!#%

!"#$%&'()*"$+,." /"!!"*

!""#$"%&'()*"+",'#-()&*"'!.0" "++"*1

2"#.%03

4#$"%&%*& "+",'#-()&*"'!.0"# "++"* (!' "!*"( 56"!7890#"*7:;.'"( <=#!->"*? *)0 >"(

@A%.07A%++B72"#.'C#,"#D#!*$!DEF"#:"*>"#!.'."+G.'"##"&'3

4!*" $%HC++!&" 4##"&%*& >". 4!*&)*&.,#"!.". <$6!.0"* I)'0->" %*> J!''"#? G#!*&' >"* )G&"7

.'!(('"*K%.&)*&.,#"!.<K*->"*,#"!.?$%(:06!*&"*E

!!"#$%&'()*+ ,-&$-."&/-'-&0120351#0%&'67%3587%##9

!" #$%%$&'%$(*+&' ,-. $(& (/01.02$/3 *040'$&$/ 5$6(+4

4(1

σ = 0

34(5%-6%'(+1'%'%-7+8*,%$'%9

B ~ = µ · H; ~ D ~ = ε · E; ~ ρ = 0; ~j = 0

: ";<

=+*.'$+8'%-5.%>+?@%$$,$%.18-,%-9

∇ × ~ H ~ = ε · E ~ ˙

: "A<

∇ × ~ E ~ = − µ · H ~ ˙

: " <

∇ · ~ H ~ = 0

: "B<

∇ · ~ E ~ = 0

: "C<

D8/E$%.18-,:F"A<%(,.6'/.16%.D-@%-58-,G)-

∂

∂t

∇ × ~ H ~ ˙ = ε · E. ~ ¨

: "F<

D8/E$%.18-,:F" <%(,.6'/.16%.D-@%-58-,G)-

( ∇× ~ )

∇ × ~ ∇ × ~ E ~ = − µ · ∇ × ~ H. ~ ˙

: "H<

I*J)(*8-,9

∇ × ~ ∇ × ~ E ~ = ∇ ~ ( ∇ · ~ E) ~

| {z }

!"#$%!&#'#()

− ∆ E ~

: "K<

I-'%(L%(@%-58-, G)-:F"F<$+8'%'E$%.18-,:F"H</1$.%M$.1

∆ E ~ = µε E. ~ ¨

: "N<

#6%-/)%(1O$'*+-58(1D-@%-58-,G)-

( ∇× ~ )

^+8JE$%.18-,:F"A<8-5G)-

∂

∂t

+8JE$%.18-,

:F" <

∆ H ~ = µε H. ~ ¨

: ";!<

=.%//.-55.%2%$$%-,$%.18-,%-J4(

E ~

^8-5

H ~

^"

!7 8$%69$:($*+&'$& (4 8;%% $9$&$. #$%%$&

#.-+$$,%*%.-%(D-/+'PJ4(%6%-%2%$$%-G)-$.-&/-+1(%1'//)@.%G)-(%1'/-+1$.-&/%-'$+-,

5%(?QD1/%$+8'%'9

U = f t − x

v + g

t + x v

: ";;<

U

^{%(J4$$'5+6%. 5.%2}%$$%-,$%.18-,

∆U = 1

v ² · U . ¨

: ";A<

3$4$)56

7$)

f ^′ = ∂f

∂ t − ^x _v

∂f

∂x = f ^′ · ∂ t − ^x _v

∂x = − 1 v · f ^′

∂ ² f

∂x ² = _v ¹ 2 · f ^′′

∂ ² f

∂t ² = f ^′′



 

 

→ ∂ ² f

∂x ² = 1

v ² · f; ¨

^8,/9%-:;0

g t + x

v

< !2 =

>),?$0-%$)*.)AB%$)*+,-<C!D=E<C!2F=$0-)'A:;0&)$G*/5$,-$5*4),&)-H$)A&$0$%$HA09./-,$I

A)5*$,1$%%$6

v = (ε · µ) ^{− 1 2}

^{< !2J=}

K)$ B%$)*+,-$, <C!"= +,& <C! = 5),& ,+0 (+ $0:;%%$,E 4$,, &)$ #$%&0)*A+,-$, 5$,H0$*A (+0

8+5'0$)A+,-50)*A+,-&$01$%%$90)$,A)$0A5),&!

