The Nonlinear Analysis of Chiral Medium

Andrey Nikolaevich Volobuev

### Abstract

The principle of calculation of a plate from a metamaterial with inductive type chiral inclusions is submitted. It is shown that distribution of an electromagnetic wave to such substance can be investigated with the help of introduction of a chiral parameter and on the basis of a detailed method of calculation. By comparison of two methods the dependence of chiral parameter from frequency of electromagnetic radiation falling on a plate is found. With the help of a detailed method the nonlinear equation for potential on the chiral plate is found. It is shown that this equation has solutions as solitary and standing waves but not running waves. The analysis of the received solutions of the nonlinear equation is carried out.

Keywords: metamaterial, chiral medium, chiral parameter, nonlinear equation, detailed method, solitary waves, standing waves

### 1. Introduction

Now the metamaterials (Greek "meta" outside), i.e. composite materials with the various inclusions allocated both chaotically, and periodically are widely applied in particular in a radio engineering, at designing space devices, in medicine, etc. [1–3]. Due to these inclusions the received materials have many useful physical, electric, optical and other properties which are not present at natural substances. Among metamaterials the substances with chiral properties [4] which capable to rotate a polarization plane of electromagnetic waves are distinguished. In optics as analogue of similar substances are optical active substances, for example, quartz, a solution of glucose etc.

However the methods of metamaterials calculation are enough limited [5]. Basically all calculations are based on the decision of the Maxwell's equations and the material equations selected according to a problem.

The existing method has restrictions since are usually used only averaged characteristics of metamaterials for example chiral parameter.

In the present work attempt of more detailed approach to properties of the chiral inclusions into metamaterials is made also the analysis of these properties on interaction of chiral elements with the electromagnetic wave falling on a plate from a metamaterial is carried out.

## 2. Standard method of calculation of metamaterial with electromagnetic wave interaction

At research of metamaterials with chiral inclusions on the basis of Maxwell's equations usually use the material equations including so-called chiral parameter χ. In [6] the material equations in the following kind are offered:

$$\mathbf{D} = \varepsilon\_d \mathbf{E} \mp i \frac{\mathcal{X}}{V} \mathbf{H},\tag{1}$$

Solving Eq. (9) with use of initial and boundary conditions it is possible to investigate processes of reflection, refraction, diffraction of an electromagnetic

3. Detailed method of calculation of metamaterial with electromagnetic

type. The plate will consist of the dielectric in which are included the current-

carrying chiral elements as spirals which axis is directed across a plate.

assumed as before that chiral inclusions have no active resistance.

inductance and capacity with an electromagnetic wave is incorrectly.

�LiSi ∂j i

Eq. (7). The density of a current through plate will look like:

j <sup>m</sup> ¼ Cm

Let us consider a plate of the metamaterial with chiral inclusions of the inductive

On Figure 1 the irradiation of a plate by an electromagnetic wave is shown. We

Feature of a plate is the capacity distributed on its surfaces at dot inductive inclusions. Therefore to examine the interaction of separate chiral element having

At the irradiation on the plate there is a potential difference submitting to the

þ φ � φ<sup>0</sup> ð Þgm, (10)

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ, (11)

∂φ ∂t

where С<sup>m</sup> there is capacity of the plate area unit, φ—potential on a plate concerning an initial level φ0, gm—electrical conductivity of units of the plate area

The first term (10) reflects a capacitor bias current, the second term—an induc-

For spiral chiral element it is possible to write down the equation of a voltage

wave in the metamaterial.

The Nonlinear Analysis of Chiral Medium DOI: http://dx.doi.org/10.5772/intechopen.80067

wave interaction

due to an inductive component.

balance:

Figure 1.

11

tive current through the chiral elements.

The plate of metamaterials irradiated by an electromagnetic wave.

$$\mathbf{B} = \mu\_a \mathbf{H} \pm i \frac{\chi}{V} \mathbf{E},\tag{2}$$

where D and B there are induction of electric and magnetic fields in the electromagnetic wave propagating in chiral medium, E and H—the strength of electric and magnetic components of wave, ε<sup>a</sup> and μa—absolute electric penetrance and magnetic permeability of chiral medium, V—velocity of an electromagnetic wave in chiral medium, χ—chiral parameter, in this case dimensionless size.

In [6], it is shown that the material Eqs. (1) and (2) can be written down in more simple kind:

$$\mathbf{D} = (\mathbf{1} \mp \chi) \varepsilon\_d \mathbf{E},\tag{3}$$

$$\mathbf{B} = (\mathbf{1} \pm \chi) \mu\_a \mathbf{H}.\tag{4}$$

In formulas (1)–(4) top signs concern to right-handed rotation chiral element bottom to left-handed rotation.

Using (3) and (4) it is possible to show [6] that if a chiral medium has only reactance the electromagnetic wave in it submits to the wave equations:

$$
\Delta \mathbf{D} = \left(\frac{\mathbf{1} \pm \chi}{V}\right)^2 \frac{\partial^2 \mathbf{D}}{\partial t^2},
\tag{5}
$$

$$
\Delta \mathbf{B} = \left(\frac{\mathbf{1} \pm \chi}{V}\right)^2 \frac{\partial^2 \mathbf{B}}{\partial t^2},
\tag{6}
$$

where t there is a time.

Further us the Eq. (5) will interest only. Substituting (3) in (5) and passing to scalar potential φ [7] we shall find:

$$
\Delta \rho = \left(\frac{\mathbf{1} \pm \chi}{V}\right)^2 \frac{\partial^2 \rho}{\partial t^2} \,. \tag{7}
$$

Let us search for the decision of the Eq. (7) as:

$$
\rho - \rho\_0 = \rho(\mathbf{r}) \exp\left(iat\right),
\tag{8}
$$

where φ<sup>0</sup> there is initial a potential level, r—set of spatial coordinates, ω—a cyclic frequency of electromagnetic wave.

