Light scattering and surface plasmons on small spherical particles

Light scattering by small particles has a long and interesting history in physics. Nonetheless, it continues to surprise with new insights and applications. This includes new discoveries, such as novel plasmonic effects, as well as exciting theoretical and experimental developments such as optical trapping, anomalous light scattering, optical tweezers, nano-spasers, and novel aspects and realizations of Fano resonances. These have led to important new applications, including several ones in the biomedical area and in sensing techniques at the single-molecule level. There are additionally many potential future applications in optical devices and solar energy technologies. Here we review the fundamental aspects of light scattering by small spherical particles, emphasizing the phenomenological treatments and new developments in this field.


Introduction
The production, control, manipulation and use of light are at the core of many technologies.
Light scattering plays key roles in all of these. Of course, the scattering of light by small particles has a long history, where it was studied in contexts such as cumulus clouds, the color of the sky and rainbows, and used in various glass artifacts and windows from the middle ages 1 . The remarkable fact is that such a classical topic is the basis of many fundamentally new and unexpected scientific and technological advances. The key is the current focus on the nanoscale and especially near-field effects at the nanoscale, while much of the older classical study was oriented towards the accessible far-field behavior.
More specifically, there have been fascinating developments in regard to the light scattering by nanosized particles, including metal particles and surfaces, where localized surface plasmons can be excited leading to optical resonance phenomena [2][3][4][5] . Small particles with surface plasmons can be used to detect the fluorescence of single molecules 6,7 , enhance Raman scattering 8 , resonantly transfer energy of excitons 9 , and create nanosized quantum amplifiers of optical energy.
Potential practical uses include 10 small-scale sensing techniques 11,12 , numerous biomedical applications 13 , manipulation of light for solar energy technologies 14 , and others.
Here we provide a short review emphasizing the nano-optics of small particles, near field effects, and the fundamental theoretical basis for their description. We begin with a review of the classical light scattering theory for spherical particles based on the quasi-static (Rayleigh) approximation and the general Mie theory. Scattering by dielectric particles is discussed along with the new topic of optical trapping. We discuss plasmon resonances and light scattering on small metallic particles, which is a subject that has been renewed by a series of new findings, including anomalous scattering with an inverted hierarchy of resonances and Fano resonances. The breakdown of the general Drude model for dielectric function at very small particle sizes and the resulting effects are discussed. Finally, we review the stimulated radiation from the surface plasmons of small particles along with concepts for new kinds of lasers based on nano-lasing related to surface plasmons coupled to an active medium (so-called spasers).
We start with a summary of the basic concepts that remain useful in understanding light scattering, focusing on the case of spherical particles. Light scattering by small particles is one of fundamental problems of electrodynamics. As mentioned, it is a classical subject for which theory 3 was developed long ago. This theory includes both the near and far field description. However, until recently the near field was inaccessible to experiments, and the interest was focused on far field effects. Now with advances in nano-technology and nano-optics, the richness of the near field theory is being exploited. This includes the production by scattering of very high light intensities with spatial variations shorter than the wavelengtha phenomenon that enables rich new physics, both linear and non-linear, at the nanoscale.
The physical understanding of light scattering by small particles began with the electric dipole concept, introduced by Lord Rayleigh in 1871 15 . One starts with the assumption that the electromagnetic phase is constant over the region of interest, which is natural since the size of small particle considered is less than the wavelength of light. Then the homogeneous field of the incident light induces a polarization, which in turn results in light scattering. Higher order scattering modes, such as quadrupole and octupole, are not considered at this level. The polarization (i.e. the induced dipole) of materials in response to electromagnetic fields is determined by the dielectric function.
