View
262
Download
0
Category
Preview:
Citation preview
∇ · D = ρ !
∇ · B = 0 !
∇× E = −∂ B∂t
"
∇× H = J + ∂ D∂t
#
D $C/m2%E ! $V/m%B ! $T V · s/m2%H ! $A/m)ρ $C/m3%J $A/m2%
&
'
∇ · D = 4πρ
∇ · B = 0
∇× E = −1c∂ B∂t
∇× H = 4πcJ + 1
c∂ D∂t
D(statcoul/cm2) E(statvolt/cm) B(gauss) H(oersted) ρ(statcoulomb)J(statamp)
! q0 ( ! E
F = q0 E
E D B H
D B !
D = ε0 E + P
B = µ0H + M
ε0 = (1/36π) × 10−9 $#)% µ0 = 4π × 10−7$*)% P M + P M ! P M ! P M ,* D B E H - J E . /
D = f( E)J = g( E)B = h( H)
0
" E H 1 D = ε0χE χ
D = ε0 E + P = ε0 E + ε0χE
= ε0(1 + χ) E = ε E
ε2 $ % 3
D = ε E = ε0n2 E
n =
√ε
ε0=
√εr
J = σ E
σ2 M = 0
B = µ0H
4 2
& $% 2 ε µ
' 2 ε µ
0 2 σ = 0 ⇒ J = 0
5 2 ρ = 0
6 72 ε µ
! 2
∇ · E = 0
∇ · B = 0
∇× E = −∂ B∂t
∇× B = εµ∂E∂t
D = ε E B = µ H
D = ε0χE + χ(2) E · E +χ(3) E · E · E + · · ·
5
7 B ! E ! 1 , E $ B% 4
∇× (∇× E) = ∇× (−∂ B∂t
) = − ∂∂t
(∇× B)
= − ∂∂t
(εµ∂E∂t
)
∇× (∇× E) = ∇(∇ · E) −∇2 E ∇ · E = 0
∇2 E = µε∂2 E∂t2
# −→E 2
∇2 B = µε∂2 B∂t2
∇2f(r, t) =1
v2
∂2f(r, t)
∂t2
v8 $ ./% f(r, t)8 $% !4
v = 1/√µε
c = 1/√µ0ε0 = 2.998 × 108m/s
µ ≈ µ0
v =1√µ0ε0εr
=c
n
6
E(r, t) = E0
ei(ωt−k·r)
k * ! 7 * E(r, t) E(r, t) 9
∇2 E = E0∇2ei(ωt−k·r)
= E0∇ · [−ikei(ωt−k·r)]= −k2 E0e
i(ωt−k·r)
= −k2 E
µε∂2 E
∂t2= −ω2µεE
k = ω√µε
"
∇ · E = 0
⇒∇ · [ E0e
i(ωt−k·r)] = E0 · ∇ei(ωt−k·r) = −ik · E0ei(ωt−k·r) = 0
k · E0 = 0
⇒ ! $ B !%#
∇× E = −∂B
∂t
−ik × E = −iω B
:
k × E = ω B
k E B E B 9 2
E
B
k
x
y
z
" B !
| B| =|k|ω| E| =
ω√µ0ε
ω| E| =
n
c| E|
7 E B 7 ! ! H B2
| E|| H| = µ
| E|| B| =
õ
ε≡ Z
Z (Ω) Z0 =
õ0
ε0= 377Ω
9 ; S 9 ! !
S = E × H
9 J/(m2 · sec) 7 !
<
U = 12( D · E + B · H)
B = ncE
U =1
2(εE2 +
B2
µ) =
1
2(ε+
n2
µc2)E2 =
1
2(ε+
εrε0µ0
µ)E2
(J/m3) # µ ≈ µ0 U !
