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We study the two-dimensional above-barrier penetration and the sub-barrier tunneling of non-relativistic particles and photons, described in the quasi-monochromatic approximation by simple plane waves. Our scheme represents the motion from the left free-motion zero-potential region to the right zero-potential region through the intermediate region with a one-dimensional rectangular potential barrier along the axis, normal to the both parallel interfaces between all three regions, and with the zero potential along the axis, parallel to the those interfaces. We have firstly obtained the analytical expressions for the infinite series of multiple internal and external reflections and also of multiple transmitted waves of particles and photons, with equal shifts between them along the interfaces for the above-barrier penetration and with various shifts between them in the case of the sub-barrier tunneling. Finally the Hartman and Fletcher effect for any transmitted wave is established.

The one-dimensional (1-D) non-relativistic-particle and photon penetration and tunneling through a potential barrier had been studied in many papers in the stationary and non-stationary descriptions (for instance, in [1-6]; see also a lot of the relevant refs therein). However, there are not very much papers with analysis of multiple internal reflections during tunneling (see, for instance, [7-11] for 1-D tunneling and [

(1)

for non-relativistic particles in the quasi-monochromatic (and initially stationary) approximation, where is the stationary wave function for a particle, m is its mass, ħ is Planck’s constant divided by 2π, is its potential (barrier), and E is its total energy. The space regions I, II and III are defined as the regions with zero potentials (I for, and III for,) and the space region III contains the barrier, (,), all three regions being infinite along the y - axis (parallel to the interfaces between I and II and also between II and III), and due to the translation symmetry the has the same y— dependence in all three regions ( permanently zero potential along the y-axis). We neglect the boundary effects in the regions with large due to the infiniteness of all three regions along the y axis. Then, using the particle-photon similarity, established in [3-5,13], we study the behavior of photons, propagating in isotropic glass media I and III, penetrating or tunneling through the isotropic air layer II. Further we briefly discuss an alternative too much simplified approach from [

In the simple 2-D geometrical schemes (Figures 1-3) all plane waves in regions I and III are represented, in a usual way for stationary pictures, by straight lines with arrows^{1}. At the bottom of

Firstly we analyze the above-barrier penetration with.

At point () the first reflected plane wave

(where is the amplitude of the firstly externally reflected wave from the left interface into I,) and the firstly penetrated (into II) plane wave (where is the amplitude of the firstly penetrated wave, ,) appear. Further, at the first exit point (x = a, y = Dy), Dy being the first shift upwards in II due to the motion with k_{y }along the y axis, the firstly transmitted plane wave

(where is the amplitude of the firstly transmitted (into II) wave) and the firstly internally reflected wave (where is the amplitude of the firstly internally reflected (into II) wave,) appear. Here Dy can be evidently evaluated as

(2a)

or

where is the phase time^{2} of particle moving with the velocity along the distance a (i.e., the time for a quasi-monochromatic particle to transfer the interval from x = 0 to x = a along the x axis inside II evaluated in the stationary-phase approximation).

Successively, at point (,) the second penetrated (into II) wave, or the second internally re-

flected (from the left interface II) wave (where is the amplitude of the second penetrated (into II) wave, or, which is the same, of the second internally reflected (from the left interface into II) wave) and the second externally reflected (into I) wave (where is the amplitude of the second externally reflected (into I) wave) appears. And so on (it can be continued up to any n-th externally reflected (into I) wave,).

From the matching conditions of the waves and their first derivatives at points (), (,), (,), (,), , we obtain, considering that we can neglect the plane waves due to the translation symmetry in the both interfaces, that

(3)

(4)

(5)

(6)

(7)

For, when (see

and

due to the flux conservation in the first passing through points () and (). By the way, here we have evidently generalized (practically repeated) the introducing of multiple internal reflections from [10,11] for our simple scheme of 2-D penetration. And in the case of 1-D penetration (namely when (see

Now let analyze the sub-barrier tunneling with E_{x }< V_{0}. We assume that the angle q is sufficiently large

(, where is defined by equality), so that and is imaginary, i.e. with and. So, in this case we have the under-barrier tunneling. In this case, instead of the above-barrier penetration, which is described by formulas (3)-(7), in order to describe the sub-barrier tunneling, we have to insert c instead, utilizing relation. So, instead of the propagating in II waves, we have the evanescent and anti-evanescent waves, and instead of the amplitudes and of the propagating waves we have the coefficients and

the evanescent and anti-evanescent waves , respectively. The correspondent scheme is represented by

And instead of the shift Dy along the y axis, defined for the above-barrier penetration by Equations (2a) and (2b) and represented in

(2c)

where

(8)

and

(9)

are the phase times (i.e., times for quasi-monochromatic particle evaluated in the stationary-phase approximation —see, for instance, [1,3] and refs therein) of the n-th step for sub-barrier tunneling through the point x = a and of the n-th step for the external reflection from the first barrier wall in the point x = 0, respectively. Of course, the shifts with the different values of are different (slightly numerically growing for the growing numbers n, but always being proportional to in the limit), and also the transmitted and externally reflected waves are quickly damping with the final vanishing, due to presence of the evanescent-wave factors of the growing order in the expressions for and with the growing number n.

In [_{x}-component inside the region II the only one usual linear combination of evanescent and anti-evanescent waves and for the k_{y}-component inside the region II only one propagating wave, and it was obtained the following expression for the only one shift along y axis at the second interface (between II and III)

which is represented in

where

,

,

,

.

And, consequently, here one obtains the only one transmitted (into region III) 2D propagating wave which moves in a parallel way to the incident wave.

So, here we compare two different approaches for 2-D sub-barrier non-relativistic-particle tunneling. Our approach, which generalizes [

Of course, it remains to confirm experimentally which of the approaches will be real for the sub-barrier tunneling. Up to now only our approach is verified by several methods, described in [

Now, starting from the strict particle-photon similarity, formulated in [2,4,5,13] (see also revelant refs therein), we can extend the established in the previous section 2 (for particles) results for the case of photon 2-D penetration and tunneling. One can see that Figures 1-3 can be also applied for photons, propagating in isotropic glass media I and III, penetrating or tunneling through the isotropic air layer II. In this case the quantity

is the index of light refraction in the glass (taking the index of light refraction in the air as 1), and

As to the light, Figures 2 and 3 can in this relation describe the frustrated total internal reflection (FTIR) of the s-polarized light tunneling through the layer II for the incident angles q > q_{crit} (frustrated-in the sense of the partial transmission through the layer II into the glass media III). They describe it differently, really in accordance with the cardinally different approaches: either by the extension of the 2D non-relativistic-particle tunneling with multiple internal reflections presented above here and also earlier in a slightly different form for light in [

1) Starting from the theoretical analysis, elaborated in [

2) But if one starts from the physical analysis, described in [

3) For the concluding analysis of such cardinal divergence between these two approaches it will be instructive to undertake decisive thorough and precise experiments for a clear description of the above-barrier penetration and sub-barrier tunneling.

4) And also it will be rather interesting to research experimentally the possibility of the photon superluminal group velocities in the parallel transmitted and externally reflected parallel propagation lines connected with the Hartman and Fletcher effect during the sub-barrier tunneling, generalizing the results of 1-D photon tunneling, described in reviews [1,3] (including refs to the experimental results).

5) In the future we intend to present additionally the results of the numerical study of 2D Gaussian wave packets incident to the first interface normally and at the angle q to the axis, normal to the interface.

We are thankful for the stimulating discussions to Prof. E. Recami, Prof. R. Mignani and Doctoressa V. Petrillo and to Dr. S. A. Maydanyuk for the useful discussions.