## Abstract

A review of nanoplasmonics is given. This includes fundamentals, nanolocalization of optical energy and hot spots, ultrafast nanoplasmonics and control of the spatiotemporal nanolocalization of optical fields, and quantum nanoplasmonics (spaser and gain-assisted plasmonics). This article reviews both fundamental theoretical ideas in nanoplasmonics and selected experimental developments. It is designed both for specialists in the field and general physics readership.

© 2011 Optical Society of America

## 1. Introduction

#### 1.1. Preamble

This is a review article on fundamentals of nanoplasmonics. Admittedly, the selection of the included subjects reflects the interests and expertise of the author.

We have made a conscious decision not to include such important and highly developed subject as SERS (Surface Enhanced Raman Scattering). The reason is that this subject is too large and too specialized for this article. There is an extensive literature devoted to SERS. This includes both reviews and original publications – see, e.g., Refs. [1–5] and a representative collective monograph [6]. Another important subject that we do not include in this review is the extraordinary transmission of light through subwavelength holes – there are extensive literature and excellent reviews on this subject – see, e.g., [7–11]. Also, due to limitations of time and space we do not cover systematically a subject of particular interest to us: the adiabatic nanoconcentration of optical energy [12]. There are many important experimental developments and promising applications of this phenomenon [12–22]. This field by itself is large enough to warrant a dedicated review. We only briefly touch this subject in Sec. 4.5.

Another important class of questions that we leave mostly outside of this review article are concerned with applications of nanoplasmonics. Among this applications are sensing, biomedical diagnostics, labels for biomedical research, nanoantennas for light-emitting diodes, etc. There exist a significant number of reviews on the applications of nanopalsmonics, of which we mention just a few below, see also a short feature article [23]. Especially promising and important are applications to cancer treatment [24, 25], sensing and to solar energy conversion [26].

Presently, nanoplasmonics became a highly developed and advanced science. It would have been an impossible task to review even a significant part of it. We select some fundamental subjects in plasmonics of high and general interest. We hope that our selection reflects the past, shows the modern state, and provides an attempt a glimpse into the future. Specifically, our anticipation is that the ultrafast nanoplasmonics, nanoplasmonics in strong field, and the spaser as a necessary active element will be prominently presented in this future. On the other hand, it is still just a glimpse into it.

#### 1.2. Composition of the article

In Sec. 2, we present an extended introduction to nanoplasmonics. Then we consider selected subfields of nanoplasmonics in more detail. Nanoplasmonics is presently a rather developed science with a number of effects and rich applications [23]. In the center of our interest and, in our opinion, the central problem of nanoplasmonics is control and monitoring of the localization of optical energy in space on the nanometer scale and in time on the femtosecond or even attosecond scale.

In Sec. 3, we consider ultimately small nanoplasmonic systems with size less or on the order of skin depth *l _{s}* where we employ the so-called quasistatic approximation to describe in an analytical form the nanolocalized optical fields, their eigenmodes and hot spots, and introduce the corresponding Green’s functions and solutions. This Section is focused on the spatial nanoconcentration of the local optical fields.

In Sec. 4 we present ideas and results of ultrafast nanoplasmonics and coherent control of nanoscale localization of the optical fields, including control in time with femtosecond resolution. We will describe both theoretical ideas and some experimental results.

One of the most important problems of the nanoplasmonics, where only recently solutions and first experimental results have been obtained, is the active and gain nanoplasmonics. Its major goal is to create nanoscale quantum generators and amplifiers of optical energy. In Sec. 5, we present theory and a significant number of experimental results available to date regarding the spaser and related polaritonic spasers (nanolasers or plasmonic lasers). We also consider a related problem of loss compensation in metamaterials.

## 2. Basics of nanoplasmonics

#### 2.1. Fundamentals

Nanoplasmonics is a branch of optical condensed matter science devoted to optical phenomena on the nanoscale in nanostructured metal systems. A remarkable property of such systems is their ability to keep the optical energy concentrated on the nanoscale due to modes called surface plasmons (SPs). It is well known [27] and reviewed below in this article that the existence of SPs depends entirely on the fact that dielectric function *ɛ _{m}* has a negative real part, Re

*ɛ*< 0. The SPs are well pronounced as resonances when the losses are small enough, i.e., Im

_{m}*ɛ*≪ −Re

_{m}*ɛ*. This is a known property of a good plasmonic metal, valid, e.g., for silver in the most of the visible region. We will call a substance a good plasmonic metal if these two properties

_{m}There is a limit to which an electromagnetic wave can be concentrated. We immediately note that, as we explain below, nanoplasmonics is about concentration of *electromechanical* energy at optical frequencies (in contrast to electromagnetic energy) on the nanoscale.

The scale of the concentration of electromagnetic energy is determined by the wavelength and can be understood from Fig. 1 (a). Naively, let us try to achieve the strongest light localization using two parallel perfect mirrors forming an ideal Fabry-Perot resonator. A confined wave (resonator mode) should propagate normally to the surface of the mirrors. In this case, its electric field **E** is parallel to the surface of the mirror. The ideal mirror can be thought of as a metal with a zero skin depth that does not allow the electric field of the wave **E** to penetrate inside. Therefore the field is zero inside the mirror and, due to the Maxwell boundary conditions, must be zero on the surface of the mirror. The same condition should be satisfied at the surface of the second mirror. Thus, the length *L* of this Fabry-Perot cavity should be equal an integer number *n* of the half-wavelengths of light in the inner dielectric, *L* = *nλ*/2. The minimum length of this resonator is, obviously *λ*/2. This implies that light cannot be confined tighter than to a length of *λ*/2 in each direction, with the minimum modal volume of *λ*^{3}/8.

One may think that it is impossible to achieve a localization of the optical energy to smaller volume than *λ*^{3}/8 by any means, because the ideal mirrors provide the best confinement of electromagnetic waves. There are two implied assumptions: (i) The optical energy is electromagnetic energy, and (ii) The best confinement is provided by ideal mirrors. Both these assumptions must be abandoned to achieve nanolocalization of optical energy.

Consider a nanoplasmonic system whose size is less than or comparable to the skin depth

*λ̵*=

*λ*/(2

*π*) =

*ω*/

*c*is the reduced vacuum wavelength. For for single-valence plasmonic metals (silver, gold, copper, alkaline metals)

*l*≈ 25 nm in the entire optical region.

_{s}For such a plasmonic nanosystem with *R* ≲ *l _{s}*, the optical electric field penetrates the entire system and drives oscillations of the metal electrons. The total energy of the system in this case is a sum of the potential energy of the electrons in the electric field and their mechanical kinetic energy. While the magnetic field is present, non-relativistic electrons’ interaction with it is weak proportional to a small parameter

*v*/

_{F}*c*∼

*α*∼ 10

^{−2}, where

*v*is the electron speed at the Fermi surface,

_{F}*c*is speed of light, and

*α*=

*e*

^{2}

*/h̄c*is the fine structure constant. Thus in this limit, which is conventionally called quasistatic, the effects of the magnetic component of the total energy is relatively small. Hence, this total energy is mostly

*electromechanical*(and not electromagnetic) energy. [At this point, it may be useful to refer to Eq. (104), which expresses the Brillouin formula for the total energy ℰ of a system in such a quasistatic case.] This is why the wavelength, which determines the length scale of the energy exchange between the electric and magnetic components of an electromagnetic wave does not define the limit of the spatial localization of energy. Because the size of the system

*R*is smaller than any electromagnetic length scale, of which smallest is

*l*, it is

_{s}*R*that defines the spatial scale of the optical energy localization. Thus the optical fields are confined on the nanoscale, and their spatial distribution scales with the system’s size. This physical picture is at the heart of the nanoplasmonics.

Consider as an example a gold nanosphere of radius *R < l _{s}*, e.g.,

*R*∼ 10 nm, subjected to a plane electromagnetic wave, as shown in Fig. 1 (b). The field penetrates the metal and causes displacement of electrons with respect to the lattice resulting in the opposite charges appearing at the opposing surfaces, as illustrated in in Fig. 1 (c). The attraction of these charges causes a restoring force that along with the (effective) mass of the electrons defines an electromechanical oscillator called a SP. When the frequency

*ω*of this SP is close to the frequency of the excitation light wave, a resonance occurs leading to the enhanced local field at the surface, as illustrated in Fig. 1 (b).

_{sp}This resonant enhancement has also an adverse side: loss of energy always associated with a resonance. The rate of this loss is proportional to Im*ɛ _{m}* [28]. This leads to a finite lifetime of SPs. The decay rate of the plasmonic field

*γ*is ∝ (Im

*ɛ*)

_{m}^{−1}. In fact, it is given below in this article as Eq. (49) in Sec. 3.4. This expression has originally been obtained in Ref. [29] and is also reproduced below for convenience,

*γ*does not explicitly depend on the system geometry but only on the optical frequency

*ω*and the permittivities. However, the system’s geometry determines the SP frequency

*ω*and, thus, implicitly enters these equations. The approximate equality in Eq. (3) is valid for relatively small relaxation rates,

*γ*≪

*ω*. Apart from

*γ*, an important parameter is the so-called quality factor

Note that another definition of the quality factor, which is often used, is

The SP quality factors *Q* calculated according to Eqs. (5) and (6) for gold and silver using the permittivity data of Ref. [30] are shown in Fig. 2. The *Q*-factors found from these two definitions agree reasonably well in the red to near-infrared (near-ir) region but not in the yellow to blue region of the visible spectrum. The reason is that these two definitions would be equivalent if metals’ permittivity were precisely described by a Drude-type formula
$\text{Re}{\varepsilon}_{m}\left(\omega \right)=-{\omega}_{p}^{2}/{\omega}^{2}$, where *ω _{p}* is the bulk plasma frequency;

*h̄ω*≈ 9 eV for one-electron metals such as silver, copper, gold, and alkaline metals. This formula is reasonably well applicable in the the red and longer wavelength part of the spectrum, but not in the yellow to blue part where the D-band transitions are important. Note that silver is a much better plasmonic metal than gold: its

_{p}*Q*-factor is several-fold of that of gold.

The finite skin depth of real metals leads to an effect related to nanoplasmonic confinement: a phase shift Δ*φ* for light reflected from a metal mirror deviates from a value of Δ*φ* = *π* characteristic of an ideal metal. As suggested in Ref. [31], this allows for ultrasmall cavities whose length *L* ≪ *λ*. While generally this is a valid idea, there two problems with Ref. [31] that affect the validity of its specific results. First, the Fresnel reflection formulas used in this article to calculate Δ*φ* are only valid for infinite surfaces but not for the “nanomirrors” in a nanocavity. Second, Eq. (1) of this article expressing *Q* is incorrect: it contains in the denominator a quantity *∂*[*ω*Im*ɛ _{m}*(

*ω*)]/

*∂ω*instead of 2Im

*ɛ*(

_{m}*ω*) as in Eq. (5). The correct expression [28] for Ohmic losses defining the

*Q*-factor, which we reproduce as Eq. (105), is proportional to Im

*ɛ*(

_{m}*ω*) as in Eq. (5) and not to

*∂*[

*ω*Im

*ɛ*(

_{m}*ω*)]/

*∂ω*, which constitutes a significant difference.

The lifetime *τ* of the SPs is related to the spectral width as

*γ*, quality factor

*Q*, and lifetime

*τ*depend explicitly only on frequency

*ω*and the type of the metal (permittivity

*ɛ*) but not on the nanosystem’s geometry or surrounding dielectric. However, this geometry and the ambient-dielectric permittivity

_{m}*ɛ*do affect the modal frequency and enter the corresponding Eqs. (3), (5), and (7) implicitly via

_{d}*ω*.

The dependence of the SP lifetime *τ* on frequency *ω* calculated for gold and silver using permittivity [30] is illustrated in Fig. 3. This lifetime is in the range 10 – 60 fs for silver and 1–10 fs for gold in the plasmonic region. These data show that nanoplasmonic phenomena are ultrafast (femtosecond).

However, the fastest linear response time *τ _{c}* of SPs, as any other linear response system, depends not on the relaxation time but solely on the bandwidth. In fact, it can be calculated as a quarter period (i.e., a time interval between zero and the maximum field) of the beating between the extreme spectral components of the plasmonic oscillations,

*ω*is the spectral bandwidth of the plasmonic spectrum. For gold and silver, this bandwidth is the entire optical spectrum, i.e.,

*h̄*Δ

*ω*≈ 3.5 eV. If aluminum is included among system’s plasmonic metals, this bandwidth is increased to

*h̄*Δ

*ω*≈ 9 eV. This yields this coherent reaction time

*τ*∼ 100 as. Thus nanoplasmonics is potentially attosecond science.

_{c}While the characteristic size of a nanoplasmonic system should be limited from the top by the skin depth, *R* ≪ *l _{s}*, it is also limited from the bottom by the so called nonlocality length

*l*– see, e.g., [32, 33]. This nonlocality length is the distance that an electron with the Fermi velocity

_{nl}*v*moves in space during a characteristic period of the optical field,

_{F}*l*, the spatial dispersion of the dielectric response function and the related Landau damping cause broadening and disappearance of SP resonances [32, 33].

_{nl}Thus, we have arrived at the basic understanding of the qualitative features of nanoplasmonics. Consider a plasmonic nanosystem whose size *R* satisfies a condition *l _{nl}* ≪

*R*≪

*l*. This nanosystem is excited by an external field in resonance. In this case, the local optical field in the vicinity of such a nanosystem is enhanced by a factor ∼

_{s}*Q*, which does not depend on

*R*. The spatial extension of the local field scales with the size of the nanosystem ∝

*R*. This is because

*R*≪

*l*, and

_{s}*l*is the smallest electromagnetic length; thus there is no length in the system that

_{s}*R*can be comparable to. When the external field changes, the local field relaxes with the relaxation time

*Q*/

*ω*that does not depend on

*R*; the lifetimes of the SP are in the femtosecond range.

In many cases of fundamental and applied significance, the size of a nanosystem can be compoarable to or even greater than *l _{s}* but still subwavelength,

*λ*≫

*R*≳

*l*. In such a case, the coupling to far-field radiation and radiative losses may greatly increase as we will discuss below in Secs. 2.2 and 2.3. Another important subfield of nanoplasmonics that is related to extended systems is the surface plasmon polaritons – see, e.g., a collective monograph [34]. We consider some polaritonic phenomena relevant to coherent control below in Sec. 4.5.

_{s}#### 2.2. Nanoantennas

Consider a molecule situated in the near-field of a metal plasmonic nanosystem. Such a molecule interacts not with the external field but with the local optical field **E**(**r**) at its location **r**. The interactions Hamiltonian of such a molecule with the optical field is *H*′ = −**E**(**r**)**d**, where **d** is the dipole operator of this molecule. Note that a modal expansion of the quantized local field operator is given below in this article by Eq. (61).

Consequently, the enhanced local fields cause enhancement of radiative and nonradiative processes in which such a molecule participate. In particular, the rates of both the excitation and emission are enhanced proportionality to the local field intensity, i.e., by a factor of ∼ *Q*^{2}. This effect is often referred to as nanoantenna effect [35–62] in analogy with the common radio-frequency antennas For the recent review of the concept and applications of optical nanoantennas see Ref. [63]. Currently, the term nanoantenna or optical antenna is used so widely that it has actually became synonymous with the entire field of nanoplasmonics: any enhancement in nanoplasmonic systems is called a nanoantenna effect.

General remarks about the terms “nanoantenna” or “optical antenna” are due. The term “antenna” has originated in the conventional radio-frequency technology where it is used in application to receivers for devices that convert the wave energy of far-field radio waves into local (near-field) electric power used to drive the input circuitry. For transmitters, antennas perform the inverse transformation: from the local field electric power to that of the emitted radio waves. Due to the general properties of time reversal symmetry there is no principal difference between the receiving and transmitting antennas: any receiving antenna can work as a transmitting one and vice versa. The mechanism of the efficiency enhancement in the radio frequency range is a combination of spatial focusing (e.g., for parabolic antennas) and resonant enhancement (e.g., for a dipole antenna). In all cases, the size of the radio antenna is comparable to or greater than the wavelength. Thus one may think that a receiving antenna collects energy from a large geometric cross and concentrates it in a small, subwalength area.

The receiving antennas in radio and microwave technology are loaded by matched impedance loads that effectively withdraw the energy from them. This suppresses the radiation by such antennas but simultaneously dampens their resonances and makes them poor resonators.

In majority of cases, the optical antennas are not matched-loaded because they are designed not to transduce energy efficiently but to create high local fields interacting with molecules or atoms, which do not load these antennas significantly. (There are exceptions though: for instance, the nanoantenna in Ref. [64] is loaded with an adiabatic nanofocusing waveguide.) The unloaded antennas efficiently loose energy to radiation (scattering), which dampens their resonances.

A question is whether this concept of collecting energy form a large geometric cross section is a necessary paradigm also in nanoplasmonics. The answer is no, which is clear already from the fact that the enhancement of the rates of both the excitation and emission of a small chromophore (molecule, rear earth ion, etc.) in the near field of a small (*R* ≲ *l _{s}*) plasmonic nanoparticle is ∼

*Q*

^{2}and

*does not depend*on the nanoparticle size

*R*. This enhancement is due to the coherent resonant accumulation of the energy of the SPs during ∼

*Q*plasmonic oscillations and has nothing to do with the size of the nanoparticle. Thus such an enhancement does not quite fit into the concept of antennas as established in the radio or microwave technology.

Another test of the nanoantenna concept is whether the efficiency of a nanoantenna is necessarily increased with its size. The answer to this question is generally no. This is because for plasmonic nanoparticles, with the increase of size there is also an increased radiative loss –see below Sec. 2.3. In contrast, for many types of radio-frequency antennas (dish antennas or microwave-horn antennas, for instance), the efficiency does increase with the size.

#### 2.3. Radiative loss

As we described above in conjunction with Fig. 1 (c), the interaction of optical radiation with a nanoplasmonic system occurs predominantly via the dipole oscillations. The radiative decay of SPs occur via spontaneous emission of photons, which is a process that does not exist in classical physics and requires a quantum-mechanical treatment. To find the radiative life time of a SP state quantum-mechanically, we need to determine the transitional dipole matrix element **d**_{0}* _{p}* between the ground state |0〉 and a single-plasmon excited state |

*p*〉. To carry out such a computation consistently, one needs to quantize the SPs, which we have originally done in Ref. [29] and present below in Sec. 5.4.1.

However, there is a general way to do it without the explicit SP quantization, which we present below in this Section. We start with the general expression for the polarizability *α* of a nanosystem obtained using quantum mechanics – see. e.g., Ref. [65], which near the plasmon frequency has a singular form,

*ω*is the frequency of the resonant SP mode. This can compared with the corresponding pole expression of the polarizability of a nanoplasmonic system, which is given below as Eq. (55), to find absolute value of the the matrix element |

_{sp}**d**

_{0}

*|.*

_{p}Here, for the sake of simplicity, we will limit ourselves to a particular case of a nanosphere whose polarizability is given by a well-known expression

*R*is the radius of the nanosphere. The SP frequency

*ω*=

*ω*corresponds to the pole of

_{sp}*α*, i.e., it satisfies an equation where we neglect Im

*ɛ*. In the same approximation, near

_{m}*ω*=

*ω*, we obtain from Eq. (11),

_{sp}Comparing the two pole approximations of Eqs. (10) and (13), we obtain the required expression for the dipole moment of a quantum transition between the ground state and the SP state,

Consider the well-known quantum-mechanical expression for the dipole-radiation rate (see, e.g., Ref. [65]),

*ɛ*(

_{m}*ω*) predict [28] that which guarantees that

*γ*

^{(}

^{r}^{)}> 0 in Eq. (16).

