˆ 0 E+ k02 ˆ  p0 E = 0,

(1.28a)

2E/x2 + k02 ˆ 0 E+ k02 ˆp0 E = 0,

(1.28b)

and the contra-propagative plane waves E et E+ are coupled. The Bloch mode that results from the coupling mode is a stationary pattern that does not carry any energy in the lossless case. We will admit without proof that, as long as the coupling exist (i.e. as long as we are in the bandgap), the Bloch-mode propagation constant kz satisfies the same phase matching as E and E, i.e. Re(kz) = p0K/2, i.e. Re(neff) = p0/(2a),

(1.29)

where Re(.) has been introduced to take into account that, in contrast to plane waves, the propagation constant of the gap Bloch mode is complex. In the (kz) diagram, the coupling between the counter-propagative plane wave results in a gap opening with an absence of Bloch mode, as shown in Fig. 1.8c. This is the classical presentation in solid state physics for the energy bands [Kit05]. We emphasize that all the periodic bandgap openings in Fig. 1.8c, which corresponds to a given band label p0 = 1,2,3 …, result from the coupling between two

18

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

red dispersion lines that are offseted by a momentum p0K, and the coupling coefficients are

ˆp0 and ˆ  p0 . In the kz() representation, the stationary pattern results in an evanescent Bloch mode with a complex propagation constant, whose real part is strictly given by Eq. (1.29). Although the phase-matching condition of Eq. (1.30) has been derived in an intuitive manner, it is exact. It is rigorously derived in [Yeh88] for bi-material thin film stacks and has been systematically observed in much complex systems, such as ridge waveguides with periodic holes in them. The dispersion relations for the real part of the effective index of the gap Bloch-modes are shown with the blue curves in Fig. 1.9.

Figure 1.9: Same as in Figs. 1.2 and 1.5. In blue is shown the phase matching condition providing the real part of the effective index of the gap Bloch-modes. The blue arrows indicate that the slope of the dispersion curve is infinite at the boundary of the gaps. Note that all closed-form expression shown in Fig. 1.9 are exact.

19

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Coupled-wave method. The previous intuitive approach can be presented more formally using a two-wave approximation. The approach is classical and can be found in many textbooks, including [Yeh88]. Let us consider the gap p0 that opens up whenever the dispersion branches that are separated by momentum of p0K intersect. In the vicinity of the point ( = 0, kz = p0/a), which corresponds to the intersection between the black line /c = kz/( ˆp0 )1/2 and red line /c = (p0Kkz)/( ˆp0 )1/2 in Fig. 1.8b. 0, the frequency at the gap centre, is then given by (

0 p   01 / 2 )1/2. To derive closed form expression for the main gap c aˆp0

properties, we assume that in the vicinity of ( = 0, kz = p0/a), the Bloch mode is mainly composed of two Fourier components, S0 et S-p0. Then the system of Eq. (1.25), (kz+mK)2 Sm + (/c)2q ˆm q Sq = 0 for any m, simplifies and we get two equations with two unknowns [(/c)2 ˆ 0  kz2] S0 + (/c)2 ˆp0 S-p0 = 0 (m = 0),

(1.30a)

(/c)2 ˆ  p0 S0 + [(/c)2 ˆ 0  (kzp0K)2] S-p0 =0 (m = p0).

(1.30b)

To simplify, let us use the variable changes kz = p0K/2+u and /c = 0/c+v, which allows us to work around the gap centre (0, p0/a) with u << p0K/2, v << 0/c. Let us remember that our approach is legitimate only for weak modulations, i.e. for ˆp0 and ˆ  p0 << ˆ 0 . The new system becomes [v( ˆ 0 )1/2u] S0 + ˆp0 K/(4 ˆ 0 ) S-p0 = 0,

(1.31a)

ˆ  p0 K/(4 ˆ 0 ) S0 + [v( ˆ 0 )1/2u] S-p0 =0.

(1.31b)

Non null solutions exist for S0 and S-p0 if the determinant is null and we obtain the hyperbolic dispersion relation in the vicinity of the gap 2

v 2  u 2 ˆ0  ˆp0 ˆ p0 K 2 16ˆ30  ˆp0 K 2 16ˆ30 ,

(1.32)

the last equality being valid if (z) is real.

20

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Figure 1.10: Hyperbolic dispersion around the boundary of the Brillouin zone. The gap width 2A is found by setting u = 0 in Eqs. (1.31a) and (1.31b). We obtain





v   ˆp0 K 4ˆ30 / 2 and the normalized gap width 2A/0 is then given by 2A / 0  ˆp0

p0ˆ0  .

(1.33)

It decreases as the gap order increases, in the limit of small modulations. It is also important to calculate the effective index in the gap. In particular for the center frequency (v = 0), where 2





the imaginary part of neff is maximum, we obtain u2   ˆp0 K 2 16ˆ02 , so that in the gap centre, neff(0) = kz/k0, is given by kz(0) = K/2[p0 + i ˆp0

2ˆ0  ].

(1.34)

We find (again with an approximate method) that the real part of the effective index of gap Bloch-modes at the boundary of the Brillouin zone are equal to p0/(2a). The coupling coefficient ˆp0 or ˆ  p0 between the counter-propagative plane waves plays a crucial role in the gap opening: the gap width and the damping rate at mid-frequency are both directly proportional to ˆp0

ˆ0  .

Let us finally consider the Bloch modes, which are defined by S0 and S-p0, Ey(z) = S0 exp(jkzz) + S-p0 exp(j(kzp0K)z). At the band edge, for kz= p0K/2, the Bloch modes

21

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

becomes Ey(z) = S0 exp(jp0Kz/2) + S-p0 exp(jp0Kz/2). At the valence band edge, S0 = S-p0 (if (z) is real) and Ey(z) = 2 S0 cos(p0Kz/2).

(1.35a)

For p0 = 1, the period is twice smaller than that of (z) and the intense electric field (in modulus) is located in the high refractive index regions. The effective index is also large, consistently with Fig. 1.9. On the contrary at the conduction band edge, S0 = S-p0 (if (z) is real) and Ey(z) = 2 S0 sin(p0Kz/2).

(1.35b)

For p0 = 1, the intense electric field in modulus is located in the low refractive index regions. The effective index is also small. 1.3.2 Penetration depth The reflectivity (especially the phase of the reflectivity coefficient) of a dielectric mirror varies rapidly with the wavelength. For this reason, one conveniently introduces an equivalent mirror with a phase independent of the wavelength, but with a reflection coefficient strictly identical to that of the dielectric mirror at the operating wavelength. Let us first denote by r(k) = |r|exp(j) the reflection coefficient of a plane wave with a wavelength  (k = 2/) impinging from a semi-infinite medium with refractive index n onto a semi-infinite periodic composite operating in the bandgap, see Fig. 1.11a. At a slightly different wavelength k+dk, if we assume that the modulus of r is independent of , the reflection coefficient can be written r(k+dk) = |r|exp(j+j’dk), where ’ denotes d/dk. Let us now consider a metal mirror with a reflection coefficient . The latter is assumed to be independent of the wavelength, like a mirror with a perfect metal but with a finite reflectivity ||2. Located at a distance L below the interface z = 0, see Fig. 11b, the mirror reflection coefficient can be written, r(metal) =  exp(2jknL) to take into account the phase delay due to the round trip over the distance L. At k+dk, the metal reflection becomes r(metal)(k+dk) =  exp(2jknL+2jnLdk), since  is assumed to be independent of the wavelength. Let us now equal the reflection expressions at k and k+dk. We obtain |r|exp(j) =  exp(2jknL),

(1.36a) 22

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

|r|exp(j+j’dk) =  exp(2jknL+2jnLdk).

(1.3b)

For this equality to be true for any dk, one needs that L = ’/(2n).

(1.37)

This expression defines the quantity L, often called «penetration length». Note that if the refractive index n depends on ng, one should replace n by the group index ng = n+dn/d. L corresponds to the distance for which the phase of the reflected beam of the periodic stack is identical (for a specific wavelength and around up to a first-order correction term) to that of a metal mirror with a reflection phase independent of the wavelength. From Eqs. (1.36a) and (1.36b), the reflectivity coefficient of the equivalent mirror is found to be  = |r| exp[j(-k’)].

(1.38)

Figure 1.11: Definition of the penetration length of a dielectric mirror by relating the intricate reflection problem of a periodic stack (a) to a much simpler problem (b) with a “metal” mirror reflectance that is independent of the wavelength. If r in (a) is given by r(k) = |r|exp(j) for a given wavelength (k = 2/), then  and L in (b) are given by L = ’/(2n) and  = |r| exp[j(k’)]. The term « penetration depth » is truly ambiguous. For metallic reflection, it refers to the spatial damping length corresponding to a 1/e attenuation of the wave that penetrates the metal. This exponential damping is the analogue for dielectric mirrors of a quantity  = /[2Im(neff)] related to the exponential decay of the Bloch-mode envelop in Fig. 1.7. It is important to realize that the penetration depth defined by L = ’/(2n) has nothing to do with neff. When periodic composites are used as Bragg mirrors for confining light in microcavities, the decay rate in the mirror (monitored by Im(neff)) is related to the mode volume V of the cavity, whereas the « penetration depth » L is related to the cavity Q factor [Sau09].

23

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

1.3.3 Structural slow light At the boundary of the bandgaps, the slopes of the dispersion relations neff() are infinite, as shown by the vertical arrows in Fig. 1.9. This implies that the phase velocity of the Bloch varies extremely (infinitely) fast with the wavelength. The consequence of this extreme dispersion is a slowdown of the light in the composite. There are several mechanisms that can generate slow light, all of which create narrow spectral regions with high dispersion. Schemes are generally grouped into two categories: material dispersion and structural dispersion. Material dispersion refers to situations in which the velocity of light pulses can be described fully in terms of the spatially uniform but frequency-dependent refractive index n of a material. A typical sort of material dispersion used in slow light leads to the establishment of a sharp dip in an absorption or gain feature. Simple saturation effects can lead to such behavior, as well as more advanced effects such as electromagnetically induced transparency (EIT) and coherent population oscillations (CPO). In general the effect takes place in a very narrow spectral region around the absorption line and is not accompanied with any electromagnetic field enhancement [Khu10,Boyd11]. Structural dispersion refers to situations in which the velocity of light pulses is slowed down because of multiple scattering. In contrast to material dispersion for which the energy is alternatively stored in photon or in atomic polarization as light propagates, all the energy is purely stored as photons that bounce forward and backward as they propagate. Thus, the structural scheme is accompanied with a field enhancement (proportional to vg1/2 as will be seen hereafter) that favors optical nonlinearities. Although both schemes lead to the same principal effect, their physics is fundamentally different. To understand what is happening at the gap edges in Fig. 1.9, let us define the group velocity vg of the Bloch mode (i.e. the speed with which a wave packet propagates in the periodic composite, it is also the energy velocity for lossless composites) by vg/c = d(/c)/dkz.

(1.36)

In the reduced units, the group velocity is also equal to dv/du and provided that we are very closed to the band edge, u/K << ˆp0

ˆ0  , we easily find that

vg/v = 4 ˆ 0 u/(K ˆp0 ),

(1.37) 24

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

with the phase velocity v = c/ ˆ 0 1/2. With the same approximation, we additionally show that S-p0/S0 = |p0|/p0[vg/v1].

(1.38)

Equation 1.38 is valid for vg  0 and essentially says that a slow Bloch mode is a standing wave, built with two counter-propagating travelling waves both propagating at the speed of light. Note that normalizing the Bloch-mode energy-flow ( |S0|2|S-p0|2) of a slow Bloch mode propagating towards the z direction to a given value (  for instance), one readily obtains that the Bloch-mode electromagnetic fields ( S0 or S-p0) scales as (vg/v)1/2, since |S0|2|S2 p0|

= |S0|2 [1S-p0/S0|2] = 2|S0|2 vg/v (according to Eq. 1.38). Slow light devices, mainly based on photonic crystal waveguides (see Fig. 1.12) or

coupled resonator optical waveguides, have received much attention because of their promise for optical signal processing applications and enhanced nonlinear light-matter interaction [Bab08]. The main motivation for using slow light on a chip is the space compression. Whether the slowing-down process is due to material or structural dispersion, delays are increased and more pulses can be stored per unit length. Therefore one immediately sees the “unique” potential offered by small group velocities for integration and miniaturization, especially if one realizes that, in contrast to electronic devices, scaling down photonic devices by reducing their dimensions is virtually impossible because of the diffraction limit. The latter can be overcome only by using metallic materials, but then one faces drastic Ohmic losses. Since the pioneer observation of slow light by a Japanese group in 2001 [Not01], many successful demonstrations underpinning this promise have already been made, such as ultra-sensitive interferometers [Vla05]. Theoretically, there is no fundamental limit on the amount to which light may be slowed down by structural dispersion. In practice however, v/vg rarely exceeds one hundred over propagation distances of 100 optical wavelengths. The main limitation is propagation losses; especially as vg approaches zero, fabrication imperfections such as inevitable roughness or errors, even kept at a very small level in the nanometer range, scatter with a scattering efficiency that increases as the inverse of the square of the group velocity. Structural slow light is highly sensitive to fabrication imperfection. This limitation is easily understood by realizing that the structural slowdown is due to a multiple scattering process, which is coupling the two counter-propagative plane waves. As a result, any imperfection, instead of scattering once, scatters many times as the light is slowed down. This physical picture can be also explained using LDOS arguments. 25

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Figure 1.12: Photonic crystal waveguide in a silicon membrane in air. This geometry offers a slow Bloch mode at telecommunication wavelength for interhole distances of 420 nm. After [OFa10]. 1.3.4 Multi-dimension Bragg mirrors So far we have considered gap opening in one-dimension for Bloch-wave propagating in the direction of periodicity. It is instructive to look at how one may generalize the concept of bandgaps to two or three dimensions. For that purpose, let us consider the geometry shown in Fig. 1.12a, where a square lattice of square cylinder with a large refractive index n2 is embedded in a low-index medium with a refractive index n1. Let us denote by a the period and by f the linear fill factor. The refractive indices being fixed, the initial problem consists in choosing the geometrical parameters, a and f, to open a bandgap. The idea, which underlines the main physics of the process, is to homogenized the 2D photonic crystal, layer by layer. The homogenization process is depicted in Fig. 1.12b, and we end up with a 1D periodic stack composed of two layers with refractive indices neff and n1, and with respective thicknesses fa and (1f)a.

Figure 1.12. Homogenization of a 2D photonic crystal lattice (a) into a much simpler 1D stack (b). 26

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Let us consider a wave impinging normally to the thin film stack (parallel to the vertical direction in the figure). Following Eq. 1.24, one may write 2 neff  0   2 a    Oa   , 2

4

(1.39)

up to the second order. Let us write the condition, such that the thin film stack in (b) is a quarter-wavelength stack: (1f)a = /(4n1),

(1.40a)

fa = /(4neff).

(1.40b)

We obtain two equations with two unknowns f and a. By eliminating a, it is easily shown that one must satisfy the equality 41(1f)2 = f [161(1f)2(0)+(2)]1/2; because the lefthand term monotonously decreases from 41 for f = 0 to 0 for f = 1 while the right-hand term is always positive or null for f = 0 and 1, there is always a solution for f. Provided that the refractive index contrast is high enough, the gap due to the effective 1D periodic composite is large at normal incidence, and resists as one departs from normal incidence, leading eventually to a full band gap in 2D for any kx and ky in the first Brillouin zone of the 2D photonic crystal. The previous analysis evidences two difficulties in relation with the dimensionality. First, is we start with a large index contrast with the 2D structure, n2n1, the effective contrast seen by the wave that propagates in the crystal is substantially smaller, neffn1, since neff is much smaller than n2. In 3D the situation is even worse. Secondly, it is noteworthy that Eq. (1.40) depends on the polarization, so that even if there is always a unique solution to realize a quarter-wave stack, the optimal parameters depend on the polarization. This implies a supplementary difficulty in relation with the band-gap opening for all polarizations. This is why following the seminal proposal by Eli Yablonovitch [Yab87], it took 3-5 years to the community to correctly design a 3D photonic crystal with a complete bandgap, for all space directions and for all polarizations. Ho, Chan and Soukoulis [Ho90] were the first theoreticians to correctly predict that a particular 3D photonic crystal would have a full bandgap. Their crystal was a diamond lattice of spheres and offer a complete bandgap at microwave frequencies, whether one embeds dielectric spheres in air or air spheres in a dielectric medium, as long as the sphere radius is chosen appropriately chosen and that the permittivity contrast is high enough. 27

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

The layer-by-layer homogenization approach can be seen an educational toy model. Real designs actually rely on full-electromagnetic band calculations using highly symmetric unit cells. However the layer-by-layer approach illustrates the fundamentals of the intricate processes leading to gap openings in 2D or 3D through multiple scattering and destructive interference and it is also at the heart of the physics of the woodpile structure, one of the most celebrated structure offering a complete 3D bandgap at near-infrared frequencies [Lin98], which can be shown to behave as a quarter-wave stack. Nowadays, 3D photonic crystals are no longer expected to bring new photonic applications in the future, but they are thought as a milestone conceptual achievement in the photonic community. However 2D photonic crystals are the subject of intense research even nowadays, because of their potential impact for on-chip optical routing and processing and because they are amenable to mass production with high accuracy. The Joannopoulos’ book [Joa95] that contains a very nice overview of the field of photonic crystal is highly recommended for future readings.

