I 1 = I 2 + G ′ U dz + C ′ · dz · ∂U

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

/"!!"*(0,1%"+$0*) (02 3"+$0*)"*

!"#$%&'%"(")*+,*-.,/.12"%3+%,45$*,)6%,-.,%$,%78%,#%"4+ %$,%797:2;,*%"%""%$1(

7),#$% '%5(%9<4$%,4'%$=%">%,#+,* -.,?%$(+,*%,@ A)'%$ +,(%"51%$#%(7),4>$51%,?%B

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1;,*$*E%$,5%(4')"@

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

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6%$(+,*#%58(".75@

!"#$"!%!&

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8&//@&,%&)7&/(6M2/-'&/(F/.@&)N),O/06P)Q4HG%

. . .

2/.)&/2&.@-4(&)6P%%(J5/%@&.0/2R$ST 3&%%&.,40F(),.0U&)0,%&%&'()*+,-.&(/02J=

!!"#$%&' ()*+ ,-#$. %&$ /012321"-&0"-1%&'-& 4%5 -"&-1 6-8-1#-"0%&' 3"0 64%59-##- :2& #"&;<

&48 1-80<)

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

E*+*-&.%M.-0@&):5&)(),-4.-0)/2(4.-,46-&5,4(&7&2&)%&/(4.-&.%,00&.0/2,46&/.&B2,%T

(4.-,40 @/0')&(&.$%&+&.(&.4+)&2.&.H D&%2& @,0:5&)(),-4.0U&)2,%(&.I)*7M.-&.&/.2&/(

@&)7&/(4.-D/&@&)-/5("

!!"#$%&'()=+>1<40?<84#0!"#$$-16-8-1#-"0%&'3"0$"<;1-0-&>#-3-&0-&"&@"-1A2#$41<0-##%&')

V/&2*+*-&.&7&/(4.-D/)@@4)2@/&I)*7M.-&.&/.2&/('*.0(,.(&.8&%M-&2,),'(&)/0/&)(=

R ^′ , L ^′ , C ^′ , G ^′

F7&/(D&)(J

W4.D&)@&.@/&X/)22*A02&.L&0&(9&,46@&.Y/&)I*%FG55"Z"[J,.-&D&.@&("B()*+04++&,+

Y&)9D&/-4.-0I4.'(=

I 1 = I 2 + G ^′ U dz + C ^′ · dz · ∂U

∂t

^F!"KJ

/01$)-)(*2

I 2 = I 1 + ∂I

∂z dz; I = I ₁ + I ₂

2

^{3 !45}

U 2 = U 1 + ∂U

∂z dz; U = U ₁ + U ₂

2

^{3 !65}

7),-$8$*9*),3:!"5$'&;(*<0,2

− ∂I

∂z = G ^′ · U + C ^′ ∂U

∂t

^{3 ! 5}

=08&$,8>0,,+,-$,*(0,-;+?$'$<@A0B2

U 1 + R ^′ Idz + L ^′ ∂I

∂t dz + U 2 = 0

− ∂U

∂z = R ^′ · I + L ^′ · ∂I

∂t

^{3 !.5}

@0'*!/)C$'$,9$,DE,3:!.5,0&

z

^$'-)1*2

∂ ² U

∂z ² = R ^′

− ∂I

∂z

+ L ^′ ∂

∂t

− ∂I

∂z

.

^{3 !F5}

=)*

∂I

∂z

^0+AG($)&+,-3:! 5$'-)1*8)&8&()$?()&B)$ !"!#$%&!'#"!(*+'# 2

∂ ² U

∂z ² = L ^′ C ^′ ∂ ² U

∂t ² + R ^′

L ^′ + G ^′ C ^′

∂U

∂t + R ^′ · G ^′ L ^′ · C ^′ · U

3 !:5

!"!# $%&'()*%+ ,-',% .&%/*'-(.,' 1,22,

H,80*92

U = U(z) · e ^iωt ; I = I(z) · e ^iωt

^{3 !I5}

J,B)$8$<K0(($'-)1*8)&0+8B$,G($)&+,-$,3:! 5+,B3:!.52

− ∂I

∂z = (G ^′ + iωC ^′ ) U

− ∂U

∂z = (R ^′ + iωL ^′ ) I

^{3 !L5}

/)$M$($-'0A$,-($)&+,-(0+*$*B0,,2

∂ ² U(z)

∂z ² = (R ^′ + iωL ^′ ) (G ^′ + iωC ^′ ) U(z)

^{3 !"N5}

/08)8*$),$/G#<)*OE,8*0,*$,PE$Q9)$,*$,!

,-.+'#.%'.%/01

U(z) = U ◦ · e ^γz

3 !""5 /+'&7),8$*9$,),3:!"N5$'&;(*<0,2

γ ² = (R ^′ + iwL ^′ ) (G ^′ + iwC ^′ ) γ = ±

iwL ^′

R ^′

iwL ^′ + 1 1 + G ^′ iwC ^′

iwC ^′

¹ 2

γ = ± iw (L ^′ C ^′ )

1 2

1 − iR ^′

wL ^{′ 1} 2

1 − iG ^′ wC ^′

¹ 2

3 !"45

9:*;')*0'617*'<5'/='/

f ≥ 10kHz

.0&)>

R ^′ wL ^′ , G ^′

wC ^′ ≪ 1

^?"#@AB

C-,0) (D//'/ E0' 6'0E'/ &')=)'/9-()+*'/0/+6'/1)'3'/E'* F&'035/. /-3G-H&+*'/)I0('&)

I'*E'/>

γ ≃ ± iw (L ^′ C ^′ )