3$)5L)$%6 >'$,$ 1$%%$ .)A #0$M+$,(

ω

^{N9, %),H5 ,/* 0$*A5! K)$ 1$%%$ 5$) ), O)*A+,-}

y

%),$/0L9%/0)5)$0A!

E _y = E _◦,y · e ^i(wt−kx)

^{< !2P=}

.)A

w = v · k; E x = E z = 0.

^{< !2C=}

8+5

∇ · ~ E ~ = 0

^:9%-A&/,,

∂E x

∂x + ∂E y

∂y + ∂E z

∂z = 0

^{< !2Q=}

K/.)A-)%A:;0+,5$0$,#/%%6

E y (y) = const;

^(+-$-!

x, z, t

^{).-$5/.A$,O/+.! < !2R=}

1)$)5A&/,,

H ~

90)$,A)$0AE+,&4)$%/+A$A&)$G*/5$,'$()$*+,-0$%/A)N(+.

E ~

^{I#$%&SK$0T+5/.I}

.$,*/,-$0-)'A5)*/+5&$.U,&+HA)9,5-$5$A(EB%$)*+,-<C! =6

∇ × ~ E ~ = − µ · H. ~ ˙

^{< !2D=}

#;0&)$$'$,$1$%%$

E _y (x, t)

N$05*4),&$,'(-%!

∇ × ~ E ~

^5949*%&)$

x

^E/%5/+*&)$VIW9.L9,$,A$!

#;0&)$

z

IW9.L9,$,A$$0-)'A5)*

( ∇ × ~ E) ~ z = ∂E y

∂x − ∂E x

∂y = ∂E y

∂x =

= E ◦,y ( − ik)e ^i(wt−kx) = − µ · ∂H z

∂t

< !"F=

T$)A%)*$U,A$-0/A)9,%)$:$0A5*%)$X%)*6

ik Z

E ◦,y e ^i(wt−kx) dt = µ

Z H ˙ z (x)dt

E ◦,y · ik

iw e ^i(wt−kx) = µH z (x, t)

→ H z (x, t) = 1

µ · v · E y (x, t).

< !"2=

4,0 2&/5(6

E ~

^7.8

H ~

^9:&%8 0(&2&. 0&.'()&2(,7;&/.,.8&)7.8 0/.8/. <2,0&6;,%%0

µ

^7.8

ε

)&&%%

0/.8 =>-%" ?,@" A" "BC" D/( E%&/27.- =A"BFC7.8 +/(

µ = µ ◦ µ r , ε = ε ◦ ε r

^{&)-/G( 0/2 ;H) 8/&}

I&)'.H@;7.-8&):&%8,+@%/(78&.

H _z (x, t) = r ε ◦

µ _◦ · r ε r

µ _r · E _y (x, t).

^{= "!!C}

4&J./(/*.8&0:&%8K&%%&.K/8&)0(,.80L

Z = E _y H z

= r µ _◦

ε ◦ · r µ _r

ε r

= "! C

M+I,'77+L

Z ◦ = r µ _◦

ε ◦

= 377Ω

^{= "!FC}

!!"#$%&'()*+,-".&/".-%&' 01&

E ~

^%&$

H ~

^23-."&..!.&.4.##."& 567"9/%&'

! "#$%$ &$''$% (% )$*'+,-#$./0-$-$% 1$2($%

! !" #$%&'()*$+,-)$)$ .$%/0))010)2) *34!5'67 .$%/$,*0&0)2)8

σ = 0

M.2*2;)&N7&.(&.3&20&%;&%8&).()/((&/.&;)&N7&.O,G2P.-/-&<2,0&.>&)02/&G7.-OK/02&.<*9

%,)/0,(/*.