1.4 Bloch modes of real photonic structures with finite lengths Bloch modes are modes of periodic media, but media are never periodic since they always have a finite extend. An often asked question is how we may use the concept of Bloch modes to analyze or design photonic devices that are never periodic. Probably, a completeness theorem for normal Bloch modes, such as the celebrated completeness theorem for optical waveguides [Vas91], would be a mathematical answer. Let us try to intuitively approach the problem. Let us start by considering the scattering problem shown in Fig. 1.13. A plane wave with a unit amplitude is incident (at normal incidence) from a semi-infinite uniform medium (refractive index n) onto a periodic composite with a finite thickness d. We assume that that the transmission region has the same refractive index n, for the sake of simplicity. This classical problem generates back-reflected and transmitted plane waves with reflection and transmission coefficients r and t. The reflected-intensity spectrum, shown in the right panel, can be calculated with the classical 22 transfer matrix method [Yeh88], a method which essentially consists in calculating all the plane wave coefficients Ap and Bp , p = 1,…N in all layers. It exhibits an oscillatory behavior with a dip around the central wavelength of the gap. In the absence of absorption, we have |r|2+|t|2 = 1.

28

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

Figure 1.13: Scattering of a normally-incident plane wave on a thin-film stack with a finite thickness d. The surrounding semi-infinite media have the same refractive index n. The right panel shows the classical reflection spectrum |r|2. Let us now consider the related problem shown in Fig. 1.14(a). The same plane wave is now impinging onto the same periodic composite, but now the composite is semi-infinite (the thickness is infinite). To solve this problem, it is necessary to abandon the classical 22 transfer matrix method. Because one needs to satisfy the outgoing wave condition into a periodic half-space (the semi-infinite composite), one needs to calculate the modes of the structure. Note that this is exactly what is done in the incident semi-infinite half-space, since the incident and the reflected plane waves are nothing else than the modes of a uniform medium with the same refractive index n. Actually, there are only two counter-propagating Bloch modes in the periodic medium as mention in the first Section, see also [Ye88]. At the interface shown with a dashed line in Fig. 1.14(a), the continuity of the tangential (parallel to the x and y vectors) electric and magnetic field leads to two equations 1 <x|PW+> + r <x|PW> = t <x|BM+>, +



(1.41a)

+

1 + r = t ,

(1.41b)

where |PW+> and |PW> are column-vectors formed by the electromagnetic components of the right- and left-propagating plane waves in the incident medium; similarly |BM+> is a column-vector formed by the electromagnetic components of the right-propagating Bloch mode at the interface (Bloch modes also depends on z indeed). These two equations are easily solved in practice for the unknown reflection and transmission coefficients, denoted by r and t to emphasize that they apply to the interface problem between two semi-infinite half spaces. A typical reflectance spectrum is shown in the right panel (b). As expected we find that |r|2 = 1 in the gap since there is no absorption loss. We also find that there is no 29

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

oscillation in the bands. Note that on the band edge |r|2 is close to 1, implying that it is difficult to couple into a slow Bloch mode. Similarly, one may also consider the scattering of a left-propagating Bloch mode (see Fig. 1.14(c)). This defines a new reflection coefficient r’ since reciprocity arguments imply that the transmission coefficients in (a) and (c) are equal under appropriate normalization.

Figure 1.14: Scattering of a normally-incident plane wave on a semi-infinite thin-film stack. (a) Sketch of the geometry. (b) “Classical” reflection spectrum |r|2. In the gap, |r|2 = 1 and in the bands there is no oscillation. (c) Sketch of the reciprocal scattering problem that defines a new scattering coefficient r’. Note that |r| = |r’| in the bands since energy conservation stipulates that |r|2+|t|2 = 1 = |r’|2+|t|2. The interface problem of Fig. 1.14a evidences that it is not required that the medium be periodic to introduce Bloch modes. It is even not required that it is semi-infinite. In fact it needs to be at least one period long. Again to evidence this statement let us consider again the finite-thickness case of Fig. 1.13a. Instead of considering a multiple scattering process between all the layers, we will assume that the periodic composite of thickness d is a “uniform” (homogenized) layer supporting two counter-propagative Bloch modes, and we will denote by A and B the excitation coefficients of the forward- and backward-propagating Bloch modes, |BM+> and |BM>. At the left and right interfaces, we have A = t + r’B exp(ik0neffd),

(1.42a)

B = r’A exp(ik0neffd).

(1.42b)

Since t = r’A exp(ik0neffd), by eliminating B from Eqs. (1.42a) and (1.42b), we finally get the classical Fabry-Perot transmission for the periodic composite with thickness d

30

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

t

t 2 expi k 0 neff d  . 1  r2 exp2i k 0 neff d 

(1.43)

Equation 1.43 shows that the periodic composite can be seen as a Fabry-Perot resonator, with a Bloch mode bouncing back and forth over a distance d between two semi-infinite half spaces that act as two mirrors with a reflection coefficient r’. It is this Fabry-Perot response that is responsible for the oscillating behavior in the “valence” and “conduction” bands, with a fringe spacing proportional to the group index ng 

dn c  neff   eff of the Bloch mode. In vg d

the gap, the Bloch mode is a standing wave with an evanescent character; as such, it does not carry any energy. However because of the finite thickness of the homogenized material, the homogenized field is a superposition of two evanescent waves (A|BM+>+ B|BM>), and this superposition carries some net power flow that tunnels through the periodic composite. Replacing the composite by a uniform medium supporting two counter-propagating represents a considerable conceptual simplification. One should keep in mind that r 

2 nneff neff  n and t   for the Fresnel coefficients at the interfaces between two n  neff n  neff

homogeneous media with refractive indices n and neff. It is only in the static limit that Fresnel formulae become valid.

Figure 1.15: Homogenized version of the scattering problem in Fig. 1.13. The reflection and transmission coefficients are exactly those of Fig. 1.13(a) as shown in panel (b).

References [Bab08] T. Baba, "Slow light in Photonic crystals", Nature Photon. 2, 465-473 (2008).

31

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

[Bou85] G. Bouchitté and R. Petit, "Homogenization techniques as applied in the electromagnetic theory of gratings", Electromagnetics 5, 17-36 (1985). [Boy11] R.W. Boyd, "Material slow light and structural slow light: similarities and differences for nonlinear optics", J. Opt. Soc. Am. B 28, A38 (2011). [Ho90] K.M. Ho, C.T. Chan and C.M. Soukoulis, "Existence of a photonic Gap in periodic dielectric structures", Phys. Rev. Lett. 65, 3152-3155 (1990). [Joa95] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic crystals (Princeton University Press, 1995). [Jac74] J.D. Jackson. Classical Electrodynamics (John Wiley & Sons eds., New York, 1974). [Kit05] C. Kittel, Introduction to solid state physics (John Wiley & Sons eds., New York, 2005). [Kro02] A.A. Krokhin, P. Halevi and J. Arriaga, "Long-wavelength limit (homogeneization) for two-dimensional photonic crystals", Phys. Rev. B 65, 115208 (2002). [Khu10] J.B. Khurgin, "Slow light in various media: a tutorial", Adv. in Opt. and Photon. 2, 287 (2010). [Lal96] P. Lalanne and D. Lalanne, "On the effective medium theory of subwavelength periodic structures", J. Mod. Opt. 43, 2063-2085 (1996). [Lal98] P. Lalanne, "Effective properties and band structures of lamellar subwavelength crystals: plane-wave method revisited", Phys. Rev B 58, 9801-07 (1998). [Lin98] S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, A three-dimensional photonic crystal operating at infrared wavelengths, Nature (London) 394, 251-253 (1998). [Mil02] G.W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002). [Not01] M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs", Phys. Rev. Lett. 87, 253902 (2001). [OFa10] L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne and T. F. Krauss, "Loss Engineered Slow Light Waveguides", Opt. Express 18, 27627-38 (2010).

32

Chapitre 1. An introduction to optical Bloch modes (lecture notes, Philippe Lalanne)

[Phe82] R.C. McPhedran, L.C. Botten, M.S. Craig, M. Nevière and D. Maystre, "Lossy lamellar gratings in the quasi-static limit", Optica Acta 29, 289-312 (1982). [Ryt56] S.M. Rytov, "Electromagnetic Properties of a Finely Stratified Medium", Soviet Physics JETP 2, 466-475 (1956). [Sau09] C. Sauvan, J.P. Hugonin and P. Lalanne, "Difference between penetration and damping lengths in photonic crystal mirrors", Appl. Phys. Lett. 95, 211101 (2009). [Smi06] D. R. Smith and J. B. Pendry, "Homogenization of metamaterials by field averaging", J. Opt. Soc. Am. B 23, 391-403 (2006). [Vas91] C. Vassallo, Optical waveguide concepts (Elsevier, Amsterdam, 1991). [Vla05] Y. A. Vlasov, M. O'Boyle, H. F. Hamann and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides", Nature 438, 65-69 (2005). [Yab87] E. Yablonovitch, "Inhibited spontaneous emission in solid state physics and electronics", Phys. Rev. Lett. 58, 2059-62 (1987). [Yab91] E. Yablonovitch, T.J. Gmitter and K.M. Leung "Photonic band structures: The face centered cubic case employing non-spherical atoms in solid state physics and electronics", Phys. Rev. Lett. 67, 2295-98 (1991). [Yeh88] P. Yeh, Optical waves in layered media, J. Wiley and Sons eds., New York, 1988. [Zha90] Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations", Phys. Rev. Lett. 65, 2650-54 (1990)

33

Chapitre 2. Equivalence between a sub- gratings and a homogeneous film (lecture notes, Philippe Lalanne)

2. Equivalence between a sub- gratings and a homogeneous film

In the limit where the wavelength of light is very much greater than the dimensions of a structure, it is possible to regard the structure as being homogeneous and possessing an appropriate effective permittivity and permeability, see chapter 1. This static limit case is worth being known but is not implementable in practice. On the opposite, when the dimensions of the structure are close to larger than the wavelength of light the optical properties are dominated by the effects of diffraction. However, there is a region between these two extremes where the dimensions are sufficiently small that no diffracted orders propagate, but where it is not possible to apply the simple approximations of an homogeneous medium. This is often referred to as the "subwavelength domain" for which homogeneization techniques do not strictly apply but give however a good physical understanding of the medium properties. Most metamaterials and subwavelength gratings described in the chapter 3 will operate in the resonance domain. In the present chapter, we shall clarify under what conditions a subwavelength grating may or may not be replaced by a homogeneous, i.e. we shall clarify in what sense grating with subwavelength periods may been seen as equivalent to thin films.

2.1 Substitution of a subwavelength grating by an artificial layer Let us consider the diffraction problem shown in the left side of Fig. 2.1, where a subwavelength grating of depth ℎ is illuminated at oblique incidence by a linearly polarized plane wave with a free-space wavelength 𝜆. The refractive index of the incident medium is 𝑛𝑖 and that of the substrate is 𝑛𝑠 . We further assume that the grating permittivity is independent of 𝑧. In this Section, we seek to answer the following question: "under what conditions may the complex diffraction problem of Fig. 1a be approximated to by a simple refractionreflection problem on a homogeneous thin film?"

Figure 2.1 : Substitution of a subwavelength grating (a) of period 𝑎 by a homogeneous thin film (b), with the same thickness ℎ . The grating

1

Chapitre 2. Equivalence between a sub- gratings and a homogeneous film (lecture notes, Philippe Lalanne) permittivity is assumed to be independent of the 𝑧-direction that is perpendicular to the substrate. For the sake of illustration, a lamellar grating with fillfactor 𝑓 and permittivities 𝜀1 and 𝜀2 is shown.

2.1.1 The short period limit To answer this question it is instructive to start by considering that the grating period is much smaller than the wavelength of the incident illumination. Following the chapter 1, we know that there is a genuine equivalence between periodic artificial media and homogeneous material in the static limit. The grating is equivalent to a uniaxial or biaxial thin film with a thickness equal to the grating depth. For instance for 1D periodic structures, the ordinary 𝑛𝑜 and extraordinary 𝑛𝑒 indices of refraction are simply 𝑛𝑜 = 〈𝜀〉1/2 and 𝑛𝑒 = 〈1⁄𝜀 〉−1/2, where 𝜀 denotes the relative permittivity of the grating and the brackets refer to spatial averaging. The short-period (static) limit is an academic case only. With the present state-of-the-art in nanofabrication, it is only possible to manufacture structures with periods slightly smaller than optical wavelengths. For these real artificial media, one does not have in hand a theorem of equivalence between periodic structures and homogeneous media. On the contrary, the physical properties of real composite materials may sometimes strongly differ from those of homogeneous media. 2.1.2 Gratings with subwavelength periods For real gratings with a finite period, it is important to bear in mind that the substitution proposed in Fig. 2.1 can only ever be an approximation because in no real situations are the two problems in (a) and (b) strictly equivalent. We will be concerned in the following by an approximate physical equivalence especially valid for the far-field patterns. Consider the diffraction shown in (a). As the incident plane wave scatters at the upper interface of the grating, it excites a countable number of reflected diffraction orders and downward-propagating Bloch modes. The latter reach the substrate interface and excite upward-propagating Bloch modes and transmitted diffraction orders as well. The dynamical diffraction results from the bouncing back and forth of the upward- and downwardpropagating Bloch modes that may absorb some energy and distribute the incident energy between the different diffraction orders. In general, three conditions are necessary to achieve an approximate physical equivalence. Condition (I). The first condition requires that only the zeroth diffraction orders propagate in the substrate and in the incident medium, all the other higher diffracted orders have to be evanescent. Whether a diffraction order propagates or not is given by the grating equation. If only the zeroth transmitted and reflected orders are to propagate, it is immediately deduced from the grating equation that the following condition |𝒌// + 𝑮𝑚 | > 𝑚𝑎𝑥(𝑛𝑖 , 𝑛𝑠 )𝑘0 , where 𝑚𝑎𝑥(. ) holds for the maximum of the arguments, 𝒌// is the parallel wavevector of the incident plane wave, 𝑮𝑚 denotes any non-null vector of the Bravais grating lattice and 𝑘0 = 𝜔⁄𝑐 = 2𝜋⁄𝜆. For instance, for a 1D grating illuminated with an angle of incidence 𝜃, the grating period 𝑎 should be smaller than a geometrical cutoff value 𝑎𝑔 𝑎 < 𝑎𝑔 = 𝜆⁄(𝑚𝑎𝑥(𝑛𝑖 , 𝑛𝑠 ) + 𝑛𝑖 𝑠𝑖𝑛(𝜃)), which does not depend on the grating geometry and materials.

2

(2.1)

Chapitre 2. Equivalence between a sub- gratings and a homogeneous film (lecture notes, Philippe Lalanne) Condition (II). The second condition is related to the number of propagating Bloch modes that are able to propagate in the grating layer. One requires that only two Bloch modes (a single if polarization degeneracy is neglected) travels backward and forward between the two grating boundaries in the same way as multiple beam interference occurs in the equivalent thin film in (b). In sharp contrast with the previous condition, the fact that for a given frequency only two Bloch modes propagate in the grating (all the other are evanescent) depends mainly on the grating geometry and weakly on the diffraction geometry. It is actually independent of 𝑛𝑖 and 𝑛𝑠 and weakly depends on 𝒌// . This defines a second cutoff that could be classed as a structural cutoff 𝑎𝑠 , which is roughly proportional to the wavelength with a proportionality factor that depends on the grating materials and geometry. Condition (III). The last condition is related to the grating depth. If the latter is large enough, all the evanescent Bloch modes that are excited at the upper and lower grating interfaces and are exponentially damped as they propagate through the grating, do not tunnel through the grating region and do not participate to the multiple beam interference. For dielectric gratings, the impact of evanescent modes on the grating effective properties is significant for grating depths smaller than a quarter wave. Figure 2.2 sketches the field distribution of a grating that satisfy the three conditions. The far field diffraction pattern is composed of the reflected and transmitted plane waves (dashed arrow). The grey areas, just above and below the grating boundaries, represent the grating near-field zones, which are composed, in addition to the specular zeroth-order, by an infinite number of evanescent diffraction orders that exponentially decay away from the grating boundaries. In the middle of the grating, the electromagnetic field is solely composed of two counter-propagative Bloch mode, which are bouncing back and forth. In the yellowish zones inside the grating near the boundaries, the electromagnetic field is much more intricate as it additionally supports many evanescent Bloch modes.

Figure 2.2 : Sketch of lamellar subwavelength grating that satisfies the three conditions. Zone A. Condition (I). The zeroth transmitted and reflected orders are shown with dashed black arrows. Zone B. The field is altered by evanescent plane waves; zone B thickness reduced with the period. Zone C. The field is altered by evanescent Bloch modes; zone C thickness reduced with the period. Zone D. Condition (II). A single Bloch mode (in red) is bouncing back and forth and monitors the phase delay of the reflected and transmitted planes

3

Chapitre 2. Equivalence between a sub- gratings and a homogeneous film (lecture notes, Philippe Lalanne) waves. Condition (II). The evanescent Bloch-mode field is negligible if the grating depth is larger than zone C thickness

2.2 Subwavelength gratings supporting two propagating Bloch modes Figure 2.3 shows the reflectance of a 1D lamellar grating made of silicon ridges (𝜀1 = 10, 𝜀2 = 1) in air (𝑛𝑖 = 𝑛𝑠 = 1) illuminated under normal incidence with a polarization parallel to the ridges. The computation is performed with a fully-vectorial method, the rigorous coupled wave analysis.