U = εE2
9 2
∂Wmech
∂t+∂U
∂t= −
∮s
S · d A
Sin
Sout
UA Wmech
9 ; $Sout−Sin% A , V ! $U% Wmech 9 |S| = v · 〈U〉 v 〈U〉 7 $ 9 S J U ρ% ; S .7/
⟨S⟩
I 7 ; $W/m2%
⟨S⟩
= Ik
k= I n
1015 1011 1015
=
S = E × H = ( E0 × H0) cos2(ωt− k · r)
# k × E = ω B ⇒ H = 1ωµk × E k = ω
√µε
S = E0 × (1
ωµk × E0) cos2(ωt− k · r) =
√ε
µE2
0 cos2(ωt− k · r) n
- $ n% $ ./ %- 7 ;
I ≡⟨S⟩
= 1T
∫ t0+Tt0
√εµE2
0 cos2(ωt− k · r) dt n= 1
T
√εµE2
0
∫ t0+Tt0
cos2(ωt− k · r) dt n
> θ = ωt− k · r cos2 θ = 12(1 + cos 2θ)
1
ωT
√ε
µE2
0
∫ ω(t0+T )
ωt0cos2 θ dθ =
1
2ωT
√ε
µE2
0
∫ ω(t0+T )
ωt0(1 + cos 2θ)dθ
=1
2ωT
√ε
µE2
0(ωT +1
2
∫ ω(t0+T )
ωt0cos 2θd2θ)
=1
2
√ε
µE2
01 +1
2ωT[sin 2(ωt0 + ωT − k · r)
− sin 2(ωt0 − k · r)]
$T ≈ 10−9 f ≈ 1 ?% $ω ≈ 3.5 × 1015*? -2@ 60' %ωT = 3.5 × 105 1 ⇒ 1
ωT 11
I ≡∣∣∣⟨S⟩∣∣∣ = 1
2
√εµE2
0 = 12( 1Z)E2
0
I = 1√εµ
· (12εE2
0) = v · 〈U〉 $ 12
7% #
〈U〉 = 12εE2
0
A
; 7 >? F = q E + qv × B + >? J = qv >? 7 $)% F1 = ρE + J × B - ρ J
∇ · E = ρ/ε0 , ∇× B = µ0J + ε0µ0∂ E/∂t
F1 = ε0(∇ · E) E +1
µ0(∇× B) × B − ε0
∂ E
∂t× B
- ∂
∂t( E × B) =
∂ E
∂t× B + E×∂
B
∂t⇒
F1 + ε0∂
∂t( E × B) = ε0(∇ · E) E − ε0 E × (∇× E) +
1
µ0(∇ · B) B − 1
µ0
B × (∇× B)
∇· B = 0 ∇× E = ∂ B/∂t V
Ftotal +∂
∂t
∫vε0( E × B)dV =
∫v[r.h.s.]dV
-
dPmechdt
+dPfielddt
=∫v[r.h.s.]dV
Pfield =
∫vε0( E × B)dV =
1
c2
∫v
SdV
! !
g = S/c2
&B
4 r.h.s. ∫v∇ · TdV →
∮s
T · ds
T = ε0
−→E−→E +
1
µ0
B B − I(ε02E2 +
B2
2µ0)
T . / 2nd r.h.s. V 4 V
P =force
area=
|∆P |∆t
A=
|g|V∆t
A
∆t c∆t V = A·c∆t
P =|g|A·c∆t
∆t
A= |g| c =
∣∣∣S∣∣∣c
C
〈P 〉 =∣∣∣⟨S⟩∣∣∣ /c = I/c
# 1.34× 103J/m2 · s # 4.46 × 10−6N/m2 ∼ 105N/m2
D ! E H
E = E0 exp i(k · r − ωt)
H = H0 exp i(k · r − ωt)
E0 H0 ? ! E H ! ? ! ?
&&
E
H
k
E
H
7 ? ?; # ? , ./ ! ? , 1 > ? ./ ,
D ? E0 ? 7 , ±π
2 !
E = E0[x exp i(kz − ωt) + y exp i(kz − ωt± π
2)]
eiπ/2 = iE = E0(x± y) exp i(kz − ωt)
E ω "
1 !
&'
E
H
k
E
H
? ! ! ?
E
H
k
E
H
!
E0 = (xE0 ± iyE′0) exp i(kz − ωt)
" ? ? ? ! ! $ ?% " D ? ? - ! E E1 E2 E1 ?
&0
E
θ
(Incident wave)
(Transmitted wave)
Transmitted axis
of polarizer
E2
E1
E θ 7 !
E1 = E cos θ
" I1 !
I1 = I cos2 θ
# ? θ ? ? ( cos2 θ 1
2
> ? ? ? P ! ?
P =Ipol
Ipol + Iunpol
? Imax = Ipol +
12Iunpol Imin = 1
2Iunpol
P =Imax − IminImax + Imin
Imax Imin ?" π
2
$ % D ? ? " π
&5
Fast
Slowπ/2
Fast
Slowπ/2
Quarter waveplate
Fast
Slowπ
Fast
Slowπ
Half waveplate
?