The dielectric function of a material (at energies above the phonon energies) is determined by its electronic structure. It is practical to calculate dielectric functions from first principles band theory and often good agreement with experiment is found 16,17 . However, for analyzing optical properties, it is often useful to approximate the optical properties of solids using the classical harmonic oscillator formalism introduced by Lorentz. In the Lorentz model, the dielectric function of non-conducting materials can be expressed as 1 Many physical phenomena can be very simply understood even in the simplest one oscillator theory. For example, the dispersion of light by prisms or water drops is explained by the frequency dependence of the refractive index. This follows the normal dispersive behavior (refractive index increases with energy) for materials like glass and water. This originates in the fact that the energy 4 ( 0 ) of the effective oscillator for transparent materials such as these is generally much larger than the frequency of visible light 19  plasmonic gold and silver nanoparticles with elongated shapes) and the Maxwell-Garnett equations 5 (providing an effective medium approach) 22 . Of course, there are also interesting questions related to light scattering by particles that cannot be directly described with Mie theory. An example is the electromagnetic hot spot between two nearby particles, which depends on coherency.
As mentioned, it is remarkable that even though more than 100 years have passed since the introduction of Mie's general theory of light scattering by a sphere, new and exciting physics associated with light scattering by small particles continues to be found [23][24][25][26] . Examples include the finding of giant optical resonances with an inverted hierarchy (e.g. the quadrupole resonance is more intense than the dipole) in scattering by small particles with negative dielectric susceptibility and weak dissipation 23 , and anomalous scattering with the complicated near-field structures, such as the vortices, unusual frequency dependence, etc. 27,28 . In addition, the Fano resonances, which are well known in quantum physics, were discovered in optics of small metallic particles [29][30][31] .
Turning to nanotechnology and nano-optics, there is an increasing focus on control of optical energy in sub-wavelength structuresa field that is leading to many new ideas and remarkable experiment results 3, 32-35 . These include nano-lenses 36 and nano-antennas 37 . The localized surface plasmons in spherical core-shell structures can result in so-called spasers, with resonantly coupled transitions with excitons from dye molecules in the shell layer 38 . The coupling between localized surface plasmons from closely separated small particles results in electromagnetic hot spots [39][40][41] .
Huge light scattering is found on anisotropic spherical particles with an active mechanism 42 . These remarkable results based on light scattering by small particles suggest many potential applications, including solar energy technologies 14,43 and nano-scale lasers 44 .

Light Scattering by Spherical Particles in the Quasi-static Approximation and Beyond
In the quasi-static approximation, light scattering by sphere particles is addressed by solving the Laplace equation for the scalar electric potential where 0 E E z  , a, ε p and ε m are the electric field along z direction, radius of particle, dielectric function of the particle and that of the medium, respectively. By comparing the scattering potential 6 from the Laplace equation with that of a dipole, the effective dipole moment can be expressed where σ geom =a 2 is the geometrical cross-section. We introduce the dimensionless cross-sections of scattering (Q sc ) and adsorption (Q abs ), the dimensionless size q=ka and the relative dielectric function ε d =ε p /ε m for convenience in the discussion that follows. In the formula, k is the wave vector in medium.
The above formula implies that as the particle size is decreased, the efficiency of adsorption will dominate over the scattering efficiency. Therefore, particles with very small size will be difficult to detect by light scattering. In addition, one may note that there is a resonant enhancement for scattering and adsorption when the condition Re(ε d )=-2 (the Fröhlich condition) is satisfied. This resonance is due to resonant excitation of the dipole surface plasmon. With the The above theory, however, doesn't capture the size effects on the spectra, including the changes of position and width of plasmon peak. Retardation effects in larger particles result in breakdown of the quasi-static approximation.
The scattering amplitudes a l and b l are defined as follows, where () Gl can be expressed as 27 , One may note that the  parameter (corresponding to dissipative losses due to electron collisions) is ignored in the derivation. The spectrum is found to have a Lorentz profile with an effective parameter  eff , which has a similar role as the dissipation parameter in the Rayleigh 8 spectrum and can be expressed as . This damping is due to the radiative losses of plasmons. This is different from the dissipative loss term (  ) in Rayleigh formula. Therefore, the singularity in the corrected scattering formula is removed due to radiative losses, even if dissipative losses are neglected.