Comparing this expression to Eq. (3) [see also Eq. (49)], we immediately conclude that, in contrast to the internal (radiationless) loss rate *γ*, the radiative rate is proportional to the volume of the system (i.e., the number of the conduction electrons in it), which is understandable. Thus for systems small enough, the radiative rate can be neglected.The quality factor of the SP resonance is actually defined by the total decay rate *γ*^{(}^{tot}^{)} [cf. Eq. (5)],

*Q*is lower for larger nanoparticles, tending to a constant for small

*R*. To quantify it, we find a ratio

We illustrate behavior of this rate ratio of the total to internal loss,
${\gamma}^{\left(\mathit{tot}\right)}/\gamma $, in Fig. 4. General conclusion is that the radiative loss for silver is not very important for nanospheres in the true quasistatic regime, i.e., for *R* < *l _{s}* ≈ 25 nm but is a dominant mechanism of loss for

*R*> 30 nm, especially in high-permittivity environments. In contrast, for gold the radiative loss is not very important in the quasistatic regime due to the much higher intrinsic losses, except for a case of a relatively high ambient permittivity,

*ɛ*= 5.

_{d}Though it is outside of the scope of this article, we would like to point out that there is a general approach to combat radiative losses in relatively large nanoparticles. This is related to the well-known Fano resonances originally discovered by Ugo Fano in atomic spectra [66]. These resonances can be described in the following way. In certain cases of optical excitation, when two quantum paths lead to the same final quantum state of the system, the resonance peaks have specific asymmetric line shapes due to the interference of these quantum paths.

An analogous phenomenon is also known in nanoplasmonics and metamaterials [67–75]. They can be explained in the following way [75]. Apart from bright plasmonic resonances with high transitional dipole moment, there are also dark ones [76], which by themselves are not very prominent in optical spectra. However, if a bright resonance and a dark resonance coexist in a certain spectral range – which is not unlikely, because the bright resonances are wide spanning relatively wide wavelength ranges – then their optical fields interfere. This interference significantly enhances the manifestation of the dark resonance: it acquires strength from the bright resonance and shows itself as an asymmetric peak-and-dip profile characteristic of a Fano resonance. An important, albeit counterintuitive, property of the Fano resonances is that, exactly at the frequency of the Fano dip, the hot spots of the nanolocalized optical fields in the nanosystem are strongest. This is because at this frequency the nanosystem emits minimal light intensity and, consequently, it does not wastefully deplete the energy of the plasmon oscillations. This leads to a decreased radiative loss and a high quality resonance quality factor.

Thus at the frequency of a Fano resonance, the radiative loss is significantly suppressed. The width of the Fano resonances is ultimately determined by the internal (Ohmic) losses described by Im*ɛ _{m}*. Summarizing, the Fano resonances enable one using relatively large nanoplasmonic particles or plasmonic metamaterials to achieve narrow spectral features with high local fields. These can be applied to plasmonic sensing and to produce spasers and nanolasers – see Sec. 5.

#### 2.4. Other important issues of plasmonics in brief

There are other very important issues and directions of investigation in plasmonics that we will not be able to review in any details in this article due to the limitations of time and space. Below we will briefly list some of them.

### 2.4.1. Enhanced mechanical forces in nanoplasmonic systems

The resonantly enhanced local fields in the vicinity of plasmonic nanoparticles lead to enhanced nanolocalized forces acting between the nanoparticles, see, e.g., Refs. [77–83]. A perspective application of plasmonically-enhanced forces is optical manipulation (tweezing) of micro- and nanoparticles [84–90].

Another direction of research is opened up by the recently introduced theoretically surface-plasmon-induced drag-effect rectification (SPIDER) [91], which is based on transfer of the linear momentum from decaying surface-plasmon polaritons (SPPs) to the conduction electrons of a metal nanowire. The SPIDER effect bears a promise to generate very high terahertz fields in the vicinity of the metal nanowire.

### 2.4.2. Interaction between electrons and surface plasmons

The surface plasmonics, as it is called today, originated by a prediction of electron energy losses for an electron beam in thin metal films below the energy of the bulk plasmons [92]. This is how coherent electronic excitations called SPPs today were predicted. Soon after this prediction, the SPP-related energy losses were experimentally confirmed [93,94]. Presently, the electron energy loss spectroscopy (EELS) in nanopalsmonics is a thriving field of research. We refer to a recent review [95] for further detail.

A distinct and original direction of research is control of mechanical motion of metal nanoparticles using electron beams [96]. It is based on the same principles as optically-induced forces. The difference in this case is that the SP oscillations in nanoparticles are excited locally, with an Angstrom precision, by a beam of fast electrons – see also Sec. 2.4.1 above.

There are other important phenomena in plasmonics based on electron-SP interaction called nonlocality [97]. One of them is dephasing of plasmons causing their decay into electron-hole pairs, which is called Landau damping, contributing to Im*ɛ _{m}*. There is necessarily a related phenomenon of spatial dispersion contributing to Re

*ɛ*. These become important for plasmonics when the size of the nanosystem become too small,

_{m}*R*≲

*l*– see Eq. (9). The nonlocality and Landau damping degrade plasmonic effects. The nonlocal effects lead to an increased decay rate of dipolar emitters at metal surfaces [32] and limits resolution of plasmonic imaging, making the so-called “perfect” lens [98] rather imperfect [33]. In aggregates, the nonlocality of dielectric responses causes reduction of local fields and widening of plasmonic resonances [99]. These broadening effects have initially been taken into account purely phenomenologically by adding an additional contribution to the width of plasmonic resonances ∼

_{nl}*A*/

*τ*, where

_{nl}*A*=

*const*[100]. Practically, if the size of a nanoparticle is less then 3 nm, the non-local broadening of the SP resonances is very significant; otherwise, it can be neglected in a reasonable approximation.

The above-mentioned publications [32, 33, 97, 99] on the nonlocality phenomena are based on a semi-phenomenological approach where the nonlocality is treated via applying additional boundary conditions stemming from the electron scattering by the boundaries of the plasmonic system. A more advanced approach to nonlocality in nanoplasmonics, albeit treatable only for very small, *R* ≲ 1 nm, nanoparticles, is based on an *ab initio* quantum-chemical approach of time-dependent density functional theory (usually abbreviated as TD-DFT) [101–107].

It shows that while for larger particles and relatively large spacing between them (≳ 1 nm), the semi-phenomenological models work quite well, for smaller nanoparticles and gaps the predicted local fields are significantly smaller. This is understandable because in *ab initio* theories there are phenomena that are important in the extremely small nanosystem such as a significant dephasing due to the stronger coupling between the collective plasmon and one-particle electron degrees of freedom, discreetness of the one-electron spectrum, spill-out of the conduction-band electrons (extension of their wave function outside of the lattice region) and the corresponding undescreening of the *d*-band electrons, and simply the discreetness of the lattice.

In the latest set of publications, e.g., [106, 107], this approach is called quantum nanoplasmonics. We would argue that this approach is traditionally called quantum chemistry because what is found from the TD-DFT quantum-mechanically is the dielectric response (susceptibility or polarizability) of the nanosystems. However, even to calculate theoretically the permittivity of a bulk method, one has to employ quantum-mechanical many-body approaches such as the random-phase approximation, self-consistent random-phase approximation (or GW-aproximation), or TD-DFT, etc. The only difference from the above-sited works is that for bulk metals the size effects are absent. Therefore permittivities can be adopted from experimental measurements such as Ref. [30, 108].

Based on the arguments of the preceding paragraph, we would reserve the therm “quantum plasmonics” for the subfields of nanoplasmonics studying phenomena related to quantum nature and behavior SPs and SPPs. This term has been proposed in our 2003 paper [29] introducing the spaser as a quantum generator of nanolocalized optical fields – see Sec. 5 and references sited therein. A related field of studies devoted to quantum behavior of single SPPs also can reasonably be called quantum plasmonics as proposed later in Refs. [109, 110].

While the decay of SP excitations is usually a parasitic phenomenon, there are some effects that completely depend on it. One of them is the SPIDER [91] mentioned above in Sec. 2.4.1. It is based on the transfer of the energy and momentum from SPPs to the conduction electrons, which microscopically occurs through the decay of the SPPs into electron-hole pairs leading to production of hot electrons.

Yet another range of phenomena associated with a plasmon-dephasing decay into incoherent electron-hole pairs (Landau damping) has come to the forefront lately. This is the plasmon-assisted and enhanced generation of a dc electric current due to rectification in Schottky diodes involving hot electrons [59, 111–113]. This phenomenon is promising for applications to photodetection and solar energy conversion. Note that the use of the Schottky contacts between the plasmonic metal and a semiconductor permits one to eliminate a requirement that the photon energy *h̄ω* is greater that the band gap. This is replaced by a much weaker requirement that *h̄ω* is greater than a significantly lower Schottky-barrier potential [114].

### 2.4.3. Nonlinear photoprocesses in nanoplasmonics

As became evident from the first steps of what now is called nanoplasmonics, the enhanced local fields in resonant metal nanosystems bring about strongly enhanced nonlinear responses [115–118].

Nonlinear nanoplasmonics is presently a very large and developed field. Some of its phenomena related to coherent control and spasing are discussed in Secs. 4, and 5. Here we will give a classification of the nonlinear nanoplasmonic phenomena and provide some examples, not attempting at being comprehensive.

Nonlinearities in nanoplasmonics can occur in the nanostructured plasmonic metal, in the embedding medium (dielectric), or in both. Correspondingly, we classify them as intrinsic, extrinsic, or combined. As an independent classification, these nonlinearities can be classified as weak (perturbative) or strong (nonperturbative). The perturbative nonlinearities can be coherent (or parametric), characterized by nonlinear polirizabilities [119] and incoherent such as nonlinear absorption, two-photon fluorescence, surface-enhanced hyper-Raman scattering (SEHRS) [120], nonlinear photo-modification, two-photon electron emission [121], etc.

Let us give some examples illustrating a variety of nonlinear photoprocesses in nanoplasmonics.

- Enhanced four wave mixing (sum- or difference frequency generation) at metal surfaces [131] is a coherent, perturbative (third-order or four-wave), intrinsic nonlinearity.
- Another four-wave mixing process in a hybrid plasmonic-photonic waveguide involves nonlinearities in both metal and dielectric [132] and, therefore, is classified as a coherent, combined, perturbative third-order nonlinear process.
- Nonperturbative (strong-field), coherent, extrinsic nonlinearity is plasmon-enhanced generation of high harmonics [136] where the enhanced nanoplasmonic fields excite argon atoms in the surrounding medium. Spaser [29] belongs to the same class where the nonlinearity is the saturation of the gain medium by the coherent plasmonic field [137]. The same is true for the loss compensation by gain [138, 139].
- Intrinsic perturbative nonlinearities in nanoplasmonics stemming from a redistribution of the electron density caused by the ponderomotive forces of nanoplasmonic fields have been predicted for surface plasmon polaritons [91, 140]. An intrinsic nonperturbative nonlinear process is the predicted plasmon soliton [141] where strong local optical fields in a plasmonic waveguide cause a significant redistribution of the conduction-electron density.
- There are also relevant strongly-nonlinear processes in non-plasmonic materials that are based on nanolocalized fields and are very similar to those in plasmonics. Among them are near-field enhanced electron acceleration from dielectric nanospheres with intense few-cycle laser fields [142]. Another such a process is a strong optical-field electron emission from tungsten nanotips controlled with an attosecond precision [143].

## 3. Nanolocalized surface plasmons (SPs) and their hot spots

#### 3.1. SPs as eigenmodes

Assuming that a nanoplasmonic system is small enough, *R* ≪ *λ̵*,*R* ≲ *l _{s}*, we employ the so-called quasistatic approximation where the Maxwell equations reduce to the continuity equation for the electrostatic potential

*φ*(

**r**),

**r**) is the so-called characteristic function of the nanosystem, which is equal to 1 when

**r**belongs to the metal and 0 otherwise. We solve this equation following the spectral theory developed in Refs. [76, 146, 147].

Consider a nanosystem excited by an external field with potential *φ*_{0}(**r**) at an optical frequency *ω*. This potential is created by external charges and, therefore, satisfies the Laplace equation within the system,

*φ*

_{1}(

**r**) is the local field.

Substituting Eq. (23) into Eq. (20) and taking Eqs. (21) and (22) into account, we obtain a second-order elliptic equation with the right-hand side that describes the external excitation source,

*s*(

*ω*) is Bergman’s spectral parameter [146] defined by Eq. (4).

As a convenient basis to solve this field equation we introduce eigenmodes (SPs) with eigen-functions *φ _{n}*(

**r**) and the corresponding eigenvalues

*s*, where

_{n}*n*is the full set of indices that identify the eigenmodes. These eigenmodes are defined by the following generalized eigen-problem,

*φ*(

_{n}**r**) satisfy the homogeneous Dirichlet-Neumann boundary conditions on a surface

*S*surrounding the system. These we set as

**n**(

**r**) denoting a normal to the surface

*S*at a point of

**r**. These boundary conditions (26) are essential and necessary to define the eigenproblem.

From Eqs. (25)–(26) applying the Gauss theorem, we find

*s*is the integral fraction of the eigenmode (surface plasmon) intensity |

_{n}*∂φ*(

_{n}**r**)/

*∂*

**r**|

^{2}that is localized within the metal.

Because the SP eigenproblem is real, and all the eigenvalues *s _{n}* are all real, the eigenfunctions

*φ*can also be chosen real, though are not required to be chosen in such a way. Physically, it means that the quasistatic nanoplasmonic eigenproblem is time-reversible.

_{n}For the eigenproblem (25)–(26), we can introduce a scalar product of any two functions *ψ*_{1} and *ψ*_{2} as

#### 3.2. Inhomogeneous localization of SPs and hot spots of local fields

One of the most fundamental properties of eigenmodes is their localization. By nature, the SP eigenmodes of small nanoplasmonic systems are localized and non-propagating. This generally follows from the fact that the eigenproblem (25) is real and has real eigenvalues, implying time-reversal invariance and, consequently, zero current carried by any eigenmode.

From the early days of nanoplasmonics, there has been keen attention paid to the localization of SP eigenmodes, because it was immediately clear that that absence of any characteristic wavelength of the localized SPs leads to the possibility of their concentration in nanoscopic volumes of the space [115,118,148]. Many early publications claimed that the SPs in disordered nanoplasmonics systems, e.g., fractal clusters, experience Anderson localization [149–155].

However, a different picture of the SP localization, named inhomogeneous localization, has been introduced [76, 156–159]. In this picture of inhomogeneous localization, eigenmodes of very close frequencies with varying degree of localization, from strongly localized at the minimum scale of the system to delocalized over the entire nanosystem coexist. This phenomenon of inhomogeneous localization has been experimentally confirmed recently [160]. The eigen-modes experiencing the Anderson localization are dark, corresponding to dipole-forbidden transitions, and thus can only be excited from the near field [76].

A related phenomenon is the formation of hot spots in local fields of nanoplasmonic system that we introduced in Refs. [156, 157, 161, 162]. As characteristic of the inhomogeneous localization, the energy is localized by different SP eigenmodes at vastly different scales. However, it is the localization at the minimum scale that gives the highest local fields and energy density; these tightly-localized modes are the most conspicuous in the near-field intensity distributions as the hot spots. The hot spots exist in all kind of nanoplasmonic system but they are especially strongly pronounced in disordered and aperiodic systems [163].

We will illustrate the hot spots and the inhomogeneous localization of the SP eigenmodes using the results of the original works that established the phenomena [156,157] using plasmonic-metal fractal clusters as objects. The model of these fractals were the so-called cluster-cluster aggregates (CCA) [164, 165]. In Fig. 5, we show two representative eigenmodes with Bergman’s eigenvalues of *s _{n}* = 0.3202 and

*s*= 0.3203, which are very close in frequency (the blue spectral range for the case of silver in water). Both the eigenmodes are highly singular and are represented by sharp peaks – hot spots – that may be separated by the distances from the minimum scale of the system to the maximum scale that is on the order of the total size of the entire system. These eigenmodes possess very different topologies but very close eigenvalues and, consequently, have almost the same frequency

_{n}*h̄ω*≈ 3.13 eV corresponding to the blue spectral range. This coexistence of the very different eigenmodes at the same frequency was called the inghomogeneous localization [156, 157].

The formation of host spots by the SP eigenmodes and the inhomogeneous localization of the eigenmodes are very pronounced for the fractal clusters. However, the same phenomena also take place in all dense random plasmonic systems. Physically, this phenomena is related to the absence of the characteristic length scale for SPs: the smallest electromagnetic scale is the skin depth *l _{s}* ≈ 25 nm, which is too large on the scale of the system to affect the SP localization. The inhomogeneous localization implies that eigenmodes can be localized on all scales but this localization is always singular. The hot spots are the concentration regions of the optical energy: sharp peaks on the minimum scale (“fine grain” size) of the system are most visible.

Note that there is a fundamental difference between the plasmonic hot spots and their counterpart in the wave optics: speckles produced by scattering of laser light from a random medium. In the speckle case, there is a characteristic size of the speckles on the order of a character distance *L _{s}* between them that is determined by diffraction:

*λ*is wavelength of light,

*A*is an aperture (cross-size of the coherent spot of light on the scattering system), and

*D*is the distance from the scatterer to the observation screen.

One of the plasmonic system models studied in significant detail is a random metal composite (RPC) also called a semi-continuous metal film [76, 126, 147, 154, 160, 166–169]. This is a planar system where metal occupies a given fill fraction *f* of the system’s volume. At a low *f*, the RPC is a system of remote randomly positioned metal particles. For high values of *f*, it is an almost continuous film with rare holes in it. For *f* ≈ 0.5, there are percolation phenomena: there is a large connected random cluster of the metal extending between the boundaries of the system [170]. This connected percolation cluster is known to possess a fractal geometry.

To consider statistical measures of the SP localization, we introdude the localization radius *L _{n}* of an eigenmode, which is is defined as the gyration radius of its electric field intensity |

**E**

*(*

_{n}**r**)|

^{2}, where

In Fig. 6 (a), we show the smoothed, discretized nanostructure of one particular sample of a RPC. This system is generated in the following way. We consider a volume of size, in our case, 32 × 32 × 32 grid steps. In the central *xz* plane of this cube we randomly fill a cell of size 2 × 2 grid steps with metal with some probability *f* (fill factor or filling factor). Then we repeat this procedure with other 2 × 2 cells in that central *xz* plane. As a result, we arrive at a thin planar layer of thickness 2 grid steps in the *y* direction and fill factor of *f* in the central *xz* plane.

