Alexander B. Khanikaev, Chihhui Wu and Gennady Shvets ...

DOI 10.1515/nanoph-2013-0009 Nanophotonics 2013; 2(4): 247–264© 2013 Science Wise Publishing &

Review article

Alexander B. Khanikaev, Chihhui Wu and Gennady Shvets*

Fano-resonant metamaterials and their applications

Abstract: New developments in the field of Fano-resonant plasmonic metamaterials are reviewed. The emphasis is on the applications of such artificial electromagnetic materials to a variety of technologically important areas: solar energy harvesting and conversion, sensing/identi-fication of ultra-small molecular contents, and extreme molding of the flow of light. Most recent theoretical tools for describing Fano-resonant metamaterials are reviewed. Both fully three-dimensional metamaterials and planar meta-surfaces are discussed, and their future prospects for applications are critically evaluated.

Keywords: Fano resonance; metamaterials; plasmonics.

*Corresponding author: Gennady Shvets, Department of Physics and Center for Nano and Molecular Science and Technology, The University of Texas at Austin, Austin, TX 78712, USA, e-mail: [email protected] B. Khanikaev: Department of Physics and Center for Nano and Molecular Science and Technology, The University of Texas at Austin, Austin, TX 78712, USA; Department of Physics, Queens College of The City University of New York, Flushing, New York 11367, USA; and The Graduate Center of The City University of New York, New York, New York 10016, USAChihhui Wu: Department of Physics and Center for Nano and Molecular Science and Technology, The University of Texas at Austin, Austin, TX 78712, USA; and Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720

Edited by Pierre Berini

1 Introduction and basics: the physics of Fano resonances

Metamaterials and meta-surfaces [1–5] have greatly expanded the range of electromagnetic properties exhib-ited by naturally occurring materials. In so doing, meta-materials have challenged our intuition by enabling such unusual phenomena as optical magnetism [6–8], negative refraction [1, 2, 9–11] and bi-anisotropy [12–14]. Numerous applications of metamaterials include super-lenses [1, 15], transformation optics and invisibility cloaks [16–20], and novel biosensors [21–25]. Because metamaterials are pro-duced by structuring their unit cells (“meta-molecules”)

on a spatial scale d which is smaller than the relevant radi-ation wavelength λ, they occupy a unique niche between photonic crystals (structured on a scale of λ) and regular materials (structured on the atomic scale). Therefore, metamaterials can be viewed as mesoscopic materials exhibiting a number of unique and unusual properties in addition to those shared with the regular materials. At the same time, metamaterials have been shown to emulate a number of basic optical phenomena observed in the nat-urally occurring atomic/molecular media. Among such familiar phenomena are electromagnetically induced transparency (EIT) [26–36], slow light (SL) [37], and even topological insulators [38]. One of the most interesting and rich in applications phenomena successfully emu-lated by optical metamaterials is the so-called Fano reso-nance [39], which is the subject of this Review.

Fano resonances were originally discovered in quantum physics [39] to describe asymmetrically shaped ionization spectral lines of atoms and molecules. Spe-cifically, it has been shown that ionization of an atom by a high-energy photon (or a fast passing electron) can proceed in two distinct ways: (i) through direct excitation of an electron from its bound state into an unbound (con-tinuum) state, or (ii) through indirect excitation of two electrons into an intermediate bound state, followed by an Auger-like process of electron ejection. The first process is non-resonant, i.e., electron ejection occurs as long as the photon’s energy exceeds the ionization threshold. It can also be described within the framework of single-particle ionization. On the other hand, the second process is inherently resonant because the two electrons must be excited into a well-defined auto-ionizing state. Moreo-ver, electron-electron interactions are responsible for the excitation and subsequent auto-ionization. The quantum-mechanical interference between these two ionization pathways results in highly asymmetric [39] dependence of the ionization cross-section on the photon energy. Of course, wave interference is not specific to quantum mechanics, and a great number of classical systems exhib-iting Fano resonance have since been identified. Most recently the concept of Fano resonances was introduced to the field of photonics [40] and then metamaterials [26]. By analogy with the original atomic system, a photonic struc-ture must possess two resonances generally classified as

248 A.B. Khanikaev et al.: Fano-resonant metamaterials © 2013 Science Wise Publishing &

“bright” (i.e., exhibiting strong coupling to incident light and short radiative lifetime) and “dark” (exhibiting weak radiative coupling and long lifetime). Typically (although not always) the near-field interaction between the two resonances is analogous to electron-electron interaction in an atom. Another analogy of metamaterials and atomic systems is that Fano resonances generally originate from the response of isolated meta-atoms whose periodic arrangement plays no crucial role.

Several comprehensive reviews of various manifesta-tions of Fano resonances in nanostructures and metama-terials [41, 42] have recently appeared. A number of recent developments motivate the present Review. First, the improved experimental diagnostic tools such as EELS [43] and near-field optical microscopy [36] enabled much more direct look at Fano interferences and resonances. Second, the improved theoretical understanding of the phenom-enon [37, 44–47] enabled the design and intuitive analytic/semi-analytic treatment of novel two and three-dimen-sional metamaterial structures in which Fano resonance is an important feature. Finally, Fano-resonant metama-terials and meta-surfaces started featuring prominently in a variety of applications. These applications include solar energy harvesting and conversion [48], refractive index sensing (especially for bio-sensing applications) [21, 23, 49–52], and broadband manipulation of light propagation [37]. Another recent important development in the field is the realization of the possibility of coupling the internal molecular degrees of freedom, such as infrared molecular vibrations, to electromagnetic metamaterial resonances [21, 23], with the resulting Fano-like interferences.

The rest of the review is organized as follows. The basic analytic theory of Fano-resonant metamaterials is introduced in Section 2. The most important properties of Fano-resonant structures, such as the large field enhance-ment at the dark resonance’s frequency, the asymmetric and spectrally steep line shapes of the reflection/transmis-sion spectral profiles, and the connection between Fano resonance and EIT, are derived. Optical properties of the single-layer metamaterials (also known as meta-surfaces) are described in Section 3. Fano-resonant meta-surfaces are the most common and most amenable to large-area fabrication examples of meta-materials. For that reason, practical applications based on meta-surfaces, such as sensing, are presently the most mature. Section 4 dis-cusses meta-surfaces with a ground plate, which is a spe-cific class of meta-materials that falls between single-layer meta-surfaces and fully three-dimensional metamateri-als. The least explored at the moment three-dimensional (multi-layer) Fano-resonant metamaterials are discussed in Section 5. Applications of 3D metamaterials to molding

ωb, τb

ωd, τb

Sin St

Sr

κ

Figure 1 Schematics of the simplest Fano resonant system described by Eq. (1). The two resonators (“bright” and “dark”) are coupled to each other, and the bright resonator is coupled to an input/output port that represents incident, reflected, and transmitted light.

light propagation, including “slow” light structures, are also reviewed. The approaches to dynamic tuning of meta-surfaces are discussed in Section 6. Section 7 discusses the opportunities for refractive index sensing and molecu-lar fingerprinting using FRAMs. The Summary and Future Outlook are presented in Section 8.

