=ρ
∇·D
!
=0
∇·B
!
" = − ∂ B
∇×E
∂t
= J +
∇×H
D
E
B
H
ρ
J
∂D
∂t
# $C/m2 %
! $V /m%
! $T V · s/m2%
! $A/m)
$C/m3 %
$A/m2 %
&
'
= 4πρ
∇·D
=0
∇·B
= − 1 ∂B
∇×E
c ∂t
=
∇×H
4π J
c
+
1 ∂D
c ∂t
2
) E(statvolt/cm)
B(gauss)
H(oersted)
ρ(statcoulomb)
D(statcoul/cm
J(statamp)
! q0
( ! E F = q0 E
E D B H
D B ! = 0 E
+ P
D
= µ0 H
+M
B
0 = (1/36π) × 10−9 $#)% µ0 = 4π × 10−7$*)% P M +
P M ! P M ! P M ,
* D B E H - J E . /
= f (E)
D
J = g(E)
= h(H)
B
0
" E H 1 D = 0 χE χ = 0 E
+ P = 0 E
+ 0 χE
D
= E
= 0 (1 + χ)E
2
$ % 3 = E
= 0 n2 E
D
n=
√
= r
0
J = σ E
σ2
M = 0
= µ0 H
B
4 2
& $
% 2 µ ' 2 µ 0 2 σ = 0 ⇒ J = 0
5 2 ρ = 0
6 72 µ ! 2
=0
∇·E
=0
∇·B
= − ∂ B
∇×E
∂t
= µ ∂ E
∇×B
∂t
D = E B = µH
χ
(3) D = 0 χE + χ(2) E · E +
·E
+ ···
E ·E
5
7
B ! E ! 1 , E $ % 4 B
= ∇ × (− ∂ B ) = − ∂ (∇ × B)
∇ × (∇ × E)
∂t
∂t
∂
= − ∂t
(µ ∂∂tE )
= ∇(∇ · E)
− ∇2 E
= 0 ∇ · E
∇ × (∇ × E)
2
= µ ∂ E2
∇2 E
∂t
→
E 2
#
−
2
= µ ∂ B2
∇2 B
∂t
∇2 f (r, t) =
1 ∂ 2 f (r, t)
v 2 ∂t2
v8 $ ./% f (r, t)8 $% !
4 √
v = 1/ µ
c = 1/õ00
µ ≈ µ0 v=√
= 2.998 × 108 m/s
1
c
=
µ0 0 r
n
6
k·r)
r , t) = E0 ei(ωt−
E(
k * ! 7
* r , t) E(
r, t)
E(
9
r)
= E0 ∇2 ei(ωt−k·
∇2 E
= E0 ∇ · [−ikei(ωt−k·r) ]
= −k 2 E0 ei(ωt−k·r)
= −k 2 E
µ
∂2E
= −ω 2 µE
∂t2
√
k = ω µ
"
=0
∇·E
⇒
∇ · [E0 ei(ωt−k·r) ] = E0 · ∇ei(ωt−k·r) = −ik · E0 ei(ωt−k·r) = 0
k · E0 = 0
⇒ ! $ B
# = − ∂B
∇×E
∂t
= −iω B
−ik × E
!%
:
k × E
= ωB
k E B B
9 2
E
z
E
k
B
y
x
" B ! √
ω µ0 |k| n |B| =
|E| =
|E| = |E|
ω
ω
c
7 E B 7 ! ! B
2
H
|E|
|E|
=µ
=
|H|
|B|
µ
≡Z
Z (Ω) Z0 = µ = 377Ω
0
0
9
; S 9
! !
=E
×H
S
9
J/(m2 · sec) 7
! <
·E
+B
· H)
U = 12 (D
B = nc E
1
n2
r 0 µ0 2
B2
1
1
) = ( + 2 )E 2 = ( +
)E
U = (E 2 +
2
µ
2
µc
2
µ
(J/m3) # µ ≈ µ0 U
! U = E 2
9
2
∂Wmech ∂U
· dA
+
=− S
∂t
∂t
s
Sin
A
U
Wmech
Sout
9
; $Sout −Sin% A , V ! $U % Wmech 9
= v · U v U 7
|S|
$ 9
S J U ρ%
; S .7
/ S I 7
; $W/m2 %
= I k = I n̂
S
k
1015 1011 1015
=
=E
×H
= (E0 × H
0 ) cos2 (ωt − k · r)
S
# k × E = ω B ⇒ H = ωµ1 k × E k = ω √µ
= E0 × ( 1 k × E0 ) cos2 (ωt − k · r) =
S
ωµ
2
E0 cos2 (ωt − k · r) n̂
µ
- $ n̂% $ .