In Fig. 6 (b), we display all of the eigenmodes (SPs) of the above-described RPC in a plot of oscillator strength *F _{n}*

*versus*localization length

*L*. These eigenmodes are strikingly unusual. First, there is a large number of eigenmodes with negligible oscillator strengths

_{n}*F*≲ 10

_{n}^{−5}. Note that the rounding-up relative error in our computations is ∼ 10

^{−6}, so these eigenmodes’ oscillator strengths do not significantly differ from zero. Such eigenmodes do not couple to the far-field electromagnetic waves, and they can be neither observed nor excited from the far-field (wave) zone. We call them

*dark modes*. They can, however, be excited and observed by NSOM (near-field scanning optical microscope) type probes in the near-field region. Such eigenmodes are also important from the computational-mathematical point of view because they are necessary for the completeness of the eigenmode set.

Second, in Fig. 6, there also are many eigenmodes with relatively large oscillator strengths, *F _{n}* ≳ 10

^{−4}, which we call

*luminous*or

*bright*modes. These do couple efficiently to the far-zone fields.

Third, both the luminous and the dark modes have localization radii *L _{n}* with all possible values, from zero to one half of the diagonal system size, and with very little correlation between

*F*and

_{n}*L*, except for the superlocalized (zero-size) eigenmodes that are all dark. This wide range of

_{n}*L*shows that

_{n}*the Anderson localization does not occur for most of the modes, including all the luminous modes*. Similar to these findings in certain respects, deviations from the simple Anderson localization have been seen in some studies of the spatial structure of vibrational modes [171, 172], dephasing rates [173] in disordered solids induced by long-range (dipole-type) interactions. A direct confirmation of this picture of the inhomogeneous localization has been obtained in experiments studying fluctuations of the local density of states of localized SPs on disordered metal films [160].

To gain more insight, we show in Fig. 7 the local electric field intensities |**E*** _{n}*(

**r**)|

^{2}for particular eigenmodes of four extreme types, all with eigenvalues very close to

*s*= 0.2. As a measure of the eigenmode oscillator strength, we show a normalized oscillator strength

_{n}*F*. The data of Fig. 7 confirm the above-discussed absence of correlation between the localization length and oscillator strength, and also show that there is no correlation between the topology of the local field intensity and the oscillator strength—compare the pairs of eigenmodes:

_{n}*s*= 0.1996 with

_{n}*s*= 0.2015, and

_{n}*s*= 0.2 with

_{n}*s*= 0.2011. Note that the large and random changes of the intensities between the close eigenmodes evident in Fig. 7 is an underlying cause of the giant fluctuations [174] and chaos [175–177] of local fields.

_{n}A fundamental property of the SP eigenmodes, whether localized or delocalized, is that they may be thought of as consisting of hot spots. While the localized eigenmodes possess a single tight hot spots, the delocalized ones consist of several or many host spots. Note that the fields in the hot spots constituting a single eigenmode are coherent. In a sense, the hot spots are somewhat analogous to speckles produced by laser light scattered from a random system. However, such speckles are limited by the half-wavelength of light and cannot be smaller than that. In contrast, there is no wavelength limitations for the SP hot spots. They are limited only by the minimum scale of the underlying plasmonic system.

#### 3.3. Retarded Green’s function and field equation solution

Retarded Green’s function *G ^{r}*(

**r**,

**r**′;

*ω*) of field equation (24), by definition, satisfies the same equation with the Dirac

*δ*-function on the right-hand side,

We expand this Green’s function over the eigenfunctions *φ _{n}* using the orthonormality Eq. (30), obtaining

This expression for Green’s function is exact (within the quasistatic approximation) and contains the maximum information on the linear responses of a nanosystem to an arbitrary excitation field at any frequency. It satisfies all the general properties of Green’s functions due to the analytical form of Eq. (36) as an expansion over the eigenmodes (surface plasmons). This result demonstrates separation of geometry of a nanosystem from its material properties and the excitation field. The eigenfunctions *φ _{n}*(

**r**) and eigenvalues

*s*in Eq. (36) depend only on geometry of the nanosystem, but not on its material composition or the optical excitation frequency. In contrast, the spectral parameter

_{n}*s*(

*ω*) depends only on the material composition and the excitation frequency, but not on the system’s geometry. One of the advantages of this approach is in its applications to numerical computations: the eigenproblem has to be solved only once, and then the optical responses of the nanosystem are determined by Green’s function that can be found by a simple summation in Eq. (36).

This Green’s function is called retarded because it describes responses that occur necessarily at later time moments with respect to the forces that cause them. (Note that this name and property have nothing to do with the electromagnetic retardation, which is due to the finite speed of light and is absent in the quasistatic approximation.) This property, also called Kramers-Kronig causality, is mathematically equivalent to all singularities of *G ^{r}*(

**r**,

**r**′;

*ω*) as a function of complex

*ω*being situated in the lower half-plane. Consequently,

*G*(

^{r}**r**,

**r**′;

*ω*) as a function of

*ω*satisfies the Kramers-Kronig dispersion relations [28]. By the mere form of the spectral expansion (36), this Green’s function satisfies all other exact analytical properties. This guarantees that in numerical simulations it will possess these properties irrespectively of the numerical precision with which the eigenproblem is solved. This insures an exceptional numerical stability of computational Green’s function approaches.

Once the Green’s function is found from Eq. (36), the local optical field potential is found as contraction of this Green’s function with the excitation potential *φ*_{0}(**r**) as

*φ*(

**r**) as a functional of the external (excitation) potential

*φ*

_{0}(

**r**),

**E**(

**r**) as a functional of the excitation (external) field

**E**

^{(0)}(

**r**) as

*α,β*,... are Euclidean vector indices (

*α,β*,··· =

*x,y,z*) with summation over repeated indices implied; the fields are

One of the exact properties of this Green’s function is its Hermitian symmetry,

If the excitation is an optical field, its wave front is flat on the scale of the nanosystem, i.e., **E**^{(0)} = const. Then from Eq. (39) we get

#### 3.4. SP modes as resonances

Each physical eigenmode is described by the corresponding pole of Green’s function (36). Close to such a pole, Green’s function and, consequently, local fields (43) become large, which describes the surface plasmon resonance of the nanosystem. A complex frequency of such a resonance can be found from the position of the corresponding pole in the complex plane of frequency,

where*ω*is the real frequency of the surface plasmon, and

_{n}*γ*is its spectral width (relaxation rate).

_{n}Note that we presume *γ _{n}* > 0, i.e., a negative sign of the imaginary part of the physical surface frequency. This a presumption, which is confirmed by the solution presented below in this Section, is based on the standard convention of the sign of an exponential in the field temporal evolution,

*t*→ +∞, as should be. The wave functions of physical surface plasmons are the familiar eigenfunctions

*φ*(

_{n}**r**), i.e., those of the geometric eigenmodes. However, their physical frequencies, of course, depend on the material composition of the system.

For weak relaxation, *γ _{n}* ≪

*ω*, one finds that this real surface plasmon frequency satisfies an equation

_{n}*γ*is a universal function of frequency

*ω*and does not explicitly depend on the eigenmode wave function

*φ*(

_{n}**r**) or system’s geometry. However, the system’s geometry does, of course, define the plasmon eigenfrequencies

*ω*. This property has been successfully used in Ref. [178] where a method of designing nanoplasmonic systems with desired spectra has been developed. Note also that the classical SPs have been quantized in Ref. [29] in connection with the prediction of spaser, a nanoscale counterpart of laser (see Sec. 5).

_{n}As follows from Eq. (28), external frequency *ω* is within the range of the physical surface plasmon frequencies and, therefore, can be close to a surface plasmon resonance [pole of Green’s function (36) as given by Eq. (45)] under the following conditions

These conditions, in fact, constitute a definition of a plasmonic system, i.e., a system where a position of surface plasmon resonance can be physically approached: the dielectric permittivity of the metal component should be negative and almost real, while the permittivity of the second constituent (dielectric) should be positive, as assumed.

It is useful to write down an expression for Green’s function (36) that is asymptotically valid near its poles, which can be obtained from Eqs. (47) and (48) as

*γ*is given above by Eqs. (48) or (49). This expression constitutes what is called the singular approximation or pole approximation of the Green’s function. When an excitation frequency is in resonance with an SP frequency, i.e.,

_{n}*ω*=

*ω*, the Green’s function (52) increases in magnitude by ∼

_{n}*ω*∼

_{n}/γ_{n}*Q*times, where the quality factor

*Q*is given by Eq. (5).

Below, for the sake of reference, we give a modal expansion for the polarizability *α* of a nanoplasmonic system as a tensor,

*α,β*denote Cartesian components, and

**M**

*is a coupling vector defined as*

_{n}Near a SP frequency, *ω* ≈ *ω _{n}*, a singular part of the polarizability (53) acquires a form

#### 3.5. Examples of local fields and their hot spots

Let us give an example of local fields computed using Eq. (39). We start with the results of the original publications Ref. [156,157] where the hot spots of the plasmonic local fields have been predicted. This prediction was made for fractal clusters because the fractals were expected to possess highly inhomogeneous and fluctuating local optical fields as was shown in pioneering papers in a subfield of physical optics that today is called nanoplasmonics [115, 148, 179].

In Fig. 8 adapted from Ref. [156], we illustrate the local-field hot spots for a silver CCA cluster of *N* = 1500 identical nanospheres embedded in water. We show local field intensity *I* = |**E**(**r***,ω*)|^{2} relative to the excitation field intensity *I*_{0} at the surface of the silver nanospheres at a relatively high frequency *h̄ω* = 3.13 eV corresponding to vacuum wavelength *λ* = 390 nm in the far blue end of the visible spectrum. We can clearly see that the local intensity is highly non-uniform, exhibiting pronounced singular hot spots. These hot spots are localized at the minimum scale of the system (on the order of the the radius of the nanospheres). The local intensity in the hot spots is greatly enhanced (by a factor of up to ∼ 600) as one would expect from an estimate *I*/*I*_{0} ∼ *Q*^{2} – cf. Fig. 2.

This hot spotting is nothing else as random nanofocusing. It is similar in this respect to the formation of speckles in the wave optics, as we have discussed above in conjunction with Fig. 5. However, reflecting the properties of the corresponding SP eigenmodes, there is no characteristic wavelength that limits this hot spot singularity by defining the characteristic size *L _{s}* of the speckles, which is also a characteristic separation between them – see Eq. (31).

Another property of the local fields of a great significance is the dramatic dependence of the intensity distribution on the polarization: the local distributions or the *x*-polarization [Fig. 5 (a)] and *y*-polarization [panel (b)] are completely different. An experimental observation of this effect has been obtained in Ref. [116] already at a very early stage of the development of nanoplasmonics.

Note that the SP eigenmode geometry is also strongly dependent on its frequency – see Fig. 5. However, in externally-excited local fields, this frequency dependence is obscured by the resonance broadening due to the losses, as is evident from the expression for the resonant part of the Green’s function

We will present below spectral and statistical properties of the local fields using a model of random planar composite (RPC). A specific RPC system used in the computation is shown in Fig. 9(a). To improve numerical accuracy, we smooth the unit-step characteristic function Θ(**r**) with a Gaussian filter with a radius of 1 grid step: this dramatically improves numerical accuracy of a grid method that we use to solve the eigenproblem. Such a smoothing is clearly seen in Fig. 9 (a).

In Fig. 9 (b)–(c), we display the spatial distribution of the local field intensity |**E**(**r**)|^{2}in the plane of the nanostructure at the surface of the metal. These computations are described in Ref. [180]. They are done for silver whose dielectric function is adopted from Ref. [30]; the embedding dielectric has permittivity is set as *ɛ _{d}* = 2.0. This intensity is plotted relative to the excitation field intensity |

**E**

_{0}|

^{2}; thus the quantity displayed is the enhancement factor of the local field intensity. Panel (b) shows the intensity computed from Eq. (38). The maximum of the local intensity enhancement of ≈ 6000 is in a reasonable agreement with the estimate ∼

*Q*

^{2}∼ 10

^{4}, where

*Q*is displayed in Fig. 2.

Dependence of the local fields on frequency is dramatic: cf. Figs. 9 (b) and (c). As frequency increases from the near-IR (1.55 eV) to visible (2.0 eV), the distribution becomes much more delocalized and its magnitude dramatically decreases, which cannot be explained by some decrease of quality factor *Q* alone. Most importantly, at all frequencies these near-field intensity distributions are dominated by the pronounced hots spots. These are manifestation of the hot spots of the SP eigenmodes – see Fig. 7.

Generally, the intensity distribution of local field intensity in Fig. 9 (b)–(c) is highly singular: it consists of relatively narrow peaks (hot spots [157,162]) separated by regions of a low intensity. This is a typical distribution of intensity in plasmonic nanosystems, which is a reflection of the inhomogeneous localization of the SP eigenmodes.

#### 3.6. Experimental examples of nanoplasmonic hot spots

There has been a significant number of experimental studies of near-field distributions of optical fields in plasmonic nanostructures. In all cases, a pronounced picture of the hots spots [156,157] has been exhibited, see, e.g., Refs. [121, 154, 167]. The inhomogeneous localization of the SP eigenmodes (see Sec. 3.2), which is inherently related to hot spots, has recently been confirmed experimentally [160].

The photoemission electron microscope (PEEM) is a powerful tool of analyzing the distribution of the local field intensity without perturbing it in any way. In the PEEM approach, the plasmonic nanosystem to be analyzed serves as a cathode and an object of an electron microscope. The electron emission is caused by the local field **E**(**r**,*ω*) of the plasmonic system. The photoelectrons are analyzed by the electron optics of the PEEM that creates a magnified image of the system in “light” of the photo-emitted electrons.

For silver, the work function *W _{f}* (i.e., the minimum energy needed to excite an electron from the Fermi surface to the zero energy that is the energy in vacuum far away from the metal) is approximately 4.2 eV. The highest energy of an optical quantum (at the vacuum wavelength of 390 nm) is 3.2 eV, i.e., it is significantly less than

*W*. Thus, a single optical photon cannot emit an electron from a silver surface. Such an emission can, however, occur through two-photon absorption, leaving for the emitted electron the kinetic energy at infinity of

_{f}*E*

_{∞}≤ 2

*h̄ω*−

*W*. Such a two-photon electron photoemission is in the foundation of the so-called two-photon photoemission PEEM (or, 2PP-PEEM). On the other hand, for ultraviolet radiation (say, from a Hg lamp), the energy of a photon is sufficient for the one-photon photoemission PEEM (1PP-PEEM). The 2PP-PEEM electron intensity mirrors the distribution of

_{f}*I*

_{2}= |

**E**(

**r**,

*ω*)|

^{4}.

A model system to illustrate the hot spots used in a 2PP-PEEM experiment of Ref. [121] is shown in Fig. 10 (a). This is a diffraction grating covered with silver layer with roughness of a < 10 nm RMS grain size, as the scanning electron micrograph (SEM) shows in the insert. The Hg lamp illumination (the energy of the quantum *h̄ω* = 4.89 eV exceeds *W _{f}* = 4.2 eV, thus allowing one-photon photoemission, 1PP-PEEM) shows a smooth image of the underlying diffraction grating with the resolution of the PEEM (≲ 100 nm).

A dramatically different picture is observed in Fig. 10 (b). Tn this case, the irradiation is with femtosecond laser pulses of *λ* = 400 nm vacuum wavelength. The corresponding energy of the quantum is below the work function, *h̄ω* = 3 eV < *W _{f}* = 4.2 eV. Thus the electron photoemission is two-photon. The corresponding 2PP-PEEM image in Fig. 10 (b) exhibits a pronounced picture of the hot spots due to the fact that in this case the optical frequency is in the plasmonic range. These hot spots are localized SPs that are excited by the

*p*-polarized radiation with a significantly greater efficiency than by an

*s*-polarized one. This suggests that SPPs excitation may play a role as an intermediate for the localized SP excitation. In a full qualitative agreement with theory (see Sec. 3.2), these hot spots are singular, highly localized, and randomly distributed in space. The local fields in these hot spots are highly enhanced as witnessed by their dominance in the 2PP process.

We emphasize again that the PEEM-based observation of the plasmonic hot spots is completely non-perturbing. The photo-emitted electrons that are used in the PEEM fly away from the metal surface naturally, no matter whether they are used for imaging or not.

There has also been a series of research dealing with the observation of the plasmonic hot spots using the scanning near-field optical microscope (NSOM or SNOM) [154, 161, 167]. A general concern about such observation is that they are perturbative: the tip of NSOM (or nanoscope, as it is often called) is typically much larger than a hot spot. Made of metal, it can, in principle, modify the host spot by both shifting its resonant frequency and decreasing the quality factor.

As an example, we present Fig. 11 adapted from Ref. [167]. This study is done on the semi-continuous metal film (random planar composite, or RPC). At relatively low values of the fill factor, *f* = 0.36 and *f* = 0.45, the local intensity distribution *I*(**r**) shows relatively delocalized regions elongated normally to the direction of propagation (vertical axis in the figure). These are analogous to the caustics of the usual 3d optics. Relatively close to the percolation point, *f* = 0.66 and *f* = 0.73, the distribution *I*(**r**) becomes highly localized exhibiting singular hot spots. The behavior of *I*(**r**) at a relatively high fill factor of *f* = 0.83 again reminds that for the low *f* showing delocalized caustics but not singular hot spots. This is understandable because in this case the system is basically a smooth film with a few defects. This film supports SPPs that are weakly scattered by the relatively few defects.

As we have discussed above in this Section, NSOM measurements of hot spots are inherently perturbative. While PEEM is nonperturbative, the spatial resolution so far has been insufficient (due to aberrations in the electron optics and large spread of the emitted electrons over their energies). Additionally, PEEM requires clean surfaces in high vacuum.

A fundamentally different non-perturbing approach to studying nanoplasmonic hot spots has been pioneered in Refs. [181, 182]. It is based on the so-called photon-localization super-resolution far-field microscopy. This method of far-field super-resolution has originally been developed in application to biological imaging [183].

This method’s fundamentals can be very briefly described as the following. Assume that there is a *single* radiating chromophore (say, fluorescing molecule) in the view field of an optical microscope. Alternatively, there may be a number of such chromophores but their concentration should be low enough so they are resolved separately by the microscope (i.e., the distance between these molecules are greater than the microscope’s resolution). The center of the emission of such a single (or separately resolved) emitter can be found with any precision that is only limited by statistical fluctuations of the number of the recorded photons but not by the resolution of the microscope provided that this microscope or the system under study do not change in the course of the observation.

After the position and brightness of a given single molecule are recorded, this molecule is naturally bleached. Then another molecule comes into the hot spot and its position and brightness are recorded until it is bleached. The process is repeated until the distribution of the brightness of emitters is built with a sufficient statistical precision.

It is assumed that the emission brightness of a single chromophore is proportional to the local field intensity of the hot spot at its position and that this chromophore exerts a negligibly weak perturbation on the local field of the hot spot. Thus this photon-localization nanoscopy is a non-perturbative method allowing one to find the intensity distribution at the hot spot on the nanoscale limited only by the statistical fluctuations (inversely proportional to the accumulation time) and the size of the chromophore itself, which is negligible in realistic situations.

The results of the hot spot local intensity-distribution measurements for an aluminum surface are shown in Fig. 12 (a). This distribution is a narrow peak with the width of ≈ 20 nm. The observed fine structure of this distribution is attributed to statistical fluctuations [181]. The cross section through this distribution displayed in Fig. 12 (b) suggests an exponential decay of this distribution function in space with the FWHM = 20 nm.