P ~

^7.8

E ~

^GOK" OK/02&. D,-.&(/0/&)7.-

M ~

^7.8

H ~

^,7;"

P ~

^7.8

M ~

&/%&. .,2 =Q&%,9 R,(/*.C"M. 8&)SP2&>*.Q&0*.,.O&.=O"T" 4/@*%02K/.-7.->*. D*%&'H%&.6M*.&.70K""""C /0(

8/& :)&N7&.O,G2P.-/-'&/(G&0*.8&)0 ,70-&@)P-(=>-%"7.(&.0(&2&.8&UGG/%87.-C"M+ :,%%0/.709

;V)+/-&)$))&-7.-',..+,.'*+@%&R&

ε r

^7.8

µ r

G/%8&.L

D ~ = ε ◦ E ~ + P ~ = ε ◦ ε r E; ~

^+/(

ε r = ε ^′ − iε ^′′

B ~ = µ ◦

H ~ + M ~

= µ ◦ µ r H; ~

^+/(

µ r = µ ^′ − iµ ^′′

^{= "!WC}

!!"#$%&'()(* !+,&'"'-."/.&01&

ε ^′

^%&$

ε ^′′

^{"& $.2}34'.!%&'."&.2!2."/.& !56) 7+49#.&:.;

71&9&5!."

f r

!!"#$%&'()<*=."7>".#?@2$". !+,&'"'-."/$.7:.9#/."#7

α

^$.2A1#92"7".2!92-."/01&$.2B2.C%.&5 DA1#92"7".2!92-."/

α

."&.7 E1#.-@#7!56) /147*

~ p = α · E ~

^F)

!"#$"%&'()*%+,+-",. /"! 012'!,3,"!('!-",.4 #$'2,.'.,5

4+5/&6708/)'7.-'*+9%&:&)

ε r , µ r

^;77.(&)072&.<=&)8&.5&.8/)0(,((>%&/27.-?@"ABCDE)

E _y

^5&.6.0,(;F

E _y = E _◦,y · e ^iωt−γx

^{? "G@C}

$/.0&(;&./.5/&3&%%&.-%&/27.-?@"HC&)-/I(

∂ ² E _y

∂x ² = γ ² E y = − εµω ² E y

γ ² = − µεω ² = − µ ◦ ε ◦ ω ² (µ ^′ − iµ ^′′ )(ε ^′ − iε ^′′ )

J&(;&

γ = α + iβ;

^{5,..&)-/I(0/2}

α = ω

c 1

2 [ | ε r | · | µ r | − (µ ^′ ε ^′ − µ ^′′ ε ^′′ )]

¹ 2

β = ω c

1

2 [ | ε r | · | µ r | + (µ ^′ ε ^′ − µ ^′′ ε ^′′ )]

¹ 2

.

^{? "GKC}

L&/09/&%FME)

µ ^′′ = ε ^′′ = 0

^&)-/I(0/2

α = 0

→

N&/.&OP+9D7.-

β = ω c

p µ ^′ ε ^′ .

^{? "GQC}

→

&.(09)/2(

k

^.,2?@"A!<@"ABCDE))&&%%&

µ, ε

^"

O,+/(&)-/I(0/2/+,%%-+&/.&.M,%%5/&-&5P+9D(&3&%%&

E y (x, t) = E ◦,y · e ^i(ωt−βx) · e ^−αx .

^{? "GHC}

6+9%/(75&.=&)%,7DF

| E _y (x) | = | E _y (0) | · e ^−αx

+/(

α = − 1 x ln

E _y (x) E y (0)

.

^{? " #C}

α

^{2,( 5/&} $/.2&/(

N p/m

?R&9&) 9)*S&(&)C" >&I)P72%/2&) /0( 5/& 6.-,I& 5&) OP+9D7.- /.

db/m

?O&;/I&%9)*S&(&)C&.(09)&2&.55&)O&T./(/*.

α ^′ = − 1

x · 20 · log ₁₀

E y (x) E y (0)

.

^{? " AC}

E y

^7.5

H z

^0/.5-&+PU>%&/27.-?@"G CEI&)5&.M&%58&%%&.8/5&)0(,.5=&)'.E9D(F

Z = E y

H _z = Z _◦ ·

r µ ^′ − iµ ^′′

ε ^′ − iε ^′′

^{? " GC}

5"2"<

E y

^7.5

H z

^0/.5-&-&.&/.,.5&)92,0&.=&)02*I&.<,7U&)8&..-/%(F

µ ^′′

µ ^′ = ε ^′′

ε ^′

?-%&/2&V&)%70(8/.'&%

tgδ

^{C ? " C}

! !" #$%%$&'()*+$,-(&. *$, $&/%,1$+ 2$,-341,.5$,-

σ

45*2($#$-*/.-$-$%7-89$:$'-$2/,;.<,.$=$,$->?