2 Mathematical description of Fano-resonant optical structures

The basic mathematics of Fano resonances in photonic systems can be best developed within the framework of temporal coupled mode theory [53–55] describing the interaction of two coupled photonic resonators with light. We further assume that only the first (“bright”) resonator with natural frequency ωb is directly coupled to incident light, while the second (“dark”) resonator with natural frequency ωd interacts with the light indirectly due to its coupling to the first resonators. A conceptual schematic of such a system is shown in Figure 1, where it is assumed that only a single channel of light reflection/transmis-sion is active. Such situation would correspond to a sub-wavelength photonic meta-surface which does not produce higher diffractive orders of light in either trans-mission or reflection. Near-field interactions between meta- molecules comprising the meta-surface could be responsible for the coupling between the two resonances. Because the two resonances are differently coupled to far-field radiation, they can have very different radiative life-times τb«τd justifying their designation as bright and dark.

The dynamics of such two-resonant system can be described by the equations of the temporal coupled mode theory [53, 55]:

A.B. Khanikaev et al.: Fano-resonant metamaterials 249© 2013 Science Wise Publishing &

-1

-1

- ( + ) ,- ( ) 0,

in j tb b b

d d

b j j b j d S ed j j d j b

ωω τ α

ω τ

+ =+ + =

�

�κκ

(1)

where b and d are the field amplitudes in the bright and dark resonators, respectively, αb is coupling strength of the bright resonator to the wave in the waveguide, Sin and ω are amplitude and frequency of the incident light wave, respectively, and κ is a real number determining coupling strength between the resonators [53]. In Eq.(1) the ampli-tudes b, d and Sin are normalized such that |b|2 and |d|2 is the energy stored by the resonators and |Sin|2 is the inci-dent power.

Note that while the simple single-input/single-output port theory of Fano interference described by this model captures most of the phenomena described in this Review, it falls short of describing polarization conversion and manipulation that occurs in double-continuum Fano (DCF) systems that are described in Section 5. Polariza-tion of light represents a second set of input/output ports that can be resonantly coupled to each other through an asymmetric Fano-resonant metamaterial. Multiple input/output ports can also represent multiple waveguides coupled to Fano resonant cavities [40] or multiple dif-fracted waves in periodic systems with the Fano reso-nance [56].

Assuming that the system is symmetric with respect to the transmission and reflection channels, the amplitudes of the scattered waves can be expressed [53, 55] in terms of the modal amplitude of the bright mode b as *r

bS bα= and * ,t inbS S bα= + giving rise to the following expressions for

the complex reflection/transmission coefficients:

2 -1

-1 -1 2

-| | ( - - ), 1 .

( - - )( - - )-

r tb d d

in inb b d d

j jS Sr t rS j j S

α ω ω τ

ω ω τ ω ω τ κ= = = = +

(2)

In general, when Ohmic losses are present in the system in addition to the radiative losses, the total lifetime τ of the modes of the system is expressed through the Ohmic τo and radiative τR lifetimes as τ τ τ= + -1[ 1/ 1/ ] .o R In the absence of the Ohmic losses, energy conservation condition (1 = |r|2+|t|2) imposes a relation between the radiative lifetime of the bright mode R

bτ and its radiative coupling parameter αb. By calculating the decay rate of the total stored energy in the absence of the excitation (Sin = 0), we find that

2 2 2 2(| | | | ) -2| | / | | ,R

bd d dW d b b bdt dt dt

τ= + = = where it was

assumed that the dark mode has infinite radiative lifetime

( )τ =1/ 0Rd in the absence of resonator coupling (κ = 0). On

the other hand, in the lossless case the energy can only be

1.0

0.8

0.6

0.4

0.2

0.8

0.6

0.4

Tra

nsm

ittan

ce |t

|2R

efle

ctan

ce |r

|2

0.2

0.80 0.85 0.90 0.95 1.00Frequency (ω/ωb)

δ=-0.1ωb

δ=0.1ωb

δ=0

1.05 1.10 1.201.150

1.0

Figure 2 Transmission and reflection spectra of the Fano-res-onant system described by Eqs. (1,2) with finite Ohmic losses. The three curves correspond to the cases of Fano resonance with δ = (ωd-ωb) = ± 0.1 ωb (red and blue curves) and electromagnetically induced transparency ωd = ωb (green curve). Parameters: τ ω=10/ ,R

b b 500/o o

b d bτ τ ω= = and κ = 0.05ωb.

radiated out:

2 2 2 2 2-2 | | / | | | | -2 | | | | ,R r tb b

d W b S S bdt

τ α= = + =

thereby providing the sought after relationship: 21/ | | .R

b bτ α=

Typical transmission/reflection spectra of the model Fano system described by Eqs. (2) are plotted in Figure 2 for different values of the spectral detuning δ = ωb-ωd between the dark and bright resonances assuming that ωb is fixed and Ohmic losses are negligible. The spectra reveal a broad reflection peak centered at ωb and a sharp spectral feature in the proximity of the dark frequency ωd. The strong asymmetry of the reflection/transmission spectra for (ωb≠ωd) are characteristic of the Fano interfer-ence between the two modes. Both the sharpness and the amplitude of the Fano feature are reduced by finite non-radiative (Ohmic) losses.

One particular case of the Fano interference corre-sponding to spectrally matched resonances, ωb = ωd, is referred to as electromagnetically induced transparency (EIT). A number of EIT implementations based on planar meta-surfaces (including the complementary ones) [26–36, 51, 57] and, more recently, on three-dimensional [37] met-amaterials have been proposed. The physical manifesta-tion of EIT shown in Figure 2 (green line) is the emergence of a narrow transmission peak due to Fano interference between the bright and dark states. Only a broad trans-mission dip due to the excitation of the bright resonance is observed in the absence (κ = 0) of such interference.