Intuitively, the red shift of the peak in the dipole resonance with increasing size is due to the weakening of the restoring force. This is because the distance between charges on opposite sides of the particle increases with size and so their interaction decreases. The red shift of resonance can be addressed directly by numerical solution of the Mie theory equations. The scattering cross-section, C sc from the dipole term neglecting dissipative losses is plotted in Figure 1a. The red shift is clearly seen. One may also note that the resonant scattering cross-section is similar for the different sizes. However, with dissipative losses, the scattering cross-section falls quickly with decreasing particle size ( Figure 1b). Clearly, the effect of dissipative losses on the scattering increases with decreasing size. The effect of the dielectric function of the medium, ε m on the plasmon resonance of small metal particles is contained in the parameters q and ε d . From equation (3), C abs is proportional to The Fröhlich resonance condition then requires a decrease in Re(ε p ), for an increase in ε m . For most metals, such as Au and Ag, Re(ε p ) decreases and Im(ε p ) increases with decreasing frequency around the resonance. Therefore, the peaks in the absorption spectra are shifted to longer wavelengths and become broader and more intense with the increase in ε m 22 .

Optical Scattering by Dielectric Particles
The light scattering by dielectric particles is easy to calculate using the Mie theory. Figure 2a and 2b shows two typical scattering curves for particles of refractive index n =1.33 and n =1.97 (that of water and glass). The curves have a series of maxima and minima. Other dielectric spheres with different refractive indices show similar behavior. Finally, we note that in the limit of an extremely large particle the scattering cross-section is twice as large as the geometrical cross-section 19 . All these results for the scattering and extinction cross-sections were given fifty years ago 19 . These results, taken in the far field, are connected with the study of light transmission in mist, fogs, cloud chambers, and so on, but the near field regions for dielectric spheres have 9 received attention much more recently.
For small size dielectric particles, the scattering cross-section will increase with increasing refractive index. This result follows from the Rayleigh theory and is indicative of the important role of dipole scattering. The squares of the intensity of the electric fields in the near field for different refractive indices are shown in Figure 3a-3c. The electric field intensity increases in the near field region with increasing refractive index. However, the near field electric field intensity from the dipolar mode doesn't become stronger as the size of dielectric particle decreases. Instead, the maximal value of electric field intensity increases along with a change in the spatial configuration, when the particle size increases in Figure 3d-3f.
Usually, Mie scattering will dominate when the particle size is larger than a wavelength. As shown in Figure 3d-3f, Mie scattering produces complex field distribution patterns reminiscent of directional antennas. This effect is sometimes referred to as a photonic nanojet. From the point of view of geometrical optics, the increase in the scattering field intensity with a more intense forward lobe reflects the fact that a dielectric sphere with large size behaves as a convex lens.
Localized regions of high intensity, such as those near the particle, can trap dielectric particles due to gradient forces (these are forces that arise because a particle with a dielectric constant higher than medium will lower the energy if it moves to the location of the maximum electric field intensity). This effect was observed by Ashkin et al. more than 30 years ago 45,46 .
The related optical tweezers technique, in which light is used to manipulate small particles, has been applied in many different fields, especially in medicine, biology and biophysics, where biologically inert particles can be functionalized and then manipulated using light [47][48][49] . Specifically, small dielectric particles can be picked up and controlled by tightly focused visible light lasers. It is interesting that this qualitative effect can also be understood just with ray optics 50 .
A photon with energy  carries the momentum . k If the photon is adsorbed by an object, a force F on the object due to the transfer of momentum will be produced and given by the formula F=n m P/c, where n m is the refractive index of the surrounding medium, P is the power of light beam and c is the light speed in vacuum. A dimensionless quantity Q et which is used to describe the efficiency of trapping light by a particular object with any shape, is defined by the formula / et m Q n P cF  . The refraction of light by a transparent object will result in the reaction 10 force acting on the object, since the momenta of the photons are changed. For the plane perpendicular to the direction of light beam, an intensity profile with high symmetry will result in a force that will tend to move the object into the center of beam, since the force from the refraction, which points to the center, will provide a restoring force when the position of the object deviates from the center, shown in Figure 4. If the focus of the beam just is above the object, a force will be generated to lift the object up towards the focus.
