1. 1 Numerical and Analytical models for various effects in models for various effects inEDFAs Inna Nusinsky-Shmuilov Supervisor:Prof. Amos Hardy TEL AVIV UNIVERSITY THE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERING Department of Electrical Engineering – Physical Engineering

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

3. 3Motivation: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)

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

5. 5 Rate equations: Energy band diagram:

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

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

8. 8 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 Boundary conditions: -known launched signal power

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

10. 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. 11 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

12. 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Good agreement between approximate analytical formula and exact numerical solution X Analytical formula is no longer valid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. 13 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

14. 14 Upconversion vs. signal power: 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

15. 15 Multichannel transmission: Assumptions for analytical solution: All previous assumptions Interactions between neighboring ions (e.g homogeneous upconversion and clustering) are ignored (C 2 =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. 16 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:

17. 17 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

18. 18 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:

19. 19 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:

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

21. 21 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. The function is the normalized distribution function of molecules, such that. Usually a Gaussian is used. 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. All energy levels are shifted manifold is shifted by the same amount from the ground ( ).

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

23. 23 Inhomogeneous gain broadening: Multichannel amplification: Aluminosilicate Germanosilicate 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. 24 Inhomogeneous gain broadening: Multichannel amplification: 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. Here the inhomogeneous broadening mixes the two signal channels and not only ASE channels, thus its influence on signal amplification is more significant.

25. 25 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:

26. 26 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

27. 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. 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. 29 Acknowledgements : Prof. Amos Hardy Eldad Yahel Irena Mozjerin Igor Shmuilov

30. 30 Appendix :

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

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

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

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

35. 35Appendix:Appendix: Homogeneous upconversion: Approximate analytical formula:

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

37. 37Appendix:Appendix: Multichannel transmission: Approximate analytical solution :

38. 38Appendix:Appendix: Definitions of parameters:

39. 39 Parameters used in the computation: Homogeneous upconversion:

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

41. 41

42. 42