Another important feature of the Fano resonance is the enhanced optical energy stored by a Fano-resonant struc-ture when illuminated by light with the frequency ω coin-cident with that of the dark resonance’s ωd [45]. Figure 3 illustrates the point by plotting the energy W = |d|2+|b|2 stored by the system described by the Eqs. (1, 2). The stored energy can reach very high values near ω = ωd when the dark mode is excited. The resulting enhancement of the optical near-field is the main enabler of the metamate-rials-based sensing of minute analyte quantities: protein monolayers [21–23, 58, 59], trace amounts of hydrogen gas [60], or single-layer graphene [61–63] to name just a few. The field enhancement peak manifests itself as the absorption peak shown in Figure 3 (right), thereby giving rise to the recently proposed [55] electromagnetically induced absorption. It is worth noting that the absorp-tion peak does not exactly coincide with either minimum or maximum of the reflectivity as has been recently noted [21, 46].

The highly simplified bi-resonant Fano model expressed by Eqs. (1, 2) has another important utility: it provides simple analytic expressions for transmission/reflection coefficients that can be used for analytic multi-parameter fitting of the experimentally obtained data. Such fitting enables numerical extraction of the most important parameters of the analytical model such as the modes’ lifetimes and spectral positions. In particular, it has been shown that the Fano model based on four-param-eter fits of the results of the electromagnetic simulation [44–46] accurately reproduce the spectra of a Fano-reso-nant meta-molecule. The bi-resonant Fano model has also been used to precisely describe experimentally observed

sharp Fano and EIT-like features of microwave metama-terials [35]. Even more importantly from the applications standpoint, the bi-resonant fits can be used for spectro-scopic characterization of small amounts of analyte added to metamaterials. For example, spectral shifts and lifetime modifications can be used for characterizing the complex conductivity of single layer graphene [63].

It follows from Eq. (3) that the Fano expression for the reflectivity can be exactly represented in the form of the double Lorenzian under the assumption that the cou-pling strength κ between the dark and bright resonances is frequency-independent:

ω

ω ω ω ω= +� �1 2

1 2

( ) ,- -A A

r

(3)

where the complex amplitudes A1,2 and frequen-cies 1,2 ,ω� respectively, are related to the origi-nal parameters of the Fano model through:

2 11

1 2

-- | | ,

-d

bA jω ω

αω ω

=

� ��

2 22

1 2

-| | ,

-d

bA jω ω

αω ω

=

� ��

and

ω ω ω ω ω κ= + ± +� � � � � 2 21,2

1 [( ) ( - ) 4 ],2 b d b d where the complex

frequencies are defined as -1b b bjω ω τ= +� and -1 .d d djω ω τ= +�

In the approximation of weak coupling corresponding to the case of high quality Fano resonance ( κ δ�| | | | /2, where -1 -1- ( - ) ( - )b d b d b djδ ω ω ω ω τ τ= ≡ +� � � is the complex-val-ued mode detuning) a simpler and more instructive ansatz can be used. In this approximation the radicals in Eq. (3) are expanded up to second order in the small parameter

κ δ2 / which gives the double Lorenzian expression of the form:

200 1.0 0.5

0.4

0.3

0.2

0.1

0

0.5

0

1.0

0.5

0

100

00.80

A B

0.85 0.90 0.95 1.0 1.101.05Frequency (ω/ωb)

0.80 0.85 0.90 0.95 1.0 1.101.05Frequency (ω/ωb)

Ref

lect

ance

|r|2

Tota

l ene

rgy

stor

ed b

y th

e re

sona

tors

|b|2 +

|d|2

Abs

orba

nce

1-|t|

2 -|r|2

Tra

nsm

ittan

ce |t

|2

Figure 3 (A) Energy of the dark mode W = |d|2+|b|2 (red line) superimposed with the reflectivity (green line) spectrum. Note that the field enhancement peak falls between the peak and dip of the reflectance spectrum. (B) Enhanced absorbance (blue line) at the Fano resonance

superimposed with the transmittance (green line). Parameters used are: ωd = 0.9ωb, 10/ ,Rb bτ ω= 500/ ,o o

b d bτ τ ω= = and κ = 0.02ωb.


φ φα αω

ω ω τ ω ω τ≈ +

+ +

ˆ ˆ2 2

-1 -1

ˆ ˆ- | | - | |( ) ,

ˆ ˆˆ ˆ( - ) ( - )

b di ib d

b b d d

j e j er

j j

(4)

where the renormalized frequencies ω /ˆ b d and life-times τ /

ˆb d of the “dressed” bright and dark modes are

2

1ˆ ( ) ,b bRe Re κω ω ω

δ

= ≈ +

� � 2

2ˆ ( ) - ,d dRe Re κω ω ω

δ

= ≈

� � 2

-1 -11

ˆ ( ) ,b bIm Im κτ ω τ

δ

= ≈ +

� � and 2

-1 -12

ˆ ( ) - ,d dIm Im κτ ω τ

δ

= ≈

� � while the effective radiative coupling parameters of the

modes are 2 2

2 22 2

ˆ| | | |2b b

δ κα α

δ κ

+=+

�� and

22 2

2 2ˆ| | | | .

2d bκ

α αδ κ

=+�

The expression (4) has simple physical meaning of the scattering of light by two independent short-living super-radiant and long-living sub-radiant modes formed by the hybridization of the original bright and dark modes. It is important to note that in addition to the corrections to the frequencies and the lifetimes due to the interaction of the modes and modification of their radiative coupling caused by their hybridization, there is also a relative phase between the new “dressed” modes ˆ ˆ ˆ- ,b dδφ φ φ= where

2 2

2 2ˆ arg

2bδ κ

φδ κ

+= +

�� and

2

2 2ˆ arg .

2dκ

φδ κ

= +� This addi-

tional phase shift ˆδφ is crucial for reproducing strongly asymmetric features seen in the experimental spectra of the Fano-resonant systems. To illustrate that Eq. (4) is indeed a good approximation to the Fano model, the cal-culation results obtained with the use of the exact expres-sion for |r|2 of Eq. (2) and the approximate expression Eq. (4) are plotted alongside in Figure 4. Examples of using Eqs. (3, 4) for multi-parameter fitting of experimental spectra are given in Section 6 in the context of investigat-ing the modification of the reflectivity spectrum by adding a thin conductive or dielectric layer to a meta-surface. In that case resonant frequencies ˆ dω and ˆ bω are shifted and modes lifetimes ˆ

dτ and ˆbτ are modified by the amount

determined by physical properties and thickness of the perturbing layers.