The concepts of geometrical optics are simple and intuitively appealing, but they are not strictly applicable to the case of particles that have sizes below the wavelength. When one considers the electric field in the near field region, one sees that the force can be separated into two parts (note that both force terms necessarily imply a rate of momentum change of the light field).
One is the force due to the intensity gradient and another one is from the scattering of light. The scattering force from the light beam with the intensity I 0 can be expressed as . The gradient force due to the intensity gradient can be given by the formula 46 where E is the electric field near the particle. Clearly, the force is directed towards the higher intensity region and does not depend on the light propagation direction.
The control of particles by optical trapping can be established when the gradient force exceeds the scattering force. In practice this is readily done for small particles, and precise three dimensional control of the particle position is possible by using optical interference or crossed beams to define a localized maximum in intensity.
An optically trapped particle in a viscose medium (viscosity η) will behave like a damped oscillator and thus the equation of motion will be 50 , 0 This type technique based on gradient forces has been broadly applied in very diverse areas 53 .
These range from disease diagnosis to gravitational detection. With new developments 54 , optical tweezers can be used to detect biological compounds at the single-molecule level 55 . Control of the motion of particles at nanometer scales with piconewton forces enables studies of molecular and 11 nanoscale dynamics, for example the investigations of molecular motors 56 . Furthermore, as mentioned and as shown in Figure 4, nanojets (directionally concentrated electromagnetic radiation) can be formed under large dielectric spheres. These nanojets are technologically important as they can be used to enhance Raman signals. Thus a dielectric particle, which is analogous to a nano-size convex lens, can induce a high light intensity under the particle (the nanojet), and this particle and its nanojet can be controllably moved and used as a nanoscale Raman probe using optical tweezers 57 (Figure 4f).

Localized Plasmons of Metal Particles
It may be noted in the above discussion that the scattering efficiency from the dipole term in Mie theory increases as the particle size decreases in the small size region around the resonance frequency (see the formula 8). This is clearly different from ordinary Rayleigh scattering. Ignoring dissipation, the high-order plasmon modes have resonant frequencies, 22 all the amplitudes, a l tends to go to unity for the corresponding frequency, the scattering cross-sections of high-order plasmon modes can be expressed as  12 In plasmonic materials, the resonant peak of each plasmon mode has a very different linewidth.
Therefore, different plasmon modes can coexist in the same frequency region. Then Fano resonances can arise due to the constructive and destructive interference of plasmon modes with different multipolarity 30 . The resonant interference does not occur in the total optical cross-section, such as the scattering and extinction cross-section for a single particle. It is seen in differential scattering cross-sections, such as forward scattering (fs) and radar back scattering (rbs), with the formulas 25    Figure 6a, where a Fano resonance near the quadrupole resonance frequency is clearly seen.
The interference of incident and re-emitted light in the scattering process generates complex patterns in the near-field region. The energy flow, as represented by the Poynting vector, from the dipole has helicoidally shaped vortices, while that from the quadrupole is still more complex with vortices and singular points 27 (Figure 6b). Higher order modes can also interfere with the broad dipole mode as the size increases. However, it is important to note that the dissipative losses of plasmonic materials must be weak for the Fano resonance to appear, since the higher-order modes are rapidly suppressed when dissipative losses increase.
The Fano resonance of a single spherical particle is generally difficult to observe due to dissipative losses. If the widths and energy positions of plasmon modes can be modulated independently, the condition about the interference between a narrow discrete mode and a broad background resonance is easier to realize. An example is a nonconcentric ring/disk cavity 59,60 . The 13 dipolar modes from disk and ring interact to result in a hybridized bonding mode and a broad higher-energy anti-bonding mode 61  important effect on the intensity in the near-field. The variation of the near-field electromagnetic intensity configurations with particle size is shown away from resonance in Figure 7 a-7d. One may note that the strength of electric field in the near-field does not increase with this parameter and that this is obviously different from the behavior illustrated for dielectric particles in Figure   3d-3f.
