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Guiding of Waves Between Absorbing Walls Journal of Modern Physics, 2012, 3, 553-560 http://dx.doi.org/10.4236/jmp.2012.37076 Published Online July 2012 (http://www.SciRP.org/journal/jmp) Guiding of Waves between Absorbing Walls Dmitrii Kouznetsov, Makoto Morinaga Institute for Laser Science, University of Electro-Communications, Tokyo, Japan Email: {dima, morinaga}@ils.uec.ac.jp Received May 5, 2012; revised June 9, 2012; accepted June 29, 2012 ABSTRACT Guiding of waves between parallel absorbing walls is considered. The principal mode is constructed; its absorption is estimated. The agreement with previous results about reflection of waves from absorbing walls is discussed. Roughly, the effective absorption of the principal mode is proportional the minus third power of the distance between walls, mi- nus 1.5d power of the wavenumber and minus 0.5d power of the local absorption of the wave in the wall. This estimate is suggested as hint for the design of the atomic waveguides, and also as tool for optimization of attenuation of the am- plified spontaneous emission (and suppression of parasitic oscillations) in high power lasers. Keywords: Zeno Effect; Atom Optics; Waveguides; Suppression of Amplified Spontaneous Emission 1. Introduction The frequency of decay is denotes with . The special system of units is used in such a way that =1 and the The consideration of reflection of waves from absorbing energy of the particle in vacuum is assumed to be square walls had been stimulated by the experiments with ridged of its wavenumber. mirrors [1] and their interpretation in terms of the Zeno In Section 3, the special case of uniform absorption is effect [2,3]. The Zeno approximation [2] showed good considered; this gives relations between parameters of agreement with experiments in wide range of parameters the wave function and the physical quantities that can be [3,4]; it describes reflection of waves of any origin. In determined experimentally. Such physical quantities are particular, it applies to the atomic waves and to the energy of the particle and its absorpfion s in the optical waves. In addition to the atom optics (discussed in [2]), the material of the wall; this quantities are assumed to be reflection and guiding of waves may affect the suppres- independent parameters. sion of the Amplified Spontaneous Emission (ASE) that In Section 4, the case of channeling is considered; is considered as serious problem [5-9]. At the scaling-up parameters of the principal mode (transversal wave- the power, the efficient suppression of ASE becomes number p, decay q and the damping ) are defined more important, and the unwanted guiding of ASE by the and expressed through the holomorphic function 1 absorber (which is supposed to suppress it) might take cosc zzz = cos , its inverse function acosc = cosc place. The estimates of the conditions of such a guiding and acosq zz = acosc exp iπ 4 ; properties of these is necessary tool for the design of powerful devices. functions are discussed and the efficient C++ imple- In this article, the wave function of a particle (atom, mentations are indicated. photon) between two absorbing walls is constructed. The In Section 5, the example of the principal mode is effective absorption of such a mode is estimated and considered; the real and imaginary parts of the wave compared to the previous results. The results are ex- functions are built through the trigonometric function of pected to have applications in both the atom optics complex argument and complex exponential for the (wanted guiding of neutral particles by the absorbing damping =1 4 . The amplitude and phase of the walls) and the design of powerful lasers (estimates of principal mode are shown for =1 4 and for conditions of the unwanted guiding of the ASE by the =1 50. walls that are supposer to absorb the ASE, and opti- In Section 6, the asymptotic behavior of parameters at mization of suppression of the ASE). small damping is considered; this is realistic case of good This article is organized in the following way: guiding conditions. In Section 2, the phenomenological absorbing Schrö- In Section 7, the estimates for the effective absorption denger equation is suggested. The case with absorbing of the principal mode are compared with the previous walls corresponds to the pure anti Hermitian potential. results; the agreement is interpreted as confirmation of Copyright © 2012 SciRes. JMP 554 D. KOUZNETSOV, M. MORINAGA the validity of the analysis. , consider the case of the uniform absorption of the In Section 8, the special case is suggested for phy- plane wave at U by (4); let does not depend on the sically realistic parameters of the