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CoompmlexpWleKBxAWpprKoxBimaAtiopnspinrGorxapimhenaetEiloecntrosn-iHnoleGWraavepguhidesnien MEaglnectictrFoielnd -Hole Waveguides in Magnetic Field 911 It is worth remarking that this is so-called system of equations in variations which describes a family of trajectories close to x0(s), that is, z(s) = n(s), and these trajectories are given by x(s) = x0 (s) + n(s)en (s), (see (20), (21), (17),(18) (19)). The crucial point of the analysis is that it is possible to choose a solution of (21) in such a way that ImΓ > 0, thus, providing asymptotic localization of ψ. Namely, if z(s) is a complex solution to (21), wronskian of Re(z(s)) and Im(z(s)) may be chosen in such a way that a(s)W(Re(z), Im(z)) = 1. Then, the following inequality holds true Im(Γ(s)) = a(s)W(Re(z), Im(z)) = 1 > 0 Re(z)2 + Im(z)2 |z(s)|2 along the trajectory. This leads to the localisation. Thus, we obtain that to the leading order the Gaussian beam asymptotic solution for electron or hole is given by ψ = e h ̄ 2z(s) It is always regular near caustics and focal points regardless its complicated geometrical structure. 3. Asymptotic expansion of the Green’s tensor for electron-hole in magnetic field in the form of integral of Gaussian beams The theoretical background of the method of Gaussian beams summation originally was developed for acoustic wave fields ((16)), and later, for electromagnetic and elastic fields ((17)). The generalisation of the method of Gaussian beams summation for electron motion in magnetic field was done in (18), (19). In this section, the approximation of electron-hole Green’s tensor near to caustics and focal points as an integral over Gaussian beams is described briefly. According to (17), (18), (19), and taking into account ray asymptotics of electron-hole Green’s tensor (see (10)), the integral over all Gaussian beams irradiated from the point source x(0) is represented as follows 2π i G ( x , x ( 0 ) , E ) = e h ̄ 0 −iθ−γ e 2 A ( γ ) d γ z(s) ( 2 3 ) i S0 (s)+S1(s)n+ p(s) n2 e−iθ/2 e1(1 + O(h ̄ 1/2)). (22) S0(s)+S1(s)n+ p(s) n2 2 z ( s ) −iθ+γ e 2 i θ+γ e2 e2 z(s) i θ−γ (1+O(h ̄1/2)), α=1,2,PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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