3 Single-layer Fano-resonant metamaterials

Planar (two-dimensional) meta-surfaces are the most common and most thoroughly studied Fano-resonant structures because of their fabrication simplicity and his-torical connection to frequency selective surfaces well known in the microwave field. Such meta-surfaces can be

0.80 0.850.70 0.75 0.90 0.95 1.00

Exact Fano result Eq.(2)

Double Lorenzian Eq.(4)

1.101.05Frequency (ω/ωb)

1.0

0.8

0.6

0.4

0.2

0

Ref

lect

ance

|r|2

Figure 4 Comparison of the reflectivities calculated with (a) the exact Fano expression Eq. (2) with the parameters ωd = 0.75ωb,

6.25/ ,Rb bτ ω= 300/ ,o o

b d bτ τ ω= = and κ = 0.05ωb, and an approxi-mate double Lorenzian Eq. (4) with the renormalized parameters ˆ 1.0071 ,b bω ω= ˆ 6.2942/ ,R

b bτ ω= ˆ 300/ ,ob bτ ω=

2ˆ| | 0.158,bα = ˆ 1.4231deg.,bφ = ˆ 0.7429 ,d bω ω= ˆ 127.0029/ ,R

d bτ ω= ˆ 300/ ,od bτ ω=

2ˆ| | 0.0044,dα = and ˆ -62.3567deg.dφ =

either periodic or aperiodic [5, 64–67]. The first studies of planar systems showing asymmetric Fano-like response date back to fundamental works of Wood and Rayleigh [68, 69] who studied metallic diffraction gratings. Reso-nant phenomena in such gratings stem from periodic pattern on the metal surface which gives rise to Bragg diffraction. Resonances appear at the frequencies and critical angles corresponding to the onset of a particular diffraction orders manifesting in reflectivity spectra as sharp asymmetric spectral features. It is interesting that Fano himself has contributed to study of diffraction grat-ings and developed rigorous theory of light scattering by these structures [56].

While metallic diffraction gratings and frequency selective surfaces in general were among common tools for optical and microwave engineers for many years, it was not until the discovery of the extraordinary optical trans-mission (EOT) [70] that these systems gained interest from the broad community of physicists working in photonics. In 1998 Ebbesen and coauthors experimentally observed the EOT of visible light through perforated silver film and attributed the effect to the excitation of the surface plasmon polaritons (SPPs). Spectrally, the EOT takes place close to the frequency corresponding to the SPP’s folded dispersion curve, suggesting that the excitation of the SPPs (or similar waves) facilitates light tunneling through sub-wavelength sized holes. The asymmetric shape of the


transmission peaks encouraged the attribution of the EOT to Fano-type interference [71, 72] between broad feature-less transmission through the holes and narrowband exci-tation of the “dark” SPPs by the periodic structure. Similar Fano-shaped reflection spectra are also observed in peri-odic arrays of plasmonic particles [73–75]. Detailed analy-sis of the EOT is beyond the scope of this Review, and in what follows we concentrate on meta-surfaces whose unit cell is comprised of one or more meta-molecules support-ing bright and dark resonances. Frequencies and lifetimes of such resonances can be engineered through meta-mol-ecules’ design.

3.1 Engineering of constituent meta-molecules

Spatial symmetry of the constituent meta-molecules plays the key role in lifetime engineering of Fano res-onances as was first demonstrated in the microwave spectral range for meta-surfaces comprised of a periodic array of asymmetrically split rings [26] shown in Figure 5. Near-field interaction between the two halves of the ring results in two modes: bright symmetric (current flowing in the same direction in both ring halves) and dark anti-symmetric (currents flowing in opposite directions in both ring halves). The lifetime of the dark mode respon-sible for transmission dip is controlled by two factors: the degree of asymmetry γ (responsible for the radia-tive lifetime) and substrate loss tangent ε (responsible for the non-radiative lifetime). For small degree of sym-metry breaking (γ ≤ 0.1), strong Fano features are shown in the inset of Figure 5A for the horizontally polarized electric field. The same design was later used to engi-neer sharp Fano resonances in the THz range [30], and similar (but complementary) design of asymmetric slits in plasmonic surface was used at visible frequencies [76]. Conceptually similar mirror symmetry breaking of a bi-layer [29] and single-layer plasmonic meta-sur-faces shown in Figure 5B, C and a single-dolmen [32] meta-molecule shown in Figure 5C were also shown to produce characteristic Fano spectra in the optical part of the spectrum. In all cases shown in Figure 5 the elec-tric field polarization is orthogonal to the remaining mirror symmetry plane.

A number of other planar Fano systems with resonant responses controllable by symmetry breaking of meta-molecules have been recently proposed, including those based on metallic truncated nanoshells [77], plasmonic Fano resonances in nano-crosses [78] and disk-ring plas-monic nanostructures with record values of quality factors

[79, 80]. Asymmetry can also be introduced by employing different materials (e.g., Au and Ag) for making different segments of an otherwise geometrically symmetric struc-ture [81, 82]. In this case the degree of asymmetry is con-trolled by the mismatch in the materials’ properties (e.g., their dielectric permittivities). Combining nanoparticles of different shapes (i.e., nanospheres, core-shell particles, etc.) into heterodimers [83] was also shown to result in Fano intereferences and asymmetric scattering. Although even single nanoparticles can exhibit Fano resonance [84], by controlling arrangement of multiple nanoparti-cles in oligomers enables unprecedented control of their resonant response.

In the structures with two elements (sometimes referred to as dimers [83, 85–89]) comprising the meta-molecules, such as the ones shown in Figure 5A, the Fano resonance stems from the interaction between individual modes of these elements. Spectral repulsion between the modes necessitates that the frequencies of the hybrid bright and dark modes are mismatched. More complex meta-molecules containing more elements and resonances enables independent control of the frequen-cies of the bright and dark resonances. When the two fre-quencies are spectrally matched, EIT-like response can be obtained [27, 28] as shown in Figure 5B. By appropriately choosing the length of the horizontal antenna, its bright electric dipolar resonance can be spectrally matched to the electric quadrupolar resonance of the two vertical antennas, which is dark for normally incident light. When incident light excites the dipole resonance of the horizon-tal antenna, the antenna’s near field excites the vertical antennas’ quadrupolar resonance. These two antennas, in turn, depolarize the horizontal antenna causing the reflectivity to drop, thereby giving rise to the EIT of the meta-surface. The details of the optical response of the meta-surface (including the quality factor of the EIT reso-nance and the depth of the reflectivity dip) are determined by the degree of near-field coupling [κ-coefficient in the Eq.(1)] between the bright dipolar and dark quadrupolar modes controlled by the antenna’s displacement s defined in Figure 5B.

