Erbium Doped Fiber Amplifiers

Erbium Doped Fiber Amplifiers are considered the most important invention of the 1990’s in the telecommunication industry. EDFAshave made WDM possible which is in turn the backbone of all wideband networks, first and foremost the Internet.

EDFAs are simple and their properties almost ideal. For example:

• High gain broad gain bandwidth and large saturation powers• Polarization independence• Low noise• easy integration with fibers

EDFA Basic Structure

Most important pump bands

Three level Model of an EDFAThe simplest EDFA model considers Erbium in a glass host as a three level system

13R

31R

3

2

121W

32NRA

21NRA

31RA 32

RA

21RA

1 Ground state 2 Meta stable state with a long life time

3 Pumping state

The important transition is 1 2 Its energy difference corresponds to ~ 1550 nm

Pumping rate from to is 1 3 13R

Stimulated emission rate from to is not important 3 1 31R

There are two transitions from level : Radiative

and Non radiative, where

The transition is mainly non radiative

3 3 32 31R R RA A A

32NRA 32 3

NR RA A

The rate of stimulated absorption and emission from tois and

2 112W 21W

The rates of spontaneous emission from are 2 2 21 21R NRA A A

21 21 211 , >> R R NRA A A

1 2 3

113 1 31 3 12 1 21 2 21 2

212 1 21 2 21 2 32 3

313 1 31 3 32

The total density of atoms is , The rate equations describing the populations in the three levels are:

N N N

dN R N R N W N W N A NdtdN W N W N A N A NdtdN R N R N Adt

3N

31 32 21 21

2 3

12 1 2 32 3

13 1 3

3 1 2

113 12 13 32

2

0

00

I n steady state, Defi ne Substitution in the steady state equation f or and yields:

using leads to

idN dta R A b W A

N NW N N b A NR N aN

N N NabN

b a R aW R A

N

13 32 12

13 12 13 32

R A aWb a R aW R A

21

21 13 321

13 3121 12 13 32 13

32

13 12 13 322

13 3121

1

1 1

1 1 1

1

1 1

Substituting the defi nitions of and , using and af ter some algebric manipulationR

a b

A

W R AN

R RW W R A RA

R W R AN

R RWA

12 13 32 1332

1W R A R

32 32

32 13 31,

Now assume and that this nonradiative transition is much f aster than the pumping rate as well as the decay f rom level to levelnamely This means that the decay f rom to

NRA A

A R R

is suffi ciently f ast so that is never occupied and that any decay f rom is only to level .

3 2

3 2 3

3

2

211

12 21

122

12 1

13

2

11

1

Defi ne

These steady state equations are very importantf or amplifi er modeling.

These equations neglect excited state absorption which reduces the

R R

WN

R W W

R WNR W W

21effi ciency of W

Three level Stark Split System

The three level model described so far is too simple for proper modeling of Erbium atoms in a glass host. A more appropriate description is obtained using a three level stark split laser model

Charge distribution in the glass induces electric fields (called Ligand fields) which initiate the Stark effect which splits each of the energy bands. Each band is split in this model into g sub levels

with

Consequently, there are no discrete levels but rather energy manifolds centered around some level. For example, the manifold at level is the reason for the ability to directly pump this level from the ground state (at 1480 nm)

1 2 3g g g

2

1N

2N

3N

1

j

g1

P 1j

1

k

g2

P 2k

1

l

g3

P 3l

Pump RjlAkj Spont.Emission

Wkj gain

32NRA NR relaxation

NRA

NRA

1 2 1 2 33 , , total dens ity in manif old N N NN N N

1

2

3

1 2 3111

,

The density of the sub levels is , , , ,

I n each manif old, thermali

n mN n m j k lj gk gl g

1,

zation processes take place mediated by non radiative processeswhose rates are and which generate orabsob phonons and acoustic vibrations.The equilibrium condition is

NR NR

NR nm NR n m

A A

A N A N

1 1

1

1

11

1

,

,

,

,

The energy diff erence between the sub bands , and

e

, -

xp /

,

ex

exp /

p

n

m m m m m

n m NR m

n

mn mg

mm

m NR

n m n n

n m

E n m E E E EN

E E kTp

E E k

A EN A kT

N N NT

Bolzman factor

1 2 3

11 3 2 2 1

I t is possible to develop rate equations f or the density of each sub bands but these are cumbersomeThe equations are reasonably compact f or , and

lj j l kj k kj k jj l j k

N N N

dN R N N A N W N Ndt

d

232 31 2

332 1 3

NRkj k

j k

NRlj j l

j l

N A N A Ndt

dN A R N Ndt

13 1 31 3

2 1

21 2

32 31 32

21 12

I t is now possible to f ormulate generalized pumpingas well as stimulated and spontaneous emission

W

lj j lj lj l j l

kj k kj jj l j l

kj kj k

NR

R p R p

W p W p

A p

A p A

A

R

W

113 1 31 3 21 2 12 2 21 2

212 1 21 2 21 2 32 3

313 1 31 3 32 3

The set of rate equations are similar to the simplifi es equations except that the coeffi cients are more complex.

dN N N W N W N A NdtdN W N W N A N A NdtdN N N A Ndt

nevertheless,known solutions of the simple equations can also serve here.

Gain CoefficientThe intensities of the signal and the pump vary as they propagate along the EDFA. The attenuation of the pump changes the degree of population inversion.

Given a signal intensity (power per unit area)

1 2

2 121 12

12

21

1

S

S S

S S

S S

at awavelength propagating through an infi nitesimal length

with densities and

absorption cross section at emission cross section at

obviously ,

S

S S

S

I

dz N N

dI N N I dz

I N

2and and dependent N z

1 1

12 112

2

1 12 2 21

For split levels with and sub levels I n reality this is not accurate.

The expression assumes equal occupation of all sub levels and also equal transition

SS S

g g

dI g N N Idz g

g g

2 1

probabilities among each pair of sub levels. I n reality this is not an accurate assumption but we defi ne phenomenological

cross sections and and defi ne e SS

a S

Se S a

a S S

S a S

e

SdI N N Idz

2 1S N N

2 1 2

2 1

2

1

,

We can now defi ne the gain coeffi cient

The exact details of are complicated. The degree of inversion is defi ne as

is dependent

means no pu

a

a S S

S

e

S

g

N N N

g N N

dI gIdz

D D z

D

1mping

means f ull inversion D