Figure 2.3 Specular reflectivity map of a 1D lamellar grating in air (𝜀1 = 10, 𝜀2 = 1, 𝑛𝑖 = 𝑛𝑠 = 1, 𝑓 = 0.4) illuminated by a normally incident plane wave under TE polarization. The scale is linear, and the maximum reflectance (bright zone) is 100%. After [Lal06].

The right side of the figure illustrates the combined effect of the three conditions. Since polarization degeneracy is lifted, for 𝜆⁄𝑎 > 1.7, the reflectance pattern corresponds to FabryPerot fringes resulting from the cycling of a single fundamental Bloch mode between the two grating interfaces. For 𝜆⁄𝑎 < 1.7 , the grating supports at least two propagative Bloch modes, and the reflection fringe pattern is much less intuitive. In particular we note that the perforated semiconductor membrane in air exhibit a broadband high reflectance for some specific values of the grating depth. In addition, for 𝜆 = 𝑎, the reflectance pattern exhibits a cut-off for all depths, which corresponds to the passing-off of the positive and negative-first diffraction orders, the Rayleigh anomaly. For 𝜆⁄𝑎 = 1.7, another cut is visible in the reflectance pattern. Note that, in contrast with the Rayleigh cut-off, the cut is less conspicuous for small values of ℎ than for large ones. The phenomenon is similar to the Rayleigh cut-off, and corresponds to the passing-off of a propagative Bloch mode at large periods, which becomes evanescent at 𝜆⁄𝑎 = 1.7. Albeit evanescent, the Bloch mode participates in the energy transfer between the two grating interfaces by tunneling. For small grating depths, many evanescent Bloch modes participates to the transfer and the reflectance pattern presents a smooth variation with the period-to4

Chapitre 2. Equivalence between a sub- gratings and a homogeneous film (lecture notes, Philippe Lalanne) wavelength ratio. In contrast, as the tunneling distance ℎ increases, the energy transfer becomes less efficient, and the cut becomes more stringent. In the present case, the refractive index of the grating ridges is larger than the substrate one, and the Rayleigh anomaly is observed for a period 𝑎𝑔 that is smaller than 𝑎𝑠 , 𝑎𝑔 < 𝑎𝑠 . This holds in practice when a layer of high refractive index is patterned on a substrate of low refractive index (TiO2/SiO2 at visible wavelength or silicon on insulator at telecommunication wavelengths. When the grating is directly etched in the substrate, the geometrical cutoff is always larger than the structural cutoff, 𝑎𝑔 > 𝑎𝑠 . Even is the three conditions are fulfilled, the substitution remains qualitative, because the propagating Bloch modes strongly differ from plane waves, and their excitation and scattering also. However the analogy may serve as guidelines for initial design of metasurfaces etched at a subwavelength scale, as it sustains a drastic reduction of complexity. We note for example, applications exploiting the form birefringence of binary gratings for fabricating wave plates, wire-grid polarizers and filters, relying on continuous grating profiles for implementing gradient-index perpendicular to substrates for broadband antireflection coating, and mimicking gradient-index parallel to substrates for beam shaping. We shall now describe some of these applications.

2.3 Exercises: grating equation and zeroth-order condition 1/ We consider a 1D grating of period 𝑎 illuminated by a plane wave at an angle 𝜃 with respect to the normal axis. The superstrate has a refractive index 𝑛1 and the substrate a refractive index 𝑛3 .

Under which condition (on 𝐾 = 2𝜋/𝑎 and 𝑘0 = 2𝜋/𝜆 ) are the zeroth reflected and transmitted orders the only propagating orders? You might help yourself using the following sketch (here, in the case of transmission) and you will define 𝑘// .

2/Conical diffraction. We consider the same 1D grating as before, but illuminated in a conical diffraction configuration: the incidence plane is no longer normal to the invariance direction of the grating.

5

Chapitre 2. Equivalence between a sub- gratings and a homogeneous film (lecture notes, Philippe Lalanne)

Calculate, as a function of the angle 𝛿, the condition under which the zeroth reflected and transmitted orders are the only propagating orders. You might help yourself using the following sketch (here, in the case of transmission).

References [Lal06] P. Lalanne, J.P. Hugonin, P. Chavel, "Optical properties of deep lamellar gratings: a coupled Bloch-mode insight", IEEE J. Lightwave Technol. 24, 2442-2449 (2006)

6

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne)

3. Metamaterials and metasurfaces

Contenu 3.1 Introduction: The ideas behind "meta" ........................................................................ 1 3.2 Metamaterials with a negative index ........................................................................... 2 3.2.1 The dream of a perfect lens .................................................................................. 2 3.2.2 Electromagnetism of negative permittivity and permeability materials ............... 2 3.2.3 Snell’s law at negative-index material interfaces ................................................. 3 3.2.4 Flat lens with negative-index metamaterials ........................................................ 4 3.2.5 Seminal negative-index metamaterial demonstrations ......................................... 5 3.2.6 Negative-index metamaterial at optical frequency ............................................... 7 3.2.7 Hyperbolic metamaterials and superesolution .................................................... 11 3.3 Applications of subwavelength gratings and metasurfaces ....................................... 13 3.3.1 Phase plates......................................................................................................... 14 3.3.2 Wire-grid polarizers and slit-array filters ........................................................... 14 3.3.3 Anti-reflection coatings ...................................................................................... 17 3.3.4 Metalenses and metagratings .............................................................................. 18 References ................................................................................................................... 23

3.1 Introduction: The ideas behind "meta" The basic idea that helped coin the word "metamaterial", and spin-off concepts such as "metasurfaces" is to generalize the idea of homogenization of a composite system with the scope of producing averaged properties that cover all possible electromagnetic responses or a general material or a general surface, even if they do not exist from any bulk piece of matter. Thus, making a metamaterial amounts in a restricted sense to producing a slab of material that behaves with respect to impinging waves in a reasonable range of frequency and wave vector (angle/direction) as though it would have some desired values of permittivity and permeability. In particular, the possibility to get any positive or negative (or complex) values for these quantities gives food for thought. It is obviously a nice bridge between engineers with demands such as steering, filtering, routing beams for given needs, and, say, electromagnetism and optics scientists.

1

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) Historically, radar and microwave community considered "frequency selective surfaces" made of several layers of metallo-dielectric composites, and made great strides towards the general idea suggested above. About forty years after these efforts, the accumulated knowledge on nearby fields (such as photonic crystals and plasmonics) and the efforts to bridge the terahertz frequency gap fostered a more unified vision of electromagnetism that made the term "metamaterial" the successful one. Also, in 1968, Viktor Veselago considered what should be the electromagnetic waves in a general medium with any signs of both 𝜀 and 𝜇 [Ves68]. He arrived to non-intuitive conclusions on what should be taken as the index of refraction (reminding that 𝑛 = √𝜀 for non-magnetic materials). These considerations spurred rich investigations at the turn of the millennium. We are thus going to start this chapter with a class of artificial materials in which the electromagnetic properties, as represented by the permittivity and permeability, can be controlled in such a way that it is possible to achieve simultaneously negative permittivity and negative permeability, but whose anisotropy is at most a side effect. We will next examine the converse case of a strong effect of anisotropy, focusing on hyperbolic metamaterials. Finally, we will give a few views at a field where metamaterial concepts are widely exploited for feasibility reasons reminiscent of that those that favored two-dimensional photonic crystal: the so-called "metasurfaces".

3.2 Metamaterials with a negative index 3.2.1 The dream of a perfect lens We have seen that the radiation of a microscopic source, a dipole, in a uniform medium, has its information along one axis, say 𝑧 , split between propagative plane waves and evanescent waves that decay along 𝑧 making it hard to see this information outside the nearfield region. Can we pick up those decaying waves and transform them into a shape that retrieves as much as possible from the initial information? This task could be assigned to a material whose "refraction" properties have to be special enough to do this uncommon job. We are going to detail below how an hypothetic material, often referred to as a "perfect lens", with negative permittivity and permeability can help achieving this task.

3.2.2 Electromagnetism of negative permittivity and permeability materials Let us study a uniform material that possesses a negative relative permittivity and a negative relative permeability, 𝜀 < 0 and 𝜇 < 0. This choice does not cause mathematical contradictions; in particular it does not change the fact that, since the material is uniform, the electromagnetic modes are plane waves 𝐴exp(𝑖𝐤𝐫 − 𝑖𝜔𝑡) with a propagation constant 𝐤 . We omit the 𝜔𝑡 dependence hereafter. Given that Maxwell equations for such a mathematically clear situation takes the form 𝐤 × 𝐇 = 𝜔𝜀0 𝜀𝐄 and 𝐤 × 𝐄 = −𝜔𝜇0 𝜇𝐇,

(3.1)

one immediately sees that the vectors 𝐄, 𝐇 and 𝐤 constitute a left-handed set when both 𝜀 and 𝜇 are negative, in contrast with the usual case where they form a right-handed set for positive values. For this reason, these materials are frequently called left-handed materials. The wave vector 𝐤 tells us the direction of the phase velocity, while the Poynting vector 𝐒=𝐄×𝐇

2

(3.2)

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) always constitutes a right-handed set together with the vectors 𝐄, 𝐇. Considering that the group velocity is along the Poynting vector, we arrive at the conclusion that since 𝐄 × 𝐇 is opposite to the usual case in left-handed materials, group velocity and phase velocity are also opposite in such systems. Solving Eq. 3.1 is easy and the positive 𝜀𝜇 product naturally appears. Retaining the vector aspect, we can write down a version of the plane wave dispersion that includes a unit vector in the direction of energy flow 𝐒⁄|𝐒| 𝜔

𝐤 = −√𝜀𝜇 𝑐 𝐒⁄|𝐒|,

(3.3)

where the minus sign is the direct consequence of the opposite directions of 𝐤 and 𝐒. We just recall that for a standard material with 𝜀 > 0 and 𝜇 > 0, the very same relation holds but without the minus sign. It is appropriate to recall that the refractive index 𝑛 is not a material property, but a wave property, which tells us how plane waves, i.e. the modes (source-free solutions) of Maxwell equations here for a uniform medium, are propagating. Thus the classical expression "a material with a refractive index" is an inappropriate wording, and conveniently referring to the minus sign in Eq. (3.3), one may refer to materials with 𝜀 < 0 and 𝜇 < 0 as negative-index metamaterials with 𝑛 = −√𝜀𝜇.

(3.4)

3.2.3 Snell’s law at negative-index material interfaces Antiparallel phase and group velocities immediately affect Snell’s law. Let us now consider an interface between a right-handed (𝜀1 , 𝜇1 ) material and a right-handed (𝜀2 , 𝜇2 ) material. To match the field evolutions on both sides along the interface, the parallel momentum 𝐤 // must be conserved at the interface. Now, we must consider what are the outgoing waves in the left-handed material. To take energy out, they still given by the direction of the Poynting vector. The change of relative signs of the 𝐤, 𝐒 pair now implies a specific change in the Poynting vector: the refracted beam must be "mirror-imaged" about the normal to the surface, i.e. there is a reversal of the component of 𝐒 along the surface, compared to the usual case with a material with 𝜀 > 0 and 𝜇 > 0. We can say that Snell’s law still applies with the negative index on side 2, √𝜀1 𝜇1 sin(𝜃1 ) = −√𝜀2 𝜇2 sin(𝜃2 ), yielding in the case of Fig. 3.1 a negative sign for 𝜃2 .

Fig. 3.1 Negative refraction at the interface between a positive index material and a negative index material. The red thick arrows represent the direction of the Poynting vector, whereas the long black arrows that are parallel to the wave vector represents the direction of the phase velocity.

3

(3.5)

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne)

3.2.4 Flat lens with negative-index metamaterials The most striking example of what we can do with a negative-index material is Veselago’s flat lens, i.e. a thin slab with 𝜀 < 0 and 𝜇 < 0 and with a thickness 𝑑 immersed in a background with a positive permittivity −𝜀 and permeability −𝜇. According to Snell’s law, the angle of refraction is equal to the negative angle of incidence, hence all the rays emanating from a point source 𝐴 located at a distance 𝑧1 from the front side of the lens are deviated so as to converge to a point 𝐴" in the lens, before diverging again. Released from the medium, the light reaches a focus for a second time at a point 𝐴′ located at a distance 𝑧2 = 𝑑 − 𝑧1 from the right side of the lens. In this traditional ray picture (we are concerned with travelling waves here, not with evanescent ones), we have already a number of important properties:  the optical path from the external focus to the internal focus is zero; it is extremal like in classical lens design,  the lens has no optical axis,  the magnification is always 1,  the geometrical aberrations are null; the point to point correspondence is perfect,  the other relevant quantity, the impedance of the medium 𝑍 = 𝑍0 √𝜇 ⁄𝜀 (𝑍0 being the impedance of vacuum) retains its positive sign so that the lens is perfectly matched to the background and the interfaces exhibit zero reflection.

Fig. 3.2 Vesselago’s flat lens (thickness 𝑑). Light formerly diverging from a point source 𝐴 located at a distance 𝑧1 from the lens is set in reverse and converges to a point 𝐴" in the lens, before diverging and finally focusing for a second time at a point 𝐴′ located at a distance 𝑧2 = 𝑑 − 𝑧1 from the right facet of the lens.

An entirely new exciting idea came with the proposal by Sir Pendry [Pen00], who revisited Veselago’s flat lens and predicted that it is able to provide infinite resolution. Pendry showed that Veselago’s lens perfectly transmits evanescent waves, implying that every detail of the object is reproduced in the image plane. Thus the major difference between the perfect lens and Veselago’s lens is that the latter relies on geometrical optics, and hence on far field properties, whereas Pendry’s lens is designed for subwavelength imaging, and must therefore be concerned with evanescent waves. There are two ways to be convinced that perfect imaging occurs. In the first way, we may consider a 2 × 2 transfer matrix approach, just like in classical thin-film stack calculation, and incorporate evanescent fields with parallel momenta larger than 𝑘0 √𝜀𝜇, see details in exercise (??). Another way is to consider that, by virtue of the correspondence between material properties and coordinate transforms, the slab with negative 𝜀 and 𝜇 implements a coordinate 4

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) transform 𝑧 ′ = −𝑧, so that the field (for instance the parallel components 𝐸𝑥 , 𝐸𝑧 , 𝐻𝑦 for TM waves) located at a distance 𝑧 from the front side of the slab is perfectly reproduced at a distance 𝑧 from the front side of the slab inside the slab with a sign change for the transverse component 𝐻𝑧 , 𝐸𝑧 . In this vision, after two sign changes for the transverse components, the lens implements two real coordinate transforms that perfectly reproduce the field created by the source on the other side of the slab. The proposition of a perfect imaging system has raised many expectations, followed by as many disappointments. The following Sections will illustrate direct spin-offs, known as superlensing devices. Figure 3.3 illustrates the perfect-lens transport of the amplitude of an evanescent plane with a parallel momentum larger than 𝑘0 √𝜀𝜇 (with a transverse momentum 𝑘⊥ such that 2 𝑘⊥2 + 𝑘// = 𝜀𝜇𝑘02 ) decaying from an initial point 𝐴 wave, before growing up exponentially in the slab and then decaying again towards its initial value at 𝐴 in the image plane at 𝐴′. Note that the amplitudes in 𝐴 and 𝐴′ are strictly identical.

Fig. 3.3 Perfect lens. The field distribution in the perfect lens when the incident plane wave with a parallel momentum larger than 𝑘0 √𝜀𝜇 is evanescent. The field is first exponentially decreasing, before growing up in the lens, and finally decaying again towards the image plane. Note that the amplitudes in A and A’ are strictly identical.

3.2.5 Seminal negative-index metamaterial demonstrations In essence, the first demonstration of negative-index material results from the association of two resonances, one with electric dipoles and the other one with magnetic dipoles. To understand it at least qualitatively, it is important to first consider what is the essence of negative permittivity, and to refer to the simplest classical picture of atom-field interactions, the Lorentz Oscillator model. Within the model, the atom is thought as a mass (the nucleus) connected to another smaller mass (the electron) by a spring that is set into motion by an electric field interacting with the charge of the electron. The displacement of an electron is associated with a dipole moment, and the cumulative effect of all individual dipole moments of all free electrons results in a macroscopic polarization per unit volume 𝐏, and finally in an effective response described by a relative polarizability, yielding a relative dielectric permittivity 𝜀eff = 1 + 𝜔𝑝2 ⁄(𝜔02 − 𝜔2 + 2𝑖𝛾𝜔),

5

(3.6)

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) where 𝛾 is the damping constant describing mainly radiative damping in the case of bound electrons, 𝜔𝑝 the analogue of the plasma frequency in free electron gazes and 𝜔0 the resonance frequency of the atomic transition. We immediately see that, on the red side of a resonance, where the loss is maximum unfortunately, the relative permittivity is negative, suggesting that any resonance response, ∝ 1⁄(𝜔 − 𝜔0 ), is indeed associated to a sign change of the its representative parameter response. In general, the higher the quality factor, the narrower the spectral region with negative permittivity and the stronger the “negativity”. The idea to implement artificial magnetism with a negative permeability directly follows up from the Lorentz dipole model, replacing the resonant electric-dipole resonance by a magnetic-dipole one [Pen99]. The latter can be implemented by a circulating current on a closed loop as is well known from elementary electromagnetic, which leads to a magnetic moment with a magnitude given by the product of the current and the area of the loop and direction perpendicular to the plane of the loop. In Fig. 3.4 is depicted the famous so-called split ring resonator, a metal loop interrupted to form a resonant 𝐿𝐶 circuit with a resonance frequency 𝜔𝐿𝐶 = (𝐿𝐶)−1/2 completely controlled by the dimensions of the “cut loop”. It then becomes intuitively apparent that a periodic array, a crystal, formed by a collection of subwavelength split ring resonators, meta-atoms, may present a Lorentz oscillator magnetic response, with a negative effective permeability 𝜇eff on the red side of its resonance, see Exercise ?3.

