# Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials

Peter Markoš
Costas M. Soukoulis
Edition: STU - Student edition
Pages: 376
https://www.jstor.org/stable/j.ctt7s2wj

1. Front Matter
(pp. I-IV)
(pp. V-VIII)
3. Preface
(pp. IX-XIV)
P. Markoš and C. M. Soukoulis
4. 1 Transfer Matrix
(pp. 1-27)

In this chapter we introduce and discuss a mathematical method for the analysis of the wave propagation in one-dimensional systems. The method uses the transfer matrix and is commonly known as thetransfer matrixmethod [7, 29].

The transfer matrix method can be used for the analysis of the wave propagation of quantum particles, such as electrons [29, 46, 49, 81, 82, 115–117, 124, 103, 108, 131, 129, 141] and of electromagnetic [39, 123, 124], acoustic, and elastic waves. Once this technique is developed for one type of wave, it can easily be applied to any other wave problem....

5. 2 Rectangular Potentials
(pp. 28-55)

The rectangular potential barrier, as shown in figure 2.1, represents one of the simplest quantum mechanical problems. We will use our transfer matrix formalism, developed in the previous chapter, to determine the transmission and reflection coefficients. Our transfer matrix results will be compared with those obtained with more traditional methods [7, 15, 25, 30]. We will show that the transfer matrix is easy to use and can be readily extended to more complicated shapes of potentials and to disordered systems.

Schrödinger’s equation is given by$-\frac{{{\hbar }^{2}}}{2m}\frac{{{\partial }^{2}}\Psi }{\partial {{x}^{2}}}+[V(x)-E]\Psi =0, \caption {(2.1)}$with a potential$V(x)=\left\{ \begin{array}{*{35}{l}} 0, & x\, \textless -a, \\ {{V}_{0}}, & -a\, \textless\, x \textless\, a, \\ 0, & a\, \textless\, x, \\ \end{array} \right. \caption {(2.2)}$and can be solved analytically; the solution can be found...

6. 3 δ-Function Potential
(pp. 56-73)

In physical applications, it is often useful to consider a simplified form of the rectangular potential, namely, theδ-function potential,$V(x)=\frac{{{\hbar }^{2}}}{2m}\Lambda \delta (x). \caption {(3.1)}$

The potential (3.1) can be obtained from the rectangular potential, defined by equation (2.2) in the limit of infinitesimally narrow barrier width, 2a→ 0, (3.2) and infinitesimally high barrier height,${{V}_{0}}=\frac{{{\hbar }^{2}}}{2m}\frac{\Lambda }{2a}\to \infty , \caption {(3.3)}$in such a way that the product 2a V0=ħ2Λ/(2m) is constant. The potential (3.1) represents either a potential barrier (Λ > 0) or a potential well (Λ < 0) [7, 15].

In this chapter, we will study first the transmission of a quantum particle through a single...

7. 4 Kronig-Penney Model
(pp. 74-97)

In section 3.4 we studied the transmission of a quantum particle (electron) throughNidenticalδ-function repulsive or attractive potentials. We calculated how the transmission coefficient depends on the parameterkℓ, whereis the distance between two neighboringδ-function potential barriers andkis the wave vector of the incident particle. We found intervals ofkℓin which the transmission coefficient decreases exponentially asNincreases and becomes infinitesimally small in the limit ofN→ ∞. These intervals were separated by other intervals in which the transmission coefficient, as a function ofkℓ, oscillates and is close to...

8. 5 Tight Binding Model
(pp. 98-119)

In this chapter, we introduce the most important ideas of electron propagation in periodic lattices, such as energy bands and gaps, the density of states, effective mass and group velocity of the electron, and the Fermi energy. We also derive the transfer matrix that enables us to find the energy of a bound state and to calculate the transmission of an electron through a system ofNparticles.

We begin by introducing and examining a very simple model, the so-calledtight binding model[11, 23, 75, 103, 115], defined by Schrödinger’s equation,$i\hbar \frac{\partial {{c}_{n}}}{\partial t}={{\varepsilon }_{n}}{{c}_{n}}+{{V}_{n}}{{c}_{n+1}}+V_{n}^{*}{{c}_{n-1}}. \caption {(5.1)}$

The tight binding model given by equation...

9. 6 Tight Binding Models of Crystals
(pp. 120-136)

In this chapter we study how the spatial periodicity of the system influences the structure of the energy spectrum. We introduce two tight binding models, the first one with a period= 2a, and the second one with a period= 4a. We show that the spectrum of the allowed energies changes considerably when the spatial period of the lattice increases. The energy band splits into subbands separated from each other by gaps [23, 39]. Also, the wave function does not have the simple form of a plane wave, but possesses more complicated spatial structure, known as Bloch...

10. 7 Disordered Models
(pp. 137-172)

In chapter 6 we studied the transmission of a quantum particle in an infinite periodic system. We found that the periodicity of the system creates bands and gaps in the energy spectrum. In the band, the particle moves freely throughout the sample for all allowed energies. This is due to the periodicity of the system, which enables successful interference of the back and forth scattered waves. In the band gap, there are no states at all and the transmission coefficient is zero.

We have also learned that a single impurity creates an isolated energy level which lies in the band...