E0 = xE0x + yE0y
E0x E0y " 7
E0x = |E0x|eiφx
E0y = |E0y|eiφy
! [
E0x
E0y
]=
[ |E0x|eiφx
|E0y|eiφy
]
? E E ? ? E # ? ? ? [
1−i
]+
[1i
]= 2
[10
]
''
>
[AB
]
[A′
B′
] E
[a bc d
]
[a bc d
] [AB
]=
[A′
B′
]
&6
E 2
[atotal btotalctotal dtotal
]=
[an bncn dn
]· · ·
[a1 b1c1 d1
]
E 7?
> 9?
?
[1 00 0
]
[0 00 1
]
±45 12
[1 ±1±1 1
]
F7
#
[1 00 −i
]
# ?
[1 00 i
]
# ±45 1√2
[1 ±i±i 1
]
*7
[1 00 −1
]
[eiφ 00 eiφ
]
G
[eiφx 00 eiφy
]
D ?G 1
2
[1 i−i 1
]
> 12
[1 −ii 1
]
&:
! "#
4 , ε µ
Ei
ki
Bi ε
i, µ
i
Et
kt
Btε
t, µ
t
# 7 ; " 7 , # ; #
!"
ki
kr
kt
θi
θr
θt
ni
nt
Incident
Reflected
Transmitted
x
z
y
D , ; 7 2
&<
exp(iki · r − iωt) exp(ikr · r − iωt) ; exp(ikt · r − iωt) $%
t φ = k · r − ωt
ki · r|z=0 = kr · r|z=0 = kt · r|z=0
" z = 0 r = xx+ yy
ki = (niωc)(kixx+ kiyy + kizz)
kr = (niωc)(krxx+ kryy + krz z)
kt = (ntωc)(krxx+ kryy + krz z)
x y z x y z (kαx , kαy , k
αz )
; α = ir t z = 0
ni(kixx+ kiyy) = ni(k
rxx+ kryy) = nt(k
txx+ ktyy)
" x y
nikix = nik
rx = ntk
tx
nikiy = nik
ry = ntk
ty
kix = krx kiy = kry
kix = (nt
ni)ktx kiy = (nt
ni)kty
; x y 4 ! ki ki
ki
kr
kt x
y
plane of incident
&=
# kx ky kz > ! x− z 4
Incident Reflected
ki k
r
kt
θi
θr
θt
Transmitted
x
z
ni
nt
(kαx , kαy , k
αz ) α = i r t
kix = sin θi kiy = 0 kiz = cos θikrx = sin θr kry = 0 krz = cos θrktx = sin θt kry = 0 ktz = cos θt
kix = krx ! ;
θi = θr
" nikix = ntk
tx
ni sin θi = nt sin θt
#
4 7
&A
∮sD · ds = 0∮
sB · ds = 0∮ E · dl = − ∂
∂t
∫sB · ds∮ H · dl = ∂
∂t
∫sD · ds
# ! "
dh
dA
i
t
n
n i t1 ds = −ndA i ds = ndA t !
Di · n = Dt · nBi · n = Bt · n
. D B / - Di Bi Dt Bt ! i n - !
dh
dA
i
t
n
t
t " dh→ 0 ; φB =∫sB · ds = 0
φD =∫sD · ds = 01
Ei · t = Et · tHi · t = Ht · t
'B
. E H / Ei Hi Et Ht ! ! i n
$" $
; , * ; ; $C ? E ! ? %2
& ? E ⊥ $ σ - H ?%
' 9 ? E ‖ $ π ?%
# ? !
ki
kr
kt
Ei
Bi
Br
Bt
Er
Et
ni
nt
θi
θt
y(y) x(x)
z(z)
! ⇒ E "#$
% ⇒ E "#$
'&
2
Ei = yEi exp i(ωt− ki · r)ki =
ωnic
(x sin θi + z cos θi)
Bi = 1ωki × Ei
Bi =nic
(−x cos θi + z sin θi)Ei exp i(ωt− ki · r)
; 2
kr =ωnic
(x sin θi − z cos θi)
Er = yEr exp i(ωt− ki · r)Br =
nic
(x cos θi + z sin θi)Er exp i(ωt− ki · r) 2
kt =ωntc
(x sin θt + z cos θt)
Et = yEt exp i(ωt− kt · r)Bt =
ntc
(−x cos θt + z sin θt)Et exp i(ωt− kt · r)- $ r = 0% # 7 ! ? D ! ! # - $z% B2
nic
sin θi(Ei + Er) =ntc
sin θtEt
$x% H 2
niµic
cos θi(Ei − Er) =ntµtc
cos θtEt
+ ! Ei + Er = Et Et
niµi
cos θi(Ei − Er) =ntµt
cos θt(Ei + Er)
" ! !"
