Jonathan Hu, Brian Marks, Qun Zhang, and Curtis Menyuk University of Maryland, Baltimore County Balt

1
Jonathan Hu, Brian Marks, Qun Zhang, and Curtis
Menyuk University of Maryland, Baltimore
CountyBaltimore, MDAlso with Laboratory for
Physical SciencesCollege Park, MD
A Shooting Algorithm to Model Backward Pumped
Raman Amplifiers
2
Raman Amplification
z 0
z L
Pump
Power (log scale)
n number of pumps m number of signals j,k
wavelength index
Signal
Distance
0
50 km
For two-point boundary value problems, the
relaxation method works well
Min et al., Photon. Technol. Lett., 12(2000)
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Gain Specification Constraint

• In practice, one often wants to fix the signal
  gain
• Distance integrated pump power is the constraint
• This is no longer a two-point boundary value
  problem

In this case, the shooting method works well
Hu et al., CLEO 2003, CThD5.
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Algorithmic Considerations

• Accuracy
• Robustness
• Speed
• Flexibility

What kind of method should we use?
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Numerical Method

• Relaxation method

Shooting method
z 0
z L
z 0
z L
Pump
Pump
input
input
Signal
Integration direction
Integration direction
Signal
input
input
The shooting method is more flexible
Morison et al., Communications of the ACM 5(1962)
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Integration Method

• Improved Euler method

Power average method
Exponential variation
Power
Zn1
Zn
Zn
Zn1
? z
? z
Both are second order accurate
Min et al., Photon. Technol. Lett., 12(2000)
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Run Time Comparison
70
Improved Euler method
Simulation time (a.u.)
10
? Power average method
5
300
5000
1000
?z step (m)
Improved Euler is preferred
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Robustness of the Shooting Method

• Requirements for convergence
• Weighted Jacobi method
• Continuation method

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Convergence Parameter
RMS difference between two consecutive iterations
m number of pumps l number of z steps Pi(n)
n-th iteration pump power Pi(n1) (n1)-th
iteration pump power
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Weighted Jacobi Method
0
10
w
0.1 0.5 0.8
c
-5
10
1
10
Iteration Number
The appropriate w speeds convergence
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Schematic for Continuation Method
Converges?
Upper tolerance
c
Lower tolerance
Input pump power
Specified input power
Iteration number
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Algorithm Comparison
Desired attributes of an algorithm
Relaxation method
Shooting method
?
?
Accuracy Robustness Speed Flexibility
?
?
?
?
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Conclusion

• Shooting method is effective for solving Raman
  amplifier equations
• Relaxation method is preferred for two-point
  boundary conditions
• Shooting method is more flexible