/ %
- 7
; I≡ S
=
=
1 t0 +T
2
E cos2 (ωt − k · r) dt n̂
T t0
µ 0
1
2 t0 +T
E
cos2 (ωt − k · r) dt n̂
T
µ 0 t0
> θ = ωt − k · r cos2 θ = 12 (1 + cos 2θ)
1
ωT
2
E
µ 0
ω(t0 +T )
ωt0
1
cos θ dθ =
2ωT
2
1
2ωT
=
2
E
µ 0
ω(t0 +T )
(1 + cos 2θ)dθ
ωt0
1 ω(t0 +T )
2
cos 2θd2θ)
E0 (ωT +
µ
2 ωt0
2
1
E0 {1 +
[sin 2(ωt0 + ωT − k · r)
µ
2ωT
− sin 2(ωt0 − k · r)]}
1
2
=
$T ≈ 10−9 f ≈ 1 ?% $ω ≈ 3.5 × 1015 *? -2@
60' %
1
ωT = 3.5 × 105 1 ⇒ ωT
11 =
I ≡ S
1
2
2
E
µ 0
= 12 ( Z1 )E02
I = √1µ · ( 12 E02) = v · U $ 12 7
% # U = 12 E02
A
; 7
>? F = qE + qv × B +
>? 7
>? J = qv
$)% F1 = ρE + J × B - ρ J = ρ/0 , ∇ × B
= µ0 J + 0 µ0 ∂ E/∂t
∇·E
- ⇒
E
+ 1 (∇ × B)
×B
− 0 ∂ E × B
F1 = 0 (∇ · E)
µ0
∂t
∂ = ∂E × B
+ E×
∂B
(E × B)
∂t
∂t
∂t
∂ = 0 (∇ · E)
× (∇ × E)
+ 1 (∇ · B)
× (∇ × B)
E
− 0 E
B
− 1B
× B)
F1 + 0 (E
∂t
µ0
µ0
∇·B = 0 ∇ × E = ∂ B/∂t
V ∂ × B)dV
Ftotal +
0 (E
= [r.h.s.]dV
∂t v
v
- dPmech dPf ield
+
= [r.h.s.]dV
dt
dt
v
Pf ield =
1
× B)dV
0 (E
= 2
c
v
SdV
v
! ! 2
g = S/c
&B
4 r.h.s. v
∇ · T dV →
s
T · ds
1 ¯ 0 2 B 2
→−
−
→
T = 0 E E + B
B − I( E +
)
µ0
2
2µ0
T . / 2nd r.h.s. V 4 V |∆P |
|g |V
f orce
P =
= ∆t = ∆t
area
A
A
∆t c∆t V
P =
|
g |A·c∆t
∆t
A
= |g | c =
= A·c∆t
S c
C /c = I/c
P = S
# 1.34 × 103J/m2 · s # 4.46 × 10−6 N/m2 ∼ 105 N/m2 D ! E H =E
0 exp i(k · r − ωt)
E
=H
0 exp i(k · r − ωt)
H
E 0 H 0 H
? ! E
! ? ! ?
&&
H
E
H
E
k
7 ? ?
; # ? ,
./ ! ? , 1 > ? ./ , D ? E0 ? 7
, ± π2 !
= E0 [x̂ exp i(kz − ωt) + ŷ exp i(kz − ωt ± π )]
E
2
eiπ/2 = i
= E0 (x̂ ± ŷ) exp i(kz − ωt)
E
E ω " 1 ! &'
H
E
H
E
k
? ! ! ?
H
E
H
E
k
! 0 = (x̂E0 ± iŷE ) exp i(kz − ωt)
E
0
" ? ?
? ! !
$ ?% "
D ? ? - ! E E 1 E 2 E 1 ?
&0
(Incident wave)
E2
E
θ
Transmitted axis
of polarizer
(Transmitted wave)
E
1
E θ 7
! E1 = E cos θ
" I1 ! I1 = I cos2 θ
# ? θ
? ? ( cos2 θ 12 >
? ?
? P ! ?
P =
Ipol
Ipol + Iunpol
? Imax = Ipol + 12 Iunpol Imin = 12 Iunpol P =
Imax − Imin
Imax + Imin
Imax Imin ?
" π2 $ % D ? ?