Substituting (8) in (7), we have:

$$
\Delta \rho(\mathbf{r}) + (\mathbf{1} \pm \chi)^2 k^2 \rho(\mathbf{r}) = \mathbf{0},\tag{9}
$$

where <sup>k</sup> <sup>¼</sup> <sup>ω</sup> <sup>V</sup> there is a module of the wave vector of a falling electromagnetic wave.

The Nonlinear Analysis of Chiral Medium DOI: http://dx.doi.org/10.5772/intechopen.80067

2. Standard method of calculation of metamaterial with electromagnetic

At research of metamaterials with chiral inclusions on the basis of Maxwell's equations usually use the material equations including so-called chiral parameter χ.

χ

χ

where D and B there are induction of electric and magnetic fields in the electromagnetic wave propagating in chiral medium, E and H—the strength of electric and magnetic components of wave, ε<sup>a</sup> and μa—absolute electric penetrance and magnetic permeability of chiral medium, V—velocity of an electromagnetic wave in

In [6], it is shown that the material Eqs. (1) and (2) can be written down in more

In formulas (1)–(4) top signs concern to right-handed rotation chiral element

D

B

φ

φ � φ<sup>0</sup> ¼ φð Þr exp ð Þ iωt , (8)

Using (3) and (4) it is possible to show [6] that if a chiral medium has only

reactance the electromagnetic wave in it submits to the wave equations:

<sup>Δ</sup><sup>D</sup> <sup>¼</sup> <sup>1</sup> � <sup>χ</sup> V <sup>2</sup> ∂<sup>2</sup>

<sup>Δ</sup><sup>B</sup> <sup>¼</sup> <sup>1</sup> � <sup>χ</sup> V <sup>2</sup> ∂<sup>2</sup>

<sup>Δ</sup><sup>φ</sup> <sup>¼</sup> <sup>1</sup> � <sup>χ</sup> V <sup>2</sup> ∂<sup>2</sup>

Further us the Eq. (5) will interest only. Substituting (3) in (5) and passing to

where φ<sup>0</sup> there is initial a potential level, r—set of spatial coordinates, ω—a

2 k2

<sup>V</sup> there is a module of the wave vector of a falling electromagnetic

Δφð Þþ r ð Þ 1 � χ

<sup>V</sup> <sup>H</sup>, (1)

<sup>V</sup> <sup>E</sup>, (2)

<sup>∂</sup>t<sup>2</sup> , (5)

<sup>∂</sup>t<sup>2</sup> , (6)

<sup>∂</sup>t<sup>2</sup> : (7)

φð Þ¼ r 0, (9)

D ¼ ð Þ 1∓χ εaE, (3) B ¼ ð Þ 1 � χ μaH: (4)

D ¼ εaE∓i

B ¼ μaH � i

In [6] the material equations in the following kind are offered:

chiral medium, χ—chiral parameter, in this case dimensionless size.

wave interaction

Chirality from Molecular Electronic States

simple kind:

bottom to left-handed rotation.

where t there is a time.

scalar potential φ [7] we shall find:

Let us search for the decision of the Eq. (7) as:

cyclic frequency of electromagnetic wave. Substituting (8) in (7), we have:

where <sup>k</sup> <sup>¼</sup> <sup>ω</sup>

wave.

10

Solving Eq. (9) with use of initial and boundary conditions it is possible to investigate processes of reflection, refraction, diffraction of an electromagnetic wave in the metamaterial.

## 3. Detailed method of calculation of metamaterial with electromagnetic wave interaction

Let us consider a plate of the metamaterial with chiral inclusions of the inductive type. The plate will consist of the dielectric in which are included the currentcarrying chiral elements as spirals which axis is directed across a plate.

On Figure 1 the irradiation of a plate by an electromagnetic wave is shown. We assumed as before that chiral inclusions have no active resistance.

Feature of a plate is the capacity distributed on its surfaces at dot inductive inclusions. Therefore to examine the interaction of separate chiral element having inductance and capacity with an electromagnetic wave is incorrectly.

At the irradiation on the plate there is a potential difference submitting to the Eq. (7). The density of a current through plate will look like:

$$j\_m = \mathbf{C}\_m \frac{\partial \rho}{\partial t} + (\rho - \rho\_0)\mathbf{g}\_{m'} \tag{10}$$

where С<sup>m</sup> there is capacity of the plate area unit, φ—potential on a plate concerning an initial level φ0, gm—electrical conductivity of units of the plate area due to an inductive component.

The first term (10) reflects a capacitor bias current, the second term—an inductive current through the chiral elements.

For spiral chiral element it is possible to write down the equation of a voltage balance:

$$-L\_i \mathbf{S}\_i \frac{\partial \mathbf{j}\_i}{\partial t} = (\boldsymbol{\varrho} - \boldsymbol{\varrho}\_0),\tag{11}$$

Figure 1. The plate of metamaterials irradiated by an electromagnetic wave.

where j <sup>i</sup> there is density of a current through the i-th chiral element, Li—inductance i-th chiral element, Si—the area of plate, falling one chiral element having inductive electrical conductivity gi .