rs ≡ ErEi
=(ni
µicos θi − nt
µtcos θt)
(ni
µicos θi +
nt
µtcos θt)
''
C nt = ni sin θi/ sin θt
niµi
cos θi(Ei − Er) =ni sin θiµt sin θt
cos θt(Ei + Er)
µt tan θt(Ei − Er) = µi tan θi(Ei + Er)
rs ≡ ErEi
=(−µi tan θi + µt tan θt)
(µi tan θi + µt tan θt)
Ei+Er = Et !
ts ≡ EtEi
=2µt tan θt
(µi tan θi + µt tan θt)
# ! C θi θt !
7 µi ≈ µt ≈ 1 4 ! I
rs = − sin(θi − θt)/ sin(θi + θt)
ts = 2 sin θt cos θi/ sin(θi + θt)
# 9 ? ! ,
& θi → 0'
'0
ki
kr
kt
Ei
Bi
Br
Bt
Er
Et
x(x)y(y)
z(z)
ni
nt
θi
θt
#
rp =−εi tan θi + εt tan θtεi tan θi + εt tan θt
tp =2εi sin θi
cos θt(εi tan θi + εt tan θt)
µi ≈ µt ≈ 1
rp = tan(θi − θt)/ tan(θi + θt)
tp = 2 cos θi sin θt/[sin(θi + θt) cos(θi − θt)]
% & '
θt
θin
i
nt
n
Ai
Ao
At
Ai
Si
Sr
St
θi
'5
# I 7 ! E ! E * ; ? I nt I ni * ! I ./ 7 . / A W ?J , ?
+ ! A0
Ai = Ar = A0 cos θi
At = A0 cos θt
9 Sα $ % 1 Wα Aα
Wα = Sα · Aα α = i r t Sα = 1
2
√εαµαE2α µα ≈ µ0
√εα = nα
√ε0
# Wi = ni
2
√ε0µoE2i A0 cos θi
# ; Wr = ni
2
√ε0µoE2rA0 cos θi
# Wt = nt
2
√ε0µoE2tA0 cos θt
$ 9?%
R ≡ Wr
Wi=E2r
E2i
= |r|2
T ≡ Wt
Wi=nt cos θtni cos θi
E2i
E2r
=nt cos θtni cos θi
|t|2
! &
# ! ; 2
'6
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
9080706050403020100
1.0
0.8
0.6
0.4
0.2
0.0
9080706050403020100
PP
SS
S
P
Brewster Angle
Brewster AngleBrewster Angle
Glancing incident
ni=1, n
r=1.5
incident angle incident angle
Pow
er R
efle
ctiv
ity (
R)
Fie
ld R
efle
ctiv
ity (
r)
# 2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
9080706050403020100
1.0
0.8
0.6
0.4
0.2
0.0
9080706050403020100
P
S
Brewster Angle
Brewster Angle
Critical Angle Critical Angle
Total Internal
ReflectionTotal Internal
Reflection
S
P
incident angle incident angle
Pow
er R
efle
ctiv
ity (
R)
Fie
ld R
efle
ctiv
ity (
r)
ni=1.5, n
r=1
*
θ = 0 ! ! 9 ? ; 9 ? θ < 10 ; I θ = 0
I θ → 0
sin(θi + θt) = sin θi cos θt + cos θi sin θt
':
ts =2 sin θt cos θi
sin θi cos θt + cos θi sin θt
+ > sin θi = nt
nisin θt
ts =2 cos θi
nt
nicos θt + cos θi
" θi = 0 θt = 0 cos θi = cos θt = 1
t = 2ni
ni+nt T = ( 2ni
ni+nt)2
+
r = ni−nt
ni+nt R = (ni−nt
ni+nt)2
? tp rp 5K ; 7
L&6 &B ; R = 0.04 ; 7; $* 7; %
; ? ? .+ / # ; I ?