~j = σ · E ~

^{@ ! AB}

C/<(-D%2$*-,(.='$(.+%:@E!FB>+

∇ × ~ H ~ = ε · E ~ ˙ + σ · E. ~

^{@ ! 3B}

C($='$(.+%:$%@E! GE!3B:$'-$%H$(-$*.(%+%)$*D%2$*-!C($,05.*->+?

∆ E ~ = µε E ~ ¨ + µσ · E ~ ˙

^{@ ! EB}

∆ H ~ = µε H ~ ¨

| {z }

!""!#$"$%

+µσ · H ~ ˙

| {z }

&'()*+(,#+-#.!("/

I$*:'$(.$F!4(J,.$,=$,$->(%2$*K.$*<82L%/<(J?

∆K = 1 D · ∂K

∂t

^{@ ! MB}

<(-

K =

N8%>$%-*/-(8%

, D =

C(9+,(8%,J8$O>($%-

"(%,$->$%2$,P%,/->$,

E _y = E _◦,y · e ^iωt−γx

^{05*$(%$$#$%$Q'(%!R8'!&$''$(%@E! EB$*:(#-}

γ ² = − µω ²

ε − i σ ω

.

^{@ ! SB}

T#$*2/,U%2+J-(8%,:$,$->#$J8<<-</%05*

Z

^?

Z = E _y

H z

= iωµ

γ ;

^{+%2<(-@E! SB?}

Z =

r µ

ε − ^iσ _ω ;

^{@ ! VB}

W$( $(%$* :$:$#$%$% 4*$X+$%>

ω

^{J/%% /',8 $(%0/.}

_ε ^σ

◦ ω

^>+< (</:(%D*$% P%-$('

ε ^′′ (ω)

^/22($*-

H$*2$%Q+<2($ &$''$%/+,#*$(-+%: >+#$,.*$(#$%! C$*4*$X+$%>:/%:

ε ^′′ (ω)

^{(,-/#$*)Y''(: )$*G}

,.($2$%)8<Z$(-+%:,/%-$('!

! ! 6$-'%%$ (&/ 75,&$8$5-$

&(* #$-*/.-$% 2($ P+,#*$(-+%: $(%$* '(%$/* R8'/*(,($*-$%Q $#$%$% &$''$ (% $(%$< [/'#*/+<Q

#$:*$%>-2+*.$(%$1$-/''R'/--$!

!!"#$%&'()*+ ,-$.01-20#-"3-&

S 1

^%&$

S 2

^4%56-5-0&%&' $-728"&-9-817)

$/-&.02,4(&.5&)+&(,%%/02&.3,.56

78)&%&'()/02&7&%5'*+9*.&.(&.9,),%%&%:;)<&(,%%*=&)>?2&-/%(

E _k = ρ · j _k

^{@ "A#B}

C,=&//0(

ρ

^5&)09&:/D02&&%&'()/02&3/5&)0(,.5"E;4-);.55&0'%&/.&.

ρ

^{&)-/=(0/25,..}

E _k

E ◦ −→ 0; H _⊥

H ◦ −→ 0.

^{@ "AFB}

@G-"/.5;:/&)(&)3/)=&%0()H+&I

U i = − φ ˙ → 0

^B

E.5&)&)0&/(0J&)%,.-&.5/&K(&(/-'&/(0=&5/.-;.-&.5/&K(&(/-'&/(J*.

E _k

^;.5

H _k

^,.5&)<&(,%%L

*=&)>?2&"

C,+/( 0(&2( /. -;(&) M?2&);.- ,. +&(,%%/02&. N=&)>?2&.

E ~

0&.')&2( ,;4 5&) +&(,%%/L 02&.3,.5I

H ~

9,),%%&%:;)3,.5*=&)>?2&"

C,0 7&%5J&)2,%(&. /. 5&) N=&)>?2&.02/2( ',.. 5;)2 E.G&.5;.- 5&0 O.5;'(/*.0-&0&(L

:&0 ,;4K2%&/4&

S ₂

^{=:G" 5&0} C;)2>;(;.-0-&0&(:&0 ,;4K2%&/4&

S ₁

&)+/((&%( G&)5&.I G*=&/

dy

4.'$(0$

S 1

⁶

∇ × ~ H ~ = ~j +

! ∂ ~ D

∂t

"

| {z }

! !"#$!%

~j

^&'(!%")+,-../ !"