The localized surface plasmon resonances of noble metal particles with the sizes of more than 10 nm have been well characterized experimentally 86 . The understanding of plasmon resonances for smaller sizes is, however, still poor. This is because both experiment and theory are challenging for small particle sizes 87,88 . In particular, both quantum effects and detailed surface interactions become important as the electrons interact more strongly with the surface including the spill-over of conduction electrons at the cluster surface, which complicates geometrical analysis 89 .
Quantitative predictions then require detailed calculations of the electronic structure for the actual atomic arrangements of the clusters of interest. For experiment, optical detection in the far-field becomes difficult for small particles due to the size dependent reduction in scattering intensity 1 .
Theoretically, time dependent density functional theory (TDDFT) based methods 35,[90][91][92] are usually limited at present to particles with the sizes below 1-2 nm 93 but still useful insights have emerged.
Methods that bring detailed quantum mechanical calculations to the longer length scales of interest would be very valuable in better understanding the size regime where quantum effects start to become important.
The first effect we mention is the red shift effect in the case of alkali-metal particles, which is due to the finite surface area 94,95 . The red shift is understood in terms of the spill-over effect 96 . At small size, the electronic density profile will extend beyond the nominal surface. This is an effect of the high kinetic energy of the s-electrons that make up the conduction states of alkali metals.
The resulting charge located outside the surface cannot be efficiently screened by the other electrons. So the polarizability is enhanced, which results in a decrease in the resonant frequency.
The effect of electron scattering at the surface may be described via a corrected dissipative loss term in the Drude model with the formula 89 ' where  bulk is the parameter describing bulk dissipative losses, R is the particle radius and υ F is the Fermi velocity. A is an empirical constant that can be set using fits of experimental data. This effect also results in a slight 16 red shift of the resonant frequency.
Next we discuss the blue shift of the plasmon resonance of small non-alkali metal particles.
This can be understood in terms d-electron contribution to the dielectric properties 84  This is an exciting time for nano-photonic applications based on light scattering by particles.
For applications, tuning of properties is important. One avenue for this is through the use of core-shell particles, including the special case of hollow core particles, instead of simple single component particles. For the spherical case, one has two dielectric functions, the core radius and the particle (core+shell) radius as parameters, instead of the single dielectric function and radius as tuning parameters for the single component case. One example of a core-shell particle used in light scattering is the case of metal particles in an aqueous solution. In this case there may be chemical effects at the surface. In particular, the interface between particle and aqueous solution can be viewed as a double layer, and furthermore anodic or cathodic polarization can induce chemical changes due to anion adsorption or desorption, alloy formation, and metal deposition including deposition of a shell with a different composition (e.g. Ag on Pd) 104 . Light scattering in such cases can be dealt with using core-shell models. Core-shell particles can be used to obtain new optical properties that single spherical particles do not exhibit [105][106][107][108] . Furthermore, techniques for producing such particles are well developed 109,110 .
The core-shell model has been studied using the full solution of Mie theory 111 and also can be solved approximately using an electrostatic solution 112 . The surface plasmon resonance condition becomes Re(ε sh ε a +ε m ε b )=0 with ε a = ε co (3-2P ra )+2ε sh P ra and ε b = ε co P ra +ε sh (3-P ra ), where ε co , ε sh and ε m are the dielectric functions of the core, the shell and the medium, respectively 112 . The parameter P ra is the ratio of the shell volume to the total volume of particle. The result is that the plasmon resonance frequency depends on the ratio of the core radius to the total radius of particle.
Core-shell structures also introduce the important concept of plasmon hybridization. This provides a powerful principle for the design of complex metallic nanostructures 113,114 . The plasmon modes of nanoshells (core-shell particles with an empty core, i.e. hollow shells) can be viewed as arising from hybridization of the plasmon modes of a nanoscale sphere and a cavity 114 .