The density of a current j <sup>i</sup> through the chiral element depends on a potential difference on a plate and electrical conductivity it chiral element gi under the formula of the Ohm's law:

$$j\_i \mathbb{S}\_i = \mathbb{g}\_i(\rho - \rho\_0). \tag{12}$$

Substituting (12) in (11) we shall find:

$$\mathbf{g}\_i = -\frac{(\boldsymbol{\rho} - \boldsymbol{\rho}\_0)}{L\_i \frac{\partial \boldsymbol{\rho}}{\partial t}}.\tag{13}$$

Electrical conductivity falling unit area of a plate it is equal:

$$\mathbf{g}\_m = -\frac{(\rho - \rho\_0)}{\mathbf{S}\_i L\_i \frac{\partial \rho}{\partial t}},\tag{14}$$

where it is taken into account gi ¼ gmSi. Having substituted (14) in (10) we shall find:

$$j\_m = \mathcal{C}\_m \frac{\partial \rho}{\partial t} - \frac{\left(\rho - \rho\_0\right)^2}{\mathcal{S}\_i L\_i \frac{\partial \rho}{\partial t}}.\tag{15}$$

Hence:

The single-row chiral plate.

The Nonlinear Analysis of Chiral Medium DOI: http://dx.doi.org/10.5772/intechopen.80067

Figure 2.

<sup>V</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> C1L<sup>1</sup>

13

djX ¼ �γ<sup>X</sup>

Having divided (20) on (18), and having reduced on γ<sup>X</sup> we shall find:

djX <sup>¼</sup> <sup>φ</sup> � <sup>φ</sup><sup>0</sup> SL<sup>1</sup> ∂φ ∂t

SdjX ¼ j

<sup>m</sup> <sup>¼</sup> <sup>φ</sup> � <sup>φ</sup><sup>0</sup> <sup>L</sup>1<sup>b</sup> <sup>∂</sup><sup>φ</sup> ∂t

presence of a cross-section current (or on the contrary) we have:

j

∂2 φ <sup>∂</sup>X<sup>2</sup> <sup>¼</sup> <sup>∂</sup><sup>φ</sup> ∂t <sup>2</sup>

> φ <sup>∂</sup>X<sup>2</sup> <sup>¼</sup> <sup>∂</sup><sup>φ</sup> ∂t <sup>2</sup>

where b there is width of a single-row plate.

Further substituting (23) in (16) we shall find:

φ � φ<sup>0</sup> CmL1b

<sup>V</sup><sup>2</sup> <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ <sup>∂</sup><sup>2</sup>

Substituting (21) in (22) we have:

∂2 φ

> ∂2 φ

> > ∂2 φ

Taking into account С<sup>1</sup> ¼ Сmb—capacity of a single-row plate unit of length and

—a square of an electromagnetic field along a plate velocity we have:

� <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ<sup>2</sup>

� <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ<sup>2</sup>

ω2

ω2

On the other hand taking into account that a longitudinal current is defined only

<sup>∂</sup>X<sup>2</sup> dX: (20)

<sup>∂</sup>X<sup>2</sup> dX: (21)

<sup>m</sup>bdX, (22)

<sup>∂</sup>X<sup>2</sup> : (23)

<sup>0</sup>: (24)

<sup>0</sup>: (25)

Using Ci ¼ CmSi—capacity of the plate falling one chiral element, and designating ω<sup>2</sup> <sup>0</sup> <sup>¼</sup> <sup>1</sup> CiLi we shall find:

$$\frac{\dot{j}\_m}{C\_m} \frac{\partial \rho}{\partial t} = \left(\frac{\partial \rho}{\partial t}\right)^2 - \left(\rho - \rho\_0\right)^2 \alpha\_0^2. \tag{16}$$

Let us consider a plate consisting of chiral elements one lines, Figure 2. Along this plate the inductive current flows.

The law of an electromagnetic induction for this current looks like:

$$-L\frac{\partial I\_X}{\partial t} = \varrho - \varrho\_0. \tag{17}$$

where IX ¼ γ<sup>X</sup> S <sup>l</sup> φ � φ<sup>0</sup> ð Þ there is a longitudinal inductive current, γX—specific inductive electrical conductivity a single-row plate, L—its inductance, S—the area of cross-section of a single-row plate, l—its length.

Hence:

$$-\gamma\_X \text{SL}\_1 \frac{\partial \rho}{\partial t} = \rho - \rho\_{0\bullet} \tag{18}$$

where <sup>L</sup><sup>1</sup> <sup>¼</sup> <sup>L</sup> <sup>l</sup> there is inductance of a single-row plate unit of length. Under the Ohm's law for density of a longitudinal current we have:

$$j\_X = -\gamma\_X \frac{\partial \rho}{\partial X}.\tag{19}$$

The Nonlinear Analysis of Chiral Medium DOI: http://dx.doi.org/10.5772/intechopen.80067

Figure 2. The single-row chiral plate.

Hence:

where j

ing ω<sup>2</sup>

<sup>0</sup> <sup>¼</sup> <sup>1</sup>

where IX ¼ γ<sup>X</sup>

where <sup>L</sup><sup>1</sup> <sup>¼</sup> <sup>L</sup>

Hence:

12

inductive electrical conductivity gi

Chirality from Molecular Electronic States

Substituting (12) in (11) we shall find:

where it is taken into account gi ¼ gmSi. Having substituted (14) in (10) we shall find:

CiLi we shall find:

S

The density of a current j

formula of the Ohm's law:

<sup>i</sup> there is density of a current through the i-th chiral element, Li—induc-

<sup>i</sup> through the chiral element depends on a potential

Si ¼ gi φ � φ<sup>0</sup> ð Þ: (12)

: (13)

, (14)

: (15)

<sup>0</sup>: (16)

tance i-th chiral element, Si—the area of plate, falling one chiral element having

difference on a plate and electrical conductivity it chiral element gi under the

gi ¼ � <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ Li ∂φ ∂t

gm ¼ � <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ SiLi ∂φ ∂t

∂φ

Let us consider a plate consisting of chiral elements one lines, Figure 2.

inductive electrical conductivity a single-row plate, L—its inductance, S—the area

∂φ

<sup>l</sup> there is inductance of a single-row plate unit of length.