→ I

S1

Hd~s ~ = I;

H z (y + dy)dz − H z (y)dz = j x (y) · dydz

→ ∂H z (y)

∂y = j x (y)

^{7 !89:}

4.'$(0$

S 2

⁶

I

S2

Ed~s ~ = − ∂φ _S2

∂t

dΦ _S2 = B _z (y) · dydx;

^;(-

B _z (y) = µ _◦ µ _r H _z (y)

− E _x (y + dy)dx + E _x (y) · dx = − ∂B z (y, t)

∂t dydx

→ ∂E _x (y)

∂y = µ ◦ µ r

∂H _z (y, t)

∂t

^{7 !8 :}

<%,/-=0>*

H z (y, t)

^(;1$-/''6

H z (y, t) = H z (y) · e ^iωt

^{7 !88:}

?($@AA*2(%/-$

x

7<+,#*$(-+%B,*(.-+%B:C(*20$,-B$./'-$%!"(%B$,$-=-(%7D!8 :+%2;(-

E x (y) = ρ · j x (y)

^$*B(#-,(.

∂j _x (y)

∂y = iωµ _◦ µ _r

ρ · H z (y).

^{7 !8E:}

F/.

y

2(G$*$%=($*-+%2;(-H'$(.+%B7D!89:$*.I'-;/%,.'($J'(.

∂ ² j x (y)

∂y ² − iωµ ◦ µ r

ρ j x (y) = 0.

^{7 !8D:}

?($/''B;$(%$KL,+%B'/+-$-6

j x (y) = j x (0) · e ⁻

q iωµ◦µr ρ ·y

( + ˆ j _x (0) · e ⁺

q iωµ◦µr ρ ·y

)

^{7 !83:}

ˆ j _x (0)

^+%22/;(-2$*$(%B$M'/;;$*-$<+,2*+M;>,,$%F+'',$(%C$B$%2$*)$*'/%B-$%"%2'(.M$(- )A%

j x (y)

^0>*

j → ∞

^!<+0-*$%%$%)A%N;/B(%I*O+%2P$/'-$(';(-

√ i = r 1

2 + i r 1

2

^$*B(#-

7 !8Q:

j x (y) = j x (0) · e ^{− y δ} · e ^{−i y δ}

^{7 !8R:}

+/(

δ = r 2ρ

ωµ _◦ µ _r ;

!"#$"%&% 4 "5#6

'%"()"%*+

ρ Cu,300K = 1.7 · 10 ⁻⁸ Ωm

f = 100M Hz; µ ◦ = 4π · 10 ⁻⁷ ; µ r = 1

→ δ = 6.6 · 10 ⁻⁶ m = 6.6 µm.

^{4 "576}

8/(9%&/2:.-4;"<=6&)-/>(0/2?:)2@.(&-),(/*.

H z (y) = − δ

1 + i j x (y).

^{4 "5=6}

A,02&/B(C

H

^DE&%?:.?F()*+?/2(&0/.?>&/&/.&)G/&H&

y

^/+8&(,%%I&J&/%0

45 ^◦

^,:B&)K2,0&"

L:H-):.??&0M2+02&.9&0&(N&00/.??,-&-&.

E x (y)

^:.?

j x (y)

I&J&/%0-%&/2O2,0/-"

FO&N/&%%,.?&)P&/(&)*>&)QR2&-/%(S

H _o , z = H _z (0) = − δ

1 + i j _x (0).

