Title: Jonathan Hu, Brian Marks, Qun Zhang, and Curtis Menyuk University of Maryland, Baltimore County Balt
1Jonathan 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
2Raman 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)
3Gain 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.
4Algorithmic Considerations
- Accuracy
- Robustness
- Speed
- Flexibility
What kind of method should we use?
5Numerical 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)
6Integration 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)
7Run 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
8Robustness of the Shooting Method
- Requirements for convergence
- Weighted Jacobi method
- Continuation method
9Convergence 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
10Weighted Jacobi Method
0
10
w
0.1 0.5 0.8
c
-5
10
1
10
Iteration Number
The appropriate w speeds convergence
11Schematic for Continuation Method
Converges?
Upper tolerance
c
Lower tolerance
Input pump power
Specified input power
Iteration number
12Algorithm Comparison
Desired attributes of an algorithm
Relaxation method
Shooting method
?
?
Accuracy Robustness Speed Flexibility
?
?
?
?
13Conclusion
- Shooting method is effective for solving Raman
amplifier equations - Relaxation method is preferred for two-point
boundary conditions - Shooting method is more flexible