1 2

1 − iR ^′

2wL ^′ 1 − iG ^′ 2wC ^′

?"#@"B

G*'//5/.J+/K'-&L5/EM,-.0/N*)'0&'*.06)E-//0/.5)'*ON3'*5/.>

γ = ± (α + ik)

^,0)

α ≃ 1

2

"

R ^′ · L ^′

C ^′ − ¹ 2

+ G ^′ L ^′

C ^{′ 1} 2 #

;

CN,P75/.1(+/1)-/)'

k ≃ ω (L ^′ C ^′ )

1 2

?4'&&'/=-3&B ?"#@ B

;'-3)'>

R ^′

⁰¹⁾ +6'*3-&6 '0/'* F*'/=7*'<5'/= ?.'+,')*0'-63N/.0. J+, 8'0)5/.1<5'*13/0))B 7*'<5'/=-63N/.0.?

∼ f ^{1 2}

^I'.'/Q(0/'R'()B#

S&&.','0/I0*EE0'G%TL4'&&'?0/16#6'0(+,P&0=0'*)'*U5'*13/0))1.'+,')*0'BE5*3QP-//5/.

5/EQ)*+,3-*-()'*0=0'*)>

U = Z L 2

L 1

Ed~s; ~

^{4'.J+/8'0)'*@=58'0)'*V}

I =

I Hds; ~

^{.'13&#4'.5,'0/'/E'*6'0E'/8'0)'* ?"#@!B}

S&&.','0/'*S/1-)=7:*E0'QP-//5/.1J'*)'0&5/.>

U(z, t) = U p e ^−αz e ^i(−kz+ωt) + U r e ^αz e ^i(kz+ωt)

^?"#@WB

C-*-51'*3N&),-/E0'Q)*+,J'*)'0&5/.,0))'&1E'*F&'035/.?W#XB>

I = − 1

R ^′ + iωL ^′

∂U

∂z

→ I(z, t) = I p · e ^−αz e ^i(−kz+ωt) + I r e ^αz · e ^i(kz+ωt)

^?"#@YB

C'Z/0)0+/E'1 !"#$%&'(!))!%("*!+'#,%*'>

I p = U p

Z L

; I r = − U r

Z L

?"#@XB

C-,0)'*.06)103>

Z L =

R ^′ + iωL ^′ G ^′ + iωC ^′

¹ 2

?"#V$B

9:*/0'E*0.'9*'<5'/='/

ω → 0

^>

Z L → r R ^′

G ^′

^?"#V@B

01'2$,03((&4&$'0'$5+$,6$,-)(*7

Z L → r L ^′

C ^′

^{8 !99:}

;,2$'<'3=)>-)(*2)$?($)&+,-8/!99:),-+*$'@A&$'+,-B$)0'$5+$,6$,4B$'&3(BC4,

∼ 1kHz

^!

D$)>E)$(F1'2$,0'$5+$,6C$'(3+FC4,

Z L

⁷

R ^′ = 4.4 · 10 ⁻³ Ω/m; L ^′ = 3.1 · 10 ⁻⁷ H/m

G ^′ = 10 ⁻¹⁰ S/m; C ^′ = 4.6 · 10 ⁻¹¹ F/m

^{8 !9G:}

H3,,$'-)B*>)&F4(-$,2$'I$'(3+F7

!!"#$%&'()(*+,-".012314%1&56!07&'"'81"9 $1.:1##1&#1"9%&'.;"$13.96&$.

Z L

⁾

J,*>E'$&$,2 2$' ?($)&+,- 8/!".: -)(* F1' 2)$ K$(($,63&(

k ≃ ω (L ^′ C ^′ ) ^{1 2}

^{+,2 23L)* F1' 2)$}

<&3>$,-$>&M),2)-N$)*

v P = ω

k = 1

√ L ^′ · C ^′ .

^{8 !9 :}

K)'B$'$&,$,$=E()6)*2$,O3E36)*A*>P+,2;,2+N*)C)*A*>B$(3-F1'$),$O43=)3(($)*+,-7

U = Z r _a

r _i

E r dr; ∇ · ~ E ~ = ρ ε ◦ ε r

?3+Q>&$'R3*6B6-(!#32+,->B$(3-

Q ^′

^3+F;,,$,($)*$'7

Q ^′ = 2πrε ◦ ε r · E r

E r = Q ^′ 2πrε ◦ ε r

8 !9.:

!!"#$%&'()*+ ,%-./01/2-"/ $/34056"5##/"2/-3

9-,0)'*.06)1037:*+60.'1;/)'.*-&<

U = Q ^′ 2πrε ◦ ε r

Z r a

r i

dr

r = Q ^′ 2πrε ◦ ε r

ln r a

r i

="#>?@

A-B-C0)D)16'&-.E'*A+-F0-&&'0)5/.<

C _K ^′ = Q ^′

U = 2πε _◦ ε r

ln

r a

r i

="#>G@

9'/;/E5()0H0)D)16'&-.

L ^′ _K

^{6'(+,,),-/:6'*}

φ ^′ = L ^′ · I.

="#>!@

φ

^{01)E-6'0E'*,-./')013'I&511B*+}8'0)5/.1&D/.'6'0J)*+,

I

^#

K0)L&'035/.=G#M?@6'(+,,),-/<

H ϕ = I 2πr φ ^′ = µ ◦ µ r

Z r _a r _i

H ϕ dr = µ ◦ µ r I 2π ln

r a

r i

="#>N@

9-,0)'*.06)1037:*E'/;/E5()0H0)D)16'&-.<

L ^′ _K = φ ^′

I = µ ◦ µ r

2π ln r a

r i

="#O$@

%0/.'1')C)0/L&'035/.=G#>"@'*.06)103E-,0)E0'P3-1'/.'13Q0/E0.('0)E'*7*'0'/4'&&'<

v P = 1

√ L ^′ C ^′ = 1

√ ε ◦ µ ◦ ε r µ r

= c

√ ε r µ r

="#OM@

90'1'1%*.'6/01.0&)7:*-&&'8'3'*&'0)5/.'/5/E01)0,R-3,'/E'*0/L&'035/.=G#M @'/)3-&)'S

/'/TD3'*5/.C5)*'U'/E#90'5/)'/1)'3'/E'V660&E5/.C'0.)WE-117:*'0/')XB013'8'3'*&'0)5/.

')Q-+6'*3-&6H+/

10kHz

^E0'L&'035/.=G#OM@.5)'*7:&&)01)WE#3#W-53E0'L*5BB'/.'13Q0/E0.S ('0)01)/-3'C57*'Y5'/C5/-63D/0.#Z-5B)6'0)*D.'C5*J0./-&H'*C'**5/.6'08'0)5/.1:6'*)*-.5/.

*:3*'/ 7+&.&03 -51 8'0)5/.1:6'*.D/.'/ 5/E 8'0)5/.10/3+,+.'/0)D)'/ =[5'*13/0))1D/E'*5/.W

ε r

S\'*)'0&5/.')#@3'*#

!!"#$%&'()*+ ,-"./"-#0%123-4%-&05-367#8-&59&

γ = ± (α + iβ)

!"!# $%&'()*+,&)*(+,)-.(/+*%., 0,& 1/%*234%+5(+//

0$)12$'(3-$'+,- 45, 45'6+,7 '89(3+:$,7$' &3';5,)<&$' =$(($ 3+: $),$' #$)*+,- 3+:-'+,7

$),$< -$-$2$,$, >2<&(+<<?)7$'<*3,7< 3; #$)*+,-<$,7$ 2$)

z = 0

^{43'))$'* 73< @}$'&A(*,)<

U(z)/I(z) = W (z)

^{;)* 7$'} B5<)*)5,

z

^{! C3,} 2$D$)&,$*

W(z)

^{3(< 7$, ,3&}

z

*'3,<:5;)$'*$,

>2<&(+<<?)7$'<*3,7

W (0)

^! =)7$'<*3,7<*'3,<:5';3*)5, $,*(3,- $),$' $),<$)*)- ;)* E;F$73,D

W(z = 0)

32-$<&(5<<$,$,#$&$'($)*+,-G

!!"#$%&'()*+,%$-&.//0$"&12-& -"&-0-"&3-"2"' 1!'-35#/33-&-&6-5-0#-"2%&')

$.(08)&2&.99&):%&/24.-&.;<#=<>4.9;<#=?>-/%(6@)&/.&2,)+*./02&3&%%&

U = U p · e ^−γz + U r e ^γz I = U p

Z L · e ^−γz − U r

Z L

e ^γz .

;"#AB>

C&D./(*.9&0E&F&G/*.06,'(*)0,+7&/(4.-0&.9&

z = 0

^H

r = U r

U p

;"#AA>

$/.-&0&(I(/.9/&:%&/24.-;<#AB>H

U = U ◦ e ^−γz + re ^γz I = U ◦

Z L

e ^−γz − re ^γz

;"#A">

J.9&)K(&%%&

z = 0

^-/%(9,..H

U

I

z=0

= W(0) = Z L · 1 + r

1 − r

;"#AL>

J46-&%M0(.,2

r

^&)2N%(+,.H

r = W (0) − Z L

W (0) + Z L

=

W (0) Z _L − 1

W (0) Z L + 1

;"#A >

K8&I/,%6N%%&H

W (0) = 0 r = − 1 W (0) → ∞ r → +1

W(0) = Z L r = 0

;"#A<>

C/& :%&/24.- ;<#A > 5&02)&/5( 9/& !"#!$%& '(()*+,"- IO/02&. 3/9&)0(,.90&5&.& 4.9

E&F&G/*.06,'(*)&5&.&# P*.6*)+& J55# 0/.9 O/.'&%()&4 4.9 5&2,%(&. 9&. C)&20/.. 5&/# $0

5&0(&2(&/.&&/.9&4(/-&J55#Q*.

W (0)

^,46

r

^+/(

| r | < 1

^#

/$00*$&,)0& )0* 1+& 2$' 3$4$5)6,0718*6' 19 :),-1,- $),$' #$)*+,- ),*$'$001,*! /)*

2$';($)&+,-<=!>?@81,,2)$;($)&+,-<=!>>@7A'B$2$C*$(($

z

D$'1((-$9$),$'*E$'2$,F

r(z) = U r e ^γz U p e ^−γz = U r

U p · e ^2γz

< !>G@

CH$I)$((19:),-1,-$),$'#$)*+,-9)*#J,-$

l, z = − l, γ = α + ik

^-)(*21,,F

r(l) = U r

U p

e ^−2γl = r · e ^−2αl · e ^−2ikl

< !>K@

| r(l) |

¹⁹#$)*+,-0$),-1,-)0*1(06+92$,L18*6'

e ^−2αl

^{8($),$'1(019M+0-1,-}<#$)*+,-1N06'O N)$'*#$)0*+,-@!M+P$'2$9N$E)'8*2)$#$)*+,-H'6#J,-$

∆l = ^λ ₂

^$),$Q&10$,2'$&+,-D6,

r(z)

+9

2π

^{! R+'&2)$} #$)*$'(J,-$

l

^{E)'2 2$'} ST)2$'0*1,2U

W(0)

^{), 2$, T$'*}

W (l)

*'1,076'9)$'*!

