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Numerical and AnalyticalNumerical and Analytical

models for various effects inmodels for various effects in

EDFAsEDFAs

Inna Nusinsky-Shmuilov

Supervisor:Prof. Amos Hardy

Tel Aviv University

TEL AVIV UNIVERSITYTHE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERING

Department of Electrical Engineering – Physical Engineering

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Outline:Outline:Outline:Outline:

Motivation

Rate equations

Homogeneous upconversion

EDFA for multichannel transmission

Inhomogeneous gain broadening

Conclusions

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Motivation:Motivation:Motivation:Motivation:Why EDFAs?

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.

Applications in the 1.55μm range wavelengths Optical power amplifiers

Low noise preamplifiers in receivers

Multichannel amplification (WDM)

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Pumping geometry:Pumping geometry:Pumping geometry:Pumping geometry:

- Forward pumping

- Backward pumping

- Bidirectional pumping

Pump

doped fiber

SignalAmplified output

tzPp , tzPs ,

0z Lz

Forward pumping

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Rate equations:Rate equations:Rate equations:Rate equations:

Energy band diagram:

4N294 I

24422114 I

ap

3N

2N2134 I

2154 I 1N

32

ea

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Rate equations:Rate equations:Rate equations:Rate equations:Second level population:

Homogeneous upconversionSignal absorption

Spontaneous emission

Pump absorption and emission

Signal emission

tzPtzPtzNtzNNhcAt

tzNppepap

pp ,,,,,

222

tzN ,2

es tzN

hcA,2

dtzPtzP ,,,,

as tzNN

hcA,2

dtzPtzP ,,,, 222 , tzNC

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Rate equations:Rate equations:Rate equations:Rate equations:Signal, ASE and pump powers:

Scattering lossesSpontaneous emission

Stimulated emission and absorption

Scattering losses

tzNNtzNdz

tzdPaes ,,

,,22

,, tzP

,,, 02 tzPPtzNes

Pump emission ,absorption and ESA

tzNNtzNdz

tzdPappepp

p ,,,

2224

tzPp ,

tzPpp ,

t

P

c

n

z

P

dz

tzdP

,

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Numerical solution of the model:Numerical solution of the model:Numerical solution of the model:Numerical solution of the model:

• Steady state solution (/ t = 0)

• The equations are solved numerically, using an iterations method

• The ASE spectrum is divided into slices of width

-known launched pump power 0zPp

• Boundary conditions:

0 zPs -known launched signal power

0,0 LzPzP ASEASE

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Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:

Schematic diagram of the process:

294 I

2114 I

2134 I

2154 I

donordonor

294 I

2114 I

2134 I

2154 I

acceptoracceptor

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Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion: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

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Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Signal and pump powers vs. position along the fiber:

Injected pump power 80mW Input signal power 1mW

Solid lines-exact solution

Circles-analytical formula

Dashed lines-exact solution without upconversion

Approximate analytical formula is quite accurate

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Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Dependence of upconversion on erbium concentration:

Good agreement between approximate

analytical formula and exact numerical

solution

X Analytical formula is no longer valid

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Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Upconversion vs. pump power:

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 power

Input signal power 1mW

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Upconversion vs. signal power:Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:

Increasing the input signal power decreases the influence of homogeneous upconversion

Approximate analytical formula’s accuracy improves with increasing the input signal power power

Injected pump power 100mW

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Multichannel transmission:Multichannel transmission:Multichannel transmission: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

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Multichannel transmission:Multichannel transmission:Multichannel transmission:Multichannel transmission:

3 channel amplifier, spectral distance 2nm: 10 channel amplifier, spectral distance 1nm:

Solid lines-exact solution

Circles-analytical formula

Good agreement between approximate

analytical formula and exact solution of rate

equations

Signal powers vs. position along the fiber:

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Multichannel transmission:Multichannel transmission:Multichannel transmission:Multichannel transmission:

The accuracy of the analytical formula improves with decreasing spectral separation between the channels

3 channel amplifier, spectral distance 4nm: 5 channel amplifier, spectral distance 2nm:

Approximate analytical formula is quiet accurate

X Analytical formula is no longer valid

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Multichannel transmission:Multichannel transmission:Multichannel transmission:Multichannel transmission:

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.

Output signal vs. signal and pump powers:

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Multichannel transmission:Multichannel transmission:Multichannel transmission:Multichannel transmission: The analytical model is used to optimize the parameters of a fiber amplifier.

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.

Optimization of fiber length:

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Inhomogeneous gain broadening:Inhomogeneous gain broadening:Energy band diagram:

is the shift in resonance frequency

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Inhomogeneous gain broadening:Inhomogeneous gain broadening:The model:

• is the number of molecules, per unit volume, whose resonant frequency has been shifted by a frequency that lies between and . ˆˆ d

• The function is the normalized distribution function of molecules, such that . Usually a Gaussian is used.

f

• A photon of wavelength , interacts with molecules with shifted cross-sections and , due to the frequency shift of .

• The width of determines the relative effect of the inhomogeneous broadening.

I f

• All energy levels are shifted manifold is shifted by the same amount from the ground ( ).

