Tel Aviv University
TEL AVIV UNIVERSITY
THE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERING
Department of Electrical Engineering – Physical Engineering
Numerical and Analytical
models for various effects in
EDFAs
Inna Nusinsky-Shmuilov
Supervisor:Prof. Amos Hardy
1
Outline:
Motivation
Rate equations
Homogeneous upconversion
EDFA for multichannel transmission
Inhomogeneous gain broadening
Conclusions
2
Motivation:
Why EDFAs?
Applications in the 1.55μm range wavelengths
Optical power amplifiers
Low noise preamplifiers in receivers
Multichannel amplification (WDM)
Why analytical models?
Insight into the significance of various parameters on the
system behavior.
Provide a useful tool for amplifier designers.
Significantly shorter computation time.
3
Pumping geometry:
- Forward pumping
- Backward pumping
- Bidirectional pumping
Pp z, t 
doped fiber
Ps  z , t 
Amplified
output
Signal
Pump
zL
z 0
Forward pumping
4
Rate equations:
Energy band diagram:
4
4
N3
 32
I 11 2
4
 ap
4
I 15 2
N4
I9 2
 42
 24
I 13 2
a
N2
e

N1
5
Rate equations:
Second level population:
Pump absorption and emission
N 2 z, t    p p 
 ap N  N 2 z, t    ep N 2 z, t  Pp  z, t   Pp  z, t 
 
t
 hcA 
Spontaneous
Signal emission
emission
N z, t   s 



 N 2  z, t   e   P z, t ,    P z, t ,   d
 2

 hcA 






  
2


  s N  N 2  z, t   a   P z, t ,    P z, t ,   d  C 2 N 2  z,t 
 hcA 
Signal absorption
Homogeneous
upconversion
6
Rate equations:
Signal, ASE and pump powers:
Stimulated emission and absorption
dP  z, t ,  


 s  e  N 2 z, t    a  N  N 2 z, t  P  z,t ,  
dz
 s e  N 2  z , t P0      P   z , t ,  
Spontaneous
emission
dPp z, t 


Scattering losses


 p  ep   24  p  N 2 z, t    ap N  N 2 z, t Pp  z, t 
dz

   p Pp z, t 

Pump emission ,absorption and ESA
Scattering losses
dP   z, t  P  n P 


dz
z
c t
7
Numerical solution of the model:
• Steady state solution (/ t = 0)
• The ASE spectrum is divided into slices of width 
• Boundary conditions:
Pp  z  0  -known launched pump power
Ps z  0 -known launched signal power