R10;($)&+,-00V0*$9<=!> @$'-)N*F

W(l) = U

I

z=−l

= Z L

e ^γl + re ^−γl

e ^γl − re ^−γl .

^{< ! W@}

/)*

r

^,1&;($)&+,-<=!>.@N$8699*91,0&()$P()&F

W(l) = Z L · W(0) + Z L tgh γl W(0) tgh γl + Z L

< ! "@

X,Y$'1((-$9$),$'+,-D6,;($)&+,-<=!>.@N$8699*91,7A'2)$Z'1,076'91*)6,IE)0&$,

r(l)

+,2

W (l)

r(l) =

W(l) Z L − 1

W(l) Z L + 1

< ! ?@

R10 !"#$%&"'()'!!0*$((* 2)$T)2$'0*1,20*'1,076'91*)6,),2$'3$4$5)6,0718*6'$N$,$21'

<-'6P$H'18*)0&$[$2$+*+,-\R1'0*$((+,-1+&1+7]$*IE$'81,1(V01*6''$1()0)$'*@!

#),)$, ), 2$'

W(l)

^{O^1(N$N$,$ 9)*}

^R(l) _Z

L = const

^{E$'2$, ), _'$)0$} 1N-$N)(2$*\ 2$'$, /)*O

*$(H+,8*$1+72$''$(($,

r

^OM&0$()$-$,!

W (l) Z L

= R(l) Z L

+ iX(l) Z L

< ! >@

!!"#$%&'()*+ ,"$-./01&$/0.1&/23.410"3& "&$-.5-6-7"3&/21803.-!-&-+94"0:;<"1'.144

!" #$%&'()*+,-%'./1% 2%$$%-+3/4(%.'3-, .- 5%($3/'$)/%-

6)1$$%.'%(-

!"!# $%&'& ()*&++&' ,-$./ $)*&++&' ,-0. 1'2 -$0)*&++&' 3' 4&5)

+167+86&' 0&23&'

9/ :-;# #! <5*=' -&1 8>15/. 7?* '0/' '6'/'@ 3+,+.'/'@ &0/'-* ;+&-*010'*)' 4'&&' AB%CD 0,

E'*&51)&+1'/F-5,.'75/='/G

E x (z, t) = E ◦,x · e ^i(wt−kz) ;

AH516*'0)5/.0/

z

^IF03)5/.D

H y (z, t) = 1

Z · E x (z, t) Z =

r µ ◦

ε _◦ · r µ r

ε r

; ε r = ε ^′ _r ; µ r = µ ^′ _r ; ε ^′′ _r = µ ^′′ _r = 0 ω = v · k ◦ ; v = (ε ◦ µ ◦ ε r µ r ) ^{− 1 2} ; k ◦ = ω √

εµ;

A"#""D

7.%8*9$.'3:%-

E ◦,x

^;

H ◦,y

^{/.-:.*,+-<%-=+3*}&)-/'+-'>?1)*),%-%2%$$%@!

A%'<' /31%- B.( -+1 %.-%( .-1)*),%-%- %4%-%- 2%$$%; B%$1% :.% C+DB%$$,$%.13-,%-

*.'E%.'+41F-,.,&%.'

e ^iωt

^3-:83/4(%.'3-,.-

z

^{G=.1'3-,%(HI$$'!}

7.%C+DB%$$,$%.13-,%-$+3'%-.-:.%/%*J+$$K

∇ × ~ E ~ = − iω ~ B

∇ × ~ B ~ = +iµεω ~ E

∇ · ~ B ~ = 0

∇ · ~ E ~ = 0

^{> ! L@}

#.-%.-1)*),%-%%4%-%2%$$%;B%$1%.*=+3*%D9)-%-'.%$$+4HF$$';$F//'/.1+$$,%*%.-H)$,%-G

:%(*+M%-/1(%.4%-K

E ~ = E ~ p · e ^{−~ γ·~ r} H ~ = H ~ p · e ^{−~ γ·~ r}

*.'

~γ = ~ α + i~k;

^>1.%(&%.-%N%($3/'$%./'3-,.-N%(4!*.'

α ~

^{@ > ! 5@}

73(1#.-/%'<%-.->O! L@%(,%4%-/.1PQ/3-,%-HI(

~γ

^K

∇ × ~ E ~ = − iµω ~ H.

^{> ! O@}

8-:%(%(/%.'/%(,.4'/.14%.:%(83/HI1(3-, R)-

∇× E ~

^*.'

E ~

^{+3/>O! 5@K}

∇ × ~ E ~ = − ~γ × E ~ p · e ^{−~ γ~ r}

→ ~γ × E ~ p = iµω ~ H p

^{> ! S@}

8-+$),4%&)**'*+-K

~γ × H ~ p = − iεω · E ~ p

^{> ! T@}

U$%.13-,>O! S@*.'

(γ × )

*3$'.9$.<.%('K

~γ × ~γ × E ~ p = +iµω~γ × H ~ p

^{> !LV@}

~γ × ~γ × E ~ p =

~γ · E ~ p

· ~γ − ~γ ² · E ~ p

^{> !LW@}

7.%(%1'%X%.'%R)->O!LV@%(,.4'*.' >O! T@K

− i ² · µεω ² E ~ p = µεω ² E ~ p (7.44)

z}|{ = k ² _◦ · E ~ p

> !L"@

U$%.1/%'<%- :%(4%.:%-(%1'%-X%.'%->O!LV@3-:>O!LW@%(,.4'/1$.%M$.1K

~γ · E ~ p

· ~γ − ~γ ² · E ~ p = k _◦ ² · E ~ p

^{> !L6@}

C3$'.9$.&+'.)-R)-U$%.13-,>O! T@*.'

