Optical fibers

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24. Fibers (guiding of propagation waves)

(2)

core (~10 µm) cladding (125 µm) poly (250 µm)

Optical fibers

' 1

2 ' 1

1

2 2

1 2

sin sin

sin , 90

sin cos

o m m

c m c

o m c

fiber

n n

NA n n

o

θ θ

φ θ φ

θ φ

=

= = −

= =

= −

core (n1) cladding (n2)

θ_m

θ’_mφ_c air (n_o)

(3)

(4)

0.05 0.1 0.5 1.0 5 10

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Lattice absorption Rayleigh

scattering

Wavelength (탆)

Illustration of a typical attenuation vs. wavelength characteristics of a silica based optical fiber. There are two communications channels at 1310 nm and 1550 nm.

OH-absorption peaks

1310 nm

1550 nm

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

9Attenuation in Optical fibers

9우측 그림은 silica를 기본 물질로 사용하 는 fiber의 파장에 따른 attenuation을 나타 낸다.

9Rayleigh scattering과 Lattice absorption 에 의해서 fundamental limit이 정해진다.

91310nm와 1550nm에서 loss의 deep이 생기는 것을 볼수 있다.

9Typical optical fiber는 1550nm에서

0.2dB/km의 attenuation coefficient를 갖는 다.

exp( )

1 1

ln ln

110 log 4.34

out in

in out

in dB

out

P P L

P P

L P L P

P

L P

α α

= −

= − =

= =

9Ex) α_dB=0.3dB/km일때 Pin=1mW이면 Pout=0.5mW 9 가 되는 거리 L은?

1 1 1

10 log 10 log

0.3 / 0.5

in

dB out

P mW

L ⁼ α P ⁼ dB km mW

=

Loss due to absorption

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Escaping wave

θ θ

θ′ < θ θθ > θc θ′

Microbending

R Cladding

Core Field distribution

Sharp bends change the local waveguide geometry that can lead to waves escaping. The zigzagging ray suddenly finds itself with an incidence angle θ′ that gives rise to either a transmitted wave, or to a greater

cladding penetration; the field reaches the outside medium and some light energy is lost.

Loss due to bending

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Scattered waves

Incident wave Through wave

A dielectric particle smaller than wavelength

Rayleigh scattering involves the polarization of a small dielectric particle or a region that is much smaller than the light wavelength.

The field forces dipole oscillations in the particle (by polarizing it) which leads to the emission of EM waves in "many" directions so that a portion of the light energy is directed away from the incident beam.

Loss due to scattering

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9Linear Polarization

9fiber내에서 빛이 통과할 때는 Ez와 Hz성분을 모두 갖는 hybrid modes를 가진다.

9그러나 굴절률차이가 아주 작은(Δ<<1)경우에는 거의 plane polarized된 traveling waves로 보이게 된다.

9이런 waves를 linearly polarized(LP)라고 부르고, 이들은 transverse electric and magnetic field 특성을 모두 갖는다.

9Lp_lm mode는 아래와같이 표현이 가능하다.

( , ) exp ( )

LP lm lm

E = E r

ϕ

j

ω

t −

β

z

E

r E₀₁

Core

Cladding

The electric field distribution of the fundamental mod in the transverse plane to the fiber axis z. The light intensity is greatest at the center of the fiber. Intensity patterns in LP₀₁, LP₁₁ and LP₂₁ modes.

(a) The electric field of the fundamental mode

(b) The intensity in the fundamental mode LP₀₁

(c) The intensity in LP₁₁

(d) The intensity in LP₂₁

92xl : max peak # along φ 9M : max peak # along r

Fiber modes

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Dielectric Waveguide (I)

(

^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

φ π

⇔ − =

TE-polarization 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

TM-polarization waveguide 의 phase 변화

2 2

1 1

1 2

2n dcos 2n dcos 2d

m θ φ θ n n

λ π λ λ

⇔ = − ≈ = − But, not simple!

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k dy − =φ mπ

(

^sin² ²

)

^{1/ 2}

:tan 2 cos

TE φ θ n

θ

  = −

  

(

^sin² ²

)

^{1/ 2}

tan cos

cos

d θ n

π θ

λ θ

  = −

 

 

cos θ

m=0

1

2

3

4 5

cosθ_m₌0 ^cosθ^m₌³

d θ

n₂ n₁ n₂

0

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Dielectric Waveguide (I)

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, )

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

( ) ( ) ( )

( ) ( ) ( ) 0

j z j z j z

y

E E

E y e E y e k E y e y

E y k E y

y

k

β β β

µεω

β β

− − −

∇ +

= ∂ − +

∂

= ∂ + − =

∂ 1424≡ 3 ⁰

2 1

)

here N k

n N n

β = ⋅

< <

N : effective index (유효굴절률) β값 결정

(11)

Dielectric Waveguide (I)

