Numerical and Analytical models for various effects in EDFAs

1. 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

2. Outline: • Motivation • Rate equations • Homogeneous upconversion • EDFA for multichannel transmission • Inhomogeneous gain broadening • Conclusions

3. Motivation: • Applications in the 1.55μm range wavelengths • Optical power amplifiers • Low noise preamplifiers in receivers • Multichannel amplification (WDM) 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.

4. doped fiber Amplified Signal output Pump Forward pumping Pumping geometry: - Forward pumping - Backward pumping -Bidirectional pumping

5. Rate equations: Energy band diagram:

6. Pump absorption and emission Spontaneous emission Signal emission Homogeneous upconversion Signal absorption Rate equations: Second level population:

7. Rate equations: Stimulated emission and absorption Spontaneous emission Pump emission ,absorption and ESA Scattering losses Scattering losses Signal, ASE and pump powers:

8. The ASE spectrum is divided into slices of width • Boundary conditions: -knownlaunched signal power -knownlaunched pump power Numerical solution of the model: • Steady state solution (/ t = 0) • The equations are solved numerically, using an iterations method

9. acceptor donor Homogeneous upconversion: Schematic diagram of the process:

10. 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

11. 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

12. 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• Good agreement between approximate analytical formula and exact numerical solution X Analytical formula is no longer valid

13. 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 power

14. 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

15. 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

16. 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 analytical formula and exact solution of rate equations Solid lines-exact solution Circles-analytical formula

17. Multichannel transmission: 3 channel amplifier, spectral distance 4nm: 5 channel amplifier, spectral distance 2nm: X Analytical formula is no longer valid • Approximate analytical formula is quiet accurate • The accuracy of the analytical formula improves with decreasing spectral separation between the channels

18. 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.

19. 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.

20. is the shift in resonance frequency Inhomogeneous gain broadening: Energy band diagram:

21. All energy levels are shifted manifold is shifted by the same amount from the ground ( ). • A photon of wavelength , interacts with molecules with shifted cross-sections and , due to the frequency shift of . • is the number of molecules, per unit volume, whose resonant frequency has been shifted by a frequency that lies between and . • The function is the normalized distribution function of molecules, such that . Usually a Gaussian is used. • The width of determines the relative effect of the inhomogeneous broadening. Inhomogeneous gain broadening: The model:

22. Inhomogeneous gain broadening: Aluminosilicate Germanosilicate Single channel amplification: • The inhomogeneous broadening is significant for germanosilicate fiber whereas aluminosilicate fiber is mainly homogeneous Solid lines-inhomogeneous model Dashed lines-homogeneous model

23. Inhomogeneous gain broadening: Germanosilicate Aluminosilicate 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.

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

25. 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

26. 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.

27. 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

28. 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

29. Acknowledgements: • Prof. Amos Hardy • Eldad Yahel • Irena Mozjerin • Igor Shmuilov

30. Appendix:

31. Appendix: Homogeneous upconversion: Assumptions for analytical solution: Strong pumping: where

32. Appendix: Homogeneous upconversion: Assumptions for analytical solution: Homogeneous upconversion not too strong: where

33. Appendix: Homogeneous upconversion: Derivation of approximate solution: We ignore the terms of second order and higher:

34. Appendix: where is derived from: Homogeneous upconversion: Rate equations solution without upconversion:

35. Appendix: Homogeneous upconversion: Approximate analytical formula:

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

37. Appendix: Multichannel transmission: Approximate analytical solution :

38. Appendix: Definitions of parameters:

39. Parameters used in the computation: Homogeneous upconversion:

40. Parameters used in the computation: Inhomogeneous gain broadening: