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SECOND HARMONIC GENERATION IN METAMATERIALS Kasumova Rena J. Baku State University, Physics Department 2012 CONTENTS The theme of my speech is SHG in metamaterials. There will be considered the following issues CONSTRUCTIONs of Metamaterial Experimental confirmation of possibility of creatiNG metamaterialS Well-known optical conseptions in metamaterials Phase effects AT Shg in a metamaterial Current State of THE Problem and prospects of development Forty five years ago prof. of MPhTI V.G.Veselago put forward first the hypothesis on possibility of existence of materials with the negative refractive index. In them the •light waves should move in direction against ray spreading and amazingly behave. As to •lenses from these materials, they must possess the unique properties and record characteristics. In like materials •the laws and notions of linear and nonlinear optics on rays behavior in a medium get broken. However, refraction index of the substances known to that time was positive. For several years of intense searches Veselago had found no material with appropriate electromagnetic properties and his idea was well forgotten. They recollected it only in the beginning of the XXI c.! Only due to the recent achievements in the field of material study, namely technology and preparation of materials, Veselago’s idea has been revived. In 1997 the scientists from the Centre of Technology of Materials af. Marconi in England have developed metamaterials which consist of macroscopic elements and disperse electromagnetic waves not in a way, any known substances do. • In perspective, the investigations on elaborations of metamaterials refracting visible light are being carried out. The practical interest to metamaterials is connected by broad applications which may be realized with their development. List two of them : •creation of invisible objects and •optical lenses with improved resolving ability. •Veselago V.G., Usp. Fiz. Nauk, 1967, v. 92, p. 517 Let’s first to speak about a nature of metamaterials. To understand how the negative refraction emerges, we’ll consider the mechanism of interaction of electromagnetic radiation with a substance. The field of electromagnetic waves, that follows from their appellation, includes both electric and magnetic components. Electrons in material move forward and back under action of electric field and on a circular orbit under the influence of magnetic one. The degree of interaction is determined by two characteristics of a substance: dielectric permittivity and magnetic permeability . The first shows the degree of electrons reaction to electric field, the second –the degree of reaction to magnetic one. The values of and of overwhelming majority of natural materials are more than zero. As a case of metamaterials shows, electromagnetic response of material may be “constructed”, by creating the tiny periodical structures. For instance, in split ring resonator magnetic flow passing through metallic ring induces in it circular current in analogy with currents connected with magnetism of some materials. As is known, optical properties of a substance are characterized by the refractive index n , which is related with dielectric permittivity and magnetic permeability by a simple ratio: n = ± √(ε∙μ). For all known materials, the sign “+” must be put before square root, and therefore their refractive index is positive. However, in 1968 Veselago showed that substances with negative and have refractive index that must be less than zero. Negative or are received in a case, when electrons in material move in direction opposite to the forces created by the electric and magnetic fields. Though such behavior n seems to be paradoxical, to compel electrons to move against the forces of electric and magnetic fields is realizable. Let’s explain it on example of the pendulum. If to push the pendulum by hand, it will begin to swing with so called resonance frequency. Pushing slightly the pendulum (in time with swinging) it is possible to increase an amplitude of swings. If to push it with higher frequency, then pushes will cease to coincide with vibrations (by phase) and in some point the pendulum will strike a hand when moving to meet it. Just as electrons in material with negative refraction index resist “the pushes” of the electromagnetic field. Making Metamaterials Metamaterials represent composition material with artificially created periodical structures. Metamaterials are synthesized by introduction to the initial natural material, of the different periodical structures with most various geometric figures, which change dielectric permittivity and magnetic permeabilities of the initial material. — In very rough approximation such introductions may be considered as artificially introduced atoms of extremely greater dimensions, i.e. macroatoms, into initial material. If the properties of material are determined by atoms and molecules forming it, properties of metamaterial are defined by macroatoms. The creator of metamaterials at their synthesizing has an opportunity of