z  0, PASE
z  L   0
PASE
• The equations are solved numerically,
using an iterations method
8
Homogeneous upconversion:
Schematic diagram of the process:
4
4
I9 2
I9 2
4
I 11 2
4
I 11 2
4
I 13 2
4
I 13 2
4
I 15 2
4
I 15 2
donor
acceptor
9
Homogeneous upconversion:
Assumptions for analytical solution:
• Signal and Pump propagate in positive direction
• Spontaneous emission and ASE are negligible compared to
the pump and signal powers
• Strong pumping (in order to neglect 1/τ)
• Loss due to upconversion is not too high
10
Homogeneous upconversion:
Signal and pump powers vs. position along the fiber:
Solid lines-exact solution
Circles-analytical formula
Approximate analytical formula
is quite accurate
Dashed lines-exact solution without upconversion
Injected pump power 80mW
Input signal power 1mW
11
Homogeneous upconversion:
Dependence of upconversion on erbium concentration:
Good agreement between approximate
X Analytical formula is no longer valid
analytical formula and exact numerical
solution
12
Homogeneous upconversion:
Upconversion vs. pump power:
Input signal power 1mW
Strong pump decreases the influence of homogeneous upconversion
If there is no upconversion (or other losses in the system), the maximum output
signal does not depend on erbium concentration
Approximate analytical formula’s accuracy improves with increasing the pump
13
power
Homogeneous upconversion:
Upconversion vs. signal power:
Injected pump power 100mW
Increasing the input signal power decreases the influence of homogeneous
upconversion
Approximate analytical formula’s accuracy improves with increasing the input
signal power power
14
Multichannel transmission:
Assumptions for analytical solution:
• All previous assumptions
• Interactions between neighboring ions (e.g homogeneous
upconversion and clustering) are ignored (C2=0)
• Spectral channels are close enough
For example:
for a two channel amplifier in the 1548nm-1558nm
band the spectral distance should be less than 4nm
For 10 channels the distance should be 1nm or less
15
Multichannel transmission:
Signal powers vs. position along the fiber:
3 channel amplifier, spectral distance 2nm:
10 channel amplifier, spectral distance 1nm:
Good agreement between approximate
Solid lines-exact solution
analytical formula and exact solution of rate
Circles-analytical formula
equations
16
Multichannel transmission:
3 channel amplifier, spectral distance 4nm:
X Analytical formula is no longer valid
5 channel amplifier, spectral distance 2nm:
Approximate analytical formula is
quiet accurate
The accuracy of the analytical formula improves with decreasing spectral
separation between the channels
17
Multichannel transmission:
Output signal vs. signal and pump powers:
The approximate solution is accurate for strong enough input signals and strong
injected power.
If input signal is too weak or injected pump is too strong, the ASE can’t be
neglected.
18
Multichannel transmission:
 The analytical model is used to optimize the parameters of a fiber
amplifier.
Optimization of fiber length:
Approximate results are less accurate for small signal powers and smaller number of
channels.
Optimum length is getting shorter when the input signal power increases and the
number of channels increases.
19
Inhomogeneous gain broadening:
Energy band diagram:
ˆ
is the shift in resonance frequency
20
Inhomogeneous gain broadening:
The model:
• All energy levels are shifted manifold is shifted by the same
amount from the ground (    ˆ ).
• A photon of wavelength  , interacts with
molecules
with shifted cross-sections
and
, due to the
ˆ
frequency shift of .
•
is the number of molecules, per unit volume,
whose resonant frequency has been shifted by a frequency that
lies between ˆ and ˆ  dˆ .
• The function f ˆ  is the normalized distribution function of
molecules, such that
. Usually a Gaussian is used.
• The width ˆ I of f ˆ  determines the relative effect of the
inhomogeneous broadening.
21
Inhomogeneous gain broadening:
Single channel amplification:
Aluminosilicate Al2 O3  SiO2
hom  11.5nm I  11.5nm
Germanosilicate GeO2  SiO2
hom  4nm I  8nm
The inhomogeneous broadening is significant Solid lines-inhomogeneous model
for germanosilicate fiber whereas aluminosilicate Dashed lines-homogeneous model
fiber is mainly homogeneous
22
Inhomogeneous gain broadening:
Multichannel amplification:
Aluminosilicate Al2 O3  SiO2
Germanosilicate GeO2  SiO2
There is significant difference between inhomogeneous broadening (solid lines)
and homogeneous one (dashed lines) for both fibers.
The channels separation is 10nm, which is larger than the inhomogeneous
linewidth of the germanosilicate fiber and smaller than the inhomogeneous
linewidth of the aluminosilicate fiber.
23
Inhomogeneous gain broadening:
Multichannel amplification:
Germanosilicate GeO2  SiO2
hom  4nm I  8nm
If we decrease the channel distance in germanosilicate fiber to 6nm (less
than  I  8nm ), we expect the effect of the inhomogeneous broadening
to be stronger.
Here the inhomogeneous broadening mixes the two signal channels and not only
ASE channels, thus its influence on signal amplification is more significant.
24
Inhomogeneous gain broadening:
Experimental verification of the model:
Germanosilicate fiber:
Circles-experimental results
Solid lines-numerical solution using inhomogeneous model
Dashed lines- numerical solution using homogeneous model
25
Conclusions:
Numerical models have been presented, for the study of erbium doped
fiber amplifiers.
Simple analytical expressions were also developed for several cases.
The effect of homogeneous upconversion, signal amplification in
multi-channel fibers and inhomogeneous gain broadening were
investigated, using numerical and approximate analytical models
Numerical solutions were used to validate the approximate
expressions.
Analytical expressions agree with the exact numerical solutions in a
wide range of conditions.
A good agreement between experiment and numerical model.
26
Suggestions for future work :
• Time dependent solution
• Modeling for clustering of erbium ions
• Considering additional pumping configurations
and pump wavelengths
• Experimental analysis of inhomogeneous broadening
27
Publications :
1. Inna Nusinsky and Amos A. Hardy, “Analysis of the effect of
upconversion on signal amplification in EDFAs”, IEEE J. Quantum
Electron.,vol.39, no.4 ,pp.548-554 Apr.2003
2. Inna Nusinsky and Amos A. Hardy, ““Multichannel amplification in
strongly pumped EDFAs”, IEEE J.Lightwave Technol., vol.22, no.8,
pp.1946-1952, Aug.2004
28
Acknowledgements :
• Prof. Amos Hardy
• Eldad Yahel
• Irena Mozjerin
• Igor Shmuilov
29
Appendix :
30
Appendix:
Homogeneous upconversion:
Assumptions for analytical solution:
Strong pumping:
 es
 as
hcA

