## Nanoplasmonics: past, present, and glimpse into future |

Optics Express, Vol. 19, Issue 22, pp. 22029-22106 (2011)

http://dx.doi.org/10.1364/OE.19.022029

Acrobat PDF (11741 KB)

### 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 OSA

## 1. Introduction

### 1.1. Preamble

1. M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. **57**, 783–826 (1985). [CrossRef]

5. H. Xu, J. Aizpurua, M. Kall, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E **62**, 4318–4324 (2000). [CrossRef]

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11. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. **82**, 729–787 (2010). [CrossRef]

12. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404–1–4 (2004). [PubMed]

12. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404–1–4 (2004). [PubMed]

22. D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. **11**, 1609–1613 (2011). [PubMed]

23. M. I. Stockman, “Nanoplasmonics: The physics behind the applications,” Phys. Today **64**, 39–44 (2011). [CrossRef]

24. S. Lal, S. E. Clare, and N. J. Halas, “Nanoshell-enabled photothermal cancer therapy: Impending clinical impact,” Accounts Chem. Res. **41**, 1842–1851 (2008). [CrossRef]

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26. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. **9**, 205–213 (2010). [PubMed]

### 1.2. Composition of the article

23. M. I. Stockman, “Nanoplasmonics: The physics behind the applications,” Phys. Today **64**, 39–44 (2011). [CrossRef]

*l*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.

_{s}## 2. Basics of nanoplasmonics

### 2.1. Fundamentals

*ɛ*has a negative real part, Re

_{m}*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 are satisfied simultaneously.

_{m}*electromechanical*energy at optical frequencies (in contrast to electromagnetic energy) on the nanoscale.

**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.

*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.

*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}*R*≲

*l*, 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

_{s}*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.

*R < l*, e.g.,

_{s}*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}*ɛ*[28]. This leads to a finite lifetime of SPs. The decay rate of the plasmonic field

_{m}*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

29. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. **90**, 027402–1–4 (2003). [PubMed]

*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 The quality factor determines how many optical periods free SP oscillations occur before field decays. It also shows how many times the local optical field at the surface of a plasmonic nanoparticle exceeds the external field.

*Q*calculated according to Eqs. (5) and (6) for gold and silver using the permittivity data of Ref. [30

30. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*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

*ω*is the bulk plasma frequency;

_{p}*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.

*τ*on frequency

*ω*calculated for gold and silver using permittivity [30

30. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*τ*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, where Δ

_{c}*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}*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.

### 2.2. Nanoantennas

**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).

*Q*

^{2}. This effect is often referred to as nanoantenna effect [35

35. A. Kramer, F. Keilmann, B. Knoll, and R. Guckenberger, “The coaxial tip as a nano-antenna for scanning near-field microwave transmission microscopy,” Micron **27**, 413–417 (1996). [CrossRef]

62. I. S. Maksymov and A. E. Miroshnichenko, “Active control over nanofocusing with nanorod plasmonic antennas,” Opt. Express **19**, 5888–5894 (2011). [PubMed]

63. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics **5**, 83–90 (2011). [CrossRef]

64. M. Schnell, P. A. Gonzalez, L. Arzubiaga, F. Casanova, L. E. Hueso, A. Chuvilin, and R. Hillenbrand, “Nanofocusing of mid-infrared energy with tapered transmission lines,” Nat. Photonics **5**, 283–287 (2011). [CrossRef]

*R*≲

*l*) plasmonic nanoparticle is ∼

_{s}*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.

### 2.3. Radiative loss

**d**

_{0}

*between the ground state |0〉 and a single-plasmon excited state |*

_{p}*p*〉. To carry out such a computation consistently, one needs to quantize the SPs, which we have originally done in Ref. [29

29. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. **90**, 027402–1–4 (2003). [PubMed]

*α*of a nanosystem obtained using quantum mechanics – see. e.g., Ref. [65], which near the plasmon frequency has a singular form, where

*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}*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}*ɛ*(

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

*γ*

^{(}

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

*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)], Therefore,

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

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

*R*<

*l*≈ 25 nm but is a dominant mechanism of loss for

_{s}*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}67. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. **99**, 147401–1–4 (2007). [PubMed]

75. M. I. Stockman, “Dark-hot resonances,” Nature **467**, 541–542 (2010). [PubMed]

75. M. I. Stockman, “Dark-hot resonances,” Nature **467**, 541–542 (2010). [PubMed]

76. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: Can one state have both characteristics?” Phys. Rev. Lett. **87**, 167401–1–4 (2001). [PubMed]

*ɛ*. 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.

_{m}### 2.4. Other important issues of plasmonics in brief

#### 2.4.1. Enhanced mechanical forces in nanoplasmonic systems

77. L. Novotny, “Forces in optical near-fields,” in *Near-Field Optics and Surface Plasmon Polaritons*, vol. 81 (Springer-Verlag, 2001), pp. 123–141. [CrossRef]

84. R. Quidant, S. Zelenina, and M. Nieto-Vesperinas, “Optical manipulation of plasmonic nanoparticles,” Appl. Phys. A **89**, 233–239 (2007). [CrossRef]

90. L. M. Tong, M. Righini, M. U. Gonzalez, R. Quidant, and M. Kall, “Optical aggregation of metal nanoparticles in a microfluidic channel for surface-enhanced Raman scattering analysis,” Lab Chip **9**, 193–195 (2009). [CrossRef]

91. M. Durach, A. Rusina, and M. I. Stockman, “Giant surface-plasmon-induced drag effect in metal nanowires,” Phys. Rev. Lett. **103**, 186801–1–4 (2009). [PubMed]

#### 2.4.2. Interaction between electrons and surface plasmons

96. A. Reyes-Coronado, R. G. Barrera, P. E. Batson, P. M. Echenique, A. Rivacoba, and J. Aizpurua, “Electromagnetic forces on plasmonic nanoparticles induced by fast electron beams,” Phys. Rev. B **82**, 235429–1–19 (2010). [CrossRef]

32. I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, “Dipolar emitters at nanoscale proximity of metal surfaces: Giant enhancement of relaxation in microscopic theory,” Phys. Rev. B **69**, 121403(R)–1–4 (2004). [CrossRef]

33. I. A. Larkin and M. I. Stockman, “Imperfect perfect lens,” Nano Lett. **5**, 339–343 (2005). [PubMed]

97. B. B. Dasgupta and R. Fuchs, “Polarizability of a small sphere including nonlocal effects,” Phys. Rev. B **24**, 554 –561 (1981). [CrossRef]

99. F. J. G. de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C **112**, 17983–17987 (2008). [CrossRef]

*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

101. V. N. Pustovit and T. V. Shahbazyan, “Quantum-size effects in SERS from noble-metal nanoparticles,” Microelectronics J. **36**, 559–563 (2005). [CrossRef]

107. P. Nordlander, J. Zuloaga, and E. Prodan, “Quantum plasmonics: Optical properties and tunability of metallic nanorods,” *ACS Nano*4, 5269–5276 (2010). [PubMed]

*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.

106. J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett. **9**, 887–891 (2009). [PubMed]

107. P. Nordlander, J. Zuloaga, and E. Prodan, “Quantum plasmonics: Optical properties and tunability of metallic nanorods,” *ACS Nano*4, 5269–5276 (2010). [PubMed]

30. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

91. M. Durach, A. Rusina, and M. I. Stockman, “Giant surface-plasmon-induced drag effect in metal nanowires,” Phys. Rev. Lett. **103**, 186801–1–4 (2009). [PubMed]

59. M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science **332**, 702–704 (2011). [PubMed]

111. P. Berini, A. Akbari, and R. N. Tait, “Surface plasmon waveguide Schottky detector,” Opt. Express **18**, 8505–8514 (2010). [PubMed]

113. U. Levy, I. Goykhman, B. Desiatov, J. Khurgin, and J. Shappir, “Locally oxidized silicon surface-plasmon Schottky detector for telecom regime,” Nano Lett. **11**, 2219–2224 (2011). [PubMed]

*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

118. V. M. Shalaev, M. I. Stockman, and R. Botet, “Resonant excitations and nonlinear optics of fractals,” Physica A **185**, 181–186 (1992). [CrossRef]

120. J. Kneipp, H. Kneipp, and K. Kneipp, “Two-photon vibrational spectroscopy for biosciences based on surface-enhanced hyper-Raman scattering,” Proc. Natl. Acad. Sci. USA **103**, 17149–17153 (2006). [PubMed]

121. A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, “Femtosecond imaging of surface plasmon dynamics in a nanostructured silver film,” Nano Lett. **5**, 1123–1127 (2005). [PubMed]

- Second-harmonic generation from nanostructured metal surfaces and metal nanoparticles [55, 122
55. S. Palomba, M. Danckwerts, and L. Novotny, “Nonlinear plasmonics with gold nanoparticle antennas,” J. Opt. A

**11**, 114030 (2009). [CrossRef]–130122. A. V. Zayats, I. I. Smolyaninov, and C. C. Davis, “Observation of localized plasmonic excitations in thin metal films with near-field second-harmonic microscopy,” Opt. Commun.