selecting the various free parameters (size of structures, forms, constant and variable period between them and etc.). There is possibility of checking light like electric current in semiconductors. Meta-atom Natural materials Metamaterials microwave Atoms optics Meta-atoms If in a natural material we deal with usual atoms, then in metamaterials macroatoms, representing periodical structures, substitute them. The negative reaction is artificially created in metamaterial by means of tiny resonance contours imitating response of a substance to the magnetic or electric field. For instance, in split ring resonator magnetic flow passing through metallic ring induces in it circular current in analogy with currents connected with magnetism of some materials. And in the lattice from the straight metallic rods electric field creates currents directed along them. In optics, in contrast to the microwave diapason, periodical structures become more simple, if in VHF there are some spirals, then in optics there are fishnet or two little sticks. Litchinitser N.M., University at Buffalo, The State University of N-Y, USA Free electrons in such outlines fluctuate with resonance frequency depending on a form and dimensions of a conductor. If there is applied field with frequency -lower than that of resonance, normal positive reaction will be observed. -However, with increasing frequency of electromagnetic wave the response becomes negative, just as in case of the pendulum moving to meet, if to push it with frequency higher than resonance one. Thus, periodical structures in some range of frequencies may react to the electric and magnetic fields as medium with negative , and the rings with splits can imitate material with negative . These conductors and rings with splits are just elementary blocks required for creation of a broad assortment of metamaterials including those which Veselago was searching. Metamaterial each element of which being less or as wave length, may be considered as an entire medium, and its structures –as metamolecules. Elyutin S.O., Maimistov А.I., and Gabitov I.R.Zh.Eksp. Teor. Fiz., 2010, v.138, p.175 Constructions of Metamaterial The scientific groups have used the different constructions of periodical structures, for instance: n<0 C-shaped nanostructures Karlsruhe group Gold nanorods, Purdue group n<0 ≠1 Fishnet structure Karlsruhe group Well-known optical conseptions in metamaterials n >0 propagating waves evanescent waves isotropic dielectrics 1 n <0 propagating waves 1 evanescent waves Now, let’s speak on the famous notions of linear and nonlinear optics. On this slide the system of coordinates, on the axes of which the values of dielectric permittivity and magnetic permeability are registered, is given. In the first quadrant there will be arranged majority of isotropic dielectrics, dielectric permittivity and -magnetic permeability of which are positive. In the second quadrant, where dielectric permittivity is less than zero, and magnetic permeability is more than zero, there will be plasm, both gaseous and plasm of solids. In the third quadrant there are arranged materials with negative refractive index, i.e. metamaterials. Note, that if waves in materials of the first and third quadrant are stable, in materials of the second and fourth quadrants waves very soon fade as for as spreading in material. 1) vector k and Poynting vector S Let’s present the components of electromagnetic wave, propagating in material in a form E(r, t)=E0 (r) exp[-i(ωt-k·r)]+c.c., H(r, t)=H0(r) exp[-i(ωt-k·r)]+c.c., k E =(ω/c)µH, k H=-(ω/c) E, (1) (2) (3) E(r, t) =- H(r, t), (4) Here c2 c S(r, t) = [E H]= [H k H]= 4 4 c 2k 2 c2k 2 E H =. = 4 4 (5) S , it follows that wave vector k is directed opposite to energy flow vector, i.e. Poynting vector S . At spreading wave flow in metamaterial from the expression for As in natural materials the vector of electric and magnetic fields form the right –handed system of coordinates with wave vector, it is accepted to call these materials right–handed materials, and in metamaterials they form the left –handed systems, such composite materials are called left –handed materials. Natural materials PIM (positive index material) Right-Handed if 0, 0 Metamaterials NIM (negative index material) Left-Handed if 0, 0 k S H k E H S E In Nims: Phase velocity is opposite to Poynting vector Veselago V.G., Usp. Fiz. Nauk, 1967, v. 92, p. 517 Interface between metamaterial - dielectric S H k E Positive Negative Index Index Material Material n0 0 0 n0 0 0 k H S E On this slide the boundary of dielectric and metamaterial and corresponding three vectors are shown. Taking into account such behaviour of vectors, let’s consider the law of refraction. 