This hybridization results in a low-energy bonding mode and high-energy antibonding mode, as mentioned in relation to the Fano effect. Many non-trivial nanostructures, such as gold nanostars 115 and nanorice 116 have plasmons that can be understood in terms the interaction of the coupled plasmons of simpler systems 117 .
The interparticle distance is another variable that can be used to produce new physics and applications. Examples are quantum tunneling 118 and large electromagnetic enhancements at the junctions 119 . The development of nanoscale fabrication methods has made possible the production of different forms of nanoparticle arrays 66,67,118,120 . These include dimers, chains, clusters, and 18 uniform arrays. The simplest prototype, which can be used as a model, is a nanoparticle dimer. The interaction between localized plasmons and the interference of the electromagnetic fields from these plasmons are the two major factors that control the electromagnetic enhancements at the junctions. Different methods, such as the coupled dipole approximation (CDA) 120 , the finite difference time domain (FDTD) method 121 and plasmon hybridization 122 , have recently been used to understand the plasmonic properties of dimers. For the practical calculation, the temporal couple-mode model as an effective method is also developed 123,124 . Within the framework of the hybridization concept, the dimer plasmons can be treated as bonding and antibonding combinations of the single particle plasmons. The shifts of the plasmons at large interparticle distance then follow the interaction between two classical dipoles, since this is the interaction that leads to the hybridization. At shorter distances, the plasmon shifts in dipolar models become stronger and vary more rapidly with distance. This is a consequence of hybridization (or mixing) coming from higher multipoles 122 . In addition, new interesting effects beyond the hybridization models, such as Young's interference, have been recently observed in the plasmonic structures 125 .
The plasmon modes for the symmetric nanoclusters can be analyzed based on plasmon hybridization with group theory 66  , where  and S are the polarization of a single particle and the structure factor of array, respectively 133 . There will be a geometric resonance when the wavelength of scattering light is commensurate with the periodicity of the particles array 134 . The study on light scattering of uniform arrays of nanoparticles is strongly connected with the fields of photonic crystals and metamaterials. A detailed review was given by Garcia de Abajo 135 , to which we refer the reader for 19 details.
Finally, we note that nonlinear optical responses can be very strongly increased using nanoparticle plasmons. This is by two main mechanisms, namely through the field enhancement near the particle surface and via the sensitivity of resonance frequency to the dielectric function of surrounding medium 136

Spasers: Surface Plasmon Lasing from Active Particles
In lasers, coherent electromagnetic energy is generally concentrated in the active region by a Fabry-Perot or similar resonator. Let us consider an ideal resonator with two perfect mirrors. The minimal distance between two mirrors for storing the electromagnetic field is half of the wavelength, as shown in Figure 8b. However, by using resonance between the electromagnetic field and a surface plasmon (described by a dipole model for a small metal particle), the electromagnetic energy can be concentrated in a much smaller region. The particle size for this can be estimated from the skin depth given by 44 2 1/2 1 where λ is the vacuum 20 wavelength. In the optical region, the size (D s , corresponding to the so-called nano-plasmon) for noble metals, such as silver, gold and copper, is roughly 25 nm 44 . The minimum size of a nano-plasmonic system is determined by the distance electrons move on the surface in a period of the optical wave. This is given by the formula 146 / n l F L  , where υ F is the effective Fermi velocity.
The resonance between a surface plasmon and the electromagnetic field also results in the loss of energy due to the dissipative losses (from the imaginary part of the dielectric function).
This induces a loss of the plasmonic field with a decay rate The key point is that by using a composite of plasmonic metal particles in a gain producing medium one may obtain a coherent radiation field in a lasing system that is smaller than the wavelength of the light.
Stockman used a quantized treatment of a gain medium and a quasi-classical treatment of the surface plasmon to construct a semi-classical dynamical equation for the spaser 148 . This gives a description of the spaser mechanism as shown in Figure 8c and  is Bergman's spectral parameter 149 . One may note that the threshold for gain just depends on the spasing frequency and the dielectric properties of system.