Very similar results are obtained for the silver colloid clusters as shown in Figs. 12 (c)–(d). Note that the aluminum surface studied is nominally smooth and contains only random roughness while the silver colloid clusters are fractals whose density fundamentally possesses large and correlated fluctuations.

## 4. Ultrafast plasmonics and coherent control on nanoscale

#### 4.1. Introduction

The nanoplasmonic processes can potentially be the fastest in optics: their shortest evolution times are defined by the inverse spectral width of the region of the plasmonic resonances and are on the order of 100 as [184], see also Sec. 2.1. The relaxation times of the SP excitations are also ultrashort, in the 10–100 fs range [185–189]. See also the SP relaxation times for gold and silver displayed in Fig. 3. The nanolocalization and such an ultrafast kinetics make plasmonic nanostructures promising for various applications, especially for the ultrafast computations, data control and storage on the nanoscale.

These and potentially many other applications require precise control over the optical excitations of the nanostructures in time and space on the femtosecond-nanometer scale. Such a control cannot be imposed by far-field focusing of the optical radiation because the diffraction limits its dimension to greater than half wavelength. In other words, the optical radiation does not have spatial degrees of freedom on the nanoscale. There is a different class of approaches to control a system on nanoscale based on plasmonic nanoparticles or waveguides brought to the *near-field* region of the system. Among these we mention: the tips of scanning near-field optical microscopes [190], adiabatic plasmonic waveguides [12], nanowires [191, 192], plasmonic superlenses [193] or hyperlenses [194]. In all these cases, massive amount of metal is brought to the vicinity of the plasmonic nanosystem, which will produce strong perturbations of its spectrum and SP eigenmodes, cause additional optical losses, and adversely affect the ultrafast dynamics and energy nanolocalization in the system. This nanowaveguide approach also may not work because of the excitation delocalization due to the strong interaction (capacitive coupling) at the nanoscale distances for optical frequencies.

We have proposed [195] a principally different approach to ultrafast optical control on the nanoscale based on the general idea of coherent control. The coherent control of the quantum state of atom and molecules is based on the directed interference of the different quantum pathways of the optical excitation [196–205], which is carried out by properly defining the phases of the corresponding excitation waves. This coherent control can also be imposed by an appropriate phase modulation of the excitation ultrashort (femtosecond) pulse [202, 206–208]. Shaping the polarization of a femtosecond pulse has proven to be a useful tool in controlling quantum systems [209].

Our idea of the coherent control on the nanoscale by the phase modulation of the excitation pulse can be explained with a schematic shown in Fig. 13. Phase modulation of the excitation pulse can be thought of as changing the frequency (color) light as the pulse progresses in time. For the sake of argument, let us assume that, as shown in Fig. 13, that initially the pulse contains blue colors that gradually change to red with the time progression. At earlier times, the dominating blue component of the pulse will excite the SP eigenmodes with corresponding high optical frequencies. As the pulse progresses, the lower-frequency eigenmodes are excited. It is assumed that the the total duration *τ _{p}* of the pulse is less than the decay (decoherence) time

*τ*=

*γ*

^{−1}of the SPs, i.e..,

*τ*≲

_{p}*τ*[for the decay rates and life times of the SPs see Eq. (3) or (49) and Fig. 3]. In such a case, the SPs of different frequencies will coexist simultaneously, and their fields will interfere. This interference depends on the relative phases and amplitudes of the SPs of different frequencies that, in turn, are determined by the relative phases of different spectral components of the excitation pulse. The ultimate goal of the spatio-temporal coherent control on the nanoscale is to have a hot spot of the local fields at a given nanosite at a given femtosecond temporal interval. Below in this article we show that this problem is solved both theoretically and experimentally.

Another approach that we have proposed [210] invokes spatial modulation of the excitation field on the microscale in a polaritonic system. This field excites SPPs whose phases are determined by those of the original field. This determines the wave fronts of the SPP waves that focus on the nanoscale at the targeted nanofoci at the required times with femtosecond temporal resolution. The spatial-phase coherent control of the SPPs has been demonstrated experimentally by different groups [211, 212].

Our initial idea [195] has been subsequently developed theoretically [147,209,213,214] and experimentally [121, 215–217]. In this coherent control approach, one sends from the far-field zone a shaped pulse (generally, modulated by phase, amplitude, and polarization) that excites a wide-band packet of SP excitations in the entire nanosystem. The phases, amplitudes, and polarizations of these modes are forced by this shaped excitation pulse in such a manner that at the required moment of time and at the targeted nanosite, these modes’ oscillations add in phase while at the other sites and different moments of time they interfere destructively, which brings about the desired spatio-temporal localization.

Theoretically, the number of the effective degrees of freedom that a shaped femtosecond pulse may apply to a nanoplasmonic system can be estimated in the following way. The number of the independent frequency bands is ∼ Δ*ω*/*γ*, where Δ*ω* is the bandwidth of the plasmonic system. For each such a band, there are two degrees of freedom: amplitude and phase. Thus, the total number *N _{DF}* of the degrees of freedom for coherent control can be estimated as

*ω*∼

*ω*, and Eq. (56) becomes where we took into account Eq. (5). In the optical region for noble metals

*Q*∼ 100 (see Fig. 2), providing a rich, ∼ 100-dimensional space of controlling parameters. The coherent control approach is non-invasive: in principle, it does not perturb or change the nanosystem’s material structure in any way.

However, how to actually determine a shaped femtosecond pulse that compels the optical fields in the nanosystem to localize at a targeted nanosite at the required femtosecond time interval is a formidable problem to which until now there has been no general and effective approach. To compare, our original chirped pulses possessed only two effective degrees of freedom (carrier frequency *ω*_{0} and chirp) which allowed one to concentrate optical energy at the tip of a V-shape structure vs. its opening [147,195]. Similarly, the two unmodulated pulses with the regulated delay *τ* between them used in the interferometric coherent control [121,213,216] also possess only two degrees of freedom (*τ* and *ω*_{0}) and can only select one of any two local-field hot spots against the other; it is impossible, in particular, to select one desired hot spot against *several* others.

There exists another method based on the adaptive genetic algorithms [202]. However, its application to the spatial-temporal localization in nanosystem is difficult due to the complexity of the problem. To date, the only example is the spatial concentration of the excitation on one arm of the three-pronged metal nanostar [215] where the obtained controlling pulses are very complicated and difficult to interpret though the nanosystem itself is rather simple. A general problem with this method is that the adaptive genetic algorithms are actually refined trial-and-error methods; they do not allow one to obtain the required controlling pulses as a result of the solution of a set of deterministic equations or an application of any regular deterministic procedure such as Green’s function integration.

#### 4.2. Time-reversal solution for coherent control

Our solution of this major problem of the coherent control, which is proposed and theoretically developed in Ref. [218], is based on an idea of time-reversal that has originally been proposed and used to control the focusing of acoustic waves and microwave radiation [219–221]. Some of these studies required use of a reverberating chamber to cause multiple interactions of the waves with the system needed to transfer the information to the far field. The electromagnetic subwavelength focusing also required a subwavelength-scale metal structure (a metal wire brush) to be positioned in the vicinity of the target system as a focusing antenna. In contrast, in nanoplasmonics there is no need for the reverberating chamber or the metal brush antenna, because the plasmonic nanosystem plays the roles of both of them. It confines the plasmonic modes for long times relative to their oscillation periods and also nano-localizes these modes.

#### 4.3. Qualitative description of time-reversal coherent control

The idea of the time-reversal solution of the nanoscale coherent control can be described using a schematic of Fig. 14. Consider a metal plasmonic nanosystem, indicated by blue in Fig. 14 (a), which may be embedded in a host dielectric (or be in vacuum). The nanosystem is excited by an external ultrafast (femtosecond) nanosource of radiation at its surface. As such we choose an oscillating dipole indicated by a double red arrow. This dipole generates a local optical electric field shown by a bold red waveform. This field excites SP oscillations of the system in its vicinity. In turn these oscillations excite other, more distant regions, and so forth until the excitation spreads out over the entire system. The relatively long relaxation time of these SP modes leads to the long “reverberations” of the plasmonic fields and the corresponding far-zone optical electric field. The latter is shown in Fig. 14 (b) where one can see that a complicated vector waveform is predicted. This waveform is time reversed, as shown in panel (c), and send back to the system as an excitation plane wave from the *far-field* zone. If the entire field, in the whole space including the near-field (evanescent) zone, were time reversed and the system would have been completely time-reversible, which would imply the absence of any dielectric losses, then the system would have been compelled by this field exactly to back-trace its own evolution in time. This would have lead to the concentration of the local optical energy exactly at the position of the initial dipole at a time corresponding to the end of the excitation pulse.

Indeed, the system is somewhat lossy, which means that it is not exactly time reversible. Nevertheless, these losses are small, and one may expect that they will not fundamentally change the behavior of the system. Another problem appear to be more significant: the evanescent fields contain the main information of the nano-distribution of the local fields in the system, and they cannot be time reversed from the far zone because they are exponentially small, practically lost there. However, our idea is that the nanostructured metal system itself plays the role of the metal brush of Ref. [221] continuously coupling the evanescent fields to the far zone. Therefore the fields in the far zone actually contain, in their reverberations, most information about the evanescent fields that will be regenerated in the process of the time reversal.

We will illustrate this idea by considering a random planar composite (RPC) whose geometry is shown in gray in the center of Fig. 15. In specific computations, as the plasmonic metal, we consider silver whose dielectric permittivity *ɛ _{m}* we adopt from bulk data [30]. This system has been generated by randomly positioning 2 × 2 × 2 nm

^{3}metal cubes on a plane, which for certainty we will consider as the

*xz*coordinate plane. The random system shown in the center of Fig. 15 has filling factor of

*f*= 0.5.

The interaction of a nanosystem with electromagnetic pulses is described in Green’s function approach using quasistatic approximation [147, 195, 222] – see Sec. 3.3. It is known that the optical excitation energy in random plasmonic nanostructures localizes in “hot spots” whose size is on the nanoscale and is determined by the minimum scale of the system inhomogeneities [76, 157, 158, 223] – see Sec. 3.5.

Initially, to find positions of these hot spots in our system, we apply an ultrashort near-infrared (near-ir) pulse whose spectral width was very large, covering a frequency band from 1.1 eV to 1.7 eV. The pulse polarization is along the *z* axis (the incidence direction is normal to the plane of the nanostructure, i.e. along the *y* axis). The resulting optical electric field **E** is expressed in terms of the external electric field of the excitation optical wave **E**_{0} and retarded dyadic Green’s function **G*** ^{r}*, as given by Eqs. (43)–(44).

The hot spots are always localized at the surface of the metal, predominantly at the periphery of the system. Their intensities found as the result of these computations are depicted by colors in the center of Fig. 15. The highest local intensity is indicated by red, and the lowest by blue in the region surrounding the metal. We have selected eight of these hot spots for our computations as denoted by letters A to H in the figure.

To generate the field in the far zone, we take a point dipole and position it at a surface of the metal at point **r**_{0} at such a hot spot, as described in the discussion of Fig. 14 above. The near-zone field **E*** ^{L}*(

**r**,

*t*) generated in response to this point dipole is found from Green’s function relation

Knowing this local electric field, we calculate the total radiating optical dipole moment of the nanosystem in the frequency domain as

*ω*and

*t*, are Fourier transforms of each other. The field in the far zone produced by this radiating dipole is given by standard electrodynamic formula – see, e.g. §67 in Ref. [224]. The time-reversed field is generated by time-reversed dipole

**D**

*(*

^{T}*t*) that is complex-conjugated in the frequency domain,

**D**

*(*

^{T}*ω*) =

**D**(

*ω*)

^{*}.

The dependence on time of the initial excitation dipole, **d**(**r**_{0},*t*) is set as an ultrashort Gaussian-shaped pulse of 12 fs duration with the carrier frequency *h̄ω*_{0} = 1.2 eV. Following the procedure described above, the fields shown in Figs. 14 and 15 have been calculated for the radiation propagating in the *y* direction (normal to the plane of the nanostructure). These fields simply copy the retarded time evolution of the emitting dipole.

At the completing stage of our calculations, the time-reversed excitation pulse is sent back to the system as a plane wave propagating along the *y* direction (normal to the nanosystem plane). To calculate the resulting local fields, we again use Green’s function Eq. (43) where the shaped excitation pulse substitutes for field **E**_{0}.

#### 4.4. Numerical results for time-reversal coherent control

The electric field of the excitation wave is chosen as a modulated waveform (including amplitude, phase, and polarization modulation) that has been computed as described above in the previous subsection. The optical excitation energy can only be concentrated at sites where SP eigenmodes localize. For the present system, these are the hot spots shown by color in the central insert of Fig. 15, labeled A-H. The corresponding calculated excitation waveforms are displayed in panels as vector plots shown as functions of time {*E _{x}*(

*t*),

*E*(

_{z}*t*)}.

There are several important features of these waveforms deserving our attention and discussion. First, these waveforms are rather long in duration: much longer than the excitation-dipole 12 fs pulses. This confirms our understanding that the initial dipole field excites local SP fields that, in a cascade manner, excite a sequence of the system SPs, which ring down relatively long time (over 200 fs, as shown in the figure). This long ring-down process is exactly what is required for the nanostructure to transfer to the far-field zone the information on the near-zone local (evanescent) fields as is suggested by our idea presented above in the introduction. The obtained fields are by shape resembling the controlling pulses for the microwave radiation [221]. However, a fundamental difference is that in the microwave case the long ringing-down is due to the external reverberation chamber, while for the nanoplasmonic systems it is due to the intrinsic evolution of the highly resonant SP eigenmodes that possess high *Q*-factors (setting a reverberation chamber around a nanosystem would have been, indeed, unrealistic).

Second, one can see that the pulses in Fig. 15 have a very nontrivial polarization properties ranging from the pure linear polarization (indicated by red as explained in the caption to Fig. 15) to the circular polarization indicated by blue, including all intermediate degrees of circularity. The temporal-polarization structure of pulses A-H in Fig. 15 is very complicated, somewhat reminding that of Ref. [215], which was obtained by a genetic adaptive algorithm. However, in our case these pulses are obtained in a straightforward manner, by applying the well-known, deterministic Green’s function of the system, which is a highly efficient and fast method.

Third, and most important, feature of the waveforms in Fig. 15 is that they are highly site-specific: pulses generated by the initial dipole in different positions are completely different. This is a very strong indication that they do transfer to the far far-field zone the information about the complicated spatio-temporal structure of the local, near-zone fields. This creates a pre-requisite for studying a possibility to use these pulses for the coherently-controlled nano-targeting.

Now we turn to the crucial test of the nanofocusing induced by the excitation pulses discussed above in conjunction with Fig. 15. Because of the finite time window (*T* = 228 fs) used for the time reversal, all these excitation pulses end and should cause the concentration of the optical energy (at the corresponding sites) at the same time, *t* = *T* = 228 fs (counted from the moment the excitation pulse starts impinging on the system). After this concentration instant, the nanofocused fields can, in principle, disappear (dephase) during a very short period on the order of the initial dipole pulse length, i.e. ∼ 12 fs. Thus this nanofocusing is a dynamic, transient phenomenon.

Note that averaging (or, integration) of the local-field intensity *I*(**r**,*t*) = |**E**(**r**,*t*)|^{2} over time *t* would lead to the loss of the effects of the phase modulation. This is due to a mathematical equality
${\int}_{-\infty}^{\infty}I\left(\mathbf{r},t\right)dt={\int}_{-\infty}^{\infty}{\left|\mathbf{E}\left(\mathbf{r},\omega \right)\right|}^{2}d\omega /\left(2\pi \right)$, where the spectral-phase modulation of the field certainly eliminates from the expression in the right-hand side. Thus the averaged intensity of the local fields is determined only by the local power spectrum of the excitation |**E**(**r**,*ω*)|^{2} and, consequently, is not coherently controllable. Very importantly, such a cancellation does not take place for nonlinear phenomena. In particular, two-photon processes such as two-photon fluorescence or two-photon electron emission that can be considered as proportional to the squared intensity *I*^{2}(**r**,*t*) = |**E**(**r**,*t*)|^{4} are coherently controllable even after time averaging (integration), as we have argued earlier [147, 213]. Note the distributions measured in nonlinear optical experiments with the detection by the PEEM [121,215,216,225] and in the fluorescence upconversion experiments [226] can be modeled as such nonlinear processes that yield distributions
$\u3008{I}^{n}\left(\mathbf{r}\right)\u3009={\int}_{-\infty}^{\infty}{I}^{n}\left(\mathbf{r},t\right)dt/T$, where *n* ≥ 2. Inspired by this, we will consider below, in particular, the coherent control of the two-photon process averaged intensity 〈*I*^{2}(**r**)〉.

Let us investigate how precisely one can achieve the spatio-temporal focusing of the optical excitation at a given nanosite of a plasmonic nanostructure using the full shaping (amplitude, phase, and polarization) of the excitation pulses found from the time-reversal method. The results for the present nanostructure, targeting sites A-H, are shown in Fig. 16. For each excitation pulse, the spatial distribution of the local field intensity is displayed for the moment of time when this local intensity acquires its global (highest) maximum. The most important conclusion that one can draw from comparing panels (a)–(h) is that for each pulse A–H this global maximum corresponds to the maximum concentration of the optical energy at the corresponding targeted nanosite A–H. This obtained spatial resolution is as good as 4 nm, which is determined by the spatial size of inhomogeneities of the underlying plasmonic metal nanosystem. It is very important that this localization occurs not only at the desired nanometer-scale location but also very close to the targeted time that in our case is *t* = 228 fs. Thus the full shaping of femtosecond pulses by the time reversal is an efficient method of controlling the spatio-temporal localization of energy at the femtosecond-nanometer scale.

Let us turn to the temporal dynamics of intensity of the nanoscale local fields at the targeted sites A–H, which is shown in Fig. 17 (a)–(h). As we can see, in each of the panels there is a sharp spike of the local fields very close to the target time of *t* = 228. The duration of this spike in most panels [(a)–(f)] is close to that of the initial dipole, i.e., 12 fs. This shows a trend to the reproduction of the initial excitation state due to the evolution of the time-reversed SP packet induced by the shaped pulses. There is also a pedestal that shows that this reproduction is not precise, which is expected due to the fact that the time reversal is incomplete: only the far-zone field propagating in one direction (along the *y* axis) is reversed. Nevertheless, as the discussion of Fig. 16 shows, this initial excitation-state reproduction is sufficient to guarantee that the targeted (initial excitation) site develops the global maximum (in time and space) of the local-field intensity. Interesting enough, the trend to reproduce the initial excitation state is also witnessed by almost symmetric (with respect to the maximum points *t* = 228 fs) shapes of all waveforms, which occurs in spite of the very asymmetric shapes of the excitation waveforms [cf. Fig. 15].

Apart from the ultrafast (femtosecond) dynamics of the nanolocalized optical fields discussed above in conjunction with Figs. 16 and 17, there is a considerable interest in its the time-integrated or averaged distributions, in particular, the mean squared intensity 〈*I*^{2}(**r**)〉. This quantity defines the nanoscale spatial distribution of the incoherent two-photon processes such as two-photon electron emission or two-photon luminescence. For example, in some approximation, the spatial distribution of the two-photon electron emission recorded by PEEM [121, 215, 216, 225] is determined by 〈*I*^{2}(**r**)〉.

