24. Fiber Optics24. Fiber Optics
Numerical aperture(N.A.) of optical fiber
Allowed modes in fibers
Attenuation
Distortion
• Modal distortion
• Material dispersion
• Waveguide dispersion
Optical fibersOptical fibers
SI(step-index) fiber
GRIN(graded-index) fiber
Optical communication systemsOptical communication systems
24-4. Optics of propagation24-4. Optics of propagation
Cladding (n2)
Core (n1)
Air (n0)
Numerical aperture (N.A.)
Step index fiber
Step-index fibersStep-index fibers
Skip distance :
(m-1)
24-5. Allowed modes : in slab (planar) waveguides24-5. 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 m cϕ ϕ=
( )1 cos . .cn N Aϕ =2 2max 1 1
2 2. . 1 1d dm N A n nλ λ
= + = − +
24-5. Allowed modes : in fibers24-5. Allowed modes : in fibers
2
max1 . .2
dm N Aπλ
⎛ ⎞= ⎜ ⎟⎝ ⎠
maxd 22
( . .)m
N Aλ π< → <Single-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
( )1/ 22 2sintan
2 cosij
i
nr e φ
θφφθ−⎛ ⎞= ↔ ⇒ =⎜ ⎟
⎝ ⎠
d θ
n2
n1
n2
core
n1>n2
1 2
1
sincnn
θ θ − ⎛ ⎞> = ⎜ ⎟
⎝ ⎠
1y yjk d jk dj je e e eφ φ− − =
이면 전반사가 일어난다
을 만족하는 wave만 살아남는다
2 2 2y
y
k d m
k d m
φ π
φ π
⇔ − =
⇔ − =
Perpendicular polarization (TE) waveguide의 phase 변화
iθn1
n2
E
( )1/ 222
2
sintan
2 2 cosij
i
nr e
nφ
θφ πφθ
−⎛ ⎞= ↔ ⇒ + =⎜ ⎟⎝ ⎠
iθn1
n2
EH
Parallel polarization (TM) waveguide의 phase 변화
y n2
n1
n2
y=0
Wave equation 으로 부터
{
22
2
2 2
2
0
0( )
EEt
E Ek y
με
μεω
∂∇ − =
∂⇔∇ + =
1 0
2 0
( ) (| | / 2, )( ) (| | / 2, )
k y n k y d corek y n k y d clading
= <⎧⎨ = >⎩
( ) j zE xE y e β−= 라 하면
{ }
2 2
22 2
2
22 2
22
( ) ( ) ( )
( ) ( ) ( ) 0
( )
j z j z j z
E EE y e E y e k E y ey
E y k E yy
k y
β β β
μεω
β
β
− − −
∇ +
∂= − +
∂
∂= + − =
∂≡14243 { }0 2 1) eff effhere N k n N nβ ≡ ⋅ < <
Neff : effective index (유효굴절률), β 값 결정
Even/odd modes in a slab (planar) waveguideEven/odd modes in a slab (planar) waveguide
z
22 2
2( ) ( ) ( ) 0E y k E y
yβ∂
+ − =∂
의 근을 구하면
2 2
2 2
: 0 ( ) sin,cos: 0 ( ) exp
In core k E yIn cladding k E y
β
β
⎧ − > ⇒ ∝⎪⎨
− < ⇒ ∝⎪⎩
2 2 2 2 22 2 0
: ( ) exp( ) exp( ) exp( )2
,
: ( ) exp( )2
| | | |
eff
dy 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 21 0
| | : ( ) sin( ) cos( )2
,
y y
y y eff
dy E y C k y D k y
k k k n N kβ
< = ⋅ + ⋅
= − = − ⋅
물리적으로 의미가 없음
Even/odd modes in a slab (planar) waveguideEven/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
ydy E y C k y
dy E y B y
dy E y B y
α
α
⎧ < =⎪⎪⎪ > = −⎨⎪⎪ < − = −⎪⎩
{ }{ }
{ }
2 2 2 2 21 0
2 2 2 2 22 2 0
| | : ( ) cos( ) ,2
: ( ) exp( ) , 2