**169**, 93–96 (1999). [CrossRef]] is a coherent, perturbative (second-order or three-wave mixing), intrinsic nonlinearity.130. M. Zdanowicz, S. Kujala, H. Husu, and M. Kauranen, “Effective medium multipolar tensor analysis of second-harmonic generation from metal nanoparticles,” N. J. Phys.

**13**, 023025–1–12 (2011). [CrossRef] - Enhanced four wave mixing (sum- or difference frequency generation) at metal surfaces [131] is a coherent, perturbative (third-order or four-wave), intrinsic nonlinearity.
131. J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett.

**104**, 046803–1–4 (2010). [PubMed] - 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.
132. T. Utikal, M. I. Stockman, A. P. Heberle, M. Lippitz, and H. Giessen, “All-optical control of the ultrafast dynamics of a hybrid plasmonic system,” Phys. Rev. Lett.

**104**, 113903–1–4 (2010). [PubMed] - An all-optical modulator consisting of a plasmonic waveguide covered with CdSe quantum dots [133] is based on a perturbative third-order, combined nonlinearity. To the same class belongs a nanoscale-thickness metamaterial modulator [134
133. D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics

**1**, 402–406 (2007). [CrossRef]].134. Z. L. Samson, K. F. MacDonald, F. De Angelis, B. Gholipour, K. Knight, C. C. Huang, E. Di Fabrizio, D. W. Hewak, and N. I. Zheludev, “Metamaterial electro-optic switch of nanoscale thickness,” Appl. Phys. Lett.

**96**, 143105–1–3 (2010). [CrossRef] - An ultrafast all-optical modulator using polaritons in an aluminum plasmonic waveguide is based on perturbative third-order, intrinsic nonlinearity [135]. There are arguments that this nonlinearity is incoherent, based on interband population transfer of carriers [135
135. K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics

**3**, 55–58 (2009). [CrossRef]].**3**, 55–58 (2009). [CrossRef] - 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
136. S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature

**453**, 757–760 (2008). [PubMed]] belongs to the same class where the nonlinearity is the saturation of the gain medium by the coherent plasmonic field [137**90**, 027402–1–4 (2003). [PubMed]]. The same is true for the loss compensation by gain [138137. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt.

**12**, 024004–1–13 (2010). [CrossRef], 139138. M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett.

**106**, 156802–1–4 (2011). [PubMed]].139. M. I. Stockman, “Loss compensation by gain and spasing,” Phil. Trans. R. Soc. A

**369**, 3510–3524 (2011). [PubMed] - 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
**103**, 186801–1–4 (2009). [PubMed]]. An intrinsic nonperturbative nonlinear process is the predicted plasmon soliton [141140. P. Ginzburg, A. Hayat, N. Berkovitch, and M. Orenstein, “Nonlocal ponderomotive nonlinearity in plasmonics,” Opt. Lett.

**35**, 1551–1553 (2010). [PubMed]] where strong local optical fields in a plasmonic waveguide cause a significant redistribution of the conduction-electron density.141. E. Feigenbaum and M. Orenstein, “Plasmon-soliton,” Opt. Lett.

**32**, 674–676 (2007). [PubMed] - 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
142. S. Zherebtsov, T. Fennel, J. Plenge, E. Antonsson, I. Znakovskaya, A. Wirth, O. Herrwerth, F. Suessmann, C. Peltz, I. Ahmad, S. A. Trushin, V. Pervak, S. Karsch, M. J. J. Vrakking, B. Langer, C. Graf, M. I. Stockman, F. Krausz, E. Ruehl, and M. F. Kling, “Controlled near-field enhanced electron acceleration from dielectric nanospheres with intense few-cycle laser fields,” Nat. Phys.

**7**, 656–662 (2011). [CrossRef]].143. M. Kruger, M. Schenk, and P. Hommelhoff, “Attosecond control of electrons emitted from a nanoscale metal tip,” Nature

**475**, 78–81 (2011). [PubMed] - Finally, a recently predicted phenomenon of metallization of dielectrics by strong optical fields [144, 145
144. M. Durach, A. Rusina, M. F. Kling, and M. I. Stockman, “Metallization of nanofilms in strong adiabatic electric fields,” Phys. Rev. Lett.

**105**, 086803–1–4 (2010). [PubMed]] belongs to a new class of highly-nonlinear phenomena where strong optical fields bring a dielectric nanofilm into a plasmonic metal-like state.145. M. Durach, A. Rusina, M. F. Kling, and M. I. Stockman, “Predicted ultrafast dynamic metallization of dielectric nanofilms by strong single-cycle optical fields,” Phys. Rev. Lett.

**107**, 086602–1–5 (2011). [PubMed]

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

### 3.1. SPs as eigenmodes

*R*≪

*λ̵*,

*R*≲

*l*, we employ the so-called quasistatic approximation where the Maxwell equations reduce to the continuity equation for the electrostatic potential

_{s}*φ*(

**r**), The systems permittivity (dielectric function) varying in space is expressed as Here Θ(

**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

76. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: Can one state have both characteristics?” Phys. Rev. Lett. **87**, 167401–1–4 (2001). [PubMed]

146. D. J. Bergman and D. Stroud, “Properties of macroscopically inhomogeneous media,” in *Solid State Physics*, vol. 46, H. Ehrenreich and D. Turnbull, eds. (Academic Press, 1992), pp. 148–270. [CrossRef]

147. M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B **69**, 054202–1–10 (2004). [CrossRef]

*φ*

_{0}(

**r**) at an optical frequency

*ω*. This potential is created by external charges and, therefore, satisfies the Laplace equation within the system, We present the field potential as where

*φ*

_{1}(

**r**) is the local field.

*s*(

*ω*) is Bergman’s spectral parameter [146

146. D. J. Bergman and D. Stroud, “Properties of macroscopically inhomogeneous media,” in *Solid State Physics*, vol. 46, H. Ehrenreich and D. Turnbull, eds. (Academic Press, 1992), pp. 148–270. [CrossRef]

*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, where the eigenfunctions

*φ*(

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

*S*surrounding the system. These we set as with

**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.

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

_{n}*∂φ*(

_{n}**r**)/

*∂*

**r**|

^{2}that is localized within the metal.

*s*are all real, the eigenfunctions

_{n}*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}*ψ*

_{1}and

*ψ*

_{2}as This construct possesses all the necessary and sufficient properties of a scalar product: it is a binary, Hermitian self-adjoined, and positive-defined operation. It is easy to show that the eigenfunctions of Eqs. (25)–(26) are orthogonal. They can be normalized as

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

118. V. M. Shalaev, M. I. Stockman, and R. Botet, “Resonant excitations and nonlinear optics of fractals,” Physica A **185**, 181–186 (1992). [CrossRef]

149. V. M. Shalaev, R. Botet, and A. V. Butenko, “Localization of collective dipole excitations on fractals,” Phys. Rev. B **48**, 6662–6664 (1993). [CrossRef]

76. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: Can one state have both characteristics?” Phys. Rev. Lett. **87**, 167401–1–4 (2001). [PubMed]

156. M. I. Stockman, “Inhomogeneous eigenmode localization, chaos, and correlations in large disordered clusters,” Phys. Rev. E **56**, 6494–6507 (1997). [CrossRef]

159. M. I. Stockman, “Giant attosecond fluctuations of local optical fields in disordered nanostructured media,” Phys. Rev. B **62**, 10494–10497 (2000). [CrossRef]

160. V. Krachmalnicoff, E. Castanie, Y. D. Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett. **105**, 183901–1–4 (2010). [CrossRef]

**87**, 167401–1–4 (2001). [PubMed]

156. M. I. Stockman, “Inhomogeneous eigenmode localization, chaos, and correlations in large disordered clusters,” Phys. Rev. E **56**, 6494–6507 (1997). [CrossRef]

157. M. I. Stockman, L. N. Pandey, and T. F. George, “Inhomogeneous localization of polar eigenmodes in fractals,” Phys. Rev. B **53**, 2183–2186 (1996). [CrossRef]

164. M. Kolb, R. Botet, and J. Julienne, “Scaling of kinetically growing clusters,” Phys. Rev. Lett. **51**, 1123–1126 (1983). [CrossRef]

165. D. A. Weitz and M. Oliveria, “Fractal structures formed by kinetic aggregation of aqueous gold colloids,” Phys. Rev. Lett. **52**, 1433–1436 (1984). [CrossRef]

*s*= 0.3202 and

_{n}*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

156. M. I. Stockman, “Inhomogeneous eigenmode localization, chaos, and correlations in large disordered clusters,” Phys. Rev. E **56**, 6494–6507 (1997). [CrossRef]

157. M. I. Stockman, L. N. Pandey, and T. F. George, “Inhomogeneous localization of polar eigenmodes in fractals,” Phys. Rev. B **53**, 2183–2186 (1996). [CrossRef]

*L*between them that is determined by diffraction: where

_{s}*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.

**87**, 167401–1–4 (2001). [PubMed]

126. M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, “Enhanced second-harmonic generation by metal surfaces with nanoscale roughness: Nanoscale dephasing, depolarization, and correlations,” Phys. Rev. Lett. **92**, 057402–1–4 (2004). [PubMed]

147. M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B **69**, 054202–1–10 (2004). [CrossRef]

154. S. Gresillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. **82**, 4520–4523 (1999). [CrossRef]

160. V. Krachmalnicoff, E. Castanie, Y. D. Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett. **105**, 183901–1–4 (2010). [CrossRef]

166. S. L. Westcott and N. J. Halas, “Electron relaxation dynamics in semicontinuous metal films on nanoparticle surfaces,” Chem. Phys. Lett. **356**, 207–213 (2002). [CrossRef]

169. E. Fort and S. Gresillon, “Surface enhanced fluorescence,” J. Phys. D **41**, 013001–1–31 (2008). [CrossRef]

*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.

*L*of an eigenmode, which is is defined as the gyration radius of its electric field intensity |

_{n}**E**

*(*

_{n}**r**)|

^{2}, where is the eigenmode electric field, as We remind that due to Eq. (30), the eigenmode fields are normalized so Eq. (33) is a standard definition of the gyration radius.

*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.

*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.

*F*≳ 10

_{n}^{−4}, which we call

*luminous*or

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

*L*with all possible values, from zero to one half of the diagonal system size, and with very little correlation between

_{n}*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

171. L. S. Levitov, “Delocalization of vibrational modes caused by electric dipole interaction,” Phys. Rev. Lett. **64**, 547–550 (1990). [PubMed]

172. D. A. Parshin and H. R. Schober, “Multifractal structure of eigenstates in the Anderson model with long-range off-diagonal disorder,” Phys. Rev. B **57**, 10232–10235 (1998). [CrossRef]

173. A. L. Burin, Y. Kagan, L. A. Maksimov, and I. Y. Polischuk, “Dephasing rate in dielectric glasses at ultralow temperatures,” Phys. Rev. Lett. **80**, 2945–2948 (1998). [CrossRef]

160. V. Krachmalnicoff, E. Castanie, Y. D. Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett. **105**, 183901–1–4 (2010). [CrossRef]

**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

_{n}174. M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Giant fluctuations of local optical fields in fractal clusters,” Phys. Rev. Lett. **72**, 2486–2489 (1994). [PubMed]

175. M. I. Stockman, L. N. Pandey, and T. F. George, “Inhomogeneous localization of polar eigenmodes in fractals,” Phys. Rev. B **53**, 2183–2186 (1996). [CrossRef]

177. M. I. Stockman, “Inhomogeneous eigenmode localization, chaos, and correlations in large disordered clusters,” Phys. Rev. E **56**, 6494–6507 (1997). [CrossRef]

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

*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,

*φ*using the orthonormality Eq. (30), obtaining

_{n}*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).

*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.

*φ*

_{0}(

**r**) as From Eqs. (23) and (37) using the Gauss theorem, we obtain an expression for the field potential

*φ*(

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

*φ*

_{0}(

**r**), Finally, differentiating this, we obtain a closed expression for the optical electric field

**E**(

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

**E**

^{(0)}(

**r**) as where

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

*α,β*,··· =

*x,y,z*) with summation over repeated indices implied; the fields are and the tensor (dyadic) retarded Green’s function is defined as

**E**

^{(0)}= const. Then from Eq. (39) we get where the local field enhancement (tensorial) factor is a contraction of the retarded dyadic Green’s function,

### 3.4. SP modes as resonances

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

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

_{n}*γ*> 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, which decays exponentially for

_{n}*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.

*γ*≪

_{n}*ω*, one finds that this real surface plasmon frequency satisfies an equation and that the surface plasmon spectral width is expressed as In terms of the dielectric permittivity as functions of frequency This expression has been given in Sec. 2.1 as Eq. (3). Importantly, the spectral width

_{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

_{n}178. P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nano-particles,” Nano Lett. **11**, 2329–2333 (2011). [PubMed]

**90**, 027402–1–4 (2003). [PubMed]

*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 are equivalent to

*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).

*α*of a nanoplasmonic system as a tensor, where the indexes

*α,β*denote Cartesian components, and

**M**

*is a coupling vector defined as*

_{n}### 3.5. Examples of local fields and their hot spots

**56**, 6494–6507 (1997). [CrossRef]

157. M. I. Stockman, L. N. Pandey, and T. F. George, “Inhomogeneous localization of polar eigenmodes in fractals,” Phys. Rev. B **53**, 2183–2186 (1996). [CrossRef]

179. V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, “Theory and numerical simulation of optical properties of fractal clusters,” Phys. Rev. B **43**, 8183 (1991). [CrossRef]

**56**, 6494–6507 (1997). [CrossRef]

*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.

*L*of the speckles, which is also a characteristic separation between them – see Eq. (31).

_{s}*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.

**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).

**E**(

**r**)|

^{2}in the plane of the nanostructure at the surface of the metal. These computations are described in Ref. [180

180. M. I. Stockman, “Electromagnetic theory of SERS,” in *Surface Enhanced Raman Scattering*, vol. 103, M. M. K. Kneipp and H. Kneipp, eds. (Springer, 2006), pp. 47 – 66. [CrossRef]

**6**, 4370–4379 (1972). [CrossRef]

*ɛ*= 2.0. This intensity is plotted relative to the excitation field intensity |

_{d}**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.

*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.

**53**, 2183–2186 (1996). [CrossRef]

162. M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Photon scanning-tunneling-microscopy images of optical-excitations of fractal metal colloid clusters - Comment,” Phys. Rev. Lett. **75**, 2450–2450 (1995). [PubMed]

### 3.6. Experimental examples of nanoplasmonic hot spots

**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.

*W*(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

_{f}*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}.

121. A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, “Femtosecond imaging of surface plasmon dynamics in a nanostructured silver film,” Nano Lett. **5**, 1123–1127 (2005). [PubMed]

*h̄ω*= 4.89 eV exceeds

*W*= 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).

_{f}*λ*= 400 nm vacuum wavelength. The corresponding energy of the quantum is below the work function,

*h̄ω*= 3 eV <

*W*= 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

_{f}*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.

154. S. Gresillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. **82**, 4520–4523 (1999). [CrossRef]

161. D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, “Photon scanning tunneling microscopy images of optical excitations of fractal metal colloid clusters,” Phys. Rev. Lett. **72**, 4149–4152 (1994). [PubMed]

167. K. Seal, A. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,” Phys. Rev. Lett. **94**, 226101–1–4 (2005). [PubMed]

167. K. Seal, A. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,” Phys. Rev. Lett. **94**, 226101–1–4 (2005). [PubMed]

*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.

