1. Wave Propagation In Amplifying Media
H. Bahlouli
Physics Department, KFUPM,
March, 2007

2. King Fahd University of Petroleum and Mineral
KFUPM

3. Photoelectron Spectroscopy: XPS, UPS and AES
VG- ESCALAB MKII Coordinator : Prof. Nouar TABET

4. The second facility ,The Dc Magnetron Sputtering system, is relatively a new facility

5. Superconductivity Lab ( Coord. Prof. Khalil Ziq )
1)Computer controlled vibrating sample magnetometer (VSM): field range 0-9 Tesla, temperature range 2–300 K.
Superconductivity Lab ( Coord. Prof. Khalil Ziq )
Low Temperature Cryostat
Control Unit of the Vibrating sample magnetometer

6. Outline INTRODUCTION Theoretical Model CONCLUSION
Stationary Wave Equations
Dynamical Wave Equations
Mapping Between the Schrödinger and the EM Wave Equations
Resonance Poles
CONCLUSION

7. Introduction
Light propagation in amplifying media is of theoretical and experimental interests.
Theoretically, they’re modeled with complex potentials to represent non-conservative amplifying systems.
Since gain systems are assumed to reach a state of equilibrium, they are usually studied using the stationary wave equation.
However, a study by Soukoulis et al. using the time-dependent wave equation showed that the transmission and reflection diverge after a critical lasing threshold.
i.e.The results obtained from the stationary wave equation are only
physical below the threshold.

8. Introduction In this work we try to:
Clarify the discrepancy between the stationary and time-dependent results.
Treat EM and Schrödinger wave equations are treated in parallel and this resulted in a mapping between them.

9. Theoretical Model Gain Media: Schrödinger Wave Equation (SE)
Complex Scattering potential
Electromagnetic Wave equation (EM)
Complex Permittivity

10. Theoretical Model

11. Stationary wave equations with GainSE EM

12. Stationary wave equations with Gain
Transmission for the Schrödinger equation vs. system length.

13. Stationary wave equations with Gain
Transmission for the wave equation vs. system length.

14. Dynamical wave equations with GainSE
numerically EM

15. Dynamical wave equations with Gain
Transmission using the Schrödinger equation vs. time for three different amplification potentials.

16. Dynamical wave equations with Gain
Transmission using the wave equation vs. time for three different amplification potentials.

17. Mapping Between the Wave Equations
Klein-Gordan (K-G)
Comparing with

18. Mapping Between the Wave Equations
The comparison yields
Using

19. The equivalent scalar potential is defined by

20. Resonance Poles
Resonance poles are obtained by setting the denominator of T equal to zero SE EM

21. Resonce Poles In support for the mapping above, when
Which is very similar to

22. Resonce Poles
Moreover
Which is very similar to

23. Resonce Poles
The resonance poles of for the Schrödinger equation on the complex E-plane with a potential directly proportional to the energy

24. Resonce Poles
The resonance poles of for the Schrödinger equation on the complex k-plane with a potential directly proportional to the energy have very similar structure to the resonance poles of for the wave equation on the complex ω-plane

25. The Discrepancy is due to the Resonance poles
The instability of stationary states is related to the time dependence factors: SE EM

26. The Discrepancy and the Resonance poles
To confirm this ansatz:
We consider a wave packet with a given energy that is close to one of the resonance poles in the lower half plane.
We put this pole in the lower half plane under scope ( k-plane or E-plane , or ω-plane).
Tune the amplification potential until the pole crosses to the upper half plane.
Study the transmission for the stationary and dynamical wave equations
Before the pole crossing
At crossing
After crossing

27. Resolving the Discrepancy
The transition of a resonance pole for different amplification potentials for the Schrödinger equation in the complex k-plane with a potential directly proportional to the energy

28. Resolving the Discrepancy
The transition of a resonance pole for different amplification potentials for the Schrödinger equation on the complex E-plane with a potential directly proportional to the energy

29. Resolving the Discrepancy
The transition of a resonance pole for different amplification potentials for the wave equation on the complex ω –plane.

30. Resolving the Discrepancy
Transmission for the stationary Schrödinger equation vs. amplification potential.

31. Resolving the Discrepancy
Transmission for the dynamical Schrödinger equation vs. time for three different amplification potentials (correspond to before crossing, at crossing, and after crossing)

32. Resolving the Discrepancy
Transmission for the stationary wave equation vs. amplification potential.

33. Resolving the Discrepancy
Transmission for the dynamical wave equation vs. time for three different amplification potentials (correspond to before crossing, at crossing, and after crossing)

34. Conclusion
This work confirms the discrepancy between the results of the stationary and dynamical wave equations as in the literature.
A mapping between the SE and the EM wave equations wave obtained through an energy dependent potential.
The drop in the transmission for the stationary wave equations was found to occur as the corresponding resonance pole crosses the real axis on the energy or frequency plane.
The divergence in the transmission for the dynamical wave equations was found to occur as the corresponding resonance pole crosses the real axis on the energy or frequency plane.