: ( ) exp( ) ; 2
y y y eff
eff
eff
dy E y D k y k k k n N k
dy E y A y k N n k
dy 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) waveguideEven/odd modes in a slab (planar) waveguide
( ), ( ) . 2d dE yAt y E y and must be continuous
dy=
22 2 2 2
22 2 2 2
0 1 2
exp( ) cos( )2 2
exp( ) sin( )2 2
(2) tan( )(1
ta
,
) 2
2n
2
2
( )2
( )
y
y
y y
y
y
y
d dle
d dA D k
d dA k D
Y X X
dX Y k
d k n n r
t X Y
k
dk k
k
α
α α
α
α
α
=
⎛ ⎞+ = +
− =
−
⎜ ⎟⎝ ⎠
⎛
− = −
==
⎞= − ≡⎜ ⎟⎝ ⎠
>
≡
=
≡
Even
...(1)
...(2)
exp( ) sin( )2 2
exp( ) cos( )2 2
y
y y
d dA C k
d dA k C k
α
α α
− =
− − =
(4) cot( )(3) 2y y
dk kα==> − =
Odd
...(3)
...(4)
2 2 2
- cot tan( - )2
(r even mode
. ,2 2
)
y
Y X X X X
d dlet Y
r
X k
X Y
α
π= =
+
≡ ≡
= 은 와동일
Even/odd modes in a slab (planar) waveguideEven/odd modes in a slab (planar) waveguide
tan : even modesY X X=
tan( / 2) : odd modesY X X π= −
r 을 크게 하면 교점이 증가하고,
이는 mode의 증가를 의미한다
r 은 굴절률 차와 d에 비례한다
Even mode와 odd mode가 번갈아 가며
나오고, 동시에 나올 수는 없다
Mode #가 커지면, β는 작아지고, Θ는 작아진다
r2ydk
2dα ⋅
Even/odd modes in a slab (planar) waveguideEven/odd modes in a slab (planar) waveguide
,2 2yd dX k Y α⎧ ⎫≡ ≡⎨ ⎬
⎩ ⎭
22 2 2 2 2 2
0 1 2( )2dX Y k n n r⎛ ⎞+ = − =⎜ ⎟
⎝ ⎠
X
Y
m = 1
m = 2
m = 3
{ }{ }
2 2 2
2 2 22
yk k
k
β
α β
= −
= −
n2
n1
n2
(m > 1) + +
( ) ( ) ??in m m mm
E y a E y a≅ ⇒∑
( )inE y
임의의 field가 waveguide로 들어오게 되면, waveguide가 guide하는mode들로 나뉘게 된다.
이때 각각의 mode들은orthogonal하다.
2
2
( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( )
( )
in n n mn
in m n n mn
m m
in mm
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 dya
E y dy
⎧ ⎫⎡ ⎤≅ ×⎨ ⎬⎢ ⎥
⎣ ⎦⎩ ⎭
= ⋅
=
=
∑∫
∑∫ ∫
∫∫∫
이렇게 나뉜 mode들은 각각 다른β을 가지고 진행하기 때문에 장거리전송시 mode간의 시간차가 생기게된다.
이를 Modal dispersion 또는Intermodal Dispersion이라고 한다.
장거리 고속통신에서는 그래서single mode waveguide를 사용한다 (X<π/2)
Mode excitationMode 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
1λ
⎛ ⎞∝⎜ ⎟⎝ ⎠
Attenuation (absorption) coefficientAttenuation (absorption) coefficient
(db/km) (db/km)
24-7. Distortion24-7. Distortion
Modal distortion
L’
L
(ps/km)
GRIN (graded index) fibersGRIN (graded index) fibers
Modal distortion
Material dispersionMaterial dispersion
Pulse broadening
Material dispersionMaterial dispersion
M : ps/nm.km: temporal pulse spread (psec)
per unit spectral width (nm)per unit length (km)
Waveguide dispersionWaveguide dispersion
Effective refractive index of guided mode : /eff gn c v≡
ϕ
1 1 1 2sin n sineffeff cn n n nnϕ ϕ= → ≥ ≥ =
<
Waveguide