The consideration of reflection of waves from absorbing walls had been stimulated by the experiments with ridged mirrors [1] and their interpretation in terms of the Zeno effect [2,3]. The Zeno approximation [2] showed good agreement with experiments in wide range of parameters [3,4]; it describes reflection of waves of any origin. In particular, it applies to the atomic waves and to the optical waves. 

In addition to the atom optics (discussed in [2]), the reflection and guiding of waves may affect the suppression of the Amplified Spontaneous Emission (ASE) that is considered as serious problem [5-9]. At the scaling-up the power, the efficient suppression of ASE becomes more important, and the unwanted guiding of ASE by the absorber (which is supposed to suppress it) might take place. The estimates of the conditions of such a guiding is necessary tool for the design of powerful devices.

In this article, the wave function of a particle (atom, photon) between two absorbing walls is constructed. The effective absorption of such a mode is estimated and compared to the previous results. The results are expected to have applications in both the atom optics (wanted guiding of neutral particles by the absorbing walls) and the design of powerful lasers (estimates of conditions of the unwanted guiding of the ASE by the walls that are supposer to absorb the ASE, and optimization of suppression of the ASE).

This article is organized in the following way:

In section 2, the phenomenological absorbing Schrö- denger equation is suggested. The case with absorbing walls corresponds to the pure anti Hermitian potential. The frequency of decay is denotes with. The special system of units is used in such a way that and the energy of the particle in vacuum is assumed to be square of its wavenumber.

In section 3, the special case of uniform absorption is considered; this gives relations between parameters of the wave function and the physical quantities that can be determined experimentally. Such physical quantities are energy of the particle and its absorpfion in the material of the wall; this quantities are assumed to be independent parameters.

In section 4, the case of channeling is considered; parameters of the principal mode (transversal wavenumber, decay and the damping) are defined and expressed through the holomorphic function , its inverse function and; properties of these functions are discussed and the efficient C++ implementations are indicated.

In section 5, the example of the principal mode is considered; the real and imaginary parts of the wave functions are built through the trigonometric function of complex argument and complex exponential for the damping. The amplitude and phase of the principal mode are shown for and for .

In section 6, the asymptotic behavior of parameters at small damping is considered; this is realistic case of good guiding conditions.

In section 7, the estimates for the effective absorption of the principal mode are compared with the previous results; the agreement is interpreted as confirmation of the validity of the analysis.

In section 8, the special case is suggested for physically realistic parameters of the experimental conditions of realization of the guiding of waves between absorbing walls.

In section 9 (Conclusion), the asymptotic estimate of the effective absorption of the guided mode through the real and imaginary parts, of the wavenumber in the wall is suggested. This main result is expected to be confirmed (or rejected) by the physical experiments with photons, atoms or any other wave of any origin.

In this case the dimension less Schrödinger equation is considered in the paraxial approximation.

For simplicity, in this section the special system of units is used such that and mass of the atom is half. Then, the Schrödinger equation for the wave function can be written as follows

where the potential depends only on the transversal coordinate (and does not depend on time in the direction of propagation).

In the simplest approximation, the entangling with numerous degrees of freedom of scattered (or relaxed) atom can be taken into account with non Hermitian potential. Such an approximation is considered, in particular, in the interpretation of the quantum reflection in therms of the Zeno effect [2]. For the wave guide, the absorption correspond to complex values of at, where is half-width of the channel between the mirrors.

Consider the quasi-monochromatic solution, assuming the exponential dependence on; let 

Then, instead of (1) we get the stationary Schrödinger equation 

below, the two cases of the solution are considered: for 

and for

where is the conventional unit step function implemented in various programming languages including Mathematica, and is constant, that has sense of half-width of the channel that confines the particle.

In order to understand the physical sense to the constant, consider the case of the uniform absorption of the plane wave at by (4); let does not depend on the first argument, id est,. Then, for we get the equation

At positive values of, the decaying in the direction solution has form 

where is solution of equation

Assuming, , wavenumber can be expressed as follows: 

For the compactness of notations, let, where and are real parameters. Then 

Alternatively, we may consider the absorption of wave in the medium, id est, as initial parameter. Making estimates for the ridged atomic mirror with distance between ridges, the absorption by intensity can be approximated as, and the absorption by amplitude is of order of 

Real part of is determined by the mass and the energy of the particle we intent to reflect or to guide, or just, where is vacuum wavelength of the waves (perhaps, ASE) that could be guided, and is the refraction index of the medium.