The dark (sub-radiant) character of the electric multi-pole and magnetic dipole resonances of meta- molecules comprising the meta-surface is predicated on their forming a regular periodic lattice which prevents side-scattering. Depending on the exact structure of constitu-ent meta-molecules, long-range interaction may be either suppressed (incoherent meta-surfaces), or enhanced (coherent meta-surfaces) [64, 90]. Coherent effects in Fano-resonant meta-surfaces are reviewed below. Even more sophisticated meta-molecules comprised of clusters


of plasmonic nanoparticles (sometimes referred to as plasmonic oligomers) have been used to design Fano reso-nant meta-surfaces with electric [33, 36, 50, 91–101] and magnetic [102] subradiant modes. Plasmonic oligomers have been fabricated using both bottom-up assembly [94] or, more commonly, top-down fabrication.

In addition to plasmonic systems, recently there was a surge of interest in so called “all-dielectric” metamaterials made of high-index dielectric nanostructures [103, 104]. Such systems can be especially promising in the near IR and visible frequency ranges were the lifetimes and quality factors of the Fano resonances are limited by strong losses in metals. In their recent work [105] Miroshnichenko and Kivshar have theoretically shown that light scattering by all-dielectric oligomers exhibits well-pronounced Fano resonances originating from the optically induced mag-netic dipole modes of individual high-dielectric nano-particles. By comparing all-dielectric structures to their

plasmonic counterparts they have shown that Fano reso-nances in all-dielectric oligomers are also less sensitive to structural variations thus offering a more practical alter-native for many nanophotonic applications.

3.2 Imaging of Fano-resonant meta- molecules in the near field

Although spectroscopic signatures of Fano interference have been conventionally observed in far-field measure-ments of transmission, reflection, or extinction, recent advances in near-field imaging enable direct charac-terization of the charge density distributions on the meta-surface. Such characterization of multi-resonant meta-surfaces yields a more complete signature of the Fano interference than its far-field counterpart because the same scattering pattern can be caused by different

0

100 0

-20

-404 6

γ=0.

06

γ=0.

11

γ=0.

23 γ=0.

33

8 10

Frequency (GHz)

80

A

60

40

20

0

ε′′=0.05

ε′′=0

0.1 0.2 0.3 0.4Asymmetry (γ)

Qua

lity

fact

or (

Q)

Tra

nsm

issi

on (

dB)

B

0.8

0.6

0.4

0.2

0120 140 160 180 200

s=50 nm

Frequency (THz)

Tra

nsm

ittan

ce/ r

efle

ctan

ce

220

0.2

C

0.1

0800 1000 1200 1400

Wavelength (nm)

Ext

inct

ion

(1-T

)

Figure 5 Lifetime engineering of Fano resonances by breaking one of the mirror symmetries of the constituent meta-molecules. (A) Micro-wave and (B) optical meta-surfaces with gradually reduced spatial symmetries. (C) Single-dolmen meta-molecule with only one remaining mirror symmetry. In all three cases Fano features are observed for the electric field orthogonal to the remaining mirror symmetry plane. Reproduced by permission from Ref. [26] (A), Ref.[29] (B), and Ref.[32].


charge distributions. To unambiguously demonstrate the root cause of Fano resonance one has to be able to spa-tially resolve the charge redistribution between differ-ent portions of the Fano-resonant meta-molecules. Two near-field imaging techniques have recently emerged. The first technique is the Electron Energy Loss Spectroscopy (EELS) [106] that enables direct mapping of the spatial distribution of all optical resonances of any structure, including plasmonic [107, 108] or dielectric-based nano-structures (see [43] for a comprehensive review and addi-tional references). Because electrons enter the near-field of a nanostructure, EELS can be used for mapping either bright or dark resonances. The resolution of EELS is fun-damentally limited the de Broglie wavelength of an elec-tron which is typically much smaller that the diffraction limit of light. One potential limitation of EELS is that its relationship to the local optical density of states (LDOS), which is of primary interest in determining optical prop-erties of nanoparticles, is clearly established [43] only for two-dimensional nanostructures. The advantage of EELS over near-field optical microscopy outlined below is that it can potentially probe the completely dark resonances

of symmetric meta-molecules that are totally decoupled from the far-field optical excitation.

The second technique illustrated by Figure 6 is the interferometric scattering-type near-field scanning optical microscopy (s-NSOM) [109]. It has been recently [36] applied to two infrared Fano-resonant meta-surfaces fabricated using top-down electron beam lithography comprised of (a) highly symmetric heptamer structures comprised of plasmonic disks shown in Figure 6A, and (b) asymmetric π-shaped nanorod antennas shown in Figure 6B. While the far-field signature of the Fano interference is a conventional asymmetric reflectivity spectrum shown in Figure 6C, its near-field signature is more interesting. Figure 6D indicates that the Fano interferences results in the phenomenon of intensity toggling: strong spatial redistribution of the high-field areas inside the metamol-ecules upon crossing the “dark” resonance. This toggling represents a direct experimental evidence of interfering electromagnetic eigenmodes of plasmonic metamolecules responsible for the phenomenon of the Fano resonance. As near-field tools such as s-NSOM become more widely available, we anticipate that more research will be done

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Figure 6 NSOM images of Fano interference and intensity toggling in infrared meta-surfaces. (A, B) Spatially-symmetric plasmonic hep-tamers (A) and asymmetric pi-structures (B) are imaged using tunable CO2 laser. (C) Polarized reflectivity from the π–structures. Vertical polarization reveals far-field Fano interference around λ≈10 μm. (D) Fano interference manifests in the near-field as intensity toggling: the left antenna is bright at the spectral point A and dark at the spectral point D. From Ref.[ 36].


on detailed mapping of charge distributions on the meta-surfaces. Thus obtained information will enable detailed comparisons with numerical simulations which, so far, have been the only reliable source of information about electric and magnetic field enhancements and charge dis-tributions on Fano-resonant meta-surfaces.

