2. OPTICAL EFFECTS (PHENOMENA) The question now arises of whether it is possible to prepare metamaterial or not? To answer this question, it is necessary to deal with a number of complicated optical effects, to separate bold forecasts of a number of scientists from actually observed optical effects. First of all, it is necessary to deal with physical essence of the observed effects, and not with their mathematical description or mathematical modeling. 2.1. Mie scattering theory Mie theory is the theory of scattering (diffraction) of plane electromagnetic wave on the homogeneous sphere of any size. Mie scattering (aerosol scattering) occurs differently from Rayleigh scattering and obeys another regularities. The following is taken for some conventional border separating both types of scattering. If the size of the scattering particles exceeds ~1/10 λ of incident light, the scattering is considered aerosol one. It is also called Mie scattering or scattering on large particles. The name is given in honor of the German scientist Gustav Mie, who was the first to create the harmonious mathematical theory of electromagnetic waves scattering on isotropic spherical particles of any size with different refractive indices in 1908. Mie himself called his work with exceptional modesty: "Contributions to the optics of turbid media, particularly of colloidal metal solutions". It was necessary to explain various coloring which the mentioned solutions obtain in different conditions, especially gold solutions. With time Mie theory has acquired significant importance for atmospheric optics and has essentially developed in the papers of the Soviet and foreign scientists (V. Shuleykin, K. Shifrin, G. Van de Hulst, D. Deyrmendzhan and many others) . However, the foundation for the theory of aerosol scattering has been laid by G. Mie. The destiny of his work is enviable: for more than 100 years the bases of Mie theory are used in many works on scattering theory.
What is the mechanism of aerosol scattering? How does it differ from Rayleigh scattering? The directional diagram of Mie scattering on the particles which sizes are comparable with λ of incident light is sphere-shaped (fig. 6, 7a). Under the influence of electric field of the incident wave, the electric charges in an obstacle (submicron spherical particle) come to oscillating motion. These excited charges radiate secondary electromagnetic waves in all directions (scattering = excitation + re-radiation, fig. 6). In addition to the secondary radiation (re-radiation) the excited elementary charges can transform a part of incident energy to other types, for example, to heat energy; such process is defined as absorption . The polar chart (fig. 7a) shows distribution of scattering intensities in the different directions by an ice particle with a diameter of 0.4 λ. Intensities are specified separately for two polarized components, i1 and i2. Figure 7b, where the scattering chart is given for a water drop with a diameter of about 4 λ, shows the extent of scattering pattern complication with the increase in the size of particles. It follows from Mie theory that, except for cases with significant conductivity or dielectric permittivity, the intensity of scattered light I reaches its maximum both in the direction coinciding with the direction of incident light (0°) and in the opposite direction (180°), and has its minimum at an angle of 90° (fig. 7b, 8) . Representative points of observation when studying the scattering process are the following (fig. 8): forward scattering area (~45°), lateral scattering area (~90°) and backscattering area (~ 180 °); forward beam area is not taken into account. The more the relation of diameter of the particle to λ is, the more energy is scattered forward in the EMR direction. Intensity shift process in the EMR direction (fig. 8) becomes pronounced already for values of d/λ = 0.3 . With further increase in the size of particles, practically all scattered light will propagate in the direction close to 0° (Fraunhofer theory, d> 2 λ). With increase in diameter of particles a number of lateral maxima and minima of intensity of scattered light appears on polar charts. Their emergence is well explained by Huygens-Kirchhoff theory. The location and amplitude of radiations of secondary waves on polar charts (fig. 7b, 8) depend on the sizes and shape of each particle taken separately. With scattering on several particles, these maxima and minima of radiation are averaged. Though the solution proposed by Mie is obtained for diffraction on one sphere, it is also applicable to diffraction on any number of spheres provided that all of them have the identical diameter and identical structure, distributed chaotically and separated by the distances large in comparison with wavelength . 2.2. Fraunhofer diffraction theory. Scattering when d> 2 λ (special case of Mie theory) Let’s consider what occurs when light falls on a particle of micron sizes. When a microparticle (over 1 micron) is captured into variable EMF of incident wave, its each molecule or atom becomes a dipole emitter. Under the influence of incident wave field, the microparticle is polarized. It is exposed not only to the incident wave field, but also to numerous fields of the elements composing the microparticle. Molecules and atoms of microparticle are "packed" densely, i. e. they are in close proximity from each other, and they cannot be considered as independent emitters of scattered light as it was accepted for Rayleigh scattering. It is necessary to consider interference of waves scattered by separate emitters, meaning that the light scattered by each molecule differs by phase, polarization