2) Refraction law , At passing the boundary of metamaterial with an usual dielectric both falling and refracting beams are along one side from normal to surface of section of two media (see b), what differs from usually case (see a) of division of natural media, when falling and refracted beams lie along different sides from normal to surface of division. k1 a kx S kx b k2 Paths of rays upon transmission through the interference between two media: a) positive and b) negative refraction; Sin, Sref and Str are the Poynting vectors; vin, vref and vtr are the phase velocities; and Ein, Eref and Etr are the amplitudes of the incident, reflected, and transmitted waves, respectively. Maimistov A.I. and Gabitov I.R. Bull. of the Rus. Acad. of Sc.; Physics, 2008, v.72, no. 5, pp. 704. 3) Paths of rays in metamaterials And, finally, let’s consider behaviour of rays in metamaterial. Following told above, we observe the following picture of behaviour of rays in a metamaterial. Flat–parallel plate from metamaterial works as focusing lens. The red dot depicts the source of light. To retain Snell’s law invariable we ought to think that relative refractive index is negative. sin 1 n2 sin 2 n1 Negative refraction is connected with that wave in such medium is reverse wave for which wave vector is directed antiparallel to the Poynting vector. Nonlinear Optics and NIMs Thus, we have considered how the basic laws of optics are breaking. What does take place at harmonic generation? At SHG wave vector and the Poynting vector are arranged in a following way: at pump frequency these two vectors are directed opposite, at frequency of SH the medium is positive and vectors are directed to one and the same direction.. In case of parametric amplification in quadratic medium at three wave interaction the picture of vectors has an appearance as presented on Fig., and in cubic medium four wave interaction is observed and there takes place the following direction of vectors. Second-Harmonic Generation Optical Parametric Amplification (2) (3) NIM NI at 1 NIM at 1 MM at PI PI 2 PIM at 2 , 3 PIM at 2 , 3 , 4 M (a) NIM at 1 k1 S1 k1 k1 S1 k 2 S2 k2 S 2 k3 S 3 k3 S 3 k4 S4 Popov, Shalaev LPL (2006), APB (2006) Shadrivov, JOSA B (2006) Agranovich, PRB (2004) S1 k2 S2 Popov et al., OL (2006), APB (2006), OL (2007) Experimental confirmation of possibility of creating metamaterialS Now, we’ll speak on experimental receiving metamaterials. The existence of media possessing the negative refraction in optical (more precisely, in IR) diapason was experimentally demonstrated. The materials with the negative refraction existing at present have high losses, however, recently there have been achieved an essential improvement of quality of these materials. Just this gives hope of receiving the transparent materials with negative refraction in future calls forth an expediency of carrying out the researches of their optical nonlinear properties. The first experimental confirmation of possibility to create material with negative refractive index was received in 2000 in California University in San Diego (UCSD) for microwave diapason. As elementary tiny bricks of metamaterial are to be much less than wavelength the researchers wave working with radiation of centimetre diapason and used the elements of size in some millimetre.(Similarly quantum electronics where at first was realized quantum generator - maser, and then was developed the laser) . • The exper. confirmation was obtained for this cube. The cube of metamaterial present three–dimensional matrix formed by copper conductors and rings with splits. Microwaves (mm, cm, dcm-diapason) with frequencies about 10GHz behave themselves in such cube unusually, because for them the cube has the negative refractive index. Lattice step is 2,68mm or about 0,1inch (depiction from the site www.sciam.ru) SuperlEns The next confirmation. Veselago used plotting the course of rays to predict that flat plate from a material with negative refractive index n 1 has to function as a lens with the unique properties. Most of us are acquainted with the lenses from the materials with positive refraction in cameras, magnifying glasses, microscopes and telescopes. They have focus distance and place, where depiction is formed, depends on combination of focal distance and the distance between an object and lens. The depictions usually differ by dimension from the object, and work best of all for the objects laying on the axis, passing through lens (on edges of lens distortions of depiction are observed). Veselago’s lens works absolutely otherwise than usual ones: its work is by far simple, Veselago’s lens operates only on the objects arranged near it, and Veselago’s lens transfers the whole optical field from one side of lens to another. All our microscopes and magnifying glasses have restrictions on solution. Not to see in any microscope the objects lesser than wavelength. 