∂φ

The law of an electromagnetic induction for this current looks like:

�<sup>L</sup> <sup>∂</sup>IX

�γXSL<sup>1</sup>

Under the Ohm's law for density of a longitudinal current we have:

j <sup>X</sup> ¼ �γ<sup>X</sup>

<sup>∂</sup><sup>t</sup> � <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ<sup>2</sup> SiLi ∂φ ∂t

� <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ<sup>2</sup>

<sup>l</sup> φ � φ<sup>0</sup> ð Þ there is a longitudinal inductive current, γX—specific

ω2

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>φ</sup> � <sup>φ</sup>0, (17)

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>φ</sup> � <sup>φ</sup>0, (18)

<sup>∂</sup><sup>X</sup> : (19)

Using Ci ¼ CmSi—capacity of the plate falling one chiral element, and designat-

.

j i

Electrical conductivity falling unit area of a plate it is equal:

j <sup>m</sup> ¼ Cm

j m Cm

Along this plate the inductive current flows.

of cross-section of a single-row plate, l—its length.

∂φ <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup><sup>φ</sup> ∂t <sup>2</sup>

$$d\dot{y}\_X = -\gamma\_X \frac{\partial^2 \rho}{\partial X^2} dX. \tag{20}$$

Having divided (20) on (18), and having reduced on γ<sup>X</sup> we shall find:

$$d\dot{\jmath}\_X = \frac{\varrho - \varrho\_0}{\mathcal{S}L\_1} \frac{\partial^2 \varrho}{\partial \mathcal{X}^2} d\mathcal{X}. \tag{21}$$

On the other hand taking into account that a longitudinal current is defined only presence of a cross-section current (or on the contrary) we have:

$$\mathbf{S}d\dot{j}\_X = j\_m b dX,\tag{22}$$

where b there is width of a single-row plate. Substituting (21) in (22) we have:

$$j\_m = \frac{\rho - \rho\_0}{L\_1 b \frac{\partial \rho}{\partial t}} \frac{\partial^2 \rho}{\partial X^2}. \tag{23}$$

Further substituting (23) in (16) we shall find:

$$\frac{\rho - \rho\_0}{C\_m L\_1 b} \frac{\partial^2 \rho}{\partial X^2} = \left(\frac{\partial \rho}{\partial t}\right)^2 - \left(\rho - \rho\_0\right)^2 \alpha\_0^2. \tag{24}$$

Taking into account С<sup>1</sup> ¼ Сmb—capacity of a single-row plate unit of length and <sup>V</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> C1L<sup>1</sup> —a square of an electromagnetic field along a plate velocity we have:

$$
\lambda V^2 (\rho - \rho\_0) \frac{\partial^2 \rho}{\partial X^2} = \left(\frac{\partial \rho}{\partial t}\right)^2 - (\rho - \rho\_0)^2 a\_0^2. \tag{25}
$$

The nonlinear Eq. (25) can be transformed to a kind correct for spatial geometry:

$$V^2 \Delta \rho + \rho\_0^2 (\rho - \rho\_0) = \frac{1}{\rho - \rho\_0} \left(\frac{\partial \rho}{\partial t}\right)^2. \tag{26}$$

Linearization of Eqs. (26) can be carried out by a ratio (8):

$$
\Delta \rho(\mathbf{r}) + k\_{\mathbb{S}}^2 \rho(\mathbf{r}) = \mathbf{0},
\tag{27}
$$

The nonlinear Eq. (25) has at least one more solution as a solitary wave:

<sup>φ</sup> � <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>φ</sup>maxexp � ð Þ <sup>k</sup>0ð Þ� <sup>X</sup> � <sup>X</sup><sup>0</sup> <sup>ω</sup>0ð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup>

a time reading, X0—coordinate of the chiral element center. The sign a minus concerns to a wave spreading from left to right, a sign plus from right to left. The illustrative graph of the solution (32) as function of coordinate X also is shown on Figure 2. Growth of potential above chiral inclusions is caused by

From the analysis of both curves it is possible to conclude that the top curve, Figure 2, concern to often enough inclusions of the chiral elements in a plate, and bottom to more rare. Therefore into the solution (32) to enter a chiral parameter it is

Obviously for the nonlinear Eqs. (25) or (26) there should be a multiwave solution. However to find such solution it is extremely difficult. Multiwave solutions are found for very much limited circle of the nonlinear wave equations [11, 12]. The multiwave solution should depend on concentration of the chiral elements in a plate. Only with its help it is possible to understand under what conditions it is possible is proved to enter the chiral parameter, i.e. to understand borders of the

Let us consider in more detail a kind of the wave arising on single-row chiral

The nonlinear Eqs. (25) and (26) can be solved a method of variables division

where φð Þ X there is function only coordinates X, T tð Þ—a function only time t.

<sup>0</sup> <sup>¼</sup> <sup>1</sup> T tð Þ

Eq. (35) breaks up to two independent equations. The equation dependent on X

<sup>0</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> V2

∂t � �<sup>2</sup>

<sup>∂</sup>X<sup>2</sup> <sup>¼</sup> <sup>φ</sup>ð Þ <sup>X</sup> <sup>∂</sup>T tð Þ

proportionality of the chiral inclusions reactance their inductivities

whereas before <sup>k</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup><sup>0</sup>

The Nonlinear Analysis of Chiral Medium DOI: http://dx.doi.org/10.5772/intechopen.80067

material Eqs. (1)–(4) applicability.