4 Meta-surfaces with a ground plateA special class of Fano-resonant metamaterials involves planar meta-surfaces separated by a dielectric spacer from a metallic ground plate shown in Figure 7A, B. The ground plate blocks the transmission and provides an ultra-broadband reflection background. In addition, the background reflection interferes with the resonances of the top meta-surface, resulting in reduced total reflection and therefore enhanced absorption. Such meta-surfaces with a ground plate have been widely used for designing

“perfect” (zero-reflectivity) plasmonic absorbers in the terahertz and mid-infrared frequency ranges [110–116]. A wide variety of resonator geometries, including circular disks [110], square and rectangular patches[114], Swiss crosses [115], split-ring resonators [112] and straight plas-monic strips [60, 111, 113, 117–119] among others have been demonstrated to achieve nearly perfect absorption. Figure 7C and D show the disk on ground plate design from Ref. [110] that exhibits more than 90% absorption in both TE and TM polarizations over a wide range of inci-dent angles. The property of “perfect” spectrally-selective wide-angle infrared absorption has been proposed for several practical applications, including ultra-sensitive refractive index sensing [60, 110] and highly efficient thermal photovoltaics [48, 115] spectrally matched to the bandgap of a photovoltaic cell.

For meta-surfaces with a ground plate perfect absorp-tion implies vanishing reflection. That has been inter-preted in the past [120, 121] as the condition for matching the complex effective dielectric permittivity εeff(ω) and

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Figure 7 Ground plate based meta-surfaces used for designing “perfect” absorbers, narrow-band infrared emitters, and ultra-sensitive refractive index sensors (from [110] and [111]). (A) When the resonators are far from the ground plate, incoming radiation is mostly reflected by the excitation of plasmonic resonances. (B) When the resonators are close to the ground plate, induced image currents in the ground plate reduce the far-field radiation from the resonators, resulting in lower radiative damping rates. (C) and (D), the plasmonic absorber design and simulated angular dispersions of the absorbance peak for both TE and TM polarizations taken from Ref. [110].


magnetic permeability μeff(ω) of the resulting layer so that reflectivity vanishes for some unique frequency ωa that sat-isfies the impedance-matching condition εeff(ωa) = μeff(ωa). However, the assignment of effective bulk parameters to a metal-terminated layer is ambiguous. For example, the bi-anisotropic response of such surface is inherently neglected. The assignment of a single quantity, frequency-dependent surface impedance z(ω) [111], circumvents this problem. Under a single-resonance model [111], the surface impedance of the ground plate terminated meta-surface can be expressed using the coupled mode theory as ( ) ,

( - )R

a Q

zi

γω

ω ω γ=

+ where ωa is the resonant frequency

of the eigenmode, and γR and γQ are the radiative and resistive damping rates, respectively. While the resistive damping rate is intrinsic to the choice of materials, the radiative damping rate can be controlled by changing the intermediate spacer thickness. When resistive and radia-tive damping rates of the resonant mode are equal (γR = γQ), the impedance matching with vacuum (z = 1) is achieved resulting in 100% absorption at the resonant frequency ω = ωa. Note that γR is controlled by the spacing between the meta-surface and the ground plate because the radia-tive damping rate of the resonant mode is reduced by the image current induced in the ground plate as illustrated in Figure 7A and B.

The capability of absorbing all of the incoming light with an ultra-thin plasmonic layer indicates strong near-field enhancement at the plasmonic resonance. Recent experimental estimates [111] of the propagation length of the surface plasmon responsible for absorption confirmed its classification as a localized surface plasmon resonance (LSRP). The strongly sub-wavelength nature of the LSRP makes the absorber’s performance virtually independ-ent of the incidence angle as shown in Figure 7C and D. Another important consequence of the sub-wavelength nature of ground plate based absorbers is that multiple absorbing meta-molecules can be placed within the same macro-cell, thereby expanding the absorber’s spectral bandwidth [68, 115, 118, 122].

5 Three-dimensional metamaterialsWhile single-layer and few-layer [29, 97, 123, 124] metama-terials (of which meta-surfaces with a ground plate is just one example) are relatively straightforward to fabricate, considerable challenges remain to fabricating genuine three-dimensional metamaterials. Several three-dimen-sional metamaterials, including a negative index metama-terial [125] and a 3D chiral polarizer [126] have been recently

fabricated, but the task of demonstrating 3D metamateri-als exhibiting Fano resonances still appears challenging with the exception of lower (microwave) frequencies [27]. Experimental challenges have not deterred theoretical explorations of such metamaterials because using the third dimension enables novel applications unattainable in planar geometries. One such application is slow light (SL): the phenomenon first realized in three-level atomic systems that relies on EIT to generate extremely sharp frequency dependence of the refractive index n(ω). The resulting group velocity ( ) / ( / )gv c n dn dω ω ω= + in the media comprised of atoms exhibiting the EIT can then be made much smaller than the speed of light c in vacuum.

As was discussed in Section 3, Fano-resonant meta-surfaces have been shown to emulate EIT. Therefore, a stack of such meta-surfaces constitutes a three-dimensional metamaterial that can be employed for SL applications. Because the reduction in group velocity is accompanied by pulse compression and energy density increase, SL pro-motes stronger light-matter interactions and finds numer-ous applications in various linear and nonlinear devices [127]. One limitation of the EIT approach is that the band-width of the resulting slow light is rather limited. Recent microwave experiments [27] have demonstrated that the SL’s bandwidth is not improved by stacking Fano-resonant microwave meta-surfaces with different resonant frequen-cies. The EIT approach to producing slow light suffers from yet another serious limitation: a rather moderate value of the product of the time delay Tdel in the medium and the bandwidth of the delayed in the medium pulse δωdel. While increasing the length of the delay medium Ldel increases Tdel, it also decreases δωdel in the same proportion because of the frequency-dependence of vg(ω), thereby keeping the product (δωdel Tdel) independent of Ldel. Three-dimensional Fano-resonant metamaterials may provide a surprising platform for engineering the bandwidth-delay product which is directly proportional to the metamaterial’s length. This is accomplished by utilizing the so-called double-continuum Fano (DCF) resonance described in the original Ugo Fano’s paper [39]. It was recently demonstrated [37] that DCF enables broad-band SL.

