**2. Metallic and dielectric waveguides; optical fibers**

Propagation of electromagnetic waves through metallic or dielectric structures, having dimensions of the order of their wavelength, is a subject of great interest for applied physics. The only practical way of generating and transmitting radio waves on a well-defined trajectory involves such metallic structures [3]. For much shorter wavelengths, i.e. for infrared radiation and light, the propagation through dielectric waveguides has produced, with the creation of optical fibers, a huge revolution in telecommunications. The main inventor of the optical fiber, C. Kao, received the 2009 Nobel Prize in Physics (together with W. S. Boyle and G. E. Smith). As one of the laurees remarks, "it is not often that the awards is given for work in applied science". [4]

The creation of optical fibers has its origins in the efforts of improving the capabilities of the existing (at the level of early '60s) communication infrastructure, with a focus on the use of microwave transmission systems. The development of lasers (the first laser was produced in May 16, 1960, by Theodore Maiman) made clear that the coherent light can be an information carrier with 5-6 orders of magnitude more performant than the microwaves, as one can easily see just comparing the frequencies of the two radiations. In a seminal paper, Kao and Hockham [5] recognized that the key issue in producing "a successful fiber waveguide depends... on the availability of suitable low-loss dielectric material", in fact - of a glass with very small < 10−<sup>6</sup> concentration of impurities, particularly of transition elements (Fe, Cu, Mn). Besides telecommunication applications, an appropriate bundle of optical fibers can transfer an image - as scientists sudying the insect eye realized, also in the early '60s. [6]

Another domain of great interest which came to being with the development of dielectric waveguides and with the progress of thin-film technology is the integrated optics. In the early '70s, thin films dielectric waveguides have been used as the basic element of all the components of an optical circuit, including lasers, modulators, detectors, prisms, lenses, polarizers and couplers [7]. The transmission of light between two optical components 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 substrate where all the thin-film optical components are deposited [7].

#### **3. 2DESs and ballistic electrons**

2 Will-be-set-by-IN-TECH

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

Propagation of electromagnetic waves through metallic or dielectric structures, having dimensions of the order of their wavelength, is a subject of great interest for applied physics. The only practical way of generating and transmitting radio waves on a well-defined trajectory involves such metallic structures [3]. For much shorter wavelengths, i.e. for infrared radiation and light, the propagation through dielectric waveguides has produced, with the creation of optical fibers, a huge revolution in telecommunications. The main inventor of the optical fiber, C. Kao, received the 2009 Nobel Prize in Physics (together with W. S. Boyle and G. E. Smith). As one of the laurees remarks, "it is not often that the awards is given for work

The creation of optical fibers has its origins in the efforts of improving the capabilities of the existing (at the level of early '60s) communication infrastructure, with a focus on the use of microwave transmission systems. The development of lasers (the first laser was produced in May 16, 1960, by Theodore Maiman) made clear that the coherent light can be an information carrier with 5-6 orders of magnitude more performant than the microwaves, as one can easily see just comparing the frequencies of the two radiations. In a seminal paper, Kao and Hockham [5] recognized that the key issue in producing "a successful fiber waveguide depends... on the availability of suitable low-loss dielectric material", in fact - of a glass with

Mn). Besides telecommunication applications, an appropriate bundle of optical fibers can transfer an image - as scientists sudying the insect eye realized, also in the early '60s. [6]

Another domain of great interest which came to being with the development of dielectric waveguides and with the progress of thin-film technology is the integrated optics. In the early '70s, thin films dielectric waveguides have been used as the basic element of all the components of an optical circuit, including lasers, modulators, detectors, prisms, lenses, polarizers and couplers [7]. The transmission of light between two optical components

concentration of impurities, particularly of transition elements (Fe, Cu,

are exposed in Section 14.

in applied science". [4]

very small

< 10−<sup>6</sup>

**2. Metallic and dielectric waveguides; optical fibers**

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 another.

#### **4. Mechanical and electrical oscillations**

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 the stiffness *k*:

$$m\frac{d^2\mathbf{x}}{dt^2} + k\mathbf{x}^2 = \mathbf{0} \tag{1}$$

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

$$L\frac{d^2q}{dt^2} + \frac{q}{C} = 0\tag{2}$$

More exactly,

More exactly,

space variation of **H**:

of the vectors **E**, **H**.

the form [3]:

permeability *μ*,

where <sup>∇</sup><sup>2</sup>

direction and to assume

∇ × **E** = −*μ*

(*�***E**) is connected with *<sup>∂</sup>***<sup>H</sup>**

∇ × **H** =*�*

**6. Propagation of electromagnetic waves in waveguides and cavities**

 =

**E** (*x*, *y*, *z*, *t*) **B** (*x*, *y*, *z*, *t*)

> ∇2 *<sup>t</sup>* +

*<sup>t</sup>* is the transverse part of the Laplacian operator:

The wave equation is reduced to two variables:

assuming that no free chages or electric current are present - a natural assumption for our approach, as we shall use Maxwell's equations only for studying the wave propagation. In this context, the only role played by (5) and (6) will be to demonstrate the transverse character

The propagation of electromagnetic waves in hollow metalic cylinders is an interesting subject, both for theoretical and practical reasons - e.g., for its applications in telecommunications. We shall consider that the metal is a perfect conductor; if the cylinder is infinite, we shall call this metallic structure *waveguide*; if it has end faces, we shall call it *cavity*. The transversal section of the cylinder is the same, along the cylinder axis. With a time dependence exp (−*iωt*), the Maxwell equations (5)-(8) for the fields inside the cylinder take

For a cylinder filled with a uniform non-dissipative medium having permittivity *�* and

The specific geometry suggests us to single out the spatial variation of the fields in the *z*

*μ�ω*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

*<sup>t</sup>* <sup>=</sup> <sup>∇</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

It is convenient to separate the fields into components parallel to and transverse the *oz* axis:

∇2

 **E B** 

<sup>∇</sup><sup>2</sup> <sup>+</sup> *μ�ω*<sup>2</sup>

*∂ ∂t*

The other one expresses Ampere's law [12], relating that the time variation of *�***E** defines the

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 93

*∂***H**

*∂***E**

∇ × **E** = *iω***B**, ∇ · **B** = 0, ∇ × **B** = −*iμ�ω***E**, ∇ · **E** = 0 (9)

**<sup>E</sup>** (*x*, *<sup>y</sup>*) exp (±*ikz* <sup>−</sup> *<sup>i</sup>ωt*)

 **E B** 

**<sup>E</sup>** <sup>=</sup> **<sup>E</sup>***z*+**E***t*, with **<sup>E</sup>***<sup>z</sup>* <sup>=</sup> **z***Ez*, **<sup>E</sup>***<sup>t</sup>* <sup>=</sup> (**<sup>z</sup>** <sup>×</sup> **<sup>E</sup>**) <sup>×</sup> **<sup>z</sup>** (14)

*<sup>∂</sup><sup>z</sup>* (say)

*<sup>∂</sup><sup>t</sup>* (7)

*<sup>∂</sup><sup>t</sup>* (8)

= 0 (10)

= 0 (12)

*<sup>∂</sup>z*<sup>2</sup> (13)

**<sup>B</sup>** (*x*, *<sup>y</sup>*) exp (±*ikz* <sup>−</sup> *<sup>i</sup>ωt*) (11)

which provides immediately an analogy between the mechanical energy:

$$\frac{1}{2}m\left(\frac{d\mathbf{x}}{dt}\right)^2 + \frac{1}{2}kx^2 = \mathcal{E}\tag{3}$$

and the electromagnetic one:

$$\frac{1}{2}L\left(\frac{d\boldsymbol{q}}{dt}\right)^2 + \frac{1}{2}\frac{q^2}{\mathbb{C}} = \mathcal{E} \tag{4}$$

The analogy between these equations reveals a much deeper fact than a simple terminological dictionary of mechanical and electromagnetic terms: it shows the inertial properties of the magnetic field, fully expressed by Lenz's law. Actually, magnetic field inertia (defined by the inductance *L*) controls the rate of change of current for a given voltage in a circit, in exactly the same way as the inertial mass controls the change of velocity for a given force. Magnetic inertial or inductive behaviour arises from the tendency of the magnetic flux threading a circuit to remain constant, and reaction to any change in its value generates a voltage and hence a current which flows to oppose the change of the flux. ([5], p.12) Even if, in the previous equations, the mechanical oscillator is a classical one, its deep connections with its quantum counterpart are wellknown ([13], vol.1, Ch. 12). Also, understanding of classical waves propagation was decissive for the formulation of quantum-wave theory [13], so the classical form of (1) and (3) is not an obstacle in the development of our arguments.

These basic remarks explain the similarities between the propagation of elastic and mechanical waves. The velocity of waves through a medium is determined by the inertial and elastical properties of the medium. They allow the storing of wave energy in the medium, and in the absence of energy dissipation, they also determine the impedance presented by the medium to the waves. In addition, when there is no loss mechanism, a plane wave solution will be obtained, but any resistive or loss term, will produce a decay with time or distance of the oscillatory solution.

Referring now to the electromagnetic waves, the magnetic inertia of the medium is provided by the inductive property of that medum, i.e. permeability *μ*, allowing storage of magnetic energy, and the elasticity or capacitive property - by the permittivity *�*, allowing storage of the potential or electric field energy. ([12], p.199)

#### **5. Maxwell's equations**

The theory of electromagnetic phenomena can be described by four equations, two of them independent of time, and two - time-varying. The time-independent ones express the fact that the electric charge is the source of the electric field, but a "magnetic charge" does not exist:

$$\nabla \cdot (\epsilon \mathbf{E}) = \rho \tag{5}$$

$$\nabla \cdot (\mu \mathbf{H}) = 0 \tag{6}$$

One time-varying equation expresses Faraday's (or Lenz's) law [12], relating the time variation of the magnetic induction, *μ***H** = **B**, with the space variation of **E** :

$$\frac{\partial}{\partial t} \left( \mu \mathbf{H} \right) \text{ is connected with } \frac{\partial \mathbf{E}}{\partial z} \text{ (say)}$$

More exactly,

4 Will-be-set-by-IN-TECH

The analogy between these equations reveals a much deeper fact than a simple terminological dictionary of mechanical and electromagnetic terms: it shows the inertial properties of the magnetic field, fully expressed by Lenz's law. Actually, magnetic field inertia (defined by the inductance *L*) controls the rate of change of current for a given voltage in a circit, in exactly the same way as the inertial mass controls the change of velocity for a given force. Magnetic inertial or inductive behaviour arises from the tendency of the magnetic flux threading a circuit to remain constant, and reaction to any change in its value generates a voltage and hence a current which flows to oppose the change of the flux. ([5], p.12) Even if, in the previous equations, the mechanical oscillator is a classical one, its deep connections with its quantum counterpart are wellknown ([13], vol.1, Ch. 12). Also, understanding of classical waves propagation was decissive for the formulation of quantum-wave theory [13], so the

classical form of (1) and (3) is not an obstacle in the development of our arguments.