^{4 "5 6}

E*)(HT2):.-?&0*>/-&.'%"()"%*(HT)&/.&,-)&%./0#1S 3/&/0(?,0U&)2R%(./0

E x (0)/E y (0)

^V

E _x (0) = ρ · j _x (0) = − ⁽¹⁺ⁱ⁾ _δ · ρ · H _z (0)

| E x (0) | = √ ρωµ ◦ µ r | H z (0) |

ET)

f = 100M Hz :

4 "5<6

| E x (0) | ≃ 3.66 · 10 ⁻³ | H z (0) | E y (0) = 377Ω · H z (0)







| E _x (0) |

E y (0) ≃ 10 ⁻⁵

^{4 "556}

A,02&/B(C?/&*>/-&L..,2+&

E k

E o → 0

^/0(/.-:(&)WR2&):.-&)HT%%("

23-"40*%#$% 5%"$(7"7$1"!%

δ

⁺

X+ ?&. M2+02&. 3/?&)0(,.? ?&) +&(,%%/02&. 3,.? :.? ?,+/( ?/& 3,.?Y&)%:0(& &/.H,2

>&)&2.&.N:'Z..&.CY&)J&.?&(+,.H*%-&.?&0$)0,(N+*?&%%S

!!"#$%&' ()*+ ,-./0123$4## 1%- ,-2"00#%&' $4- 5378-49%4&1:4-#%.0#4".0%&' ;-3 ,"&74"0.<=74

/&>4"04-3!4-<=74&)

4/,"*,/-5672$''#$,.*$(#-*$.-9$%/+2($-/-,:.'(./%0/''$%2$%;.6,.$%<$*'+,-$,7=($

2$% !"#$%&'() >

I _x ^′ = Z ∞

y=0

j x (y)dy = δ

1 + i j x (0)

^{? !@AB}

= − H z (0)

^{? !@CB}

#$( <$*=$%2+%9 2$* D'$(.+%9$% ?A!E3B +%2 ?A!@FB! 4($,$* (,- 2/6(- /+. $%-,G*$.$%2 2$6

4+*.H+-+%9,9$,$-5(%I./,$6(-

H _z

^2$*&$''$J+%2#$-*/9,9'$(.>

I ^′ = H z (0)

^{? !@KB}

L',,G$5(M,.$%N':.$%=(2$*,-/%2

ρ ^′ /Ω

^{2$M%($*-6/%2$%}&(2$*,-/%2$(%$,;#$*H:.$%,-OP$, 6(-"(%.$(-,':%9$J"(%.$(-,#*$(-$+%24(P$

δ

^>

ρ ^′ = ρ · l A = ρ

δ ρ ^′ =

r ρ · ω · µ _◦ µ _r

2

^{? !@3B}

4/%% 9('- 0O* 2($ $%-,-$.$%2$ <$*'+,-'$(-+%9 (% $(%$6 ;#$*H:.$%,-OP

dx, dz

^{#$( L+0-*$Q$%}

$(%$*./*67%(,.$%&$''$6(- 6/9%!N$'2/6G'(-+2$

H ◦

^>

dP = 1

2 H _◦ ² · ρ ^′ dxdz.

^{? !ARB}

*#+,&-.&+/&+0

•

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∇ × ~ H ~ = ~j + D; ~ ˙ |· − E ~

46)2J6(6.-0-&0&(D

− E ~ · ∇ × ~ H ~ = − ~j · E ~ − D ~ ˙ · E ~

∇ × ~ E ~ = − B; ~ ˙ |· H ~

K.:6'(/*.0-&0&(D

H ~ · ∇ × ~ E ~ = − H ~ · B ~ ˙

^E"#LMH

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| {z } = − H ~ · B ~ ˙ − ~j · E ~ − D ~ ˙ · E ~

∇ · ~

E ~ × H ~

= − H ~ · B ~ ˙ − σ · E ~ ² − D ~ ˙ · E ~

?/(

D ~ = ε · E; ~ B ~ = µ · H ~

^-/%(:,..

∇ · ~ E ~ × H ~

+ σ ~ E ² + µ ~ H · H ~ ˙ + ε ~ E · E ~ ˙ = 0;

·

Z dV

→ I

A

E ~ × H ~ d ~ A

| {z }

1

+ σ Z

V

E ~ ² dV

| {z }

2

+ ∂

∂t Z

V

1

2 µ ~ H ² + 1 2 ε ~ E ²

dV

| {z }

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S ~ = E ~ × H; ~ /W/m ² .

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U(~x, t) = ~ Re ~u(~x) · e ^iωt

= 1 2

~u · e ^iωt + ~u ^∗ · e ^−iωt

~u(x) · e ^iωt

^{);%(&A$)()$5+19.$B$}8&';%$../,2C+,

U . ~

=D'*'+(/5%$

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U ~ · V ~ = 1

2 Re[~u ^∗ · ~v

| {z }

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+~u · ~v · e ^2iwt ].

