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