These basic remarks explain the similarities between the propagation of elastic and mechanical waves. The velocity of waves through a medium is determined by the inertial and elastical properties of the medium. They allow the storing of wave energy in the medium, and in the absence of energy dissipation, they also determine the impedance presented by the medium to the waves. In addition, when there is no loss mechanism, a plane wave solution will be obtained, but any resistive or loss term, will produce a decay with time or distance of the

Referring now to the electromagnetic waves, the magnetic inertia of the medium is provided by the inductive property of that medum, i.e. permeability *μ*, allowing storage of magnetic energy, and the elasticity or capacitive property - by the permittivity *�*, allowing storage of the

The theory of electromagnetic phenomena can be described by four equations, two of them independent of time, and two - time-varying. The time-independent ones express the fact that the electric charge is the source of the electric field, but a "magnetic charge" does not exist:

One time-varying equation expresses Faraday's (or Lenz's) law [12], relating the time variation

(*μ***H**) is connected with *<sup>∂</sup>***<sup>E</sup>**

of the magnetic induction, *μ***H** = **B**, with the space variation of **E** :

*∂ ∂t* ∇ · (*�***E**) = *ρ* (5)

∇ · (*μ***H**) = 0 (6)

*<sup>∂</sup><sup>z</sup>* (say)

*kx*<sup>2</sup> <sup>=</sup> <sup>E</sup> (3)

*<sup>C</sup>* <sup>=</sup> <sup>E</sup> (4)

which provides immediately an analogy between the mechanical energy:

1 2 *m dx dt* <sup>2</sup> + 1 2

1 2 *L dq dt* <sup>2</sup> + 1 2 *q*2

and the electromagnetic one:

oscillatory solution.

**5. Maxwell's equations**

potential or electric field energy. ([12], p.199)

$$\nabla \times \mathbf{E} = -\mu \frac{\partial \mathbf{H}}{\partial t} \tag{7}$$

The other one expresses Ampere's law [12], relating that the time variation of *�***E** defines the space variation of **H**:

$$\frac{\partial}{\partial t} \left( \epsilon \mathbf{E} \right) \text{ is connected with } \frac{\partial \mathbf{H}}{\partial z} \text{ (say)}.$$

More exactly,

$$\nabla \times \mathbf{H} = \varepsilon \frac{\partial \mathbf{E}}{\partial t} \tag{8}$$

assuming that no free chages or electric current are present - a natural assumption for our approach, as we shall use Maxwell's equations only for studying the wave propagation. In this context, the only role played by (5) and (6) will be to demonstrate the transverse character of the vectors **E**, **H**.

#### **6. Propagation of electromagnetic waves in waveguides and cavities**

The propagation of electromagnetic waves in hollow metalic cylinders is an interesting subject, both for theoretical and practical reasons - e.g., for its applications in telecommunications. We shall consider that the metal is a perfect conductor; if the cylinder is infinite, we shall call this metallic structure *waveguide*; if it has end faces, we shall call it *cavity*. The transversal section of the cylinder is the same, along the cylinder axis. With a time dependence exp (−*iωt*), the Maxwell equations (5)-(8) for the fields inside the cylinder take the form [3]:

$$
\nabla \times \mathbf{E} = i\omega \mathbf{B}, \quad \nabla \cdot \mathbf{B} = 0, \quad \nabla \times \mathbf{B} = -i\mu \epsilon \omega \mathbf{E}, \quad \nabla \cdot \mathbf{E} = 0 \tag{9}
$$

For a cylinder filled with a uniform non-dissipative medium having permittivity *�* and permeability *μ*,

$$
\left(\nabla^2 + \mu\epsilon\omega^2\right) \begin{Bmatrix} \mathbf{E} \\ \mathbf{B} \end{Bmatrix} = 0\tag{10}
$$

The specific geometry suggests us to single out the spatial variation of the fields in the *z* direction and to assume

$$\begin{Bmatrix} \mathbf{E}\left(\mathbf{x}, y, z, t\right) \\ \mathbf{B}\left(\mathbf{x}, y, z, t\right) \end{Bmatrix} = \begin{Bmatrix} \mathbf{E}\left(\mathbf{x}, y\right) \exp\left(\pm ikz - i\omega t\right) \\ \mathbf{B}\left(\mathbf{x}, y\right) \exp\left(\pm ikz - i\omega t\right) \end{Bmatrix} \tag{11}$$

The wave equation is reduced to two variables:

$$
\left[\nabla\_l^2 + \left(\mu\epsilon\omega^2 - k^2\right)\right] \begin{Bmatrix} \mathbf{E} \\ \mathbf{B} \end{Bmatrix} = 0\tag{12}
$$

where <sup>∇</sup><sup>2</sup> *<sup>t</sup>* is the transverse part of the Laplacian operator:

$$
\nabla\_t^2 = \nabla - \frac{\partial^2}{\partial z^2} \tag{13}
$$

It is convenient to separate the fields into components parallel to and transverse the *oz* axis:

$$\mathbf{E} = \mathbf{E}\_2 + \mathbf{E}\_{l\prime} \text{ with } \mathbf{E}\_2 = \hat{\mathbf{z}} \mathbf{E}\_{\prime\prime}. \ \mathbf{E}\_l = (\hat{\mathbf{z}} \times \mathbf{E}) \times \hat{\mathbf{z}} \tag{14}$$

where **n** is a normal unit at the surface *S*. From the first equation of (23):

so:

Also, from the second one:

themselves into two distinct categories:

Transverse magnetic (TM) waves:

Transverse electric (TE) waves:

situation).

by:

**7. Waveguides**

**<sup>n</sup>** <sup>×</sup> **<sup>E</sup>** <sup>=</sup> **<sup>n</sup>**<sup>×</sup> (−**n***Et* <sup>+</sup> **<sup>z</sup>***Ez*) <sup>=</sup> **<sup>n</sup>**<sup>×</sup>**<sup>z</sup>***Ez* <sup>=</sup> <sup>0</sup>

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 95

**<sup>n</sup>** · **<sup>B</sup>** <sup>=</sup> **<sup>n</sup>**·(−**n***Bt* <sup>+</sup> **<sup>z</sup>***Bz*) <sup>=</sup> <sup>−</sup>*Bt* <sup>=</sup> <sup>0</sup>

where *∂*/*∂n* is the normal derivative at a point on the surface. Even if the wave equation for *Ez* and *Bz* is the same ((eq. (12)), the boundary conditions on *Ez* and *Bz* are different, so the eigenvalues for *Ez* and *Bz* will in general be different. The fields thus naturally divide

*Bz* = 0 everywhere; boundary condition, *Ez* |*<sup>S</sup>* = 0 (26)

With this value for *Bt* in the component of the first equation (16) parallel to **n**, we get:

*∂Bz*

*Ez* <sup>=</sup> 0 everywhere; boundary condition, *<sup>∂</sup>Bz*

For a given frequency *ω*, only certain values of wave number *k* can occur (typical waveguide situation), or, for a given *k*, only certain *ω* values are allowed (typical resonant cavity

The variuos TM and TE waves, plus the TEM waves if it can exist, constitute a complete set of fields to describe an arbitrary electromagnetic disturbance in a waveguide or cavity [3].

For the propagation of waves inside a hollow waveguide of uniform cross section, it is found from (18) and (19) that the transverse magnetic fields for both TM and TE waves are related

*�* (*TM*)

*�* (*TE*)

*Ht* <sup>=</sup> <sup>±</sup> <sup>1</sup>

*Z* =

 *<sup>k</sup> �ω* <sup>=</sup> *<sup>k</sup> k*0 *μ*

> *μω <sup>k</sup>* <sup>=</sup> *<sup>k</sup>*<sup>0</sup> *k μ*

and *k*<sup>0</sup> is given by (21). The ± sign in (28) goes with *z* dependence, exp (±*ikz*) [3]. The transverse fields are determined by the longitudinal fields, according to (18) and (19):

where *Z* is called the *wave impedance* and is given by

*Ez* |*<sup>S</sup>* = 0 (24)

*<sup>∂</sup><sup>n</sup>* <sup>|</sup>*<sup>S</sup>* <sup>=</sup> <sup>0</sup> (25)

*<sup>∂</sup><sup>n</sup>* <sup>|</sup>*<sup>S</sup>* <sup>=</sup> <sup>0</sup> (27)

(29)

*<sup>Z</sup>***<sup>z</sup>** <sup>×</sup> **<sup>E</sup>***<sup>t</sup>* (28)

*<sup>z</sup>* is as usual, a unit vector in the *<sup>z</sup>*−direction. Similar definitions hold for the magnetic field **B**. The Maxwell equations can be expressed in terms of transverse and parallel fields as [3]:

$$\frac{\partial \mathbf{E}\_t}{\partial z} + i\omega \hat{\mathbf{z}} \times \mathbf{B}\_t = \nabla\_t \mathbf{E}\_{\mathcal{Z}} \quad \hat{\mathbf{z}} \cdot (\nabla\_t \times \mathbf{E}\_t) = i\omega B\_{\mathcal{Z}} \tag{15}$$

$$\frac{\partial \mathbf{B}\_{l}}{\partial z} - i\mu \epsilon \omega \hat{\mathbf{z}} \times \mathbf{E}\_{l} = \nabla\_{l} \mathbf{B}\_{\overline{z}} \cdot \hat{\mathbf{z}} \cdot (\nabla\_{l} \times \mathbf{B}\_{l}) = -i\mu \epsilon \omega \mathbf{E}\_{z} \tag{16}$$

$$\nabla\_{\mathbf{f}} \cdot \mathbf{E}\_{\mathbf{f}} = -\frac{\partial E\_z}{\partial z}, \ \nabla\_{\mathbf{f}} \cdot \mathbf{B}\_{\mathbf{f}} = -\frac{\partial B\_{\overline{z}}}{\partial z} \tag{17}$$

According to the first equations in (15) and (16), if *Ez* and *Bz* are known, the transverse components of **E** and **B** are determined, assuming the *z* dependence is given by (11). Considering that the propagation in the positive *z* direction (for the opposite one, *k* changes it sign) and that at least one *Ez* and *Bz* have non-zero values, the transverse fields are

$$\mathbf{E}\_{\mathbf{f}} = \frac{\dot{\mathbf{i}}}{\mu \epsilon \omega^2 - k^2} \left[ k \nabla\_{\mathbf{f}} \mathbf{E}\_{\mathbf{z}} - \omega \hat{\mathbf{z}} \times \nabla\_{\mathbf{f}} \mathbf{B}\_{\mathbf{z}} \right] \tag{18}$$

$$\mathbf{B}\_{l} = \frac{i}{\mu \epsilon \omega^{2} - k^{2}} \left[ k \nabla\_{l} \mathbf{B}\_{\overline{z}} + \omega \epsilon \omega \hat{\mathbf{z}} \times \nabla\_{l} \mathbf{E}\_{\overline{z}} \right] \tag{19}$$

Let us notice the existence of a special type of solution, called the *transverse electromagnetic* (TEM) wave, having only field components transverse to the direction of propagation [6]. From the second equation in (15) and the first in (16), results that *Ez* = 0 and *Bz* = 0 implies that **E***<sup>t</sup>* = **E***ETM* satisfies

$$\nabla\_t \times \mathbf{E}\_{ETM} = 0, \ \nabla\_t \cdot \mathbf{E}\_{ETM} = 0 \tag{20}$$

So, **E***ETM* is a solution of an *electrostatic* problem in 2D. There are 4 consequences:

1. the axial wave number is given by the infinite-medium value,

$$k = k\_0 = \omega \sqrt{\mu \epsilon} \tag{21}$$

as can be seen from (12).