11. 8 Numerical Solution of the Schrödinger Equation
(pp. 173-180)

The one-dimensional Schrödinger equation$-\frac{{{\hbar }^{2}}}{2m}\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}\Psi +V(x)=E\Psi (x) \caption {(8.1)}$can be solved analytically only for a few elementary problems [15]. In most of the applications, we have to find the transmission and the energy spectrum numerically. In this chapter, we describe the simplest numerical algorithm for solution of the Schrödinger equation, and discuss the accuracy of the results obtained. We describe a simple numerical algorithm that enables us to treat various scattering problems numerically. Applying this algorithm to the simplest problem—that of the free particle—enables us to discuss the accuracy of the numerical algorithm and to estimate the numerical error of our...

12. 9 Transmission and Reflection of Plane Electromagnetic Waves on an Interface
(pp. 181-204)

In this chapter, we will investigate the very basic phenomena of transmission and reflection of electromagnetic wave propagating through the interface between two media. From the requirements of the continuity of the tangential components of the electric and the magnetic fields, we derive the transfer matrix for a single interface between two media. Its elements determine the transmission and reflection amplitudes for both electric and magnetic fields.

Next, we study the behavior of the electromagnetic waves incident on the surface of a dielectric and a metal. We learn how different electromagnetic properties of these materials influence the transmission and reflection...

13. 10 Transmission and Reflection Coefficients for a Slab
(pp. 205-224)

Consider now a slab of thicknesswith permittivityε2and permeabilityμ2, located between two semi-infinite media with electromagnetic parameters (ε1,μ1) and (ε3,μ3), respectively. We want to calculate transmission and reflection amplitudes for a plane wave arriving from the left for both TE and TM polarizations. As in section 9.2, we assume that the permittivity and permeability of incoming and outgoing media are real. This might not be true for the parameters of the slab.

Transmission through a planar slab is schematically shown in figure 10.1. We see that the problem is more complicated than that of...

14. 11 Surface Waves
(pp. 225-242)

In this chapter we study an interesting phenomenon, namely, the excitation of surface waves. We will see that for an appropriate choice of the electromagnetic parameters an interface between two media can support the excitation of surface waves [2, 58]. Surface waves can propagate along the interface and decay exponentially as a function of the distance from the surface, shown in figure 11.1. This phenomenon has no analogy in quantum physics.

We analyze first the surface waves on a single interface between two media. We find that surface waves propagate only along the interface separating two media with opposite signs...

15. 12 Resonant Tunneling through Double-Layer Structures
(pp. 243-248)

In chapter 10 we learned that the transmission through a dielectric slab is determined by the relation of the slab thickness to the wavelength of the electromagnetic wave inside the slab. In particular, the transmission coefficient is close to 1 when the wavelength of the electromagnetic wave in thez-direction is proportional to even multiples of the slab thickness.

Now, we will study the transmission of the electromagnetic wave through a system of two slabs, embedded in homogeneous material, as shown in figure 12.1 [39]. We concentrate on the case when there is no wave propagation in the layera,...

16. 13 Layered Electromagnetic Medium: Photonic Crystals
(pp. 249-274)

Previously, in chapters 9 and 10, we analyzed the transmission of electromagnetic waves through a single interface and through a thin slab of width. In chapter 12 we learned that the transmission through two slabs of the same material leads to resonant transmission, even if resonant transmission through one layer is not possible. This indicates that more complicated structures could have new transmission properties not observable in single components of the structure. To investigate this problem in more detail, in this chapter we will apply the transfer matrix formalism to analysis of the transmission coefficient through layeredperiodicmedia,...

17. 14 Effective Parameters
(pp. 275-285)

Up to now, we have analyzed the transmission of electromagnetic waves in homogeneous media. The only inhomogeneities were given by the interfaces between two homogeneous materials. We assumed that the distance between two adjacent interfaces is larger than, or at least comparable to, the wavelength of the propagating electromagnetic wave.

Now we will analyze structures that possess inhomogeneities much smaller than the wavelength. A simple example of such a structure is the layered medium shown in figure 13.2, where the thickness of each slab,aandb, is much smaller thanλ. In such a case, the propagating electromagnetic wave...

18. 15 Wave Propagation in Nonlinear Structures
(pp. 286-297)

In this chapter, we will investigate the nonlinear response of wave propagation in one-dimensional structures.

When dielectric materials are arranged periodically, electromagnetic waves at some frequencies are forbidden to propagate. This was discussed in detail in chapter 13. Most of the interest in multilayer structures focuses on thelinearregime in which the dielectric constant is independent of the field strength. However, the presence of opticalnonlinearityin a system leads to a much richer and more complex response to radiation. We will see that the transmission coefficient is a function of the intensity of the incoming electromagnetic wave. This...

19. 16 Left-Handed Materials
(pp. 298-320)

In this chapter, we summarize the electromagnetic properties of so-called left-handed materials. This name was given to man-made composites that possess, in a certain frequency region, negative real parts of both the permittivity and permeability.

We have discussed some properties of left-handed materials in previous chapters. In chapter 9, we found that an interface between a vacuum and a left-handed medium might allow perfect transmission. In chapter 11, we analyzed in detail the existence of surface electromagnetic waves localized at the interface between vacuum and a left-handed material. In chapter 13, we used left-handed materials in construction of infinite layered...

20. Appendix A Matrix Operations
(pp. 321-326)
21. Appendix B Summary of Electrodynamics Formulas
(pp. 327-340)
22. Bibliography
(pp. 341-348)
23. Index
(pp. 349-352)