~γ

^%(,.4'K

~γ ·

~γ × H ~ p

| {z }

0

= − i · ε · w · ~γ · E ~ p

→ ~γ · E ~ p = 0

^{> !L @}

8,+/(&)29%(+,.,40:&);%&/24.-<=">?@+/(<=">!@A

~γ ² = − k ² _◦

~γ ² =

~ α + i~k 2

= α ~ ² + 2i~ α~k − ~k ²

<!">>@

8,),406*%-(A

α ² − k ² = − k _◦ ² ; k ² = k _◦ ² + α ²

~

α · ~k = 0

<!"> @

B*%-&)4.-&.A

•

^8/&3&%%&.%9.-&:&)/.2*+*-&.&.3&%%&/0('%&/.&),%0/+2*+*-&.&.B,%%A

k ≥ k ◦

<="!C@6D):&.E.0,(F.,2;%&/24.-<="! @

•

^8/&89+G64.-

α ~

^:&)3&%%&.,+G%/(4:&%/&-(0&.')&2(F4)E405)&/(4.-0)/2(4.-"

• ~γ

^0(&2(0&.')&2(,46

E ~ p

^H:"2"

~

α · E ~ p = ~k · E ~ p = 0

<!">=@

$.(0G)&2&.::/&0&)I,.:5&:/.-4.-'J..&.6*%-&.:&K)/&.(/&)4.-&.-&L92%(L&):&.A

~k = k · ~e z

<E405)&/(4.-:&)3&%%&/.

z

^MI/2(4.-@

~

α = α · ~e y

E ~ p = E 0 · ~e x

<!">N@

8,..&)-/5(0/2,40;%&/24.-<="!N@6D):,0

H

^MB&%:A

H ~ p = 1

iµω (α~e y + ik~e z ) × E ◦ · ~e x

= 1

µω E ₀ (iα~e z + k~e y )

<!">O@

$)-&5./0A8/&0&/.2*+*-&.&3&%%&2,(.4)&/.&.0&.')&2(F4)E405)&/(4.-0)/2(4.-0(&2&.:&.

E

MB&%:,.(&/%" 8,-&-&. -/5( &0 0*L*2% &/.&. 0&.')&2(H L/& &/.&. G,),%%&% F4

~k

0(&2&.:&.

H

MB&%:,.(&/%"$02,.:&%(0/2:,+/(4+&/.&

H

^*:&)

T E

^M3&%%&<P-%"Q,G"?4.:E55"="O@"

8/& $ *:&) RSM3&%%& &)-5/( 0/2 ,.,%*-H L&.. :/& ;%&/24.-&. <="!C@ 4.: <="!!@ .,2

H ~ p

,46-&%J0(L&):&.A

H ~ p = H ◦ · ~e y

E ~ p = 1

εω H ◦ (iα~e z − k~e x )

$M3&%%& <!" #@

!!"#$%&'()*+ ,%-./#$0-"/&1"/-%&'/"&/-"&2030'/&/&4 /!/&/& 567/##/)

7.%8#9:2%$$%%(,.4'/.1+$//;%<.%$$%=>/3-,?@(A%-3-4%,(%-<'%-B+3*C1)*),%-%%4%-%

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#/?+$$%%.-%%4%-%H1)*),%-%H$.-%+(;)$!2%$$%%.- *.'L

ω = v · k ◦ = c

√ ε r µ r · k ◦ ; k ◦ = ω √ ε r µ r

c

C !5KD

!!"#$%&' ()89+:"&;<##%&$ =/>/?"0& /"&/-/!/&/&7/##/ <&"$/<##/"1/&$/-.#@2/!/"

z = 0

⁾

8&)+4(4.-9

:&.')&2( ,4;,%%&.<&) 3&%%&.,.(&/% =/)< (*(,% )&>&'(/&)(" ?,),%%&%,.(&/% %@46( =&/(&)

→

A405)&/(4.-/.

y

^{BC/2(4.-DE,%%F,%0}GB3&%%&H=/&/.A55"HI"J#K-&L&/2.&(KDE,%%FF,%0$B3&%%&

H=&..+,.</&M)/&.(/&)4.-&.N*.

E ~ ₁

^4.<

H ~ ₁

-&-&.&/.,.<&)N&)(,402(K"

!"#$%$"$%& '%() #"+,-./0#) 1"..2 !#$0(3!+$4

!!"#$%&'()**+,"&-.##/&$/ 0/##/&-12&3/&%&$ 4125/63"2&$/10/##/&#7&'/"& %8!1/"3%&'81":;

3%&'

y

⁾

k _◦ = w · √ εµ

E)&/),4+=&%%&.L,2%<&)&5&.&.D2*+*-&.&.3&%%& H!" OK P/&&/.6,%%&.<&. 3&%%&.-)QR&.(),-&.F.<&SJDF.<&SO0(&2(6T)</& ,40%,46&.<&3&%%&"

E 1,x = E 1,0 · e ^{−ik ◦ (ysinφ 1 +zcosφ 1 )} H 1,y = E _1,0

Z cosφ 1 e ^{−iko(ysinφ 1 +zcosφ 1 )} H _1,z = − E 1,0

Z sinφ ₁ e ^{−iko(ysinφ 1 +zcosφ 1 )}

^{H!" UK}

A+V&(,%%()/(('&/.&W)&24.-D0*.<&).&/.&X*(,%)&>&S/*.&/."P,0E&%<',.../2(&/.<)/.-&."