2. the magnetic field, deduced from the first eq. in (16), is

$$\mathbf{B}\_{ETM} = \pm \sqrt{\mu \epsilon \hat{\mathbf{z}}} \times \mathbf{E}\_{ETM} \tag{22}$$

for waves propagating as exp (±*ikz*). The connection between **B***ETM* and **E***ETM* is just the same as for plane waves in an infinite medium.

3. the TEM mode cannot exist inside a single, hollow, cylindrical conductor of infinite conductivity. The surface is an equipotential; the electric field therefore vanishes inside. It is necessary to have two or more cylindrical surfaces to support the TEM mode. The familiar coaxial cable and the parallel-wire transmission line are structures for which this is the dominant mode.

4. the absence of a cutoff frequency (see below): the wave number (21) is real for all *ω*.

In fact, two types of field configuration occur in hollow cylinders. They are solutions of the eigenvalue problems given by the wave equation (12), solved with the following boundary conditions, to be fulfilled on the cylinder surface:

$$
\mathbf{n} \times \mathbf{E} = 0, \ \mathbf{n} \cdot \mathbf{B} = 0 \tag{23}
$$

where **n** is a normal unit at the surface *S*. From the first equation of (23):

$$\mathbf{n} \times \mathbf{E} = \mathbf{n} \times (-\mathbf{n}E\_l + \hat{\mathbf{z}}E\_z) = \mathbf{n} \times \hat{\mathbf{z}}E\_z = 0$$

so:

6 Will-be-set-by-IN-TECH

*<sup>z</sup>* is as usual, a unit vector in the *<sup>z</sup>*−direction. Similar definitions hold for the magnetic field **B**. The Maxwell equations can be expressed in terms of transverse and parallel fields as [3]:

According to the first equations in (15) and (16), if *Ez* and *Bz* are known, the transverse components of **E** and **B** are determined, assuming the *z* dependence is given by (11). Considering that the propagation in the positive *z* direction (for the opposite one, *k* changes it

Let us notice the existence of a special type of solution, called the *transverse electromagnetic* (TEM) wave, having only field components transverse to the direction of propagation [6]. From the second equation in (15) and the first in (16), results that *Ez* = 0 and *Bz* = 0 implies

for waves propagating as exp (±*ikz*). The connection between **B***ETM* and **E***ETM* is just the

3. the TEM mode cannot exist inside a single, hollow, cylindrical conductor of infinite conductivity. The surface is an equipotential; the electric field therefore vanishes inside. It is necessary to have two or more cylindrical surfaces to support the TEM mode. The familiar coaxial cable and the parallel-wire transmission line are structures for which this is

In fact, two types of field configuration occur in hollow cylinders. They are solutions of the eigenvalue problems given by the wave equation (12), solved with the following boundary

4. the absence of a cutoff frequency (see below): the wave number (21) is real for all *ω*.

<sup>∇</sup>*<sup>t</sup>* · **<sup>E</sup>***<sup>t</sup>* <sup>=</sup> <sup>−</sup>*∂Ez*

**<sup>E</sup>***<sup>t</sup>* <sup>=</sup> *<sup>i</sup>*

**<sup>B</sup>***<sup>t</sup>* <sup>=</sup> *<sup>i</sup>*

1. the axial wave number is given by the infinite-medium value,

2. the magnetic field, deduced from the first eq. in (16), is

same as for plane waves in an infinite medium.

conditions, to be fulfilled on the cylinder surface:

sign) and that at least one *Ez* and *Bz* have non-zero values, the transverse fields are

So, **E***ETM* is a solution of an *electrostatic* problem in 2D. There are 4 consequences:

<sup>+</sup> *<sup>i</sup><sup>ω</sup>***<sup>z</sup>** <sup>×</sup> **<sup>B</sup>***<sup>t</sup>* <sup>=</sup> <sup>∇</sup>*tEz*, **<sup>z</sup>** · (∇*<sup>t</sup>* <sup>×</sup> **<sup>E</sup>***t*) <sup>=</sup> *<sup>i</sup>ωBz* (15)

*μ�ω*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> [*k*∇*tEz* <sup>−</sup> *<sup>ω</sup>***<sup>z</sup>** <sup>×</sup> <sup>∇</sup>*tBz*] (18)

*μ�ω*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> [*k*∇*tBz* <sup>+</sup> *ω�ω***<sup>z</sup>** <sup>×</sup> <sup>∇</sup>*tEz*] (19)

∇*<sup>t</sup>* × **E***ETM* = 0, ∇*<sup>t</sup>* · **E***ETM* = 0 (20)

*<sup>k</sup>* <sup>=</sup> *<sup>k</sup>*<sup>0</sup> <sup>=</sup> *<sup>ω</sup>*√*μ�* (21)

**<sup>B</sup>***ETM* <sup>=</sup> <sup>±</sup>√*μ�***<sup>z</sup>** <sup>×</sup> **<sup>E</sup>***ETM* (22)

**n** × **E** = 0, **n** · **B** = 0 (23)

*<sup>∂</sup><sup>z</sup>* (17)

*<sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>i</sup>μ�ω***<sup>z</sup>** <sup>×</sup> **<sup>E</sup>***<sup>t</sup>* <sup>=</sup> <sup>∇</sup>*tBz*, **<sup>z</sup>** · (∇*<sup>t</sup>* <sup>×</sup> **<sup>B</sup>***t*) <sup>=</sup> <sup>−</sup>*iμ�ωEz* (16)

*<sup>∂</sup><sup>z</sup>* , <sup>∇</sup>*<sup>t</sup>* · **<sup>B</sup>***<sup>t</sup>* <sup>=</sup> <sup>−</sup>*∂Bz*

*∂***E***t ∂z*

*∂***B***t*

that **E***<sup>t</sup>* = **E***ETM* satisfies

as can be seen from (12).

the dominant mode.

$$E\_z \mid\_S = 0 \tag{24}$$

Also, from the second one:

$$\mathbf{n} \cdot \mathbf{B} = \mathbf{n} \cdot \left(-\mathbf{n}B\_l + \hat{\mathbf{z}}B\_z\right) = -B\_l = 0$$

With this value for *Bt* in the component of the first equation (16) parallel to **n**, we get:

$$\frac{\partial B\_{\mathbb{Z}}}{\partial n}|\_{\mathbb{S}} = 0 \tag{25}$$

where *∂*/*∂n* is the normal derivative at a point on the surface. Even if the wave equation for *Ez* and *Bz* is the same ((eq. (12)), the boundary conditions on *Ez* and *Bz* are different, so the eigenvalues for *Ez* and *Bz* will in general be different. The fields thus naturally divide themselves into two distinct categories:

Transverse magnetic (TM) waves:

$$B\_2 = 0 \text{ everywhere; boundary condition, } E\_z \mid\_S = 0 \tag{26}$$

Transverse electric (TE) waves:

$$E\_2 = 0 \text{ everywhere;} \quad \text{boundary condition,} \frac{\partial B\_2}{\partial n} \mid\_S = 0 \tag{27}$$

For a given frequency *ω*, only certain values of wave number *k* can occur (typical waveguide situation), or, for a given *k*, only certain *ω* values are allowed (typical resonant cavity situation).

The variuos TM and TE waves, plus the TEM waves if it can exist, constitute a complete set of fields to describe an arbitrary electromagnetic disturbance in a waveguide or cavity [3].

#### **7. Waveguides**

For the propagation of waves inside a hollow waveguide of uniform cross section, it is found from (18) and (19) that the transverse magnetic fields for both TM and TE waves are related by:

$$H\_{\rm f} = \pm \frac{1}{Z} \hat{\mathbf{z}} \times \mathbf{E}\_{\rm f} \tag{28}$$

where *Z* is called the *wave impedance* and is given by

$$Z = \begin{cases} \frac{k}{\epsilon \omega} = \frac{k}{k\_0} \sqrt{\frac{\mu}{\epsilon}} & (TM) \\\frac{\mu \omega}{k} = \frac{k\_0}{k} \sqrt{\frac{\mu}{\epsilon}} & (TE) \end{cases} \tag{29}$$

and *k*<sup>0</sup> is given by (21). The ± sign in (28) goes with *z* dependence, exp (±*ikz*) [3]. The transverse fields are determined by the longitudinal fields, according to (18) and (19):

TM waves:

$$E\_t = \pm \frac{ik}{\gamma^2} \nabla\_t \psi \tag{30}$$

with the same result for *kmn*.

Defining a cutoff frequency *ωλ*,

**7.2 Modes in a resonant cavity**

of the fields having the form

So, the wavenumber *k* is restricted to:

and the condition a(35) impose a quantization of *ω* :

then the wave number can be written:

In a more general geometry, there will be a spectrum of eigenvalues *γ*<sup>2</sup>

*k*2

*ωλ* <sup>=</sup> *γλ* √*μ�* 

*<sup>k</sup><sup>λ</sup>* <sup>=</sup> <sup>√</sup>*μ�*

*k* = *p π*

*μ�ω*<sup>2</sup> *<sup>p</sup><sup>λ</sup>* = *p π d* 2 + *γ*<sup>2</sup>

So, the existence of quantized values of *k* implies the quantization of *ω*.

**8. Electromagnetic wave propagation in optical fibers**

step profile (as also *ncl* = *const*.), or may be graded, for instance:

*nco* (*r*) = *nco* (0)

In a resonant cavity - i.e., a cylinder with metallic, perfect conductive ends perpendicular to the *oz* axis - the wave equation is identical, but the eigenvalue problem is somewhat different, due to the restrictions on *k*. Indeed, the formation of standing waves requires a *z*−dependence

Optical fibers belong to a subset (the most commercially significant one) of dielectric optical waveguides [6]. Although the first study in this subject was published in 1910 [14], the explosive increase of interest for optical fibers coincides with the technical production of low loss dielectrics, some six decenies later. In practice, they are highly clindrical flexible fibers made of nearly transparent dielectric material. These fibers - with a diameter comparable to a human hair - are composed of a central region, the *core* of radius *a* and reffractive index *nco*, surrounded by the *cladding*, of refractive index *ncl* < *nco*, covered with a protective jacket [15]. In the core, *nco* may be constant - in this case, one says that the refractive-index profile is a

> 1 − Δ

 *r a α*

frequency *ω*, the wave number *k* is determined for each value of *λ* :

solutions *ψλ*, with *λ* taking discrete values (which can be integers or sets of integers, see for instance (37)). These different solutions are called the modes of the guide. For a given

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 97

*<sup>λ</sup>* <sup>=</sup> *μ�ω*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

*<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

*<sup>λ</sup>* and corresponding

*<sup>λ</sup>* (40)

*<sup>λ</sup>* (41)

*<sup>λ</sup>* (42)

*<sup>λ</sup>* (45)

, *r* < *a* (46)

*A* sin *kz* + *B* cos *kz* (43)

*<sup>d</sup>* , *<sup>p</sup>* <sup>=</sup> 0, 1, ... (44)

TE waves:

$$H\_{\rm f} = \pm \frac{ik}{\gamma^2} \nabla\_{\rm f} \psi \tag{31}$$

where *<sup>ψ</sup>* exp (±*ikz*) is *Ez* (*Hz*) for TM (TE) waves, and *<sup>γ</sup>*<sup>2</sup> is defined below. The scalar function *ψ* satisfies the 2D wave eq (12):

$$\left(\nabla\_t + \gamma^2\right)\psi = 0\tag{32}$$

where

$$
\gamma^2 = \mu \epsilon \omega^2 - k^2 \tag{33}
$$

subject to the boundary condition,

$$
\psi|\_{\mathbb{S}} = 0 \text{ or } \frac{\partial \psi}{\partial n}|\_{\mathbb{S}} = 0 \tag{34}
$$

for TM (TE) waves.

