Floquet’s theorem

This is the parabolic curve shown in Fig. 4.4(a). A linear atomic lattice will provide a periodic rather than a constant potential, i.e. V(x) = V[x + a), where a is the repeat distance of the array. Bloch s theorem, also known as Floquet s theorem, states that possible solutions of the Schrodinger equation with a periodic potential are ... 

The RWA is appropriate when the Rabi frequency is much smaller than the other frequencies in the problem, viz. the transition frequency and the detuning. The first of these conditions limits the intensity, and so ultimately RWA breaks down it then becomes necessary to include the effects of the counterrotating terms, and one returns to solving the time-dependent Schrodinger equation, with a Hamiltonian which is periodic in time this is done in a more general way by applying Floquet s theorem for differential equations with periodically varying coefficients. 

If we limit ourselves to scan the yz plane, the orthogonal terms x P are either zero or cancel when k varies. We then note that (3.27) and (3.28) are similar as far as variation in the z direction is concerned, which should not surprise us since Floquet s theorem is valid for both cases in this direction. However, their behavior in the xy plane clearly indicates a planar wave in the double infinite case and a cylindrical wave in the collinear case. 

Note that the element orientation can vary from stick to stick while the interstick spacing can be arbitrary. Only the element spacing in the z direction must be the same from stick to stick. Otherwise, Floquet s Theorem will be violated. 

When analyzing the finite array, it is advisable first to review the infinite array case that is, the finite number of columns shown in Fig. 4.1 becomes infinite as shown earlier in Fig. 1.1. As discussed for example in reference 62 or Chapter 3, infinite arrays are significantly simpler to analyze than finite arrays. In particular, if exposed to an incident plane wave with direction of propagation equal to s, the element cnrrents are all related to each other by Floquet s Theorem ... 

However, when the array has only a finite number of columns as shown in Fig. 4.1, the column currents differ from column to column in amplitude as well as phase. Only along the infinite Z dimension does Floquet s Theorem apply as indicated in Fig. 4.1. 

When the array is infinite, these Floquet currents are the only ones present. This is a direct consequence of Hoquet s Theorem, which basically states that for a periodic strucmre, the phases of the element currents must match those of the incident field. However, when the structure becomes finite, it is no longer periodic and Floquet s Theorem simply does not apply. 

Let us assume that a surface wave with a phase velocity depending only on the infinite structure was indeed present. Since the phase velocity of the Floquet currents depend only on the angle of incidence, the two waves would produce an interference wave with a wavelength unrelated to the periodicity of the array as shown, for example, in Fig. 4.19a. That would violate Floquet s Theorem, which is valid for an infinite array only, not a finite. Thus, the new kind of surface wave can exist only on the finite array. 

The second problem is actually more complex. When considering an infinite array, the terminal impedance will be the same from element to element in accordance with Floquet s Theorem. However, when the array is finite, it is well known that the terminal impedance will differ from element to element in an oscillating way around the infinite array value (sometimes denoted as jitter). We postulated that this phenomenon was related to the presence of surface waves of the same type as encountered in Chapter 4. However, there is a significant difference in amplitude of these surface waves in the passive and active cases. This is due to the fact that the elements in the former case in general are loaded with pure reactances (if any), while the elements in the latter case are (or should be) connected to individual amplifiers or generators containing substantial resistive components (as encountered when conjugate matched). 

We also note that the angles of the transmission coefficient for the inner meander lines are approximately twice that for the outer meander lines, as they should be (see discussion above). The meander-line dimensions are given in the respective Smith charts. We observe that both of these designs have the same dimensions and D, and so on. This is done so as not to violate Floquet s Theorem (see reference 26). The differences in impedances are obtained by using different line widths w in the two cases. 

Fig. D.2 A finite x infinite array comprised of 2Q+1 infinitely long stick arrays. All elements are driven with voltage generators and the currents in the reference elements in row 0 are denoted Iq. For Q -> oo we obtain an infinite x infinite array where Floquet s Theorem yields...

Floquet s Theorem applies [26], yielding Substituting (D.3) into (D.2), we obtain... 

The multiple reflection terms / 2 in Eq. (4.89) are added in accordance with our findings on the selective interaction of modes showing similar angular wave number, yielding dispersion relation (4.84). An explicit proof of Eq. (4.89) for arbitrary orders I may be based on the fact that reflection fields which cyclicly reproduce themselves after I reflections can be represented in terms of fields which gain a phase factor exip(—2nin/l at each reflection (Floquet s theorem). This was reported by Langbein in 1971 [95,99]. 

According to the Floquet theorem [Arnold, 1978], this equation has a pair of linearly independent solutions of the form x(z, r) = u(z, r) exp( 27rizT//3), where the function u is /3-periodic. The solution becomes periodic at integer z = n, so that the eigenvalues e we need are en = e( n). To find the infinite product of e s we exploit the analytical properties of the function e(z). It has two simple zeros in the complex plane such that... 

S.-I. Chu, D.A. Telnov, Beyond the Floquet theorem Generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields, Phys. Rep. 390 (2004) 1. 

The title of this subsection refers to a recent paper by Kylstra and Joachain investigating double poles of the S matrix [25]. Their paper is based on the time-dependent Lippmann-Schwinger equation. Since the Hamiltonian is periodic in time the use of the Floquet theorem permits one to apply the time-independent theory of scattering. As in Ref. [25] we consider two quasi bound states (n = 2). Instead of starting from an Hamiltonian periodic in time (semi-classical approximation) we use a time-independent model. The laser field is assumed to be quantized and as a result the total Hamiltonian describing the atom in the laser field is time-independent (see chapter VI of Ref. [13]) Our aim is twofold To reproduce the results of Ref. [25] and, more generally, to illustrate the relevance of simple models to describe collision... 

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