&
'
./ ! 9 F (x) = 0
x1 x2 F (x) = 01 ax1 + bx2 a b (∇2 + k2)E = 0 1 ! E1 E2 aE1 + bE2 3
4 #%% 4 ?1 ! , " ! 1 = E
01 ei(k1 ·r−ωt+θ1 )
E
2 = E
02 ei(k2 ·r−ωt+θ2 )
E
θ1 θ2 ! !
φ = k · r − ωt + θ ω
k1 k2 $ k = n ωc % 4 !
total = E
01 ei(k1 ·r+θ1 ) + E
1 + E
2 = E
02 ei(k2 ·r+θ2 ) e−iωt
E
==
=A
" D & ! .7/ 7
Itotal =
=
=
2
E 1 + E
2 ) · (E
∗ + E
∗)
(E
1
2
2
2
1
E1 + E
+ E
2
2
E1 ·E
∗
= E
= I1
and
2
E2 1 · E
01 · E
∗ = E
∗ ei[(k1 −k2 )·r+(θ1 −θ2 )] =
E
2
02
2 · E
∗ + E
∗
·E
2
1
∗ = E
∗ e−i[(k1 −k2 )·r−(θ1 −θ2 )] =
2 · E
02 · E
E
1
01
= I2
I1 I2 ei[(k1 −k2 )·r+(θ1 −θ2 )]
I1 I2 e−i[(k1 −k2 )·r+(θ1 −θ2 )]
cos δ = 12 (eiδ + e−iδ ) 7
! Itotal = I1 + I2 + 2 I1 I2 cos δ
δ = (k1 − k2 ) · r + (θ1 − θ2 )
7
r 7
. / ) δ . / δ = 0, ±2π, ±4π, · · ·
. / δ = ±π, ±3π, ±5π, · · ·
I1 = I2 = I Itotal = 2I + 2I cos δ = 4I cos2
δ
2
I
4I
AB
total
2I
0
π
2π
3π
4π
5π
6π
7π
, δ .
/ , , ? " ! (k1 − k2) · r , 7
, ,
1 , , (a) Phase difference induced by material
material (n)
mirror
combiner
Laser
Beam
Splitter
detector
(b) Phase difference induced by path difference
mirror
combiner
Laser
Beam
Splitter
detector
A&
, δ δ=
c2
k2 (r) · ds −
c1
k1 (r) · ds + (θ2 − θ1 )
C1
S
3
P
C2
5% 4
@
! @
! $ ! % " Light source
Slit
Double Slit
Screen
! 7 ! θ1 = θ2 . / A'
x
P
S
x
1
h
θ
θ
z
S2
D
D h θ z , S1 S2 P OP D = h sin θ hθ
θ tan θ =
x
D
" , 2π δ=
2π hx
·
= 2πm
λ D
m = 0, ±1, ±2, · · ·
xm = m λD
h
m mth , OP D = mλ
, I = 4I1 cos2
δ
πhx
= 4I1 cos2 (
)
2
λD
- 7 C ? ∆λ0 λ0
A0
Isource
∆λ
λ
λ0
∆λ λ0 .7/ ? x = 0 $
m = 0% |m| > 0 , Idifferent color
x
Itotal
x
D m m = 0
, xm = m λDh |δx| = |m|
D
δλ
h
4 $
|∆x| =
D
λ0
h
δm = 1%
A5
δx
m
m+1
∆x
" |δx|
1
|∆x|
|m| λ0
δλ0
+ 7 # 5BB ;BBnm |m| <
550nm
∼ 1.8
300nm
& x = 0 m = 0 ' x m = ±1, ±2 0 x m 2 , 3
A6
/ ,
" . /
< < d 4 < . ! </ 7
< 5K
n1
D
θ
i
A
C
θt
n2
d
B
n3
3 < DC , $C9J% OP D = n2 (AB + BC) − n1 AD
AB = BC =
AD = 2AB sin θt cos(
d
cos θt
π
− θi ) = 2AB sin θt sin θi
2
+ n1 sin θ1 = n2 sin θ2 2d
(n2 − n1 sin θi sin θt )
cos θt
2n2 d
(1 − sin2 θt )
=
cos θt
= 2n2 d cos θt
OP D =
, , δ =
2π
(2n2 d cos θt )
λ0
λ0 * < D &
# 7? ni nt
ni < nt ⇒ π phase shift
ni > nt ⇒ 0 phase shift
A:
n1 < n2 > n3 < A δ=
2π
(2n2 d cos θt ) + π = 2mπ
λ0
n2 d cos θt =
λ0
(2m
4
+ 1)
m = 0, 1, 2 · · ·
, 7
" <
1 5K < d
Er2
E
r1
Ei
n1=1
Ei
n
2
n3
anti-reflection coating
< 7< n2 n3 n1 = 1 < n2 < n3 π , δ=
2π
(2n2 d cos θt )
λ0
π ? < Er1
Er2 , (2m + 1)π $θi = θt = 0%
4πdn2
= (2m + 1)π
λ0
A;
7< d = (2m + 1)
λ0
4n2
m = 0, 1, 2, · · ·
Er1 Er2 < # < I r = nn −n
+n
i
i
t
t
n1 − n2
n2 − n3
=
n1 + n2
n2 + n3
# n1 = 1 n2 =
3
√
n3
& 4
7
@
"
Mirror 1
Compensation Plate
Mirror 2
Beam Splitter
Light source
Screen
P
& ' < & < 6BK & 6BK ' < ' 0
& & ,
A=
& & ( & & ' - 9 & d 2d n = 1
, δ = 2d
ω
c
9 IP = I1 + I2 + 2 I1 I2 cos(
2ωd
)
c
6BK I1 = I28 12 I0 IP = I0 [1 + cos(
2ωd
)] = I0 [1 + cos ωτ ]
c
τ = 2dc . / d 9 τ IP
-4π/ω
-2π/ω
0
2π/ω
4π/ω
τ
AA
S
2d
S
θ
t
θ'
d
t
Mirror 2
Mirror 1
}
off-axis waves
d θt θt θt ,7 δ=
2π
· 2d cos θt = 2πm
λ
m d1 2d = 4λ0 m=4
2
1
3
,
, * 7
&BB
Reference
Mirror
S
Measured
Mirror
screen
7 < 7
< .</ * , " 7 7
&)'B 30
4 6 4 ' cos δ δ , δ $δ = 2π
· OPD% λ
!
