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MICHAŁ SZOTA

Transmission in aperiodic Severin superlattices

INTRODUCTION

Multilayers are materials with very interesting physical properties [1÷22], particularly interesting is the propagation of electromagnetic waves (EW) in superlattices [20]. It should be noted, that the internal design of the system and the medium in which it is immersed has a significant influence on the transmission of electromagnetic waves [17÷21]. Therefore it is important to understand the behaviour of an electromagnetic wave for the widest range of structures, which will allow to design systems with given properties.

Calculation of superlattice transmission was made using the matrix method [17÷20, 22] for the Severin's superlattice [23].

TRANSMISSION CALCULATION

Electromagnetic wave propagation can be described by the matrix equation:

outin in in 1 1

1 0

JT

, j j , jj

EE E D P D

(1)

where inE is the intensity of the electric field of an

electromagnetic wave incident on the structure, inE describes the

electric field of the reflected electromagnetic wave, and outE is

the intensity of the electromagnetic field of wave passing through the multilayer system. The was defined, hereinafter referred to as the characteristic matrix of 22 dimension, as:

in 1 11

J

, j j , jj

D P D

(2)

what inserted into the equation (1) gives

outin in

0

T EE E

. (3)

Matrixes of electromagnetic wave propagation P and transmission on the medium boundaries D were defined as:

in 1in 1

1in 1

11

11

2

2

111

111

0

0

j j j

j j j

, j,

in, j, j

j , jj , j

j , jj , j

id n cos

jid n cos

rD

rt

rD

rt

eP

e

, (4)

where jn is the refractive index of layer j , and jd is layer thickness; j describes angle of direction of incidence of electromagnetic wave to a normal of multilayer structure; is

a wavelength. The variables t and r define Fresnel amplitude coefficients respectively for the transmission and reflectance of the polarization P and S

P1

11

1

11

1P1

11

S1

11

1

S1

2sign

sign sign

sign sign

sign sign

2sign

sign sign

sig

jj

jj , j

j jj j

j j

j jj j

j jj , j

j jj j

j j

jj

jj , j

j jj j

j j

jj

j , j

cosn

nt

cos cosn n

n n

cos cosn n

n nr

cos cosn n

n n

cosn

nt

cos cosn n

n n

cosn

r

11

1

11

1

n sign

sign sign

jj

j j

j jj j

j j

cosn

n n

cos cosn n

n n

. (5)

In equation (5) function sign x was defined as:

1 0

sign 0 0

1 0

, x

x , x

, x

. (6)

Using the characteristic matrix (2) we can determine the transmission T of given superlattice

2

out out

in in 11

1n cosT

n cos

(7)

and reflectance R

2

11

21

R (8)

Calculation of superlattice transmission were performed using the matrix method [17÷20] for the Severin's superlattice.

SEVERIN'S STRUCTURE

Creation of aperiodic Severin's structure requires the use the following substitution rule

A BB

B AB

(9)

The condition for the initial creation of the Severin's structure Xk is

Prof. asssoc. Michał Szota (mszota@wip.pcz.pl) – Institute of MaterialsEngineering, Czestochowa University of Technology, Czestochowa, Poland

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0 B (10)

Using the formula (9) and (10) we create the next generation k of Severin's structure

0

1

2

B

AB

BBAB

(10)

Next generations of Severin's structure are showed in Table 1.

Table 1. Structure of first six generations k of Severin's structure Tabela 1. Struktura sześciu pierwszych pokoleń k łańcucha Severina

k Xk

1 AB 2 BBAB 3 ABABBBAB 4 BBABBBABABABBBAB 5 ABABBBABABABBBABBBABBBABABABBBAB 6 BBABBBABABABBBABBBABBBABABABBBAB

ABABBBABABABBBABBBABBBABABABBBAB 7 ABABBBABABABBBABBBABBBABABABBBAB

ABABBBABABABBBABBBABBBABABABBBAB BBABBBABABABBBABBBABBBABABABBBAB ABABBBABABABBBABBBABBBABABABBBAB

RESEARCH

In this study the propagation of electromagnetic waves thru Severin superlattice built with lossless and non-dispersive materials. For computation matrix method was used which takes into account the use of structure constructed from materials with a negative refractive index. The results of determining the transmission in superlattice were collected in Figures 1÷6. The horizontal axis defines the transmission map of the incident wave length λ of visible light. The vertical axis indicates the angle of electromagnetic wave to the normal vector of multilayer structure. By the white color complete transmission of electromagnetic wave has been marked, and by the black, when there is no transmission.

Refractive index of the layers for the investigated structures were established on A 1 544n . (NaCl) and B 3 4n . (GaAs) respectively [22], except Figure 2, where a change in a trans-mission was studied, when material B is converted to its metamaterial equivalent [7], for which the refractive index is

B 3 4n . . Figure 1 shows the map of transmission for the future

generations of Severin's structure, calculated for the polarization P and S. Structure was then placed in vacuum ( in out 1n n ). Thickness of single layer was B A 200 nmd d .

