**14. Conclusions**

22 Will-be-set-by-IN-TECH

In 2DESs, there exists a unique oportunity to control ballistic carriers via electrostatic gates, which can act as refractive elements for the electron path, in complete analogy to refractive elements in geometrical optics. [8], [25], [26]. The refraction of a beam of ballistic electrons can be simply described, using elementary considerations. If in the "left" half-space, the potential has a constant value *V*, and in the "right" one, a different (but also constant) one, *V* + Δ*V*, an

<sup>E</sup> <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*k*<sup>2</sup>

<sup>E</sup>� <sup>=</sup> <sup>E</sup> <sup>+</sup> *<sup>e</sup>*Δ*<sup>V</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*k*�<sup>2</sup>

Translational invariance along the interface preserves the parallel component of electron

*<sup>k</sup>* <sup>=</sup>

 *n*� *el nel*

Considering that the energies E, E� are Fermi energies, proportional (in a 2D system) to the electron densities *nel* (not to be confused with refractive index!), the Snell's law takes the form:

An electrostatic lens for ballistic electrons was set up in [8] and its focusing action was demonstrated in the GaAs / AlGaAs heterostructure. In this way, the close analogy between the propagation of ballistic electrons and geometrical optics has been put in evidence.

Another nice experiment used a refractive electronic prism to switch a beam of ballistic

The quantum character of ballistic electrons is clearly present, even they are regarded as beams of particles. Transmission and reflection of electrons on a sharp (compared to *λF*), rectangular barrier, induced via a surface gate at *T* = 0.5*K*, follow the laws obtained in quantum mechanics, for instance the following expression for the transmission coefficient:

> *<sup>k</sup> <sup>k</sup>*<sup>0</sup> <sup>−</sup> *<sup>k</sup>*<sup>0</sup> *k* 2

with *k*, *k*<sup>0</sup> - the wavevectors inside and outside the barrier, and *a* - its width (compare the

sin<sup>2</sup> (*ka*)

*<sup>T</sup>* <sup>=</sup> <sup>1</sup> 1 + 0.25

<sup>E</sup>�

sin *θ* sin *<sup>θ</sup>*� <sup>=</sup> *<sup>k</sup>*�

> sin *θ* sin *<sup>θ</sup>*� <sup>=</sup>

electrons between different collectors in the same 2DES [25]

previous formula with the results of III.I.7 of [13]). [27]

<sup>2</sup>*m*<sup>∗</sup> (154)

<sup>2</sup>*m*<sup>∗</sup> (155)

<sup>E</sup> (157)

(158)

, with the kinetic energy

*k* sin *θ* = *k*� sin *θ*� (156)

**13. Optics experiments with ballistic electrons**

emerges in the "right" half-plane under the angle *θ*�

momentum and thus

or

incident electron arriving with an incident angle *θ* and kinetic energy

Several analogies between electromagnetic and quantum-mechanical phenomena have been analyzed. They rely upon the fact that both wave equation for electromagnetic and electric field with well defined frequency, obtained from the Maxwell equations, and the time-independent Schrodinger equation, have the same form - which is a Helmholtz equation.

However, the description of these analogies is by no way a simple dictionary between two formalisms. On the contrary, their physical basis has been discussed in detail, and they have been developed for very modern domains of physics - optical fibers, 2DESs, electron waveguides, electronic transport in mesoscopic and nanoscopic regime. So, the analogies examined in this chapter offers the opportunity of reviewing some very exciting, new and rapidely developing fields of physics, interesting from both the applicative and fundamental perspective. It has been stressed that the analogies are not simple curiosities, but they bear a significant cognitive potential, which can stimultate both scientific understanding and technological progress in fields like waveguides, optical fibers, nanoscopic transport.