Now we test the spatial concentration of time-averaged mean-squared intensity 〈*I*^{2}(**r**)〉 for all sites, which is displayed in Fig. 18. As clearly follows from this figure, in all cases, there are leading peaks at the targeted sites. Thus the two-photon excitation, even after the time averaging, can be concentrated at desired sites using the coherent-control by the time-reversed shaped pulses.

We point out that there has recently been an experimental demonstration of a coherent spatiotemporal control on the nanoscale by polarization and phase pulse shaping [217]. The optical energy concentration at a given site on a ∼ 50 nm spatial scale at a given time on a ∼ 100 fs temporal scale has been demonstrated. Since this time scale is comparable to or longer than the SP dephasing time, the time-reversal method could not be employed.

#### 4.5. Coherent control by spatiotemporal pulse shaping

For coherent control on the nanoscale, as we have described above in Sec. 4, the phase of the excitation waveform along with its polarization provide functional degrees of freedom to control the nanoscale distribution of energy [121,147,195,213–215,217,225,227]. Spatiotemporal pulse shaping permits one to generate dynamically predefined waveforms modulated both in frequency and in space to focus ultrafast pulses in the required microscopic spatial and femtosecond temporal domains [228, 229].

Here we follow Ref. [210] that has introduced a method of full coherent control on the nanoscale where a temporally and *spatially* modulated waveform is launched in a graded nanostructured system, specifically a wedge – see schematic of Fig. 19. Its propagation from the thick (macroscopic) to the thin (nanoscopic) edge of the wedge and the concurrent adiabatic concentration provide a possibility to focus the optical energy in nanoscale spatial and femtosecond temporal regions.

This method unifies three components that individually have been developed and experimentally tested. The coupling of the external radiation to the surface plasmon polaritons (SPPs) propagating along the wedge occurs through an array of nanoobjects (nanoparticles or nanoholes) that is situated at the thick edge of the wedge. The phases of the SPPs emitted (scattered) by individual nanoobjects are determined by a spatio-temporal modulator. The nanofocusing of the SPPs occurs due to their propagation toward the nanofocus and the concurrent adiabatic concentration [12, 230, 231].

The coupling of the external radiation to SPPs and their nanofocusing have been observed – see, e.g., Refs. [232,233]. The second component of our approach, the spatio-temporal coherent control of such nanofocusing has been developed [228,229]. The third component, the adiabatic concentration of SPPs also has been observed and extensively studied experimentally [13–16, 18, 19, 22].

The adiabatic concentration (nanofocusing) is based on adiabatic following by a propagating SPP wave of a graded plasmonic waveguide, where the phase and group velocities decrease while the propagating SPP wave is adiabatically transformed into a standing, localized SP mode. A new quality that is present in this approach is a possibility to arbitrary move the nanofocus along the nanoedge of the wedge. Moreover, it is possible to superimpose any number of such nanofoci simultaneously and, consequently, create any distribution of the nanolocalized fields at the thin edge of the wedge.

To illustrate this idea of the full spatiotemporal coherent control, now let us turn to a wedge that contains a line of nanosize scatterers (say, nanoparticles or nanoholes) located at the thick edge and parallel to it, i.e. in the *x* direction in Fig. 19. Consider first monochromatic light incident on these nanoparticles or nanoholes that scatter and couple it into SPP wavelets. Every such a scatterer emits SPPs in all directions; there is, of course, no favored directionality of the scattering.

At this point, we assume that the excitation radiation and, correspondingly, the scattered wavelets of the SPP are coherent, and their phases smoothly vary in space along the thick edge, i.e., in the *x* direction. Then the SPP wavelets emitted by different scatterers will interfere, which in accord with the Huygens-Fresnel principle leads to formation of a smooth wavefront of the SPP wave at some distance from the scatterers in the “far SPP field”, i.e., at distances much greater than the SPP wavelength 2*π*/*k _{SPP}*.

Such wavefronts are shown in Fig. 19 with concave black curves. The energy of the SPP is transferred along the rays, which are the lines normal to the wavefronts, shown by the colored lines. By the appropriate spatial phase modulation of the excitation radiation along the line of scatterers (in the *x* direction) over distances of many SPP wavelengths, these wavefronts can be formed in such a way that the rays intersect at a given point, forming a nanofocus at the thin (sharp) edge of the wedge, as shown schematically in Fig. 19. Diffraction of the SPP waves will lead to a finite size of this focal spot.

By changing the spatial phase profile of the excitation radiation, this focal spot can be arbitrarily moved along the thin edge. This focusing and adiabatic concentration, as the SPPs slow down approaching the sharp edge, will lead to the enhancement of the intensity of the optical fields in the focal region. This dynamically-controlled concentration of energy is a plasmonic counterpart of a large phased antenna array (also known as an aperture synthesis antenna), widely used in radar technology (synthetic aperture radar or SAR) and radio astronomy [234].

Now we can consider excitation by spatiotemporally shaped ultrashort pulses independently in space. Such pulses are produced by spatio-temporal modulators [228, 229]. The field produced by them is a coherent superposition of waves with different frequencies whose amplitudes and phases can arbitrarily vary in space and with frequency. This modulation can be chosen so that all the frequency components converge at the same focal spot at the same time forming an ultrashort pulse of the nanolocalized optical fields.

As an example we consider a silver [30] nanowedge illustrated in Fig. 19 whose maximum thickness is *d _{m}* = 30 nm, the minimum thickness is

*d*= 4 nm, and whose length (in the

_{f}*y*direction) is

*L*= 5

*μ*m. Trajectories calculated by the Wentzel-Kramers-Brillouin (WKB) method in Ref. [210] for

*h̄ω*= 2.5 eV are shown by lines (color used only to guide eye); the nanofocus is indicated by a bold red dot. In contrast to focusing by a conventional lens, the SPP rays are progressively bent toward the wedge slope direction.

Now consider the problem of coherent control. The goal is to excite a spatiotemporal waveform at the thick edge of the wedge in such a way that the propagating SPP rays converge at an arbitrary nanofocus at the sharp edge where an ultrashort pulse is formed. To solve this problem, we use the idea of back-propagation or time-reversal [220, 221, 235]. We generate rays at the nanofocus as an ultrashort pulse containing just several oscillations of the optical field. Propagating these rays, we find amplitudes and phases of the fields at the thick edge at each frequency as given by the complex propagation phase (eikonal) Φ(*ρ*), where *ρ* is a 2-d coordinate vector in the plane of the wedge. Then we complex conjugate the amplitudes of frequency components, which corresponds to the time reversal. We also multiply these amplitudes by exp(2ImΦ), which pre-compensates for the Ohmic losses. This provides the required phase and amplitude modulation at the thick edge of the wedge.

We show an example of such calculations in Fig. 20. Panel (a) displays the trajectories of SPPs calculated [210] by the WKB method. The trajectories for different frequencies are displayed by colors corresponding to their visual perception. There is a very significant spectral dispersion: trajectories with higher frequencies are much more curved. The spatial-frequency modulation that we have found succeeds in bringing all these rays (with different frequencies and emitted at different *x* points) to the same nanofocus at the sharp edge.

The required waveforms at different *x* points of the thick edge of the wedge are shown in Fig. 20 (b)–(d) where the corresponding longitudinal electric fields are shown. The waves emitted at large *x*, i.e., at points more distant from the nanofocus, should be emitted significantly earlier to pre-compensate for the longer propagation times. They should also have different amplitudes due to the differences in the adiabatic compression along the different rays. Finally, there is clearly a negative chirp (gradual decrease of frequency with time). This is due to the fact that the higher frequency components propagate more slowly and therefore must be emitted earlier to form a coherent ultrashort pulse at the nanofocus.

In Fig. 20 (e) we display together all three of the representative waveforms at the thick edge to demonstrate their relative amplitudes and positions in time. The pulse at the extreme point in *x* (shown by blue) has the longest way to propagate and therefore is the most advanced in time. The pulse in the middle point (shown by green) is intermediate, and the pulse at the center (*x* = 0, shown by red) is last. One can notice also a counterintuitive feature: the waves propagating over longer trajectories are smaller in amplitude though one may expect the opposite to compensate for the larger losses. The explanation is that the losses are actually insignificant for the frequencies present in these waveforms, and the magnitudes are determined by adiabatic concentration factor.

Figure 20 (e) also shows the resulting ultrashort pulse in the nanofocus. This is a transform-limited, Gaussian pulse. The propagation along the rays completely compensates the initial phase and amplitude modulation, exactly as intended. As a result, the corresponding electric field of the waveform is increased by a factor of 100. Taking the other component of the electric field and the magnetic field into account, the corresponding increase of the energy density is by a factor ∼ 10^{4} with respect to that of the SPPs at the thick edge.

To briefly conclude, an approach [210] to full coherent control of spatiotemporal energy localization on the nanoscale has been presented. From the thick edge of a plasmonic metal nanowedge, SPPs are launched, whose phases and amplitudes are independently modulated for each constituent frequency of the spectrum and at each spatial point of the excitation. This pre-modulates the departing SPP wave packets in such a way that they reach the required point at the sharp edge of the nanowedge in phase, with equal amplitudes forming a nanofocus where an ultrashort pulse with required temporal shape is generated. This system constitutes a “nanoplasmonic portal” connecting the incident light field, whose features are shaped on the microscale, with the required point or features at the nanoscale.

#### 4.6. Experimental demonstrations of coherent control on the nanoscale

The ideas of the coherent control of the nanoscale distribution of ultrafast optical fields both space and in time, which have been introduced theoretically in Refs. [147, 195, 210, 214, 218, 236, 237], have been investigated and confirmed experimentally. Using the full phase and amplitude modulation of the excitation-pulse wavefront in both polarizations (the so-called polarization pulse shaping), the experiments have achieved both spatial control [121, 215] and spatiotemporal control [217] on nanometer-femtosecond scale.

Recently spatiotemporal nanofocusing via the adiabatic concentration along the lines of ideas presented above in Sec. 4.5 has been successfully demonstrated experimentally [21]. In this work, a shaped femtosecond pulse has been coupled by a grating to a TM_{0} SPP mode on the surface of an adiabatically-tapered nanocone. The spatiotemporal concentration of optical energy in space to a ∼ 10 nm region and in time to a 15 fs duration (Fourier-transform limited, i.e., the shortest possible at a given bandwidth). Indeed the position of the nanofocus in Ref. [21] is always the the tip of the nanocone; so the possibility of moving the nanofocus in space is not available.

The ideas of employing the spatial modulation of the excitation wavefront [210] described above in Sec. 4.5 have been experimentally tested and confirmed for continuous wave (CW) excitation [211,212]. We will present some of these experimental results below in this Section.

We start with experiments on polarization-shaping coherent control that we adapt from Ref. [215]. The corresponding experimental approach is schematically illustrated in Fig. 21. Polarization-shaped ultrashort laser pulses illuminate a planar nanostructure, with two-photon photoemission electron microscopy (PEEM) [238] providing the feedback signal from the nanoscale field distribution that is essential for adaptive near-field control.

The spatial resolution of two-photon PEEM (∼ 50 nm) is determined by its electron optics and is, thus, independent of the electromagnetic light-field diffraction limit. The sensitivity of the two-photon PEEM patterns to the optical field intensities arises from the nonlinear two-photon photoemission process whose intensity is proportional to the time-integrated fourth power of the local electric-field amplitude. With these elements in place, a user-specified nanoscopic optical field distribution is realized by processing recorded photoemission patterns in an evolutionary algorithm that directs the iterative optimization of the irradiating laser pulse shape.

The basic idea of the experiment is that the measured PEEM pattern identifies the origin of ejected photoelectrons and hence the regions of high local field intensity. A controlled variation of the PEEM pattern then proves the spatial control over the nanoscopic field distribution. We have already discussed such an approach above – see Fig. 10 [121] and the corresponding discussion in Sec. 3.6.

The nanostructure used consists of circular Ag disks with 180 nm diameter and 30 nm height, fabricated by electron-beam lithography on a conductive, 40-nm-thick indium-tin oxide (ITO) film grown on a quartz substrate. The disks are arranged into three dimers that form the arms of a star-like shape (Fig. 21 (a), lower right). The whole nanostructure is about 800 nm across, while the gap between two of the dimer disks is ∼ 10 nm wide. After inspection by scanning-electron microscopy (SEM), the sample is mounted in the ultrahigh-vacuum PEEM set-up. The deposition of a small amount of caesium (∼ 0.1 monolayers) reduces the work function of the Ag nanostructure to about 3.1 eV, that is, just below the threshold for two-photon photoemission with 790 nm photons.

The PEEM pattern obtained after maximization of the photoemission from the upper two arms of the Ag nanostructure in shown in Fig. 21 (c). It shows strong emission from these two upper arms and almost no emission from the bottom arm. Analogously, the photoemission after minimization of the upper part PEEM brightness [Fig. 21 (e)] occurs mainly in the lower area while the contribution from the upper two arms is extremely weak. The adaptively determined solution to each optimization problem has been proven to be robust with respect to slight imperfections in the experimental nanostructures. These successful optimizations demonstrate that polarization pulse shaping allows adaptive control of the spatial distribution of photoelectrons on a subwavelength scale, and thus of the nanoscopic optical fields that induce photoemission.

The optimally polarization-shaped laser pulses after adaptive maximization and minimization described above are shown in Figs. 21 (b) and (d), respectively, as determined by dual-channel spectral interferometry [239, 240]. In this representation, the shape of the quasi-three-dimensional figure indicates the temporal evolution of the polarization state of the electric field, with the color representing the instantaneous oscillation frequency. Contributions from both transverse polarization components are visible in each of the two cases. Whereas the upper-region photoemission maximization is achieved with a comparatively simple time evolution, the corresponding minimization requires a more complex field with varying degrees of ellipticity, orientation and temporal amplitudes.

Our idea [210] of the coherent control on the nanoscale by spatial modulation (shaping) of the excitation waveform has been developed theoretically [237] and experimentally [211,212]. The coherent control of nanoscale distribution of local optical fields based on CW excitation aimed at achieving a deterministic control of plasmonic fields by using the spatial shaping of high order beams such as Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams has been carried out in Ref. [211]. It has been shown experimentally that the spatial phase shaping of the excitation field provides an additional degree of freedom to drive optical nanoantennas and consequently control their near field response.

An example of such a deterministic coherent control is illustrated in Fig. 22. It shows a double gap antenna formed by three 500 nm aligned gold bars forming two identical 50 nm air gaps separated by 500 nm. For reference, in panel (a) it displays a measured two-photon luminescence (TPL) map when driving the whole antenna with a Gaussian beam linearly polarized along the *x*-axis. Note that similar to what has been discussed above in Sec. 4.4, in particular, in conjunction with Fig. 18, the TPL reflects the time-averaged distribution of the local field intensity 〈*I*^{2}(**r**)〉. As we see from Fig. 22 (a) and as expected, a field concentration is observed in both gaps. Figures 22 (b) and (c) show TPL maps recorded when the *π*-phase shift of a HG10 beam coincides, respectively, with the right and left gaps. These data demonstrate how a suitable positioning of the phase jump over the double antenna enables us to selectively switch on and off one of the two hot-spot sites.

Even closer to the original idea [210] that a plasmonic wavefront can be shaped and focused at a predetermined spot by a spatial phase modulation of the excitation waveform incident on optically-addressable launch pads is a recent publication [212]. This article achieves controlled launching and propagation of SPPs by spatially designing the amplitude and phase of the incident light. The chosen amplitude profile, consisting of four bright (“on”) SPP launching platforms and one central dark (“off”) arena, fully separates plasmonic effects from photonic effects and in addition is the necessary starting point for later focusing and scanning experiments. Any intensity detected inside the arena is purely plasmonic.

Adapting from Ref. [212], we present the achieved SPP focusing in Fig. 23. A phase optimization loop is used to focus SPPs at a pre-chosen target. This loop yields the optimal phase for each launching pad (“superpixel”) as well as the relative intensity to focus. The amplitude profile is the same in all cases including the bare gold case, with four launching areas and a central dark arena where only SPPs can propagate. The incident polarization is diagonal in relation to the grating lines so as to have all available angles (2*π* range) contributing to the focus, thereby maximizing the numerical aperture and resolution.

Successful focusing at the center of the SPP arena is shown in Fig. 23 (a). The structured SPP wavefront produces an intensity in the designated target that is at least 20 times higher than the average SPP background of an unstructured wavefront. The measured size of the plasmonic focus is 420 nm, consistent with the diffraction limit of the SPPs. The flexibility of the method (scanning the focus) is demonstrated in Figs. 23 (b) and (c), which shows the SPP focus relocated without mechanical motion to controlled positions in the plasmonic arena.

The work of Ref. [212] has fully implemented the idea of Ref. [210] on the spatial-phase-modulation control of the SPP wavefronts to position a SPP nanofocus at a desired location at the surface. However, it employs only CW excitation and does not exploit a potential femtosecond temporal degree of freedom to achieve such a nanofocusing at a predetermined moment of time as in Ref. [210].

## 5. Quantum nanoplasmonics: Spaser and nanoplasmonics with gain

#### 5.1. Introduction to spasers and spasing

Not just a promise anymore [241], nanoplasmonics has delivered a number of important applications: ultrasensing [242], scanning near-field optical microscopy [190, 243], SP-enhanced photodetectors [51], thermally assisted magnetic recording [244], generation of extreme uv [136], biomedical tests [242, 245], SP-assisted thermal cancer treatment [246], plasmonic enhanced generation of extreme ultraviolet (EUV) pulses [136] and extreme ultraviolet to soft x-ray (XUV) pulses [247], and many others – see also Ref. [23].

To continue its vigorous development, nanoplasmonics needs an active device – near-field generator and amplifier of nanolocalized optical fields, which has until recently been absent. A nanoscale amplifier in microelectronics is the metal-oxide-semiconductor field effect transistor (MOSFET) [248,249], which has enabled all contemporary digital electronics, including computers and communications and enabled the present day technology as we know it. However, the MOSFET is limited by frequency and bandwidth to ≲ 100 GHz, which is already a limiting factor in further technological development. Another limitation of the MOSFET is its high sensitivity to temperature, electric fields, and ionizing radiation, which limits its use in extreme environmental conditions and nuclear technology and warfare.

An active element of nanoplasmonics is the spaser (Surface Plasmon Amplification by Stimulated Emission of Radiation), which was proposed [29, 250] as a nanoscale quantum generator of nanolocalized coherent and intense optical fields. The idea of spaser has been further developed theoretically [137–139, 251]. Spaser effect has recently been observed experimentally [252]. Also a number of SPP spasers (also called nanolasers) have been experimentally observed [253–256].

Spaser is a nanoplasmonic counterpart of laser: it is a quantum generator and nanoamplifier where photons as the generated quanta are replaced by SPs. Spaser consists of a metal nanoparticle, which plays a role of the laser cavity (resonator), and the gain medium. Figure 24 schematically illustrates geometry of a spaser introduced in the original article [29], which contains a V-shaped metal nanoparticle surrounded by a layer of semiconductor nanocrystal quantum dots.