181. H. Cang, A. Labno, C. G. Lu, X. B. Yin, M. Liu, C. Gladden, Y. M. Liu, and X. Zhang, “Probing the electromagnetic field of a 15-nanometre hotspot by single molecule imaging,” Nature **469**, 385–388 (2011). [PubMed]

182. A. McLeod, A. Weber-Bargioni, Z. Zhang, S. Dhuey, B. Harteneck, J. B. Neaton, S. Cabrini, and P. J. Schuck, “Nonperturbative visualization of nanoscale plasmonic field distributions via photon localization microscopy,” Phys. Rev. Lett. **106**, 037402 (2011). [PubMed]

183. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science **300**, 2061–2065 (2003). [PubMed]

*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.

181. H. Cang, A. Labno, C. G. Lu, X. B. Yin, M. Liu, C. Gladden, Y. M. Liu, and X. Zhang, “Probing the electromagnetic field of a 15-nanometre hotspot by single molecule imaging,” Nature **469**, 385–388 (2011). [PubMed]

## 4. Ultrafast plasmonics and coherent control on nanoscale

### 4.1. Introduction

184. M. I. Stockman, M. F. Kling, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic field microscope,” Nat. Photonics **1**, 539–544 (2007). [CrossRef]

185. T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldman, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. **80**, 4249–4252 (1998). [CrossRef]

189. T. Zentgraf, A. Christ, J. Kuhl, and H. Giessen, “Tailoring the ultrafast dephasing of quasiparticles in metallic photonic crystals,” Phys. Rev. Lett. **93**, 243901–1–4 (2004). [CrossRef]

*near-field*region of the system. Among these we mention: the tips of scanning near-field optical microscopes [190], adiabatic plasmonic waveguides [12

12. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404–1–4 (2004). [PubMed]

191. A. Ono, J. Kato, and S. Kawata, “Subwavelength optical imaging through a metallic nanorod array,” Phys. Rev. Lett. **95**, 267407–1–4 (2005). [CrossRef]

192. G. Shvets, S. Trendafilov, J. B. Pendry, and A. Sarychev, “Guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays,” Phys. Rev. Lett. **99**, 053903–1–4 (2007). [PubMed]

193. J. B. Pendry, “Perfect cylindrical lenses,” Opt. Express **11**, 755–760 (2003). [PubMed]

194. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science **315**, 1686–1686 (2007). [PubMed]

195. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. **88**, 067402–1–4 (2002). [PubMed]

196. D. J. Tannor and S. A. Rice, “Control of selectivity of chemical reaction via control of wave packet evolution,” J. Chem. Phys. **83**, 5013–5018 (1985). [CrossRef]

205. M. Shapiro and P. Brumer, “Quantum control of bound and continuum state dynamics,” Physics Reports **425**, 195–264 (2006). [CrossRef]

202. H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Chemistry - whither the future of controlling quantum phenomena?” Science **288**, 824–828 (2000). [PubMed]

206. A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science **282**, 919–922 (1998). [PubMed]

208. N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature **418**, 512–514 (2002). [PubMed]

209. T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. **92**, 208301–1–4 (2004). [PubMed]

*τ*of the pulse is less than the decay (decoherence) time

_{p}*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.

210. M. Durach, A. Rusina, K. Nelson, and M. I. Stockman, “Toward full spatio-temporal control on the nanoscale,” *Nano Lett.*7, 3145–3149 (2007). [PubMed]

211. G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the optical near field of nanoantennas with spatial phase-shaped beams,” Nano Lett. **9**, 3608–3611 (2009). [PubMed]

212. B. Gjonaj, J. Aulbach, P. M. Johnson, A. P. Mosk, L. Kuipers, and A. Lagendijk, “Active spatial control of plasmonic fields,” Nat. Photonics **5**, 360–363 (2011). [CrossRef]

195. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. **88**, 067402–1–4 (2002). [PubMed]

147. M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B **69**, 054202–1–10 (2004). [CrossRef]

209. T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. **92**, 208301–1–4 (2004). [PubMed]

213. M. I. Stockman and P. Hewageegana, “Nanolocalized nonlinear electron photoemission under coherent control,” Nano Lett. **5**, 2325–2329 (2005). [PubMed]

214. M. Sukharev and T. Seideman, “Phase and polarization control as a route to plasmonic nanodevices,” Nano Lett. **6**, 715–719 (2006). [PubMed]

121. A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, “Femtosecond imaging of surface plasmon dynamics in a nanostructured silver film,” Nano Lett. **5**, 1123–1127 (2005). [PubMed]

215. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. d. Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature **446**, 301–304 (2007). [PubMed]

217. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Struber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” *Proc. Natl. Acad. Sci. USA*107, 5329–5333 (2010). [PubMed]

*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*of the degrees of freedom for coherent control can be estimated as For a plasmonic system with the maximum bandwidth Δ

_{DF}*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.

*ω*

_{0}and chirp) which allowed one to concentrate optical energy at the tip of a V-shape structure vs. its opening [147

**69**, 054202–1–10 (2004). [CrossRef]

195. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. **88**, 067402–1–4 (2002). [PubMed]

*τ*between them used in the interferometric coherent control [121

**5**, 1123–1127 (2005). [PubMed]

213. M. I. Stockman and P. Hewageegana, “Nanolocalized nonlinear electron photoemission under coherent control,” Nano Lett. **5**, 2325–2329 (2005). [PubMed]

216. M. Bauer, C. Wiemann, J. Lange, D. Bayer, M. Rohmer, and M. Aeschlimann, “Phase propagation of localized surface plasmons probed by time-resolved photoemission electron microscopy,” Appl. Phys. A **88**, 473–480 (2007). [CrossRef]

*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.

202. H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Chemistry - whither the future of controlling quantum phenomena?” Science **288**, 824–828 (2000). [PubMed]

215. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. d. Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature **446**, 301–304 (2007). [PubMed]

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

218. X. Li and M. I. Stockman, “Highly efficient spatiotemporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time reversal,” Phys. Rev. B **77**, 195109–1–10 (2008). [CrossRef]

219. A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with time-reversal antennas,” Phys. Rev. Lett. **90**, 014301–1–4 (2003). [PubMed]

221. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science **315**, 1120–1122 (2007). [PubMed]

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

*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.

221. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science **315**, 1120–1122 (2007). [PubMed]

*ɛ*we adopt from bulk data [30

_{m}**6**, 4370–4379 (1972). [CrossRef]

^{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.

**69**, 054202–1–10 (2004). [CrossRef]

**88**, 067402–1–4 (2002). [PubMed]

222. M. I. Stockman, “Electromagnetic theory of SERS,” in *Surface Enhanced Raman Scattering – Physics and Applications*, K. Kneipp, M. Moskovits, and H. Kneipp, eds. (Springer-Verlag, 2006), pp. 47–66. [PubMed]

**87**, 167401–1–4 (2001). [PubMed]

**53**, 2183–2186 (1996). [CrossRef]

158. M. I. Stockman, “Chaos and spatial correlations for dipolar eigenproblems,” Phys. Rev. Lett. **79**, 4562–4565 (1997). [CrossRef]

223. M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Optical-absorption and localization of eigen-modes in disordered clusters,” Phys. Rev. B **51**, 185–195 (1995). [CrossRef]

*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**

*, as given by Eqs. (43)–(44).*

^{r}**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

*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*)

^{*}.

**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.

*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

*E*(

_{x}*t*),

*E*(

_{z}*t*)}.

221. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science **315**, 1120–1122 (2007). [PubMed]

*Q*-factors (setting a reverberation chamber around a nanosystem would have been, indeed, unrealistic).

*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.

*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

**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

**69**, 054202–1–10 (2004). [CrossRef]

213. M. I. Stockman and P. Hewageegana, “Nanolocalized nonlinear electron photoemission under coherent control,” Nano Lett. **5**, 2325–2329 (2005). [PubMed]

**5**, 1123–1127 (2005). [PubMed]

215. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. d. Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature **446**, 301–304 (2007). [PubMed]

216. M. Bauer, C. Wiemann, J. Lange, D. Bayer, M. Rohmer, and M. Aeschlimann, “Phase propagation of localized surface plasmons probed by time-resolved photoemission electron microscopy,” Appl. Phys. A **88**, 473–480 (2007). [CrossRef]

225. A. Kubo, N. Pontius, and H. Petek, “Femtosecond microscopy of surface plasmon polariton wave packet evolution at the silver/vacuum interface,” Nano Lett. **7**, 470–475 (2007). [PubMed]

226. E. Verhagen, L. Kuipers, and A. Polman, “Enhanced nonlinear optical effects with a tapered plasmonic waveguide,” Nano Lett. **7**, 334–337 (2007). [PubMed]

*n*≥ 2. Inspired by this, we will consider below, in particular, the coherent control of the two-photon process averaged intensity 〈

*I*

^{2}(

**r**)〉.

*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.

