24. Fiber Optics 24. Fiber Optics
Numerical aperture(N.A.) of optical fiber Allowed modes in fibers
Attenuation Distortion
• Modal distortion
• Material dispersion
• Waveguide dispersion
Optical fibers Optical fibers
SI(step-index) fiber
GRIN(graded-index) fiber
Optical communication systems
Optical communication systems
24-4. Optics of propagation 24-4. Optics of propagation
Cladding (n2)
Core (n1) Air (n0)
Numerical aperture (N.A.)
Step index fiber
Step-index fibers Step-index fibers
Skip distance :
(m-1)
24-5. Allowed modes : in slab (planar) waveguides 24-5. Allowed modes : in slab (planar) waveguides
Modes (self-sustaining waves) must satisfy the condition.
Total number of propagating modes is the value of m when
ϕ
m =ϕ
c(
n1cosϕ
c = N A. .)
2 2
max 1 1
2 2
. . 1 1
d d
m N A n n
λ λ
= + = − +
24-5. Allowed modes : in fibers 24-5. Allowed modes : in fibers
2 max
1 . .
2
m π d N A λ
⎛ ⎞
= ⎜ ⎝ ⎟ ⎠
max
d 2
2
( . .) m < →
λ π
< N ASingle-mode (mono-mode) fiber :
For single-mode, In fibers, a little different
(참고) Even/odd modes in a slab (planar) waveguide (참고) Even/odd modes in a slab (planar) waveguide
(
sin2 2)
1/ 2tan 2 cos
j i
i
r e φ φ φ θ n
θ
⎛ ⎞ −
= ↔ ⇒ ⎜ ⎟⎝ ⎠ =
d θ
n2 n1 n2
core
n1>n2
1 2
1 c sin
n θ θ> = − ⎛⎜ n ⎞⎟
⎝ ⎠
1
y y
jk d j jk d j
e
−e e
φ −e
φ=
이면 전반사가 일어난다
을 만족하는 wave만 살아남는다
2 y 2 2
y
k d m
k d m
φ π
φ π
⇔ − =
⇔ − =
Perpendicular polarization (TE) waveguide의 phase 변화
θi
n1 n2
E
(
2 2)
1/ 22
tan sin
2 2 cos
j i
i
r e n
n
φ φ φ π θ
θ
⎛ ⎞ −
= ↔ ⇒ ⎜⎝ + ⎟⎠ = θi
n1 n2
E H
Parallel polarization (TM) waveguide의 phase 변화
y n2 n1 n2
y=0
Wave equation 으로 부터
{
2 2
2
2 2
2
0 0 ( )
E E
t
E E
k y με μεω
∇ − ∂ =
∂
⇔∇ + =
1 0
2 0
( ) (| | / 2, )
( ) (| | / 2, )
k y n k y d core k y n k y d clading
= <
⎧⎨ = >
⎩ ( ) j z
E = xE y e− β 라 하면
{ }
2 2
2
2 2
2
2
2 2
2
2
( ) ( ) ( )
( ) ( ) ( ) 0
( )
j z j z j z
E E
E y e E y e k E y e y
E y k E y
y
k y
β β β
μεω
β β
− − −
∇ +
= ∂ − +
∂
= ∂ + − =
∂ ≡14243 here) β ≡ Neff ⋅k0
{
n2 < Neff < n1}
Neff : effective index (유효굴절률), β 값 결정
Even/odd modes in a slab (planar) waveguide Even/odd modes in a slab (planar) waveguide
z
2
2 2
2
( ) ( ) ( ) 0
E y k E y
y β
∂ + − =
∂ 의 근을 구하면
2 2
2 2
: 0 ( ) sin,cos
: 0 ( ) exp
In core k E y
In cladding k E y β
β
⎧ − > ⇒ ∝
⎪⎨
− < ⇒ ∝
⎪⎩
2 2 2 2 2
2 2 0
: ( ) exp( ) exp( ) exp( )
2
, : ( ) exp( ) 2
| | | |
eff
y d E y A y B y A y
k N n k
d
B A or B y
A
E y B y
A B
α α α
α β α
α
> = − + = −
= − = − ⋅
<
=
=
= ⇒ = −
−
Symmetric anti-symmetric even odd
2 2 2 2 2
1 0
| | : ( ) sin( ) cos( )
2
,
y y
y y eff
y d E y C k y D k y
k k β k n N k
< = ⋅ + ⋅
= − = − ⋅
물리적으로 의미가 없음
Even/odd modes in a slab (planar) waveguide Even/odd modes in a slab (planar) waveguide
d θ
n2 n1 n2
y=0 y=d/2
y=-d/2
| | : ( ) sin( ) 2
: ( ) exp( )
2
: ( ) exp( )
2
y
y d E y C k y
y d E y B y
y d E y B y
α α
⎧ < =
⎪⎪
⎪ > = −
⎨⎪
⎪ < − = −
⎪⎩
{ }
{ }
{ }
2 2 2 2 2
1 0
2 2 2 2 2
2 2 0
| | : ( ) cos( ) ,
2