^{F !G6H}

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U ~ · V ~ = 1

2 Re (~u ^∗ · ~v) = 1

2 Re (~u · ~v ^∗ ) U · V = 1

2 Re (u ^∗ · v) = 1

2 Re (u · v ^∗ ).

^{F !G"H}

K9$-)$..4D'($,*+:,%),2LM$5%+'E

~ ¯ S = 1

2

E ~ × H ~

F !GGH

F()$-$)%.! I)%%$./,2DA$'

e ^iwt

^);%0)$'&.;+$,%0&.%$,HN1)%($13)'5&,%$).

Re S ~ ¯ = 1 2 Re

E ~ × H ~ ^∗

.

^{F !GOH}

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Re S ¯ = 1 2 Re

E ~ × H ~ ^∗ = 1

2 E ◦ H ◦ = 1 2

E _◦ ² Z ◦

= E _◦ ² 754Ω

F

E ~

^/,(

H ~

^{;),(),*0&;$H F !GRH}

P,($'SQ0$C+,#/,(4/,5;$,($',E

E _◦ ∼ 100 V m Re S ¯ ∼ 10 ⁴

754 ≃ 13 W m ²

=$.((/'0;0.&2),T/4%E

∼ 300 kV

m ; → Re S ¯

)*+%,-./#

≃ 10 ⁸ W

m ²

^{F !GUH}

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S ¯ = 1, 35 kW/m ²

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^)1M&5//1

; D ~ = ε ◦ E; ~ B ~ = µ ◦ H ~

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E ~ = − ∇ ~ φ − A ~ ˙

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∇ · ~ E ~ = ρ ε ◦

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Z ρ(~x ^′ )

| ~x − ~x ^′ | d ³ x ^′ .

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4πε ◦ · φ(~x, t) = Z

V 1

ρ ~x ^′ , t − ^r _c r d ³ x ^′ +

Z

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ρ ~x ^′ , t − ^r _c

r d ³ x ^′ .

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∆φ = ∆φ ₁ + ∆φ ₂ .

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f t − ^r _c

∂ ²

∂r ² f = 1 c ²

∂ ² f

∂t ² .

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∆f = 1 r · ∂ ²

∂r ² (r · f ).

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4,+/(&)-/5(0/267)

∆φ 2

^,809:";<=>

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Z

V 2

1 r

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= 1

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+ 1

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¨ ρ

r d ³ x ^′ ;

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8.A+/(9:";!=JA/K"+/(

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1

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j = j _z · e ^iwt ;

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r dz

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A z (r, t) = µ ◦ I ◦ l

4πr e ^−ikr e ^iωt .

^{9 !::;}

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r = const

^'0/A-03

A z

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r

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| {z }

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H r = − k · F · I ◦

λ cosϑ

r i

kr + 1 k ² r ²

e ^−ikr · e ^iωt H _ϑ = − k · F · I ◦

2λ sinϑ

r i

kr + 1 k ² r ² − 1

e ^−ikr · e ^iωt

E _ϕ = − Z _◦ · H _ϑ

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@61)*-3&3-*-()'*01)0(8'1'&#50:+&10,='*/L'&8E

Re ~ S = 1 2 Re

E ~ _ϑ × H ~ _ϕ ^∗

=

= Z ◦ I _◦ ² l ² 8λ ² · sin ² ϑ

r ² · ~e r

F"#NNOH

50' @61)*-3&3-*-()'*01)0( 01) *+)-)0+/119,,')*013 >, 80'

z

^{?@31'# =P* 80'} *-80-&' @6?

3B/.0.('0)8'1='*/L'&8'1.0&)E

E ϑ , H ϕ ∼ 1

r

F"#NNQH

!!"#$%&'()**+ ,-%.#"01234561##%&'$143!'156430#61&71"56%&'5$"0618 &94."1463%:

ϑ = 90 ^◦

⁾

2"1'1564"01#61;%4<1=1"'6$1&>"&?%551"&14@#3&'1& A &61&&1 B

l = λ/2

^C)

6*078/*79)*0/,7%-/*:*'0/;()+;,%*'(*<;)*-=>%*?