A.<&)M5&)>@2&-/%(9

E _k = E x = 0;

=&-&.9

(σ → ∞ )

H ⊥ = H z = 0; (

=&-&.

∇ × ~ E ~ = − µ · H; ~ ˙

^5&/

σ → ∞ )

^{H!" !K}

"#$%&'()*+,-*-.-/&,-*&-0-11-23-15-6,&0,#/-1

φ 2

^{%7$1%7(-#$8119}

E _2,x = E _2,0 · e ^{−ik ◦ (ysinφ 2 −zcosφ 2 )} H 2,y = − E 2,0

Z cosφ 2 e ^{−iko(ysinφ 2 −zcosφ 2 )} H 2,z = − E 2,0

Z sinφ 2 e ^{−iko(ysinφ 2 −zcosφ 2 )}

:;< =>

?)*

z = 0

^{6)$$-#$,5-,#@ 7#+}%7$(%11-#+-

E x

^@0-11-A-*%+-%7$1B$5-#<C%6,&D-/866&6%#

+7*5"++,&,8#E8#

E ₁ , x

^7#+

E ₂ , x

^#%5F1-,57#A:!< G>9

E 1,0 · e ^{−ik ◦ ysin φ 1} + E 2,0 · e ^{−ik ◦ ysin φ 2} = 0 sin φ ₁ = sin φ ₂ ; φ ₁ = φ ₂ = φ

E 1,0 = − E 2,0 .

:;< >

H#$A-$%6&-*A,D&$,5()*+%$?-1+,6I%1D*%769

S y







E x = − iE _1,0 · 2sin (k _◦ zcosφ) · e ^{ik ◦ ysinφ} H z = i ^E _Z ^1,0 2sin (k ◦ zcosφ) sinφe ^{ik ◦ ysinφ}

H y = ^E _Z ^{1 , 0} 2sin (k ◦ zcosφ) cosφe ^{ik ◦ ysinφ}

:;< !>

C% #7* -,#-

E x

@J86K8#-#&-2 %D-* $83851-,#-

H z

^{2 %1$ %75 -,#-}

H y

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&7#A$*,5&7#A

y

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H

^{@0-11-< ?)*+-#} M8N#&,#A@

O-/&8*A,1&9

Re ~ S y = 1 2 · Re

E ~ x × H ~ _z ^∗

:;< P>

Q-+,#A7#A%7(?1R5-#6,&,+-%1-*S-,&(R5,A/-,&9

E k = H ⊥ = 0.

^{C%6,&(81A&,#7#$-*-6?%119}

E x = H z = 0

^()*

z = 0.

:;< T>

C%$5-,U&26%#/%##,#+%$0-11-#(-1+#%5F1-,57#A:!<=!@!<=P>-,#-'3-,&-M1%&&-,6"D$&%#+

z = b

^{-,#D*,#A-#%#+-#}V&-11-#6,&

sin (k _◦ b cos φ) = 0

(k ◦ b cos φ) = m · π; m = 0, +1, +2, . . .

:;<!W>

b = m · λ _◦ 2 · cosφ

λ _◦

^{,$&+%D-,+,-,"!+!0-11-#1R#A-,6X-+,766,&}

ε r

²

µ r

^<

C-*"7$+*7/

k 0 · cos(φ)

D-'-,5#-&+-#M5%$-#E8*$57D,#'@Y,5&7#A7#+3,*+6,&

k z

D-'-,5#-&

:"DD<!<Z[>9

k z = k 0 · cos(φ).

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

!!"#$%&'()*+, -"&./'0&0"&0120"3010&"$04##0"30&$0&-!0&0

∆z = z 2 − z 1 = b

⁾

9:*;-1<=4'&&'/7'&;/-3>&'035/.?@# @A.0&)B

C0'4'&&'/-516*'0)5/.'/)&-/.

y

'*7+&.)'/)1D*'3'/;;',E'*,

e ^{−ik y ·y} = e ^{ik ◦ sin·φ·y}

^?"#@FA

,0)

k y = k ◦ · sinφ.

G/ ;0'1', 9-&& 01)-&1+ ;0' 4'&&'/&H/.' 6I.#;'* J516*'0)5/. 0/K=L03)5/..*MN'*-&1

λ ◦

^#O5/

(-//

sinφ

,0))'&1;'*P';#?@#@$A0/J63H/.0.('0)Q+/

λ _◦

^5/;

b

^{-51.';*:()R'*;'/B}

k y = k ◦ · p

1 − cos ² φ = k ◦ · s

1 − mλ ◦

2b 2

k y = 2π λ y

; λ y = λ ◦

sin(φ)

^?"#@SA

9:*

λ ◦ < ^2b _m

^6&'06)

k

*''&&T ;#3#'1 U/;');0'J516*'0)5/. '0/'* 4'&&'1)-))#9:*

λ ◦ > ^2b _m

^R0*;

k

^{0,-.0/H*V;#3#;0'}4'&&'R0*;.';H,D7)#W0'(-//103IR013'/;'/X&-))'/ /03)-516*'0)'/#

C-13'0N)V'1.06)'0/'5/)'*'>*'/I7*'Y5'/I7:*;0'4'&&'/-516*'0)5/.#

J&&.','0/.0&)B

c = f · λ

^!"##$

= f · λ ◦ √ ε r µ r

f = c

λ _◦ √ ε r µ r

?"#@ZA

J51?@#@ZA7+&.);-//7:*;0' 5/)'*'>*'/I7*'Y5'/I,0)

k y → 0

^B

λ ◦c = 2b

m ;

^5/;,0)?@#@"A

f _◦c = c · m

2b · √ ε r µ r

; k _◦c = mπ b ;

λ c → ∞ ; v pc → ∞

?"#@"A

X3-1'/.'13R0/;0.('0) 5/; 4'&&'/&H/.'