Equation (32) for *ψ*, together with boundary condition (34), specifies an eigenvalues problem. The similarity with non-relativistic quantum mechanics is evident.

#### **7.1 Modes in a rectangular waveguide**

Let us illustrate the previous general theory by considering the propagation of TE waves in a rectangular waveguide (the corners of the rectangle are situated in (0, 0),(*a*, 0),(*a*, *b*),(0, *b*)). In this case, is easy to obtain explicit solutions for the fields [3]. The wave equation for *ψ* = *Hz* is

$$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \gamma^2\right)\psi = 0\tag{35}$$

with boundary conditions *∂ψ*/*∂n* = 0 at *x* = 0, *a* and *y* = 0, *b*. The solution for *ψ* is easily find to be:

$$\psi\_{mn}\left(\mathbf{x},y\right) = H\_0 \cos\left(\frac{m\pi\mathbf{x}}{a}\right) \cos\left(\frac{n\pi y}{b}\right) \tag{36}$$

with *γ* givem by:

$$
\gamma\_{mn}^2 = \pi^2 \left(\frac{m^2}{a^2} + \frac{n^2}{b^2}\right) \tag{37}
$$

with *m*, *n* - integers. Consequently, from (33),

$$k\_{mn}^2 = \mu \epsilon \omega^2 - \gamma\_{mn}^2 = \mu \epsilon \left(\omega^2 - \omega\_{mn}^2\right), \text{ }\omega\_{mn}^2 = \frac{\gamma\_{mn}^2}{\mu \epsilon} \tag{38}$$

As only for *ω* > *ωmn*, *kmn* is real, so the waves propagate without attenuation; *ωmn* is called cutoff frequency. For a given *ω*, only certain values of *k*, namely *kmn*, are allowed.

For TM waves, the equation for the field *ψ* = *Ez* will be also (39), but the boundary condition will be different: *ψ* = 0 at *x* = 0, *a* and *y* = 0, *b*. The solution will be:

$$
\psi\_{mn}\left(\mathbf{x},\mathbf{y}\right) = E\_0 \sin\left(\frac{m\pi\mathbf{x}}{a}\right) \sin\left(\frac{n\pi y}{b}\right) \tag{39}
$$

with the same result for *kmn*.

8 Will-be-set-by-IN-TECH

*<sup>γ</sup>*<sup>2</sup> <sup>∇</sup>*t<sup>ψ</sup>* (30)

*<sup>γ</sup>*<sup>2</sup> <sup>∇</sup>*t<sup>ψ</sup>* (31)

*ψ* = 0 (32)

*<sup>∂</sup><sup>n</sup>* <sup>|</sup>*<sup>S</sup>* <sup>=</sup> <sup>0</sup> (34)

*ψ* = 0 (35)

(36)

(37)

(39)

*μ�* (38)

*<sup>γ</sup>*<sup>2</sup> <sup>=</sup> *μ�ω*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> (33)

*Et* <sup>=</sup> <sup>±</sup> *ik*

*Ht* <sup>=</sup> <sup>±</sup> *ik*

<sup>∇</sup>*<sup>t</sup>* <sup>+</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>ψ</sup>* <sup>|</sup>*<sup>S</sup>* <sup>=</sup> 0 or *∂ψ*

Equation (32) for *ψ*, together with boundary condition (34), specifies an eigenvalues problem.

Let us illustrate the previous general theory by considering the propagation of TE waves in a rectangular waveguide (the corners of the rectangle are situated in (0, 0),(*a*, 0),(*a*, *b*),(0, *b*)). In this case, is easy to obtain explicit solutions for the fields [3]. The wave equation for *ψ* = *Hz*

with boundary conditions *∂ψ*/*∂n* = 0 at *x* = 0, *a* and *y* = 0, *b*. The solution for *ψ* is easily

 *m*<sup>2</sup> *<sup>a</sup>*<sup>2</sup> <sup>+</sup>

As only for *ω* > *ωmn*, *kmn* is real, so the waves propagate without attenuation; *ωmn* is called

For TM waves, the equation for the field *ψ* = *Ez* will be also (39), but the boundary condition

 *mπx a* sin

*<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *mn* , *ω*<sup>2</sup>

 *mπx a* cos

*n*2 *b*2   *nπy b* 

 *nπy b* 

*mn* <sup>=</sup> *<sup>γ</sup>*<sup>2</sup> *mn*

*∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*<sup>2</sup>

The similarity with non-relativistic quantum mechanics is evident.

 *∂*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

*ψmn* (*x*, *y*) = *H*<sup>0</sup> cos

*γ*2 *mn* = *<sup>π</sup>*<sup>2</sup>

will be different: *ψ* = 0 at *x* = 0, *a* and *y* = 0, *b*. The solution will be:

*ψmn* (*x*, *y*) = *E*<sup>0</sup> sin

*mn* = *μ�*

cutoff frequency. For a given *ω*, only certain values of *k*, namely *kmn*, are allowed.

where *<sup>ψ</sup>* exp (±*ikz*) is *Ez* (*Hz*) for TM (TE) waves, and *<sup>γ</sup>*<sup>2</sup> is defined below. The scalar function

TM waves:

TE waves:

where

is

find to be:

with *γ* givem by:

*ψ* satisfies the 2D wave eq (12):

subject to the boundary condition,

**7.1 Modes in a rectangular waveguide**

with *m*, *n* - integers. Consequently, from (33),

*k*2

*mn* <sup>=</sup> *μ�ω*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

for TM (TE) waves.

In a more general geometry, there will be a spectrum of eigenvalues *γ*<sup>2</sup> *<sup>λ</sup>* and corresponding solutions *ψλ*, with *λ* taking discrete values (which can be integers or sets of integers, see for instance (37)). These different solutions are called the modes of the guide. For a given frequency *ω*, the wave number *k* is determined for each value of *λ* :

$$k\_{\lambda}^{2} = \mu \epsilon \omega^{2} - \gamma\_{\lambda}^{2} \tag{40}$$

Defining a cutoff frequency *ωλ*,

$$
\omega\_{\lambda} = \frac{\gamma\_{\lambda}}{\sqrt{\mu \varepsilon}} \sqrt{\omega^2 - \omega\_{\lambda}^2} \tag{41}
$$

then the wave number can be written:

$$k\_{\lambda} = \sqrt{\mu \epsilon} \sqrt{\omega^2 - \omega\_{\lambda}^2} \tag{42}$$

#### **7.2 Modes in a resonant cavity**

In a resonant cavity - i.e., a cylinder with metallic, perfect conductive ends perpendicular to the *oz* axis - the wave equation is identical, but the eigenvalue problem is somewhat different, due to the restrictions on *k*. Indeed, the formation of standing waves requires a *z*−dependence of the fields having the form

$$A\sin kz + B\cos kz\tag{43}$$

So, the wavenumber *k* is restricted to:

$$k = p \frac{\pi}{d}, \quad p = 0, 1, \dots \tag{44}$$

and the condition a(35) impose a quantization of *ω* :

$$
\mu \epsilon \omega\_{p\lambda}^2 = \left( p \frac{\pi}{d} \right)^2 + \gamma\_\lambda^2 \tag{45}
$$

So, the existence of quantized values of *k* implies the quantization of *ω*.

#### **8. Electromagnetic wave propagation in optical fibers**

Optical fibers belong to a subset (the most commercially significant one) of dielectric optical waveguides [6]. Although the first study in this subject was published in 1910 [14], the explosive increase of interest for optical fibers coincides with the technical production of low loss dielectrics, some six decenies later. In practice, they are highly clindrical flexible fibers made of nearly transparent dielectric material. These fibers - with a diameter comparable to a human hair - are composed of a central region, the *core* of radius *a* and reffractive index *nco*, surrounded by the *cladding*, of refractive index *ncl* < *nco*, covered with a protective jacket [15]. In the core, *nco* may be constant - in this case, one says that the refractive-index profile is a step profile (as also *ncl* = *const*.), or may be graded, for instance:

$$n\_{co}\left(r\right) = n\_{co}\left(0\right)\left[1 - \Delta\left(\frac{r}{a}\right)^{a}\right], \quad r < a \tag{46}$$

field components, assuming that *∂n*2/*∂z* = 0, we find generalizations of the 2D scalar wave

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 99

In contrast to (32) for ideal metallic guides, the equations for *Ez*, *Hz* are coupled. In general, thee is no separation into purely TE and TM modes. The only simplification occurs in the case of a step-profile refractive index, where we can solve the equation (54) or (55) in each domain of constant refractive index, and match the two solutions, using appropriate boundary conditions. In this case, the radial part of the electric field (for the first mode) in the core is [6]:

· <sup>∇</sup>*tHz* <sup>=</sup> <sup>−</sup>*ωkz�*<sup>0</sup>

· <sup>∇</sup>*tEz* <sup>=</sup> *<sup>ω</sup>kzμ*<sup>0</sup>

<sup>0</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> (*r*/*a*)

<sup>0</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> (*r*/*a*)

These solutions are identical (using an appropriate "dictionary") with the solution of the Schrodinger equation for a particle moving in a potential with cylindrical symmetry, the radial part of the potential being a rectangular well of finite depth. However, this kind of analogies can be more easely developed for planar dielectric waveguides, namely for "step-index" dielectrics, consisting of a central slab of finite thickness and of higher refractive index (core), and two lateral, half-space medium of lower refractive index (cladding). Indeed, in such a situation, the quantum counterpart of the dielectric guide is much more extensively studied,

**9. An optical-quantum analogy: the three-layer slab optical waveguide and the**

We shall calculate in detail the TE modes of a three-layer slab optical waveguide, with a 1D structure, and the bound states of a particle in a rectangular well, and we shall find that these problems have identical solutions. Of course, the physical meaning of the parameters entering in each solution are different, but the mathematical structure of the solutions is identical.

We consider a three-layer slab optical waveguide, with a 1D structure [16]. The electromagnetic wave propagates along the *x* axis, and the slabs are: a semi-infinite medium of refractive index *n*1, having as right border the *yz* plane; a slab of refractive index *n*2, having as left border the plane *yz* and as left border a plane paralel to it, cutting the *ox* axis at *x*<sup>0</sup> = *W*;

*<sup>γ</sup>*<sup>2</sup> **<sup>z</sup>** ·

*<sup>γ</sup>*2*n*<sup>2</sup> **<sup>z</sup>** ·

<sup>∇</sup>*tn*<sup>2</sup> × ∇*tEz*

<sup>∇</sup>*tn*<sup>2</sup> × ∇*tHz*

, *<sup>r</sup>* <sup>&</sup>lt; *<sup>a</sup>* (56)

, *<sup>r</sup>* <sup>&</sup>gt; *<sup>a</sup>* (57)

(54)

(55)

equation (32):

and

∇2

∇2

and in the cladding:

*<sup>t</sup> Hz* <sup>+</sup> *<sup>γ</sup>*2*Hz* <sup>−</sup>

*<sup>t</sup> Ez* <sup>+</sup> *<sup>γ</sup>*2*Ez* <sup>−</sup>

 *ω γc*

 *k*<sup>2</sup> *z γn*

*R* (*r*) =

*R* (*r*) =

in almost any textbook of quantum mechanics.