*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].

*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

**5**, 1123–1127 (2005). [PubMed]

**446**, 301–304 (2007). [PubMed]

216. M. Bauer, C. Wiemann, J. Lange, D. Bayer, M. Rohmer, and M. Aeschlimann, “Phase propagation of localized surface plasmons probed by time-resolved photoemission electron microscopy,” Appl. Phys. A **88**, 473–480 (2007). [CrossRef]

225. A. Kubo, N. Pontius, and H. Petek, “Femtosecond microscopy of surface plasmon polariton wave packet evolution at the silver/vacuum interface,” Nano Lett. **7**, 470–475 (2007). [PubMed]

*I*

^{2}(

**r**)〉.

*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.

217. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Struber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” *Proc. Natl. Acad. Sci. USA*107, 5329–5333 (2010). [PubMed]

### 4.5. Coherent control by spatiotemporal pulse shaping

**5**, 1123–1127 (2005). [PubMed]

**69**, 054202–1–10 (2004). [CrossRef]

**88**, 067402–1–4 (2002). [PubMed]

**5**, 2325–2329 (2005). [PubMed]

**446**, 301–304 (2007). [PubMed]

217. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Struber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” *Proc. Natl. Acad. Sci. USA*107, 5329–5333 (2010). [PubMed]

225. A. Kubo, N. Pontius, and H. Petek, “Femtosecond microscopy of surface plasmon polariton wave packet evolution at the silver/vacuum interface,” Nano Lett. **7**, 470–475 (2007). [PubMed]

227. M. Sukharev and T. Seideman, “Coherent control of light propagation via nanoparticle arrays,” J. Phys. B **40**, S283–S298 (2007). [CrossRef]

228. M. M. Wefers and K. A. Nelson, “Programmable phase and amplitude femtosecond pulse shaping,” *Opt. Lett.*18, 2032–2034 (1993). [PubMed]

229. T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal coherent control of lattice vibrational waves,” Science **299**, 374–377 (2003). [PubMed]

210. M. Durach, A. Rusina, K. Nelson, and M. I. Stockman, “Toward full spatio-temporal control on the nanoscale,” *Nano Lett.*7, 3145–3149 (2007). [PubMed]

*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.

**93**, 137404–1–4 (2004). [PubMed]

230. A. J. Babajanyan, N. L. Margaryan, and K. V. Nerkararyan, “Superfocusing of surface polaritons in the conical structure,” J. Appl. Phys. **87**, 3785–3788 (2000). [CrossRef]

231. D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” *J. Appl. Phys.*104, 034311–1–8 (2008). [CrossRef]

232. W. Nomura, M. Ohtsu, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. **86**, 181108–1–3 (2005). [CrossRef]

233. L. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. **5**, 1399–1402 (2005). [PubMed]

228. M. M. Wefers and K. A. Nelson, “Programmable phase and amplitude femtosecond pulse shaping,” *Opt. Lett.*18, 2032–2034 (1993). [PubMed]

229. T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal coherent control of lattice vibrational waves,” Science **299**, 374–377 (2003). [PubMed]

13. E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express **16**, 45–57 (2008). [PubMed]

16. F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. **5**, 67–72 (2009). [PubMed]

18. C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nano-confined light source,” Nano Lett. **7**, 2784–2788 (2007). [PubMed]

19. C. C. Neacsu, S. Berweger, R. L. Olmon, L. V. Saraf, C. Ropers, and M. B. Raschke, “Near-field localization in plasmonic superfocusing: A nanoemitter on a tip,” Nano Lett. **10**, 592–596 (2010). [PubMed]

22. D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. **11**, 1609–1613 (2011). [PubMed]

*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.

*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}*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.

228. M. M. Wefers and K. A. Nelson, “Programmable phase and amplitude femtosecond pulse shaping,” *Opt. Lett.*18, 2032–2034 (1993). [PubMed]

229. T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal coherent control of lattice vibrational waves,” Science **299**, 374–377 (2003). [PubMed]

**6**, 4370–4379 (1972). [CrossRef]

*d*= 30 nm, the minimum thickness is

_{m}*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

210. M. Durach, A. Rusina, K. Nelson, and M. I. Stockman, “Toward full spatio-temporal control on the nanoscale,” *Nano Lett.*7, 3145–3149 (2007). [PubMed]

*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.

220. G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. **92**, 193904–1–3 (2004). [PubMed]

**315**, 1120–1122 (2007). [PubMed]

235. G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, “Time reversal of wideband microwaves,” Appl. Phys. Lett. **88**, 154101–1–3 (2006). [CrossRef]

*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.

*Nano Lett.*7, 3145–3149 (2007). [PubMed]

*x*points) to the same nanofocus at the sharp edge.

*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.

*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.

^{4}with respect to that of the SPPs at the thick edge.

*Nano Lett.*7, 3145–3149 (2007). [PubMed]

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

**69**, 054202–1–10 (2004). [CrossRef]

**88**, 067402–1–4 (2002). [PubMed]

*Nano Lett.*7, 3145–3149 (2007). [PubMed]

214. M. Sukharev and T. Seideman, “Phase and polarization control as a route to plasmonic nanodevices,” Nano Lett. **6**, 715–719 (2006). [PubMed]

218. X. Li and M. I. Stockman, “Highly efficient spatiotemporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time reversal,” Phys. Rev. B **77**, 195109–1–10 (2008). [CrossRef]

236. M. I. Stockman, “Ultrafast nanoplasmonics under coherent control,” New J. Phys. **10**, 025031–1–20 (2008). [CrossRef]

237. T. S. Kao, S. D. Jenkins, J. Ruostekoski, and N. I. Zheludev, “Coherent control of nanoscale light localization in metamaterial: Creating and positioning isolated subwavelength energy hot spots,” Phys. Rev. Lett. **106**, 085501–1–4 (2011). [PubMed]

**5**, 1123–1127 (2005). [PubMed]

**446**, 301–304 (2007). [PubMed]

*Proc. Natl. Acad. Sci. USA*107, 5329–5333 (2010). [PubMed]

21. S. Berweger, J. M. Atkin, X. G. Xu, R. L. Olmon, and M. B. Raschke, “Femtosecond nanofocusing with full optical waveform control,” Nano Lett. , p. doi: [CrossRef] (2011). [PubMed]

_{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

21. S. Berweger, J. M. Atkin, X. G. Xu, R. L. Olmon, and M. B. Raschke, “Femtosecond nanofocusing with full optical waveform control,” Nano Lett. , p. doi: [CrossRef] (2011). [PubMed]

*Nano Lett.*7, 3145–3149 (2007). [PubMed]

211. G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the optical near field of nanoantennas with spatial phase-shaped beams,” Nano Lett. **9**, 3608–3611 (2009). [PubMed]

212. B. Gjonaj, J. Aulbach, P. M. Johnson, A. P. Mosk, L. Kuipers, and A. Lagendijk, “Active spatial control of plasmonic fields,” Nat. Photonics **5**, 360–363 (2011). [CrossRef]

**446**, 301–304 (2007). [PubMed]

238. M. Bauer, O. Schmidt, C. Wiemann, R. Porath, M. Scharte, O. Andreyev, G. Schonhense, and M. Aeschlimann, “Time-resolved two photon photoemission electron microscopy,” Appl. Phys. B **74**, 223–227 (2002). [CrossRef]

**5**, 1123–1127 (2005). [PubMed]

239. T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. **26**, 557–559 (2001). [CrossRef]

240. T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B **74**, S133–S144 (2002). [CrossRef]

*Nano Lett.*7, 3145–3149 (2007). [PubMed]

237. T. S. Kao, S. D. Jenkins, J. Ruostekoski, and N. I. Zheludev, “Coherent control of nanoscale light localization in metamaterial: Creating and positioning isolated subwavelength energy hot spots,” Phys. Rev. Lett. **106**, 085501–1–4 (2011). [PubMed]

211. G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the optical near field of nanoantennas with spatial phase-shaped beams,” Nano Lett. **9**, 3608–3611 (2009). [PubMed]

212. B. Gjonaj, J. Aulbach, P. M. Johnson, A. P. Mosk, L. Kuipers, and A. Lagendijk, “Active spatial control of plasmonic fields,” Nat. Photonics **5**, 360–363 (2011). [CrossRef]

**9**, 3608–3611 (2009). [PubMed]

*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.

