∇ · 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%