Документ взят из кэша поисковой машины. Адрес оригинального документа : http://shg.phys.msu.ru/educat/vinogradov/Lesson3.pdf Дата изменения: Wed Mar 12 16:11:34 2008 Дата индексирования: Mon Oct 1 20:38:16 2012 Кодировка: Поисковые слова: far side

, pc j =e rot e = 0 div j = 0 -- , (x,y), 1 2 (A.M. 1970 v. 59, pp. 110-115), , = 1 , = 2 . , .


. , , < J >=
eff



eff , .



j = 1
2

[

nвe

] ]

e = 1/ 1

2

[

nв j

n , xy. , .

[ [

j в n ] = 1 2 [ n в e ] в n = - 1 2 n в [ n в e ] = =- 1
2

( n ( ne ) - e ( nn ) )
2

= 1 2 e = 1/ 1 2 j
2

e в n ] = 1/ 1 2 [ n в j ] в n = -1/ 1 2 n в [ n в j ] = =-1/ 1
j = 1
2

( n ( nj ) - j ( nn ) )

[

e в n ] ,

e = 1 / 1

[

jвn

]


j e j' e'
j =e rot e = 0 div j = 0

j' e'

j = e, rot div e = 0 j = 0

1 =

2


( x, y ) ( x, y ) . ( x, y ) , 1 2 , ( x, y ) 2 . ( x, y ) 2 , 1 . , , ,

< J >=

eff

< E >


< E >= 1/ 1 2 n в J < J >= 1 2 n в E

< J >=

eff

< E >

1 2 n в E =

eff

{1/

1 2 n в J

}
eff

< J >=



1 2 n в E =

eff

2 eff

/ 1 2 n в E
2

= 1


. , . 1 , 2 , 1 2 . , , 1 2 ( , c1,2 ), 3 c3 , eff = 1 2 , 1 , 2 , , . , c1,2 + c 3 > 0.5 . . , 1 , , , 0.5, . , 0.5 .

2


, . . , , : .


. , . < j >=< E >=< > E =< >< E >



eff

=< >

. . , E = j / . < E >=< j / >=< 1 / > j =< 1 / >< j >



eff

= 1/ < 1/ >


- jx xx xy Ex 1 +2 1 +2 j = E = y yx yy y 1 +2 1 +2 1 +2 jx = jx , E y j y - jx =
1 1 + eff xy = eff ee 1 +
2



Ex E y

Ex - E
E
y

y



eff xx eff yx

- 1 +

2

2

- 1 +

2 2





xy 1 jx = Ex + E xx xx j y = yx Ex + yy E
y y

jx = jy


xx yx

xy Ex yy E y
y

xy 1 Ex = jx - E xx xx j y = Ex =
xy xx xy



yx xx

jx -

yx
xx

xy

E y + yy E

y

1 Ex = jx - xx j 1 jx = 1/ j
y y

E

y

=


xx

yx xx

jx - E

yx
xx

-

yy

E

y

Ex +
xx xx

xy / 1/

y

xx xx

=

yx / 1/

xx

yx / xx xy / Ex + 1/ xx

-

yx xy - yy
xx

xx

E

y


.
1 0 1 = 0 1 , . e ( x, y ) 2 0 2 = 0 2
eff

=

11 21

12 22

n -- , t -- , ,

--
2

j ( x, y

)

--
2

e ( x, y ) = e1 1 + e2

j ( x, y ) = j1 1 + j2

1 , 2 ,
e1 , e2 j1 , j2






(

e1 - e2 ) t = 0

(

j1 - j2 ) n = 0

e ! ! !
e0 = m1e1 + m2 e2

m1 e2 = e0 - en m2 e j1 - j2 =1 e1 - 2 e2 = (1 - 2 ) e0 + ( m21 + m1 2 ) n m2 e1 = e0 + en ,

n ,

e =-m2 n ( m21 + m1 2 ) n

-1

[

n (1 - 2 ) e0

]



j0 = m1 j1 + m2 j2 = = m11 e1 + m2 2 e2 = m1 = m11 ( e0 + en ) + m2 2 e0 - en m2 = (m11 + m2 2 ) e0 + m1e ( 1 - 2 ) n

, e = -m2 n ( m2 1 + m1 2 ) n

-1

[

n (1 - 2 ) e0

]

[ n (1 - 2 )] 1 - 2 ) n e0 j0 = m11 + m2 2 - m1m2 ( n ( m2 1 + m1 2 ) n


,
j0 = eff e0



eff ij

=
2 ij

= ( m11 ij + m2 +

)

-

nl ( m21 lm + m1

mm2 1

2 lm

)

nm

(

1 ik -

2 ik

)

nk n p (1 pj -

2 pj

)

c


-

, core = 1 , shell = 2 . ext = 0

, .