P = Z π

0

Z 2π 0

Re S ~

r ² sinϑdϑdϕ;

^{8'/ 6-*'%;()}@"!AABC

P = Z ◦ I _◦ ² l ² 8λ ²

Z π 0

Z 2π 0

sin ³ ϑdϑ

| {z }

4 3

dϕ

|{z}

2π

P = πZ ◦ l ²

3λ ² · I _◦ ²

^{@ !AADC}

!"!# $%&'()*+,-./01&-%'+0

1'*#2*'0;()+*,1'23-7(/*((**,E3-)*8'/*'(*,:*%*,-*'/;()@<37F'7--*'/;()GH&*'+,7%/-*'/;()

*/!C

!!"#$%&'()*D+ &E9FF#%&'1"&14 &61&&13& 1"&171014#1"6%&')

4.-&5*(&.&)6&,%(&/%7&04.(&..&.8/7&)0(,.709

P = 1

2 R s · I _◦ ² = πZ ◦ l ² 3λ ² · I _◦ ²

→ R s = 2π 3 Z ◦

l λ

2

| {z }

!"#$%&'()*+,-")!#',

;

R s = 790 Ω · l

λ 2

.

:"#;;<=

>/&0/0(7&) !"#$%&'()*+,-")!#',&/.&)4.(&..&/+?,'@@+#

!"!# $%&'()*+ , -./(001* 012 3'%0(/ 4)55('

A&/ $/.8/)'@.- &%&'()*+,-.&(/02&) 3&%%&. ,@B 4(*+& *7&) C*%&'D%& 8&)7&. >/E%+*+&.(&

&)F&@-(#

G+*E(/02&.3&%%&.5&)&/2028/.-&.7/& $%&'()*.&.-&-&.D5&)7&.E*0/(/H&. I&).&.#

A&/'%&/.&. J()&@*5K&'(&.+/(

φ ≪ λ

^2,(+,..#/%-+($)!"-&&'(#

0-1-"2&'(3

L)*M&J()&@*5K&'(&:F#A#6,@2N6@ME,)(/'&%N

. . .

^=+,2&.C/&OJ()&@@.-#

A&/0&2)2*2&.,.)&-&.7&.P)&Q@&.F&.',..+,.7/&

e ⁻

^,%0Q@,0/B)&/5&(),2(&.O

→

^R2*+0*.OJ()&@@.-

3/) 8*%%&. 7/& P)&Q@&.F,52S.-/-'&/( 7&) -&0()&@(&. T/2(%&/0(@.- 5&/ $/.B,%% &/.&0 P)&O

Q@&.F0E&'()@+0:J*..&.%/2(=5&)&2.&.#

G+0()&@&.7&.U5K&'(&.(0(&2(7,0>/E*%+*+&.(

p = q · l

˙

p = q ˙ · l = I · l = I ◦ · e ^iωt · l.

→ p = 1

iω I ◦ e ^iωt · l = I ◦ · l

ω e ⁱ ( ^{ωt− π} 2 ) .

:"#;;V=

>,02&/M(

p = p ◦ · e ⁱ ( ^{ωt− π} 2 )

+/(

p ◦ = I _◦ · l

ω ; I ◦ = ωp _◦

l .

:"#;W!=

>/&0&.J()*+8&)(0&(F&.8/)/.7/&L%&/2@.-:X#;;Y=&/.#>,..&)-/5(0/2

P = Z _◦ p ² _◦ · ω ⁴

12πc ² .

:"#;W;=

A&/$/.B,%% H*.8&/M&+T/2(8/)7,%0*7&)'@)F8&%%/-&4.(&/%&)2&5%/20(S)'&)-&0()&@(#

λ

.%#&

∼ 400 nm

λ

^"/!

∼ 800 nm

:"#;WW=

167'/83-)/89,+60:*,%;-/('0+*,)*0/,*</*(='%/-*'0/<()*(>

P

!"#

P

^$%&

∼ 16

1

^{? !5@ A}

1'*0*,)'B/+'*,3/)*8;,B/*C3,)*(D<(+EB*(+03((*03&'*+*(B-6<*(F6)*0%'77*-?#/,*<-'%/A!