*Nano Lett.*7, 3145–3149 (2007). [PubMed]

**5**, 360–363 (2011). [CrossRef]

**5**, 360–363 (2011). [CrossRef]

*π*range) contributing to the focus, thereby maximizing the numerical aperture and resolution.

**5**, 360–363 (2011). [CrossRef]

*Nano Lett.*7, 3145–3149 (2007). [PubMed]

*Nano Lett.*7, 3145–3149 (2007). [PubMed]

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

### 5.1. Introduction to spasers and spasing

241. H. A. Atwater, “The promise of plasmonics,” Sci. Am. **296**, 56–63 (2007). [PubMed]

242. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. **7**, 442–453 (2008). [PubMed]

51. L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photonics **2**, 226–229 (2008). [CrossRef]

244. W. A. Challener, C. Peng, A. V. Itagi, D. Karns, W. Peng, Y. Peng, X. Yang, X. Zhu, N. J. Gokemeijer, Y. T. Hsia, G. Ju, R. E. Rottmayer, M. A. Seigler, and E. C. Gage, “Heat-assisted magnetic recording by a near-field transducer with efficient optical energy transfer,” Nat. Photonics **3**, 220–224 (2009). [CrossRef]

136. S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature **453**, 757–760 (2008). [PubMed]

242. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. **7**, 442–453 (2008). [PubMed]

245. N. Nagatani, R. Tanaka, T. Yuhi, T. Endo, K. Kerman, Y. T. Tamiya, and E Tamiya, “Gold nanoparticle-based novel enhancement method for the development of highly sensitive immunochromatographic test strips,” Sci. Technol. Adv. Mater. **7**, 270–275 (2006). [CrossRef]

246. L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. USA **100**, 13549–13554 (2003). [PubMed]

**453**, 757–760 (2008). [PubMed]

247. I.-Y. Park, S. Kim 1, J. Choi, D.-H. Lee, Y.-J. Kim, M. F. Kling, M. I. Stockman, and S.-W. Kim, “Plasmonic generation of ultrashort extreme ultraviolet light pulses,” Nat. Photonics (2011) (In Press). [CrossRef]

23. M. I. Stockman, “Nanoplasmonics: The physics behind the applications,” Phys. Today **64**, 39–44 (2011). [CrossRef]

**90**, 027402–1–4 (2003). [PubMed]

137. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. **12**, 024004–1–13 (2010). [CrossRef]

139. M. I. Stockman, “Loss compensation by gain and spasing,” Phil. Trans. R. Soc. A **369**, 3510–3524 (2011). [PubMed]

251. M. I. Stockman, “Spasers explained,” Nat. Photonics **2**, 327–329 (2008). [CrossRef]

252. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Sutee-wong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature **460**, 1110–1112 (2009). [PubMed]

253. M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Ntzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express **17**, 11107–11112 (2009). [PubMed]

256. R. A. Flynn, C. S. Kim, I. Vurgaftman, M. Kim, J. R. Meyer, A. J. Mkinen, K. Bussmann, L. Cheng, F. S. Choa, and J. P. Long, “A room-temperature semiconductor spaser operating near 1.5 micron,” Opt. Express **19**, 8954–8961 (2011). [PubMed]

**90**, 027402–1–4 (2003). [PubMed]

### 5.2. Spaser fundamentals

**90**, 027402–1–4 (2003). [PubMed]

251. M. I. Stockman, “Spasers explained,” Nat. Photonics **2**, 327–329 (2008). [CrossRef]

253. M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Ntzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express **17**, 11107–11112 (2009). [PubMed]

254. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**, 629–632 (2009). [PubMed]

257. M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Noetzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics **1**, 589–594 (2007). [CrossRef]

**90**, 027402–1–4 (2003). [PubMed]

**87**, 167401–1–4 (2001). [PubMed]

33. I. A. Larkin and M. I. Stockman, “Imperfect perfect lens,” Nano Lett. **5**, 339–343 (2005). [PubMed]

*l*∼ 1 nm. So, the spaser is truly nanoscopic – its minimum total size can be just a few nanometers.

_{nl}*R*is much less than the wavelength

*λ*and whose metal thickness is between

*l*and

_{nl}*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

_{n}**90**, 027402–1–4 (2003). [PubMed]

**12**, 024004–1–13 (2010). [CrossRef]

251. M. I. Stockman, “Spasers explained,” Nat. Photonics **2**, 327–329 (2008). [CrossRef]

258. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express **15**, 2622–2653 (2007). [PubMed]

*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

**87**, 167401–1–4 (2001). [PubMed]

259. D. J. Bergman and D. Stroud, “Theory of resonances in the electromagnetic scattering by macroscopic bodies,” Phys. Rev. B **22**, 3527–3539 (1980). [CrossRef]

258. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express **15**, 2622–2653 (2007). [PubMed]

*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.

**90**, 027402–1–4 (2003). [PubMed]

260. E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: Controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express **17**, 8548–8551 (2009). [PubMed]

261. J. Seidel, S. Grafstroem, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. **94**, 177401–1–4 (2005). [PubMed]

262. M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated emission of surface plasmon polaritons,” Phys. Rev. Lett. **101**, 226806–1–4 (2008). [PubMed]

258. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express **15**, 2622–2653 (2007). [PubMed]

253. M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Ntzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express **17**, 11107–11112 (2009). [PubMed]

257. M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Noetzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics **1**, 589–594 (2007). [CrossRef]

*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.

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

**90**, 027402–1–4 (2003). [PubMed]

263. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B **71**, 115409–1–4 (2005). [CrossRef]

**90**, 027402–1–4 (2003). [PubMed]

263. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B **71**, 115409–1–4 (2005). [CrossRef]

**15**, 2622–2653 (2007). [PubMed]

264. Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Resonance amplification of left-handed transmission at optical frequencies by stimulated emission of radiation in active metamaterials,” Opt. Express **16**, 20974–20980 (2008). [PubMed]

267. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. **105**, 127401–1–4 (2010). [PubMed]

**12**, 024004–1–13 (2010). [CrossRef]

**12**, 024004–1–13 (2010). [CrossRef]

268. S. W. Chang, C. Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express **16**, 10580–10595 (2008). [PubMed]

269. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics **2**, 351–354 (2008). [CrossRef]

270. I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys Rev A **71**, 063812– (2005). [CrossRef]

*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

270. I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys Rev A **71**, 063812– (2005). [CrossRef]

**12**, 024004–1–13 (2010). [CrossRef]

261. J. Seidel, S. Grafstroem, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. **94**, 177401–1–4 (2005). [PubMed]

262. M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated emission of surface plasmon polaritons,” Phys. Rev. Lett. **101**, 226806–1–4 (2008). [PubMed]

271. M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. **8**, 3998–4001 (2008). [PubMed]

274. P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. **35**, 1197–1199 (2010). [PubMed]

260. E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: Controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express **17**, 8548–8551 (2009). [PubMed]

275. M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “The effect of gain and absorption on surface plasmons in metal nanoparticles,” Appl. Phys. B **86**, 455–460 (2007). [CrossRef]

276. M. A. Noginov, “Compensation of surface plasmon loss by gain in dielectric medium,” J. Nanophotonics **2**, 021855–1–17 (2008). [CrossRef]

*λ*= 882 nm in a gold nanostrip waveguide with a dyes solution as a gain medium has been observed [277

277. I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics **4**, 382–387 (2010). [CrossRef]

**17**, 11107–11112 (2009). [PubMed]

254. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**, 629–632 (2009). [PubMed]

*μ*m

^{2}[255

255. R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. **10**, 110–113 (2010). [PubMed]

*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

256. R. A. Flynn, C. S. Kim, I. Vurgaftman, M. Kim, J. R. Meyer, A. J. Mkinen, K. Bussmann, L. Cheng, F. S. Choa, and J. P. Long, “A room-temperature semiconductor spaser operating near 1.5 micron,” Opt. Express **19**, 8954–8961 (2011). [PubMed]

252. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Sutee-wong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature **460**, 1110–1112 (2009). [PubMed]

### 5.4. Equations of spaser

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

*φ*(

_{n}**r**) are described by a wave equation (25) [29

**90**, 027402–1–4 (2003). [PubMed]

**87**, 167401–1–4 (2001). [PubMed]

**90**, 027402–1–4 (2003). [PubMed]

*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

_{n}**90**, 027402–1–4 (2003). [PubMed]

*π*by 4

*π*.