V2

where α there is a constant.

looks like:

15

<sup>φ</sup>ð Þ <sup>X</sup> <sup>T</sup><sup>2</sup>

<sup>V</sup><sup>2</sup> <sup>1</sup> φð Þ X

plate at falling on it of an electromagnetic wave.

[13]. We search for the solution of Eq. (25) as:

Having substituted (33) in (25) we shall find:

ð Þt ∂2 φð Þ X

Let us divide both parts of the equation on <sup>φ</sup><sup>2</sup>ð Þ <sup>X</sup> <sup>T</sup><sup>2</sup>

∂2 φð Þ X <sup>∂</sup>X<sup>2</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup>

∂2 φð Þ X <sup>∂</sup>X<sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

φ � φ<sup>0</sup> � XLi ¼ ωLi.

irrational.

2 !

<sup>V</sup> , φmax there is a peak value of potential, t0—an initial of

φ � φ<sup>0</sup> ¼ φð Þ X T tð Þ: (33)

� <sup>φ</sup><sup>2</sup>

<sup>∂</sup>T tð Þ ∂t � �<sup>2</sup> ð Þ <sup>X</sup> <sup>T</sup><sup>2</sup>

¼ �α<sup>2</sup>

� �φð Þ¼ <sup>X</sup> <sup>0</sup>: (36)

ð Þ<sup>t</sup> <sup>ω</sup><sup>2</sup>

ð Þt . In result we shall receive:

<sup>0</sup>: (34)

: (35)

2

: (32)

where is k<sup>2</sup> <sup>S</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> 0þω<sup>2</sup> <sup>V</sup><sup>2</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> .

## 4. Various kinds of the equation solution of a metamaterial and an electromagnetic wave interaction

Eqs. (27) and (9) reflect the same physical process – propagation of electromagnetic fluctuations on the chiral plate. Distinction consists that at a deduction (27) as against (9) was not necessity to use the material Eqs. (1)–(4) i.e. the chiral parameter was not used.

On the basis of Eqs. (27) and (9) identity it is possible to put down:

$$k\_{\mathbb{S}}^2 = k\_0^2 + k^2 = (1 \pm \chi)^2 k^2. \tag{28}$$

Hence the chiral parameter can be written down as:

$$
\pm \chi = \sqrt{\mathbf{1} + \frac{k\_0^2}{k^2}} - \mathbf{1}.\tag{29}
$$

If k0<<k or ω0<<ω formula (29) becomes simpler:

$$
\chi = \pm \frac{k\_0^2}{2k^2} = \pm \frac{\alpha\_0^2}{2\alpha^2}.\tag{30}
$$

Let us notice that quantum calculation of an optical active substance [8, 9] results in the formula for chiral parameter:

$$\chi = \frac{2V\delta}{3\hbar} \frac{a\nu\mathbb{0}}{a\nu\_{0\circ}^2 - a^2},\tag{31}$$

where ћ there is Planck's reduced constant, δ - size proportional to product of the real parts electric and magnetic dipole moments of an optical active molecule power transition excited by a light of the wave given length, ω0<sup>j</sup>—in this case the frequency corresponding to power transition 0 ! j [10].

The increase in a degree of frequency dependence ω<sup>0</sup> up to square-law in formula (30) in comparison with (31) is characteristic at transition from quantum area in classical.

On Figure 2 the illustrative graph of the potential fluctuations on the chiral plate is shown according to the oscillatory solutions satisfying Eqs. (9) and (27). Character of fluctuations will be investigated below.

The nonlinear Eq. (25) can be transformed to a kind correct for spatial geometry:

φ � φ<sup>0</sup>

∂φ ∂t � �<sup>2</sup>

<sup>S</sup>φð Þ¼ r 0, (27)

: (26)

: (28)

� 1: (29)

<sup>2</sup>ω<sup>2</sup> : (30)

<sup>0</sup><sup>j</sup> � <sup>ω</sup><sup>2</sup> , (31)

<sup>0</sup> <sup>φ</sup> � <sup>φ</sup><sup>0</sup> ð Þ¼ <sup>1</sup>

V2

electromagnetic wave interaction

where is k<sup>2</sup>

eter was not used.

area in classical.

14

<sup>S</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> 0þω<sup>2</sup> <sup>V</sup><sup>2</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup>

Chirality from Molecular Electronic States

<sup>Δ</sup><sup>φ</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup>

<sup>0</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> .

> k2 <sup>S</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup>

Hence the chiral parameter can be written down as:

If k0<<k or ω0<<ω formula (29) becomes simpler:

quency corresponding to power transition 0 ! j [10].

ter of fluctuations will be investigated below.

results in the formula for chiral parameter:

Linearization of Eqs. (26) can be carried out by a ratio (8):

<sup>Δ</sup>φð Þþ <sup>r</sup> <sup>k</sup><sup>2</sup>

4. Various kinds of the equation solution of a metamaterial and an

On the basis of Eqs. (27) and (9) identity it is possible to put down:

�χ ¼

χ ¼ �

<sup>χ</sup> <sup>¼</sup> <sup>2</sup>V<sup>δ</sup> 3ћ

Eqs. (27) and (9) reflect the same physical process – propagation of electromagnetic fluctuations on the chiral plate. Distinction consists that at a deduction (27) as against (9) was not necessity to use the material Eqs. (1)–(4) i.e. the chiral param-

<sup>0</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>χ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ k2 0 k2

s

k2 0 <sup>2</sup>k<sup>2</sup> ¼ � <sup>ω</sup><sup>2</sup>

Let us notice that quantum calculation of an optical active substance [8, 9]