Fig. 3.4 Illustration of the analogy between a usual 𝐿𝐶 circuit and a cut metallic loop, also called a split ring resonator formed with a plate capacitor 𝐶 = 𝜇𝜀0 𝑤𝑡⁄𝑑 (𝑤, 𝑑 and 𝑡 being the width of the metal ring, the width of the gap of the capacitor, and the metal thickness) and an inductance 𝐿 = 𝜇0 𝑙 2 ⁄𝑡, (𝑙 being the size of the ring); 𝜔𝐿𝐶 = (𝐿𝐶)−1/2. See exercise ?? for details.

In Fig. 3.5 is shown the first negative-index metamaterials demonstrated at microwave frequencies. The material idea consists in superimposing in a single structure two metamaterials known for providing negative permittivity and negative permeability at microwave frequencies, a metallic wire structure, which provides a predominantly freeelectron response to electromagnetic fields, and the split-ring-resonator structure, which provides a predominantly magnetic response to electromagnetic fields. Thus, at the heart of negative-index concept is a fundamental physics approach in which a continuous material, described by the relatively simple electromagnetic parameters 𝜀eff and 𝜇eff , conceptually replaces an inhomogeneous collection of scattering objects. In general, the continuous material parameters are tensors and are frequency dependent, but nevertheless represent a considerable reduction in complexity for describing wave-propagation behavior. The effective parameter response stems from the universal resonant response of a harmonic oscillator to an external frequency-dependent perturbation, in which the details of a material are replaced conceptually by a collection of harmonically bound charges, either electric or fictitious magnetic.

6

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) The microwave experiments were performed with an incident wave confined between two parallel metallic plates (the upper one is removed in the photograph of Fig. 3.6). In order to couple to the metal loops, the magnetic field has to be perpendicular to the plane of the loops, and in order to couple to the rods, the electric field has to be parallel with the rods, and of course the direction of propagation is perpendicular to the electric and magnetic fields. The main experimental results are shown in Fig. 3.6a. If only the spit-ring resonators are present (all metal rods are removed), the transmission spectrum shown with a solid curve reveals the existence of a stop band between the frequencies 4.7 and 5.2 GHz. Following the interpretation in [Smi00], the ring array acts as a material with a negative permeability in the stop band and the propagation is forbidden since 𝑛eff = √𝜀eff 𝜇eff with 𝜀eff ≈ 1 and 𝜇eff < 0 is imaginary. The metamaterial behaves as a magnetic metal, lower-right quadrant in Fig. 3.6c. When the rods are included (dotted curve), the stop band turns into a passband. Despite the considerable amount of absorption, the attenuation declines from 50 dBm to about 32 dBm. Again the interpretation is simple: the rod array acts as a material with a negative effective permittivity over the entire spectral range of interest, upper-left quadrant in (c), and when the effective permittivities and permeabilities are both negative, then 𝑛eff = −√𝜀eff 𝜇eff becomes real. The stop band turning into a pass band appears to provide a convincing support that a material with both material constants negative can propagate electromagnetic waves [Smi00]. As discussed in Fig. 3.1, if a metamaterial has a negative permittivity and a negative permeability, one should observe negative refraction. This beautiful initial demonstration was followed with a negative refraction Snell’s law experiment performed on a wedge-shaped (prism-like) metamaterial with a structure similar to that in Fig. 3.6b, see [She01] for details, and by other experiments where material losses were minimized and the structure presented a better impedance match to free space. These additional measurements have sufficed to convince most that materials with negative refractive index are indeed a reality.

Fig. 3.6 (a) First observation of a negative-index-material at microwave frequencies (dotted curve). The solid curve represents the transmission of the split-ring-resonator array only. (b) The metamaterial consists of a splitring-resonator array, created lithographically on a circuit board, and metallic post array. The period is 𝑎 = 8.0 mm, which corresponds to 1/8 of the resonance wavelength of the split rings. (c) The 4-quadrants of operation for a metamaterial, with either positive or negative 𝜀eff and 𝜇eff . After [Smi00].

3.2.6 Negative-index metamaterial at optical frequency In the late 2000, there has been a sustained effort in the community to push the operation frequency of the metamaterials deeper and deeper into the terahertz region, ultimately to reach 7

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) optical frequencies. However, in practice this has proved difficult. While it is easy to find negative permittivity materials (e.g., ordinary metals), the same is not true for negative permeability. Natural materials do not exhibit any magnetic response at such high frequencies, and the creation of artificial magnetism at optical frequencies is challenging. Although the lump-element model developed in exercise 2 shows that the magnetic resonance wavelength 𝜆𝐿𝐶 scales with the structural size, 𝜆𝐿𝐶 = 2𝜋√𝜀𝑐 𝑙(𝑤 ⁄𝑑 )1/2 , this linear scaling breaks down at higher frequencies. The mean reason comes from the scaling of the other typical characteristic length that should be considered with metals as one scales the dimensions down to the visible, the frequency dependence of the skin depth, 𝛿 = 𝜆⁄(2𝜋√𝜀𝑚 ) . Inserting the expression of the metal relative permittivity 𝜀𝑚 (𝜔) = −𝜔𝑝2 ⁄𝜔2 given by the Drude model at high frequencies with 𝛾(𝜔) ≪ 𝜔 < 𝜔𝑝 , we find that the skin depth becomes a constant, independent of the wavelength. Thus, as we scale down all the dimensions of metallic-loop resonators down to the visible, the electromagnetic field largely penetrates in the metal and the resonator no longer works as in the long wavelength regime with a field essentially null inside the metal. We already mentioned issues related to homogenization for metal structures in Section 15.3.4(??) of the previous(??) chapter. Perhaps the most convincing demonstration of negative-index metamaterials operating in the visible and telecom ranges of the spectrum with a thick sample (a real metamaterial, not an optically thin sample) is the one achieved in 1998 in Berkeley [Val08], see Fig. 3.6. Abandoning the traditional split-ring-resonator geometry, the demonstration is based a 3D optical metamaterial made of cascaded ‘fishnet’ structures obtained by etching a hole array in alternating layers of silver and magnesium fluoride. Tested in a prism-like experiment (Fig. 3.6b), negative refraction is unambiguously observed over a broad spectral range for 𝜆 > 1500 nm, see Fig. 3.6c. What is remarkable in this experiment is that, in sharp contrast with earlier experiments mainly limited to optically thin samples because of significant fabrication challenges and strong energy dissipation in metals, this observation is made from transmission measurements with a thick 21-layer sample. As shown in the right panel of Fig. 3.6c, the experimental measurements of the negative index 𝑛 (as deduced from the Snell’s law) pretty well match fully-vectorial computational results of the normalized propagation constant 𝑛𝑒𝑓𝑓 = 𝑘𝑧 ⁄𝑘0 (𝑘𝑥 = 𝑘𝑦 = 0) of the fundamental Bloch-mode of the fishnet. Another important feature is the remarkably-high figure of merit, 𝐹𝑂𝑀 = 2Re(𝑛𝑒𝑓𝑓 )⁄Im(𝑛𝑒𝑓𝑓 ) = 3.5 (large FOM corresponds to low loss) that is predicted numerically. Because of Ohmic losses, it has been challenging to design NIMs for optical frequencies (visible and telecom windows) and initial efforts have been mainly limited to optically thin samples of SRRs because of significant fabrication challenges and strong energy dissipation in metals, see the review article [Sha07]. Such thin structures are analogous to a few layers of atoms, making it difficult to assign bulk properties such as the index of refraction.

8

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne)

Fig. 3.6 Demonstration of negative refraction at telecom frequencies with fishnet metamaterials. (a) Diagram and SEM image of the 21-layer fishnet structure fabricated in [Val08], 𝑝 = 860 nm, 𝑎 = 565 nm and 𝑏 = 265 nm. The structure consists of alternating layers of 30-nm silver (Ag) and 50-nm magnesium fluoride (MgF2). (b) Experimental setup used to observe negative refractive results obtained with a wedged prism-like fishnet metamaterial. (c) Left: Images of the beam on the CCD camera for various wavelengths. The horizontal axis corresponds to the beam shift, and positions of 𝑛 = 1 and 𝑛 = 0 are denoted with white lines. Right panel: Measurements and simulation of the fishnet refractive index. The circles show the results of the experimental measurement. The solid black curves shows the real part of the effective index (the normalized propagation constant along the normal to the layers) of the fundamental Bloch-mode of the fishnet for 𝒌// = 0. After [Val08].

It is instructive to understand the physical origin of the negative effective permittivity in the experiment. The fishnet is composed of two intersecting subwavelength channels that couple and exchange energy. The negative effective permittivity is attributable to the longitudinal (𝑧-direction) channel consists of air holes in a metal film, which transport energy through resonant tunneling. Conversely, the negative effective permeability is attributable to the transversal ( 𝑥 - or 𝑦 -direction) channel formed by metal-insulator-metal (MIM) waveguides that support the propagation of gap plasmon modes. This mode that was studied in detail in the slow-light Section of chapter ?? plays a crucial role in nanophotonics; we will meet again at the end of the chapter when studying wire-grid polarizers. Figure 3.7 shows an 9

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) enlarged view of a MIM stack with a finite length. We also depict the symmetric profile of the 𝑦-component of the magnetic field of the gap plasmon mode. Directly from the Maxwell curl equations, it follows that the 𝑥-component of the electric field is antisymmetric, so that the electric current due to the electron motion is also antisymmetric. Either using a Fabry-Perot picture in which the gap plasmon mode is bouncing back and forth between the facets, or with a lumped-elements model associating a capacitance to the two facets and an inductance to the cut loop, it becomes intuitively clear that the transversal channels of the Fishnet structures behaves as split-ring resonators producing negative artificial magnetism.

Fig. 3.7 Implementation of current-loop resonators with thin metalinsulator-metal stacks. The black curve shows the symmetric 𝑦-component of the magnetic field of the gap plasmon mode formed by the coupling of the surface plasmons of the two metal-dielectric interfaces. The red arrows 𝜕𝐻

show the associated 𝑥 -component 𝐽𝑥 ∝ 𝑦 of the induced current, which 𝜕𝑧 is antisymmetric with respect to the plane 𝑧 = 0. The finite length MIM is also called “cut wire pairs” in the related literature.

However great and convincing it may be, the demonstration of negative-index metamaterials at telecom or visible frequencies has not produce a genuine impact on applications. The literature has extensively mentioned the critical issue of Ohmic losses in metals at high frequencies, often trying to dope metamaterials to compensate absorption with amplification, but this is not the sole reason. Perhaps more fundamentally, it should be noted that the design of a metamaterial has to follow challenging requirements, e.g. large resonance are needed inside the unit cell to build a large magnetic response from materials that are nonmagnetic by essence, the interaction between every individual atom should remain weak so that effective properties can be simply inferred by spatially averaging at a characteristic length much larger than meta-atom size and weaker than the wavelength. This is difficult (maybe impossible) to meet in practice, and somehow has never been realized. For instance, even for the beautiful fishnet structure of Fig. ?6a, it can be shown that the magnetic resonance arises from a transversal MIM resonance which involves 4-5 unit cells. If doping to compensate absorption, it is not 4-5 cells that are involved, but 20. In other words, the fishnet structure is far from what can be described as an artificial media. Its optical properties are more related to those of subwavelength gratings, with a huge spatial dispersion, rather than a collection of meta-atoms that reproduce the properties of every individual atom by ensemble averaging. Similar observations will be reached for some metasurfaces studied at the end of the Chapter.

10

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne)

3.2.7 Hyperbolic metamaterials and superesolution In this Section, we consider highly anisotropic metamaterials, known as hyperbolic metamaterials. They display hyperbolic dispersion, which originates from one of the principal components of their electric or magnetic effective tensor having the opposite sign to the other two principal components. We will consider the special case of non-magnetic metamaterials for which all the diagonal components of the effective permeability tensor are 𝜇0 and the effective permittivity tensor is thus 𝜀𝑒𝑓𝑓

𝜀// =[0 0

0 𝜀// 0

0 0], 𝜀⊥

(3.7)

with 𝜀⊥ > 0 and 𝜀// < 0 for instance. Here, the subscripts // and ⊥ indicate components parallel and perpendicular to the anisotropy axis. These media have been called indefinite materials, a denomination that emphasizes that not all the eigenvalues of the constitutive tensors have the same sign. In vacuum, the linear dispersion and isotropic behavior of propagating modes implies a spherical isofrequency surface given by the equation 𝑘𝑥2 + 𝑘𝑦2 + 𝜔2

𝑘𝑧2 = 𝑐 2 , where 𝑘𝑥 , 𝑘𝑦 and 𝑘𝑧 are respectively the 𝑥, 𝑦 and 𝑧 components of the wave vector of the mode. The interesting properties of indefinite materials stem from the isofrequency surface of extraordinary (transverse magnetic polarized, 𝐸𝑧 ) modes, which is given by 2 𝑘𝑥2 +𝑘𝑦

𝜀⊥

𝑘2

𝜔2

//

𝑐2

+ 𝜀𝑧 =

.

(3.8)

The spherical isofrequency surface of vacuum usually distorts to an ellipsoid for anisotropic cases. However for 𝜀⊥ 𝜀// < 0, we have extreme anisotropy and the isofrequency surface opens into an open hyperboloid, see Fig. 3.8a. Such a phenomenon requires the material to behave like a metal in two directions of polarization and a dielectric (insulator) in the other. This does not readily occur in nature at optical frequencies but can be achieved using artificial nanostructured electromagnetic media, as shown by the results of the static-limit homogenization of composite materials in Chapter 15?3. The most important property of such media, called hyperbolic media, is related to the behavior of waves with large wave vector magnitudes. In vacuum, such large wave vector waves are evanescent and decay exponentially. However, in hyperbolic media the open form of the isofrequency surface allows for propagating waves with infinitely large wave vectors. For instance if 𝑘𝑥 or 𝑘𝑦 becomes increasingly large, much larger than 𝜔⁄𝑐 for instance, we find that 𝑘𝑧2 > 0 in Eq. 3.8 since 𝜀⊥ 𝜀// < 0 , implying that a mode may propagate in the anisotropic direction with infinitely large transverse momenta. Thus there are no evanescent waves in such a medium. This idealistic property of propagating high- 𝑘 modes has led to a multitude of device studies, especially for near-field imaging in relation with the perfect lens.

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Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne)

Fig. 3.8 Hyperbolic metamaterials. (a) Hyperboloid isofrequency surface for a uniaxial medium with an extremely anisotropic dielectric response, 𝜀⊥ > 0 and 𝜀// < 0 . Hyperboloid surface are also obtained for the reciprocal case of 𝜀⊥ < 0 and 𝜀// > 0. (b) Simple and classical example of hyperbolic metamaterials implemented with alternate layers of metals and insulators. For any arbitrarily large in plane wave vectors, 𝑘𝑧 remains positive in the perpendicular direction. The whole figure assumes an idealistic case without Ohmic losses.

It is considerably easier to produce hyperbolic structures than isotropic metamaterials that possess both negative permittivity and negative permeability, as the only essential criterion for hyperbolic structures is that the motion of free electrons be constrained in one or two spatial directions. Perhaps the simplest hyperbolic structure consists in a stack of alternate layers of dielectric and metallic materials, with 𝜀// = < 𝜀 > and 𝜀⊥ = < 1⁄𝜀 >1/2 behaving as a metal and a dielectric, respectively. Due to the ease of fabrication, they gave rise to many studies for radiative decay enhancement, heat transport, and enhance absorption. One particular implementation of hyperbolic metamaterials is their use for super-resolution imaging. Figure 3.9 shows the first demonstration of near-field imaging with an elementary hyperbolic metamaterial, an insulator/(35-nm-thick)silver/insulator stack [Fan05]. In the experiment, series of narrow slits forming arbitrary patterns are inscribed onto a 50-nm-thick chrome (Cr) mask. The mask is covered by a 40-nm PMMA spacer layer and the silver film. It is illuminated from the rear side by a mercury lamp emitting at 365 nm. The image of the object is recorded in the photoresist (PM) on the other side of the silver superlens, and is revealed by development. The pattern inscribed in the developed photoresist is then observed with an AFM. Figure 3.9b shows the AFM image obtained on the front side of the photoresist with the silver superlens for an arbitrary object ‘‘NANO’’ is shown in Fig. and reproduced with a high fidelity. In [Fan05], a resolution of 90 nm (𝜆⁄4) is claimed.

