! ! C 7
! ! " ./
! < " !
! 7
$ %
! ! %
" <
./ .
/ d n1 n2 n3 = n2 " x
n1
B
A
E
d
n2
θ
C
y
5&
n3=n1
z
5'
4 k 3
. / " ! ? $ ?% λ = λ0 /n1 k = nk0 c = c0 /n1 ? y x − z θ z θ θ̄c = 90◦ − θc < φr < 4 7 <
#
! ? # < ! x
7 A B , 2π < A C < φr < AC = d/ sin θ AB = AC · cos 2θ AC − AB = AC(1 − cos 2θ) =
d
(2 sin2 θ)
sin θ
= 2d sin θ
, 2π
(2d sin θ) − 2φr = 2πm
λ
m = 0, 1, 2 · · ·
50
# φr ( $ <% 7
θ = θm m = 1, 2 · · ·
λ
sin θm = m 2d
H m θm ! mth m = 1 θ1 = sin−1 (λ/2d)1
m 7 x kx = nk0 sin θm = (2π/λ) sin θm ,
kxm ≡ hm = m
π
d
m = 1, 2 · · ·
±θ
z y − z (0, ky , kz ) (0, −ky , kz )1 y z βm z k kzm ≡ βm = k cos θm
= k 1 − sin2 θm
=
k2 −
m2 π 2
d2
*
θm kym βm , - k
, θm &
φr m sin θ0 = λφ
2πd r
55
kxm
x
xm
! "
! Am exp(−ikx − iβm z) Am exp(ikx − iβm z) 7 ! Ex (x, z) ∝ um (x) exp(iβm z)
um (x) =
⎧ 2
⎨
cos mπx
d
d
2
mπx
⎩
sin
d
d
m = 1, 3, 5 · · ·
m = 2, 4, 6 · · ·
um(x) H x z 56
x
#" $
< θ ≤ θ̄c # m θi H θi > θ̄c < 1 ! M ? M = Int(
sin θc
)
λ/2d
$% "
0 " µ = 1 √
nj = r,j j = 1, 2, 3 (x) = r (x)0 n1 = n3 = −µ(x)
= ∇(∇ · E)
− ∇2 E
∇ × (∇ × E)
∂2E
r (x) ∂ 2 E
=
−
∂t2
c2 ∂t2
∇ · D = 0 - r = r (x) = 0
∇·D
+ (∇) · E
= (∇ · E)
+ 0 ( ∂r )Ex
= ∇ · E
∂x
⇒
= − 1 ( ∂r )Ex
∇·E
r ∂x
∇2 E ∇ · E ≈ 0 - 9 ? ? ! E
5:
E = Ey ŷ ⇒ Ex = 0 ⇒ ∇ · E = 0 $3 9
? !
? H %
2
- ? H -
∂ 2 E/∂y
=0
y 7 (
∂2
∂2 r ∂ 2 E(x,
z, t)
+
)
E(x,
z,
t)
=
2
2
2
2
∂x
∂z
c
∂t
D z, t) ! E(x,
z
ω i(ωt−βz)
E(x,
z, t) = E(x)e
9
∂ 2 E(x)
ω2 2
E(x)
=
−
−
β
r 2 E(x)
∂x2
c
2 2
nj ω = − 2 E(x)
c
nj j (j = 1, 2, 3) k02 = ωc j
2
2
∂ 2 E(x)
∂x2
+ (n2j k02 − β 2 )E(x)
=0
- β z 2 β 2 n2j k02 J!
h2 = n2j k02 − β 2
% h2 > 0 h "
n2j k02 > β 2
∂ 2 E(x)
=0
+ h2 E(x)
∂x2
E(x) = A cos hx + B sin hx
% h2 < 0 h = iγ "
5;
n2j k02 < β 2
∂ 2 E(x)
=0
− γ 2 E(x)
2
∂x
E(x) = Aeγx + Be−γx
& & ' β n2 > n3 > n1 , β %
β > n2 k0 > n3 k0 > n1 k0
0 " %
n2 k0 > β > n3 k0 > n1 k0
$
& 0% $ e−x 5=
& ex 0 x < 0% E ∂ E/∂x
! .! / .
/
%
n2 k0 > n3 k0 > β > n1 k0
%
n2 k0 > n3 k0 > n1 k0 > β
- & ' 0
. / . /
β k ' k0n2 n1
n2
θi
k0n2
θi
h=kx
β
D
sin θi =
β
k0 n2
$% $% β = k0 n1
sin θi =
k0 n1
n1
=
k0 n2
n2
G
< ( sin θc = nn ! < D $% $% G
$% < ! 0 4 H 1
2
5A
! ( ' x
n1
a
d=2a
n2
z
0
-a
n3=n3
H ? ! E = E ŷ - n3 = n1 E
& 0
γ =
( )%
β 2 − n21 k02 > 0
h = n22k02 − β 2
Ea *%
Es
E = Er e−γx
E = Es cos hx + Ea sin hx
B
x
z
y
E
k
# & E H !
, - E
= Ey ŷ $ ? H
H % ! B = Bx x̂ + By ŷ + Bz ẑ = µHx x̂ + µHy ŷ + µHz ẑ - ! k E
# = − ∂B
∇×E
∂t
⇒
=
∇×E
x̂
∂
∂x
0
ŷ
ẑ ∂
∂y
∂
∂z
Ey
0
=−
∂Ey
∂Ey
x̂ +
ẑ = −iωµHx x̂ − iωµHz ẑ
∂z
∂x
" ! x z Hz H ∂Ey /∂x 6B
? Ey ∂Ey /∂x $ 4 E ∂E/∂x
> ! ' x = a
Es cos ha = Er e−γa
E
∂E/∂x
hEs sin ha = γEr e−γa
J
ha tan ha = γa
$%
h γ a " 7
−ha cot ha = γa
$%
!