0,5 micron –is a green light, blue is 0,4. Lesser than 0,4 micron you won’t merely see in microscope. In order to be able to see more tiny objects it is necessary to settle electronic microscopes, to infringe nature, destroy an object which we want to examine. But such lens permits to reproduce precisely the depiction. We have not only flat lens, but it also gives hundred percent reproduction. On this slide Veselago’s lens work is shown. The right-angled bar from material with negative refractive index forms superlens. Light (yellow lines) from an object refracts (on the left) on a surface of lens and again meets, forming turning over depiction inside the bar. Going out from it, light refracts once more and creates the second depiction (on the right). For some metamaterials depiction contains the details more small than wavelength of used light, what is impossible for lenses with positive refraction. Optical elements with positive refractive index are restricted by diffraction limit –they can permit the details, size of which is equal or more than light wave length, reflected from the object. “In World of Science” However, in 2004 Anthony Grbic and George V.Eleftheriades from Toronto University experimentally showed that metamaterial, constructed so in order to have 1, and 1 in diapason of radio frequencies (kHz-hundreds of MHzcm, dc, mm), may really allow the objects on a scale lesser than diffraction limit. It should be noted that difficulty of creating metamaterials in a field of optical lengths of waves has two aspects. Above all, metallic conducting elements forming micro schemes of metamaterial of conductor and rings with split type are to be lessened to the scale of nanometers to make them smaller than wavelength of visible light (400-700nm). Secondly, short wavelengths correspond to higher frequencies, while metals at such frequencies possess the worst conductivity, thus suppressing resonances, on which metamaterials properties are grounded. Despite difficulty of making of metamaterials refracting visible light: In 2005 Costas Soukoulis from IOWa St. University and Martin Wegener from Karlsrue University in Germany experimentally demonstrated that it was possible to make rings with splits which worked at wavelength of 1,5mkm=1500nm in all. On this picture experimentally realized superlens is depicted. Superlens suggested by the authors presents a plate of silver Ag, separated from depicted object (small nanowires from chromium Cr) by layer of polymethyemetacrylate (PMMA) of thickness equal to plate thickness (~40nm). Over silver plate is covered by the layer of photoresist (PR), in which depiction is registered. Silver plate functions as material strengthening damping modes of high spatial harmonics on the account of resonance excitement of surface plasmons. In spite of modesty of the given value, note, that it is nevertheless, more than length of damping which for the dimension of details in 60nm, made up 11nm. Resolution of depiction in 60nm at wavelength of UV radiation of 365nm has been achieved. • Superlens, flatparalllel plate Ag of thickness in 40nm, creates depiction of nanowires of chromium (diameter 60nm) in a layer of photoresist (PR). Fang N., Lee H., Sun C, Zhang X, 2005, Science, v.308, p.534 As well here three photos are presented. Thin layer of silver at very small distances functions as superlens. Here images of the word NANO are received by means of focused beam of ions (on the left), by optical way without superlens (in the centre) and optically, but with silver layer of 35nm thickness (on the right). Dimension of scale section is 2000nm long. Superlens gives the best resolution than wavelength of used light of 365nm. Following told above, we can see that refractive index for one and the same medium may be positive in one spectral field and negative in the other one. Therefore, features of negative refraction may become apparent at interaction of wave packages, bearing frequencies of which lie in the different spectral fields, in which refractive index has the various signs. From this point of view such a famous nonlinear effect as generation of harmonic is an example of unusual interaction of electromagnetic pump wave with the metamaterial. In the present work the results of studies of some processes, where the negative refraction leads to the new effects lacking in usual media, are reported. An efficiency of parametric processes depends on condition of phase match –equality of phase velocities of interacting waves. If one among waves has bearing frequency in a region of negative refraction, energy flow corresponding it will be directed to the side opposite the flows of other waves. On case of SHG it is shown that pump and harmonic amplitudes damp with past distance by waves, so nonlinear medium functions as distributed mirror. “In world of science” Let’s note two important features of metamaterials. First, interaction of electromagnetic waves with medium with negative refractive index differs from usually observed picture of interaction. Interaction of electromagnetic wave with a substance is determined by response of medium to electric and magnetic components of wave. This reaction of the substance depends on material constant media characterizing its electric and magnetic properties, i.e. dielectric permittivity and magnetic permeability . If in natural materials coefficient of magnetic permeability was taken equal to 1 (for majority of the transparent substances in optical range, it practically, does not differ from 1), then, in metamaterials plays the same important and determining role, as does. Therefore, propagation, refraction or reflection of electromagnetic waves in