12

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne)

Fig. 3.9 Superlensing with hyperbolic metamaterials (in the present case, simply, an insulator/metal/insulator stack). (a) Optical experiment. (b) Image formed by the insulator/silver/insulator “superlens” of an arbitrary object “NANO’’ etched in a chrome (Cr) mask, as revealed by the AFM scan of the photoresist (PR) after development. After [Fan05].

Most of the applications of hyperbolic media may be understood in terms of Fermi’s Golden Rule which claims the scattering rate into a particular mode is proportional to that mode’s density of state. Hence, for a scatterer placed in the vicinity of an hyperbolic medium, e.g. the chrome mask in the previous experiment, light will preferentially excite the metamaterial. Fundamentally, the mode that plays a fundamental role is the dominant Bloch mode of the metamaterial, the supermode formed by the coupling of the gap plasmon modes (see Fig. 3.7) of every dielectric layer in the stack. For the superlensing experiment, the dominant mode is the fundamental mode of the insulator/silver/insulator stack, which is formed by the coupling of the surface plasmon modes of the two insulator/metal interface. This mode has properties much similar to those of gap modes. It is symmetric like in Fig. 3.7, has no cut-off and high- 𝑘// values as the metal thickness decreases. However, we should emphasize that, because of Ohmic losses in the metal, the high- 𝑘// modes all possess a large imaginary part. In Chapter ??, we showed that both Re(𝑘// ) and Im(𝑘// ) scale with the dielectric gap thickness for metal/insulator/metal stacks, or with the metal thickness for insulator/silver/insulator stacks. Thus the most attractive property of hyperbolic metamaterial is upset by Ohmic losses, as the high- 𝑘// modes do not propagate; they are actually confined by loss. Actually what’s happen is that, when considering loss in Eq. 3.8 with realistic values for 𝜀// and 𝜀⊥ (taking into account absorption), 𝑘// ≡ 𝑘𝑥 , 𝑘𝑦 becomes complex and consequently, so does 𝑘𝑧 . The idealistic transfer of high- 𝑘// waves, as suggested by the hyperboloid dispersion diagram, is not observed in reality.

3.3 Applications of subwavelength gratings and metasurfaces The basic principles of optical design and the physics of reflection, refraction and diffraction on which it is based, have been well understood for a very long time. This knowledge has enabled the successful development of optical science and technology over the last couple of centuries and in recent years has been developed to new levels of sophistication 13

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) with the availability of computer programs for the optimization of all types of systems. However, until relatively recently the whole of optical technology has been limited by the very reasonable constraint that optical systems can only be designed to be made from materials that are actually available. This limitation can be overcome by considering composites, periodic or not, with subwavelength features, which synthesize new materials with properties that are not naturally available. The possibilities offered by these new materials have been explored as early as the 1980’s with the first success of microtechnologies in optics, and has been a boosted under the name of metasurfaces for various applications with the deployment of nanotechnologies . However, in electromagnetism, the question of the use of subwavelength features to tailor the propagation of electromagnetic waves for instance at microwave frequencies in high effective-index artificial dielectrics obtained by doping polystyrene foam sheets with subwavelength metallic insets, has a venerable history, which started well before the advent of nanophotonics. In this Section, we shall describe a set of tutorial applicative examples starting from subwavelength gratings to gradually move to metasurfaces: form birefringence with binary gratings for manufacturing wave plates, wire-grid polarizers and filters, vertically gradedindex gratings implementing broadband antireflection coating, and laterally graded-index metasurfaces for lensing, beam shaping and near-perfect absorption.

3.3.1 Phase plates The form birefringence of artificial dielectrics is ways larger than that of naturally available ones. Hence, in polarization phase-plate settings, binary subwavelength gratings composed of tiny dielectric wires may provide phase retardation that are substantial even for subwavelength grating depths. This is easily realized from the averaging discussed in chapter 1 with the static-limit formulations: for a two-medium lamellar with relative permittivities 𝜀1 and 𝜀2 and fillfactor 𝑓, one gets 1/2

𝑛𝑜 = ((1 − 𝑓)𝜀1 + 𝑓𝜀2 )

, and

(3.9a)

𝑛𝑒 = ((1 − 𝑓)⁄𝜀1 + 𝑓/𝜀2 )−1/2.

(3.9b)

A phase plate producing a phase birefringence 𝜑 can be implemented with a grating depth ℎ 𝜑 𝜆 given by ℎ = 2𝜋 𝑛 −𝑛 . For quarter-wave elements, 𝜑 = 𝜋⁄2, the birefringence achieved for 𝑜

𝑒

UV-molding materials of high refractive indices, 𝜀1 = 1 (air) and 𝜀2 ≈ 3 is large, and the grating thickness is small enough so that phase plates can be manufactured at very low cost. The use of high-index semiconductor materials lead to highly-chromatic dispersion for 𝑛𝑜 and 𝑛𝑒 . By clever design, however, it is possible for the phase retardation of a given 2𝜋ℎ composite slab of thickness h, Δ𝜑 = 𝜆 (𝑛𝑜 (𝜆) − 𝑛𝑒 (𝜆)), to remain constant over a broad spectral interval thanks to a good use of the composite 𝑛𝑜,𝑒 (𝜆) dispersion. For instance, by etching a binary subwavelength grating into a silicon substrate, quarter-wave phase elements exhibiting a phase retardance of 𝜋/4 can be manufactured for operation over the 3.5–5.0 µm wavelength range [Deg01].

3.3.2 Wire-grid polarizers and slit-array filters The wire-grid polarizer was probably the earliest device to exploit the form birefringence of subwavelength metallic gratings as it was used by Hertz to test the properties of the newly discovered radio wave in the late 19th century. It consists of a fine grid of parallel metal wires 14

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) with a spacing smaller than the wavelength of light. Thus, the electrons are free to oscillate along the wires but not perpendicularly because the wire width is smaller than the wavelength. Therefore for light polarized parallel to the wires (TE polarization case), the grid behaves as a conductor and it reflects, whereas for light polarized normal to the wires, it cannot behave so and tends to transmit. This result may simply be derived by inserting into Eqs. (3.9a) and (3.9b) a large negative value for the relative permittivity 𝜀2 =𝜀𝑚 ≪ 0 of the metal and assuming 𝜀1 = 1. Then one gets 𝑛𝑜 ≈ (𝑓𝜀m )1/2, and

(3.10a)

𝑛𝑒 ≈ ((1 − 𝑓)⁄𝜀1 )−1/2.

(3.10b)

Taking 𝑓 = 0.5 for instance, it appears that waves polarized along the wires see a good metal with a large and negative permittivity 𝑛𝑜2 ≈ 𝜀𝑚 ⁄2, whereas waves polarized perpendicular to the wires see a low index material with a refractive index ≈ 1.4, 𝑛𝑒2 ≈ (1⁄2)−1/2 . There are various ways of producing wire-grid polarizers. For radio waves and microwaves up to the far infrared, it is possible to wind a wire around a suitable frame (like a painting frame) and produce a free-standing grid. For shorter wavelengths in the near infrared, obtaining a device of substantial area makes it necessary to form a series of very fine parallel wires on a transparent substrate. The shortest wavelength at which a grid works as a polarizer being dictated by its period, progresses in lithography of finer patterns have led to wire grids being manufactured in the visible (10), and even in the ultraviolet. In this spectral window, dichroic materials providing a preferential absorption of light in particular polarization directions are rather missing. In reality, the principle of operation of wire-grid polarizers is substantially different from the naive homogenization picture. To be approximately valid, Eqs. 3.10a and 3.10b require that the dominant Bloch modes resemble plane waves. For dielectric materials, such as polymers or even semiconductors, this is approximately valid for wavelength-to-period ratio 𝜆/𝑎 equal or greater than ≈ 2 − 3, a statement to be refined in the next sections. For metallodielectric composites, it is additionally required that Bloch modes penetrate the metal almost uniformly. Another scale then plays a key role: the skin depth, which turns out to be equal to ≈ 20 nm for noble metal in the visible and near infrared. The metal wire cross-sections are much larger, even for wire grid polarizers operating in the visible, so that the genuine Bloch-mode homogenization regime is only reached for very tenuous wires with widths smaller than the skin depths, which in practice does not occur. So for most wire-grids, the electromagnetic field is essentially null in the metal and light is effectively funneled into the slits between the wires. Figure 3.10 shows the transmission spectra of a gold metal plate perforated with tiny subwavelength slits (0.5 μm) in the infra-red (period 𝑎 = 3.5 µm) for TM polarization (magnetic field parallel to wires). The data, computed with the RCWA, confirm that the electromagnetic field is indeed null in the metal ridges, which are much larger than the skin depth, and exhibit transmission peaks that are much larger than the normalized aperture of the slit, evidencing that the incident light is efficiently funneled into the slit as sketched by the chicane-shaped arrows. What happens is that the incident light excites the fundamental mode of the slit (the gapplasmon mode with a normalized propagation constant close to 1, see chapter ?? and Fig. 3.6 above). The mode bounces back and forth between the upper and lower interfaces, just like in a Fabry-Perot resonance [Ada79]. Therefore every individual slit behaves as a resonator with 15

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) a cross-section larger than the slit aperture, on the order of one wavelength at resonance (in 1D cross-section are a length). Then it is not surprising that an array with a subwavelength slit-to-slit distance almost fully transmits light at resonance. This is exactly what is happening at point A of Fig. 3.10(c).

Fig. 3.10 Principle of operation of subwavelength wire gratings. (a) Sketch of the geometry. The grating is composed of a periodic array of tiny slits in a metal film. (b) Funneling effect at resonance (point A in (c)) illustrated by the distribution of the electromagnetic fields intensity (in grey) at the entrance of an illuminated slit. The white arrows indicate the direction and norm of the Poynting vector. (c) Transmission spectrum for a TM-polarized normally incident plane wave. The computation is performed for gold wires of width 3 µm and height 4 µm. The slit width is 0.5 µm. The resonance in A does not shift as the angle of incidence is varied.

We further understand why the resonance wavelength does not disperse when the incident plane wave is tilted, or when the position of the slits are randomly and independently shifted. The slits act as independent waveguides that capture the incident energy at resonance, and collectively they almost fully transmit light. For TE polarization, the fundamental slit mode is cutoff for slit widths smaller than half the wavelength so that the dominant Bloch mode is evanescent. The incident light is exponentially damped (as predicted by Eq. 3.4, but with a wrong attenuation), and light is essentially reflected. In general, the larger the wavelength-to-period ratio and the thicker the grating depth, the better the wire-grid polarizer performs. Large thicknesses guaranty a high contrast since TE-polarization transmission becomes very small. However, the TMtransmission is degraded because of Ohmic losses in the metal, especially in the visible and the ultraviolet regions of the spectrum. The other feature seen in the spectrum is resonance peak B. Unlike peak A, this peak B does evolve (it “disperses”) as the angle of incidence of the incident plane wave is varied. What is happening is the following: Part of the incident energy that is scattered by the slit aperture is coupled into the slit, but not all of it, leaving a significant part not reflected but coupled onto the surface, where it propagates up to the adjacent slits and couples into them. This happens also at point A. However, when this light that is re-coupled from the surface is in phase with the light that is directly coupled from the incident wave (this happens only at 16

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) point B when the grating equation is approximately satisfied, 𝑘// ± 2𝜋/𝑎 = ±𝑘0 ), the slits becomes coherently coupled via the field scattered by the interface. This coupling has been attributed extensively to surface plasmons in the literature, but surface plasmons only play a casual role, which is even not predominant in the visible and negligible at infrared frequencies, like in the example of Fig. 3.10.

3.3.3 Anti-reflection coatings The feature of an optical surface which gives rise to unwanted Fresnel reflections is the sudden transition, or impedance mismatch, from one optical medium to another. If the transition can be made more gradual and extended over at least a significant fraction of a wavelength, the reflection can be significantly reduced. This has been achieved in glass by treating the surface with acid. Under the appropriate circumstances, material may be leached out of the glass to leave a more open structure in the region of the surface and a gradually more dense structure as one penetrates into the glass. As this occurs at a molecular level it is on a scale much finer than the wavelength of light and is equivalent to a gradual change of refractive index and the reflection of the surface is reduced to a level as low as that achieved with very complex multilayer antireflection coatings. An analogue of the leached glass antireflection surface is to be found on the eye of the night-flying moth. The cornea is covered with a fine regular hexagonal array of "egg box" protuberances which have a period of about 200 nm and a similar depth and a cross section that is approximately sinusoidal. This natural corrugation was discovered by Bernhard in 1967 who postulated [Ber67] that it results in a gradual change of refractive index which reduced the reflection over a wide spectral and angular bandwidth and significantly improved the moth's camouflage. The problem of surface reflection is particularly acute for semiconductors which have very high values of refractive index in the visible and near infrared spectral regions. The reflection losses at normal incidence are  40%; to reduce them would be of interest to maximize absorption for solar-cell applications and other optoelectronic devices. Figure 3.11a shows the reflectivity of a silicon substrate with high aspect-ratio pyramid arrays etched on the front interface. The reflectivity under illumination at normal incidence does not exceed 4% over 4 octaves starting from 200 nm, a level as low as that achieved with very complex multilayer antireflection coatings. Note that the abrupt reflectivity increase at 𝜆 = 1100 nm is not due to the pyramids. For incident beams with energies above 1.1 eV, the silicon becomes transparent and the beam is reflected on the rear interface of the silicon substrate. For structured films with periods only slightly smaller than the wavelength, accurate design should consider reflection due to effective refractive index mismatch like in gradedindex films, but also the reflection due mode profile mismatch, which is often the dominant effect (Fig. 3.11b). Quite apart from their optical performance, structures of this type offer the additional advantages that there are no problems with adhesion or with diffusion of one material into another. Since they are monolithic and introduce no foreign material, they also tend to be more stable and durable than multilayer dielectric materials particularly when used with highpowered lasers.

17

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne)

Fig. 3.11 (a) Antireflection behavior of a square array of 350-nm-high pyramids etched into silicon. The grating period is 150 nm. The inset shows a scanning electron micrograph of an antireflection surface generated with an electron beam writer. For comparison, the reflectivity of bare silicon substrate is shown with the dashed curve. (after [Kan99]). (b) Progressive Bloch-mode profile transformation in pyramid arrays computed with the RCWA, from plane wave in air to plane wave in silicon (blue curves, with corresponding effective index values indicated).

An analogue of the leached glass antireflection surface is to be found on the eye of the nightflying moth. The cornea is covered with a fine regular hexagonal array of “egg box” protuberances that have a period of about 200 nm with a similar depth, and a cross-section that is approximately sinusoidal. This natural corrugation was discovered by Bernhard in 1967 [Ber67] who postulated that it results in a gradual change of refractive index that reduced the reflection over a wide spectral and angular bandwidth and significantly improved the moth’s camouflage. The problem of surface reflection is particularly acute for semiconductors on account of their very large index of refraction in the visible and near infrared ranges. The reflection losses at normal incidence are up to ≈ 40% for Si, Ge or GaAs. Reducing them would be of interest to maximize absorption for solar-cell applications and other optoelectronic devices. Figure 3.11(a) shows the reflectivity of a silicon substrate with high aspect-ratio pyramid arrays The reflectivity under illumination at normal incidence does not exceed 4% over 4 octaves starting from 200 nm, a level as low as that achieved with very complex multilayer antireflection coatings. For structured films with periods only slightly smaller than the wavelength, accurate design should consider reflection due to effective refractive index mismatch like in gradedindex films, but also the reflection due mode profile mismatch, which is often the dominant effect (Fig. 3.11(b)). Quite apart from their optical performance, structures of this type offer the additional advantages that there are no problems with adhesion or with diffusion of one material into another. Since they are monolithic and introduce no foreign material, they also tend to be more stable and robust than multilayer dielectric materials particularly when used with highpower lasers and high fluence pulses.

3.3.4 Metalenses and metagratings Index-gradients may also be manufactured parallel to substrate. We then enter the field of the so-called 'flat optics' that shapes the phase of an incident wave in free space through subwavelength structures, a concept that is parent to that of holograms. We shall start by 18

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) briefly recalling the principle of operation of échelette-type diffractive optical elements with their limitations, and then switch to metalenses and metagratings composed of nanoelements, considering first those based on propagation phase-delays with nanowaveguide arrays and eventually those based on resonance phase-delays with nanoresonator arrays. Echelette diffractive optical elements. Conventional diffractive optical components with “échelette” profiles, such as blazed échelette gratings depicted in Fig. 3.12 or Fresnel lenses, implement a 2𝜋-phase delay across the component surface by a 𝜆0 ⁄(𝑛 − 1) excursion of the diffractive surface thickness. Nowadays, échelette-type diffractive optical elements are manufactured at low-cost by replication technologies such as embossing, molding and casting. Those technologies offer very high resolution, typically in the nanometer range, and allow the fabrication of large area, complex microstructures with minimal light loss in durable materials through high volume industrial production processes. However, they suffer three main limitations: 

The discontinuities of the wrapped phase introduce a shadow that wastes light into spurious orders. The effect is stringent for short period échelettes, to an extent that the échelette approach is essentially useless for high-NA lenses, see Fig. 3.12a.