and emergence location . The polar charts (fig. 9) show angular dependence of distribution of scattered light intensity. The chart gives the following information: • numbers on external border of the chart signify scattering angles; • distance between the center of the chart and color distribution curve shows intensity of light scattered in this direction; • radial intensity axis (logarithmic scale) and concentric circles show that intensity changes by 10 times upon transition from one circle to the other one. For example, the light wave 650 nm long (red light) falls on a water drop with a diameter of 20 µm (fig. 9a). The drop of such diameter will "keep within" about 30 λ of red light . Therefore a lot of secondary waves of scattered light will be excited as a result of scattering on a microparticle. The amplitudes of these waves depend on the size of the scattering particle. When light is scattered on a gold particle with a diameter of 1.5 µm (fig. 9b, red curve), the intensity for small scattering angles, i. e. from 0° to ~ 15°, is approximately 100 times more than in the opposite direction. This difference is much less for blue curve (particle diameter is 0.5 µm) . Fraunhofer diffraction (static scattering of laser light, laser diffraction, laser diffractometry) is used for determination of the size of particles by measuring angular dependence of scattering intensity (fig. 10). The term "static scattering" has something in common with the term "diffuse reflection". The falling beam is reflected at several angles as a result of diffuse reflection (fig. 6), and not at one angle as with specular reflection. Diffuse reflection is observed, when the surface irregularities have wavelength order (or exceed it) and are located randomly. If particle diameter is more than λ of incident light (particles up to several microns in size, fig. 10a, 11-Fraunhofer), then diffraction process occurs primarily. If the size of particles is the same or less than λ of incident light, much more light is scattered on the particles at large angles (fig. 10b) and reflected back (fig. 9b, Au 0.5 µm). The sizes of particles are determined by laser diffraction method measuring intensity of propagation in forward direction for small angles (<35°). Interpretation of light scattering model of according to Mie theory is applied to a whole range of the sizes of particles, including Rayleigh scattering theory and Fraunhofer diffraction as special cases. If the size of particles in a sample exceeds λ of light, Fraunhofer theory dominates. In order to understand diffraction, imagine a beam of light as wide wave front hitting the particle and partially surrounding it, similar to water wave hitting rather big obstacle. When superposition of different parts of the broken wave front (interference) occurs behind the particle, the characteristic diffraction pattern (fig. 11, Fraunhofer) described by Fraunhofer theory and depending on diameter of particles (the more densely diffraction rings are located, the more particle is, and vice versa) is revealed . Along with diffraction, light dispersion is observed when natural light come across submicron particle. Red and orange colors are in the center, light-blue and blue colors are on diffraction rings (fig. 11, Fraunhofer). The shown illustration is the image of light scattering intensity for spherical particle which can be accurately described by means of Bessel function . Bessel functions are applied when solving a number of wave propagation tasks. In the final Mie theory, light scattered by particle is represented as the sum of infinite slowly converging series. Each addend of the series represents a complex function. The lesser particle size is, the lesser addends can be considered for series summing . If the particle size becomes less than 1/10 λ, Mie theory transits into Rayleigh theory (Rayleigh scattering), if more than 2 λ – into Fraunhofer scattering theory (fig. 11). If the sphere diameter is very large in comparison with λ (d >> λ), laws of geometrical optics are applied, while the most part of incident light is reflected. 2.3. Surface Plasmon Resonance, SPR When EMR interacts with Me nanoparticles, mobile conduction electrons of particles are displaced relative to positively charged grid Me ions. This shift has a collective nature: the movement of electrons is coordinated by phase. If the size of particle is much less than λ of incident light, movement of electrons leads to dipole emergence. The result is the force aiming to return electrons in balance position. The value of the returning force is proportional to displacement value, similar to typical oscillator; therefore one may speak about availability of natural frequency of collective oscillations of electrons in a particle. If oscillation frequency of incident light coincides with the natural oscillation frequency of free electrons near Me particle surface, the sharp increase in oscillation amplitude of "electronic plasma", quantum analog of which is plasmon, will be observed. This phenomenon was called Surface Plasmon Resonance, SPR . SPR is the effect caused by collective oscillations of conduction electrons on ME nanoparticle surface (Ag, Au, Cu), and consequently by EMF oscillations (fig. 12). SPR is followed by significant light absorption in visible spectrum. Wavelength, where absorption peak maximum is observed, depends on nanoparticle metal, its size and shape [29–32]. When Me particle is too large (submicron), and also in case of radiation by light of Me surface, SPR is not observed. EMR energy can: • be reflected (like from mirror); • be reradiated (for example, IR-radiation from black Me surface); • be absorbed in Me array (the same black surface). 