" π
&5
Fast
Fast
Slow
Slow
Fast
Fast
Slow
π
π/2
Slow
π/2
π
Half waveplate
Quarter waveplate
? 0 = x̂E0x + ŷE0y
E
E0x E0y "
7
iφ
E0x = |E0x |e x
E0y = |E0y |eiφy
! E0x
E0y
|E0x |eiφx
|E0y |eiφy
=
? E E ? ? E # ? ? ? 1
−i
+
1
i
=2
1
0
'' A
> B
a b
c d
A
B
A
B
E =
A
B
a b
c d
&6
E 2
atotal btotal
ctotal dtotal
=
an bn
cn d n
···
a1 b1
c1 d 1
E 7
?
?
> 9?
±45◦
1
0
1
0
# ?
# ±45◦
*7 G D ?
G
>
√1
2
1 0
0 −1
eiφx 0
0 eiφy
0
0 0
1
1 ±1
±1 1
0
−i
0
i
1 ±i
±i 1
eiφ 0
0 eiφ
1
0
0
0
1
2
# F7 1 i
1
2
−i 1 1 −i
1
2
i 1
&:
! "#
4 ,
µ
E
E
t
i
ki
Bi
kt
εi, µi
Bt
εt, µt
# 7 ;
" 7
,
# ; # !" Incident
θ
k
i
ni
r
θi
kr
Reflected
x
kt
Transmitted
θt
y
z
nt
D , ; 7 2
&<
; $% t φ = k · r − ωt exp(iki · r − iωt)
exp(ikr · r − iωt)
exp(ikt · r − iωt)
" z = 0 ki · r|z=0 = kr · r|z=0 = kt · r|z=0
r = xx̂ + y ŷ
ki = (ni ω )(k i x̂ + k i ŷ + k i ẑ)
x
y
z
c
kr = (ni ω )(k r x̂ + k r ŷ + k r ẑ)
x
y
z
c
kt = (nt ω )(k r x̂ + k r ŷ + k r ẑ)
x
c
y
z
x̂ ŷ ẑ x y z (kxα , kyα, kzα)
; α = i
r t z = 0
ni (kxi x + kyi y) = ni (kxr x + kyr y) = nt (kxt x + kyt y)
" x y ni kxi = ni kxr = nt kxt
ni kyi = ni kyr = nt kyt
kxi = kxr
kxi = ( nnti )kxt
kyi = kyr
kyi = ( nnti )kyt
;
x
y 4 ! ki ki y
ki
kt
x
kr
plane of incident
&=
# kx ky kz > ! x − z 4
z
ki
kr
θi
Incident
n
i
θr
Reflected
x
n
t
θt
Transmitted
kt
(kxα , kyα, kzα) α = i r t
kxi = sin θi
kxr = sin θr
kxt = sin θt
kyi = 0
kyr = 0
kyr = 0
kzi = cos θi
kzr = cos θr
kzt = cos θt
kxi = kxr ! ;
θi = θr
"
ni kxi = nt kxt ni sin θi = nt sin θt
#
4 7 &A
· ds = 0
D
s=0
s B · d
· dl = − ∂ B
· ds
E
∂t s
∂
· dl =
· ds
H
D
s
∂t s
# ! " dA
i
dh
t
n
i t1 ds = −n̂dA i ds = n̂dA t ! n̂
i · n̂ = D
t · n̂
D
i · n̂ = B
t · n̂
B
. D B / - D i B i D t B t ! i n - ! dA
i
dh
t
t
n
t̂ φD = s D · ds = 01 " dh → 0 ; φB =
i · t̂ = E
t · t̂
E
i · t̂ = H
t · t̂
H
s
· ds = 0
B
'B
. E H / E i H i E t H t !
! i n $" $
; , * ; ; $C ? E ! ? %2
⊥ E
& ?
$ σ - H ?%
E
' 9 ?
$ π ?%
# ? ! Ei
ki
Bi
Er
θi
ni
kr
Br
y(y) θ
t
nt
x(x)
Et
Bt
kt
z(z)
%! ⇒
⇒
"#$ E
"
#$ E
'&
2
i = ŷEi exp i(ωt − ki · r)
E
ki = ωni (x̂ sin θi + ẑ cos θi )
c
B i = ω1 ki × E i i = ni (−x̂ cos θi + ẑ sin θi )Ei exp i(ωt − ki · r)
B
c
; 2
kr = ωni (x̂ sin θi − ẑ cos θi )
c
r = ŷEr exp i(ωt − ki · r)
E
r = ni (x̂ cos θi + ẑ sin θi )Er exp i(ωt − ki · r)
B
c
2
kt = ωnt (x̂ sin θt + ẑ cos θt )
c
t = ŷEt exp i(ωt − kt · r)
E
t = nt (−x̂ cos θt + ẑ sin θt )Et exp i(ωt − kt · r)
B
c
$ r = 0% # - 7
! ? D ! ! # - $ẑ% B 2
$x̂% H 2
nt
ni
sin θi (Ei + Er ) =
sin θt Et
c
c
ni
nt
cos θi (Ei − Er ) =
cos θt Et
µi c
µt c
+ ! Ei + Er = Et Et ni
nt
cos θi (Ei − Er ) =
cos θt (Ei + Er )
µi
µt
"
! ( µnii cos θi −
Er
rs ≡
= ni
Ei
( µi cos θi +
nt
µt
nt
µt
cos θt )
cos θt )
!"