rp = tan(θi−θt)tan(θi+θt)
→ 0 θi + θt = 90
ntni
=sin θi
sin(90 − θi)=
sin θicos θi
+ ! + θB
θB = tan−1(nt
ni)
+ 9 ? ni > nt nt > ni
'<
; ./ . / ni > nt $# % #
θt = sin−1(nint
sin θi)
4 ni > nt ni
ntsin θi > 1 > ! θc
ni
ntsin θc = 1
θc = sin−1(nt
ni)
" θt = 90 4
Et ∝ e−ik·r = e−ikt(x sin θt+z cos θt)
4
sin θt =nint
sin θi =sin θisin θc
cos θt = ±√
1 − sin2 θt = ±√
1 − (sin θisin θc
)2
- θi > θc #
cos θt = −i√
(sin θisin θc
)2 − 1 ≡ −iα θc < θ <π
2
α =
√(sin θisin θc
)2 − 1
Et ∝ e−ikt(x sin θt+z cos θt)
Et = e−ktαze−iktx√
1+α2
$ x % z $ % H
'=
z
1 ; ! ! 9 # ! E H $7 9 %
I · n =⟨S⟩
=1
2Re( E × H∗)
# ; kt ⊥ Et
I =⟨St⟩· n =
1
2Re[( Et × H∗
t ) · n]
=1
2Re[ Et × (
1
µtωkt × Et
∗)] · n
=1
2µtωRe[E2
t (kt · n)]
kt · n = kt cos θt = −iαkt
⟨St⟩· n = 0
4 ! z = 1/γ z
1/e 1 γ
1
γ=
1
ktα
4 7 ! ; + ; 7 ;
Variable attenuator Prism coupler
'A
! ; I 9 ? cos θt = iα sin θt =
√1 + α2
# ?
rs = −sin(θi − θt)
sin(θi + θt)
= −sin θi cos θt − cos θi sin θtsin θi cos θt + cos θi sin θt
=
√1 + α2 cos θi − iα sin θi√1 + α2 cos θi + iα sin θi
= eiφs
# 9 ? ! sinα cosβ =12sin(α−
β) + sin(α + β)
rp =tan(θi − θt)
tan(θi + θt)
=sin(θi − θt) cos(θi + θt)
sin(θi + θt) cos(θi − θt)
=− sin 2θt + sin 2θisin 2θt + sin 2θi
=sin θi cos θi − sin θt cos θtsin θi cos θi + sin θt cos θt
=sin θi cos θi − iα
√1 + α2
sin θi cos θi + iα√
1 + α2= eiφp
; & Rs = Rp = 1 ! ; 4 !
; ! 9?
φs = 2 tan−1 α sin θi√1 + α2 cos θi
φp = 2 tan−1 α√
1 + α2
sin θi cos θi
0B
180
160
140
120
100
80
60
40
20
0
Ph
ase
ch
an
ge (
ϕ)
908070605040
Incident angle (θ)
P
S
; 1 ./ ;
4 90 ; & ni nt ! ; 9 ? D 4 90 ! 7 ; ! ; ; 7 #
& $ &M% 7 ( ; C 7 $ θi & ' 90% 7 ; . / 7
0&
. / D = ε E B = µ H ε µ ! 4 J = σ E $ σ %-
∇ · E = 0
∇ · B = 0
∇× E = −∂B
∂t
∇× B = µσ E + µε∂ E
∂t
∇2 E = µε∂2 E
∂t2+ µσ
∂ E
∂t
! .7 / 4 ! 7 # ω
E(r, t) = E(r)eiωt
∇2 E(r) + ω2µ(ε− iσ
ω) E(r) = 0
( - !
ε = ε− iσ
ω
∇2 E(r) + ω2µε E(r) = 0
4 1
∇2 E(r) + k2 E(r) = 0
k = ω
√µ(ε− iσ
ω) =
nω
c
0'
n = n(1 − iκ) κ ! ? z
E(r) = E0 exp(−ikz) = E0 exp(−inωcz) exp(−z
d)
d = c/nωκ . / n k ε µ σ
n2 =c2
2[
√µ2ε2 + (
µσ
ω)2 + µε]
n2κ2 =c2
2[
√µ2ε2 + (
µσ
ω)2 − µε]
# σω ε n2κ2 c2µσ
2ω
d √
2µσω
1/√σ
10N
4 # ; &% '% ε µ # ; # ! ;
θt
sin θt =nint
sin θi
; I ; ? ? ; 4 ;
R = |r|2 = n−1n+1
· n∗−1n∗+1
= (n−1)2+(nκ)2
(n+1)2+(nκ)2
= 1 − 4n(n+1)2+(nκ)2
00
- n = inκ
R =(inκ− 1)(−inκ− 1)
(inκ + 1)(−inκ + 1)= 1
. ;/ $k = ω
√µ(ε− iσ
ψ)% " ; $AB A6K%
Recommended