2

2 2

2

( ) ( ) ( ) 0

E y k E y

y β

∂ + − =

∂ ^{의 근을 구하면}

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

| | | |

y d E y A y B y A y

k N n k

y d E y B y

A B B A or B A

α α α

α β

α

> = − + = −

= − = − ⋅

< − =

= ⇒ = = −

Symmetric anti-symmetric even odd

2 2 2 2 2

1 0

| | : ( ) sin( ) cos( )

2

,

y y

y d E y C k y D k y

k k β k n N k

< = ⋅ + ⋅

= − = ⋅

물리적으로 의미가 없음

(12)

d θ

n₂ n₁ n₂

| | : ( ) sin( ) 2

: ( ) exp( ) 2

y

y d E y C k 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

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

 < = = − = ⋅



 > = − = − = − ⋅



 < − =



Even

solutions:

Odd

solutions:

y=0 y=d/2

y=d/2

(13)

. 2 at y = d

2

2 2 2 2

2

2 2 2 2

0 1 2

exp( ) cos( )

2 2

exp( ) sin( )

2 2

(2) tan( )

(1) 2

. ,

2 2

tan

( )

2

( )

2

y

y y

y

d d

A D k

d d

A k D k

k k d

d d

let X k Y

Y X X

X Y d k

d k n n r

α

α α

α

− =

− − = −

==> =

= =

=

+ =    +

=    − Even

E(y) and dE(y)/dy are continuous

...(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

. 2

at y = d E(y) and dE(y)/dy are continuous

...(3) ...(4)

2 2 2

. ,

2 2

cot tan( )

2

(r even mode )

y

d d

let X k Y

Y X X X X

X Y r

α

π

= =

= − = −

+ = 은 와 동일

(14)

tan : even mode Y = X X

tan( / 2) : odd mode Y = X X −π

9r 을 크게 하면 교점이 증가하고, 이는 mode의 증가를 의미한다

9r 은 굴절률 차와 d 에 비례한다

9Even mode와 odd mode가 번갈아 가며 나오 고, 동시에 나올 수는 없다

9Mode가 커지면, β는 작아지고, Θ는 작아진다 2

r

y

k d 2

α ⋅d

(15)

Guidance

condition k d^y − =

φ

m

π

2 2 1/ 2

1 0

2 2

1 0 2 0

(sin )

tan( )

2 cos

cos

( sin ) ( )

y

n

k k n k

n k n k

φ θ α

θ θ

α θ

= − =

=

= −

1

2 tan ( )

tan( )

2

y

y y

y

k

k d m

k k d m

k

φ α

α π

π α

−

=

− =

9m : even =>

9m : odd =>

tan( ) 2

y

y y

k d

k k k d

α

=

2 ) cot(kyd ky

− α =

(16)

n₂ n₁ n₂

+

m>1

+

( ) ( ) ??

in m m m

m

E y ≅

∑

a E y ⇒ a

in( ) E y

9임의의 field가 waveguide로 들어 오게 되면, waveguide가 guide하는 mode들로 나뉘게 된다. 이때 각각 의 mode들은 orthogonal하다.

2

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

in n n m

n

in m n n m

n

m m

in 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<pi/2)

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0 1 2 3 4 5 6 V b

1

0 0.8

0.6

0.4

0.2

LP₀₁ LP₁₁ LP₂₁ LP₀₂

2.405

Normalized propagation constant b vs. V-number for a step index fiber for various LP modes.

9Modes in step index fiber

9V-number (normalized frequency)

9Normalized propagation constant,b

2 2 1/ 2 1/ 2

1 2 1

1 2

2 2 2

1 2 1 1 2 1

2 2

( ) (2 )

( ) / 2

( ) / ( ) / 2

a a

V n n n n

n n n

n n n n n n

π π

λ ⁻ ⁼ λ ^∆

= +

∆ = − ≈ −

2 2

2

2 2

1 2

2

1

( / )

0 1

k n

b n n

k n b k n b

β β

β

= −

−

= ⋅ ⇔ =

9우측 그림에서 볼 수 있듯이 V-number가 2.405보다 작으면 single mode가 된다.

(18)

Dispersions

(19)

(20)

0

1.2 1.3 1.4 1.5 1.6

1.1 -30

20 30

10

-20 -10

λ (µm)

Dm

Dm + Dw

D_w λ₀

Dispersion coefficient (ps km^-1nm^-1)

Material dispersion coefficient (D_m) for the core material (taken as SiO₂), waveguide dispersion coefficient (D_w) (a = 4.2 µm) and the total or chromatic dispersion coefficient D_ch (= Dm + D_w) as a function of free space wavelength, λ.

9Material dispersion

9Fiber core에서 guide된 wave의 진행속도 는 n1에 의해서 결정되는데, n1이

wavelength에 따라서 달라지기 때문에 생기 는 dispersion

9Material의 wavelength에 따른 특성에 의 해서 생기는 dispersion이다.

2

1 2

m

g

L D

D c

v

τ λ

β π

λ λ

∆ = ∆

 

∂  

= ∂   = − 

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