Numerical and Analytical

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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
SupervisorProf. Amos Hardy
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Outline

• Motivation
• Rate equations
• Homogeneous upconversion
• EDFA for multichannel transmission
• Inhomogeneous gain broadening
• Conclusions

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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.

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Pumping geometry

• - Forward pumping
• - Backward pumping
• - Bidirectional pumping

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Rate equations
Energy band diagram
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Rate equations
Second level population
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Rate equations
Signal, ASE and pump powers
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Numerical solution of the model

• Steady state solution (?/ ?t 0)

• The equations are solved numerically, using an
  iterations method

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Homogeneous upconversion
Schematic diagram of the process
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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/t)

• Loss due to upconversion is not too high

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Homogeneous upconversion
Signal and pump powers vs. position along the
fiber
Solid lines-exact solution Circles-analytical
formula Dashed lines-exact solution without
upconversion

• Approximate analytical formula is quite accurate

Injected pump power 80mW Input signal power
1mW
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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X Analytical formula is no longer valid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
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 formulas accuracy
  improves with increasing the pump power

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Homogeneous upconversion
Upconversion vs. signal power
Injected pump power 100mW

• Increasing the input signal power decreases the
  influence of homogeneous upconversion
• Approximate analytical formulas accuracy
  improves with increasing the input signal power
  power

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Multichannel transmission
Assumptions for analytical solution

• All previous assumptions

• Interactions between neighboring ions (e.g
  homogeneous
• upconversion and clustering) are ignored
  (C20)

• 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
Signal powers vs. position along the fiber
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

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Multichannel transmission
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

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

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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 cant be neglected.

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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.

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Inhomogeneous gain broadening
Energy band diagram
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Inhomogeneous gain broadening
The model
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Inhomogeneous gain broadening
Single channel amplification
Solid lines-inhomogeneous model Dashed
lines-homogeneous model

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

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Inhomogeneous gain broadening
Multichannel amplification

• 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
Multichannel amplification

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

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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
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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.

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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

• 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

• Prof. Amos Hardy
• Eldad Yahel
• Irena Mozjerin
• Igor Shmuilov

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Appendix
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Appendix
Homogeneous upconversion
Assumptions for analytical solution
Strong pumping
where
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Appendix
Homogeneous upconversion
Assumptions for analytical solution
Homogeneous upconversion not too strong
where
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Appendix
Homogeneous upconversion
Derivation of approximate solution
We ignore the terms of second order and higher
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Appendix
Homogeneous upconversion
Rate equations solution without upconversion
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Appendix
Homogeneous upconversion
Approximate analytical formula
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Appendix
Multichannel transmission
Assumptions for analytical solution
Strong pumping
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Appendix
Multichannel transmission
Approximate analytical solution
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Appendix
Definitions of parameters
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Parameters used in the computation
Homogeneous upconversion
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Parameters used in the computation
Inhomogeneous gain broadening
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