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
n0
 0
0
n0
 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  21 ( 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
8112
i
k1c 2
(6)
( 2) 
 eff
A1 A2 exp( iz )
dA2
4 222 ( 2) 2
  2 A2  i
 eff A1 exp(  iz )
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( iz )
dz
dA2
  2 A2  i 2 A12 exp(  iz )
dz
(7)
where the following notation is introduced
 22 (2)
1 12 (2)
112 (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=21, we employ the system (6) under the following boundary conditions
A1 ( z  0)  A10 exp( i10 ),
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 210  iz / 2)
i
   tg l
2
2 
where
2
 22 ,
4
2   1 2 I10,
,
(9)
I j  Aj Aj
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
22
(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 expi (2  21  z 
dz
dz
(16)
In the CIA we obtain
d1
  1a2 cos(2  21  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 2z )
 (1  tan 2 l ) sin c 2z .
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.
.