A typical meta-molecule comprising a DCF meta-surface shown in Figure 8C possesses no mirror reflec-tion symmetries, thereby coupling the dark (quadrupolar) resonance to both (x- and y-polarized) in-plane dipoles. Thus, the dark (discrete) resonance of a DCF meta-surface is coupled to two electromagnetic continua, x- and y-polar-ized electromagnetic waves propagating in the z-direc-tion, thus justifying its designation. A stack of identical DCF meta-surfaces supports a propagating mode which is spectrally broad, yet possesses a “flat” ω=∂ ∂ �( / )gv k c


segment as schematically shown in Figure 8A [37]. The low group velocity spectral region occurs for ω≈ωQ, where ωQ is the frequency of the dark resonance. By stacking N meta-surfaces with adiabatically varying Fano resonance frequencies ( 1 ) ( ),j N

Q Q zω ω= ⋅⋅⋅ ≡ as shown in Figure 8A, one can ensure that different spectral components of the wave-packet are slowed down at different locations inside the 3D meta-material, as illustrated in Figure 8B.

The result of such adiabatic dispersion engineering is that a propagating wave packet can be slowed down over a broad frequency range with spectrally uniform group velocity and transmission coefficients as shown in Figure 8C. Therefore, by stacking more meta-surfaces one can enhance the spectral bandwidth ω0-Δω < ω < ω0+Δω over which the pulse can be uniformly slowed down, thereby indefinitely increasing the bandwidth-delay product (δωdel Tdel). Note that such increase is impossible

to accomplish with EIT by increasing the number of iden-tical stacked meta-surfaces because the dependence of vg(ω) and |T|2(ω) on the frequency shown in Figure 8D reduces the bandwidth.

6 Dynamic tuning of Fano-resonant metamaterials and meta-surfaces

Although Fano-resonant metamaterials can strongly enhance light-matter interactions, this comes at the expense of broadband response that may be desirable for some applications that require broad spectral range. To circumvent this limitation, large arrays of Fano-res-onant “pixels” with different spectral responses can be utilized. For example, arrays of individual pixels tuned

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Figure 8 Double-continuum Fano (DCF) approach to making slow-light metamaterials. (A) Photonic band structures of three sequentially stacked DCF meta-surfaces. (B) Different frequency components of the wave packet are slowed down inside different meta-surfaces. Inset shows unit cell of the metamaterial with red dot indicating position of the field intensity probe. (C) Spectrally-flat profiles of transmission, loss, and group velocity. (D) Same, but for a stack of meta-surfaces supporting EIT. From Ref. [37].


to various vibrational (fingerprint) resonances of biologi-cal molecules have been utilized for fingerprinting and orientation-sensing of protein monolayers [21]. However, for many other applications real-time control of Fano reso-nances is desirable, and several avenues for such dynamic tuning recently suggested for various (not necessarily Fano-resonant) metamaterials and meta-surfaces include using phase-change media [128, 129], liquid crystals [130–132], carrier manipulation in semiconductor substrates [133], electrical modulation with graphene [134–136], and electromechnical reconfiguration [137]. Some of these and other approaches are now applied to Fano-resonant structures. For example, one approach to dynamic tuning relies on controlling the symmetry of Fano-resonant meta-molecules embedded into elastic materials. It has been suggested that mechanically tunable plasmonic nano-structures can serve as a platform for dynamically tunable nanophotonic devices such as tunable filters and sensors. Gold heptamers have been embedded into a flexible matrix [138] to show that the reduction of the spatial symmetry of these plasmonic molecules caused by the mechanical stress can control the strength of the Fano interference.

Another promising approach is to utilize the optical response of single-layer graphene (SLG) for dynamic tuning. In particular, due to its controllable optical response, graphene was shown to allow modulation of the spectral position and quality factors of the Fano reso-nances. Strong optical absorption of SLG due to interband transitions was used to experimentally demonstrate the red-shifting and lifetime modification of Fano resonances [61] in the near-infrared part of the spectrum. Graphene is

also attractive for tuning the response of THz metamateri-als. SLG’s response at THz frequencies is predominantly resistive, and the resulting Ohmic loss was shown to be broadly tunable by electrostatic doping of graphene. Sub-stantial gate-induced switching and linear modulation of terahertz waves can thus be achieved in Fano-resonant meta-surfaces at the room temperature [62].

In contrast to resistive THz response, in the mid-IR spectral range graphene has a distinct plasmonic response which enables its application for spectral tuning of the Fano resonances as opposed to amplitude tuning in THz. Indeed, it has been recently shown [63] that inductive coupling of Fano resonant meta-surface with graphene enables a unique blue-shift tunability of the Fano reso-nances. It was experimentally shown that just by placing graphene on top of a meta-surface as shown in Figure 9, the Fano resonance can be blue-shifted as much as 30 cm-1

and reflectivity changed by more than 10%. Dynamic tun-ability can be achieved by electrostatic gating of graphene in a slightly different geometry [136].

Hybridization between Fano-resonant metama-terials and graphene has an additional appealing application: infrared spectroscopy of graphene. Spe-cifically, by using Eq.(3) for the reflectivity coefficient from the meta-surface and realizing that the addition of graphene to the meta-surface results in the spectral shift of the complex reflectivity poles from 1,2ω ω= � to

1,2 1,2 ,ω ω ω= +∆� we can use the complex-valued spectral shifts Δω1,2 to estimate the ratio of the imaginary and real parts of graphene’s conductivity. Specifically, it can be shown that for small spectral shift the following relation

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Figure 9 Single-layer graphene can be used to tune metamaterial resonances. (A) SLG laid on top of a Fano-resonant metamaterial. (B) Blue shifting of the metamaterial resonance by increasing the Fermi level EF of the SLG. From Ref. [62].


holds: 1,2 1,2( ) / ( ) ( ) / ( ),g gRe Im Im Reω ω σ σ∆ ∆ = where the complex conductivity of graphene σg is evaluated at the corresponding frequency of the first (bright) or second (dark) resonance. Graphene can be characterized as being low-loss, and its response as primarily inductive whenever Im(σg)>>Re(σg). Such spectral shift is illus-trated in Figure 9. Note that by using several metama-terial pixels tuned to different resonant frequencies it is possible to optically characterize graphene’s parameters (such as the Fermi level energy EF) at mid-infrared fre-quencies [63]. The importance of Fano-resonant meta-material for such infrared spectroscopy of graphene is underlined by generally weak response of graphene to mid-IR radiation. Strong optical field enhancement in close proximity of the meta-surface is responsible for considerable spectral shift of metamaterial’s resonance due to graphene’s conductivity.