ω0<sup>j</sup> ω2

where ћ there is Planck's reduced constant, δ - size proportional to product of the real parts electric and magnetic dipole moments of an optical active molecule power transition excited by a light of the wave given length, ω0<sup>j</sup>—in this case the fre-

On Figure 2 the illustrative graph of the potential fluctuations on the chiral plate is shown according to the oscillatory solutions satisfying Eqs. (9) and (27). Charac-

The increase in a degree of frequency dependence ω<sup>0</sup> up to square-law in formula (30) in comparison with (31) is characteristic at transition from quantum

2 k2

0

The nonlinear Eq. (25) has at least one more solution as a solitary wave:

$$
\rho - \rho\_0 = \rho\_{\text{max}} \exp\left(-\frac{\left(k\_0(X - X\_0) \pm a\_0(t - t\_0)\right)^2}{2}\right). \tag{32}
$$

whereas before <sup>k</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup><sup>0</sup> <sup>V</sup> , φmax there is a peak value of potential, t0—an initial of a time reading, X0—coordinate of the chiral element center. The sign a minus concerns to a wave spreading from left to right, a sign plus from right to left.

The illustrative graph of the solution (32) as function of coordinate X also is shown on Figure 2. Growth of potential above chiral inclusions is caused by proportionality of the chiral inclusions reactance their inductivities φ � φ<sup>0</sup> � XLi ¼ ωLi.

From the analysis of both curves it is possible to conclude that the top curve, Figure 2, concern to often enough inclusions of the chiral elements in a plate, and bottom to more rare. Therefore into the solution (32) to enter a chiral parameter it is irrational.

Obviously for the nonlinear Eqs. (25) or (26) there should be a multiwave solution. However to find such solution it is extremely difficult. Multiwave solutions are found for very much limited circle of the nonlinear wave equations [11, 12]. The multiwave solution should depend on concentration of the chiral elements in a plate. Only with its help it is possible to understand under what conditions it is possible is proved to enter the chiral parameter, i.e. to understand borders of the material Eqs. (1)–(4) applicability.

Let us consider in more detail a kind of the wave arising on single-row chiral plate at falling on it of an electromagnetic wave.

The nonlinear Eqs. (25) and (26) can be solved a method of variables division [13]. We search for the solution of Eq. (25) as:

$$
\rho - \rho\_0 = \rho(X)T(t). \tag{33}
$$

where φð Þ X there is function only coordinates X, T tð Þ—a function only time t. Having substituted (33) in (25) we shall find:

$$\left(\mathbf{V}^{2}\boldsymbol{\varrho}(\mathbf{X})\mathbf{T}^{2}(t)\frac{\partial^{2}\boldsymbol{\varrho}(\mathbf{X})}{\partial\mathbf{X}^{2}} = \left(\boldsymbol{\varrho}(\mathbf{X})\frac{\partial\mathbf{T}(t)}{\partial t}\right)^{2} - \boldsymbol{\varrho}^{2}(\mathbf{X})\mathbf{T}^{2}(t)\boldsymbol{\varrho}\_{0}^{2}.\tag{34}$$

Let us divide both parts of the equation on <sup>φ</sup><sup>2</sup>ð Þ <sup>X</sup> <sup>T</sup><sup>2</sup> ð Þt . In result we shall receive:

$$V^2 \frac{1}{\rho(X)} \frac{\partial^2 \rho(X)}{\partial X^2} + a\_0^2 = \left(\frac{1}{T(t)} \frac{\partial T(t)}{\partial t}\right)^2 = -a^2. \tag{35}$$

where α there is a constant.

Eq. (35) breaks up to two independent equations. The equation dependent on X looks like:

$$\frac{\partial^2 \rho(X)}{\partial X^2} + \left(k\_0^2 + \frac{a^2}{V^2}\right) \rho(X) = \mathbf{0}.\tag{36}$$

Comparing (36) and (27) we notice that k<sup>2</sup> <sup>S</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>V</sup>2. Hence <sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup> <sup>V</sup>2, and hence α ¼ ω.

The solution of Eq. (36) we shall write down as:

$$
\rho(X) = \rho(\mathbf{0}) \exp\left(ik\_S X\right). \tag{37}
$$

where φð Þ 0 there is value of function φð Þ X in the beginning of coordinates. The second equation of equality (35) looks like:

$$\frac{\partial T(t)}{\partial t} = i\alpha T(t). \tag{38}$$

Solving this equation we shall find:

$$T(t) = T(\mathbf{0}) \exp\left(iat\right),\tag{39}$$

Author details

Samara, Russia

17

Andrey Nikolaevich Volobuev

The Nonlinear Analysis of Chiral Medium DOI: http://dx.doi.org/10.5772/intechopen.80067

provided the original work is properly cited.

Department of Medical and Biological Physics, Samara State Medical University,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: volobuev47@yandex.ru

where Tð Þ 0 there is initial value of function T tð Þ.

Using (33) and the real parts of solutions (37), (39) we shall find the solution of Eq. (25):

$$
\rho - \rho\_0 = \rho\_A \exp\left(i\alpha t\right) \exp\left(ik\_S X\right) = \rho\_A \cos\alpha t \cos k\_S X = \rho\_A \cos\alpha t \cos \frac{2\pi X}{\lambda}.\tag{40}
$$

where it is designated φ<sup>A</sup> ¼ Tð Þ 0 φð Þ 0 —a peak value of potential on a plate, λ—a wave length.

Formula (40) represents the equation of a standing wave.

Condition of the nodes occurrence in a standing wave <sup>X</sup>ns ¼ �ð Þ <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>λ</sup> 4, where n ¼ 0, 1, 2, ….