In order to avoid additional guiding by the step of the refraction index, the real part of the refraction index of the wall is supposed to be matched to that of the central region; but some reflection still may happen due to the imaginary part. Then, the real and imaginary parts of the swiare o wavenumber can be expressed as follows: 

Parameter has sense of the energy of the particle, and also sense of the square of wavenumber, while has sense of the decay rate at the time scale, if a wave would be uniform in the space. For the estimates, the term with in (13) can be neglected; however it is kept in the deduction that may be applied not only to atoms, but also to other kinds of waves (optical, acoustical, waves on the surface of a liquid, etc.) For optics (both atom optics and conventional optics), the typical case is of low absorption; waves rather propagate than absorb, to,

In this case, for the estimate of the primary parameters, we may use the approximation 

and treat and as initial parameters of the model describing the absorption of the wave inside the walls.

On the other hand, one may consider as “given” the energy of the particle and the absorption. Then,

In the following consideration, parameters and and are supposed to be known. These parameters determine behavior of the wave inside the absorbing wall.

For the atom optics, id est, for the atom wave, the absorption should be positive. For the optical wave, in principle, the absorption may be negative (gain medium), but the amplified spontaneous emission (unavoidable in the gain medium) limits the application of the formalism to very short distance of propagation. For this reason, the channeling in the pumped region may have more applications. In such a way, this section gives the sense to the parameters and that appear in the equations (3) and (5) , that describe the channeling of a particle by the potential by (5). This channeling is considered in the next section. Then, the effective propagation constant is estimated in terms of and half-width of the channel.

For the case of potential by (5), search the solution in the following form:

where is constant.

For the experimental realization, is expected to be of order of, determined in the previous section. As for the imaginary part, is expected to be small compared to, and is expected to decrease as increases, allowing the interpretation in terms of the Zeno effect [2]: the stronger is the absorption in the region of the “observation” (id est,), the better is the channeling.

Substitution of (19) to (3) gives the equation for in the following form: 

Search the solution of (20) as the combination of the cosinusoidal and the exponential, let 

where, , , are constant parameters. From the physical reasons (almost free propagation inside the channel), parameter is expected to be of order of wavenumber from the previous section.

The substitution of (21) into (20) determines that

From the physical reasons, it is expected that is small compared to and, therefore,.

The continuity of at and the continuity of give the relations 

The four equations (22), (23), (24), (25) allows to express new parameters in terms of the already defined parameters and.

Subtraction of (23) from (22) gives 

Dividing of (25) by (24) gives

The combination of (26) and (27) gives 

Using the relation, equation (28) can be written as 

then

This equation can be written as follows: 

where for all complex,

Complex map of function cosc is shown in the left hand side of figure 0; properties of this function and its inverse function are described in TORI [10]. The name of function cosc is chosen in analogy with well established name of function sinc [11], defined with

As usually, the name of the inverse function is created adding prefix “a” or “arc”.

Equation for one of solutions with can be written as follows:

Equation (34) can be “inverted”, giving

where is inverse function of cosc. Damping is dimensionless parameter determining the efficiency of channeling,

Properties of function acosc are described at TORI; the efficient implementation in C++ is suggested [10]. The complex map of acosc is show in the right hand side of figure 1 with levels of constant real part and levels of constant imaginary part. Behavior of real and imaginary parts of function acosq of real argument is shown in fthe left hand side of figure 2.

The decay rate of mode in the region with absorption is determined with parameter. Once is determined, from equation (27),

Function acosqq is shown in the right hand part of figure 2.

The damping by (36) determines the properties of the mode with given distance between walls and given real and imaginary parts of the wavenumber that determines the propagation of wave in the material of the wall. With tools defined above, the principal mode of wave guided between absorbing walls is expressed with equation (21). Parameters, and are defined with equations (35), (37), (24).

As an example of the assembling of such a mode, the case is presented in figure 3; the real and imaginary parts of the components of function are plotted versus dimensionless product. In this case,

The amplitude and phase of the mode for and are shown in figure 4. for, the parameters are

With functions ArcCosq and ArcCosqq implemented in TORI through [10], one can easy assemble the principal mode for other values of the damping parameter with minimal modification of the codes supplied there.

The effective absorption of a guided mode is one of most important parameters of any waveguide. This section consider the case of low damping and, correspondently, strong channeling.

According to (19) the effective absorption is determined by parameter, 

From equation (22), 

The real and imaginary parts of determine the effective wavenumber and absorption of the guided mode “exactly” in the mathematical sense. As for the physical applications, the case of strong guiding (and low absorption) is of interest. This case corresponds to the small values of the damping parameter, and the asymptotic behavior of the absorption of the guided mode is considered in this section.

For the strong guiding, the propagation constant can be expanded as follows: 

then, the absorption of the mode can be expressed as follows:

In order to provide the flux of probability from the center of mode to the absorbing walls, the imaginary part of the transversal wavenumber should be negative,. The expansion of funciton acocq at zero gives:

This gives the approximation for the transversal wavenumber in the following form: 

and the estimate for the effective absorption 

Then, by (36) and should be used, giving. Then, the effective absorption of the mode 

where has sense of wavenumber, and is the absorption in the wall.