2.4. Surface Plasmon Polaritons, SPPs Research of SPPs has begun in connection with radiowave propagation research. The concept of "surface electromagnetic waves" (SEW) was introduced by A. Sommerfeld when in 1899 he considered a task about axial current in long straight wire and obtained the solutions of Maxwell equations from which it follows that amplitude of electromagnetic oscillations quickly attenuates upon moving away from wire surface. He interpreted these solutions as the proof of SEW existence. Experimental demonstration of SEW on the border with metal was given by R. Wood in 1912 with scattering of electrons in thin Me foil. At that time the phenomenon has not been understood and remained known as "Wood anomaly" up to 60th. After A. Sommerfeld, the German theorist V. Kohn established that the flat interface of dielectric and good conductor made directing impact on propagation of bulk wave and that SEW was possible on flat interface of media with insignificant intensity losses. SEW interpretation in terms of surface plasmon polaritons was given by Ugo Fano [33, 44]. Let’s define SPP. SPP is the compound particle emerging when EMR interacts with elementary media excitations, the interaction, leading to their coupling, becomes especially strong when frequencies ω and wave vectors k coincide (resonance). The coupling waves (polaritons) possessing characteristic law of dispersion ω (k) are formed in this area. The polariton consists partially of EMR energy and energy of own medium excitation. Let’s consider what occurs when light passes through thin (~ 50 nm) Me foil (fig. 13). When EMR hits Me surface, the local excitation of conduction Me electrons occurs, in addition to classical reflection; excited electrons, in turn, creating spherically divergent secondary EMR with the same wavelength (according to Huygens). EMR is attenuated due to interference almost in all directions, except one direction, namely along media interface. That is what SPPs are. At the expense of the Me small thickness, secondary plasmon polaritons occur on foil surface opposite to light source (perturbation of electronic density on the foil opposite side causes emergence of secondary SPPs, energy is transmitted by means of EMF through conduction electrons, fig. 13a). This results in the original signal recovery on the opposite side of foil (similar to secular bird). The signal is weakened, of course. This signal extends further, beyond the foil. Human eye perceives this effect as partial passing of light through foil though actually the signal was almost completely absorbed, transferred by means of EMF and was newly generated on the opposite side of the obstacle (foil). It is possible to formulate what the SPPs are as follows: SPPs are a kind of SEW and represent the complex of heterogeneous p-polarized wave and wave of induced free charges propagating along the conductive surface . Field intensity of SPPs is the most at the media interface and exponentially decreases with moving away from it. The principle of light passing through obstacle due to SPPs effect is the same, as when EMF passes through Me ungrounded screen (EMF absorption – orientation of electrons in Me – repeated EMF generation). 2.5. Negative relative dielectric permittivity of metal Metal refractive index is less than one. Authors of articles about metamaterials [1–8, 19] operate with such concepts as relative dielectric permittivity (RDP), refractive index and magnetic conductivity, and all these values allegedly can accept negative values or to be <1. How it can be explained from the physical point of view? We shall note at once that for metals themselves RDP concept does not exist, it is only applicable to dielectrics. EMR in Me does not extend, it attenuates. However, there are some exceptions. Energy in the form of SPPs can propagate along Me-dielectric interface or in the form of EMF in Me. It is necessary to consider, what EMF we are talking about. If constant one, then Me are excluded, and if variable one, then Me acquires RDP (RDP decreases with high frequencies). When visible light falls on Me surface, variable EMF is formed in thin near-surface layer (at the expense of the frequency inherent to EMR) (EMR itself can be considered as special case of variable EMF). This is the case that we are interested in. It is important, whether Me foil is grounded (we interested in the thin Me layer not exceeding 30–50 nm). If it is grounded, all energy will go to the earth, RDP will be incredibly high (more than 1010), and, therefore, considerations of the RDP value will lose meaning. If the foil is not grounded, energy (in the form of variable EMF of light wave) will pass through foil, and we can record Me RDP. Relative dielectric permittivity ε is closely connected with refractive index n. , where µ is magnetic conductivity. RDP contains imaginary component () in the absorbing media, therefore refractive index
becomes complex . In the field of optical frequencies where µ = 1, the real part of refractive index describes the refraction itself