On the ends of the single-row chira plate, Figure 2, should be nodes of a standing wave. If excitation of a wave occurs in the center of a plate the number of the maximal node can be found under the formula � <sup>l</sup> <sup>2</sup> ¼ �ð Þ <sup>2</sup>nmax <sup>þ</sup> <sup>1</sup> <sup>λ</sup> <sup>4</sup> or <sup>n</sup>max <sup>¼</sup> <sup>l</sup> <sup>λ</sup> � <sup>1</sup> 2 .

It is necessary to note that running waves <sup>φ</sup> � <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>φ</sup><sup>А</sup> <sup>2</sup> cosð Þ kSX � ωt are not the solution of Eq. (25) therefore formula (40) from the physical point of view cannot be presented as the sum of the direct, and reflected from borders plate waves though mathematical this procedure is simple for making. It is consequence of Eq. (25) nonlinearity.

### 5. Conclusion

Distribution of potential to a plate from a metamaterial with inductive chiral inclusions is investigated as with use of the material equations together with the Maxwell's equations, and on the basis of a detailed method of calculation of the chiral elements and an electromagnetic wave interaction. Comparison of two approaches has allowed to find out that introduction of the chiral parameter is correct only at enough high concentration of the chiral inclusions. On the basis of comparison of two methods results the frequency dependence of chiral parameter is found. At use of a detailed method of calculation the nonlinear equation for the potential having solutions as standing waves and solitary waves is received. Running waves are not the solution of this equation. Necessity of the multiwave solution existence of the nonlinear equation which should depend on concentration of the chiral elements in a metamaterial is marked.

The Nonlinear Analysis of Chiral Medium DOI: http://dx.doi.org/10.5772/intechopen.80067

Comparing (36) and (27) we notice that k<sup>2</sup>

Chirality from Molecular Electronic States

The solution of Eq. (36) we shall write down as:

The second equation of equality (35) looks like:

where Tð Þ 0 there is initial value of function T tð Þ.

Solving this equation we shall find:

α ¼ ω.

Eq. (25):

wave length.

<sup>n</sup>max <sup>¼</sup> <sup>l</sup>

n ¼ 0, 1, 2, ….

<sup>λ</sup> � <sup>1</sup> 2 .

Eq. (25) nonlinearity.

5. Conclusion

16

<sup>S</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup>

where φð Þ 0 there is value of function φð Þ X in the beginning of coordinates.

Using (33) and the real parts of solutions (37), (39) we shall find the solution of

where it is designated φ<sup>A</sup> ¼ Tð Þ 0 φð Þ 0 —a peak value of potential on a plate, λ—a

Condition of the nodes occurrence in a standing wave <sup>X</sup>ns ¼ �ð Þ <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>λ</sup>

On the ends of the single-row chira plate, Figure 2, should be nodes of a standing wave. If excitation of a wave occurs in the center of a plate the number of

solution of Eq. (25) therefore formula (40) from the physical point of view cannot be presented as the sum of the direct, and reflected from borders plate waves though mathematical this procedure is simple for making. It is consequence of

Distribution of potential to a plate from a metamaterial with inductive chiral inclusions is investigated as with use of the material equations together with the Maxwell's equations, and on the basis of a detailed method of calculation of the chiral elements and an electromagnetic wave interaction. Comparison of two approaches has allowed to find out that introduction of the chiral parameter is correct only at enough high concentration of the chiral inclusions. On the basis of comparison of two methods results the frequency dependence of chiral parameter is found. At use of a detailed method of calculation the nonlinear equation for the potential having solutions as standing waves and solitary waves is received. Running waves are not the solution of this equation. Necessity of the multiwave solution existence of the nonlinear equation which should depend on concentration of the

<sup>∂</sup>T tð Þ

φ � φ<sup>0</sup> ¼ φАexp ð Þ iωt exp ð Þ¼ ikSX φ<sup>А</sup> cosωt cos kSX ¼ φ<sup>А</sup> cosωt cos

Formula (40) represents the equation of a standing wave.

the maximal node can be found under the formula � <sup>l</sup>

chiral elements in a metamaterial is marked.

It is necessary to note that running waves <sup>φ</sup> � <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>φ</sup><sup>А</sup>

<sup>0</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup>

φð Þ¼ X φð Þ 0 exp ð Þ ikSX : (37)

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>i</sup>ωT tð Þ: (38)

T tðÞ¼ Tð Þ 0 exp ð Þ iωt , (39)

<sup>V</sup>2. Hence <sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup>

<sup>V</sup>2, and hence

2πX

<sup>4</sup> or

<sup>2</sup> cosð Þ kSX � ωt are not the

<sup>2</sup> ¼ �ð Þ <sup>2</sup>nmax <sup>þ</sup> <sup>1</sup> <sup>λ</sup>

<sup>λ</sup> : (40)

4, where

## Author details

Andrey Nikolaevich Volobuev Department of Medical and Biological Physics, Samara State Medical University, Samara, Russia

\*Address all correspondence to: volobuev47@yandex.ru

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

[1] Slusar V. Metamaterials in the Antennas Techniques: A History and Main Principles. Vol. 7. Minsk: Electronics, NTB; 2009. pp. 70-79

[2] Capolino F. Theory and Phenomena of Metamaterials. Boca Raton: Taylor & Francis; 2009. p. 992

[3] Vendik IB, Vendik OG. Metamaterials and its application in technique of ultrahigh frequencies. Technical Physics. 2013;58(1):1-24

[4] Neganov VA, Osipov OV. Reflecting, Waveguiding, and Emitting Structures with Chiral Elements. Moscow: Radio i Svyaz; 2006. 280p