*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.

*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 where the dot denotes temporal derivative. We use the standard rotating wave approximation (RWA), which only takes into account the resonant interaction between the optical field and chromophores. We denote |1〉 and |2〉 as the ground and excited states of a chromophore, with the transition |2〉 ⇌ |1〉 resonant to the spasing plasmon mode

*n*. In this approximation, the time dependence of the nondiagonal elements of the density matrix is

*p*th chromophore of the gain medium.

_{12}to describe the polarization relaxation and a difference

*p*th chromophore, and

*Ĥ*. To provide conditions for the population inversion (

*g*. In this way, we obtain the following equation of motion: where

*γ*

_{2}is the decay rate |2〉 → |1〉.

*γ*as

_{n}*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.

**r**

*can be found in a standard way using the quantized field (61) as As in Schawlow-Towns theory of laser-line width [278], this spontaneous emission of SPs leads to the diffusion of the phase of the spasing state. This defines width*

_{p}*γ*of the spasing line as This width is small for a case of developed spasing when

_{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*

_{s}*g*= 0.

*ɛ*= 2; the permittivity of silver is adopted from Ref. [30

_{d}**6**, 4370–4379 (1972). [CrossRef]

*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

**90**, 027402–1–4 (2003). [PubMed]

260. E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: Controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express **17**, 8548–8551 (2009). [PubMed]

#### 5.4.2. Equations for CW regime

*p*th chromophore

*ω*, which generally does not coincide with either the gain transition frequency

_{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 This system defines the stationary (CW-generation) number of SPs per spasing mode,

_{n}*N*.

_{n}**90**, 027402–1–4 (2003). [PubMed]

*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

_{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

**87**, 167401–1–4 (2001). [PubMed]

**90**, 027402–1–4 (2003). [PubMed]

*Q*, a high density of the chromophores, and a large transition dipole (oscillator strength) of the chromophore transition. The small modal volume

*V*(at a given number of the chromophores

_{n}*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.

*ω*=

_{n}*ω*

_{21}. As a result, we obtain from Eq. (75), where the integral is extended over the volume

*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

*Q*≫ 1, where definition (6) is used. Taking into account Eqs. (47), (48) and (78), (79), we obtain from Eq. (77) a necessary condition of spasing at a frequency

*ω*as

*g*(whose dimensionality is cm

^{−1}) is given by a standard expression Substituting it into Eq. (80), we obtain the spasing criterion in terms of the gain as where

*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}*ɛ*= 2 (simulating a light glass) and

_{d}*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

*N*on the excitation rate

_{n}*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.

^{18}– 10

^{20}, while in the spaser the SP population is

*N*≲ 100. This is due to the much stronger feedback in spasers because of the much smaller modal volume

_{n}*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}252. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Sutee-wong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature **460**, 1110–1112 (2009). [PubMed]

*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.

*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*

_{s}*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

_{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}**15**, 2622–2653 (2007). [PubMed]

265. M. Wegener, J. L. Garcia-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express **16**, 19785–19798 (2008). [PubMed]

265. M. Wegener, J. L. Garcia-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express **16**, 19785–19798 (2008). [PubMed]

**15**, 2622–2653 (2007). [PubMed]

265. M. Wegener, J. L. Garcia-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express **16**, 19785–19798 (2008). [PubMed]

267. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. **105**, 127401–1–4 (2010). [PubMed]

### 5.6. Spaser as ultrafast quantum nanoamplifier

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

277. I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics **4**, 382–387 (2010). [CrossRef]

**12**, 024004–1–13 (2010). [CrossRef]

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

*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.

*N*, rather rapidly, within a few hundred femtoseconds approaches the same stationary (“logical”) level. At this level, an SP population of

_{n}*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.

*N*) is in the interval of time it is required for the spaser to reach the final (CW) state. For very small

_{n}*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}*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).

*N*∼ 100). (An exception is a case of little practical importance when the initial

_{n}*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

*n*= 0.66

_{a}*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}*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*= 5 × 10

_{n}^{−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.

*N*of the SP population large enough [for instance, for

_{n}*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.

*N*≈ 1. For a value of the initial

_{n}*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}*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.

_{n}*n*

_{21}= 0.65) is created by a short pulse at

*t*= 0 and simultaneously initial SP population

*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).

_{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)].

_{n}*N*is less than the critical one (i.e.,

_{n}*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

- 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
**64**, 39–44 (2011). [CrossRef], 1916. F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol.

**5**, 67–72 (2009). [PubMed]], and solar energy conversion [2619. C. C. Neacsu, S. Berweger, R. L. Olmon, L. V. Saraf, C. Ropers, and M. B. Raschke, “Near-field localization in plasmonic superfocusing: A nanoemitter on a tip,” Nano Lett.

**10**, 592–596 (2010). [PubMed]]. 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) [2326. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.

**9**, 205–213 (2010). [PubMed]**64**, 39–44 (2011). [CrossRef], 242180. M. I. Stockman, “Electromagnetic theory of SERS,” in

*Surface Enhanced Raman Scattering*, vol. 103, M. M. K. Kneipp and H. Kneipp, eds. (Springer, 2006), pp. 47 – 66. [CrossRef], 279**7**, 442–453 (2008). [PubMed], 280279. K. Kneipp, M. Moskovits, and H. Kneipp, eds.,

*Surface Enhanced Raman Scattering: Physics and Applications*(Springer-Verlag, Heidelberg New York Tokyo, 2006). [CrossRef]].280. J. Kneipp, H. Kneipp, B. Wittig, and K. Kneipp, “Novel optical nanosensors for probing and imaging live cells,” Nanomedicine: Nanotechnology, Biology and Medicine

**6**, 214–226 (2010). [CrossRef] - Another promising idea is to use superconducting plasmonics to dramatically reduce losses [72,281
72. V. A. Fedotov, A. Tsiatmas, J. H. Shi, R. Buckingham, P. de Groot, Y. Chen, S. Wang, and N. I. Zheludev, “Temperature control of Fano resonances and transmission in superconducting metamaterials,” Opt. Express

**18**, 9015–9019 (2010). [PubMed]–283281. F. J. Dunmore, D. Z. Liu, H. D. Drew, S. Dassarma, Q. Li, and D. B. Fenner, “Observation of below-gap plasmon excitations in superconducting YBa

_{2}Cu_{3}O_{7}films,” Phys. Rev. B**52**, R731–R734 (1995). [CrossRef]]. However, this is only applicable for frequencies below the superconducting gaps, i.e., in the terahertz region.283. A. Tsiatmas, A. R. Buckingham, V. A. Fedotov, S. Wang, Y. Chen, P. A. J. de Groot, and N. I. Zheludev, “Superconducting plasmonics and extraordinary transmission,” Appl. Phys. Lett.

**97**, 111106–1–3 (2010). [CrossRef] - 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
284. A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science

**331**, 290–291 (2011). [PubMed]*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
285. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics

**1**, 41–48 (2007). [CrossRef]]. 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.286. N. I. Zheludev, “A roadmap for metamaterials,” Optics and Photonics News

**22**, 30–35 (2011). [CrossRef]

138. M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. **106**, 156802–1–4 (2011). [PubMed]

**369**, 3510–3524 (2011). [PubMed]

#### 5.7.2. Permittivity of nanoplasmonic metamaterial

*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).

*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 for

**r**belonging to the surface

*S*of the volume under consideration.

287. M. I. Stockman, K. B. Kurlayev, and T. F. George, “Linear and nonlinear optical susceptibilities of Maxwell Garnett composites: Dipolar spectral theory,” Phys. Rev. B **60**, 17071–17083 (1999). [CrossRef]

*V*of the metamaterial is given by a formula where

*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 where we took into account the Maxwell continuity equation ∇ [

*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 From the last equality, we obtain the required homogenization formula as an expression for the effective permittivity of the metamaterial:

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

*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

**87**, 167401–1–4 (2001). [PubMed]

**69**, 054202–1–10 (2004). [CrossRef]

218. X. Li and M. I. Stockman, “Highly efficient spatiotemporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time reversal,” Phys. Rev. B **77**, 195109–1–10 (2008). [CrossRef]

**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 The first term in this equation corresponds to the mean (macroscopic) field and the second one describes the deviations of the local field from the mean field containing contributions of the hot spots [157

_{n}**53**, 2183–2186 (1996). [CrossRef]

*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).