h2 = n22 k02 − β 2
γ 2 = β 2 − n21 k02
h β a2 γ
γ 2 a2 + h2 a2 = (n22 − n21 )k02 a2 ≡ V2a2
$%
V = k0 n22 − n21 γ h $ β % P$% $% $% $% Q
# 2
ha tan ha =
√
V2 a2 − h2 a2
! h γa ha $% $%
6&
γa
5
4
3
s
a
s
a
2
1.7
1
4
1
0
1
ha
2
3
4
5
6
Graphical construction to find the modes in a planar waveguide. The
curves labelled "s" and "a" represent symmetric and antisymmetric
modes respectively and the circle, radius aV, is shown for a=1, 1.7 and 4.
2
& h γ β β .
/ β .
/ ' $ % 7
$n1 = n3 %
0 " h $ γ % " ! !
5 *
7 $ aV% γ . !/ 6 + $5% , , 1 . / - 7 ! I
, λ
a< 4 n22 − n21
6'
λ : 1 + Int(
2aV
)
π
; m = 0, 1, 2, 3, · · · $ % ! E(x,
y)e−iβmz
./ β $ β % !
$! % ! = $ % ! E(x,
y, z) =
m (x, y)e−iβmz
am E
m
am .
/ ! - # ( , ! $ %
(# )*
C ! ! ! x y " ! n2 n1 cladding
core
n1
n2
2a
60
∇2 E + n2j k02 E = 0
j = 1, 2
! 1 ∂ ∂E
1 ∂2E ∂2E
+ n2j k02 E = 0
(r
)+ 2 2 +
2
r ∂r ∂r
r ∂θ
∂z
" z7 e−iβz E(r, θ, z) = E(r, θ)e−iβz
1 ∂2E
1 ∂ ∂E
(r
) + 2 2 + (n2j k02 − β 2 )E = 0
r ∂r ∂r
r ∂θ
2
E(r, θ) = R(r)Θ(θ)
∂2E
d2 Θ
=
R
∂θ2
dθ2
dR
∂E
=Θ
∂r
dr
Θ dR
d2 R
1 ∂ ∂E
(r
)=
+Θ 2
r ∂r ∂r
r dr
dr
d2 R R d2 Θ
Θ dR
+ Θ 2 + 2 2 + (n2j k02 − β 2 )RΘ = 0
r dr
dr
r dθ
r2/RΘ r 2 d2 R
r dR
1 d2 Θ
2
2 2
2
+ r (nj k0 − β ) = −
+
≡ l2
2
2
R dr
R dr
Θ dθ
l2 ?
# ? d2 Θ
+ l2 Θ = 0
dθ2
Θ(θ) = A cos lθ + B sin lθ
! ? ! $ %
l=0
65
l=3
# l2
d2 R 1 dR
2 2
2
+
k
−
β
−
)R = 0
+
(n
j 0
dr 2
r dr
r2
β < n2 k0
β > n1 k0
"
!
κ2 = n22 k02 − β 2
γ 2 = β 2 − n21 k02
4 d2 R 1 dR
l2
2
+
(κ
+
−
)R = 0
dr 2
r dr
r2
core :
cladding :
d2 R 1 dR
l2
2
−
(γ
+
+
)R = 0
dr 2
r dr
r2
+ •
4 κ2 − rl > 0 .+ ! l/
Jl (κr) + Jl (κr) r ( κr 1 r Jl (κr) 2
2
Jl (κr) 1 π
2
cos[κr − (l + ) ]
πκr
2 2
κr 1
•
66
4 κ2 + rl > 0 .+ l/ Kl (γr) Kl (γr) κr 1
2
2
e−γr
Kl (γr) √
2πγr
- $l > 0% " β ! F
κ γ κ2 + γ 2 = k02 (n22 − n21 )
? $V % !2
V = k0 a n22 − n21
6:
$ !%
7 ! 2 • V
number of modes = 4
•
V
(polarization degeneracies included)
π2
? 7 ! V ∝ a V 2 V < 2.405 !
D ! , 1 7 ! ! ! , λc =
2πa 2
n2 − n21
2.405
.
7 !/ 7
! 7
? $
? ! 7 ! % ( 7
? x̂ ŷ ! β .
/ ! ! J0 + - + 7 ! E(r) = E0 e−r
2 /w 2
w !
1.619
w
2.87
= 0.65 + −3/2 + 6
a
V
V
A:K + λ = 0.86λc 2λc &
V () a * -w
0
6;
w
)
" ! " ! θ ! θ̄c "
7 θa θ̄c 2 1 · sin θa = n1 sin θ̄c sin θa = n1 (1 − cos θ̄c )1/2 = n1 [1 − (
n2 2 1/2
) ] = (n21 − n22 )1/2
n1
θa = sin−1 NA
NA = (n1 − n2 )1/2
! θa !
! G θa ! 7
! 4 ! θa ! C ! 7
!
6=