Analytical Unit Pulse Propagation in an Active Single Resonance Lorentzian Medium

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1 1568 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 7 3, 1 Analytical Unit Pulse Propagation in an Active Single Resonance Lorentzian Medium Ali Abdolali, Maziar Hedayati, and Shahram Hedayati Department of Electrical Engineering, Iran University of Science and Technology, Iran Abstract In a causally dispersive medium the signal arrival appears in the dynamical field evolution as an increase in the field amplitude from that of the precursor fields to that of the steady-state signal. The classical theory of Sommerfeld and Brillouin of pulse propagation in a Lorentz medium is reexamined. While many rigorous studies of wave propagation in passive Lorentzian media have been performed, the corresponding problem in active media has remained theoretically unexplored. We use analytical approximations for describe the correct saddle-point. In this method used the refractive index for the determination of the response of an active Lorentzian medium. In this paper illustrated that do not exist distance saddle points and then the near saddle points used for the determination of the response of an active Lorentzian medium to a step modulated pulse. There are Two types of wave dispersion, (Temporal dispersion and Spatial dispersion). Here used the time dispersion and surveyed in the frequency domain. 1. INTRODUCTION Wave propagation through a linear, temporally dispersive medium has been a complex and sometimes controversial research topic since the 19th century. Hamilton did the first study on the dispersive medium in the 1839 which the time that Group velocity is introduce. Subsequently, Rayleigh describe the difference between Group velocity and phase velocity. Classical Lorentz models, Debye and Drude has been the turning point feature in speech the liner Nonconductor medium. Sommerfeld and Brillouin were among the early researchers studying the wave propagation in linear, homogeneous, isotropic, causally dispersive media. With Sommerfeld s proof within the classical Maxwell-Lorentz theory that an electromagnetic signal could not propagate faster than the vacuum speed of light in a Lorentz model dielectric, Brillouin then introduced the signal and energy velocities within that medium as a replacement for the group velocity This theory is valuable because it provides analytic expressions for the dynamics of pulses under mature dispersion conditions in a medium that accurately models several real lossy dispersive materials 1]. Their study led to the discovery of two Precursor signal. In the classical theory of dispersive pulse propagation. Both a high-frequency (above resonance) Sommerfeld precursor and a low-frequency (below resonance) Brillouin precursor are present in the propagated field structure when the input pulse is ultra wideband. That s the approach the refractive index were replaced with Mac Lauren series of refraction depending on the frequency. The result was a simple phrase that was associated with complex frequency, distance and time. At after time Handelsman and Bleistein removed the shortcoming of this method byuniform asymptotic approach. But Einstein did the newest method for describing propagation wave in this media ]. In this paper we consider the propagation of wave in an active Lorentzian medium. Recently, there has been experimental observation of superluminal velocities in active media. This paper is organized as follows. In Section, we present the active Lorentzian medium, which is a model for an inverted two level atomic medium. In Section 3, presented we achieve saddle points by analytical method. Section 4 includes the asymptotic analysis of the wave propagation in an active Lorentzian medium.. THE COMPLEX INDEX OF REFRACTION FOR THE ACTIVE LORENTZIAN AND THE LOCATIONS OF BRANCH POINT LOCATION The analysis is presented here. The analysis is presented for the case of a single-resonance Lorentz model dielectric with dielectric permittivity where is the undammed resonance frequency, ε() b = 1 + ε + iδ (1) b = πn f θ m ()

2 Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 7 3, where is the frequency difference between the two levels p is the plasma frequency with number density N of Lorentz oscillators, me the mass of an electron, f and e, are the oscillator strength, electron charge and δ is the line width of the resonance. With relative magnetic permeability µ = µ, the complex index of refraction is given by n() = ɛ()/ɛ = 1 + b + iδ (3) Based on these inequalities an arbitrary set of the parameters for the active Lorentzian medium is chosen as, = Hz, b = Hz, δ = Hz. Figure 1 shows the plot of the real and imaginary part of the refractive index of the medium in active and passive case and Fig. the imaginary of them. The branch point locations are given by ± = ± 1 δ δi ± = ± δ δi where 1 = b. The branch lines chosen here consists of the line segments + + and as shown in Fig. 3. We can analyze n() in the complex w-plane expect in the + +,. (4) Figure 1: The real part of the refractive index for passive and active media. Figure : The imaginary part of the refractive index for passive and active media. Figure 3: Branch points and branch cuts for the active single resonance Lorentzian medium. Figure 4: A general picture of the two distance saddle points path.