λ _◦

^{.'3'/ 7:*}

f c

^{-&1+ .'.'/} 5/'/;&03# G/ 5/1'*'*

P')*-3)5/..'3)6'0

f c

^;'*%0/7-&&R0/('&

φ → 0 ^◦

^#

73/ 8%- 9:%. ;+(+$$%$ <%($+3=%-8%-> $%.'%-8%- #4%-%- %-'/'%1' /1$.%?$.1 83(1 #.-9.%1%-

9:%.%(:%.'%(%(;+(+$$%$%(>$%.'%-8%(@$A1%- *.'

x = x 1

^3-8

x = x 2

^%.- B%1'%&1)1$$%.'%(!73=

8.%/%- @$A1%- /.-88.%B+-84%8.-,3-,%-

E y = E z = 0

^/):.%

H x = 0

^{=C( 3-/%(%}4%'(+1'%'%-

H _m,0

^{D2%$$%-+3')*+'./1%(=C$$'=C(+$$%}2%$$%-$%.'%(1E1%-

∆x = | x ₁ − x ₂ |

^!

73/ 8.%/%-F(C-8%- 8%(F(3-8*)8%-/'+4.$.'A' G

H 1,0

^{DH)8%I :A1$'*+- 4%. ,%,!}

b = ∆z

^*%./'

a = ₂ ^b

^{! 7$$%.-}

b

^{8%J-.%(' 8+4%. 8.% 3-'%(%} F(%-9=(%K3%-9 4%. 8%( ,%,%4%-%- L(.%-'.%(3-, 8%/

H 1,0

^DH)8%!

!!"#$%&' ()*+, -"./01."2&.!03"04%&' "5 624##0"701 891

m = 1

^%&$

m = 2

⁾ :1%&$52$0".7 $"0

H _1,0

^;<0##05"7:10&3810=%0&3

w c 1

⁾

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E m,n

^:.%+31

H m,n

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+-,%(%,' :%(8%-! @C( 4%$.%4.,% H)8%-

H m,n

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E m,n

^{,.$' 8+-- .- N}%(+$$,%*%.-%(3-, <)- F$%.13-,GO!5 IP

k _0c,mn ² = mπ b

2

+ nπ h

2

;

^{G !OQI}

@C(8.%H)8%-.*B%1'%&1)1$$%.'%(,.$'G1.%(4%./.-8R%'9'

x, y

^8.%S(+-/<%(/+$&))(8.-+'%-3-8

z

^8.%73/4(%.'3-,/(.1'3-,IP

H mn

!"#$"%

H z = H _z,◦ cos mπx b cos nπy

h e ^{−j 2 λh π z} 2π λ k

= k E y = H _z,◦ k

k _◦c ² Z ₀ r µ r

ε r

mπ

b sin mπx b cos nπy

h e ^−j

“ ₂

π λh z+ ^π ₂ ”

H x = H _z,◦ 2π λ h k ² _◦c Z ₀ mπ

b sin mπx b cos nπy

h e ^−j

“ _{2 π}

λh z− ^π 2

”

E x = H z,◦

k k _◦c ² Z 0

r µ r

ε r

nπ

h cos mπx b sin nπy

h e ^−j

“ _2π

λh z− ^π 2

”

H y = H z,◦

2π λ h k ² _◦c

nπ

b cos mπx α sin nπy

b e ^−j

“ ₂

π λh z− ^π 2

”

+/(

E y = − Z F,H H y E x = Z F,H H y

^{8"# 9:}

;<)

E mn

^!"#$"%-/%(=

E z = E _z,◦ sin mπx α sin nπy

b e ^{−j 2 λh π z} E x = E z,◦

2π λ h k _c ²

mπ

α cos mπx α sin nπy

b e ^−j

“ _2π

λh z+ ^π 2

”

H y = E _z,◦ k k _c ²

1 Z 0

q _µ

r

ε r

mπ

α cos mπx α sin nπy

b e ^−j

“ ₂

π λh z+ ^π 2

”

E y = E _z,◦ 2π λ h k _c ²

nπ

b sin mπx α cos nπy

b e ^−j

“ _2π

λh z+ ^π 2

”

H x = E _z,◦ k k _c ²

1 Z 0

q _µ

r

ε r

nπ

b sin mπx α cos nπy

b e ^−j

“ ₂

π λh z− ^π ₂ ”

+/(

E x = Z F,E H y E y = − Z F,E H x

^{8"# :}

>,5&/',..?&-&.@&.A,.@5&@/.-4.-&.?&@&)

E m0

^.*2

E on

&B/0(/&)&.8?&-&.