''
C nt = ni sin θi / sin θt ni sin θi
ni
cos θi (Ei − Er ) =
cos θt (Ei + Er )
µi
µt sin θt
µt tan θt (Ei − Er ) = µi tan θi (Ei + Er )
rs ≡
Er
(−µi tan θi + µt tan θt )
=
Ei
(µi tan θi + µt tan θt )
Ei + Er = Et ts ≡
! Et
2µt tan θt
=
Ei
(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
Ei
Bi
Er
ki
kr
Br
θi
ni
y(y)
nt
x(x)
θt
Et
Bt
kt
z(z)
#
rp =
tp =
µi ≈ µt ≈ 1
−i tan θi + t tan θt
i tan θi + t tan θt
2i sin θi
cos θt (i tan θi + t tan θt )
rp = tan(θi − θt )/ tan(θi + θt )
tp = 2 cos θi sin θt /[sin(θi + θt ) cos(θi − θt )]
% & '
Ai
Ai
Sr
Si
ni
θi θi
Ao
nt
n
θt S
t
At
'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 = 12 µ Eα2 µα ≈ µ0 √α = nα√0
α
α
# Wi =
ni
2
# ; Wr =
ni
2
# Wt =
nt
2
?%
0 2
E A
µo i 0
0 2
E A
µo r 0
cos θi
0 2
E A
µo t 0
cos θt
R≡
T ≡
cos θi
$ 9
Wr
E2
= r2 = |r|2
Wi
Ei
Wt
nt cos θt Ei2
nt cos θt 2
=
=
|t|
2
Wi
ni cos θi Er
ni cos θi
!
& # ! ; 2
'6
n =1, n =1.5
i
r
0.2
1.0
Brewster Angle
0.0
0.8
Glancing incident
Power Reflectivity (R)
Field Reflectivity (r)
P
-0.2
S
-0.4
0.6
S
0.4
-0.6
P
0.2
-0.8
-1.0
Brewster Angle
0.0
0
10
20
30
40
50
60
70
80
90
0
10
20
incident angle
30
40
50
60
70
80
90
incident angle
# 2
ni=1.5, nr=1
1.0
1.0
0.8
0.8
Total Internal
Reflection
Total Internal
Reflection
Power Reflectivity (R)
Field Reflectivity (r)
0.6
0.4
S
0.2
0.6
0.4
Brewster Angle
P
S
0.2
0.0
Brewster Angle
Critical Angle
P
Critical Angle
-0.2
0.0
0
10
20
30
40
50
60
70
incident angle
80
90
0
10
20
30
40
50
60
70
80
90
incident angle
* θ = 0◦ ! ! 9 ? ; 9 ? θ < 10◦
; I θ = 0◦ I θ → 0◦ sin(θi + θt ) = sin θi cos θt + cos θi sin θt
':
2 sin θt cos θi
sin θi cos θt + cos θi sin θt
ts =
+ > sin θi = nn
t
i
sin θt ts =
nt
ni
2 cos θi
cos θt + cos θi
" θi = 0 θt = 0 cos θi = cos θt = 1
t=
2ni
ni +nt
T = ( n2n+n )2 i
i
t
+ r=
ni −nt
ni +nt
R = ( nn −n
)2 +n
i
t
i
t
? tp rp 5K ; 7
L&6 &B ; R = 0.04 ;
7; $* 7; %
; ? ? .+ / # ; I ? −θ )
rp = tan(θ
→0
θi + θt = 90◦
tan(θ +θ )
i
t
i
t
nt
sin θi
sin θi
=
=
◦
ni
sin(90 − θi )
cos θi
+ !
