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24. Fiber Optics 24. Fiber Optics

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(1)

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

(2)

Optical fibers Optical fibers

SI(step-index) fiber

GRIN(graded-index) fiber

(3)

Optical communication systems

Optical communication systems

(4)

24-4. Optics of propagation 24-4. Optics of propagation

Cladding (n2)

Core (n1) Air (n0)

Numerical aperture (N.A.)

Step index fiber

(5)

Step-index fibers Step-index fibers

Skip distance :

(m-1)

(6)

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

(

n1cos

ϕ

c = N A. .

)

2 2

max 1 1

2 2

. . 1 1

d d

m N A n n

λ λ

= + = − +

(7)

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 A

Single-mode (mono-mode) fiber :

For single-mode, In fibers, a little different

(8)

(참고) Even/odd modes in a slab (planar) waveguide (참고) Even/odd modes in a slab (planar) waveguide

(

sin2 2

)

1/ 2

tan 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/ 2

2

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

(9)

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

(10)

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

(11)

| | : ( ) 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

(12)

, ( ) ( ) . 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

+

= 와 동일

Even/odd modes in a slab (planar) waveguide

Even/odd modes in a slab (planar) waveguide

(13)

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 nn = r

X Y

m = 1

m = 2

m = 3

{ }

{ }

2 2 2

2 2 2

2

k

y

k k

β α β

= −

= −

(14)

n2 n1

n2

(m > 1)

+ +

( ) ( ) ??

in m m m

m

E y

a E ya

in( ) 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

(15)

24-6. Attenuation (loss) 24-6. Attenuation (loss)

Extrinsic losses : bending, defects, …

(16)

24-6. Attenuation (loss) 24-6. Attenuation (loss)

Intrinsic losses : absorption, Rayleigh scattering, …

Silica glass fiber

4

1 λ

(17)

Attenuation (absorption) coefficient Attenuation (absorption) coefficient

(db/km) (db/km)

(18)

24-7. Distortion 24-7. Distortion

Modal distortion

L’

L

(ps/km)

(19)

GRIN (graded index) fibers GRIN (graded index) fibers

Modal distortion

(20)

Material dispersion Material dispersion

Pulse broadening

(21)

Material dispersion Material dispersion

M : ps/nm.km

: temporal pulse spread (psec) per unit spectral width (nm) per unit length (km)

(22)

Waveguide dispersion Waveguide dispersion

Effective refractive index of guided mode :

n

eff

c v /

g

ϕ

1

sin n

1 eff 1

sin

2

eff c

n = n ϕ → ≥ nn ϕ = n

<

Waveguide

참조

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