metamaterials is accompanied by another character of interaction, what takes place in natural materials. Secondly, the unique properties of metamaterials become apparent in certain frequency range. From this point of view, such widely known effects as generation of harmonics is a striking example of non traditional result of interaction of electromagnetic wave of pumping with the given artificial medium. As show researches absorption is one of the most great problems which is observed at practical application of NIMs. Owing to SHG transfer of depiction of near IR diapason to the area of visible diapason (VD), where absorption is, usually, much smaller, was suggested as a means for overcoming dissipating losses and thus, elaborating the superlenses. Tagiev Z.H., Kasumova R.J., Gadjieva L.S. J. Rus. Las. Research. 2011, v. 32, 152. I. Phase Effects at Second-harmonic generation in a metamaterial USED APPROXIMATION Several words about the applied by us approximation for theoretical study of nonlinear optical processes in metamaterials, i.e. about the constant -intensity approximation (CIA). In conditions of phase-matched and phase-mismatched harmonics generation in metamaterials was investigated in a series of articles, mainly in constant-field approximation (CFA). In this approximation both the real amplitude and the phase of the fundamental wave (FW) are considered the constant. The CFA describes rightly only the initial stage of nonlinear interaction of waves when it is possible to ignore the reverse reaction of generated waves to intensive pumping wave at all. At such approach information on a series of qualitatively important features of the nonlinear process is lost. The CIA of FW in contrast to the CFA does not impose any restrictions on phase of interacting waves. This allows to make more strict analysis of nonlinear interaction of waves in a material with account of the change of the phases of all interacting waves. The given approximation partially takes into regard the reverse effect of excited wave on exciting wave. Tagiev Z.H., Kasumova R.J., Salmanova R.A., Kerimova N.V.. Opt. B: Quantum Semiclas. Opt. v. 3, (2001), 84. We'll consider supposing that a metamaterial medium has simultaneously the negative values of dielectric permittivity and magnetic permeability at pump frequency 1 (1 0, 1 0) and the positive values of dielectric permittivity and magnetic permeability at harmonic frequency 2 21 ( 2 0, 2 0 ). We think, that radiation flow of pump S1 falls normally on the left side surface of a metamaterial of length l and spreads along the positive direction of z axis. It is known that under these conditions in a metamaterial (the following picture for vectors takes place) wave vectors of pump wave and harmonics k1,2 , respectively, as well vector of energy flow wave of harmonic S2 are identically directed along the negative axis z . These three vectors are opposite to Poynting vector S1 . In this case these four vectors are arranged in a following way: NIMs PIMs S1 k1 k2 S2 k1 S1 k2 S2 We'll follow the method applied in [Shadrivov I.V. et al], supposing that nonlinear response of medium is, mainly, bound with magnetic constituent of waves. In this case the process of SHG in a metamaterial is described by reduced equations of a form dA1 1 A1 dz 8112 i k1c 2 (6) ( 2) eff A1 A2 exp( iz ) dA2 4 222 ( 2) 2 2 A2 i eff A1 exp( iz ) 2 dz k2c Here A1,2 are complex amplitudes of pump and second harmonic wave at frequencies 1,2 correspondingly, 1,2 are the wave absorption coefficients in metamaterial at frequencies 1,2 respectively, =k2-2k1 is phase mismatch between interacting waves, k1,2 (k1,2>0) are modulus's of wave vectors k1,2 directed towards opposite to z axis, (2)eff is an efficient quadratic susceptibility of material. Analogously, we can write the system of reduced equations for electric components of the waves, in view of the corresponding substitution of dielectric permittivity 1,2 by the magnetic permeability 1,2 and vice versa. Taking into account to the different signs dielectric permittivity 1,2 in the metamaterial, from Eq. (1) we obtain dA1 1 A1 i 1 A1 A2 exp( iz ) dz dA2 2 A2 i 2 A12 exp( iz ) dz (7) where the following notation is introduced 22 (2) 1 12 (2) 112 (2) 4 1 8 2 eff 8 2 eff 2 2 eff , , k2c k1c k1c 2 with 1,2 are the nonlinear coupling coefficients of interacting waves in a metamaterial at frequencies 1,2 respectively. When the pump wave is propagating in the metamaterial along the z axis, it follows from the expression for S1 that wave vector k1 directs towards the side opposite to vector of energy flow S1 , i.e. opposite to z axis. Under the conditions of phase matching, harmonic wave generated in nonlinear medium has a wave vector k2, coinciding with k1 in direction. Since 2 >0, 2 >0, both vectors k2 and S2 coincide in their direction and go in the opposite direction with respect to the z axis. Therefore, in the case where the negative values of dielectric permittivity and magnetic permeability at pump frequency 1 and the positive values of dielectric permittivity and magnetic permeability are considered