**quantum rectangular well**

**9.1 The optical problem**

<sup>2</sup>

2 <sup>∇</sup>*tn*<sup>2</sup> 

> *J*0 *n*2 *cok*<sup>2</sup>

*K*0 *n*2 *clk*2

*J*0 *n*2 *cok*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup>

*K*0 *n*2 *clk*2 <sup>0</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup>

<sup>∇</sup>*tn*<sup>2</sup> 

For *α* = 2, the profile is called parabolic. One of the main parameters characterizing an optical fiber is the profile hight parameter Δ,

$$\Delta = \frac{1}{2} \left( 1 - \frac{n\_{cl}^2}{n\_{c0}^2} \right) \simeq 1 - \frac{n\_{cl}}{n\_{c0}}, \quad n\_{c0} = \max n\_{c0} \left( r \right)|\_{r \le a} \tag{47}$$

Besides Δ, one usually also defines the fiber parameter *V* :

$$V = ka\sqrt{2\Lambda} \tag{48}$$

Assimilating the propagating light with a geometric ray, it must be incident on the core-cladding interface at an angle smaller than the critical angle *θ<sup>c</sup>* :

$$\theta\_{\mathcal{L}} = \arcsin \frac{n\_{\mathcal{cl}}}{n\_{\mathcal{co}}} \tag{49}$$

in order to be totally reflected at this interface, and therefore to remain inside the core. However, due to the wave character of light, it must satisfy a self-interference condition, in order to be trapped in the waveguide [6]. There are only a finite number of paths which satisfy this condition, and therefore a finite number of modes which propagate through the fiber. The fiber is multimode if 12.5*μm* < *r* < 100*μm* and 0.01 < Δ < 0.03, and single-mode if 2*μm* < *r* < 5*μm* and 0.003 < Δ < 0.01 [15]. By far the most popular fibers for long distance telecommunications applications allow only a single mode of each polarization to propagate [6].

#### **8.1 Modes in circular fibers**

We consider a fiber of uniform cross section with relative magnetic permeability = 1 and *n* varying only on transverse directions [3]. Assuming a *z*− and *t*− dependence exp (*ikzz* − *iωt*), the Maxwell equations can be combined, to yield the Helmholtz wave equations for **H** and **E**:

$$
\nabla^2 \mathbf{H} + \frac{n^2 \omega^2}{c^2} \mathbf{H} = i\omega \epsilon\_0 \left(\nabla n^2\right) \times \mathbf{E} \tag{50}
$$

$$
\nabla^2 \mathbf{E} + \frac{n^2 \omega^2}{c^2} \mathbf{E} = -\nabla \left[ \frac{1}{n^2} \left( \nabla n^2 \right) \cdot \mathbf{E} \right] \tag{51}
$$

where we have written *�* = *n*2*�*0. Just as in Sect. 6, the transverse components of E and H can be expressed in terms of the longitudinal fields *Ez*, *Hz*, *i*.*e*.

$$E\_l = \frac{i}{\gamma^2} \left[ k\_z \nabla\_t E\_z - \omega \mu\_0 \hat{\mathbf{z}} \times \nabla\_t H\_z \right] \tag{52}$$

and

$$H\_{\rm t} = \frac{\dot{i}}{\gamma^2} \left[ k\_z \nabla\_{\rm t} H\_z + \omega \varepsilon\_0 n^2 \hat{\mathbf{z}} \times \nabla\_{\rm t} E\_z \right] \tag{53}$$

where *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> *<sup>n</sup>*2*ω*2/*c*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> *<sup>z</sup>* is the radial propagation constant, as for metallic waveguides. If we take the *z* component of the eqs (54), (55) and use (52) to eliminate the transverse field components, assuming that *∂n*2/*∂z* = 0, we find generalizations of the 2D scalar wave equation (32):

$$
\nabla\_t^2 H\_z + \gamma^2 H\_z - \left(\frac{\omega}{\gamma c}\right)^2 \left(\nabla\_t n^2\right) \cdot \nabla\_t H\_z = -\frac{\omega k\_z \epsilon\_0}{\gamma^2} \hat{\mathbf{z}} \cdot \left[\nabla\_t n^2 \times \nabla\_t E\_z\right] \tag{54}
$$

and

10 Will-be-set-by-IN-TECH

For *α* = 2, the profile is called parabolic. One of the main parameters characterizing an optical

, *nco* = max *nco* (*r*)|*r*≤*<sup>a</sup>* (47)

2Δ (48)

× **E** (50)

(49)

(51)

(53)

� <sup>1</sup> <sup>−</sup> *ncl nco*

*<sup>V</sup>* <sup>=</sup> *ka*<sup>√</sup>

Assimilating the propagating light with a geometric ray, it must be incident on the

*<sup>θ</sup><sup>c</sup>* <sup>=</sup> arcsin *ncl*

in order to be totally reflected at this interface, and therefore to remain inside the core. However, due to the wave character of light, it must satisfy a self-interference condition, in order to be trapped in the waveguide [6]. There are only a finite number of paths which satisfy this condition, and therefore a finite number of modes which propagate through the fiber. The fiber is multimode if 12.5*μm* < *r* < 100*μm* and 0.01 < Δ < 0.03, and single-mode if 2*μm* < *r* < 5*μm* and 0.003 < Δ < 0.01 [15]. By far the most popular fibers for long distance telecommunications applications allow only a single mode of each polarization to propagate

We consider a fiber of uniform cross section with relative magnetic permeability = 1 and *n* varying only on transverse directions [3]. Assuming a *z*− and *t*− dependence exp (*ikzz* − *iωt*), the Maxwell equations can be combined, to yield the Helmholtz wave

*<sup>c</sup>*<sup>2</sup> **<sup>H</sup>** <sup>=</sup> *<sup>i</sup>ω�*<sup>0</sup>

where we have written *�* = *n*2*�*0. Just as in Sect. 6, the transverse components of E and H can

If we take the *z* component of the eqs (54), (55) and use (52) to eliminate the transverse

*<sup>c</sup>*<sup>2</sup> **<sup>E</sup>** <sup>=</sup> −∇

 <sup>∇</sup>*n*<sup>2</sup> 

*<sup>γ</sup>*<sup>2</sup> [*kz*∇*tEz* <sup>−</sup> *ωμ*0**<sup>z</sup>** <sup>×</sup> <sup>∇</sup>*tHz*] (52)

*<sup>z</sup>* is the radial propagation constant, as for metallic waveguides.

 1 *n*2 <sup>∇</sup>*n*<sup>2</sup> · **E** 

*kz*∇*tHz* <sup>+</sup> *ω�*0*n*2**<sup>z</sup>** <sup>×</sup> <sup>∇</sup>*tEz*

*nco*

fiber is the profile hight parameter Δ,

[6].

and

**8.1 Modes in circular fibers**

equations for **H** and **E**:

where *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> *<sup>n</sup>*2*ω*2/*c*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

<sup>Δ</sup> <sup>=</sup> <sup>1</sup> 2  <sup>1</sup> <sup>−</sup> *<sup>n</sup>*<sup>2</sup> *cl n*2 *co*

Besides Δ, one usually also defines the fiber parameter *V* :

core-cladding interface at an angle smaller than the critical angle *θ<sup>c</sup>* :

<sup>∇</sup>2**<sup>H</sup>** <sup>+</sup>

*Et* <sup>=</sup> *<sup>i</sup>*

*Ht* <sup>=</sup> *<sup>i</sup> γ*2 

<sup>∇</sup>2**<sup>E</sup>** <sup>+</sup>

be expressed in terms of the longitudinal fields *Ez*, *Hz*, *i*.*e*.

*n*2*ω*<sup>2</sup>

*n*2*ω*<sup>2</sup>

$$
\nabla\_t^2 \mathbf{E}\_z + \gamma^2 \mathbf{E}\_z - \left(\frac{k\_z^2}{\gamma n}\right)^2 \left(\nabla\_l n^2\right) \cdot \nabla\_l \mathbf{E}\_z = \frac{\omega k\_z \mu\_0}{\gamma^2 n^2} \hat{\mathbf{z}} \cdot \left[\nabla\_l n^2 \times \nabla\_l H\_z\right] \tag{55}
$$

In contrast to (32) for ideal metallic guides, the equations for *Ez*, *Hz* are coupled. In general, thee is no separation into purely TE and TM modes. The only simplification occurs in the case of a step-profile refractive index, where we can solve the equation (54) or (55) in each domain of constant refractive index, and match the two solutions, using appropriate boundary conditions. In this case, the radial part of the electric field (for the first mode) in the core is [6]:

$$R\left(r\right) = \frac{J\_0\left(\sqrt{n\_{co}^2 k\_0^2 - \beta^2} \left(r/a\right)\right)}{J\_0\left(\sqrt{n\_{co}^2 k\_0^2 - \beta^2}\right)}, \quad r < a \tag{56}$$

and in the cladding:

$$R\left(r\right) = \frac{K\_0\left(\sqrt{n\_{cl}^2 k\_0^2 - \beta^2} \left(r/a\right)\right)}{K\_0\left(\sqrt{n\_{cl}^2 k\_0^2 - \beta^2}\right)}, \quad r > a \tag{57}$$

These solutions are identical (using an appropriate "dictionary") with the solution of the Schrodinger equation for a particle moving in a potential with cylindrical symmetry, the radial part of the potential being a rectangular well of finite depth. However, this kind of analogies can be more easely developed for planar dielectric waveguides, namely for "step-index" dielectrics, consisting of a central slab of finite thickness and of higher refractive index (core), and two lateral, half-space medium of lower refractive index (cladding). Indeed, in such a situation, the quantum counterpart of the dielectric guide is much more extensively studied, in almost any textbook of quantum mechanics.

## **9. An optical-quantum analogy: the three-layer slab optical waveguide and the quantum rectangular well**

We shall calculate in detail the TE modes of a three-layer slab optical waveguide, with a 1D structure, and the bound states of a particle in a rectangular well, and we shall find that these problems have identical solutions. Of course, the physical meaning of the parameters entering in each solution are different, but the mathematical structure of the solutions is identical.

#### **9.1 The optical problem**

We consider a three-layer slab optical waveguide, with a 1D structure [16]. The electromagnetic wave propagates along the *x* axis, and the slabs are: a semi-infinite medium of refractive index *n*1, having as right border the *yz* plane; a slab of refractive index *n*2, having as left border the plane *yz* and as left border a plane paralel to it, cutting the *ox* axis at *x*<sup>0</sup> = *W*;

With (64), (66) in (68):

With *k*<sup>0</sup> = 2*π*/*λ*, (73) becomes:

quantum to classical mechanics.

field is *Hy***<sup>y</sup>** <sup>+</sup> *Hz***<sup>z</sup>** <sup>=</sup> *Hz***<sup>z</sup>**. But *Hz* <sup>=</sup> <sup>−</sup> *<sup>i</sup>*

the continuity of *dEy*

derivative, *dEy*

coincides with (1).