7 Applications of Fano-resonant meta-surfaces to biological sciences

Fano-resonant metamaterials enable strong spectrally-selective enhancement of electromagnetic fields in their immediate vicinity, thereby considerably boosting the interaction of light with matter placed in close proximity of (or in direct contact with) the meta-surface. Therefore, even weak perturbation in the electromagnetic environ-ment of the metamaterial can significantly alter its scat-tering characteristics [79]. Small changes of the refractive index are particularly easy to detect optically using meta-materials. One of the most common applications the Fano resonant plasmonic metamaterials is, therefore, refractive index sensing [22, 31, 49–51, 139, 140]. Such sensors can be used for determining bulk concentrations of the analyte which contains chemically or biochemically relevant mol-ecules, such as glucose or proteins. The main consequence of altered dielectric environment on the Fano spectra is spectral shifts of the resonant peaks and change in their intensity. Absolute values of spectral shift can be directly related to mass accumulation of an analyte in the vicin-ity of the metamaterial. At the same time, for the case of absorptive media, strong field enhancement can result in broadening and decrease of the amplitude of sharp spec-tral features associated with Fano resonances. Analyzing this information enables the quantitative characterization of the analyte.

If the optical fields are localized within a distance Lloc from metamaterial’s surface, it can be used as an

attractive biosensor [23, 37, 141] which is particularly sen-sitive to binding events involving small molecules such as, for example, proteins. Moreover, spectral selectiv-ity of Fano-resonant meta-surfaces enables researchers to match the spectral positions of the “dark” resonances with those of the vibrational fingerprints of various bio-molecules such as proteins, DNA/RNA, and others. Using a diversity of Fano-resonant “pixels”, some of them tuned to molecular fingerprint, it is possible [37] to carry out in-depth characterization of the bio-molecules attached to metamaterial-based sensors. This is done by measur-ing sensor reflectivities |rbare|2(ω) and |rprot|2(ω) without/with the protein, followed by the analysis of the difference spectrum ΔR(ω) = |rbare|2-|rprot|2. In general, ΔR(ω) peaks at the frequency of the dark resonance ω≈ωQ. However, the spectral shape of ΔR(ω) can become considerably more complex whenever a vibrational resonance occurs at the frequency ωi close to ωQ. The difference spectrum contains detailed information Such measurement can be carried out either with completed mono and bi-layers of proteins [37], or even in real time during the buildup of protein monolayers. The resulting difference spectra are sensitive not only to the mass density of the protein, but also to pro-teins’ orientation on the surface of the sensor. Obtaining the information on protein’s orientation is important for designing high capacity biosensors that rely on the spe-cific orientation of antibodies and attachment proteins for capturing the antigens.

A typical example of such biosensor based on an array of Fano-resonant asymmetric metamaterials (FRAMMs) is shown in Figure 10. Spectral positions of Fano resonances are varied by scaling the dimensions of the individual meta-molecules inside each pixel. The FRAMM-based biosensor was applied to studying the sequential attach-ment of two protein monolayers. The first fusion protein A/G was used as an attachment monolayer for the second monolayer of antibodies (IgG) as shown in Figure 10. The resulting difference spectra ΔR(ω) for one (dashed lines) and two (solid lines) monolayers are shown in Figure 10 for three characteristic pixels. One of the pixels (green line) is tuned to the so-called Amide-I (C = O vibration of the protein backbone) resonance while the other two pixels are detuned from all vibrational resonances. By comparing the maximum values of ΔR for the detuned pixels to theo-retical predictions, the thicknesses of the monolayer and bi-layer were shown to be h1≈2.7 nm and H≡h1+h2≈7.9 nm, respectively. Owing to the enhanced interaction of light with the protein at the meta-surface, < 3 nm thick protein A/G monolayer causes easily detectable 4% reflectivity changes, indicating that molecular monolayer thickness can be reliably measured with nm-scale accuracy.


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Figure 10 Schematic representations of proteins’ mono- and bilayers binding to the metal surface and the equivalent dielectric model (left panel). Experimental spectra before (dashed lines) and after (solid lines) binding of IgG antibodies to three different FRAMM sub-strates immobilized by the protein A/G (right panel). Indicated reflectivity ratios vary with the spectral position of the FRAMMs’ resonant frequencies.

Further information about proteins’ orientation on the surface can be obtained by analyzing the ratios

( ) ( )( ) /bi monoQ max maxT R Rω =∆ ∆ of the peak reflectivities for a bi-

layer and a monolayer. Using the detailed information about the proteins’ structure from the protein data bank [37], it is possible to demonstrate that the attachment protein A/G is connected to the FRAMM with its helical axis almost normal to the FRAMM’s surface. This orien-tation of the attachment protein is indeed conducive to high-capacitance antigen sensing because the Y-shaped antibody attaches its stem (the Fc region) to protein A/G while leaving its Fab region available for antigen binding. Such detailed information about binding and orientation of proteins can be used for studying real-time dynamics and buildup of protein monolayers. Because FRAMM-based biosensing and fingerprinting provides more spe-cific information about attached biomolecules than the more common surface plasmon resonance (SPR) sensor, we can speculate that metamaterials-based biosensor will eventually become commonplace tools for biochemistry and biosensing.

8 Future outlookTremendous advances in theoretical understanding, experimental characterization, and sheer variety of Fano-resonant metamaterials and meta-surfaces that occurred

during this decade transformed these electromagnetic structures from objects of curiosity into powerful explora-tory tools. Two properties of Fano resonances, their sharp spectral features and strong field enhancements, have made them ideally suited for developing novel spectro-scopic and sensing tools. In the next few years we will most likely witness the expansion of Fano-resonant media into the nonlinear regime, where these two properties will be essential for enhancing light-matter interactions. Heterogeneous integration of Fano metamaterials with nanoscale objects such as quantum dots, quantum wells, graphene, and bio-molecules will be used for developing new active and switchable devices as well as for character-izing these inherently quantum structures. Experimental demonstrations of introducing gain media into metamate-rials will result in low-threshold lasers. And so the history will make a full circle: the phenomenon that originated in the quantum mechanical world and was most recently adopted by classical physicists will be used for enhancing and modifying quantum phenomena

Acknowledgements: The authors would like to acknowl-edge the support of the grant from the Office of Naval Research and the National Science Foundation.

Received April 5, 2013; accepted July 23, 2013; previously published online September 21, 2013


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