The wrapping relies on a nominal design wavelength 𝜆0 for which the 2𝜋 -phase excursion is achieved, implying that the imprinted phase varies with the wavelength of the incident light and thus the efficiency 𝜂 decreases as the illumination wavelength 𝜆 departs from 𝜆0 , 𝜂 = sinc 2 (1 − 𝜆0 ⁄𝜆).



Not only does the efficiency of a diffractive lens varies with the wavelength, but so does its focal length, 𝑓(𝜆2 )𝜆2 = 𝑓(𝜆1 )𝜆1 . Actually, the focal length decreases with wavelength, in sharp contrast with refractive lenses based on various glasses for which the refractive index (resp. focal length) decreases (resp. increases) with wavelength. It is this complementarity that is exploited in achromatic hybrid doublets to reduce the weight and size of optical systems [Sto88].

Fig. 3.12 Metagratings with efficiencies larger than those of échelette gratings. (a) Ray tracing picture of the shadowing effect in blazed échelettetype gratings. The effect illustrated for normally incident beams worsens at oblique incidence. (b) First-order transmission efficiencies of metagratings as a function of the deviation angle in air (the grating period is varying) for a normally-incident and unpolarized plane wave at 𝜆 = 633 nm. Circles:

19

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) experimental results. Red curve: theoretical results. Blue solid curve: theoretical efficiency of échelette gratings in glass, illustrating the drastic effect of the shadow on the efficiency as the shadow zone on the diffraction increases. The lower inset is an electron micrograph of a section of the grating operating for a 20° deviation angle; the period is 1.9 µm, the pillar aspect ratio reaches 4.6. After [Lal17].

Optical phase arrays with metasurfaces. Overall, two main physical effects have been exploited to implement the required 2𝜋-phase excursion with flat optics. In the first approach explored in the 90’s, the phase excursion is imprinted by modulating the effective index 𝑛𝑒𝑓𝑓 of artificial dielectrics at a constant thickness ℎ 𝛥𝜑 = 𝑘0 (𝑛𝑒𝑓𝑓 (𝑥, 𝑦) − 1)ℎ.

(3.11)

Such gradients that are parallel to substrates can be implemented with metasurfaces composed of subwavelength nanopillars or nanoholes of graded widths. Figure 3.12b summarizes the main results obtained in the 90’s for a series of gratings operating in the visible with increasing deviation angles (i.e. decreasing periods). A remarkable property is that the efficiency drop of échelette gratings at large deviation angles (solid blue curve) no longer occurs. Indeed every TiO2 nanopillar behaves as an independent monomode waveguide that confines light and guides it vertically throughout the metasurface, thus removing the shadow effect. The waveguiding effect also guaranties that the imprinted phase is independent of the parallel wave vector of the incident beam, implying that the same pillar structure can be used for various incidence of the illumination, effectively focusing obliquely incident plane waves or collimating diverging beams [Lal17]. In the second approach explored in the 2010’s, nanoresonators instead of nanowaveguides are used to imprint the phase excursion. As light excites a resonator, the is always proportional to (𝜔 − 𝜔𝑅 )−1, 𝜔𝑅 being the complex resonance frequency of the resonator, see Chapter ?. This implies that by progressively varying the shape of a series of subwavelength resonators etched on a surface, one may implement a gradual resonance frequency shift and imprint a controllable phase shift on the scattered wave, from 0 to 𝜋 and not 2𝜋.

Fig. 3.13 High-NA micro-lens made of resonant amorphous silicon nanopost. Half of the 400-µm-diameter lens is shown on the left side. The lens offers a focusing spot size (FWHM) of 1.2 𝜆 for unpolarized light under normal incidence at 𝜆 = 1550 nm. The efficiency of the focusing spot is

20

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) 75%. Left panels: Scanning electron microscope images of the silicon posts forming the micro-lens. Red scale bars, 1 µm. After [Arb15].

The idea of metasurfaces based on resonance-phase nanoelements was introduced in a celebrated article with tiny V-shape plasmonic nanoantennas [Yu11], and then further developed with dielectric nanoresonators, to avoid Ohmic losses in the metal. To achieve the required 2𝜋 phase excursion rather than 𝜋, the idea is to combine two resonances, for instance by exploiting semiconductor nanorods with strong electric and magnetic dipole resonances. Both types of dipoles can be excited simultaneously within simple nanoparticle geometries (such as spheres or rods). Additionally, since the two dipole radiations interfere, the radiation pattern in backward direction can be strongly diminished, providing enhanced transmission and making it possible to fabricate high performance metasurfaces with modest aspect ratios at visible and near-infrared wavelengths, see Fig. 3.13. The two approaches seem to lead to two conflicting visions of the design of metalenses and metagratings. In the first approach (Fig. 3.14a), the scatterers are not independent and the light that occupies a transverse cross section of ≈ 𝜆2 sees a few scatterers. The optical properties (polarization, phase ) result from grating-like properties due to locally periodic artificial dielectrics. In the second approach (Fig. 3.14b), the metasurface is seen as an array of independent scatterers. Every scatterer has an effective cross-section, is potentially resonant, preferentially scattering forward/backward, polarization selective, wavelength selective,. . . . The conflict is only shallow. Indeed, both approaches face the same fundamental issue: the intrinsic difficulty of confining and controlling light properties at deep subwavelength scales without drastic absorption loss,– and finally adopt similar balancing solutions. On the one hand, one would prefer to have small spacing between nanostructures (left panel in Fig. 3.14c) so that the phase would be finely sampled, which is important especially for achieving high efficiency. What happens is that the nanostructures (either nanowaveguides or nanoresonators) become electromagnetically coupled with their neighbors, so that the phase at the local control points is no longer really monitored locally at the level of every individual nanostructure, but on an extended area covering a few neighbors. On the other hand, large spacings (right panel in Fig. 3.14c) that decouple the evanescent tails of the nanostructure fields, are thus preferable, but what happens is that higher-order modes, especially the first antisymmetric modes, come into play and the propagation or resonance phase-delay is not accurately controlled. It is essentially difficult to imprint rapidly varying phases by exploiting subwavelength confinement. Indeed reaching a good balance between the two extreme sampling cases requires nanostructures with a strong transversal confinement of light, otherwise the sampling points do not independently control the phase. How to implement such confinements? The whole literature on nanowaveguide or nanoresonator metalenses has consistently provided one and the same response over the years: nanostructures with a high-index contrasts obtained by etching high-index films, e.g. titanium dioxide or semiconductors, are preferable. This guarantees that the evanescent electromagnetic tails of every nanostructure decrease rapidly and are essentially damped before reaching the neighbors (central panel in Fig. 3.14c).

21

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne)

Fig. 3.14 Optical phase arrays. (a) Metasurfaces based on propagation phase-delays. (b) Metasurfaces based on resonance phase-delays. (c) General requirement for optical phase arrays. Right panel: The phase sampling is fine, the local nanostructures (represented as nanoposts) are coupled electromagnetically and the supermode propagation constant varies with the parallel wave vector 𝒌// of the incident plane wave. Central panel: The nanoposts are uncoupled. Their normalized propagation constant is equal to the effective index of the isolated post, and the imprinted phase is independent of 𝒌// . Right panel: The second-order supermode is no longer evanescent for large samplings and the imprinted phase that depends on both modes is not accurately controlled. After [Lal17].

14.4.5 Metasurfaces Metasurfaces are often considered as the two dimensional versions of 3D metamaterials, but they are not. By metasurfaces, one generally means any nanostructured surfaces capable of manipulating light into the subwavelength regime to achieve a macroscopic function, without any reference to homogenization or artificial media. Because of the developments in nanoscale technologies, metasurfaces have received increasing attention in the 2010’s, with the perspective to implement various functions by controlling the amplitude, phase, polarization, orbital momentum, absorption, reflectance, emissivity of light at high spatial resolution [Gen17]. This leads scientists and engineers to revisit various traditional applications in optics from a different angle. Generally speaking, metasurfaces are more conceived around the concept of nanoresonator arrays, being plasmonic or dielectric, than artificial materials with a prescribe effective property. They are also known as frequency selective surfaces, or reflect- and transmit-arrays in the microwave community. Initially introduced with metallic nanoresonators, metasurface studies in optics have rapidly evolved towards the manipulation of optically induced Mie-like resonances in dielectric and semiconductor nanoparticles with high refractive indices. Such 22

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) particles offer better opportunities for reduced dissipative losses and large resonant enhancement of both electric and magnetic near-fields. Semiconductor nanostructures also offer longer excited-carrier lifetimes and can be electrically doped and gated to realize subwavelength active devices. It is presently difficult to anticipate the genuine added value of this booming field. Nevertheless Fig. 3.15 depicts a tantalizing and unique opportunity offered by metasurfaces by using the funnel effect of resonances for spectral multiplexing metasurface functionalities at the unit cell level. That cannot be performed over a continuum of wavelengths, but it can be envisioned for a discrete set of wavelengths, at blue, green and red wavelengths (RGB) for instance. Each individual nanostructure should be replaced by three even-smaller colored nanostructures with effective cross sections that are more than 10 times larger than their physical cross sections, so that all the light incident on every unit cell (dashed-line box in (a) or (b)) is funneled onto a single nanostructure. In the spectral domain, see (c), each colored nanostructure should be wavelength-selective to avoid inevitable spectral crosstalk due to the spurious scattering by neighbor nanostructures designed for operation at a given wavelength and illuminated at another. Fulfilling that dual requirement for every nanostructure places a real challenge for the design, requiring deep subwavelength nanoresonators that efficiently funnel light in the spatial domain and provide high Q resonance in the spectral domain. Metalenses similar to those presented in Fig. 3.15 have been fabricated using this principle with the same RGB focal length, but only a weak efficiency per polarization has been achieved. However, design constraints are relaxed for the realization of absorbing surfaces, and ultrathin super-absorbers have been successfully achieved, pointing out the intriguing possibility of achieving nearunity and broadband optical absorption in a deep-subwavelength layers with a variety of device technologies, including subwavelength arrays of metallic or semiconductor structures.

Fig. 3.15 Spectral multiplexing with metasurfaces. (a) Initial metasurface layout, possibly with a spatially graded variation. (b) Spatial multiplexing using the funnel effect of nanoresonators. (c) Corresponding spectral RGB multiplexing. To avoid crosstalk, the multiplexing should fulfill drastic requirements in space (antenna funneling) and spectral (high Q) domains.

References [Ves68] V.G. Veselago, "The electrodynamics of substances with simultaneously negative values of  and µ", Soviet. Phys. Usp 10, 509-514 (1968). [Pen99] J. B. Pendry, A.J. Holden, D.J. Robbins and W.J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena", IEEE Trans; Microwave Theory Tech. 47, 2075-2084 (1999). [Pen00] J. B. Pendry, "Negative refraction makes a perfect lens", Phys. Rev. Lett. 85, 3966-3969 (2000).

23

Chapitre 3. Metamaterials and metasurfaces (lecture notes, Philippe Lalanne) [Smi00] D. R. Smith, W. J. Padila, D. C. Vier, S. C. Nemat-Nasser and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity", Phys. Rev. Lett. 84, 4184-4187 (2000). [Val08] J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. Genov, G. Bartal and X. Zhang, "Threedimensional optical metamaterial with a negative refractive index", Nature 455, 376 (2008). [Fan05] N. Fang, H. Lee, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens", Science 308, 534-537 (2005). [Lal06] P. Lalanne, J.P. Hugonin, P. Chavel , Optical properties of deep lamellar gratings: a coupled Bloch-mode insight , IEEE J. Lightwave Technol. 24, 2442- 2449 (2006) [Deg01] P.C. Deguzman and G.P. Nordin, Stacked subwavelength gratings as circular polarization filters, Appl. Opt. 40, 5731-37 (2001). [Tam97] H. Tamada, T. Doumuki, T. Yamaguchi and S. Matsumoto, Al wire-grid polarizer using the spolarization resonance effect at the 0.8-µm-wavelength band, Opt. Lett. 22, 419-421 (1997). [Ada79] J.T. Adams, L.C. Botten, Double Gratings and Their Application as Fabry-Perot Interferometers, J. Opt. (Paris) 10, 109 (1979). [Ber67] G. Bernhard, Structural and functional adaptation in a visual system, Endeavor 26, 79-84 (1967). [Kan99] Y. Kanamori, M. Sasaki and K. Hane, Broadband antireflection gratings fabricated upon silicon substrate, Opt. Lett. 24, 1422-24 (1999). [Lal17] P. Lalanne and P. Chavel, Metalenses at visible wavelengths: past, present, perspectives, Laser Photonics Rev. 11, 1600295 (2017). [Yu11] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, Z. Gaburro, Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science 334, 333–337 (2011). [Ker83] M. Kerker, D.S. Wang, C. Giles, Electromagnetic scattering by magnetic spheres, J. Opt. Soc. Am. 73, 765 (1983). [Arb15] A. Arbabi, Y. Horie, A.J. Ball, M. Bagheri, A. Faraon, Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays, Nature Commun. 6, 7069 (2015). [Gen17] P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, Recent advances in planar optics: from plasmonic to dielectric metasurfaces Optica 4, 139-152 (2017). [Sto88] T. Stone and N. George, Hybrid diffractive-refractive lenses and achromats, Appl. Opt. 27, 2960-2971 (1988).

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Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

Surface plasmon polaritons

Content 6.1. Drude model ............................................................................................................. 2 6.2. Surface plasmon polaritons ......................................................................................... 3 6.3. Surface plasmons with large effective indices (k// >> k0) ................................................. 6 6.4. Metallic nanoparticles ................................................................................................ 8 6.5. Launching surface plasmons with isolated slits and slit arrays ........................................10 6.6. Field scattered by a single tiny slit on the metal surface .................................................12 6.6.1. One-dimensional subwavelength indentations: the quasi-cylindrical wave .................13 6.6.2. 0D subwavelength indentations ............................................................................17 6.7. Wood anomaly .........................................................................................................18 6.8. The plasmon absorption ............................................................................................21 References .....................................................................................................................22

From the point of view of optics, surface plasmons are electromagnetic modes of metallodielectric interfaces. They have been extensively studied in the seventies and eighties [Rae88]. From the point of view of solid state physics, a plasmon is a collective excitation of free electrons [Kit74]. They are both; they are polaritons. Plasmonics has regained much attention in the beginning of the XXI century due to its potential to manipulate "light" at a subwavelength scale: plasmon is the obligatory passage for light to dialog with the quantum or classical nanoworld. The reason is simple. With dielectric materials, non-evanescent electromagnetic modes have spatial variations of the order of /2 at most. At metal interfaces, the free metal electrons come into play and the electromagnetic energy splits into an oscillation of electrons coherently coupled to classical electromagnetic fields in dielectrics … and the resolution of the polariton (the hybrid electronphoton mode) becomes limited by Ohmic losses (absorption) in the metal. How far can we go into electromagnetic confinement before loss becomes unacceptable and for what applications? We then enter the topic of metal nanostructures [Bar03], such as plasmonic waveguides, slits and holes in metal films, metal nanoparticles, metal tips … 1

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

6.1. Drude model The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in metals. It considers the metal to be formed of a mass of positively-charged ions from which a number of "free electrons" are detached. The model was extended in 1905 by Hendrik Antoon Lorentz (and hence is also known as the Drude– Lorentz model) and was supplemented with the results of quantum theory in 1933 by Arnold Sommerfeld and Hans Bethe. In the classical description adopted by Drude, we denote by v the speed of a free electron, by m its effective mass, by e its charge (e < 0) and by  the mean free time between two collisions with the ions of the crystal (dissipative term). The result of Drude's work was the simplified equation describing the averaged motion of electrons dv/dt+v/ = e/m E(t),

(6.1)

where E(t) = Re[E0exp(it)] is the electric field. The effect of the magnetic field can be omitted because its force is minuscule compared to the driving electric force. Finally, if we have n electrons per unit volume, the net flux of charge carriers per unit time is given by nv and the current density J is given by J = nev = E.

(6.2)

By solving Eq. (6.1) in the forced oscillation regime, we readily obtain that the electrical conductivity  is given by  = ne2/m(+i)1. ne2/m is often called the DC ( = 0) conductivity and is denoted by 0. The displacement r(t) of an electron being associated with a dipole moment p = er, the cumulative effect of all individual dipole moments of all free electrons results in a macroscopic polarization per unit volume P = np. Defining the displacement vector D and the relative permittivity r as D  0E  P  0r E , we obtain for the metal permittivity r = 1(ne2/m0) / (2i/).