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Inhomogeneous gain broadening:Inhomogeneous gain broadening:Single channel amplification:

Aluminosilicate 232 SiOOAl Germanosilicate 22 SiOGeO

Solid lines-inhomogeneous model

Dashed lines-homogeneous model

The inhomogeneous broadening is significant for germanosilicate fiber whereas aluminosilicate fiber is mainly homogeneous

nm5.11hom nmI 5.11 nm4hom nmI 8

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Inhomogeneous gain broadening:Inhomogeneous gain broadening:Multichannel amplification:Aluminosilicate 232 SiOOAl Germanosilicate 22 SiOGeO

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.

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Inhomogeneous gain broadening:Inhomogeneous gain broadening:Multichannel amplification:Germanosilicate 22 SiOGeO

If we decrease the channel distance in germanosilicate fiber to 6nm (less than ), we expect the effect of the inhomogeneous broadening to be stronger.

nmI 8

Here the inhomogeneous broadening mixes the two signal channels and not only ASE channels, thus its influence on signal amplification is more significant.

nm4hom nmI 8

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Inhomogeneous gain broadening:Inhomogeneous gain broadening:Experimental verification of the model:

Circles-experimental results

Solid lines-numerical solution using inhomogeneous model

Dashed lines- numerical solution using homogeneous model

Germanosilicate fiber:

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Conclusions:Conclusions:Numerical models have been presented, for the study of erbium doped fiber amplifiers.

Simple analytical expressions were also developed for several cases.

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.

The effect of homogeneous upconversion, signal amplification in multi-channel fibers and inhomogeneous gain broadening were investigated, using numerical and approximate analytical models

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Suggestions for future work :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

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Publications :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

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Acknowledgements :Acknowledgements :

• Prof. Amos Hardy

• Eldad Yahel

• Irena Mozjerin

• Igor Shmuilov

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Appendix :Appendix :

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Homogeneous upconversion:Homogeneous upconversion:Assumptions for analytical solution:

as

es

ppapp P

hcA

0

apepppapp

BP

hcA

110 2

Strong pumping:

where

0

~0

~

2

p

s

epappp

esasss

P

PB

Appendix:Appendix:Appendix:Appendix:

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Homogeneous upconversion:Homogeneous upconversion:Assumptions for analytical solution:

Homogeneous upconversion not too strong:

where

Appendix:Appendix:Appendix:Appendix:

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0042

2

P

PQPQNCC sasppeff

zPQQzPQzP sesaspp

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Appendix:Appendix:Appendix:Appendix:Homogeneous upconversion:Homogeneous upconversion:Derivation of approximate solution:

zNzNzN 222

~

zPzPzP ppp ~

zPzPzP sss ~

We ignore the terms of second order and higher:

22 , zPzP ps zPzP ps ,

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Rate equations solution without upconversion:

zPQQzPQQ

zPQzPQNzN

sesaspepap

saspap

2

~

zqRzPzP ss 1exp0~~

RzPzPPzPq

sspp exp0~~

0~~

Homogeneous upconversion:Homogeneous upconversion:Appendix:Appendix:Appendix:Appendix:

wherewhere is derived from:is derived from: z

3

421

143

431

1

1 B

BB

q

q

zBB

BBz

zqB 1exp 4

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Appendix:Appendix:Appendix:Appendix:Homogeneous upconversion:Homogeneous upconversion:Approximate analytical formula:

RzzPzPPPqzP s

q

sq

spp

exp~

001

ases

q

qq

qs QQD

Dq

DDq

UNCzP

11

121

61

5

12

2 111

1

ln1ln 41

11

3 qDQQD

QQDD

ases

asesq

3

2

222

zPQQzPQ

zPQzPQNCzN

sesaspp

saspp

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Appendix:Appendix:Appendix:Appendix:Multichannel transmission:Multichannel transmission:Assumptions for analytical solution:Strong pumping:

I

i

ias

I

i

ies

ppapp P

hcA

1

1

0

appappp

is

I

i

ias

ies

is

is

ppapp P

P

P

hcA

11

0

0

01

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Appendix:Appendix:Appendix:Appendix:Multichannel transmission:Multichannel transmission:Approximate analytical solution :

zRzPAzP i

qis

ip

i exp Ii ...1

I

i

is

ies

iaspp

I

i

is

iaspp

zPQQzPQ

zPQzPQNzN

1

12

81

1

expˆˆexpˆˆ

98

91

1811

BKq

Mq

TzBB

TzBB

TMz 1exp

z

q

vvzzvPzP

i

iqqi

is

is

i 11 expexp0 1

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esass

epappq

hcAQ asssas

hcAQ essses

0

~0

~

2

p

s

epappp

esasss

P

PB

23 1 BqRB s

sess qRNB 14

papp NR qN sass

320 2 hcP

Appendix:Appendix:Appendix:Appendix:Definitions of parameters:

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Parameters used in the computation:Parameters used in the computation:

Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:Homogeneous upconversion:

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Parameters used in the computation:Parameters used in the computation:

Inhomogeneous gain broadening:Inhomogeneous gain broadening:Inhomogeneous gain broadening:Inhomogeneous gain broadening:

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