E z = 0

^:#

!" #$%&'(')*+ ,-').$/0 1,-').$/0 2)3 4)$5'&%&'('6

>/& A&0*.,.C'4)D& @&)5&(),2(&(&.E2?/.-4.-0+*@& &/.&)F,D/(G(?/)@@4)2@/& H<(&C,2%

2,),'(&)/0/&)(=

Q = ω r · W

P

^{8"# I:}

ω r =

A&0*.,.C6)&J4&.C

W =

-&0K&/2&)(&;&%@&.&)-/& /+A&0*.,(*)

P =

^L&)%40(%&/0(4.-/+A&0*.,(*)5C?#,46@&. 3G.@&.#

M&/0K/&%=

E 010

^NO*@&P

W = 100J; f = 100M Hz; P = 100kW ; Q = 10 ⁵ , ω · W = 60GW

^QQ#

ωW

^{0(&%%( @/&} M%/.@%&/0(4.- @,)R ?&%2& /+ A&0*.,(*) %,46&.@ D*. &%&'()/02&) /. +,-.&N (/02&);&%@&.&)-/&0*?/&C4)<'-&?,.@&%(?/)@#

9)&:;</$0')*2)3>?@A33)BC*BB&0'D-32$<:C<</E&*B)*2)3#$%&'('F

P = − dW

dt

^{1 !7G6}

H&'2)3,0)&/C*B17!776B&0'F

ω r W

Q = − dW

dt

^{1 !IJ6}

K*')B3$'&L*)3B&;'F

W (t) = W ◦ · e ^{− ωr Q ·t}

^{1 !I86}

KMN$5CCMB&0'D-32&)B)<O)&/)3')?)02)*)3B&)2)<4)<L*$'L3<F

W(t) = 1 2 ε ◦

Z

V

E(~x, t) ~

2

d ³ x + µ ◦

2 Z

V

H(~x, t) ~

2

d ³ x

^{1 !IP6}

H$* 5$** Q)2)M 4)<L*$'L3$C/ )&* A3<$'.</$0';&02 1R/E&*B53)&<M&' 2&<53)')M A0)M)*')*

L, C, R

^{6.L32*)*!S$**B&0'D-32&)}B)<O)&/)3')A*)3B&)F

W(t) = 1

2 C · U(t) ² + 1

2 L · I (t) ² .

^{1 !I"6}

?-32&)T)&'OC*5')M$U&M$0)3)0)5'3&</)3;.E!M$B*)'&</)3A*)3B&)B&0'F

W ◦ = 1

2 CU _◦ ² = 1

2 LI _◦ ² .

^{1 !I 6}

?-32&).)&'0&/):;/(*B&B5)&'2)3 )0)5'3&</)*?)02<'(35))3B&;'<&/2$M&'$*Q)2)MV3'*$/

17!I86+17!IP6F

E(t) = E ◦ · e ^{− ωr 2 Q t} · e ^{−iω r t}

^{1 !IW6}

S)3)UOL*)*'&)00):;D$00 2)3R/E&*BC*B$MO0&'C2)0(X' <&/ 2C3/)&*)RCO)3OL<&'&L*%L* ?3)@

YC)*.)*&*2)3ZMB);C*B%L*

w r

;)</3)&;)*F

E(t) = 1

√ 2π Z +∞

−∞

E(ω)e ^−iωt · dω.

E(ω) = 1

√ 2π Z ∞

0

E ◦ e ^{− ωr 2 Q t} · e ^{i(ω−ω r )t} · dt

^{1 !I[6}

S$3$C<)3B&;'<&/2C3/K*')B3$'&L*C*2\C$23&)3)*F

| E(w) | ² ∼ 1 (ω − ω r ) ² +

ω r

2Q

2

^{1 !I76}

!"# !#$%!"&"#'()(*+,-."/ ),1 2'-"($*."-$"!3,(4 4"()(($5

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

!!"#$%&' ()*,+ -.//0%1!0% 2%3 4./5"66%&' $.3 7./8&0583'95. 9!.3 $". 7./8&0&2!3."5.

∆f

2:"/<.& $.&

3db

=>%&?5.&)

Q = f r

∆f

^9"#::;

8*9/$;0&/)<);';*=2)3,-').$/0>-3#$%&'(')*?/*)@-00;*=1

ε = ε ◦ , µ = µ ◦

^6A

Q = V

A · δ · g

^{1 !BC6}

V =

4)9?*$'?3%?0;D)*

A =

4)9?*$'?3?E)3F(/)

δ =

G5&*'&)>)

g =

,)?D)'3&)>$5'?3 !"#$%&!'()*+$&*,HHI!JK/.

Q

^{&9'$09?E&9$;>2)*},)?D)'3&)>$5'?3=0)&/2)DL)3/(0'*&9$;94)9?*$'?3%?0;D)*1%)30;9'>3)&)

@)029M)&/)3;*=6;*2 >)022;3/23;*=)*)D N$*2%?0;D)*! O$/9')/)*29&*2 .P)& 4)9?*$'?3)*

=).)&='+P)0/).;3Q)9/0);*&=;*==)0$2)*)3R)&0/)*=))&=*)'9&*2!RSM&9/))T)5'&%)Q)9/0);U

*&=;*=9=3$2&)*')*0&)=)*.P&9/)*

2M V /m

^;*2

20M V /m

^!

!!"#$%&'()*+, -./0"/#12(345/#63718.&/6.29%2:.&1&!18<#1%&"'%&')="1!1"$1&/>#"&?1&:&3

&1&#1"612 !1@186"'61& =2"@62A<21& #/$1& 8"< '1'1&B!12 $1& C/#6$2"@62A<21& /> %D1&#1"612 %&$

$12>"66"'1& =2"@62A<21/%@)="1>/'&16"8<1&E1#$#"&"1& %>8<#"1D1&$"1 :.&1&#1"612)

!!"#$%&' ()*+, -./0&1203.&1%/ '.4055.#2.& .##"52"/7.& 8"&9.#9.##.& ":

π

^;<0$., =.&17!132.

>.##.& /"&$":?.'.&2142.33.'2)