#### 5.2. Spaser fundamentals

As we have already mentioned, the spaser is a nanoplasmonic counterpart of the laser [29, 251]. The laser has two principal elements: resonator (or cavity) that supports photonic mode(s) and the gain (or active) medium that is population-inverted and supplies energy to the lasing mode(s). An inherent limitation of the laser is that the size of the laser cavity in the propagation direction is at least half wavelength and practically more than that even for the smallest lasers developed [253,254,257]. In the spaser [29] this limitation is overcome. The spasing modes are surface plasmons (SPs) whose localization length is on the nanoscale [76] and is only limited by the minimum inhomogeneity scale of the plasmonic metal and the nonlocality radius [33] *l _{nl}* ∼ 1 nm. So, the spaser is truly nanoscopic – its minimum total size can be just a few nanometers.

The resonator of a spaser can be any plasmonic metal nanoparticle whose total size *R* is much less than the wavelength *λ* and whose metal thickness is between *l _{nl}* and

*l*, which supports a SP mode with required frequency

_{s}*ω*. This metal nanoparticle should be surrounded by the gain medium that overlaps with the spasing SP eigenmode spatially and whose emission line overlaps with this eigenmode spectrally [29]. As an example, we consider a model of a nanoshell spaser [137, 251, 258], which is illustrated in Fig. 25. Panel (a) shows a silver nanoshell carrying a single SP (plasmon population number

_{n}*N*= 1) in the dipole eigenmode. It is characterized by a uniform field inside the core and hot spots at the poles outside the shell with the maximum field reaching ∼ 10

_{n}^{6}V/cm. Similarly, Fig. 25 (b) shows the quadrupole mode in the same nanoshell. In this case, the mode electric field is non-uniform, exhibiting hot spots of ∼ 1.5 × 10

^{6}V/cm of the modal electric field at the poles. These high values of the modal fields is the underlying physical reason for a very strong feedback in the spaser. Under our conditions, the electromagnetic retardation within the spaser volume can be safely neglected. Also, the radiation of such a spaser is a weak effect: the decay rate of plasmonic eigenmodes is dominated by the internal loss in the metal. Therefore, it is sufficient to consider only quasistatic eigenmodes [27, 76] and not their full electrodynamic counterparts [259].

For the sake of numerical illustrations of our theory, we will use the dipole eigenmode [Fig. 25 (a)]. There are two basic ways to place the gain medium: (i) outside the nanoshell, as shown in panel (c), and (ii) in the core, as in panel (d), which was originally proposed in Ref. [258]. As we have verified, these two designs lead to comparable characteristics of the spaser. However, the placement of the gain medium inside the core illustrated in Fig. 25 (d) has a significant advantage because the hot spots of the local field are not covered by the gain medium and are sterically available for applications.

Note that any *l*-multipole mode of a spherical particle is, indeed, 2*l* + 1-times degenerate. This may make the spasing mode to be polarization unstable, like in lasers without polarizing elements. In reality, the polarization may be clamped and become stable due to deviations from the perfect spherical symmetry, which exist naturally or can be introduced deliberately. More practical shape for a spaser may be a nanorod, which has a mode with the stable polarization along the major axis. However, a nanorod is a more complicated geometry for theoretical treatment, and we will consider it elsewhere.

The level diagram of the spaser gain medium and the plasmonic metal nanoparticle is displayed in Fig. 25 (e) along with a schematic of the relevant energy transitions in the system. The gain medium chromophores may be semiconductor nanocrystal quantum dots [29, 260], dye molecules [261, 262], rare-earth ions [258], or electron-hole excitations of an unstructured semiconductor [253,257]. For certainty, we will use a semiconductor-science language of electrons and holes in quantum dots.

The pump excites electron-hole pairs in the chromophores [Fig. 25 (e)], as indicated by the vertical black arrow, which relax to form excitons. The excitons constitute the two-level systems that are the donors of energy for the SP emission into the spasing mode. In vacuum, the excitons would recombine emitting photons. However, in the spaser geometry, the photoemission is strongly quenched due to the resonance energy transfer to the SP modes, as indicated by the red arrows in the panel. The probability of the radiativeless energy transfer to the SPs relative to that of the radiative decay (photon emission) is given by the so-called Purcell factor

where*R*is a characteristic size of the spaser metal core. Thus this radiativeless energy transfer to the spaser mode is the dominant process whose probability is by orders of magnitude greater than that of the free-space (far-field) emission.

The plasmons already in the spaser mode create the high local fields that excite the gain medium and stimulate more emission to this mode, which is the feedback mechanism. If this feedback is strong enough, and the life time of the spaser SP mode is long enough, then an instability develops leading to the avalanche of the SP emission in the spasing mode and spontaneous symmetry breaking, establishing the phase coherence of the spasing state. Thus the establishment of spasing is a non-equilibrium phase transition, as in the physics of lasers.

#### 5.3. Brief overview of latest progress in spasers

After the original theoretical proposal and prediction of the spaser [29], there has been an active development in this field, both theoretical and experimental. There has also been a US patent issued on spaser [250].

Among theoretical developments, a nanolens spaser has been proposed [263], which possesses a nanofocus (“the hottest spot”) of the local fields. In Refs. [29, 263], the necessary condition of spasing has been established on the basis of the perturbation theory.

There have been theories published describing the SPP spasers (or, “nanolasers” as sometimes they are called) phenomenologically, on the basis of classic linear electrodynamics by considering the gain medium as a dielectric with a negative imaginary part of the permittivity, e.g., [258]. Very close fundamentally and technically are works on the loss compensation in metamaterials [264–267]. Such linear-response approaches do not take into account the nature of the spasing as a non-equilibrium phase transition, at the foundation of which is spontaneous symmetry breaking: establishing coherence with an arbitrary but sustained phase of the SP quanta in the system [137]. Spaser is necessarily a deeply-nonlinear (nonperturbative) phenomenon where the coherent SP field always saturates the gain medium, which eventually brings about establishment of the stationary (or, continuous wave, CW) regime of the spasing [137]. This leads to principal differences of the linear-response results from the microscopic quantum-mechanical theory in the region of spasing, as we discuss below in conjunction with Fig. 27.

There has also been a theoretical publication on a bowtie spaser (nanolaser) with electrical pumping [268]. It is based on balance equations and only the CW spasing generation intensity is described. Yet another theoretical development has been a proposal of the lasing spaser [269], which is made of a plane array of spasers.

There have also been a theoretical proposal of a spaser (“nanolaser”) consisting of a metal nanoparticle coupled to a single chromophore [270]. In this paper, a dipole-dipole interaction is illegitimately used at very small distances *r* where it has a singularity (diverging for *r* → 0), leading to a dramatically overestimated coupling with the SP mode. As a result, a completely unphysical prediction of CW spasing due to single chromophore has been obtained [270]. In contrast, our theory [137] is based on the full (exact) field of the spasing SP mode without the dipole (or, any multipole) approximation. As our results of Sec. 5.5 below show, hundreds of chromophores per metal nanoparticle are realistically requited for the spasing even under the most favorable conditions.

There has been a vigorous experimental investigation of the spaser and the concepts of spaser. Stimulated emission of SPPs has been observed in a proof-of-principle experiment using pumped dye molecules as an active (gain) medium [261]. There have also been later experiments that demonstrated strong stimulated emission compensating a significant part of the SPP loss [262, 271–274]. As a step toward the lasing spaser, the first experimental demonstration has been reported of a partial compensation of the Joule losses in a metallic photonic metamaterial using optically pumped PbS semiconductor quantum dots [260]. There have also been experimental investigations reporting the stimulated emission effects of SPs in plasmonic metal nanoparticles surrounded by gain media with dye molecules [275, 276]. The full loss compensation and amplification of the long-range SPPs at *λ* = 882 nm in a gold nanostrip waveguide with a dyes solution as a gain medium has been observed [277].

At the present time, there have been a number of the successful experimental observations of the spaser and SPP spasers (the so-called nanolasers). An electrically-pumped nanolaser with semiconductor gain medium has been demonstrated [253] where the lasing modes are SPPs with a one-dimensional confinement to a ∼ 50 nm size. A nanolaser with an optically-pumped semiconductor gain medium and a hybrid semiconductor/metal (CdS/Ag) SPP waveguide has been demonstrated with an extremely tight transverse (two-dimensional) mode confinement to ∼ 10 nm size [254]. This has been followed by the development of CdS/Ag nanolasers generating a visible single mode at a room temperature with a tight one-dimensional confinement (∼ 20 nm) and a two-dimensional confinement in the plane of the structure to an area ∼ 1 *μ*m^{2} [255]. A highly efficient SPP spaser in the communication range (*λ* = 1.46 *μ*m) with an optical pumping based on a gold film and an InGaAs semiconductor quantum-well gain medium has recently been reported [256].

Finally, an observation has been published of a nanoparticle spaser [252]. This spaser is a chemically synthesized gold nanosphere of radius 7 nm surrounded by a dielectric shell of a 21 nm outer radius containing immobilized dye molecules. Under nanosecond optical pumping in the absorption band of the dye, this spaser develops a relatively narrow-spectrum and intense visible emission that exhibits a pronounced threshold in pumping intensity. The observed characteristics of this spaser are in an excellent qualitative agreement and can be fully understood on the basis of the corresponding theoretical results described below in Sec. 5.5.

#### 5.4. Equations of spaser

### 5.4.1. Quantum density matrix equations (optical Bloch equations) for spaser

The SP eigenmodes *φ _{n}*(

**r**) are described by a wave equation (25) [29, 76]. The electric field operator of the quantized SPs is an operator [29]

*â*are the SP creation and annihilation operators, −∇

_{n}*φ*(

_{n}**r**) =

**E**

*(*

_{n}**r**) is the modal field of an

*n*th mode, and

*s*′

*= Re [*

_{n}*ds*(

*ω*)/

_{n}*dω*]. Note that we have corrected a misprint in Ref. [29] by replacing the coefficient 2

_{n}*π*by 4

*π*.

The spaser Hamiltonian has the form

*Ĥ*is the Hamiltonian of the gain medium,

_{g}*p*is a number (label) of a gain medium chromophore,

**r**

*is its coordinate vector, and*

_{p}**d̂**

^{(p)}is its dipole moment operator. In this theory, we treat the gain medium quantum mechanically but the SPs quasiclassically, considering

*â*as a classical quantity (c-number)

_{n}*a*with time dependence as

_{n}*a*=

_{n}*a*

_{0n}exp(−

*iωt*), where

*a*

_{0n}is a slowly-varying amplitude. The number of coherent SPs per spasing mode is then given by

*N*= |

_{p}*a*

_{0n}|

^{2}. This approximation neglects the quantum fluctuations of the SP amplitudes. However, when necessary, we will take into account these quantum fluctuations, in particular, to describe the spectrum of the spaser.

Introducing *ρ*^{(p)} as the density matrix of a *p*th chromophore, we can find its equation of motion in a conventional way by commutating it with the Hamiltonian (62) as

*n*. In this approximation, the time dependence of the nondiagonal elements of the density matrix is ${\left({\rho}^{\left(p\right)}\right)}_{12}={\overline{\rho}}_{12}^{\left(p\right)}\text{exp}\left(i\omega t\right)$, and ${\left({\rho}^{\left(p\right)}\right)}_{21}={\overline{\rho}}_{12}^{\left(p\right)*}\text{exp}\left(-i\omega t\right)$, where ${\overline{\rho}}_{12}^{\left(p\right)}$ is an amplitude slowly varying in time, which defines the coherence (polarization) for the |2〉 ⇌ |1〉 spasing transition in a

*p*th chromophore of the gain medium.

Introducing a rate constant Γ_{12} to describe the polarization relaxation and a difference
${n}_{21}^{\left(p\right)}={\rho}_{22}^{\left(p\right)}-{\rho}_{11}^{\left(p\right)}$ as the population inversion for this spasing transition, we derive an equation of motion for the non-diagonal element of the density matrix as

*p*th chromophore, and ${\mathbf{d}}_{12}^{\left(p\right)}$ is the corresponding transitional dipole element. Note that always ${\mathbf{d}}_{12}^{\left(p\right)}$ is either real or can be made real by a proper choice of the quantum state phases, making the Rabi frequency ${\tilde{\Omega}}_{12}^{\left(p\right)}$ also a real quantity.

An equation of motion for
${n}_{21}^{p}$ can be found in a standard way by commutating it with *Ĥ*. To provide conditions for the population inversion (
${n}_{21}^{p}>0$), we imply existence of a third level. For simplicity, we assume that it very rapidly decays into the excited state |2〉 of the chromophore, so its own populations is negligible. It is pumped by an external source from the ground state (optically or electrically) with some rate that we will denote *g*. In this way, we obtain the following equation of motion:

*γ*

_{2}is the decay rate |2〉 → |1〉.

The stimulated emission of the SPs is described as their excitation by the coherent polarization of the gain medium. The corresponding equation of motion can be obtained using Hamiltonian (62) and adding the SP relaxation with a rate of *γ _{n}* as

As an important general remark, the system of Eqs. (64), (66), and (67) is highly nonlinear: each of these equations contains a quadratic nonlinearity: a product of the plasmon-field amplitude *a*_{0n} by the density matrix element *ρ*_{12} or population inversion *n*_{21}. Altogether, this is a six-order nonlinearity. This nonlinearity is a fundamental property of the spaser equations, which makes the spaser generation always an essentially nonlinear process that involves a noneqilibrium phase transition and a spontaneous symmetry breaking: establishment of an arbitrary but sustained phase of the coherent SP oscillations.

A relevant process is spontaneous emission of SPs by a chromophore into a spasing SP mode. The corresponding rate
${\gamma}_{2}^{\left(p\right)}$ for a chromophore at a point **r*** _{p}* can be found in a standard way using the quantized field (61) as

*γ*of the spasing line as

_{s}*N*≫ 1. However, for

_{p}*N*∼ 1, the predicted width may be too high because the spectral diffusion theory assumes that

_{p}*γ*≲

_{s}*γ*. To take into account this limitation in a simplified way, we will interpolate to find the resulting spectral width Γ

_{n}*of the spasing line as ${\Gamma}_{s}={\left({\gamma}_{n}^{-2}+{\gamma}_{s}^{-2}\right)}^{-1/2}$.*

_{s}We will also examine the spaser as a bistable (logical) amplifier. One of the ways to set the spaser in such a mode is to add a saturable absorber. This is described by the same Eqs. (64)–(67) where the chromophores belonging to the absorber are not pumped by the external source directly, i.e., for them in Eq. (66) one has to set *g* = 0.

Numerical examples are given for a silver nanoshell where the core and the external dielectric have the same permittivity of *ɛ _{d}* = 2; the permittivity of silver is adopted from Ref. [30]. The following realistic parameters of the gain medium are used (unless indicated otherwise):

*d*

_{12}= 1.5×10

^{−17}esu,

*h̄*Γ

_{12}= 10 meV,

*γ*

_{2}= 4×10

^{12}s

^{−1}(this value takes into account the spontaneous decay into SPs), and density of the gain medium chromophores is

*n*= 2.4×10

_{c}^{20}cm

^{−3}, which is realistic for dye molecules but may be somewhat high for semiconductor quantum dots that were proposed as the chromophores [29] and used in experiments [260]. We will assume a dipole SP mode and chromophores situated in the core of the nanoshell as shown in Fig. 25 (d). This configuration are of advantage both functionally (because the region of the high local fields outside the shell is accessible for various applications) and computationally (the uniformity of the modal fields makes the summation of the chromophores trivial, thus greatly facilitating numerical procedures).

### 5.4.2. Equations for CW regime

Physically, the spaser action is a result of spontaneous symmetry breaking when the phase of the coherent SP field is established from the spontaneous noise. Mathematically, the spaser is described by homogeneous differential Eqs. (64)–(67). These equations become homogeneous algebraic equations for the CW case. They always have a trivial, zero solution. However, they may also possess a nontrivial solution describing spasing. An existence condition of such a nontrivial solution is

*p*th chromophore ${n}_{21}^{\left(p\right)}$ is explicitly expressed as

From the imaginary part of Eq. (71) we immediately find the spasing frequency *ω _{s}*,

*ω*

_{21}or the SP frequency

*ω*, but is between them (this is a frequency walk-off phenomenon similar to that of laser physics). Substituting Eq. (72) back into Eqs. (71)–(72), we obtain a system of equations

_{n}*N*.

_{n}Since ${n}_{21}^{\left(p\right)}\le 1$, from Eqs. (73), (74) we immediately obtain a necessary condition of the existence of spasing,

*Q*=

*ω/γ*is the quality factor of SPs,

_{n}*V*is the volume of the spasing SP mode, and

_{n}*N*is the of number of the gain medium chromophores within this volume. Deriving this estimate, we have neglected the detuning, i.e., set

_{c}*ω*

_{21}–

*ω*= 0. We also used the definitions of

_{n}*A*of Eq. (61) and ${\tilde{\Omega}}_{12}^{\left(p\right)}$ given by Eq. (65), and the estimate |∇

_{n}*φ*(

_{n}**r**)| ∼ 1/

*V*following from the normalization of the SP eigenmodes ∫|∇

*φ*(

_{n}**r**)|

^{2}

*d*

^{3}

*r*= 1 of Ref. [76]. The result of Eq. (76) is, indeed, in agreement with Ref. [29] where it was obtained in different notations.

It follows from Eq. (76) that for the existence of spasing it is beneficial to have a high quality factor *Q*, a high density of the chromophores, and a large transition dipole (oscillator strength) of the chromophore transition. The small modal volume *V _{n}* (at a given number of the chromophores

*N*) is beneficial for this spasing condition: physically, it implies strong feedback in the spaser. Note that for the given density of the chromophores

_{c}*n*=

_{c}*N*, this spasing condition does not explicitly depend on the spaser size, which opens up a possibility of spasers of a very small size limited from the bottom by only the nonlocality radius

_{c}/V_{n}*l*∼ 1 nm. Another important property of Eq. (76) is that it implies the quantum-mechanical nature of spasing and spaser amplification: this condition essentially contains the Planck constant

_{nl}*h̄*and, thus, does not have a classical counterpart. Note that in contrast to lasers, the spaser theory and Eqs. (75), (76) in particular do not contain speed of light, i.e., they are quasistatic.

Now we will examine the spasing condition and reduce it to a requirement for the gain medium. First, we substitute all the definitions and assume the perfect resonance between the generating SP mode and the gain medium, i.e., *ω _{n}* =

*ω*

_{21}. As a result, we obtain from Eq. (75),

*V*of the system, and the Θ-function takes into account a simplifying realistic assumption that the gain medium occupies the entire space free from the core’s metal. We also assume that the orientations of the transition dipoles ${\mathbf{d}}_{12}^{\left(p\right)}$ are random and average over them, which results in the factor of 3 in the denominator in Eq. (77).

From Eqs. (27) and (34), it follows that

*Q*≫ 1,

*ω*as

For the sake of comparison, consider a continuous gain medium comprised of the same chromophores as the gain shell of the spaser. Its gain *g* (whose dimensionality is cm^{−1}) is given by a standard expression

*g*has a meaning of the threshold gain needed for spasing. Importantly, this gain depends only on the dielectric properties of the system and spasing frequency but not on the geometry of the system or the distribution of the local fields of the spasing mode (hot spots, etc.) explicitly. However note that the system’s geometry (along with the permittivities) does define the spasing frequencies.