However, the spasing frequency is determined by the geometry of the system.
Progress in spasers, including both theory and experiment, has been rapid 44 . The original theoretical concept of the spaser was proposed with V-shaped metallic structures and semiconductor quantum dots 10 . Following this, a nanolens spaser was proposed with a linear chain structure of metal nanospheres combined with an active medium 150 . A proposal for a spaser based on metal cores with an active shell was considered on basis of linear electrodynamics 151 . A narrow-diversion coherent radiation on based on the combination of a metamaterial and a spaser was proposed by Zheludev et al. 152 . The combination of the plasmon of an anisotropic spherical particle and an active medium was also proposed to result in a spaser 42 . In experiments, a spaser was realized on a conjugate structure based on a metallic core and a dye-doped dielectric shell 38 ( Figure 8). Spasers have also been realized in other nano-structures, such as in CdS nanowires combined with a silver substrate and separated by a MgF 2 layer 153 . The development of active sub-wavelength optical elements such as in spasers, is expected to lead not only to diverse applications, but also to new fundamental insights into non-linear light matter interactions. 22

Conclusions
We have briefly reviewed the theory of light scattering by small spherical particles and aspects of the important progress on light scattering on small spherical particles. It is remarkable that although many of the fundamental aspects of the theory are more than 100 years old, there continue to be new, surprising and useful developments, such as spasers and optical tweezers based on it. The interest 100 years ago was in the far field. While the formalism showed fascinating near field behavior, specifically giant concentrations of electromagnetic energy in regions much smaller than the wavelength and with complex spatial distributions, this was not explored until much more recently. Now these effects are being exploited to yield remarkable new nanoscale effects and potential applications in different science areas, such as high-resolution optical imaging, small-scale sensing techniques, light-activated cancer treatments, enhanced light absorption in photovoltaics and photocatalysis, and numerous biomedical applications. We expect that many more applications will be developed exploiting optical scattering by small particles, and especially nanophotonics applications based on the near field, and far field applications using linear and non-linear plasmonic effects.  Figure 2 Relative scattering cross-sections of a water particle with reflective index n =1.33(a), glass particle with reflective index n =1.97 (b), and ideal particle with very high reflective index n =200 (c) as the functions of dimensionless size parameter q. The first maximum of Q sc happens for the quantity 2q(n-1) ~ 4. The case of refractive index n =200 represents n~∞. Q sc approaches 2a 2 at large q. Near-field distribution of the square of electric field density on a silver particle scattering light λ=496 nm away from resonance with different sizes (a-d), where the dielectric constant is ε= -9.56 + 0.31i (the radius R=2 nm, E max 2 =14 E 0 2 ; R=20 nm, E max 2 =16 E 0 2 ; R=200 nm, E max 2 =17 E 0 2 ; R=500 nm, E max 2 =16 E 0 2 ), dielectric function and near-field distribution of electric field density on a small particle at the resonance condition for gold (e, f) and silver (g, h). In (f), λ=481 nm, the radius of sphere is R=1.6 nm, dielectric constant is ε=-2.0 +4.4i and E max 2 =11 E 0 2 . In (h), λ=354 nm, the radius of sphere is R=1.6 nm, the dielectric constant is ε=-2.0 + 0.28i and E max 2 =457 E 0 2 . 37 Fig. 8 Figure 8 Quality factor of plasmons of gold and silver (a), schematic presentation of the electromagnetic wave in a Fabry-Perot resonator with the minimum possible distance between two mirrors (b), and the spasing mechanism (c-f). Schematic structure of a core-shell with a silver nanoshell on a dielectric core surrounded by the dense nanocrystal quantum dots (c), schematic presentation of the mechanism for energy transfers between e-h pairs and a resonant plasmon model via an exciton level (d), transmission electron microscope image of a Au core in a Au/silica/dye core-shell nanoparticle structure (e), and spaser mode with λ = 525 nm and Q = 14.8; the inner and outer circles represent the core and the shell layers, respectively.