hcA
 B2  11   ep  ap 
p ap  p Pp 0
p ap  p Pp 0
where
     es  Ps 0
B2  s s as
~
 p p  ap   ep  Pp 0
~
31
Appendix:
Homogeneous upconversion:
Assumptions for analytical solution:
Homogeneous upconversion not too strong:
Ceff 
4C 2 N Q p Pp 0  Qas Ps 0
P 0
2
1
where
P z   Q p Pp  z   Qas  Qes Ps  z 
32
Appendix:
Homogeneous upconversion:
Derivation of approximate solution:
~
N 2  z   N 2  z   N 2  z 
~
Pp  z   Pp  z   Pp  z 
~
Ps z   Ps z   Ps z 
We ignore the terms of second order and higher:
Ps z 2 , Pp z 2 , Ps z   Pp z 
33
Appendix:
Homogeneous upconversion:
Rate equations solution without upconversion:
~
N 2 z   N
Q P z   Q P z 
Q  Q P z   Q  Q P z 
ap p
ap
ep
as s
p
as
es
s
~
~
Ps z   Ps 0 exp Rz 1  q  z 


q
~
~
~
~
Pp z   Pp 0 Ps z  Ps 0 exp  Rz 
where  z  is derived from:
  z 1q
 B2 B4
 1

B3

1  B3 B4
1  B3 B4   z 
1 q




 expB4 1  qz 
34
Appendix:
Homogeneous upconversion:
Approximate analytical formula:

Pp z   q Pp 0 Ps 0
q


q 1
~
 Ps z  Ps z  exp  Rz 


q 1

C 2 N 2U 1
D
1


q 1
2


Ps z  

1

q


1

1  q  D5 q1  D6 
D1 q 1  Qes  Qas


 D1 q 1  Qes  Qas
 D3 ln 
 D1  Qes  Qas
N 2 z   C 2 N 2

p
Q P z   Q
p
p



  D4 1  q  ln  



Q P z   Q
p
 
as
as Ps  z 
2

 Qes Ps z 
3
35
Appendix:
Multichannel transmission:
Assumptions for analytical solution:
Strong pumping:
I
hcA
 p ap  p Pp 0


i
es

i
as
i 1
I
i 1
 I i i i

   s s  es   asi Psi 0 
hcA
  i 1
 11   ap 

p ap  p Pp 0 
 p p ap Pp 0






36
Appendix:
Multichannel transmission:
Approximate analytical solution :


Pp z   Ai Psi z  exp  Ri z 
i  1...I
qi
Psi z   Psi 0 exp vi z 1 z  1
q qi
v v
exp  1 i
 qi

z 

I
N 2 z   N
Q p Pp z    Qasi Psi z 
i 1
I


Q p Pp z    Qasi  Qesi Psi z 
i 1
1q 1
1
M
Bˆ 81q1 1  Bˆ 9 exp Tz 
Bˆ  Bˆ exp Tz 

8
9

K B8
 exp zM  1T 
37
Appendix:
Definitions of parameters:
P0    2hc 2 3
q

p  ap   ep

s  as   es 
R   p ap N    p   s as N   s   q
Qes 
 s s es
hcA
Qas 
s s as
hcA
 s s  as   es  Ps 0
B2 
~
 p p  ap   ep  Pp 0
~
B3    s   R 1  q   B2
B4  s es N  R 1  q     s 
38
Parameters used in the computation:
Homogeneous upconversion:
39
Parameters used in the computation:
Inhomogeneous gain broadening:
40
41
42