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

King Fahd University of Petroleum and Mineral KFUPM

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

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

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

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

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.

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.

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

Theoretical Model

Stationary wave equations with Gain SE EM

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

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

Dynamical wave equations with Gain SE numerically EM

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

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

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

Mapping Between the Wave Equations The comparison yields Using

The equivalent scalar potential is defined by

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

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

Resonce Poles Moreover Which is very similar to

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

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

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

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

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

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

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

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

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)

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

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)

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.