Fig. 1. Charts for transmission T for the Severin's superlattice, depending on the incident wavelength λ and the angle of incidence Θ. The parameter k specifies the number of superlattice generations. The thickness of a single layer was 200 nm. The refractive index of the material A was 1.544, material B was equal to 3.4, and for the input and output layers were respectively nin = nout = 1 Rys. 1. Wykresy transmisji T dla supersieci Severina w zależności od długości fali padającej λ i kąta padania Θ. Parametr k określa numer pokolenia supersieci. Grubość pojedynczej warstwy wynosiła 200 nm. Współczynnik załamania materiału A wynosił 1,544, materiału B był równy 3,4, a dla warstw wejściowej i wyjściowej wynoszą odpowiednio nin = nout = 1

Fig. 2. Charts for transmission T for the Severin's superlattice, depending on the incident wavelength λ and the angle of incidence Θ. The parameter k specifies the number of superlattice generations. The thickness of a single layer was 200 nm. The refractive index of the material A was 1.544, material B was equal to –3.4, and for the input and output layers were respectively nin = nout = 1 Rys. 2. Wykresy transmisji T dla supersieci Severina w zależności od długości fali padającej λ i kąta padania Θ. Parametr k określa numer pokolenia supersieci. Grubość pojedynczej warstwy wynosiła 200 nm. Współczynnik załamania materiału A wynosił 1,544, materiału B był równy –3,4, a dla warstw wejściowej i wyjściowej wynoszą odpowiednio nin = nout = 1

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Fig. 3. Charts for transmission T for the Severin's superlattice, depending on the incident wavelength λ and the angle of incidence Θ. The parameter k specifies the number of superlattice generations. The thickness of a single layer was 100 nm. The refractive index of the material A was 1.544, material B was equal to 3.4, and for the input and output layers were respectively nin = nout = 1 Rys. 3. Wykresy transmisji T dla supersieci Severina w zależności od długości fali padającej λ i kąta padania Θ. Parametr k określa numer pokolenia supersieci. Grubość pojedynczej warstwy wynosiła 100 nm. Współczynnik załamania materiału A wynosił 1,544, materiału B był równy 3,4, a dla warstw wejściowej i wyjściowej wynoszą odpowiednio nin = nout = 1

Fig. 4. Charts for transmission T for the Severin's superlattice, depending on the incident wavelength λ and the angle of incidence Θ. The parameter k specifies the number of superlattice generations. The thickness of a single layer was 200 nm. The refractive index of the material A was 1.544, material B was equal to 3.4, and for the input and output layers were respectively nin = nout = 2 Rys. 4. Wykresy transmisji T dla supersieci Severina w zależności od długości fali padającej λ i kąta padania Θ. Parametr k określa numer pokolenia supersieci. Grubość pojedynczej warstwy wynosiła 100 nm. Współczynnik załamania materiału A wynosił 1,544, materiału B był równy 3,4, a dla warstw wejściowej i wyjściowej wynoszą odpowiednio nin = nout = 2

Fig. 5. Charts for transmission T for the Severin's superlattice, depending on the incident wavelength λ and the angle of incidence Θ. The parameter k specifies the number of superlattice generations. The thickness of a single layer was 100 nm. The refractive index of the material A was 1.544, material B was equal to 3.4, and for the input and output layers were respectively nin = nout = 3 Rys. 5. Wykresy transmisji T dla supersieci Severina w zależności od długości fali padającej λ i kąta padania Θ. Parametr k określa numer pokolenia supersieci. Grubość pojedynczej warstwy wynosiła 100 nm. Współczynnik załamania materiału A wynosił 1,544, materiału B był równy 3,4, a dla warstw wejściowej i wyjściowej wynoszą odpowiednio nin = nout = 3

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Fig. 6. Charts for transmission T for the Severin's superlattice, depending on the incident wavelength λ and the angle of incidence Θ. The parameter k specifies the number of superlattice generations. The thickness of a single layer was 100 nm. The refractive index of the material A was 1.544, material B was equal to 3.4, and for the input and output layers were respectively nin = nout = 4 Rys. 6. Wykresy transmisji T dla supersieci Severina w zależności od długości fali padającej λ i kąta padania Θ. Parametr k określa numer pokolenia supersieci. Grubość pojedynczej warstwy wynosiła 100 nm. Współczynnik załamania materiału A wynosił 1,544, materiału B był równy 3,4, a dla warstw wejściowej i wyjściowej wynoszą odpowiednio nin = nout = 4

Charts from Figure 2 show the transmission for the parameters of Figure 1 with the revised value of the refractive index for the B layer. In Figure 3 you can see how the thickness of a single layer affects the transmission system from Figure 1, which is

B A 100 nmd d . Figures 4 to 6 show the effect of the external medium for the

propagation of electromagnetic wave. Refractive indexes of medium for Figures 4÷6 were respectively in out 2,n n in out 3n n and

in out 4n n

CONCLUSIONS

On the basis of the studies it can be concluded that the structure of transmission depends strongly on the polarization of the incident wave. The transmission has a band structure where for the consecutive numbers generations k there is an increase of the number of bands and reduction of their width at half maximum.

A change of material B for left-handed material material, changes the shape of the transmission bands (Fig. 1 and 2).

Decrease in the thickness of the single layer from B A 200 nmd d to B A 100 nmd d reduces the number of

bands and their movement (Fig. 1 and 3). Figures 4÷6 show the large impact of the change in refractive index of the environment on the map of transmission of Severin's superlattice. A shift of transmission bands towards lower angles appears. There should also be noted the phenomenon of tunneling of electromagnetic waves, noticeable in Figures 4÷6.

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