at the harmonic frequency 2=21, we employ the system (6) under the following boundary conditions A1 ( z 0) A10 exp( i10 ), A2 ( z l ) 0 (8) Here z=0 corresponds to the entrance in metamaterials, 10 is an initial phase of pump wave at the entry to nonlinear medium. Solving the system (7) in the constant-intensity approximation of basic radiation (I1(z)=I1(z=0)=I10) with account of boundary conditions (8), for complex amplitude of harmonic wave on length z of nonlinear medium we receive (1,2=0) A2 ( z ) 2 i 2 A10 (sin z tg l cos z ) exp( i 210 iz / 2) i tg l 2 2 where 2 22 , 4 2 1 2 I10, , (9) I j Aj Aj The given analytical expression received for frequency conversion in the metamaterial differs from the analogous expression in case of conversion in ordinary material. The difference lies in the additional members in numerator and denominator, proportional to parameter tg’l what is explained by boundary conditions in metamaterial (8). As a result of that dynamic process of frequency conversion in the negative index materials depends directly on full length of metamaterial l, i.e. on extending in the nonlinear medium in its any point the process of conversion depends on the value l. From the received expression it is seen that harmonic wave amplitude depends on factor, taking into regard reverse reaction of excited harmonic wave on pump wave (1 /=0). This factor influences on harmonic wave phase. In addition, from (4) it follows that in contrast to the result in the constant-field approximation wave harmonic phase A2(z) depends on intensity of pump wave. At 1=0 from (9) there is obtained the result in the absence of pump exhaustion, i.e. the result of the constant field approximation. For efficiency of pump wave energy conversion to harmonic wave energy (or reduced intensity of second harmonic wave) in a metamaterial we receive from (9) (sinh z tanh l cosh z )2 ~ I2 (z) 2 2 ( z ) I 2 2 I10 I10 2 2 2 (tanh 2 l 1) 4 at 2 8 2 , где. 2 2 2 2 4 ~ 2 ( z ) I 2 (10) 2 . In case 8 2 I2 (z) (sinh z tanh l cosh z)2 22 I10 I10 2 (1 tanh 2 l ) 2 2 4 , (11) From expression (11) obtained for the efficiency of frequency conversion, it follows that there exist optimum values of intensity of the fundamental radiation and a phase mismatch at which the efficiency of frequency conversion is maximum. opt The optimum value of the phase mismatch according to (11) can be found from the numerical solution of the following equation under the condition 2 2 8 2 2 1 l (sin z tan l cos z ) 4 2 l cos z 2 2 cos z tan l sin z 2 2 z 4 cos l cos l (12) opt and optimum value of intensity I10 is obtained by solving numerically the equation 2 2 4 4 2 sin z 1 ( l z ) tan l z tan l 2 4 cos l 4 4 l 2 2 2 cos z tan l ( l z ) z 2 cos 2 l 4 cos l (13) opt opt From Eqs. (12) and (13) it follows that the values и I10 depend not only on interaction length but opt also the coefficients of nonlinear interaction in the metamaterials. The value of depends on the opt pump intensity, while I10 of FW depends on the phase mismatch. The efficiency of frequency conversion 2 under the conditions of phase matching 2 2 ( 0 ) reads (where 2 ) I 2 ( z ) 2 (sinh z tanh l cosh z )2 2 ( z ) 2 I10 I10 22 (14) The efficiency of frequency conversion 2 at / 4 2 can be found by expression of the functions in (11) in a Taylor series in the vicinity of zero. As a result, we arrive at the following analytical expression: 2 2 ( z )2 2 ( z ) 1 2(l )2 2 (15) From which it follows that, at low values of reduced pump intensity I10 ( z 1), the efficiency of frequency conversion 2 is directly proportional to I10 . At greater of the pump intensity ( z 1 ) the efficiency of frequency conversion 2 does not depend on I10 . The results of numerical calculations of analytical expressions (10) и (11), for efficiency obtained in the CIA for different values of reduced phase mismatch / 2 are shown in Figs. below. Dependences of reduced pump intensity I1 I1 / I10 (curves 1) and SH wave I2 I2 / I10 (curve 2) on the reduced length of the metamaterial z z , calculated in the CIA at different values of /2 equal to 1.4 (dotted curves), 0.8 (dashed curves), and (solid curves). The behaviors of the curves strongly differ from the usual dependence of the efficiency of frequency conversion observed in natural materials, maximum efficiency of frequency conversion takes place at the input, 2 ( z) This effect can be explained by the fact that the metamaterial plays the role of a mirror reflecting the SH wave at the input to the metamaterial, i.e. the maximum efficiency of frequency conversion takes place at the input (not output) of the metamaterial. In other words, excited SH radiation is directed towards the exciting pump wave. Comparing the curves corresponding to the same value , we see that a greater efficiency of frequency conversion corresponds to the case of phase matching (compare solid, dotted, and dashed curves 1 and 2). Dependences of I1 (curves 3 и 4) and SH wave I 2 (curves 1 and 2) on the reduced length of metamaterial calculated in the CIA (curves 1 and 3) and CFA (curves 2 and 4) for the parameter / 2 =0.8. z The dependences