*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 If so, the second (local) term of the field (89) dominates and, with a good precision, the local field is approximately the eigenmode’s field: Substituting this into Eq. (87), we obtain a homogenization formula where

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

_{n}### 5.8. Conditions of loss compensation by gain and spasing

*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.

*ω*=

*ω*is Taking Eq. (96) into account, this reduces to Finally, taking into account Eqs. (28), (47) and that Im

_{n}*ɛ*(

_{m}*ω*) > 0, we obtain from Eq. (99) the condition of the loss (over)compensation as where the strict inequality corresponds to the overcompensation and net amplification. In Eq. (97) we have assumed non-polarized gain transitions. If these transitions are all polarized along the excitation electric field, the concentration

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

_{c}**87**, 167401–1–4 (2001). [PubMed]

**53**, 2183–2186 (1996). [CrossRef]

*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

*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.

*k*≲

*l*, can be described by these conditions because they are, basically, quasistatic. For instance, the SPPs on a thin metal wire of a radius

_{s}*R*≲

*l*are described by a dispersion relation [12

_{s}**93**, 137404–1–4 (2004). [PubMed]

*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

*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] Note that for the energy of the system to be definite, it is necessary to assume that the loss is not too large, |Re

*ɛ*̄| ≫ Im

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

*Q*(i.e., the rate of the heat-density production in the system) is [28] Assume that the internal (Ohmic) loss dominates over other loss mechanisms such as the radiative loss, which is also a realistic assumption since the Ohmic loss is very large for the experimentally studied systems and the system itself is very small (the radiative loss rate is proportional to the volume of the system). In such a case of the dominating Ohmic losses, we have

*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

*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

**87**, 167401–1–4 (2001). [PubMed]

*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.

*∂*(

*ω*Re

*ɛ̄*)

*/∂ω*> 0. Thus, Im

*ɛ̄*< 0 in Eq. (106) would lead to a negative decrement, implying that the initial small fluctuation starts exponentially grow in time in its field and energy, which is an instability. Such an instability is indeed not impossible: it will result in spasing that will eventually stabilize |

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

*the net gain to be precisely zero*[137

**12**, 024004–1–13 (2010). [CrossRef]

*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.

**6**, 4370–4379 (1972). [CrossRef]

_{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*= 6 × 10

_{c}^{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

267. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. **105**, 127401–1–4 (2010). [PubMed]

*n*= 6 × 10

_{c}^{18}cm

^{−3}used in Ref. [267

**105**, 127401–1–4 (2010). [PubMed]

**105**, 127401–1–4 (2010). [PubMed]

*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

_{h}**105**, 127401–1–4 (2010). [PubMed]

*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

**105**, 127401–1–4 (2010). [PubMed]

**105**, 127401–1–4 (2010). [PubMed]

**105**, 127401–1–4 (2010). [PubMed]

288. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature **466**, 735–738 (2010). [PubMed]

289. Z. Gryczynski, O. O. Abugo, and J. R. Lakowicz, “Polarization sensing of fluorophores in tissues for drug compliance monitoring,” Anal. Biochem. **273**, 204–211 (1999). [PubMed]

*σ*= 2×10

^{−16}cm

^{2}at 690 nm in concentration

*n*= 1.2×10

_{c}^{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

288. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature **466**, 735–738 (2010). [PubMed]

288. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature **466**, 735–738 (2010). [PubMed]

**466**, 735–738 (2010). [PubMed]

*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.

**466**, 735–738 (2010). [PubMed]

*a*≈ 280 nm, is not accidental. As follows from theoretical consideration of Ref. [297

297. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. USA **106**, 1693–1698 (2009). [PubMed]

*d*of the nanowire) is greater then the skin depth,

*d*≳

*l*≈ 25 nm, which allows one to circumvent Landau-Lifshitz’s limitation on the existence of optical magnetism [28,297

_{s}297. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. USA **106**, 1693–1698 (2009). [PubMed]

*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

_{s}**466**, 735–738 (2010). [PubMed]

**17**, 8548–8551 (2009). [PubMed]

290. N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Expr. **18**, 24140–24151 (2010). [CrossRef]

262. M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated emission of surface plasmon polaritons,” Phys. Rev. Lett. **101**, 226806–1–4 (2008). [PubMed]

*μ*m polymer film containing Rodamine 6G dye in the

*n*= 1.2 × 10

_{c}^{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.

**101**, 226806–1–4 (2008). [PubMed]

**101**, 226806–1–4 (2008). [PubMed]

*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*= 4 × 10

_{e}^{−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

274. P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. **35**, 1197–1199 (2010). [PubMed]

277. I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics **4**, 382–387 (2010). [CrossRef]

*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.

**460**, 1110–1112 (2009). [PubMed]

**17**, 11107–11112 (2009). [PubMed]

256. R. A. Flynn, C. S. Kim, I. Vurgaftman, M. Kim, J. R. Meyer, A. J. Mkinen, K. Bussmann, L. Cheng, F. S. Choa, and J. P. Long, “A room-temperature semiconductor spaser operating near 1.5 micron,” Opt. Express **19**, 8954–8961 (2011). [PubMed]

254. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**, 629–632 (2009). [PubMed]

255. R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. **10**, 110–113 (2010). [PubMed]

292. J. K. Kitur, V. A. Podolskiy, and M. A. Noginov, “Stimulated emission of surface plasmon polaritons in a micro-cylinder cavity,” Phys. Rev. Lett. **106**, 183903–1–4 (2011). [PubMed]

*μ*m (i.e., in the communication near-ir range) has been reported [256

**19**, 8954–8961 (2011). [PubMed]

**105**, 127401–1–4 (2010). [PubMed]

293. O. Hess, S. Wuestner, A. Pusch, K. L. Tsakmakidis, and J. M. Hamm, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. Royal Soc. A **369**, 3525–3550 (2011). [CrossRef]

**466**, 735–738 (2010). [PubMed]

**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*of these modes is zero, and these modes are localized, i.e., they are SPs. Their wave function is periodic with period 2

_{g}*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

_{s},λ̵295. H. Ghafouri-Shiraz, *Distributed Feedback Laser Diodes and Optical Tunable Filters* (Wiley, 2003). [CrossRef]

*in plane*of the structure, unlike the signal that propagates normally to it as in Ref. [288

**466**, 735–738 (2010). [PubMed]

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286. | N. I. Zheludev, “A roadmap for metamaterials,” Optics and Photonics News |

287. | M. I. Stockman, K. B. Kurlayev, and T. F. George, “Linear and nonlinear optical susceptibilities of Maxwell Garnett composites: Dipolar spectral theory,” Phys. Rev. B |

288. | S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature |

289. | Z. Gryczynski, O. O. Abugo, and J. R. Lakowicz, “Polarization sensing of fluorophores in tissues for drug compliance monitoring,” Anal. Biochem. |

290. | N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Expr. |

291. | E. Kretschmann and H. Raether, “Radiative decay of nonradiative surface plasmons excited by light,” Z. Naturforsch. |

292. | J. K. Kitur, V. A. Podolskiy, and M. A. Noginov, “Stimulated emission of surface plasmon polaritons in a micro-cylinder cavity,” Phys. Rev. Lett. |

293. | O. Hess, S. Wuestner, A. Pusch, K. L. Tsakmakidis, and J. M. Hamm, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. Royal Soc. A |

294. | F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. A |

295. | H. Ghafouri-Shiraz, |

296. | P. W. Anderson, “Absence of diffusion in certain random lattices,” |

297. | R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. USA |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(320.7120) Ultrafast optics : Ultrafast phenomena

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Plasmonics

**History**

Original Manuscript: September 6, 2011

Revised Manuscript: October 13, 2011

Manuscript Accepted: October 20, 2011

Published: October 24, 2011

**Virtual Issues**

(2011) *Advances in Optics and Photonics*

Vol. 6, Iss. 11 *Virtual Journal for Biomedical Optics*

Collective Phenomena (2011) *Optics Express*

**Citation**

Mark I. Stockman, "Nanoplasmonics: past, present, and glimpse into future," Opt. Express **19**, 22029-22106 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-22029

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