(6.3)

Two time scales are included in the model. On one hand, the plasmas frequency, p = (ne2/m0)1/2, is the mode frequency of the plasma composed of free electron gazes. The plasma frequency lies in the near ultraviolet for most metals; for gold, we have n = 5.9 10 cm3 and  = 3.0 10 s, and thus p = 1.37 10 rd/s and p = 2c/p = 138 nm. The second time scale is the relaxation time . It is of the order of 10 fs for noble metals. The relaxation time describes the relaxation processes for excited electrons. A major source of confusion is

2

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

that in most references, the dependence of  on the frequency is neglected. Care must be taken as the relaxation of an electron with an energy of a few eV has little in common with the relaxation of an electron with a energy of few meV. The scattering processes are completely different. It follows that it is not correct to insert Eq. (6.3) the value of  derived from the conductivity at zero frequency. In particular, it is known that 1/ at zero frequency decays when the temperature decays. However, this is not a valid conclusion for electrons with a larger energy. Indeed, even at low temperature, the electron-electron interaction remains an efficient relaxation channel that is almost not dependent on the electron temperature. One of the practical conclusions of this paragraph is that metal losses cannot be significantly reduced when reducing the temperature [Kit74], unfortunately. The Drude model for noble metals is valid in the IR and in the red part of the visible range. At smaller frequencies, especially in the blue, photons can excite electrons in the valence band (usually a d-band) so that new absorption channels are available. This introduces serious deviations from the Drude that can be modelled with Drude-Lorentz models. Figure 6.1 shows tabulated data for the permittivity of gold at optical frequencies.

Figure 6.1: Real and imaginary part of the relative permittivity of gold at optical frequencies. Data are taken from [Pal85].

6.2. Surface plasmon polaritons We now study the interface between a metal (medium 2, x < 0 and relative permittivity m) and a dielectric material (medium 1, x > 0 and relative permittivity d). We look for the existence of a bounded surface mode on the interface with a field that is exponentially decaying on both side of the interface, in media 1 and 2, see Fig. 6.2.

3

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

Figure 6.2: Surface plasmon polaritons are localized electromagnetic modes/charge density oscillations bounded to a metal interface between dielectric and metallic materials. (a) Charges and magnetic field Hy oscillations. (b) The plasmon field decays exponentially in both media, kp  (pk0  kSP)1/2, with p = 1 or 2. 1/ For simplicity, we assume that the metal is lossless (this represents a drastic approximation especially for frequencies  close to p) with a relative permittivity m, purely real and negative. What does it imply for the mean free time between ionic collisions in the Drude model? 2/ Without loss of generality, we consider a transverse electromagnetic field (TM polarization), H = (0, Hy, 0) and E = (Ex, 0, Ez). In both media, the permittivity is independent of space (uniform). Derive the Helmholtz equations satisfied by Hy oscillating at the frequency  in the metal and in the dielectric (note that since each media is uniform,   D  0 leads to

  E  0 ). 3/ We look for a solution with a propagation direction along the z axis k = (0, 0, k//) For x < 0 (metal)

Hy exp[(ik//z+2x)],

(6.4a)

For x > 0 (dielectric) Hy exp[(ik//z1x)],

(6.4b)

where it is assumed that k// is real (because the metal is lossless). k // is the same in both media. Why? 4/ From the Helmholtz equations, derive the expressions of 1 and 2, as a function of k//, , c, m and d. The expressions involve square-roots and thus a sign indetermination. How to choose the good sign for the solution to be physical? What is the condition on k// for the solution to be a bound surface mode? 5/ By using the field continuity equation for Ez, derive the dispersion relation

d  2  m 1  0 .

(6.5)

4

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

6/ Then derive the dispersion relation linking the frequency  to the parallel momentum k// of the plasmon

k // 

 d m . c d   m

(6.6)

7/ An important step: The metal is described by a lossless Drude model (m < 0). Inserting the Drude's form of the metal permittivity dielectric constant into the dispersion relation of Eq. 6.6, we obtain for the lossless approximation case ( = 0)

kSP

n  1 c

2  2p

1  1 2  2p

.

(6.7)

Show that the general shape of the dispersion relation (k//) is given by Fig. 6.3.

Figure 6.3: Lower branch of the dispersion of surface plasmon polaritons at metal-dielectric interfaces. For that purpose, first consider low frequencies ( << p) in the far infrared and microwaves and show that kSP is essentially equal to the dielectric propagation constant in the dielectric material, kSP = n1/c. Show that the surface mode is then completely expelled from the metal and becomes a “regular” plane wave in the dielectric material grazing the surface of the metal. Then consider high frequencies close to the plasma frequencies and show that kSP/(/c) diverges for an asymptotic frequency a and express a as a function of p and d. As a, explain why the plasmon is more and more confined on the interface. 8/ In the 2A course on guided waves, we have shown that the group velocity is equal to the energy velocity for lossless modes, like in the present study. The group velocity d/dk// (and thus the energy velocity) vanishes for a (in the absence of loss). To understand how this is possible, derive the direction of the Poynting vectors Sd and Sm in the dielectric and in the metal close to the interface, S 

1 Re E  H *  . Show the directions by a graph, 2

5

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

without omitting to additionally represent the direction of the phase velocity /k// of the surface plasmon. What is happening as a? 9/ Helped again by the Drude model, indicate on the same graph the direction of the electric current Jz.

6.3. Surface plasmons with large effective indices (k// >> k0) The existence of the horizontal asymptote in Fig. 6.3 is an intriguing phenomenon that is not encountered with dielectric structures. One should remember that we use a lossless Drude model. In reality, large propagation constants in metal-dielectric waveguides are systematically accompanied by large losses, because the plasmon energy becomes mostly electronic at large k//’s. We note that Eq. (6.6) remains valid when loss ( Im  m   0 ) are considered in the model. The propagation constant becomes a complex number, whose imaginary part is related to the damping of the mode. Dielectric fibres and slabs can guide electromagnetic modes in a different way. Decreasing the diameter of a fibre (or the thickness of a slab) reduces the number of supported guided modes (“the density of states”). The fundamental mode in an optical fibre/slab is the only mode with no cut-off diameter or thickness; when the fibre diameter d is decreased, the fundamental guided mode penetrates deeper into the surrounding medium and eventually (at d = 0) becomes a bulk plane-wave in the medium surrounding the fibre (Fig. 6.4a). At the same time, as d → 0, the wavelength λ of the guided mode monotonically increases from λ0, the wavelength of the bulk wave in the fibre medium (as d → ∞), to the wavelength in the surrounding medium. As a result, the mode size (shown by the dashed lines) decreases when the diameter of the fibre is decreased to ~ λ0, and then increases to infinity when the fibre diameter is reduced further. Thus, decreasing the diameter of an optical fibre or the thickness of a guiding slab to zero cannot lead to subwavelength localization of the guided mode. This feature indicates the diffraction limit of light in dielectric waveguides and constitutes the main hurdle in achieving higher degrees of miniaturization and integration of optical devices. The situation for guiding surface plasmon modes in metallic slits (metal-insulatormetal or MIM stacks) or metal films (insulator-metal-insulator or IMI) or metallic nanorod is drastically different. When the width t of a metal film or of the slit is reduced below the wavelength λ0sp of the SPP on the flat metal interface, the fundamental SPP mode (whose magnetic field has axial symmetry and is perpendicular to the propagation axis) experiences a strong monotonic increase in localization and a significant reduction in its phase and group velocities [Yan12]

6

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

neff  

1 d  .  m t

(6. 7)

Again, strong dispersion (neff is proportional to the wavelength) is associated to small group velocities, implies that the mode is extremely damped. The propagation distance is subwavelength in general for group velocities smaller than  c/20. As a result of the large dispersion, the diameter of the guided surface plasmon mode can be decreased to just a few tens of nanometers, limited only by the increased dissipative losses, the atomic structure of matter and spatial dispersion. This feature is the principle behind using plasmonic nanostructures as subwavelength optical devices for highly integrated nano-optics circuits and components, as well as for the delivery of light to the nanoscale, towards quantum dots and individual molecules.

Figure 6.4: Guided modes: dielectric fibres versus metal nanowires. a,b, Typical field structures, localization and wavelengths of the fundamental modes guided by dielectric fibres (a) and cylindrical metal nanowires (b) for different core diameters. λ0 and λ0sp are the mode wavelengths for infinitediameter fibres or metal nanowires, respectively. The dashed horizontal

7

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

lines show the localization of the mode at the 1/e level of the field. After [Gra10].

6.4. Metallic nanoparticles To start with, let us consider a tiny metallic box with a transverse dimension t and a length L. The box is placed in a uniform background of relative permittivity  d . To understand the origin of the fundamental (electric dipolar resonance) resonance of the box, we use a FabryPerot model (see [Yan12] for a complete description) in which the fundamental resonance is seen as resulting from the bouncing back and forth of the IMI plasmon mode of an infinitely long wire with the same transverse dimensions. If we assume that the phase of the modal reflectance of the IMI plasmons at the facets is 0 or , the resonance wavelength R satisfies the Fabry-Perot condition

2 neff L  2 . If the transverse dimension is small enough, the R

effective index is given by Eq. (6.7) and the condition becomes m  

d L .  t

(6. 8)

Amazingly and in strong contrast with dielectric resonances which require a longitudinal length comparable to the resonant wavelength, there is no explicit dependence on the wavelength in Eq. (6.8). The resonance wavelength of tiny metal particles is colorless? Not really, the wavelength dependence is in the metal permittivity. The latter (its real part) is negative and is equal to a few units at visible frequencies. Thus provided that L is only a little bit larger than t, Eq. (6.8) always admits a solution in the visible range. As one shrinks down all the dimension of the metallic nanoparticle, in the static limit indeed, the resonance wavelength remains unchanged. This is a unique property not encountered with dielectric nanoparticle.

Figure 6.5: Fundamental (dipolar electric) resonance of a metal box seen as a Fabry Perot of a IMI surface plasmon bouncing back and forth along the longer dimension of the box.

8

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

For the sake of understanding, it is interesting to consider the related famous textbook case of a metallic nanosphere in a uniform background of relative permittivity  d . If the diameter D is much smaller than the wavelength (k0D << 1, static limit), under an incident uniform illumination with an electric field E0, the electric field E inside the nanosphere is also uniform and is given by E 

3 d E0 [Jac74]. Thus the resonance wavelength of tiny  m  2 d

metal nanospheres occurs when  m  2d , the famous “colorless” condition which is the analogue of Eq. (6.8) for a specific geometry. The Fabry-Perot analysis evidences that the existence of localized surface plasmon resonances (LSPR) in small deep-subwavelength volumes is fundamentally related to the existence of surface plasmon polaritons with small cross-sections. It also evidences that the resonance wavelength can be easily tuned by varying the dimensions, the parameter allowing the largest change being the transverse dimension t for small t’s indeed. Figure 6.7 illustrates the broad tunability (over one octave) that can be achieved with Au-shell/silicacore spherical nanoshells, just by varying the gold layer thickness by a few nanometers. The basic mechanism behind the large change is the divergence of the effective index of IMI layers as the thickness vanishes.

Figure 6.7: Visual demonstration of the tunability of metal nanoshells (top), and optical spectra of Au-shell/silica-core nanoshells (the labels indicate the corresponding Au shell thickness t). Taken from [Loo04]. In the static limit (i.e. when the nanoparticle dimensions are so small that radiation loss can be neglected), we may derive analytically many important properties. For instance, it can

9

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

be shown that the quality factor Q of any metallo-dielectric resonant nanoparticle composed of a single lossy metal is completely determined from the relative permittivity of the metal Q  R

Re m   . 2Im m 

(6. 9)

In this limit, the specific structure of the particle is not important and the Q-factor depends neither on the geometric shape nor on the dielectric media, provided that the latter are dispersionless and lossless. This universal result has been derived in [Wan06] by assuming a nanoparticle built in a noble metal with  Re m   Imm  that supports a purely electrostatic resonance   E  0 , whose damping is solely due to absorption (no radiation losses). This last assumption can be justified by the fact that the scattering of small particles scales as the particle volume squared whereas the absorption scales only as the particle volume. Metal nanoparticles have drawn much research efforts and also witnessed successful applications because of their capability to support strong and highly-confined resonances. In particular metal nanoparticles can effectively convert local changes of refractive index into frequency shifts of the resonance [Lal07]. This property has driven considerable developments in sensing technologies based on localized surface plasmon resonance (LSPR) of metallic nanoparticles, benefiting from significant advance in the detection of single metal nanoparticles. Thanks to their small mode volume, LSPRs are suitable for achieving detection of ultra-small refractive-index changes [Ste08]. Even single-molecule sensitivity has been achieved [Ame12].

6.5. Launching surface plasmons with isolated slits and slit arrays The interaction of light with isolated tiny holes, slits, boxes or grooves, etched or deposited on a metal film is fundamental importance for exciting surface plasmons from free space. In this section, we summarize the main results known for the excitation of plasmons by isolated slits illuminated by an incident plane wave, see Fig. 6.8a. We define the total plasmon excitation efficiency  as the ratio between the plasmon power flux (launched on both sides of the indentation) and the incident power flux that directly impinges onto the slit. The efficiency may exceed unity for subwavelength indentations, implying that more light than is geometrically incident upon the indentation is converted into plasmons. In general, η depends on the slit width, on the dielectric constant of the metal, and slightly on the angle of incidence of the illumination beam. The latter is assumed to be a plane wave in the following.

10

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

Figure 6.8: Computed total SPP excitation efficiency by a subwavelength semi-infinite slit perforated in a metal substrate as a function of the normalized slit width w/ for different wavelengths (black: 0.6 µm, red: 1 µm, green: 3 µm and blue: 10 µm). The air/gold interface is illuminated by a normally incident plane wave. The gold frequency-dependent permittivity is taken from tabulated data [Pal85]. The total efficiency corresponds to the efficiency of the two SPP modes launched in the two opposite directions on the air-gold interface. From [Lal06].

Figure 6.8b summarizes the main useful results predicted for slits at an airgold interface, but other noble metals used at frequencies close to the plasma frequency exhibit a similar behavior. First, the SPP excitation efficiency strongly depends on the slit width, with a maximum value reached for a slit width equal to w ≈ 0.23. Second, at visible wavelength, the efficiency is fairly large; it may become of order 0.5, implying that of the power scattered by the slit half goes into heat, if no other corrugation is present to decouple the launched SPPs to radiation modes. Third the efficiency rapidly decreases for large wavelengths; it scales as d  m 

1/ 2

, reaching only 2.8% for =10 μm.

The computational data of Fig. 6.8 can be quantitatively explained with an approximate model [Lal06] that provides a closedform expression for the efficiency

  d  m 

1/ 2

F w   ,

(6.10)

where F w   is proportional to a sinclike function of the normalized slit width, which weakly depends on the dielectric constants of the problem. The sinclike behavior of the SPPexcitation efficiency has been qualitatively verified experimentally at  = 780 nm, by detecting the SPPs launched at the transmission facet of a single slit with a near-field probe [Kih08], and a quantitative agreement showing a maximum and almost-null efficiencies for w ≈ 0.23 and w ≈ , respectively, has been obtained with far-field measurements performed

11

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

on a slit doublet when only a single slit is illuminated at  = 810 nm [Rav09]. Equation (6.10) shows two important trends. It predicts that the plasmonic excitation can be enhanced by immersing the sample in a dielectric material (it increases proportionally to nd), and that the efficiency scales as  m

1/ 2

(as the inverse of the wavelength for a Drude model). This

scaling law that is valid for any subwavelength indentation indicates that the visible part of the electromagnetic spectrum is the most exciting region for studying SPPs. There are two ways to increase the SPP-generation efficiency of single slits, either by using highly-oblique incidence or grooves instead of slits. For Fabry-Perot groove resonances, the light converted into SPPs can be two to three times larger than that which is directly

incident

onto

the

aperture

area

(  2-3)

[Lal06].

Additionally,

an

almostunidirectional SPPlaunching may be achieved for certain incidences and for slit width larger than /2, a value above which the first antisymmetric slit mode becomes propagative. Indeed, the SPPgeneration efficiency can be further increased by arraying the grooves or ridges like in a gratingplasmon coupler. The efficiency increase is however achieved at the expense of miniaturization.

6.6. Field scattered by a single tiny slit on the metal surface The field scattered by subwavelength indentations or emitted by subwavelength emitters in the vicinity of interfaces has been of longstanding interest in electromagnetism. In the 1900’s, the rapid development of radio-wave technology prompted theoretical studies to explain why very long-distance (over-ocean transmission have been achieved in 1907 by Marconi) transmission could be achieved with radio waves above the earth. The solution is indeed linked to guiding by the ionosphere layers, but at the beginning of the 20th century, the explanation was thought to be due to the nature of the surface waves launched on the flat earth by the emitting antennas acting as a dipole. Sommerfeld was the first to determine the complete electromagnetic field radiated by a subwavelength antenna (a 0D vertical dipole) at the interface between two semi-infinite half spaces. He verified that his complicated solution [Som26] is composed of a “direct contribution”, which decays algebraically as r 2 at asymptotically long-distance from the antenna [Nor35], and of a bounded Zenneck mode [Zen07], the analogue of the surface plasmon polariton for metals at optical frequencies, with an exponential decay. The direct contribution, known as the Norton wave [Col04], was therefore believed to be responsible for long-distance radio transmission. In plasmonics, the field scattered on metallic surfaces by subwavelength indentations is also essential, since it is responsible for the electromagnetic interaction between nearby indentations on metal surfaces. From a mathematical point of view, the solution of this 12

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

photonic problem is identical to that of the radio-wave problem [Lal09, Nik09]. However, there are also differences. We are mainly concerned by short-distance (rather than longdistance) electromagnetic interactions, since the distance between two neighboring indentations on subwavelength optical surfaces rarely exceeds 10λ. The second important difference concerns the fact that the dipole orientation cannot be chosen in nanophotonics. For instance, for a 1D subwavelength indentation under illumination of transverse-magnetic (TM) polarization, two coherent equivalent electrical dipoles of different polarizations are generally excited with different strengths. 6.6.1. One-dimensional subwavelength indentations: the quasi-cylindrical wave The field scattered on the metal surface by a 1D subwavelength indentation illuminated by a TM plane wave has been the subject of intense research [Gay06, Lal06b, Che06, Aig07, Dai09, Ung08]. Hereafter we simply summarize the main results, which are documented in a review article [Lal09]. Referring to Fig. 6.9, a subwavelength indentation invariant along the y-axis (the z-axis being perpendicular to the surface) and illuminated with a plane wave polarized in the x-z plane (Fig. 6.9a), can be replaced by two electric line sources in the dipolar approximation (Fig. 6.9b), one Jz being polarized perpendicularly to the interface (along the z-axis) and the other one Jx parallel to the interface (along the x-axis). Concerning the field scattered on the surface (this is the field that is responsible for the electromagnetic interaction between the indentations on the surface), three important properties are worth mentioning here.