_{th}In Figs. 26 (a) and (b), correspondingly, we illustrate the analytical expression (82) for gold and silver embedded in a dielectric with *ɛ _{d}* = 2 (simulating a light glass) and

*ɛ*= 10 (simulating a semiconductor), correspondingly. These are computed from Eq. (82) assuming that the metal core is embedded into the gain medium with the real part of the dielectric function equal to

_{d}*ɛ*. As we see from Fig. 26, the spasing is possible for silver in the near-ir communication range and the adjacent red portion of the visible spectrum for a gain

_{d}*g*< 3000 cm

^{−1}(regions below the red line in Fig. 26), which is realistically achievable with direct band-gap semiconductors (DBDSs).

#### 5.5. Spaser in CW mode

The “spasing curve” (a counterpart of the light-light curve, or L-L curve, for lasers), which is the dependence of the coherent SP population *N _{n}* on the excitation rate

*g*, obtained by solving Eqs. (73), (74), is shown in Fig. 27 (a) for four types of the silver nanoshells with the frequencies of the spasing dipole modes as indicated, which are in the range from near-ir (

*h̄ω*= 1.2 eV) to mid-visible (

_{s}*h̄ω*= 2.2 eV). In all cases, there is a pronounced threshold of the spasing at an excitation rate

_{s}*g*∼ 10

_{th}^{12}s

^{−1}. Soon after the threshold, the dependence

*N*(

_{n}*g*) becomes linear, which means that every quantum of excitation added to the active medium with a high probability is stimulated to be emitted as a SP, adding to the coherent SP population.

While this is similar to conventional lasers, there is a dramatic difference for the spaser. In lasers, a similar relative rate of the stimulated emission is achieved at a photon population of ∼ 10^{18} – 10^{20}, while in the spaser the SP population is *N _{n}* ≲ 100. This is due to the much stronger feedback in spasers because of the much smaller modal volume

*V*– see discussion of Eq. (76). The shape of the spasing curves of Fig. 27 (a) (the well-pronounced threshold with the linear dependence almost immediately above the threshold) is in a qualitative agreement with the experiment [252].

_{n}The population inversion number *n*_{21} as a function of the excitation rate *g* is displayed in Fig. 27 (b) for the same set of frequencies (and with the same color coding) as in panel (a). Before the spasing threshold, *n*_{21} increases with *g* to become positive with the onset of the population inversion just before the spasing threshold. For higher *g*, after the spasing threshold is exceeded, the inversion *n*_{21} becomes constant (the inversion clamping). The clamped levels of the inversion are very low, *n*_{21} ∼ 0.01, which again is due to the very strong feedback in the spaser.

The spectral width Γ* _{s}* of the spaser generation is due to the phase diffusion of the quantum SP state caused by the noise of the spontaneous emission of the SPs into the spasing mode, as described by Eq. (69). This width is displayed in Fig. 27 (c) as a function of the pumping rate

*g*. At the threshold, Γ

*is that of the SP line*

_{s}*γ*but for stronger pumping, as the SPs accumulate in the spasing mode, it decreases $\propto {N}_{n}^{-1}$, as given by Eq. (69). This decrease of Γ

_{n}*reflects the higher coherence of the spasing state with the increased number of SP quanta and, correspondingly, lower quantum fluctuations. As we have already mentioned, this is similar to the lasers as described by the Schawlow-Townes theory [278].*

_{s}The developed spasing in a dipole SP mode will show itself in the far field as an anomalously narrow and intense radiation line. The shape and intensity of this line in relation to the lines of the spontaneous fluorescence of the isolated gain medium and its SP-enhanced fluorescence line in the spaser is illustrated in Figs. 27 (d)–(f). Note that for the system under consideration, there is a 20 meV red shift of the gain medium fluorescence with respect to the SP line center. It is chosen so to illustrate the spectral walk-off of the spaser line. For one percent in the excitation rate above the threshold of the spasing [panel (d)], a broad spasing line (red color) appears comparable in intensity to the SP-enhanced spontaneous fluorescence line (blue color). The width of this spasing line is approximately the same as of the fluorescence, but its position is shifted appreciably (spectral walk-off) toward the isolated gain medium line (green color). For the pumping twice more intense [panel (e)], the spaser-line radiation dominates, but its width is still close to that of the SP line due to significant quantum fluctuations of the spasing state phase. Only when the pumping rate is an order of magnitude above the threshold, the spaser line strongly narrows [panel (f)], and it also completely dominates the spectrum of the radiation. This is a regime of small quantum fluctuations, which is desired in applications.

These results in the spasing region are different in the most dramatic way from previous phenomenological models, which are based on linear electrodynamics where the gain medium that has negative imaginary part of its permittivity plus lossy metal nanosystem, described purely electrodynamically [258, 265]. For instance, in a “toy model” [265], the width of the resonance line tends to zero at the threshold of spasing and then broadens up again. This distinction of the present theory is due the nature of the spasing as a spontaneous symmetry breaking (nonequilibrium phase transition with a randomly established but sustained phase) leading to the establishment of a coherent SP state. This non-equilibrium phase transition to spasing and the spasing itself are contained in the present theory due to the fact that the fundamental equations of the spasing (64), (66), and (67) are nonlinear, as we have already discussed above in conjunction with these equations – see the text after Eq. (67). The previous publications on gain compensation by loss [258, 265, 267] based on linear electrodynamic equations do not contain spasing. Therefore, they are not applicable in the region of the complete loss compensation and spasing, though their results are presented for that region.

#### 5.6. Spaser as ultrafast quantum nanoamplifier

### 5.6.1. Problem of setting spaser as an amplifier

As we have already mentioned in Sec. 5.1, a fundamental and formidable problem is that, in contrast to the conventional lasers and amplifiers in quantum electronics, the spaser has an inherent feedback that typically cannot be removed. Such a spaser will develop generation and accumulation of the macroscopic number of coherent SPs in the spasing mode. This leads to the the population inversion clamping in the CW regime at a very low level – cf. Fig. 27 (b). This CW regime corresponds to the net amplification equal zero, which means that the gain exactly compensates the loss, which condition is expressed by Eq. (73). This is a consequence of the nonlinear gain saturation. This holds for any stable CW generator (including any spaser or laser) and precludes using them as amplifiers.

There are several ways to set a spaser as a quantum amplifier. One of them is to reduce the feedback, i.e., to allow some or most of the SP energy in the spaser to escape from the active region, so the spaser will not generate in the region of amplification. Such a root has successfully been employed to build a SPP plasmonic amplifier on the long-range plasmon polaritons [277]. A similar root for the SP spasers would be to allow some optical energy to escape either by a near-field coupling or by a radiative coupling to far-field radiation. The near-field coupling approach is promising for building integrated active circuits out of the spasers.

Following Ref. [137], we consider here two distinct approaches for setting the spasers as quantum nanoamplifiers. The first is a transient regime based on the fact that the establishment of the CW regime and the consequent inversion clamping and the total gain vanishing require some time that is determined mainly by the rate of the quantum feedback and depends also on the relaxation rates of the SPs and the gain medium. After the population inversion is created by the onset of pumping and before the spasing spontaneously develops, as we show below in this Section, there is a time interval of approximately 250 fs, during which the spaser provides usable (and as predicted, quite high) amplification – see Sec. 5.6.2 below.

The second approach to set the spaser as a logical quantum nanoamplifier is a bistable regime that is achieved by introducing a saturable absorber into the active region, which prevents the spontaneous spasing. Then injection of a certain above-threshold amount of SP quanta will saturate the absorber and initiate the spasing. Such a bistable quantum amplifier will be considered in Sec. 5.6.3.

The temporal behavior of the spaser has been found by direct numerical solution of Eqs. (64)–(67). This solution is facilitated by the fact that in the model under consideration all the chromophores experience the same local field inside the nanoshell, and there are only two types of such chromophores: belonging to the gain medium and the saturable absorber, if it is present.

### 5.6.2. Monostable spaser as a nanoamplifier in transient regime

Here we consider a monostable spaser in a transient regime. This implies that no saturable absorber is present. We will consider two pumping regimes: stationary and pulse.

Starting with the stationary regime, we assume that the pumping at a rate (per one chromophore) of *g* = 5 × 10^{12} s^{−1} starts at a moment of time *t* = 0 and stays constant after that. Immediately at *t* = 0, a certain number of SPs are injected into the spaser. We are interested in its temporal dynamics from this moment on.

The dynamical behavior of the spaser under this pumping regime is illustrated in Figs. 28 (a), (b). As we see, the spaser, which starts from an arbitrary initial population *N _{n}*, rather rapidly, within a few hundred femtoseconds approaches the same stationary (“logical”) level. At this level, an SP population of

*N*= 67 is established, while the inversion is clamped at a low level of

_{n}*n*

_{21}= 0.02. On the way to this stationary state, the spaser experiences relaxation oscillations in both the SP numbers and inversion, which have a trend to oscillate out of phase [compare panels (a) and (b)]. This temporal dynamics of the spaser is quite complicated and highly nonlinear (unharmonic). It is controlled not by a single relaxation time but by a set of the relaxation rates. Clearly, among these are the energy transfer rate from the gain medium to the SPs and the relaxation rates of the SPs and the chromophores.

In this mode, the main effect of the initial injection of the SPs (described theoretically as different initial values of *N _{n}*) is in the interval of time it is required for the spaser to reach the final (CW) state. For very small

*N*, which in practice can be supplied by the noise of the spontaneous SP emission into the mode, this time is approximately 250 fs (cf.: the corresponding SP relaxation time is less then 50 fs). In contrast, for the initial values of

_{n}*N*= 1 – 5, this time shortens to 150 fs.

_{n}Now consider the second regime: pulse pumping. The gain-medium population of the spaser is inverted at *t* = 0 to saturation with a short (much shorter than 100 fs) pump pulse. Simultaneously, at *t* = 0, some number of plasmons are injected (say, by an external nanoplasmonic circuitry). In response, the spaser should produce an amplified pulse of the SP excitation. Such a function of the spaser is illustrated in Figs. 28 (c) and (d).

As we see from panel (c), independently from the initial number of SPs, the spaser always generates a series of SP pulses, of which only the first pulse is large (at or above the logical level of *N _{n}* ∼ 100). (An exception is a case of little practical importance when the initial

*N*= 120 exceeds this logical level, when two large pulses are produced.) The underlying mechanism of such a response is the rapid depletion of the inversion seen in panel (d), where energy is dissipated in the metal of the spaser. The characteristic duration of the SP pulse ∼ 100 fs is defined by this depletion, controlled by the energy transfer and SP relaxation rates. This time is much shorter than the spontaneous decay time of the gain medium. This acceleration is due to the stimulated emission of the SPs into the spasing mode (which can be called a “stimulated Purcell effect”). There is also a pronounced trend: the lower is initial SP population

_{n}*N*, the later the spaser produces the amplified pulse. In a sense, this spaser functions as a pulse-amplitude to time-delay converter.

_{n}### 5.6.3. Bistable spaser with saturable absorber as an ultrafast nanoamplifier

Now let us consider a bistable spaser as a quantum threshold (or, logical) nanoamplifier. Such a spaser contains a saturable absorber mixed with the gain medium with parameters indicated at the end of Sec. 5.4.1 and the concentration of the saturable absorber *n _{a}* = 0.66

*n*. This case of a bistable spaser amplifier is of a particular interest because in this regime the spaser comes as close as possible in its functioning to the semiconductor-based (mostly, MOSFET-based) digital nanoamplifiers. As in the previous Subsection, we will consider two cases: the stationary and short-pulse pumping.

_{c}We again start with the case of the stationary pumping at a rate of *g* = 5×10^{12} s^{−1}. We show in Figs. 28 (e), (f) the dynamics of such a spaser. For a small initial population *N _{n}* = 5 × 10

^{−3}simulating the spontaneous noise, the spaser is rapidly (faster than in 50 fs) relaxing to the zero population [panel (e)], while its gain-medium population is equally rapidly approaching a high level [panel (f)]

*n*

_{21}= 0.65 that is defined by the competition of the pumping and the enhanced decay into the SP mode (the purple curves). This level is so high because the spasing SP mode population vanishes and the stimulated emission is absent. After reaching this stable state (which one can call, say, “logical zero”), the spaser stays in it indefinitely long despite the continuing pumping.

In contrast, for initial values *N _{n}* of the SP population large enough [for instance, for

*N*= 5, as shown by the blue curves in Figs. 28 (e) and (f)], the spaser tends to the “logical one” state where the stationary SP population reaches the value of

_{n}*N*≈ 60. Due to the relaxation oscillations, it actually exceeds this level within a short time of ≲ 100 fs after the seeding with the initial SPs. As the SP population

_{n}*N*reaches its stationary (CW) level, the gain medium inversion

_{n}*n*

_{21}is clamped down at a low level of a few percent, as typical for the CW regime of the spaser. This “logical one” state salso persists indefinitely, as long as the inversion is supported by the pumping.

There is a critical curve (separatrix) that divide the two stable dynamics types (leading to the logical levels of zero and one). For the present set of parameters this separatrix starts with the initial population of *N _{n}* ≈ 1. For a value of the initial

*N*slightly below 1, the SP population

_{n}*N*experiences a slow (hundreds fs in time) relaxation oscillation but eventually relaxes to zero [Fig. 28 (e), black curve], while the corresponding chromophore population inversion

_{n}*n*

_{21}relaxes to the high value

*n*

_{21}= 0.65 [panel (f), black curve]. In contrast, for a value of

*N*slightly higher than 1 [light blue curves in panels (e) and (f)], the dynamics is initially close to the separaratrix but eventually the initial slow dynamics tends to the high SP population and low chromophore inversion through a series of the relaxation oscillations. The dynamics close to the separatrix is characterized by a wide range of oscillation times due to its highly nonlinear character. The initial dynamics is slowest (the “decision stage” of the bistable spaser that lasts ≳ 1 ps). The “decision time” is diverging infinitesimally close to the separatrix, as is characteristic of any threshold (logical) amplifier.

_{n}The gain (amplification coefficient) of the spaser as a logical amplifier is the ratio of the high CW level to the threshold level of the SP population *N _{n}*. For this specific spaser with the chosen set of parameters, this gain is ≈ 60, which is more than sufficient for the digital information processing. Thus this spaser can make a high-gain, ∼ 10 THz-bandwidth logical amplifier or dynamical memory cell with excellent prospects of applications.

The last but not the least regime to consider is that of the pulse pumping in the bistable spaser. In this case, the population inversion (*n*_{21} = 0.65) is created by a short pulse at *t* = 0 and simultaneously initial SP population *N _{n}* is created. Both are simulated as the initial conditions in Eqs. (64)–(67). The corresponding results are displayed in Figs. 28 (g) and (h).

When the initial SP population exceeds the critical one of *N _{n}* = 1 (the blue, green, and red curves), the spaser responds with generating a short (duration less than 100 fs) pulse of the SP population (and the corresponding local fields) within a time ≲ 100 fs [panel (g)]. Simultaneously, the inversion is rapidly (within ∼ 100 fs) exhausted [panel (h)].

In contrast, when the initial SP population *N _{n}* is less than the critical one (i.e.,

*N*< 1 in this specific case), the spaser rapidly (within a time ≲ 100 fs) relaxes as

_{n}*N*→ 0 through a series of realaxation oscillations – see the black and magenta curves in Fig. 28 (g). The corresponding inversion decays in this case almost exponentially with a characteristic time ∼ 1 ps determined by the enhanced energy transfer to the SP mode in the metal – see the corresponding curves in panel (h). Note that the SP population decays faster when the spaser is above the generation threshold due to the stimulated SP emission leading to the higher local fields and enhanced relaxation.

_{n}#### 5.7. Compensation of loss by gain and spasing

### 5.7.1. Introduction to loss compensation by gain

A problem for many applications of plasmonics and metamaterials is posed by losses inherent in the interaction of light with metals. There are several ways to bypass, mitigate, or overcome the detrimental effects of these losses, which we briefly discuss below.

- The most common approach consists in employing effects where the losses are not fundamentally important such as surface plasmon polariton (SPP) propagation used in sensing [23], ultramicroscopy [16, 19], and solar energy conversion [26]. For realistic losses, there are other effects and applications that are not prohibitively suppressed by the losses and useful, in particular, sensing based on SP resonances and surface enhanced Raman scattering (SERS) [23, 180, 242, 279, 280].
- Yet another proposed direction is using highly doped semiconductors where the Ohmic losses can be significantly lower due to much lower free carrier concentrations [284]. However, a problem with this approach may lie in the fact that the usefulness of plasmonic modes depends not on the loss
*per se*but on the quality factor*Q*, which for doped semiconductors may not be higher than for the plasmonic metals. - One of the alternative approaches to low-loss plasmonic metamaterials is based on our idea of the spaser: it is using a gain to compensate the dielectric (Ohmic) losses [285, 286]. In this case the gain medium is included into the metamaterials. It surrounds the metal plasmonic component in the same manner as in the spasers. The idea is that the gain will provide quantum amplification compensating the loss in the metamaterials quite analogously to the spasers.

We will consider theory of the loss compensation in the plasmonic metamaterials using gain [138, 139]. Below we show that the full compensation or overcompensation of the optical loss in a dense resonant gain metamaterial leads to an instability that is resolved by its spasing (i.e., by becoming a generating spaser). We further show analytically that the conditions of the complete loss compensation by gain and the threshold condition of spasing – see Eqs. (80) and (82) – are identical. Thus the full compensation (overcompensation) of the loss by gain in such a metamaterial will cause spasing. This spasing limits (clamps) the gain – see Sec. 5.5 – and, consequently, inhibits the complete loss compensation (overcompensation) at any frequency.

### 5.7.2. Permittivity of nanoplasmonic metamaterial

We will consider, for certainty, an isotropic and uniform metamaterial that, by definition, in a range of frequencies *ω* can be described by the effective permittivity *ɛ*̄(*ω*) and permeability *μ*̄(*ω*). We will concentrate below on the loss compensation for the optical electric responses; similar consideration with identical conclusions for the optical magnetic responses is straightforward. Our theory is applicable for the true three-dimensional (3d) metamaterials whose size is much greater than the wavelength *λ* (ideally, an infinite metamaterial).

Consider a small piece of such a metamaterial with sizes much greater that the unit cell but much smaller than *λ*. Such a piece is a metamaterial itself. Let us subject this metamaterial to a uniform electric field **E**(*ω*) = −∇*ϕ* (**r**,*ω*) oscillating with frequency *ω*. Note that **E**(*ω*) is the amplitude of the macroscopic electric field inside the metamaterial. We will denote the local field at a point **r** inside this metamaterial as **e**(**r**,*ω*) = −∇*φ*(**r**,*ω*). We assume standard boundary conditions

**r**belonging to the surface

*S*of the volume under consideration.