3 157 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 7 3, 1 3. ANALYSIS OF THE PHASE FUNCTION AND ITS SADDLE POINTS For achieve the response of wave that propagating in the dispersive media we must solve this integration 3] A(Z, t) = 1 f() exp i(k()z t)] Z > (5) π where f() is the is the Laplace transform of the initial time behavior of the wave at the plane boundary z =, In preparation for the asymptotic analysis in a temporally dispersive medium, it is necessary first to determine the location of saddle point. If ϕ(, θ) is the complex phase function which is defined as follows Φ(, θ) = in() θ] To obtain the asymptotic expansion of the propagated field A(z, t) we must study the behavior of ϕ(,θ) in the saddle points. The condition that ϕ(,θ) be in the saddle point n() + n () θ = (6) The roots of this equation then give the desired saddle-point locations. So ( 1 + jδ ) b ( ( + iδ) + jδ ( + jδ) = θ 1 + jδ ) + jδ (7) where 1 = b We can the roots of this equation to be calculated numerically the and gain the saddle point but we use the analytical method and with good approximation for this work. The degree of this equation is eight so we have eight root but the four of this them is acceptable High-frequency Approximation To solve the Eight-degree equation use of both high and low frequency approximation ] 1 θ ( + δi) 1 + δi b ( + δi) + δi = 1 + δi therefore to achieve the distant saddle points we use the bellow approximation b ( ( + iδ) ( + jδ) b 1 δi ( )) 1 + O 3 + δi ( b θ ) θ iδb 1 θ 1 = 3.. Low-frequency Approximation To obtain description of the near saddle point locations, We use the Equation (7) with eight degree which may be rewritten as that used Taylor expansion If θ ( + jδ ) = 1 + jδ b ( + δi) + jδ +b 4 ( + δi) ( 1 + jδ)( + jδ) (9) ( + δi) ( + jδ) ( 1 + jδ) = 1 4 ] δ + δi δ3 i ( 1 + ) 1 (8) And ( + δi) o + jδ = 1 δi + (1 δ ) + δi ) ] (3 4 δ 3

4 Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 7 3, (a) (b) Figure 5: Total sampled field at 1 µm inside the active media for (a) 1.38 < θ < 1; (b) 1 < θ < By replacing this approximation and if δ b,, Equation (8) simplified + δi θ θ ( b θ θ 3b α θ θ) θ θ 3b α = α = 1 δ 3 1 (4 1 b ) With solving the equation obtained the near saddle points. sp ± N = ±ψ(θ) jδζ(θ)ψ(θ) = (θ θ ) θ θ 3b α δ ζ(θ) = 3 θ θ b θ θ 3b α ( θ θ b θ θ 3b α ) 1/ (1) (11) Fig. 4 shows a general picture of the two near saddle points path that we achieve by analytical method. 4. ASYMPTOTIC CALCULATION OF THE FIELD A(Z, θ) After obtaining the dynamics of saddle points and surviving of the topography of the phase function Φ(, θ) it is now possible to calculate A(z, θ). For our modulated unit step function we use the method that mentioned in 4]. Fig. 5 shows the analytical evaluated propagated pulse for 1 < θ < 1.38 and 1.38 < θ < 1. The values of, δ and b used here are = Hz, b = Hz, δ = and c = Hz. In the active medium against the passive medium when time increase, the response of the unit pulse is not damped and dominate of the wave remain Constance. 5. CONCLUSION We use analytical approximations for describe the correct saddle-point. In this method, we used the refractive index for the determination of the response of an active Lorentzian medium and we illustrated that do not exist distance saddle points and then the near saddle points used for the determination of the response of an active Lorentzian medium to a step modulated pulse REFERENCES 1. Wait, J. R., Propagation of pulses in dispersive media, Radio Sci. D, Vol. 69, , 1965.

5 157 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 7 3, 1. Loudon, R., The propagation of electromagnetic energy through an absorbing dielectric, J. Phys. A, Vol. 3, 33 45, Chu, S. and S. Wong, Linear pulse propagation in an absorbing medium, Phys. Rev. Lett., Vol. 48, , Oughstun, K. E. and G. C. Sherman, Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media, Proceedings of the International Union of Radio Science 198 International Symposium on Electromagnetic Waves, Paper 113, C/1 C/5, International Union of Radio Science, Munich, Federal Republic of Germany, 198.