Figure 6.9: Equivalence of a subwavelength indentation under TM illumination (a) with two electric line (Dirac (x,z)) sources (b). Property 1: The field radiated on the surface by each individual line source, either Jx or Jz, can be decomposed into a surface plasmon mode and a quasi-cylindrical wave (quasi-CW), which represents a “direct” contribution from the source. Figure 6.10a shows the magnetic field emitted at =1 µm by a line source Jz polarized vertically and placed on an air/gold interface. Far away from the surface (in the far-field), the magnetic field behaves as a cylindrical wave, with an x 2  z 2

1 / 2

damping. Close to the

surface (z << ), the emitted field shown in Fig. 6.10b is much more complex. It is composed 13

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

of a plasmon with an exponential damping (dashed curve) and of a quasi-CW (solid curve). The latter takes two asymptotic forms. It is very intense and behaves as a cylindrical wave (dotted blue line) with a 1/x1/2 decay rate at small propagation distances. Then it is dropping at a faster rate for intermediate distances <x<10, before reaching an asymptotic regime behavior with an x3/2 damping rate at large propagation distances. It is the analogue of the Norton wave (shown with the dotted red line) discovered for radio communication.

Figure 6.10: Magnetic field radiated at =1 µm on an air/gold interface (z = 0) by a line source Jz(x,y) polarized vertically. The field is composed of a plasmon (dashed curve) and of a quasi-CW (solid curve). The gold permittivity is m = 46.8 + 3.5i. The existence and importance of the quasi-cylindrical wave at optical frequencies on metals has been first observed with a very elegant slit-groove experiment [Gay06], in which the groove acts as a line source and the slit as a local detector of the field scattered by the groove. By systematically varying the groove-slit separation-distance in a series of samples, the field pattern is recorded. The experimental data, which were probably contaminated by an undesired ad-layer on the silver film, have been initially interpreted in a confusing manner as shown in [Lal06], but they had the merit to unambiguously reveal the existence and importance of a direct wave (different from the SPP) that is initially dominant for |x| < 2 .

Property 2: The quasi-cylindrical waves radiated on the surface by each individual line source, Jx or Jz, although they differ in amplitude and phase, are almost identical in shape. 14

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

Figure 6.10 illustrates this nice property. In the left graph, the magnetic fields of the quasi-cylindical waves radiated by vertical (blue curve) and horizontal (red curve) line sources (Jx = Jz =1) are shown as a function of the distance x from the source. The calculation is performed for a gold substrate at  = 800 nm. In the right panel, the source Jx has been optimized (Jx  3i) so that its associated quasi-cylindrical wave is similar to that generated by the vertical source. It turns out that, although a slight difference remains, the two fields are almost superimposed. It can be shown that this difference becomes smaller and smaller as the metal conductivity increases (or  increases) [Lal09].

Figure 6.10: Illustration of property 2 for a gold/air interface at  = 800 nm. Left: The blue and red curves represent the magnetic field of the quasicylindrical waves radiated on the surface by two line sources polarized vertically and horizontally, respectively, with Jx = Jz = 1. The two quasicylindrical waves seem completely different a priori. Right: In reality, the two quasi-cylindrical waves are almost identical in shape and only differ by a constant, as show by the new red curves obtained for Jx  3i. The frequency-dependent Au permittivity takes value from [Pal85]. This property has an important consequence if one neglects the small residual difference. When a subwavelength indentation is illuminated by a TM polarized light, the scattered field can be seen as the total field radiated by two line sources, Jx and Jz, and the relative amplitudes of the line sources are arbitrary: they depend on the incident illumination (its angle of incidence for instance if it is a plane wave), on the actual geometry of the indentation… A priori two independent radiation problems should be considered, but since the quasi-cylindrical waves associated to the two line source polarizations are identical in shape, any arbitrary sub-λ indentation illuminated by any incident electromagnetic field will launch a unique field (the quasi-cylindrical wave) on a metallic surface, in addition to the surface plasmon.

15

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

Property 3: As one moves from the visible to longer wavelengths, the surface plasmon is less attenuated, but it is also less and less efficiently excited, whereas the quasi-CWs are equally excited at all energies. Figure 6.11 represents the magnitude of the total magnetic field H(x, z = 0) radiated on the interface (z = 0) by a z-polarized line source located at x = z = 0. The total field results from the sum of two contributions, the surface plasmon (blue-dotted) and quasi-CW (redsolid) contributions, respectively. We first note that the initial quasi-CW contribution at short distances is nearly independent of the metal dielectric properties, whereas the initial plasmon contribution rapidly drops as the metal conductivity increases, |HSP|  |m|–1/2. At visible wavelengths ( = 0.633 µm), the plasmon contribution dominates even at relatively short distances, the plasmon and quasi-CW being actually equal for xc  /6. At thermal-infrared wavelengths ( = 9 µm), the quasi-CW is preponderant until distances as large as 100. It can be shown that the initial crossing distance xc below which the quasi-CW wave dominates increases with the metal conductivity, xc ≈ |m|/(2d3/2) [Lal09].

Figure 6.11: Magnetic field, H(x), radiated by a vertically-polarized line source Jz at an Ag/air interface (inset on the top) for wavelengths ranging from the visible to thermal infrared. The blue dashed curves correspond to the surface plamson and the red-solid curves to quasi-cylindrical wave. Thin black lines show a damping scaling as 1/x1/2. The calculations are performed for silver with a frequency-dependent permittivity taken from [Pal85]; similar results have been obtained for gold. Note the logarithmic scales used in both the horizontal and vertical axes, which are all identical for the sake of comparison. After [Lal06b].

16

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

6.6.2. 0D subwavelength indentations As mentioned above, the field radiated by an electrical-dipole source or sub- antennas in the vicinity of metallic surfaces is well documented in the literature on antennas. Hereafter we simply show an example for the sake of illustration. For a vertical dipole, perpendicular to the interface, the in-plane component of the radiated field is radially polarized and isotropic. The situation is more interesting for an in-plane dipole (let us say parallel to the x-axis). Both the plasmon and quasi-CW fields on the surface are anisotropic. Figure 6.12 shows the radial electric fields radiated by such a dipole at  = 800 nm. The plasmon field is proportional to r 1 / 2 expikSP r  and its electric vector is mainly perpendicular to the surface with a small in-plane component. Along any direction (different from =/2) the in-plane plasmon field tends to be radially polarized, as E/Er  tan()/r, where the subscript  and r are used to denote the orthoradial and radial components of the fields. The quasi-CW wave initially varies as r 1 expik0 r  at small distances from the dipole, then at longer distances, its electric field amplitude decays algebraically with distance as r 2 expik0 r  , like the Norton radio wave. The electric field of the quasi-CW points mainly along the direction perpendicular to the interface, and its in-plane components satisfy E/Er  tan(). However, for the vertical dipole the in-plane component of the quasi-CW field is radially polarized and isotropic.

Figure 6.12: Radial electric field radiated on an Au/air surface by a dipole point source polarized parallel to the surface (along x-axis) for =800 nm. The left panels show a surface map for the plasmon (top) and quasi-CW (bottom) fields. The same units are used in both plots. As the metal conductivity increases, the perfect-conductor limit is reached and the quasi-

cylindrical wave becomes a spherical wave with a r 1 expik0 r  behavior. In the right panel, the radial field (Ex) is plotted as a function of x for y = 0. The blue dashed curve corresponds to the plasmon mode and the red-solid curve to the quasi-CW. The frequency-dependent value of Au permittivity is taken from [Pal85].

17

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

6.7. Wood anomaly In 1902, R.W. Wood, when observing the spectrum of a continuous light source reflected by an array of metallic grooves (a diffraction grating) when the incident wave is polarized with its magnetic vector parallel to the grooves (TM polarization), noticed a surprising phenomenon: “I was astounded to find that under certain conditions, the drop from maximum illumination to minimum, a drop certainly of from 10 to 1, occurred within a range of wavelengths not greater than the distance between the sodium lines” [Woo02]. Since then, grating anomalies have fascinated specialists of optics and physics, and nowadays with the progress of nanofabrication, metallic surfaces patterned with subwavelength indentations are studied for a variety of interesting properties. By considering the metal as perfectly conducting and using a complicated mathematical derivation, Lord Rayleigh proposed the first explanation to the existence of the anomalies [Ray07]: an anomaly in a given spectrum occurs at a wavelength corresponding to the passing-off of a spectrum of higher order, in other words, at the wavelength given by the grating equation for which a scattered wave emerges tangentially to the grating surface, i.e. for

nk0  k x  m2  a ,

(6.11)

where m is a relative integer and the propagation constant nk0 is matched to the parallel wave vector kx of the incident plane wave through a wave vector modulus 2/a of the 1D reciprocal lattice associated to the grating (a being the periodicity). But this interpretation was erroneous; it is nowadays known as the Rayleigh anomaly, associated to a redistribution of the energy into the different diffractions orders when a higher order is passing-off or –on. Profoundly, Wood anomaly is a grating resonance. U. Fano was the first to realize it in his seminal article published in 1941 [Fan41] (40 years after Wood’s observation). Fano’s model is much less mathematically involved than the theoretical work by Lord Rayleigh. It rather relies on a Huygens-type very intuitive interpretation, and importantly suggests that a surface mode with a parallel momentum greater than the free space momentum be involved in the energy transport between adjacent grooves. It is retrospectively interesting and amazing to see how the surface wave, which is nowadays known as the surface plasmon polariton of a flat interface, is introduced in Fano’s model. U. Fano first considered the modes of a glass layer sandwiched between a metal and a vacuum (see Fig. 6.13a) and asks himself “Is there left any mode when the thickness of the glass layer vanishes?”. Then by solving analytically the bi-interface problem, he calculated the parallel propagation constants of the modes and showed that one and only one bound mode (the plasmon mode that Fano called a “superficial wave”) exists in the limit of vanishingly small glass thickness for TM polarization. 18

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

He found that this particular mode has a complex propagation constant whose real part, actually kSP (surprisingly Fano does not give any analytical expression for kSP), is always slightly larger than the modulus k0 of the wave-vector in a vacuum. Then Fano made the ansatz that Wood’s anomaly originates from a collective resonance of the subwavelength surface (see Fig. 6.13b), in which the part of the wave scattered by groove A excites the bound mode travelling along the surface and reaches the neighboring groove B in phase with the incident wave (phase-matching condition). Fano’s phase matching condition reads as

Re k SP   k x  m2  a ,

(6.12)

Figure 6.13: Fano’s microscopic model of Wood’s anomaly (from [Fan41]). (a) Geometry used by Fano to predict the existence of SPPs at a metal-air interface by considering light guided by TIR at the glass-air interface and by metallic reflection at the glass-metal interface and by studying the limit case for which the glass-slab thickness tends to zero. This bounded mode is nothing else than the SPP of the flat metallic surface, which will be discovered 16 years after by Ritchie [Rit57]. (b) In Fano’s interpretation, resonance occurs whenever the SPP that is scattered by groove A and that is traveling along the grating interface with the plasmon phase velocity kSP reaches the neighboring groove B in phase with the incident wave. Fano interpretation can be set down more quantitatively. Let us introduce the reflection and transmission coefficients,  and , of a surface plasmon that is reflected or transmitted by a single groove, see Fig. 6.14a. Let us also introduce the coefficients of right- and left-going surface plasmons, Ap and Bp, excited in between the grooves in the array, Fig. 6.14b. In the absence of any illumination, if one neglects the quasi-cylindrical wave, the mode of periodic surface can be found by finding the self-consistent solution of the set of surfaceplasmon equations

19

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

An 1  An expikSP a  Bn 1 expikSP a ,

(6.13a)

Bn 1  Bn  2 expikSP a  An 1 expikSP a ,

(6.13b)

where the same coefficient  is used in the equations because we consider symmetric grooves.

Introducing

the

Bloch-Floquet

condition,

An 1  An expikx a

and

Bn 1  Bn expikx a , the system of Eqs. (6.13a) and (6.13b) admits a solution if the determinant is null

1   exp ikx a expikSP a1   expikx a expikSP a  2 expi 2kSP a  0 .

(6.14)

We further assume that   1 , i.e. that the subwavelength indentation weekly scatter the incident plasmon mode (it is a non-resonant indentation), and neglect the  2 term. Each parenthesis can be independently equal to zero for a complex frequency that verifies

  ~ 1/ 2  ~       d m   ~   1   exp ikx a  exp i  ~  a   0 .  c     m    d 

(6.15)

~  , we obtain two phase matching condition Neglecting the variation of  and  m with Im  ~ for Re 

Re kSP   k x  arg a  m2  a ,

(6.16) 1/ 2

and the mode lifetime

    d  m    Im ~ 1  a  c  1     d   m 

. Equation (6.16) is a refined

version of the classical Eq. (6.12). For normal incidence, because the coupling is bidirectional, the mode is a stationary pattern and the plasmon reflectance has to be considered. The two parentheses in Eq. (6.14) become equal and the system of Eqs. (6.13a) and (6.13b) admits two solutions, 1     expikSP a , at frequencies such that

Re kSP   arg   a  m2  a ,

(6.17)

for which the “+” sign corresponds to dark modes and the “”sign to leaky modes. The microscopic model presented here neglects the quasi-cylindrical waves launched in between the indentations [Liu08]. Taking them into account provides can be performed fully analytically [Zha14], leading to an exact equation, in which the kSP term in Eqs. (6.16) and (6.17) is replaced by a more complex term that additionally takes into account the quasi20

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

cylindrical wave. However, it is worth mentioning that the plasmon scattering coefficients  and  both remain in the rigorous model. They are therefore found to play a fundamental role in the electromagnetic properties of subwavelength metallic surfaces in general.

Figure 6.14: Microscopic model of Wood’s anomalies. (a) Elementary scattering coefficients for Wood’s anomaly, where  and  characterize the reflection and transmission of the plasmon incident onto a single groove. (b) Multiple scattering of surface plasmon polaritons on a periodic array of grooves. Ap and Bp denote the coefficients of right- and left-going surface plasmons excited in between the grooves in the array.

6.8. The plasmon absorption So far we have considered the temporal response of surface plasmons. Actually the mode lifetime is very short, a few femtoseconds. In this Section, we rapidly overview what is happening in the metal after the damping of the plasmon. If no radiation channel is available, such as for surface plasmon polaritons on flat surfaces, the initial decay of surface plasmons occurs mainly via the creation of an electron-hole pair. This rapid process (10 fs) is detrimental for most plasmonic applications and devices, since it destroys the coherence of the collective electron oscillations, limits the lifetime and propagation length of surface plasmons and prevents the realization of drastic local field enhancements. The generated electrons, sometimes called hot electrons, have an energy level above Fermi level, and their cooling proceeds by electron-phonon coupling and by diffusive motion. The temperature relaxation in time and sample depth can be modeled by two coupled diffusion equations, one describing the heat conduction of electrons and the other that of the lattice, both equations being connected by a term that is proportional to the electron-phonon coupling constant and to the temperature difference between electrons and lattice [Hoh00]. The following figure well summarizes the different effects and timescales that occur after absorption of plasmons (or photons) in metals. In the experiment, a first laser pump pulse (λ = 800 nm, 100-fs full-width-at-half-maximum) excites surface plasmons at an air-gold interface from the glass substrate in the so-called “Kretschmann” configuration with a defocused beam. A second focused laser pulse (λ = 532 nm, 100-fs FWHM) probes the airgold interface and records the reflectance variations ΔR induced by small changes of the metal permittivity Δ of metal, with a controlled delay between the pump and the probe pulses. Figure 1c shows a typical temporal ΔR/R response with a three-step process with 21

Chapitre 6. Surface plasmon polaritons (lecture notes, Philippe Lalanne)

different time scales. First, the energy is absorbed by the free electron gas. The latter is much lighter and reactive than the ion lattice and can thermalize very rapidly towards the equilibrium Fermi-Dirac distribution with a time scale of  100 fs. Then, the hot electron gas relaxes through internal electron-phonon collisions characterized by a time scale  1ps, before a classic heat diffusion transport takes place with a much longer characteristic time. Thus the plasmon depth is accompanied by decoherence, from an initial collective oscillation to a hot electron, and finally phonons.

Figure 6.15: Plasmon death. Typical temporal ΔR/R response recorded on flat metal interfaces when a plasmon is absorded. Inset: Schematic of the experimental setup (adapted from [Bro87]).

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