of the reduced intensities of the pump wave and the SH wave on the reduced length of metamaterial z for the parameter 0.8 calculated in the CIA are shown in Fig. One can see that, if the reverse reaction of the excited wave on the FW is taken into account, decreases in the intensities of the pump and SH waves take place (compare curve 1 with curves 2, 3 и 4). In this Fig. we show reduced efficiencies of frequency conversion 2 versus I10 z for six values of . With increase in the phase mismatch, the efficiency of frequency conversion falls. At the same time, the optimum value of the FW intensity at which the efficiency of frequency conversion is maximum increases. Dependences of the efficiency of the frequency conversion 2 on the reduced pump intensity I10 z , calculated in the CIA at z for six values z / 2 , namely, 0 ( 1), 0.1 (2), 0.5 (3), 0.8 ( 4), 1.2 (5) and 1.5 (кривая 6). Dependences of 2 ( z) on the reduced phase mismatch between the waves at the fundamental and SH frequencies z / 2 calculated in the CIA for four values of z equal to 0.2 (1), 0.35 (2), 0.5 (3) и 0.85 (4). In Fig., we show the dependences of 2 ( z) on reduced phase mismatch for four values of parameter z . The curves demonstrate that with increase in the pump intensity a redistribution of the intensity from the central maximum to the lateral maxima takes place. The dependences of 2 ( z) on full reduced length l of metamaterial is offered in Fig. Dependence of maximum efficiency of conversion 2, max ( z) on the reduced full length of metamaterial l at z / 2 0.08 , 1,2 0 It is seen that maximum value of the efficiency acquires the constant value, i.e. in dependence the regime of saturation is observed. Hence practically an important conclusion follows. In CIA it is possible to calculate the optimum value of full length l of metamaterial for the purpose of obtaining the efficient frequency conversion on this length of similar media. Kasumova R.J., Mater. of the XXIV Int. Sc. and Prac. Conf. "Theory and practice in the physical, mathematical and technical sciences," London, G.B., May 3 – 13, 2012 Taking into account the reverse reaction of the SH wave on the phase of the fundamental-radiation leads to qualitatively new effects (not present in the CFA). • First, the location of minima of the SH intensity depends on the pump intensity. Second, with increase in the FW intensity the zeroes of the curves are shifted to the direction of greater values of mismatch, and the width of the curve for increases. This effect allows to use efficiently the materials with critical synchronism. Third, with increase in the intensity of the FW, the central maximum of the curves decreases, but that of the lateral maxima increases. Tagiev Z.H., Kasumova R.J., Gadjieva L.S. J. Rus. Las. Research. 2011, v. 32, 152. II. Self-action effect The self-action effect is usually connected with cubic nonlinearity of media; however, in the media without inversion centers, a strong self-action can take place as well. It is not possible to take into account self-action effect in quadratic-nonlinear media in the CFA, where the pump-wave phase is assumed to be constant, but it can be performed in the CIA. Assuming Aj ( z ) a j ( z )exp i j ( z ) , where j 1, 2 , the system of Eq. (7) can be rewritten as follows: da1 d ia1 1 i 1a1a2 expi (2 21 z dz dz (16) In the CIA we obtain d1 1a2 cos(2 21 z ) dz (17) After some algebra, we obtain for change of the pump-wave phase in the process of light propagation inside the metamaterial 1 ( z ) 1 (0) z 2 8 cos l tan l cos 2 l (1 cos 2z ) (1 tan 2 l ) sin c 2z . z (18) For comparison, we present the expression for the change in the pump-wave phase in the process of light propagation in a homogeneous quadratic medium 1vol ( z ) 1vol (0) where vol vol z 2 vol 8 1 sin c 2 z vol (19) (vol )2 is the phase mismatch at the SHG in usual quadratic media, 2( ) , 4 vol vol 2 (vol )2 2 1vol 2vol I10 . Comparing obtained formulas (18) and (19), one can see that the expression for the FW phase in the metamaterial includes the positive addition to the initial value 1(0) , whereas in the case of the homogeneous quadratic medium the addition has the negative sign. Also numerical calculations of these expressions show that these additions differ in value. Tagiev Z.H., Kasumova R.J.. Opt. Commun., 2006, v. 261, p.258. From obtained expression (18) follows that in the metamaterial the phase velocity of the excitation wave and the refractive index of the medium depend on the pump intensity, i.e., the self-action effect of the light wave is observed. At 1 0 ( 2 2 1 2 I10 ) and 1vol 0 , we obtain the result known in the CFA, vol vol namely, 1( z) 1(0) и 1 ( z ) 1 (0) , i.e., the pump-wave phase is constant. Dependences of the phase shift of the pump wave 1 1 (0) in the metamaterial on the reduced pump intensity Гz calculated in the CIA (solid 1 and 2)and in the homogeneous medium (dotted 1 and 2) for two values of z / 2 , namely, 0.1 (1) and 0.8 (2). In Fig., the dependences of the phase shift of the pump wave 1 1(0) in the metamaterial are shown versus the intensity of the