In the similar way, the highest modes can be constructed. For the mth transversal mode, the transversal wavenumber scales proportionally to; and the absorption of mode scales proportionally to.

The absorption of mode can be interpreted also in terms of the multiple reflection of guided wave from the walls. The coefficient of reflection is estimated in the description of the ridged mirrors in terms of the Zeno effect [2],

where

is wavenumber and can be replaced to; while is distance between idealized absorbers that can be approximated as, and is the grazing angle. (Notation of [2] is not used here, to keep letter denoting the transversal wavenumber; so, in (52) and (53) , notation is used instead.)

For the good channeling conditions, the effective absorption can be approximated with 

where is half-width of the channel and is ratio of the transversal wavenumber 

to the wavenumber. At the reflection of wave from a ridged mirror, plays role of the grazing angle.

Substitution of (52) and (55) into (54) gives the following expression for the absorption 

The grazing angle can be approximated with 

giving the estimate for the efficient absorption 

For the comparison to the previous result, should be replaced to and should be replaced to, giving the absorption by probability 

This expression should be compared to (51).

In the first approximation, the consideration of the multiple reflection from absorbing walls and the consideration of mode guided between the absorbing walls give the same prediction about effective absorption of this mode. The consideration of the multiple reflection from absorbing walls and the consideration of mode guided between the absorbing walls give the same prediction about effective absorption of this mode.

Consider the application of the estimate (59) for the guiding of the realistic laser beam. Assume, the absorption in the walls 

Following the ideology of the Zeno interpretation of absorbing walls [2], such an absorption may be approximated with series of slits separated by distance 35 mm.

Let the wavenumber is 

Let the halfwidth of the channel

This gives the estimate for the absorption of guided modes,

For the principal mode (), after to propagate distance, the attenuation factor is of order of

that means, that the most of the initial power of the guided mode is still delivered. As for the second mode, its attenuation 

that means significant dicrimination of the second mode.

Using the approximation of the set of absorbers as a continuous medium [2], the example above may correspond to transfer of near infra-red light through the set of 10 slits separated with distance 35 mm. Roughly, the amplitude of field after the set of slits can be approximates with the cosinusoidal profile. However, the presence of the highest modes, as well as the diffraction of the tails of the mode on the edges should make the similarity qualitative. Similar result one may expect to observe at propagation of light through the set of pinholes of radius. The similarity with the idealized cosinusoidal or Besseliean mode should improve at the increase of number of slits or pinholes; a hundred of silts or pinholes may be sufficient to get the quantitative agreement with the idealized cosinusoidal or Besselian profile.

The accurate consideration of the discrete character of the absorbing walls, as well as construction of the mode for the case with circular symmetry may be continuation of this work. For the paraxial case, the estimates are universal and are not sensitive to the origin of waves. In particular, the results are expected to apply to the electromagnetic waves as well as to the cold atoms, exhibiting the wave properties.

Guiding of wave of any origin between absorbing walls is considered. Wavenumber and the amplitude absorption of wave in the wall, and the half-width of the channel are considered as given parameters.

The dimensionless damping parameter by (36) is suggested to characterize the scale of the effect.

The first (principal) mode (21) with lowest absorption is explicitly constructed. The transversal wavenumber of the mode is expressed through the function acosc of complex argument; properties of this function are described and the numerical implementation is supplied [10]. The propagation constant is expressed with equation (45); the asymptotic estimate (51) of the absorption of the mode is suggested. The estimate agrees with that on the base of the Zeno reflection of the waves from the absorbing medium reported earlier [2].

The estimates above are important in the design of the suppression of the amplified spontaneous emission (ASE) in the high power lasers. The guiding of modes by the absorption walls happens whenever the engineers want this effect or not. Similar estimate is valid for the highest modes. For the mth transversal mode, the asymptotic estimate is suggested for the absorption 

through the half-width of the channel, wavenumber and absorption of wave in the walls.

Similar estimate (with slightly higher absorption) correspond to the case with circular symmetry, that can be treated in the similar way; the mode is expressed with the Bessel function, parameter plays role of the radius of the channel. At small value of damping by (36), the transversal wavenumber is almost real.

The result should be useful in both, wanted guiding of cold neutral particles by their detection (absorption) and the efficient suppression of the unwanted guiding of waves, for example, ASE in powerful optical amplifiers, and optimization of the ASE absorbers.

Authors are grateful to Dr. Hilmar Oberst and Prof. Fujio Shimizu and Prof. Kazuko Shimizu for the collaboration.