Defining:

we have:

in a potential *V* :

*n*2

*d*2*Ey dx*<sup>2</sup> <sup>+</sup>

*d*2*Ey dx*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup> 0 *�<sup>r</sup>* <sup>−</sup> *<sup>n</sup>*<sup>2</sup> *eff*

*d*2*Ey dx*<sup>2</sup> <sup>+</sup>

*d*2*ψ dx*<sup>2</sup> <sup>+</sup>

surfaces, the boundary conditions for the electromagnetic fields are:

2. the tangential components of the magnetic field are continuous

3. the normal components of the electric flux density **D** = *ε***E** are continuous 4. the normal components of the magnetic flux density **B** = *μ***H** are continuous

continuity of the normal components of the magnetic flux, *Hx* = *<sup>β</sup>*

*neff* <sup>=</sup> *<sup>β</sup> k*0 4*π*<sup>2</sup> *λ*2 *�<sup>r</sup>* <sup>−</sup> *<sup>n</sup>*<sup>2</sup> *eff*

4*π*<sup>2</sup>

It is interesting to compare (74) with the Schrodinger equation for a particle of mass *m* moving

For bound states, E = − |E| < 0. In (75), the energy is subject of quantization, similar to

Let us discuss now the boundary conditions. In the absence of charges and current flow on

The tangential electric field at the boundary is *Ey***<sup>y</sup>** <sup>+</sup> *Ez***<sup>z</sup>** <sup>=</sup> *Ey***<sup>y</sup>** and the tangential magnetic

to (70), so the condition (3) is automatically fulfilled. As *μ* = *μ*0, condition (4) claims the

*dx* . So, the conditions (1) and (2) impose the continuity of *Ey* and of its

*dx* . The normal component of the electric field *Ex* is identically zero, according

*μ*0*ω dEy*

1. the tangential components of the electric field are continuous while crossing the border

*eff* in (73) - with appropriate boundary conditions, see below. So, the quantum-mechanical energy is proportional to *�r*, confirming the analogy stated in Sect.4. The opposite of the potential is proportional to the square of the refractive index - the so-called "upside-down correspondence" [1] between optical and mechanical propagation: a light wave tends to concentrate in the area with maximum refractive index, while a particle tends to propagate on the bottom of the potential. Also, the wavelength *λ* corresponds to the Planck constant *h* : when *λ* → 0, the wave optics is replaced by geometrical optics, similarly with transition ftrom

*�μ*0*ω*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup>

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 101

*Ey* = 0 (71)

*Ey* = 0 (73)

*Ey* = 0 (74)

<sup>→</sup> *<sup>β</sup>* <sup>=</sup> *neff <sup>k</sup>*0, *<sup>k</sup>*<sup>0</sup> <sup>=</sup> *<sup>ω</sup>*√*�*0*μ*<sup>0</sup> (72)

*<sup>h</sup>*<sup>2</sup> (−2*mV* <sup>+</sup> <sup>2</sup>*m*E) *<sup>ψ</sup>* <sup>=</sup> <sup>0</sup> (75)

*dx* , so the continuity of *Hz* is equivalent to

*<sup>μ</sup>*0*<sup>ω</sup> Ey*, so this condition

and a semi-infinite medium of refractive index *n*3, for the remaining space. The inner slab corresponds to the core, and the outer ones - to the cladding.

It is instructive to obtain the wave equation for the electric and magnetic field in this simple geometry, starting directly from the Maxwell equations (7), (8). Assuming that *μ* = *μ*<sup>0</sup> throughout the entire system and that the *t*−dependence is:

$$\mathbf{E} = \mathbf{E}\left(t=0\right)\exp\left(-i\omega t\right), \quad \mathbf{H} = \mathbf{H}\left(t=0\right)\exp\left(-i\omega t\right).$$

the equations for the field components are:

$$\frac{\partial E\_z}{\partial y} - \frac{\partial E\_y}{\partial z} = i\mu\_0 \omega H\_\text{x} \tag{58}$$

$$\frac{\partial E\_{\mathbf{x}}}{\partial z} - \frac{\partial E\_{\mathbf{z}}}{\partial \mathbf{x}} = \dot{\imath}\mu\_{0}\omega H\_{\mathbf{y}} \tag{59}$$

$$\frac{\partial E\_y}{\partial x} - \frac{\partial E\_x}{\partial y} = i\mu\_0 \omega H\_z \tag{60}$$

Also,

$$\frac{\partial H\_z}{\partial y} - \frac{\partial H\_y}{\partial z} = -i\epsilon\omega E\_x \tag{61}$$

$$\frac{\partial H\_{\text{X}}}{\partial z} - \frac{\partial H\_{\text{z}}}{\partial \mathbf{x}} = -i\epsilon\omega\mathbf{E}\_{\text{y}} \tag{62}$$

$$\frac{\partial H\_{\text{y}}}{\partial \mathbf{x}} - \frac{\partial H\_{\text{x}}}{\partial \mathbf{y}} = -i\epsilon\omega E\_{\text{z}} \tag{63}$$

#### **TE mode**

We shall look for the TE mode. By definition, in this mode there is no electric field in longitudinal direction, *Ez* = 0, there is no space variation in the *y* direction, so *∂*/*∂y* → 0, and the *z*-dependence is exp (−*iβz*), so *∂*/*∂z* → −*iβ*. The Maxwell equations (58)-(63) become:

$$H\_{\rm X} = \frac{\beta}{\mu\_0 \omega} E\_{\rm y} \tag{64}$$

$$-\beta E\_x = \mu\_0 \omega H\_y \tag{65}$$

$$H\_z = -\frac{i}{\mu\_0 \omega} \frac{\partial E\_y}{\partial x} \tag{66}$$

$$
\beta H\_y = -\epsilon \omega \varepsilon\_x \tag{67}
$$

$$-i\mathfrak{H}H\_{\mathfrak{X}} - \frac{\partial H\_{\mathfrak{z}}}{\partial \mathfrak{x}} = -i\epsilon\omega\varepsilon E\_{\mathfrak{y}} \tag{68}$$

$$\frac{\partial H\_y}{\partial \mathbf{x}} = \mathbf{0} \tag{69}$$

From (69), *Hy* = *const* and we can put *Hy* = 0, so from (65), (67), *Ex* = 0. So,

$$E\_X = E\_z = H\_{\bar{y}} = 0 \tag{70}$$

With (64), (66) in (68):

$$\frac{d^2 E\_y}{dx^2} + \left(\epsilon\mu\_0\omega^2 - \beta^2\right) E\_y = 0\tag{71}$$

Defining:

12 Will-be-set-by-IN-TECH

and a semi-infinite medium of refractive index *n*3, for the remaining space. The inner slab

It is instructive to obtain the wave equation for the electric and magnetic field in this simple geometry, starting directly from the Maxwell equations (7), (8). Assuming that *μ* = *μ*<sup>0</sup>

**E** = **E** (*t* = 0) exp (−*iωt*), **H** = **H** (*t* = 0) exp (−*iωt*)

We shall look for the TE mode. By definition, in this mode there is no electric field in longitudinal direction, *Ez* = 0, there is no space variation in the *y* direction, so *∂*/*∂y* → 0, and the *z*-dependence is exp (−*iβz*), so *∂*/*∂z* → −*iβ*. The Maxwell equations (58)-(63) become:

> *Hx* <sup>=</sup> *<sup>β</sup> μ*0*ω*

*Hz* <sup>=</sup> <sup>−</sup> *<sup>i</sup>*

*∂Hy*

<sup>−</sup> *<sup>i</sup>βHx* <sup>−</sup> *<sup>∂</sup>Hz*

From (69), *Hy* = *const* and we can put *Hy* = 0, so from (65), (67), *Ex* = 0. So,

*μ*0*ω*

*∂Ey*

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> *<sup>i</sup>μ*0*ωHx* (58)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> *<sup>i</sup>μ*0*ωHy* (59)

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> *<sup>i</sup>μ*0*ωHz* (60)

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> <sup>−</sup>*i�ωEx* (61)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>−</sup>*i�ωEy* (62)

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> <sup>−</sup>*i�ωEz* (63)

*Ey* (64)

*<sup>∂</sup><sup>x</sup>* (66)

− *βEx* = *μ*0*ωHy* (65)

*βHy* = −*�ωEx* (67)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>−</sup>*i�ωEy* (68)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>0</sup> (69)

*Ex* = *Ez* = *Hy* = 0 (70)

corresponds to the core, and the outer ones - to the cladding.

throughout the entire system and that the *t*−dependence is:

*∂Ez <sup>∂</sup><sup>y</sup>* <sup>−</sup> *<sup>∂</sup>Ey*

*∂Ex <sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>∂</sup>Ez*

*∂Ey <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>∂</sup>Ex*

*∂Hz <sup>∂</sup><sup>y</sup>* <sup>−</sup> *<sup>∂</sup>Hy*

*∂Hx <sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>∂</sup>Hz*

*∂Hy <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>∂</sup>Hx*

the equations for the field components are:

Also,

**TE mode**

$$
\hbar n\_{eff} = \frac{\not\!\!\!\!\!\!\/}{k\_0} \to \not\!\!\!\/ = n\_{eff} k\_0 \,, \ k\_0 = \omega \sqrt{\epsilon\_0 \mu\_0} \tag{72}
$$

we have:

$$\left(\frac{d^2 E\_y}{d\mathbf{x}^2} + k\_0^2 \left(\epsilon\_r - n\_{eff}^2\right)\right) E\_y = 0\tag{73}$$

With *k*<sup>0</sup> = 2*π*/*λ*, (73) becomes:

$$\left(\frac{d^2E\_y}{dx^2} + \frac{4\pi^2}{\lambda^2}\left(\epsilon\_r - n\_{eff}^2\right)\right)E\_y = 0\tag{74}$$

It is interesting to compare (74) with the Schrodinger equation for a particle of mass *m* moving in a potential *V* :

$$\frac{d^2\psi}{d\mathfrak{x}^2} + \frac{4\pi^2}{h^2} \left(-2mV + 2m\mathcal{E}\right)\psi = 0\tag{75}$$

For bound states, E = − |E| < 0. In (75), the energy is subject of quantization, similar to *n*2 *eff* in (73) - with appropriate boundary conditions, see below. So, the quantum-mechanical energy is proportional to *�r*, confirming the analogy stated in Sect.4. The opposite of the potential is proportional to the square of the refractive index - the so-called "upside-down correspondence" [1] between optical and mechanical propagation: a light wave tends to concentrate in the area with maximum refractive index, while a particle tends to propagate on the bottom of the potential. Also, the wavelength *λ* corresponds to the Planck constant *h* : when *λ* → 0, the wave optics is replaced by geometrical optics, similarly with transition ftrom quantum to classical mechanics.