[5] Katsenelenbaum BZ, Korshunova EN, Sivov AN, Shatrov AD. Chiral electrodynamics objects. Moscow, Uspekhi Fizicheskikh Nauk. 1997; 167(11):1201-1212

[6] Volobuev AN. Electrodynamics of circular dichroism and its application in the construction of a circular polaroid. Technical Physics. 2016;61(3):337-341

[7] Levich VG. Course of Theoretical Physics. Vol. 1. Moscow: Fizmatlit; 1962. p. 41

[8] Condon E. Theory of optical rotating ability. Moscow, Uspekhi Fizicheskikh Nauk. 1938;19(3):380-430

[9] Volobuev AN. Quantum Electrodynamics through the Eyes of a Biophysics. New York: Nova Science Publishers, Inc.; 2017. p. 252

[10] Volkenchtein MV. Biophysics. Lan: St. Petersburg; 2008. p. 576

[11] Ablovits М, Sigur H. Solitons, Methods of Reverse Task. Moscow: Mir; 1987. 480p

[12] Dodd R, Eilbek G, Ghibbon G, Morris G. Solitons and Nonlinear Wave Equations. Moscow: Mir; 1988. 696p

Chapter 3

Abstract

ters such as (H2O)n, K<sup>+</sup>

ture 1 K; (2) (H2O)n, K<sup>+</sup>

1. Introduction

19

Systems

Chirality Properties of Modeling

The research addresses the problem of chirality existence in modeling water with various impurity molecules using new numerical algorithm of chirality determination. It is based on asymmetry analysis of molecular system composed of water molecules. The following molecular systems are investigated: (1) small water clus-

300 K; and (3) chiral biological molecules of L-valine, D-valine, L-glycerose, and D-glycerose and left or right water clusters (H2O)4 with water molecule's shell with

thickness varied from 4 to 14 Å with a step of 2 Å. The systems (1), (2) are investigated by Monte Carlo method and the interaction is simulated with Poltev-Malenkov potentials. Systems (3) are initiated using Solvate software, and then aqueous systems are optimized by the conjugate gradient algorithm using the MMFF94 potential. It is revealed that there is no predominance of right-handed or lefthanded substructures in all studied configurations of water molecules. But in small aqueous systems (2), (3), the number of types of water structures, taking into

account chirality, depends on the presence of impurity ion and its type.

numerical simulation, computer modeling, Monte Carlo procedure, Poltev-Malenkov potential, MMFF94 potential, conjugate gradients

twisting abilities of nematic liquid crystals [3].

Keywords: chirality, water structure, small water clusters, biological molecules,

Chirality is the structural characteristic of molecules, which determines their physical, chemical, and biological properties. Chiral molecules differ not only in properties associated with rotation of the polarization plane for plane-polarized light but also in being involved in processes of metabolism and catabolism and also in pharmacological activity [1]. Besides, the presence of some chiral impurity allows one to control chemical reaction and to change its rate [2] and also influences the

In biological systems, fundamental role is played by homochirality of chiral compounds, such as amino acids, phospholipids, sugars, etc. This property influences formation of macromolecular prebiological systems [4–6].

(H2O)m (n = 4÷8, m = 5÷10) at tempera-

(H2O)p (n = 4÷9, p = 5÷8) at temperature

Water in Different Aqueous

(H2O)m, and Na+

(H2O)p, and Na+

Khakhalin Andrey Vladimirovich and

Gradoboeva Olga Nikolaevna

[13] Tikhonov AN, Samarski AA. Equipments of Mathematical Physics. Moscow: Nauka; 1972. p. 82

## Chapter 3

References

Francis; 2009. p. 992

Svyaz; 2006. 280p

167(11):1201-1212

p. 41

[3] Vendik IB, Vendik OG.

[1] Slusar V. Metamaterials in the Antennas Techniques: A History and Main Principles. Vol. 7. Minsk: Electronics, NTB; 2009. pp. 70-79

Chirality from Molecular Electronic States

[2] Capolino F. Theory and Phenomena of Metamaterials. Boca Raton: Taylor & [12] Dodd R, Eilbek G, Ghibbon G, Morris G. Solitons and Nonlinear Wave Equations. Moscow: Mir; 1988. 696p

[13] Tikhonov AN, Samarski AA. Equipments of Mathematical Physics.

Moscow: Nauka; 1972. p. 82

Metamaterials and its application in technique of ultrahigh frequencies. Technical Physics. 2013;58(1):1-24

[4] Neganov VA, Osipov OV. Reflecting, Waveguiding, and Emitting Structures with Chiral Elements. Moscow: Radio i

[5] Katsenelenbaum BZ, Korshunova EN, Sivov AN, Shatrov AD. Chiral electrodynamics objects. Moscow, Uspekhi Fizicheskikh Nauk. 1997;

[6] Volobuev AN. Electrodynamics of circular dichroism and its application in the construction of a circular polaroid. Technical Physics. 2016;61(3):337-341

[7] Levich VG. Course of Theoretical Physics. Vol. 1. Moscow: Fizmatlit; 1962.

[8] Condon E. Theory of optical rotating ability. Moscow, Uspekhi Fizicheskikh

Electrodynamics through the Eyes of a Biophysics. New York: Nova Science

[10] Volkenchtein MV. Biophysics. Lan:

Nauk. 1938;19(3):380-430

[9] Volobuev AN. Quantum

Publishers, Inc.; 2017. p. 252

St. Petersburg; 2008. p. 576

1987. 480p

18

[11] Ablovits М, Sigur H. Solitons, Methods of Reverse Task. Moscow: Mir;
