О числе энергетических уровней частицы в гребенчатой структуре

Multilayer model in optics.
New analitic results.
M.D.Kovalev BMSTU
mdkovalev@mtu-net.ru
Planar multilayer waveguide
For ТЕ-waves propagating along Oz axis this is a
boundary-value problem for the equation
2
2
c d Ey
2
2
 n j Ey   Ey ,
2
2
 dx
ñ óñëîâèÿìè
Ey (x j  0)  Ey (x j  0),
Ey (x j  0)  Ey (x j  0),
Ey ()  0.
Reduced variables
2
dE
y
2
2


E


E
j y
y
2
d




2
n
1
m

1
÷èñëî
ñëî¸â,
n

n
,
max
n

n

n
,

x
1
m

1
j k1

 
0
n
j
, 
ïðèâåä
é
ýôô
é
j
n
n
1
1
ïîêàçàòåëü
ïðåëîìëåí
ÿ
,



1



,t

ïðèâå
ÿ
òîë
k
j
First example. 7 layers. The
number of ТЕ-modes: K=6.
t

3
,
t

t

1
,
t

5
,
t

4
2
3
5
4
6




1
,




0
,
6
,


1
,
3
,


1
,
6
,


2
;
17

35
2
4
6

1
,
9039

;
1
,
6003
...;
1
,
5196
...;
1
,
280
...;
1
,
14
...
1
,
0
.
Traditional dispersion equations –equations
for the eigenvalues of the propagation
constant

• Type 1 – equation, is obtained by equating to
zero of the determinant of homogenius linear
system due to boundary conditions.
• Type 2 ― equation, obtained by the known
method of characteristic matrices
• This equations have too many terms if the
number of layers is more then 4.
• Investigation of waveguides with many layers is
now actual.
The properties of the dispersion equations
• Th.1. Type 1 equation has roots, coinsiding with the
refraction indexes of the inner layers of the waveguide.
This roots may not be the eigenvalues of propagation
constant.
Th.2. The set of roots of type 2 equation is exactly the set
of the eigenvalues of propagation constant.
• We propose a new one form of the dispersion equation.
This equation in some known cases have no parasitic
roots, and moreover it may be treated geometrically.
Multilayer equation


2 2
q


,
õàðàê
òèêà
ñë
j
j




Q
j

1





Q

q
è
Q

q
th
q
t

arth
,
3

j

m

1
,
21
j j

1j

1
j

1 



q
j

1








P
j

1




P

q
è
P

q
th
q
t

arth
,1

j

m

2
,
m

1
m
j
j

1
j

1
j

1




q
j

1







 

Q
P
j
j






th
q
t

arth

arth

0
.
jj

 



q
q
j
j
  



Homogenius variables,vectors a j
a
j
Q

;Q
,
2
2
b

j
*
j
Q
j


*
j

2
2
2
a


1
,
b


2
2
a
a


 
j

1
j
 

a


V
(

)
j

1
j




b
b
1
j

j
Vectors
A
j
P

; P
,
2
2
B

j
*
j
P
j

Aj


*
j

2 2
2 2
A


,
B


m
m

1
m
 
A
A


j

1
j
 

A


V
(

)
j

1
j




B
B
1
j

j


aj(
)è A
)
j(
Theorem 3. Vectors
• rotate counter-clockwise when
is decreasing.
  0
• Theorem 4. If

• then the directions of this vectors are
converging to the direction of Ox axis.
The multilayer equation in
vector form
o

q j (a j 1 , A j )
0


2
2
r
   (a j 1 , A j )
The formulae for the number of
TE-modes.




O
a
(

),
A

)
0
j
1
j(
P
p

m

1
m
K








P

arctg
lim



2
m

1

0
2
2
2
m

1
V j ( )
Transform
ch j

sh j
 r
 j
rj sh j 


ch j





r

, 
t


.



2
j
2
2
j
2
2
j
j
2
j
The second example, K=7.
The difference from the first example is
only
3 5 1,8


1
,
9232
...,
1
,
7046
...,
1
,
5802
...,
1
,
46
..
1
,
3068
...,
1
,
1263
...,
1
,
0022
...
Thank you for the attention
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Ковалев М.Д., Число TE- и TM-мод в многослойном
планарном волноводе со слоями двух типов. Электромагнитные волны
и электронные системы, 2009, т. 14, № 2, С.4--17.
Ковалев М.Д. Многослойное уравнение. Чебышевский сборник.
Тула, т. 7, выпуск 2 (18), 2006, С. 99—106.
Майер А.А., Ковалев М.Д. . Дисперсионное уравнение для
собственных значений эффективного показателя преломления в
многослойной волноводной структуре. ДАН, 2006, т. 407, №6, С.
766--769.
Ковалев М.Д. , Многослойная модель в оптике и квантовой
механике. ЖВМ и МФ, 2009, т. 49, №8, С. 1 -- 14.
Ковалев М.Д., Об энергетических уровнях
частицы в гребенчатой структуре. ДАН, 2008, Том 419,
№ 6, С. 1--5.