: ( ) exp( ) ,
2
: ( ) exp( ) ;
2
y y y eff
eff
eff
y d E y D k y k k k n N k
y d E y A y k N n k
y d E y A y N effective index
β
α α β α
α
⎧ < = = − = − ⋅
⎪⎪
⎪ > = − = − = − ⋅
⎨⎪
⎪ < − =
⎪⎩
Even
solutions:
Odd
solutions:
d θ
n2 n1 n2
y=0 y=d/2
y=-d/2
Even/odd modes in a slab (planar) waveguide
Even/odd modes in a slab (planar) waveguide
, ( ) ( ) . 2
d dE y
At y E y and must be continuous
= dy
2
2 2 2 2
2
2 2 2 2
0 1 2
exp( ) cos( )
2 2
exp( ) sin( )
2 2
(2) tan( )
(1
ta
,
) 2
2 n
2
2
( )
2
( )
y
y
y y
y
y
y
d d
le
d d
A D k
d d
A k D
Y X X
X Y d k
d k n n r
t X Y
k k k d
k α
α α
α
α α
=
+ =⎛ ⎞ +
− =
−
⎜ ⎟⎝ ⎠
⎛
− = −
==
=⎜ ⎟⎝ ⎠⎞ − ≡
>
≡
=
≡ Even
...(1) ...(2)
exp( ) sin( )
2 2
exp( ) cos( )
2 2
y
y y
d d
A C k
d d
A k C k
α
α α
− =
− − =
(4) cot( )
(3) y y 2
k k d α
==> − =
Odd
...(3) ...(4)
2 2 2
- cot tan( - ) 2 (r even mode
. ,
2 2
)
y
Y X X X X
d d
let Y
r X k
X Y
α
= = π
+
≡ ≡
= 은 와 동일
Even/odd modes in a slab (planar) waveguide
Even/odd modes in a slab (planar) waveguide
tan : even modes Y = X X
tan( / 2) : odd modes Y = X X − π
9 r 을 크게 하면 교점이 증가하고, 이는 mode의 증가를 의미한다 9 r 은 굴절률 차와 d에 비례한다
9 Even mode와 odd mode가 번갈아 가며 나오고, 동시에 나올 수는 없다
9 Mode #가 커지면, β는 작아지고, Θ는 작아진다
r
y 2 k d 2
α ⋅d
Even/odd modes in a slab (planar) waveguide Even/odd modes in a slab (planar) waveguide
2 , 2
y
d d
X k Y α
⎧ ≡ ≡ ⎫
⎨ ⎬
⎩ ⎭
2
2 2 2 2 2 2
0
(
1 2)
2
X + Y = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ d k n − n = r
X Y
m = 1
m = 2
m = 3
{ }
{ }
2 2 2
2 2 2
2
k
yk k
β α β
= −
= −
n2 n1
n2
(m > 1)
+ +
( ) ( ) ??
in m m m
m
E y ≅
∑
a E y ⇒ ain( ) E y
9임의의 field가 waveguide로 들어 오게 되면, waveguide가 guide하는 mode들로 나뉘게 된다.
9이때 각각의 mode들은 orthogonal하다.
2
2
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
in n n m
n
in m n n m
n
m m
in m
m
m
E y a E y E y dy
E y E y dy a E y E y dy a E y dy
E y E y dy
a E y dy
⎧⎡ ⎤ ⎫
≅ ×
⎨⎢⎣ ⎥⎦ ⎬
⎩ ⎭
= ⋅
=
=
∫ ∑
∫ ∫ ∑
∫
∫ ∫
9이렇게 나뉜 mode들은 각각 다른 β을 가지고 진행하기 때문에 장거리 전송시 mode간의 시간차가 생기게 된다.
이를 Modal dispersion 또는
Intermodal Dispersion이라고 한다.
9장거리 고속통신에서는 그래서 single mode waveguide를 사용한 다 (X<π/2)
Mode excitation
Mode excitation
24-6. Attenuation (loss) 24-6. Attenuation (loss)
Extrinsic losses : bending, defects, …
24-6. Attenuation (loss) 24-6. Attenuation (loss)
Intrinsic losses : absorption, Rayleigh scattering, …
Silica glass fiber
4
1 λ
⎛∝ ⎞
⎜ ⎟
⎝ ⎠
Attenuation (absorption) coefficient Attenuation (absorption) coefficient
(db/km) (db/km)
24-7. Distortion 24-7. Distortion
Modal distortion
L’
L
(ps/km)
GRIN (graded index) fibers GRIN (graded index) fibers
Modal distortion
Material dispersion Material dispersion
Pulse broadening
Material dispersion Material dispersion
M : ps/nm.km
: temporal pulse spread (psec) per unit spectral width (nm) per unit length (km)
Waveguide dispersion Waveguide dispersion
Effective refractive index of guided mode :
n
eff≡ c v /
gϕ
1
sin n
1 eff 1sin
2eff c
n = n ϕ → ≥ n ≥ n ϕ = n
<
Waveguide