E = 1.
0



core = acore r cos( )
shell

= (a

shell

r + bshell / r 2 ) cos( )

extern =-r cos( ) + (bextern / r 2 ) cos( )
acore = ashell + bshell / r13

-1 + bextern / r23 = a

shell

+ bshell / r23





core

acore =

extern

ashell - 2bshell / r13 shell -1 - 2bextern / r23 = shell ashell - 2bshell / r23


bext 1 1 1 m2 = + ( 0 - 2 ) m1 ( 1 - 2 ) 3m1 2

ern

=0

m1 , m2 : r13 m1 = 1 - m2 = 3 r2 , . , ­ . -, . , . , .


:

1 1 1 m2 = + ( 0 - 2 ) m1 ( 1 - 2 ) 2m1

2

(2 HS) = 0 . (1) HS ,

1 1 1 m1 = + (1) ( HS - 1 ) m2 ( 2 - 1 ) 2m2

1



A raytracing program has been usedtocalculate ray trajectories in the cloak assuming that R2 >> . The rays essentially following the Poynting vector. (A) A 2D cross section of rays striking our system, diverted within the annulus of cloaking material contained within R1 < r < R2 to emerge on the far side undeviated from their original course. (B) A 3D view of the same process.


A point charge located near the cloaked sphere. We assume that R2 << , the near field limit, and plot the electric displacement field. The field is excluded from the cloaked region, but emerges from the cloaking sphere undisturbed. We plot field lines closer together near the sphere to emphasize the screening effect.


, . . , 1928, (Voigt)



(V ) eff

=< >= mi
i

i

1929 (Reuss) , , :



( R) eff

=<

-1

> = mi i
-1 -1 i

-1


, . -.
(V ) eff



(R) eff

.


,
E0 i
eff ij

E0 j =

E = -; < E >= E0

inf

< Ei ij E j >

E ( x ) = E0 , , inf E =-; < E >= E < Ei ij E j >
0

E0i

eff ij

E0 j =

E =-; < E >= E0

inf

Ei ij E

j

E0i ij E0

j

= E0i < ij > E0

j



eff ij



ij

=

(V ) eff ij


, j0i
-1 eff ij 0 j

j=

divj = 0; < j >= j0

inf

- ji ij1 j

j

= (

- j0i ij1 j0

j

= j0i

-1 ij

j0

j



-1 eff ij



-1 ij

( R ) -1 ij

)

. 1) ( ). , . 2) , ­ . , . , , .


, , W01 = e01 eff ( i ) e01 eff ( i ) , . , W01 . , , e01 . . , " e01 " e02, e01 W02 = e02
eff

( i )

e02


"- " We = inf
i : i = pi

(

e01

eff

( i )

e01 + e02

eff

( i )

e02

)

. .
. F ( v0 ,
F ( v0 ) F ( v0 +

)



)

2 :{ = 0, a jk x x = 0 j k


F ( v ) = det ( v ) = e11e22 - e12 e22 F (v ) = E T E

e11 v= e12

e21 e22

0

0

0

1

E = {e11 , e12 , e21 , e22

}

0 0 -1 0 T= 0 -1 0 0 0 0 0

, . 1 e11 v= e12 e21 1 / x1 2 / x1 = 1 / x2 2 / x2 e22

, F (v) = E0 T E0 + k11k det vk = k21k k1 k2
2k 2k


k 0

Ek T Ek = E0 T E0 E T E = E T E

=0


E0 eff E0 = = inf
E:< E >= E0 E =-

E ( - t T ) E + tE T E
E

inf E ( - tT ) E + tE0 T E0 :< E >= E
E =-
0

inf f ( x ) + ( x ) inf f ( x ) + inf ( x ) f ( x ) = f 0 ( x ) + inf f ( x ) ( x ) = 0 ( x ) + inf ( x )

f0 ( x ) 0

0 ( x ) 0

inf f ( x ) + ( x ) = inf f 0 ( x ) + 0 ( x ) + inf f + inf = = inf f 0 ( x ) + 0 ( x ) + inf f + inf inf f + inf


E0 eff E0 = = inf E ( - tT ) E + tE T E E:< E >= E0
E =- E: E =-

inf E ( - tT ) E + tE0 T E < E >= E0

0

,
E: E =-

inf E ( - tT ) E E0 ( - tT < E >= E0

)

-1 - 1

E

0



eff



(

- tT

)

-1 -1

+ tT

, . .. tmax ­ , - tT


Y-

Y ( eff ) = -( p1 2 + p21 ) - p1 p2 (1 - 2 ) (
Y ( eff , 1 , 2 ) = Y -1 (
-1
eff

eff

- p11 - p2
-1 2

2

)

-1

(1 - 2 )

,
i

-1 1

,

)

Y (i , 1 , 2 ) = -

eff

= p11 + p2 2 - p1 p2 (1 - 2 ) (Y + p1 2 + p21 Y-

)

-1

(1 - 2 )



eff



(

- tT

)