o
&B&
*+,+- . ) r
1
Pr1
r
P
P
2
θ
r2
incident
plane wave
dsinθ
d
r
P
N
rN
D
observe scene at infinity
4 $Nd% ! $" % ! P ( N 2
E = E0 (Pr1 )ei(ωt−kr1 ) + E0 (Pr2 )ei(ωt−kr2 ) + · · · + E(PrN )ei(ωt−krN )
rN N th P PrN N th E0 = E0 (Pr1 ) = · · · = E0 (PrN ) E = E0 [ei(ωt−kr1 ) + ei(ωt−kr2 ) + · · · + ei(ωt−krN ) ]
= E0 ei(ωt−kr1 ) [1 + e−ik(r2 −r1 ) + · · · + e−ik(rN −r1 ) ]
-
r2 − r1 = r21 = d sin θ =C9J ( r3 − r1 =
2r21 , · · · , rN − r1 = (N − 1)r21 ." , ( δ ≡ k(r2 − r1) = kd sin θ d (
! 9 E = E0 ei(ωt−kr1 ) 1 + e−iδ + (e−iδ )2 + · · · + (e−iδ )(N −1)
1 + x + x2 + · · · xN −1 =
xN − 1
x−1
! i(ωt−kr1 )
E = E0 e
i(ωt−kr1 )
= E0 e
e−iN δ − 1
e−iδ − 1
e−iN δ/2 (e−iN δ/2 − eiN δ/2 )
e−iδ/2 (e−iδ/2 − eiδ/2 )
= E0 ei{ωt−kr1 −(N −1)δ/2}
sin N2δ
sin 2δ
- &B'
1
Dc = r1 + (N − 1)d sin θ
2
P # i(ωt−kDc )
E = E0 e
sin Nδ
2
sin 2δ
|E|2 sin2 ( N kd2sin θ )
I = I0
θ
sin2 ( kd sin
)
2
I = I0 [
θ
sin2 (N π d sin
)
λ
]
d sin θ
2
sin (π λ )
- sin2 (Nφ) sin2(φ) φ = π d sinλ θ ? π
d sin θ
= mπ
λ
d sin θ = mλ
m = 0, ±1, ±2, · · ·
m = 0, ±1, ±2, · · ·
m $ ) % ? 2π φ → 0, or 2πN sin θ θ1
⇒
θ
θ 2
(Nπ d sin
sin2 (Nπ d sin
)
)
2
λ
λ
2
d sin θ
d sin θ 2 = N
sin (π λ )
(π λ )
I = N 2 I0
d sin θ = mλ
# θ I 4 φ = mπ I = N 2 I0 3 N &B0
Principal
maximum
Secondary
maximum
0
π/2
π
(θ)
3π/2
2π
4 Nd sin θ = mλ
θ λ dm
d sin θ
= cos θ =
dθ
Nd dθ
> dθ ∼ ∆θ dm ∼ ∆m = 1 λ
Nd cos θ
θ = 0 ∆θ =
∆θ =
λ
# ! = λ/grating size Nd
∆θ ? ? P 4
P , ! ?
*+,+/ 0 & & :0 < < 7
< < ,
&B5
n
E0t2r4
E0t2r3
E0tr4
3
E0tr
2
E t r
E0tr2
0
E0tr
E0r
E0t2r2
E0t2
E0t
θ
E0
d
+ D7
< n # <
7
7<
< ! < # :0 , δ δ=
4π
nd cos θ
λ0
- r I < t I 97
<
< E0 t E0tr E0 tr2, · · · !
E0 7
E0t2 E0 t2r2 E0 t2 r4, · · · , δ Et = E0 t2 + E0 t2 r 2 eiδ + E0 t2 r 4 ei2δ + · · ·
= E0 t2 (1 + r 2 eiδ + r 4 ei2δ + · · ·)
E0 t2
=
1 − r 2 eiδ
(1 − a)−1 = 1 + a + a2 + · · · It = I0
|t|4
|1 − r 2 eiδ |2
- r t < r = |r|eiδ /2 δr < # r
&B6
0 π 1 ! R < T r t R = |r|2 = rr∗ and T = |t|2 = tt∗
It = I0
T2
|1 − Rei∆ |2
∆ = δ + δr , -
|1 − Rei∆ |2 = (1 − Rei∆ )(1 − Re−i∆ ) = 1 − 2R cos ∆ + R2
4R
2 ∆
= (1 − R)2 [1 +
sin
]
(1 − R)2
2
* I0 T 2
It =
(1 − R)2
1
1 + F sin2
∆
2
1/(1 + F sin2 ∆2 ) , F =
4R
(1 − R)2
# ! 7
1 < R , < R ∆/2 " (1 + F sin2 ∆2 )−1 π F ∆/2 = Nπ 2Nπ =
4π
nd cos θ + δr
λ0
N N = 0, ±1, ±2, · · ·