Let us discuss now the boundary conditions. In the absence of charges and current flow on surfaces, the boundary conditions for the electromagnetic fields are:


The tangential electric field at the boundary is *Ey***<sup>y</sup>** <sup>+</sup> *Ez***<sup>z</sup>** <sup>=</sup> *Ey***<sup>y</sup>** and the tangential magnetic field is *Hy***<sup>y</sup>** <sup>+</sup> *Hz***<sup>z</sup>** <sup>=</sup> *Hz***<sup>z</sup>**. But *Hz* <sup>=</sup> <sup>−</sup> *<sup>i</sup> μ*0*ω dEy dx* , so the continuity of *Hz* is equivalent to the continuity of *dEy dx* . So, the conditions (1) and (2) impose the continuity of *Ey* and of its derivative, *dEy dx* . The normal component of the electric field *Ex* is identically zero, according to (70), so the condition (3) is automatically fulfilled. As *μ* = *μ*0, condition (4) claims the continuity of the normal components of the magnetic flux, *Hx* = *<sup>β</sup> <sup>μ</sup>*0*<sup>ω</sup> Ey*, so this condition coincides with (1).

it coincides with eq. (20) Ch.III, vol.1, [13]. It is the energy eigenvalue equation for the

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 103

�

*V*1, *x* > *a V*2, *a* > *x* > *b V*3, *b* > *x*

*ψ* (*x*) = 0 (91)

<sup>2</sup> (93)

<sup>2</sup> (94)

<sup>2</sup> (95)

<sup>2</sup> (99)

= tan Γ<sup>2</sup> (100)

*eff* = Γ<sup>2</sup> (96)

(92)

*<sup>h</sup>*¯ <sup>2</sup> (−*<sup>V</sup>* (*x*) <sup>+</sup> <sup>E</sup>)

*V*<sup>2</sup> < *V*<sup>1</sup> < *V*<sup>3</sup> It is useful to consider a particular situation, when *n*<sup>1</sup> = *n*<sup>3</sup> in the optical waveguide, respectively when *V*<sup>1</sup> = *V*<sup>3</sup> = 0, *V*<sup>2</sup> < 0, in the quantum mechanical problem. In (91), −*V* (*x*) 0 is given, and we have to find the eigenvalues of the energy E < 0. For the optical waveguide (71), *�μ*0*ω*<sup>2</sup> is given, and the eigenvalues of the quantity <sup>−</sup>*β*<sup>2</sup> (essentially, the propagation constant *β*) must be obtained. Let us note once again that the refractive index in the optical waveguide corresponds to the opposite of the potential, in the quantum mechanical

Let us investigate in greater detail the consequences of the particular situation just mentioned,

<sup>+</sup> *<sup>q</sup><sup>π</sup>*

<sup>=</sup> tan *<sup>γ</sup>*2*<sup>W</sup>*

<sup>=</sup> <sup>−</sup> tan *<sup>γ</sup>*2*<sup>W</sup>*

<sup>1</sup> = Γ1, *γ*2*a* = *ak*<sup>0</sup>

� *n*2 <sup>2</sup> <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

= tan Γ<sup>2</sup> (*q odd*) (97)

= − tan Γ<sup>2</sup> (*q even*) (98)

<sup>2</sup> <sup>=</sup> <sup>−</sup>*γ*2*<sup>W</sup>*

arctan *<sup>γ</sup>*<sup>2</sup> *γ*1

> *γ*1 *γ*2

*γ*2 *γ*1

� *n*2 *eff* <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

> Γ1 Γ2

Γ2 Γ1

Γ2

*<sup>K</sup>*<sup>2</sup> <sup>−</sup> <sup>Γ</sup><sup>2</sup> 2

Γ2

�

<sup>1</sup> <sup>=</sup> *<sup>K</sup>*<sup>2</sup> <sup>−</sup> <sup>Γ</sup><sup>2</sup>

Schrodinger equation of a particle of mass *m*, moving in the potential *V* (*x*) :

2*m*

⎧ ⎨ ⎩

*V* (*x*) =

� *d*<sup>2</sup> *dx*<sup>2</sup> <sup>+</sup>

where *V* (*x*) is a piecewise-defined function ([13], III.1.6):

*n*<sup>1</sup> = *n*3. With *q* → −*q*, eq. (90) becomes:

problem.

It gives, for *q* odd:

and for *q* even:

*W*

Defining *K* through the equation:

we get, instead of (94), (95):

<sup>2</sup> <sup>=</sup> *<sup>a</sup>*, *<sup>γ</sup>*1*<sup>a</sup>* <sup>=</sup> *ak*<sup>0</sup>

the eigenvalue conditions (97), (98) take the form:

Putting:

Consequently, in our case, the boundary conditions request the continuity of *Ey* and of its derivative *dEy*/*dx* , at the slab boundaries. The equation (73), together with these boundary conditions, define a Sturm-Liouville problem, which determines the eigenvalues of *neff* or, equivalently, of *β*.

For the physics of optical fibers, the most interesting situation is that corresponding to an oscillatory solution inside the core and exponentially small ones outside the core (in the cladding):

$$E\_{\mathcal{Y}}\left(\mathbf{x}\right) = \mathbb{C}\_1 \exp\left(\gamma\_1 \mathbf{x}\right), \quad \gamma\_1 = k\_0 \sqrt{n\_{eff}^2 - n\_1^2} \tag{76}$$

$$\mathbf{x} = \mathbf{C}\_2 \sin \left(\gamma\_2 \mathbf{x} + \mathbf{a}\right), \qquad \gamma\_2 = k\_0 \sqrt{n\_2^2 - n\_{eff}^2} \tag{77}$$

$$=\mathcal{C}\_3 \exp\left(-\gamma\_3 \left(\mathbf{x} - \mathbf{W}\right)\right), \quad \gamma\_3 = k\_0 \sqrt{n\_{eff}^2 - n\_3^2} \tag{78}$$

As we just have seen, the boundary conditions are equivalent to the continuity of *Ey* (*x*) and of its derivative, *dEy* (*x*) /*dx* :

$$\frac{dE\_y\left(\mathbf{x}\right)}{d\mathbf{x}} = \gamma\_1 \mathbf{C}\_1 \exp\left(\gamma\_1 \mathbf{x}\right), \quad \gamma\_1 = k\_0 \sqrt{n\_{eff}^2 - n\_1^2} \tag{79}$$

$$\mathbf{x} = \gamma\_2 \mathbf{C}\_2 \cos \left(\gamma\_2 \mathbf{x} + \mathfrak{a}\right) \tag{80}$$

$$=-\gamma\_3 \mathbb{C}\_3 \exp\left(-\gamma\_3 \left(\mathbf{x} - \mathbf{W}\right)\right) \tag{81}$$

So, the continuity at *x* = 0 means:

$$\mathbf{C}\_1 = \mathbf{C}\_2 \sin \alpha \tag{82}$$

$$
\gamma\_1 \mathbb{C}\_1 = \gamma\_2 \mathbb{C}\_2 \cos \mathfrak{a} \tag{83}
$$

Similarily, at *x* = *W*:

$$\mathbf{C}\_2 \sin \left( \gamma\_2 \mathcal{W} + \mathfrak{a} \right) = \mathbf{C}\_3 \tag{84}$$

$$
\gamma\_2 \mathbb{C}\_2 \cos \left( \gamma\_2 W + \mathfrak{a} \right) = -\gamma\_3 \mathbb{C}\_3 \tag{85}
$$

Dividing (83) by (82), we get:

$$\frac{\gamma\_1}{\gamma\_2} = \cot \mathfrak{a},\tag{86}$$

$$\alpha = \operatorname{arccot} \frac{\gamma\_1}{\gamma\_2} + q\_1 \pi\_\prime \; q\_1 = 0, 1, 2, \dots \tag{87}$$

Dividing (85) by (81), we get:

$$\frac{\gamma\_3}{\gamma\_2} = -\cot\left(\gamma\_2 W + \mathfrak{a}\right),$$

$$-\operatorname{arccot}\frac{\gamma\_3}{\gamma\_2} - \mathfrak{a} + q\_2 \pi = \gamma\_2 W \tag{88}$$

Substitution of *α* from (87) into (88) gives:

$$-\operatorname{arccot}\frac{\gamma\_3}{\gamma\_2} - \operatorname{arccot}\frac{\gamma\_1}{\gamma\_2} + q\pi = \gamma\_2 W \tag{89}$$

Written in the form:

$$-\arctan\frac{\gamma\_2}{\gamma\_3} - \arctan\frac{\gamma\_2}{\gamma\_1} + q\pi = \gamma\_2 W \tag{90}$$

it coincides with eq. (20) Ch.III, vol.1, [13]. It is the energy eigenvalue equation for the Schrodinger equation of a particle of mass *m*, moving in the potential *V* (*x*) :

$$
\left[\frac{d^2}{d\mathbf{x}^2} + \frac{2m}{\hbar^2} \left(-V\left(\mathbf{x}\right) + \mathcal{E}\right)\right] \psi\left(\mathbf{x}\right) = \mathbf{0} \tag{91}
$$

where *V* (*x*) is a piecewise-defined function ([13], III.1.6):

$$V(\mathbf{x}) = \begin{cases} V\_{1\prime} & \mathbf{x} > a \\ V\_{2\prime} & a > \mathbf{x} > b \\ V\_{3\prime} & b > \mathbf{x} \end{cases} \tag{92}$$
 
$$V\_2 < V\_1 < V\_3$$

It is useful to consider a particular situation, when *n*<sup>1</sup> = *n*<sup>3</sup> in the optical waveguide, respectively when *V*<sup>1</sup> = *V*<sup>3</sup> = 0, *V*<sup>2</sup> < 0, in the quantum mechanical problem. In (91), −*V* (*x*) 0 is given, and we have to find the eigenvalues of the energy E < 0. For the optical waveguide (71), *�μ*0*ω*<sup>2</sup> is given, and the eigenvalues of the quantity <sup>−</sup>*β*<sup>2</sup> (essentially, the propagation constant *β*) must be obtained. Let us note once again that the refractive index in the optical waveguide corresponds to the opposite of the potential, in the quantum mechanical problem.

Let us investigate in greater detail the consequences of the particular situation just mentioned, *n*<sup>1</sup> = *n*3. With *q* → −*q*, eq. (90) becomes:

$$
\arctan \frac{\gamma\_2}{\gamma\_1} + \frac{q\pi}{2} = -\frac{\gamma\_2 W}{2} \tag{93}
$$

It gives, for *q* odd:

$$\frac{\gamma\_1}{\gamma\_2} = \tan \frac{\gamma\_2 W}{2} \tag{94}$$

and for *q* even:

$$\frac{\gamma\_2}{\gamma\_1} = -\tan\frac{\gamma\_2 W}{2} \tag{95}$$

Putting:

14 Will-be-set-by-IN-TECH

Consequently, in our case, the boundary conditions request the continuity of *Ey* and of its derivative *dEy*/*dx* , at the slab boundaries. The equation (73), together with these boundary conditions, define a Sturm-Liouville problem, which determines the eigenvalues of *neff* or,

For the physics of optical fibers, the most interesting situation is that corresponding to an oscillatory solution inside the core and exponentially small ones outside the core (in the

As we just have seen, the boundary conditions are equivalent to the continuity of *Ey* (*x*) and

 *n*2 *eff* <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

 *n*2 <sup>2</sup> <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

> *n*2 *eff* <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

 *n*2 *eff* <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

= *γ*2*C*<sup>2</sup> cos (*γ*2*x* + *α*) (80) = −*γ*3*C*<sup>3</sup> exp (−*γ*<sup>3</sup> (*x* − *W*)) (81)

*C*<sup>1</sup> = *C*<sup>2</sup> sin *α* (82) *γ*1*C*<sup>1</sup> = *γ*2*C*<sup>2</sup> cos *α* (83)