-1 -1

+ tT



Y ( eff ) + t
max

max

T 0

t det[ - t
max



T] = 0




eff

eff 1

0

eff = 0

0 = eff

0 0 0



eff 2

0 0

0 0



eff 1

0 0 0

0



eff 2

Y ( eff 1 ) 0 0 0 Y ( eff 2 ) 0 Y ( eff ) == 0 0 Y ( eff 1 ) 0 0 0 Y

0 0 0 (
eff 2

)
2

det(Y ( eff ) + tT ) = (Y 2 (

eff 1

)+t

2

)(

Y (

eff 1

)Y (

eff 2

)-t

)
2

0



- tT

12 - t 2 0

2 2 - t2 0


2 min

t = min [ 1 ,

]

Y ( 1 )Y ( 2 ) -

0



W j = inf
i :( i ( = pi

(

j01

-1 eff

( i )

j01 + j02

-1 eff

( i )

j02

)



Y (1 / 1 )Y (1 / 2 ) - (1 /
-1 eff

)

2 min

0

Y ( eff , 1 , 2 ) = Y -1 (

,

-1 1

,

-1 2

)

Y ( 1 )Y ( 2 ) -

2 max

0




eff

= p11 + p2 2 - p1 p2 (1 - 2 ) (Y + p1 2 + p21

)

-1

(1 - 2 )


Y ( 1 )Y ( 2 ) =
2 min

(0, ), ( , 0 ) , ( 1 ,

1

)

1 1 -

eff 1

1 + 2 -

eff 1

1 1 = m2 2 -
HS 1

eff 1

m1 1 + m2 2 ef

f1

( R , V ) , ( V , R ) , (

,

HS 1

)


Y ( 1 )Y ( 2 ) -
2 max

0

( R , V ) , ( V , R ) , (

HS 2

,

HS 2

)

1 1 1 1 p1 + = +1 1 - 2 2 - 2 p1 2 - 2 p1 2

2




1

A
V



B
R


R HS1

2



HS2



V






rot E = 0

,
=- grad E



div D = 0



div ( grad ) = 0 -- r'=ar , r n . , .


G-, , . , , / L , L -- . . , ( ), a ( x ) . , x=x/ .



d dx u
()


=0





d a ( x / ) dx

0 . = u0 ( x, ) + u1 ( x, ) + 2u2 ( x, ) + ... , = x / (1)

ui ( x, ) =x/.


,
d dx

x x a ( ) F x, = a ( ) x F x,

-1 a ( ) F ( x, ) +

d2 dx 2

x a ( ) F x ,

x -1 d = dx a() x F x, + a() F ( x, ) =

d = a ( ) dx x 2 = a ( ) 2 F x -2 a ( ) +

x -1 d F x, + a ( ) F ( x , ) = dx 2 x -1 x, + a() F ( x, ) + -1a() F ( x, ) + x x F ( x, )


(1) , , 2 . -,




-2

i

u0 (x, ) aij ( ) =0 j



-1

u0 ( x , ) u0 ( x , ) u1 ( x, ) a ( ) + a ( ) x + x a ( ) =0



0



u 2 ( x, ) a ( ) u1 ( x, + a ( ) x

+ ) +

u1 ( x, ) a ( ) + x x u0 ( x , ) a ( ) - f ( x) = 0 x






-2

i

u0 (x, ) aij ( ) =0 j

(2)

u0 ( x, ) = C ( x) / a () , ,

(
0

1

d u0 ( x, ) / ) d = u0 ( x, ) = 0 = C ( x ) C( x) 0 a () 0 0

1

1

x: u0 ( x, ) = v0 ( x )


-1



u ( x, ) u ( x, ) u ( x, ) a ( ) 1 a ( ) 0 a ( ) 0 + + =0 x x

u0 ( x, ) = v0 ( x ), u1 ( x, ) v0 ( x ) + a () = 0 x

u1 ( x, ) v0 ( x ) + = C1 ( x ) / a ( x

)

(4)


C1 ( x )
v ( x) v ( x) d C1 ( x) = 0 / =0 1/ a x a () x 0
1 -1



u1 ( x, ) v0 ( x ) + = C1 ( x ) / a ( x
-1

)

u1 ( x, ) v0 ( x ) 1/ a = a ( x

)

- 1

u1 ( x, ) =


0



a -1 a (

-1

)

d v0 ( x ) + D( x ) -1 x


0 u2 ( x, ) u ( x, ) a () + a () 1 + x u1 ( x, ) u ( x, ) a () + a () 0 =0 x x (5)

+ x

(5) 0 u1 ( x, ) u0 ( x, ) x a ( ) + x a ( ) x d = 0 1
u1 ( x, ) v0 ( x ) 1/ a = x a (
-1

)

- 1

u0 ( x, ) = v0 ( x )





1

0

a -1 -1 2 a () v0 ( x) -1 a () x 2 a -1 -1 2 0 v0 ( x) = a ( ) a ( ) x 2 1 a
-1 -1

+

2 v0 ( x) a () d = 2 x -1 -1 2 v0 ( x) 0 d = a d = 0 2 x 1

2 v0 ( x ) =0 2 x
a
eff



=a

-1 -1


. j / a ( ) = E ( ) ,
n

1 E= a (

)

jn

, . , .