fundamental-radiation for different values of the phase mismatch. The results in case of homogeneous quadratic medium (dotted curves) and in case of metamaterials are cited. A greater change in the phase of pump-wave takes place for the metamaterial The comparison of the dependences for the both types of media shows that a greater change in the phase of pump-wave takes place for the metamaterial (compare the solid of both figures). For instance, if in the metamaterial at Гz=1.2 a change in the phase mismatch from z / 2 = 0.1 (solid 2) to z / 2 z / 2 = 0.8 (solid 1) leads to an increase in 1 1(0) by approximately eight times, in the homogeneous quadratic medium this change makes up the lesser value by ten times. The change in the sign of the difference 1 1(0) upon switching from the homogeneous medium to the metamaterial can be explained, as was mentioned above, by the negative refractive index of the metamaterial. Dependences of the phase shift of the pump wave on the reduced length of the metamaterial calculated in the CIA (solid 1-4) and in the homogeneous medium (dotted 1 и 2) for the value of / 2 equal to 0.2 (1), 0.5 (2), 0.8 (3) и 1.2 (4). Numerical calculations in a wide range of parameters for both cases show that the behavior of the curves 1 1(0) calculated from the reduced intensity and length is the same. In conclusion we’ll offer the table reflecting a modern state of the researches on metamaterials in the following scientific groups. As is seen, owing to the achievements in elaboration of metamaterials we have managed to come close to the visible diapason. So, Parduaye’s group has received a metamaterial which worked on a wavelength of 710 nanometers. Current State of affairs Litchinitser N. Presentation. Development stages At present there are being elaborated three perspective directions. • Optical properties of metal are well depicted by the model of free electrons. Assume that light acts on a metal, i.e. variable electric field with frequency influences of a medium. That is, we sent electron in one side, but it moves to the opposite one. You might draw attention to that in optics metamaterials become more simple, if in VHF there are some spirals, then in optics there are fishnet or two little sticks. It is connected with that in optics we have negative response free of charge, as a property of metal. We have only to receive the negative magnetic response from our metals. Both in metals and optics the electric current is directed against the field. Metall in optics is all the same as inductance, but if to consider metallic particle, it behaves itself as inductance, the surrounding does as a capacity, and all this together makes up LC-a chain, and resonance in it –plasmonic resonance on metal. A wave will run along it. The length at wave running by chain is in no way connected with that of what length is linht, it is determined by geometry of a wire itself. Thus metallic nanowires are ideal source, as wavelength in excitement may reach 1-2nanometres. The research of such nanowires present the first perspective direction of studies in metamaterials. The second direction is related with plasmoniic superlens. The famous researcher Zang affirms that he will make nozzle for every optical microscope and you’ll have opportunity to see how virus of cancer inculcates in a cell and what takes place in a real object. The people can scrutinize the objects smaller than wavelength. For these purposes there exists scanning microscopy. New let’s make comparison. Well, I’ll close my eyes and begin to touch upon the object studied, one, the other, out then register and treat consequently the information. But we may open eyes and to see the given object at once, i.e. in parallel or simultaneously all the features of the object. This includes, Zang affirms, the difference between consecutive treatment and parallel one. Let’s estimate the low physical limit of such device. 2nm is the upper limit, and it is already X- ray region of spectrum. Of course, we cannot see an atom, but we have a chance to see virus, and this is already a great achievement. The third perspective direction is plasmonic nanolaser. The materials with negative refraction, existing at present, have great losses, however, there is hope of receiving the transparent NIMs in future, and therefore, it is expedient to research theoretically their optical properties. Sarichev А.К. Seminar. Conclusion o We have shown that, account of phase changes of interacting waves leads to decrease of conversion efficiency in contrast to the CFA. This observation appears to be useful for correcting the calculations performed for the metamaterials. o We have demonstrated that the light propagation in metamaterials with quadratic nonlinearity is accompanied by self-action effect. o We show that it is possible to change the phase velocity of the pump wave by varying such parameters as the pump intensity, nonlinear medium length, and phase mismatch between the interacting waves. o The phase change of the pump wave in metamaterials is higher than that in the usual quadratic media. Thank You for attention. .