View source for Guiding of waves between absorbing walls

'''Guiding of waves between absorbing walls''' is phenomenon based on the reflection of waves coming at the [[grazing angle]] to a plane separating the media with different value of the [[index of refraction]], whenever the step of index of refraction is real or complex. The imaginary part of the index of refraction represents the absorption. The guiding can be interpreted as the multiple reflection.

'''Warning!''' this article is under construction.

==Absorbing Schroedinger==
In the simple case, the propagation of scalar waves between absorbing waves can be described with the [[Absorbing Schroedinger]] equation:
:$ \displaystyle \!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \mathrm i \hbar \dot \Psi = \hat H ~ \Psi -\mathrm i ~ \hat V ~ \Psi $
where 
$\Psi$ is wave function, the dot differentiats it with respect to time,
$\hat H$ is the conventional, Hermitian part of [[Hamiltonian]], and
$\hat V$ is the Hermitian operators that represents the entanglement of the state with some other degrees of freedom, that are taken into account only phenomenologocally. As usually, $\hbar$ is the [[Planck constant]]; with the appropriate set of physical unite, this constant can be set to unity.

The Absorbing Schroedinger describes the evolution of the quantum state of a particle slightly coupled to many other particles, [[environment]], giving evolution of the component of the [[wave function]] of the particle that "did not yet interact" with the environment.

Assume the empty space between parallel walls, and uniform absorption with rate $\gamma$ within walls. In the [[coordinate representation]], for a particle of mass $m$, this corresponds to 
:$ \displaystyle \!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \hat H = \frac{-\hbar^2}{2m} \nabla^2$
and
:$ \displaystyle \!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ \hat V = \hbar \gamma~ \mathrm{UnitStep} \left( |x|-d \right)$
where $2d>0$ is distance betwen walls, and $x$ is transversal coordinate.

In the [[quasimonochromatic approximation]], the exponential dependence of wave function on time $t$ is assumed: 
:$ \displaystyle \!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ \Psi = \psi ~ \exp(\mathrm i \omega t)$
where $\omega$ (or $\hbar \omega$, if the Planxk constant is drawn through the deduction) has sense of energy of the particle, and $\psi$ is spacial part of the wave function, that is supposed to depend on time very slowly if at all; in the simple approximation $\psi$ is function of only few spacial coordinates; for example, $x$ and $z$. <!-- Often the [[Decart system of coordinates]] is assumed, and it is supposed that mainly the particle propagates along $z$ axis, and $x$ is called the transversal coordinate.!-->

The substitution of (2),(3),(4) into (1) gives the "monochromatic" equation
:$ \displaystyle \!\!\!\!\!\!\!\!\!\! (5) ~ ~ ~ \hbar\, \omega\, \psi = \frac{-\hbar^2}{2m} \nabla^2 \psi - \mathrm i ~\hbar ~\gamma~ \mathrm{UnitStep} \left( |x|-d \right) ~ \psi$
where $\psi= \psi(x,z)$ and $\nabla$ differentiates with respect to $x$ and $z$.

At propagation of wave in the uniform absorbing medium with absorption rate $\gamma$ (case $d\rightarrow 0$) the solution of equation (5) can be expressed as follows:
:$ \displaystyle \!\!\!\!\!\!\!\!\!\! (6) ~ ~ ~ \psi = \exp( \mathrm i k z )$ 
The substitution to (5) gives
:$ \displaystyle \!\!\!\!\!\!\!\!\!\! (7) ~ ~ ~ \hbar \omega = \frac{(\hbar k)^2}{2m} - \mathrm i \hbar \gamma$
which means leads to
:$ \displaystyle \!\!\!\!\!\!\!\!\!\! (8) ~ ~ ~ k = \sqrt{  \frac{2m \omega}{\hbar} + \frac{2 \mathrm i \gamma m }{ \hbar} } =  
\sqrt{  b^2 + \frac{2 \mathrm i \gamma m }{ \hbar} }  $
where 
: $ \displaystyle \!\!\!\!\!\!\!\!\!\! (9) ~ ~ ~ b= \sqrt{  \frac{2m \omega}{\hbar}}$
is real quantity, that can be  interpreted as wavenumber of the particle in the empty space.

==Relation to the Zeno effect==
The absorbing walls can be considered as detectors, that check, wether the particle has left from the region between walls or not yet. In [[Wuantum mechanics]], the detection can be interpreted as absorption annihilation of a particle localized between walls (perhaps, wiht creation of a particle(s) in some other states. The detection of the transition suppress it. The [[absorbing Schroedinger]] above describes this process, giving the shape of th eguided mode and its effective absorption. Such an estimate can be applied also to the case of discrete periodic absorbers. Fore the reflection of cold atoms from ridged mirrors, the estimate gives good agreement with experiments
<ref name="zeno">
http://tori.ils.uec.ac.jp/PAPERS/optrevri.pdf (99k) 
D.Kouznetsov, H.Obrest. Reflection of waves from a ridged surface and the Zeno effect. -- Optical Review, 2005, v.12, No.5, p.363-366.
</ref>. The good guiding can be interpreted as multiple reflection, giving the same estimate for the absorption of the giuded mode
<ref name="morinaga">
http://tori.ils.uec.ac.jp/PAPERS/2012morinaga.pdf 
D.Kouznetsov, M.Morinaga. Guiding of waves between absorbing walls. Journal of Modern Physics, under consideration. 2012.
</ref>.

==References==
<references/>

==Keywords==
[[Zeno effect]],
[[Quantum reflection]],

[[Category:Optics]]
[[Category:Quantum mechanics]]
[[Category:Zeno effect]]
[[Category:Quantum reflection]]