and imaginary part describes absorption. For massive Ag when λ = 589.3 nm = 0.20 + 3.44 i. Me refractive index including real part < 1 and imaginary part can be interpreted as exponential-attenuating wave incapable to pass through Me (incapable directly, but capable to pass by means of EMR – EMF – EMR transformation, chapter 2.4). Ag imaginary part is minimum in comparison with other Me (for example, for Au = 0.188 + 5.39 i ), therefore Ag is the most perspective material from the point of view of metamaterial creation. Let’s compare passing of light through dielectric (fig. 2a), Me foil (fig. 13) and through Me-dielectric laminated structure (fig. 14). In dielectric the wave refracts deviating towards normal to the surface in direct proportion to the magnitude of dielectric constant (fig. 14a). Meanwhile, SPPs are emerging in Me (fig. 14b), energy is directed perpendicular to Me surface what can be formally described, within mathematical model, as negative RDP. 2.6. Evanescent waves (evanescent field) Evanescent waves (from lat. evanescentis "disappearing, ephemeral") are attenuating waves arising near interface of two dielectrics and moving along the interface (fig. 15). Features of evanescent waves (distinctions from SPPs) are as follows: • they propagate near plane interface in one of the media (less dense); • they arise on border of two dielectrics; • angle θ at which light falls on the interface must exceed or be equal to the angle of total internal reflection (fig. 15). Modulation of intensity profile of standing evanescent wave with the period of 239.2 nm along prism surface was recorded in experiment . The exciting argon laser generates radiation at wavelength (in vacuum) of 514.5 nm. Efficiency of collecting photons of evanescent field with dielectric tip of SOTM is 63% that corresponds to the tip effective diameter of 80 nm. The attenuation distance along z axis (height of evanescent wave) is 103.9 nm. The beam of light falling from the optically denser medium (glass prism) on plane interface with optically less dense medium (air) cannot leave the glass and is completely reflected from the interface (fig. 15). Nevertheless EMF in optically less dense medium is other than zero, though it exponentially decreases when moving away from the interface. It is not a real free electromagnetic wave since this field cannot exist on its own without the interface. However, it possesses all properties of traveling surface wave, i. e. electromagnetic wave propagating along the surface and attenuating when moving away from it. This is the evanescent wave that always accompanies processes of total internal reflection. Its amplitude is proportional, and its phase coincides with amplitude and phase of real electromagnetic wave propagating in prism [50, 51]. 2.7. Overcoming the diffraction limit using SOTM as an example The investigated sample is located on prism surface in the scope of evanescent field (fig. 16). Thin optical fiber is brought directly to the scanned surface (at a distance smaller than λ) in the area of evanescent field. The evanescent field cannot move away from the interface, however if there is another optically dense medium nearby, evanescent field will "jump" into it, having turned into normal traveling wave. This "jumping" is no other than tunneling of light radiation from optic element into scanning fiber. Evanescent field is converted into propagating mode of optical fiber in the probe and is further sent to a detector. Thanks to small sizes of needle tip (about 50–100 nm), it is possible to take measurements with submicron accuracy (SOTM resolution capacity is about 100 nm). 2.8. Frustrated Total Internal Reflection, FTIR Let’s consider EMR passing through two plane parallel plates of the same transparent material A (let’s call them A1 and A2) separated by a layer of other transparent material, for instance, air. Since light speed in air is almost equal to its speed in vacuum, total internal reflection is possible upon transition from A to air. If width of air layer considerably exceeds λ, the presence of plate A2 will change nothing. Light will attenuate exponentially at the exit from A1 as before. If width of air layer is less than λ, the situation will change. In this case the wave leaving A1 enters into plate A2 and "revives" there in the form of normal, but not attenuating wave, only with smaller amplitude . The probability of quantum tunneling of particle is always less than one. Similarly, amplitude of light wave which has reached A2 is inferior to its amplitude in A1. However I. Hooper and his colleagues  have shown in the theory and have confirmed in experiment that in principle it is possible to achieve 100% of revival of initial wave. For this purpose it is necessary to separate A1 and A2 not just with air layer, but with several layers: first, thin film of other transparent material B1, then air layer, then same film B2, and, at last, A2 plate. Furthermore, both films and air layer should be much thinner than the length of light wave. Calculations show that with correct selection of incidence angle of initial beam and other parameters of experiment it is possible to realize an ideal and to achieve that light with unchanged amplitude will come out of plate A2. It is due to the fact that B1 and B2 work as the optical resonators amplifying light passing through them. Similar effects in physics of semiconductor heterostructures have been described in the 70th as resonant tunneling (the largest contribution to their research was made by Nobel laureate in 1973 Leo Esaki), but they have not been observed in optics so far (at least as claimed by Hooper and his coauthors). In experiment of English physicists  polarized laser beam with the wavelength of 700 nm was leaving quartz prism, crossing film of transparent zinc sulfide 209 nm thick, 131 nm air gap, another same zinc sulfide film and was captured in the second quartz prism. However, the experimenters did not manage to achieve 100% of passing of radiation which was slightly absorbed in zinc sulfide. However, the real level of light transmission was rather high, about 85%. If the gap between quartz prisms had been filled only with air alone, transmission degree would not exceed 30%. To be continued