+ θB θB = tan−1 ( nnti )
+ 9 ? ni > nt nt > ni
'<
; ./
. / ni > nt $# % # θt = sin−1 (
4 ni > nt nn
nn sin θc = 1 i
t
i
t
ni
sin θi )
nt
sin θi > 1
> ! θc θc = sin−1 ( nnti )
" θt = 90◦ 4
Et ∝ e−ik·r = e−ikt (x sin θt +z cos θt )
4 sin θt =
ni
sin θi
sin θi =
nt
sin θc
cos θt = ± 1 − sin2 θt = ± 1 − (
sin θi 2
)
sin θc
- θi > θc # cos θt = −i (
sin θi 2
) − 1 ≡ −iα
sin θc
α=
(
θc < θ <
π
2
sin θi 2
) −1
sin θc
Et ∝ e−ikt (x sin θt +z cos θt )
Et = e−kt αz e−ikt x
√
1+α2
$ x % z $ %
H '=
z 1 ; ! ! 9
# ! E H $7
9
% = 1 Re(E
×H
∗)
I · n̂ = S
2
# ; kt ⊥ E t
1
t × H
t∗ ) · n̂]
Re[(E
I = St · n̂ =
2
1
t ∗ )] · n̂}
t × ( 1 kt × E
=
Re{[E
2
µt ω
1
=
Re[Et2 (kt · n̂)]
2µt ω
kt · n̂ = kt cos θt = −iαkt
St · n̂ = 0 4 ! z = 1/γ z 1/e 1 γ 1
1
=
γ
kt α
4 7
! ; + ; 7
; Variable attenuator
Prism coupler
'A
! ; I √ 9 ? cos θt = iα sin θt = 1 + α2
# ?
sin(θi − θt )
sin(θi + θt )
sin θi cos θt − cos θi sin θt
= −
sin θi cos θt + cos θi sin θt
√
1 + α2 cos θi − iα sin θi
= √
= eiφs
1 + α2 cos θi + iα sin θi
rs = −
# 9 ? ! sin α cos β = 12 {sin(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 2θi
sin 2θt + sin 2θi
sin θi cos θi − sin θt cos θt
sin θi cos θi + sin θt cos θt
√
sin θi cos θi − iα 1 + α2
√
= eiφp
sin θi cos θi + iα 1 + α2
; & Rs = Rp = 1 !
; 4 ! ; ! 9
? α sin θi
1 + α2 cos θi
√
α 1 + α2
sin θi cos θi
φs = 2 tan−1 √
φp = 2 tan−1
0B
180
160
Phase change (ϕ)
140
P
120
100
S
80
60
40
20
0
40
50
60
70
80
90
Incident angle (θ)
; 1 .
/ ; 4 90◦ ; & ni nt ! ; 9 ? D 4 90◦ ! 7 ; !
;
; 7 #
& $ &M% 7
( ; C 7 $ θi & ' 90◦% 7 ;
.
/ 7
0&
. / D = E B = µH µ ! 4 J = σE $ σ %
- = 0
∇·E
= 0
∇·B
= −
∇×E
∂B
∂t
= µσ E
+ µ
∇×B
∂E
∂t
= µ
∇2 E
∂E
∂2E
+
µσ
∂t2
∂t
! .7
/ 4 ! 7 # ω r, t) = E(
r)eiωt
E(
r) + ω 2 µ( −
∇2 E(
iσ )E(r) = 0
ω
( - ! ˜ = −
iσ
ω
r) = 0
r) + ω 2 µ˜E(
∇2 E(
4 r) + k̃ 2 E(
r) = 0
∇2 E(
k̃ = ω µ( −
ñω
iσ
)=
ω
c
1
0'
ñ = n(1 − iκ)
!
κ ? z r) = E
0 exp(−ik̃z) = E
0 exp(−in ω z) exp(− z )
E(
c
d
d = c/nωκ . / n k̃ µ σ
n2 =
c2
µσ
[ µ2 2 + ( )2 + µ]
2
ω
c2
µσ
n κ = [ µ2 2 + ( )2 − µ]
2
ω
2 2
# ωσ n2κ2 c2ωµσ 2
d
2
µσω
√
1/ σ 10N
4 # ; &% '% µ # ;
# ! ; θt ni
sin θ˜t =
sin θi
ñt
; I ; ? ? ; 4 ; ñ−1 ñ −1
R = |r|2 = ñ+1
· ñ∗ +1 =
4n
= 1 − (n+1)2 +(nκ)2
∗
(n−1)2 +(nκ)2
(n+1)2 +(nκ)2
00
- R=
ñ = inκ
(inκ − 1)(−inκ − 1)
=1
(inκ + 1)(−inκ + 1)
. ;/ $k̃ = ω µ( − iσψ )% " ; $AB A6K%