experimental condi- first argument, id est, x,=zfz. Then, for tions of realization of the guiding of waves between f = fx we get the equation absorbing walls. ff i=0 (6) In Section 9 (Conclusion), the asymptotic estimate of the effective absorption A of the guided mode through At positive values of , the decaying in the direction the real and imaginary parts c , s of the wavenumber z solution has form in the wall is suggested. This main result is expected to f =expikz (7) be confirmed (or rejected) by the physical experiments with photons, atoms or any other wave of any origin. where kkk : 0 , 0 is solution of equa- tion 2. Schrödinger Wave and the Absorption k 2 =i (8) In this case the dimension less Schrödinger equation is Assuming 0 , 0 , wavenumber k can be considered in the paraxial approximation. expressed as follows: For simplicity, in this section the special system of units is used such that =1 and mass of the atom is k =i (9) half. Then, the Schrödinger equation for the wave func- For the compactness of notations, let kcs=i , tion =x,,zt can be written as follows where c and s are real parameters. Then 2222 it = x zUx (1) cs22 = (10) where the potential UU x depends only on the 2=cs (11) transversal coordinate x (and does not depend on time Alternatively, we may consider the absorption of wave z in the direction of propagation). in the medium, id est, k as initial parameter. Mak- In the simplest approximation, the entangling with ing estimates for the ridged atomic mirror with distance numerous degrees of freedom of scattered (or relaxed) T between ridges, the absorption by intensity can be atom can be taken into account with non Hermitian approximated as 1 T , and the absorption by amplitude is potential. Such an approximation is considered, in par- of order of 12T ticular, in the interpretation of the quantum reflection in therms of the Zeno effect [2]. For the wave guide, the s ==12kT (12) absorption correspond to complex values of Ux at Real part of k is determined by the mass and the x > X , where X is half-width of the channel between energy of the particle we intent to reflect or to guide, or the mirrors. just 2πn , where is vacuum wavelength of the Consider the quasi-monochromatic solution, assuming waves (perhaps, ASE) that could be guided, and n is the exponential dependence on t ; let the refraction index of the medium. x,,zt =expit xz , (2) In order to avoid additional guiding by the step of the Then, instead of (1) we get the stationary Schrödinger refraction index, the real part of the refraction index of equation the wall is supposed to be matched to that of the central region; but some reflection still may happen due to the = 222 x zUx2 (3) imaginary part. Then, the real and imaginary parts of the below, the two cases of the solution are considered: for swiare o wavenumber k can be expressed as follows: Ux= i = const (4) =kkkcs222 = 22 = (13) and for ==kkksc2 2)= 2 (14) Ux=iUn itStepx d (5) Parameter has sense of the energy of the particle, where UnitStep is the conventional unit step function and also sense of the square of wavenumber, while implemented in various programming languages includ- has sense of the decay rate at the time scale, if a wave ing Mathematica, and d is constant, that has sense would be uniform in the space. For the estimates, the of half-width of the channel that confines the particle. term with k in (13) can be neglected; however it is 3. Uniform Absorption kept in the deduction that may be applied not only to atoms, but also to other kinds of waves (optical, acous- In order to understand the physical sense to the constant tical, waves on the surface of a liquid, etc.) For optics Copyright © 2012 SciRes. JMP D. KOUZNETSOV, M. MORINAGA 555 (both atom optics and conventional optics), the typical the “observation” (id est, x d ), the better is the case is of low absorption; waves rather propagate than channeling. absorb, to, Substitution of (19) to (3) gives the equation for f fx in the following form: s k =1 (15) 2 ck fxdf i UnitStep = 0 (20) In this case, for the estimate of the primary parameters, Search the solution of (20) as the combination of the we may use the approximation cosinusoidal and the exponential, let 2 2 kc (16) cos px , x d fx= (21) rqxdxdexp , and treat ck= and s = k as initial parameters of the model describing the absorption of the wave inside the walls. where r , , p , q are constant para- On the other hand, one may consider as “given” the meters. From the physical reasons (almost free propa- energy of the particle and the absorption k . gation inside the channel), parameter is expected to Then, be of order of wavenumber k from the previous sec- tion.
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