To present our results in a closed form, we first derive a homogenization formula used in Ref. [287] (see also references cited therein). By definition, the electric displacement in the volume *V* of the metamaterial is given by a formula

*ɛ*(

**r**,

*ω*) is a position-dependent permittivity. This can be identically expressed (by multiplying and dividing by the conjugate of the macroscopic field

*E*

^{*}) and, using the Gauss theorem, transformed to a surface integral as

*ɛ*(

**r**,

*ω*)

**e**(

**r**,

*ω*)] = 0. Now, using the boundary conditions of Eq. (83), we can transform it back to the volume integral as

### 5.7.3. Plasmonic eigenmodes and effective resonant permittivity of metamaterials

This piece of the metamaterial with the total size *R* ≪ *λ* can be treated in the quasistatic approximation. The local field inside the nanostructured volume *V* of the metamaterial is given by the eigenmode expansion [76, 147, 218]

**E**(

*ω*) is the macroscopic field. In the resonance,

*ω*=

*ω*, only one term at the pole of in Eq. (88) dominates, and it becomes

_{n}*E*∼

_{n}*V*

^{−1/2}, and where

*f*is the metal fill factor of the system, and

*Q*is the plasmonic quality factor. Deriving expression (90), we have also taken into account an equality Im

*s*(

*ω*) =

_{n}*s*(1 –

_{n}*s*)

_{n}*/Q*, which is valid in the assumed limit of the high quality factor,

*Q*≫ 1 (see the next paragraph).

For a good plasmonic metal *Q* ≫ 1 – see Fig. 2. For most metal-containing metamaterials, the metal fill factor is not small, typically *f* ≳ 0.5. Thus, keeping Eq. (28) in mind, it is very realistic to assume the following condition

*b*> 0 is a real positive coefficient whose specific value is

_{n}Using Eqs. (94) and (27), (34), it is straightforward to show that the effective permittivity (94) simplifies exactly to

#### 5.8. Conditions of loss compensation by gain and spasing

In the case of the full inversion (maximum gain) and in the exact resonance, the host medium permittivity acquires the imaginary part describing the stimulated emission as given by the standard expression

*ɛ*= Re

_{d}*ɛ*,

_{h}**d**

_{12}is a dipole matrix element of the gain transition in a chromophore center of the gain medium, Γ

_{12}is a spectral width of this transition, and

*n*is the concentration of these centers (these notations are consistent with those used above in Secs. 5.4.1–5.6.3). Note that if the inversion is not maximum, then this and subsequent equations are still applicable if one sets as the chromophore concentration

_{c}*n*the inversion density:

_{c}*n*=

_{c}*n*

_{2}–

*n*

_{1}, where

*n*

_{2}and

*n*

_{1}are the concentrations of the chromophore centers of the gain medium in the upper and lower states of the gain transition, respectively.

The condition for the full electric loss compensation in the metamaterial and amplification (overcompensation) at the resonant frequency *ω* = *ω _{n}* is

*ɛ*(

_{m}*ω*) > 0, we obtain from Eq. (99) the condition of the loss (over)compensation as

*n*should be multiplied by a factor of 3.

_{c}Equation (100) is a fundamental condition, which is precise [assuming that the requirement (92) is satisfied, which is very realistic for metamaterials] and general. Moreover, it is fully analytical and, actually, very simple. Remarkably, it depends only on the material characteristics and does not contain any geometric properties of the metamaterial system or the local fields. (Note that the system’s geometry does affect the eigenmode frequencies and thus enters the problem implicitly.) In particular, the hot spots, which are prominent in the local fields of nanostructures [76, 157], are completely averaged out due to the integrations in Eqs. (87) and (94).

The condition (100) is completely non-relativistic (quasistatic) – it does not contain speed of light *c*, which is characteristic of also of the spaser. It is useful to express this condition also in terms of the total stimulated emission cross section *σ _{e}*(

*ω*) (where

*ω*is the central resonance frequency) of a chromophore of the gain medium as

We see that Eq. (100) *exactly* coincides with a spasing condition expressed by Eq. (80). This brings us to an important conclusion: the full compensation (overcompensation) of the optical losses in a metamaterial [which is resonant and dense enough to satisfy condition (92)] and the spasing occur under precisely the same conditions.

We have considered above in Sec. 5.4.2 the conditions of spasing, which are equivalent to (101). These are given by one of equivalent conditions of Eqs. (80), (82), (100). It is also illustrated in Fig. 26. We stress that exactly the same conditions are for the full loss compensation (overcompensation) of a dense resonant plasmonic metamaterial with gain.

We would like also to point out that the criterion given by the equivalent conditions of Eqs. (80), (82), (100), or (101) is derived for localized SPs, which are describable in the quasistatic approximation, and is not directly applicable to the propagating plasmonic modes (SPPs). However, we expect that very localized SPPs, whose wave vector *k* ≲ *l _{s}*, can be described by these conditions because they are, basically, quasistatic. For instance, the SPPs on a thin metal wire of a radius

*R*≲

*l*are described by a dispersion relation [12]

_{s}*γ*≈ 0.57721 is the Euler constant. This relation is obviously quasistatic because it does not contain speed of light

*c*.

### 5.8.1. Discussion of spasing and loss compensation by gain

This fact of the equivalence of the full loss compensation and spasing is intimately related to the general criteria of the the thermodynamic stability with respect to small fluctuations of electric and magnetic fields – see Chap. IX of Ref. [28],

*strict*inequalities for all frequencies for electromagnetically stable systems. For systems in thermodynamic equilibrium, these conditions are automatically satisfied.

However, for the systems with gain, the conditions (103) can be violated, which means that such systems can be electromagnetically unstable. The first of conditions (103) is opposite to Eqs. (98) and (100). This has a transparent meaning: the electrical instability of the system is resolved by its spasing.

The significance of these stability conditions for gain systems can be elucidated by the following *gedanken* experiment. Take a small isolated piece of such a metamaterial (which is a metamaterial itself). Consider that it is excited at an optical frequency *ω* either by a weak external optical field **E** or acquires such a field due to fluctuations (thermal or quantum). The energy density ℰ of such a system is given by the Brillouin formula [28]

*ɛ*̄| ≫ Im

*ɛ*̄. This condition is realistic for many metamaterials, including all potentially useful ones.

The internal optical energy-density loss per unit time *Q* (i.e., the rate of the heat-density production in the system) is [28]

*dℰ/dt*=

*Q*. Then Eqs. (104) and (105) can be resolved together yielding the energy ℰ and electric field |

**E**| of this system to evolve with time

*t*exponentially as

We are interested in a resonant case when the metamaterial possesses a resonance at some eigenfrequency *ω _{n}* ≈

*ω*. For this to be true, the system’s behavior must be plasmonic, i.e., Re

*ɛ̄*(

*ω*) < 0. Then the dominating contribution to

*ɛ̄*comes from a resonant SP eigenmode

*n*with a frequency

*ω*≈

_{n}*ω*. In such a case, the dielectric function [76]

*ɛ̄*(

*ω*) has a simple pole at

*ω*=

*ω*. As a result,

_{n}*∂*(

*ω*Re

*ɛ̄*)/

*∂ω*≈

*ω∂*Re

*ɛ̄/∂ω*and, consequently, Γ =

*γ*, where

_{n}*γ*is the SP decay rate given by Eqs. (3) or (48), and the metal dielectric function

_{n}*ɛ*is replaced by the effective permittivity

_{m}*ɛ̄*of the metamaterial. Thus, Eq. (106) is fully consistent with the spectral theory of SPs – see Sec. 3.4.

If the losses are not very large so that energy of the system is meaningful, the Kramers-Kronig causality requires [28] that *∂*(*ω*Re*ɛ̄*)*/∂ω* > 0. Thus, Im*ɛ̄* < 0 in Eq. (106) would lead to a negative decrement,

**E**| and ℰ at finite stationary (CW) levels of the spaser generation.

Note that the spasing limits (clamps) the gain and population inversion making *the net gain to be precisely zero* [137] in the stationary (continuous wave or CW) regime see Sec. 5.6 and Fig. 27 (b). Above the threshold of the spasing, the population inversion of the gain medium is clamped at a rather low level *n*_{21} ∼ 1%. The corresponding net amplification in the CW spasing regime is exactly zero, which is a condition for the CW regime. This makes the complete loss compensation and its overcompensation impossible in a dense resonant metamaterial where the feedback is created by the internal inhomogeneities (including its periodic structure) and the facets of the system.

Because the loss (over)compensation condition (100), which is also the spasing condition, is geometry-independent, it is useful to illustrate it for commonly used plasmonic metals, gold and silver whose permittivity we adopt from Ref. [30]. For the gain medium chromophores, we will use a reasonable set of parameters: Γ_{12} = 5 × 10^{13} s^{−1} and *d*_{12} = 4.3 × 10^{−18} esu. The results of computations are shown in Fig. 29. (Note that this figure expresses a condition of spasing equivalent to that of Fig. 26). For silver as a metal and *n _{c}* = 6 × 10

^{18}cm

^{−3}, the corresponding lower (black) curve in panel (a) does not reach the value of 1, implying that no full loss compensation is achieved. In contrast, for a higher but still very realistic concentration of

*n*= 2.9 × 10

_{c}^{19}cm

^{−3}, the upper curve in Fig. 29 (a) does cross the threshold line in the near-infrared region. Above the threshold area, there will be the instability and the onset of the spasing. As Fig. 29 (b) demonstrates, for gold the spasing occurs at higher, but still realistic, chromophore concentrations.

### 5.8.2. Discussion of published research on spasing and loss compensations

Now let us discuss the implications of these results for the research published recently on the gain metamaterials. To carry out a quantitative comparison with Ref. [267], we turn to Fig. 29 (a) where the lower (black) curve corresponds to the nominal value of *n _{c}* = 6 × 10

^{18}cm

^{−3}used in Ref. [267]. There is no full loss compensation and spasing. This is explained by the fact that Ref. [267] uses, as a close inspection shows, the gain dipoles parallel to the field (this is equivalent to increasing

*n*by a factor of 3) and the local field enhancement [this is equivalent to increasing

_{c}*n*by a factor of (

_{c}*ɛ*+ 2)/3. Because the absorption cross section of dyes is measured in the appropriate host media (liquid solvents or polymers), it already includes the Lorentz local-field factor. To compare to the results of Ref. [267], we increase in our formulas the concentration

_{h}*n*of the chromophores by a factor of

_{c}*ɛ*+ 2 to

_{h}*n*= 2.9 × 10

_{c}^{19}cm

^{−3}, which corresponds to the upper curve in Fig. 29 (a). This curve rises above the threshold line exactly in the same (infra)red region as in Ref. [267].

This agreement of the threshold frequencies between our analytical theory and numerical theory [267] is not accidental: inside the region of stability (i.e., in the absence of spasing) both theories should and do give close results, provided that the the gain-medium transition alignment is taken into account, and the local field-factor is incorporated.

However, above the threshold (in the region of the overcompensation), there should be spasing causing the population inversion clamping and zero net gain, and not a loss compensation. To describe this effect, it is necessary to invoke Eq. (67) for coherent SP amplitude, which is absent in Ref. [267]. Also fundamentally important, spasing, just like the conventional lasing, is a highly-nonlinear phenomenon, which is described by nonlinear equations – see the discussion after Eq. (67).

The complete loss compensation is stated in a recent experimental paper [288], where the system is actually a nanofilm rather than a 3d metamaterial, to which our theory would have been applicable. For the Rhodamine 800 dye used with extinction cross section [289] *σ* = 2×10^{−16} cm^{2} at 690 nm in concentration *n _{c}* = 1.2×10

^{19}cm

^{−3}, realistically assuming

*ɛ*= 2.3, for frequency

_{d}*h̄ω*= 1.7 eV, we calculate from Eq. (101) a point shown by the magenta solid circle in Fig. 29 (a), which is significantly above the threshold. Because in such a nanostructure the local fields are very non-uniform and confined near the metal similar to the spaser, they likewise cause a feedback. The condition of Eq. (92) is likely to be well-satisfied for Ref. [288]. Thus, the system may spase, which would cause the the clamping of inversion and loss of gain.

In contrast to these theoretical arguments, there is no evidence of spasing indicated in the experiment – see Ref. [288], which can be explained by various factors. Among them, the system of Ref. [288] is a gain-plasmonic nanofilm and not a true 3d material. This system is not isotropic. Also, the size of the unit cell *a* ≈ 280 nm is significantly greater than the reduced wavelength *λ*, which violates the quasistatic conditions and makes the possibility of homogenization and considering this system as an optical metamaterial problematic. This circumstance may lead to an appreciable spatial dispersion. It may also cause a significant radiative loss and prevent spasing for some modes.

We would also like to point out that the fact that the unit cell of the negative-refracting (or, double-negative) metamaterial of Ref. [288] is relatively large, *a* ≈ 280 nm, is not accidental. As follows from theoretical consideration of Ref. [297], optical magnetism and, consequently, negative refraction for metals is only possible if the minimum scale of the conductor feature (the diameter *d* of the nanowire) is greater then the skin depth, *d* ≳ *l _{s}* ≈ 25 nm, which allows one to circumvent Landau-Lifshitz’s limitation on the existence of optical magnetism [28,297]. Thus, a ring-type resonator structure would have a size ≳ 2

*l*(two wires forming a loop) and still the same diameter for the hole in the center, which comes to the total of ≳ 4

_{s}*l*≈ 100 nm. Leaving the same distance between the neighboring resonator wires, we arrive at an estimate of the size of the unit cell

_{s}*a*≳ 8

*l*= 200 nm, which is, indeed, the case for Ref. [288] and other negative-refraction “metamaterials” in the optical region. This makes our theory not directly applicable to them. Nevertheless, if the spasing condition (80) [or (82), or (101)] is satisfied, the system still may spase on the hot-spot defect modes.

_{s}In an experimental study of the lasing spaser [260], a nanofilm of PbS quantum dots (QDs) was positioned over a two-dimensional metamaterial consisting of an array of negative split ring resonators. When the QDs were optically pumped, the system exhibited an increase of the transmitted light intensity on the background of a strong luminescence of the QDs but apparently did not reach the lasing threshold. The polarization-dependent loss compensation was only ∼ 1 %. Similarly, for an array of split ring resonators over a resonant quantum well, where the inverted electron-hole population was excited optically [290], the loss compensation did not exceed ∼ 8 %. The relatively low loss compensation in these papers may be due either to random spasing and/or spontaneous or amplified spontaneous emission enhanced by this plasmonic array, which reduces the population inversion.

A dramatic example of possible random spasing is presented in Ref. [262]. The system studied was a Kretschmann-geometry SPP setup [291] with an added ∼ 1*μ*m polymer film containing Rodamine 6G dye in the *n _{c}* = 1.2 × 10

^{19}cm

^{−3}concentration. When the dye was pumped, there was outcoupling of radiation in a range of angles. This was a threshold phenomenon with the threshold increasing with the Kretschmann angle. At the maximum of the pumping intensity, the widest range of the outcoupling angles was observed, and the frequency spectrum at every angle narrowed to a peak near a single frequency

*h̄ω*≈ 2.1 eV.

These observations of Ref. [262] can be explained by the spasing where the feedback is provided by roughness of the metal. At the high pumping, the localized SPs (hots spots), which possess the highest threshold, start to spase in a narrow frequency range around the maximum of the spasing criterion – the left-hand side of Eq. (100). Because of the sub-wavelength size of these hot spots, the Kretschmann phase-matching condition is relaxed, and the radiation is outcoupled into a wide range of angles.

The SPPs of Ref. [262] excited by the Kretschmann coupling are short-range SPPs, very close to the antisymmetric SPPs. They are localized at subwavelength distances from the surface, and their wave length in the plane is much shorter the *ω*/*c*. Thus they can be well described by the quasistatic approximation and the present theory is applicable to them. Substituting the above-given parameters of the dye and the extinction cross section *σ _{e}* = 4 × 10

^{−16}cm

^{2}into Eq. (101), we obtain a point shown by the black diamond in Fig. 29, which is clearly above the threshold, supporting our assertion of the spasing. Likewise, the amplified spontaneous emission and, possibly spasing, appear to have prevented the full loss compensation in a SPP system of Ref. [274].

Note that the long-range SPPs of Ref. [277] are localized significantly weaker (at distances ∼ *λ*) than those excited in Kretschmann geometry. Thus the long-range SPPs experience a much weaker feedback, and the amplification instead of the spasing can be achieved. Generally, the long-range SPPs are fully electromagnetic (non-quasistatic) and are not describable in the present theory.

As we have already discussed in conjunction with Fig. 26, the spasing is readily achievable with the gain medium containing common DBGSs or dyes. There have been numerous experimental observations of the spaser. Among them is a report of a SP spaser with a 7-nm gold nanosphere as its core and a laser dye in the gain medium [252], observations of the SPP spasers (also known as nanolasers) with silver as a plasmonic-core metal and DBGS as the gain medium with a 1d confinement [253,256], a tight 2d confinement [254], and a 3d confinement [255]. There also has been a report on observation of a SPP microcylinder spaser [292]. A high efficiency room-temperature semiconductor spaser with a DBGS InGaAS gain medium operating near 1.5 *μ*m (i.e., in the communication near-ir range) has been reported [256].

The research and development in the area of spasers as quantum nano-generators is very active and will undoubtedly lead to further rapid advances. The next in line is the spaser as an ultrafast nanoamplifier, which is one of the most important tasks in nanotechnology.

In contrast to this success and rapid development in the field of spasing and spasers, there has so far been a comparatively limited progress in the field of loss compensation by gain in meta-materials, which is based on the same principles of quantum amplification as the spaser. This status exists despite a significant effort in this direction and numerous theoretical publications, e.g., [267,293]. There has been so far a single, not yet confirmed independently, observation of the full loss compensation in a plasmonic metamaterial with gain [288].

In large periodic metamaterials, plasmonic modes generally are propagating waves (SPPs) that satisfy Bloch theorem [294] and are characterized by quasi-wavevector **k**. These are propagating waves except for the band edges where **ka** = ±*π*, where **a** is the lattice vector. At the band edges, the group velocity *v _{g}* of these modes is zero, and these modes are localized, i.e., they are SPs. Their wave function is periodic with period 2

*a*, which may be understood as a result of the Bragg reflection from the crystallographic planes. Within this 2

*a*period, these band-edge modes can, indeed, be treated quasistatically because 2

*a*≪

*l*. If any of the band-edge frequencies is within the range of compensation [where the condition (80) [or, (82)] is satisfied], the system will spase. In fact, at the band edge, this metamaterial with gain is similar to a distributed feedback (DFB) laser [295]. It actually is a DFB spaser, which, as all the DFB lasers, generates in a band-edge mode.

_{s},λ̵Moreover, not only the SPPs, which are exactly at the band edge, will be localized. Due to unavoidable disorder caused by fabrication defects in metamaterials, there will be scattering of the SPPs from these defects. Close to the band edge, the group velocity becomes small, *v _{g}* → 0. Because the scattering cross section of any wave is
$\propto {v}_{g}^{-2}$, the corresponding SPPs experience Anderson localization [296]. Also, there always will be SPs nanolocalized at the defects of the metamaterial, whose local fields are hot spots – see Fig. 10 and, generally, Sec. 3.5 and the publications referenced therein. Each of such hot spots within the bandwidth of conditions (80) or (82) will be a generating spaser, which clamps the inversion and precludes the full loss compensation.

Note that for a 2d metamaterial (metasurface), the amplification of the spontaneous emission and spasing may occur in SPP modes propagating *in plane* of the structure, unlike the signal that propagates normally to it as in Ref. [288].

## Acknowledgments

This work was supported by Grant No. DEFG02-01ER15213 from the Chemical Sciences, Biosciences and Geosciences Division and by Grant No. DE-FG02-11ER46789 from the Materials Sciences and Engineering Division of the Office of the Basic Energy Sciences, Office of Science, U.S. Department of Energy, and by a grant from the U.S.Israel Binational Science Foundation.

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