Victor Barsan

*Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest, Magurele Romania* 

## **1. Introduction**

18 Will-be-set-by-IN-TECH

88 Trends in Electromagnetism – From Fundamentals to Applications

Saha, M. N. (1949). Note on Dirac's theory of magnetic monopoles. *Physical Review*, Vol. 75,

Schwinger, J.; Deraad Jr., L. L.; Milton, K. A.; Tsai, W. Y. & Norton, J. (1998). *Classical*

Thomson, J. J. (1904). *Elements of the Mathematical Theory of Electricity and Magnetism*,

Wilson, H. A. (1949). Note on Dirac's theory of magnetic poles. *Physical Review*, Vol. 75,

*Electrodynamics*, Westview Press, Cambridge.

Cambridge University Press, Cambridge.

(1968-1968).

(309-309).

In this chapter, we shall expose several analogies between oscillatory phenomena in mecahanics and optics. The main subject will be the analogy between propagation of electromagnetic waves in dielectrics and of electrons in various time-independent potentials. The basis of this analogy is the fact that both wave equations for electromagnetic monoenergetic waves (i.e. with well-defined frequency), obtained directly from the Maxwell equations, and the time-independent Schrodinger equation are Helmholtz equations; when specific restrictions - like behaviour at infinity and boundary conditions - are imposed, they generate similar eigenvalues problems, with similar solutions. The benefit of such analogies is twofold. First, it could help a researcher, specialized in a specific field, to better understand a new one. For instance, they might efficiently explain the fiber-optics properties to people already familiar with quantum mechanics. Also, even if such researchers work frequently with the quantum mechanical wave function, the electromagnetic modal field may provide an interesting vizualization of quantum probability density field [1]. Second, it provides the opportunity of cross-fertilization between (for instance) electromagnetism and optoelectronics, through the development of ballistic electron optics in two dimensional (2D) electron systems (2DESs); transferring concepts, models of devices and experiments from one field to another stimulate the progress in both domains.

Even if in modern times the analogies are not credited as most creative approaches in physics, in the early days of developement of science the perception was quite different. "Men's labour ... should be turned to the investigation and observation of the resemblances and analogies of things... for these it is which detect the unity of nature, and lay the fundation for the constitution of the sciences.", considers Francis Bacon, quoted by [1]. Some two centuries later, Goethe was looking for the "ultimate fact" - the Urphänomenon - specific to every scientific discipline, from botanics to optics [2], and, in this investigation, attributed to analogies a central role. However, if analogies cannot be considered anymore as central for the scientific investigation, thay could still be pedagogically useful and also inspiring for active scientific research.

Let us describe now shortly the structure of this chapter. The next two sections are devoted to a general description of the two fields to be hereafter investigated: metallic and dielectric waveguides (Section 2) and 2DESs (Section 3). The importance of these topics for the development of optical fibers, integrated optics, optoelectronics, transport

became a problem of interconnecting of two waveguides. So, the traditional optical circuit, composed of separate devices, carefully arranged on a rigid support, and protected against mechanical, thermal or atmospheric perturbations, has been replaced with a common

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 91

Electronic transport in conducting solids is generally diffusive. Its flow follows the gradient in the electrochemical potential, constricted by the physical or electrostatic edges of the specimen or device. So, the mean free path of electrons is very short compared to the dimension of the specimen. [8] One of the macroscopic consequences of this behaviour is the fact that the conductance of a rectangular 2D conductor is directly proportional to its width (*W*) and inversely proportional to its length (*L*). Does this ohmic behaviour remain correct for arbitrary small dimensions of the conductor? It is quite natural to expect that, if the mean free path of electrons is comparable to *W* or *L* - conditions which define the ballistic regime of electrons - the situation should change. Although the first experiments with ballistic electrons in metals have been done by Sharvin and co-workers in the mid '60s [9] and Tsoi and co-workers in the mid '70s [10], the most suitable system for the study of ballistic electrons is the two-dimensional electron system (2DES) obtained in semiconductors, mainly in the *GaAs* − *AlxGa*1−*xAs* heterostructures, in early '80s. In such 2DESs, the mobility of electrons are very high, and the ballistic regime can be easily obtained. The discovery of quantum conductance is only one achievement of this domain of mesoscopic physics, which shows how deep is the non-ohmic behaviour of electrical conduction in mesoscopic systems. In the ballistic regime, the electrons can be described by a quite simple Schrodinger equation, and electron beams can be controlled via electric or magnetic fields. A new field of research, the classical ballistic electron optics in 2DESs, has emerged in this way. At low temperatures and low bias, the current is carried only by electrons at the Fermi level, so manipulating with such

electrons is similar to doing optical experiments with a monochromatic source [11].

The propagation of ballistic electrons in mesoscopic conductors has many similarities with electromagnetic wave propagation in waveguides, and the ballistic electron optics opened a new domain of micro- or nano-electronics. The revealing of analogies between ballistic electrons and guided electromagnetic waves, or between optics and electric field manipulation of electron beams, are not only useful theoretical exercises, but also have a creative potential, stimulating the transfer of knowledge and of experimental techniques from one domain to

It is useful to begin the discussion of the analogies presented in this chapter with some very general considerations [12]. The most natural starting point is probably the comparison between the mechanical equation of motion of a mechanical oscillator having the mass *m* and

*dt*<sup>2</sup> <sup>+</sup> *kx*<sup>2</sup> <sup>=</sup> <sup>0</sup> (1)

*<sup>C</sup>* <sup>=</sup> <sup>0</sup> (2)

*m d*2*x*

> *<sup>L</sup> <sup>d</sup>*2*<sup>q</sup> dt*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*

and the electrectromagnetical equation of motion of a *LC* circuit [12]:

substrate where all the thin-film optical components are deposited [7].

**3. 2DESs and ballistic electrons**

another.

the stiffness *k*:

**4. Mechanical and electrical oscillations**

phenomena in mesoscopic and nanoscopic systems, is explained. In Section 4, some very general considerations about the physical basis of analogies between mechanical (classical or quantum) and electromagnetic phenomena, are outlined. Starting from the main experimental laws of electromagnetism, the Maxwell's equations are introduced in Section 5. In the next one, the propagation of electromagnetic waves in metalic and dielectric structures is studied, and the transverse solutions for the electric and magnetic field are obtained. These results are applied to metalic waveguides and cavities in Section 7. The optical fibers are described in Section 8, and the behaviour of fields, including the modes in circular fibers, are presented. Although the analogy between wave guide- and quantum mechanical- problems is treated in a huge number of references, the subject is rarely discussed in full detail. This is why, in Section 9, the analogy between the three-layer slab optical waveguide and the quantum rectangular well is mirrored and analyzed with utmost attention. The last part of the chapter is devoted to transport phenomena in 2DESs and their electromagnetic counterpart. In Section 10, the theoretical description of ballistic electrons is sketched, and, in Section 11, the transverse modes in electronic waveguides are desctibed. A rigorous form of the effective mass approach for electrons in semiconductors is presented in Section 12, and a quantitative analogy between the electronic wave function and the electric or magnetic field is established. Section 13 is devoted to optics experiments made with ballistic electrons. Final coments and conclusions are exposed in Section 14.