= cot *α*, (86)

+ *q*1*π*, *q*<sup>1</sup> = 0, 1, 2, ... (87)

− *α* + *q*2*π* = *γ*2*W* (88)

+ *qπ* = *γ*2*W* (89)

+ *qπ* = *γ*2*W* (90)

*C*<sup>2</sup> sin (*γ*2*W* + *α*) = *C*<sup>3</sup> (84) *γ*2*C*<sup>2</sup> cos (*γ*2*W* + *α*) = −*γ*3*C*<sup>3</sup> (85)

<sup>1</sup> (76)

*eff* (77)

<sup>3</sup> (78)

<sup>1</sup> (79)

*Ey* (*x*) = *C*<sup>1</sup> exp (*γ*1*x*), *γ*<sup>1</sup> = *k*<sup>0</sup>

= *C*<sup>2</sup> sin (*γ*2*x* + *α*), *γ*<sup>2</sup> = *k*<sup>0</sup>

= *C*<sup>3</sup> exp (−*γ*<sup>3</sup> (*x* − *W*)), *γ*<sup>3</sup> = *k*<sup>0</sup>

*dx* <sup>=</sup> *<sup>γ</sup>*1*C*<sup>1</sup> exp (*γ*1*x*), *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> *<sup>k</sup>*<sup>0</sup>

*γ*1 *γ*2

*γ*2

*γ*2

= − cot(*γ*2*W* + *α*),

<sup>−</sup> arccot *<sup>γ</sup>*<sup>1</sup>

<sup>−</sup> arctan *<sup>γ</sup>*<sup>2</sup>

*γ*2

*γ*1

*<sup>α</sup>* <sup>=</sup> arccot *<sup>γ</sup>*<sup>1</sup>

*γ*3 *γ*2

<sup>−</sup> arccot *<sup>γ</sup>*<sup>3</sup>

<sup>−</sup> arctan *<sup>γ</sup>*<sup>2</sup>

<sup>−</sup> arccot *<sup>γ</sup>*<sup>3</sup>

*γ*2

*γ*3

equivalently, of *β*.

of its derivative, *dEy* (*x*) /*dx* :

So, the continuity at *x* = 0 means:

Similarily, at *x* = *W*:

Dividing (83) by (82), we get:

Dividing (85) by (81), we get:

Written in the form:

Substitution of *α* from (87) into (88) gives:

*dEy* (*x*)

cladding):

$$\frac{W}{2} = a, \ \gamma\_1 a = a k\_0 \sqrt{n\_{eff}^2 - n\_1^2} = \Gamma\_1, \ \gamma\_2 a = a k\_0 \sqrt{n\_2^2 - n\_{eff}^2} = \Gamma\_2 \tag{96}$$

we get, instead of (94), (95):

$$\frac{\Gamma\_1}{\Gamma\_2} = \tan \Gamma\_2 \qquad (q \, odd) \tag{97}$$

$$\frac{\Gamma\_2}{\Gamma\_1} = -\tan \Gamma\_2 \quad (q \, even) \tag{98}$$

Defining *K* through the equation:

$$
\Gamma\_1^2 = \mathcal{K}^2 - \Gamma\_2^2 \tag{99}
$$

the eigenvalue conditions (97), (98) take the form:

$$\frac{\sqrt{K^2 - \Gamma\_2^2}}{\Gamma\_2} = \tan \Gamma\_2 \tag{100}$$

16 Will-be-set-by-IN-TECH 104 Trends in Electromagnetism – From Fundamentals to Applications Waveguides, Resonant Cavities, Optical Fibers and their Quantum Counterparts <sup>17</sup>

$$\frac{\Gamma\_2}{\sqrt{\mathcal{K}^2 - \Gamma\_2^2}} = -\tan\Gamma\_2\tag{101}$$

and substituting (115) in (116), we get:

*k* <sup>κ</sup> <sup>=</sup> tan

the conditions (118), (119) become respectively:

For *n*<sup>2</sup> even:

and for *n*<sup>2</sup> odd:

Putting

the well:

and we get:

defined parity.

*k* <sup>κ</sup> <sup>=</sup> tan

<sup>κ</sup> <sup>=</sup> <sup>−</sup> tan

−*ka* <sup>2</sup> <sup>+</sup> *π* 2 

*ka*

*<sup>ξ</sup>* <sup>=</sup> *ka*

tan *<sup>ξ</sup>* <sup>=</sup> <sup>−</sup> *<sup>ξ</sup>* 

cot *<sup>ξ</sup>* <sup>=</sup> *<sup>ξ</sup>* 

*kx* + arctan

<sup>2</sup> (*x*) <sup>=</sup> *<sup>B</sup>* sin

<sup>2</sup> (*x*) <sup>=</sup> *<sup>B</sup>* sin

Let us write now the wavefunction (106) using the expression (115) for *α*:

*<sup>u</sup>*<sup>2</sup> (*x*) <sup>=</sup> *<sup>B</sup>* sin

With (118) and (119), the equation (123) splits in two equations:

*u*∗

*u*∗∗

*U*∗

<sup>2</sup> (*y*) = *u*∗∗ 2 *y* + *a* 2 

*U*∗∗

<sup>2</sup> (*y*) = *u*<sup>∗</sup> 2 *y* + *a* 2 

*k*

−*ka* <sup>2</sup> <sup>+</sup> *n*2 2 *π* 

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 105

<sup>2</sup> , arctan

, arctan

<sup>2</sup> , *<sup>C</sup>* <sup>=</sup> *<sup>k</sup>*0*<sup>a</sup>*

*<sup>C</sup>*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>2</sup>

*<sup>C</sup>*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>n</sup><sup>π</sup>*

= (−1)

<sup>2</sup> (126)

*<sup>n</sup> B* sin *ky* (127)

*<sup>n</sup> B* cos *ky* (128)

*k* <sup>κ</sup> <sup>+</sup> *<sup>n</sup><sup>π</sup>*

*kx* <sup>−</sup> *ka*

*a*

= *B* sin (*ky* + *nπ*) = (−1)

*ky* + *π* <sup>2</sup> <sup>+</sup> *<sup>n</sup><sup>π</sup>*

So, the wavefunctions corresponding to the eigenvalues obtained from (121), (122) have well

Let us stress once more that the core of the optical-quantum analogy consists in eqs. (74), (75), which can be formulated as follows: the refractive index for the propagation of light plays a similar role to the potential, for the propagation of a quantum non-relativistic particle, and both the electric (or magnetic) field and the wave function are the solution of essentially

*kx* <sup>−</sup> *ka* <sup>2</sup> <sup>+</sup> *π* <sup>2</sup> <sup>+</sup> *<sup>n</sup><sup>π</sup>*

We translate now the coordinate *x*, so that the origin of the new axis is placed in the center of

*x* = *y* +

<sup>=</sup> *<sup>B</sup>* sin

*k* <sup>κ</sup> <sup>=</sup> <sup>−</sup>*ka*

> *k* <sup>κ</sup> <sup>=</sup> <sup>−</sup>*ka*

<sup>2</sup> <sup>+</sup> *π*

<sup>2</sup> (120)

, 0 < *x* < *a* (123)

(117)

(121)

(122)

(124)

(125)

<sup>2</sup> (118)

<sup>2</sup> (119)

So, the eigenvalue equation (90) splits into two simpler conditions (100), (101), carracterizing states with well defined parity, as we shall see further on (of course, the parity of *q*, mentioned just after (93), has nothing to do with the parity of states).

We shall analyze now the same problem, starting from the quantum mechanical side.

#### **9.2 The quantum mechanical problem: the particle in a rectangular potential well**

We discuss now the Schrodinger equation for a particle in a rectangular potential well ([17], v.1, pr.25), one of the simplest problems of quantum mechanics:

$$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V\left(x\right)\right]\psi\left(x\right) = \mathcal{E}\psi\left(x\right)\tag{102}$$

$$V(\mathbf{x}) = \begin{cases} -\mathcal{U}, & 0 < \mathbf{x} < a \\ 0, & else were \end{cases} \tag{103}$$

Let be:

$$\mathcal{E} = -\frac{\hbar^2 \varkappa^2}{2m};\ \mathcal{U} = \frac{\hbar^2 k\_0^2}{2m};\ \ k^2 = k\_0^2 - \varkappa^2 \tag{104}$$

We are looking for bound states inside the well:

$$
\mu\_1(\mathbf{x}) = A \exp\left(\varkappa \mathbf{x}\right), \quad \mathbf{x} < \mathbf{0} \tag{105}
$$

$$
\mu\_2\left(\mathbf{x}\right) = B\sin\left(k\mathbf{x} + \mathbf{a}\right), \quad 0 < \mathbf{x} < a \tag{106}
$$

$$u\_3\left(\mathbf{x}\right) = D \exp\left(\varkappa \left(a - \mathbf{x}\right)\right), \qquad a < \mathbf{x} \tag{107}$$

The wavefunction and its derivatives:

$$u\_1' \left( \mathbf{x} \right) = \varkappa A \exp \left( \varkappa \mathbf{x} \right), \quad \mathbf{x} < \mathbf{0} \tag{108}$$

$$
\mu\_2' \left( \mathbf{x} \right) = kB \cos \left( k \mathbf{x} + \mathbf{a} \right), \quad \mathbf{0} < \mathbf{x} < \mathbf{a} \tag{109}
$$

$$\mu\_{\mathfrak{Z}}'(\mathbf{x}) = -\varkappa \mathcal{D} \exp \left( \varkappa \left( a - \mathbf{x} \right) \right), \qquad a < \mathbf{x} \tag{110}$$

must be continuous in *x* = 0:

$$A = B \sin \mathfrak{a} \tag{111}$$

$$k \varkappa A = kB \cos \mathfrak{a} \tag{112}$$

and in *x* = *a*:

$$B\sin\left(ka+\mathfrak{a}\right)=\mathbb{C}\tag{113}$$

$$kB\cos\left(ka+\varkappa\right)=-\varkappa\mathbb{C}\tag{114}$$

Dividing (111), (112):

$$\frac{1}{\varkappa} = \frac{1}{k} \tan \kappa, \qquad \kappa = \arctan \frac{k}{\varkappa} + n\pi \tag{115}$$

and (113), (114):

$$\frac{1}{k}\tan\left(ka+\pi\right)=-\frac{1}{\varkappa}, \qquad ka+\pi=-\arctan\frac{k}{\varkappa}+n\_1\pi\tag{116}$$

and substituting (115) in (116), we get:

$$\frac{k}{2\varkappa} = \tan\left(-\frac{ka}{2} + \frac{n\_2}{2}\pi\right) \tag{117}$$

For *n*<sup>2</sup> even:

16 Will-be-set-by-IN-TECH

So, the eigenvalue equation (90) splits into two simpler conditions (100), (101), carracterizing states with well defined parity, as we shall see further on (of course, the parity of *q*, mentioned

We discuss now the Schrodinger equation for a particle in a rectangular potential well ([17],

<sup>−</sup>*U*, 0 <sup>&</sup>lt; *<sup>x</sup>* <sup>&</sup>lt; *<sup>a</sup>*

0 2*m*

; *k*<sup>2</sup> = *k*<sup>2</sup>

*u*<sup>1</sup> (*x*) = *A* exp (κ*x*), *x* < 0 (105)

<sup>1</sup> (*x*) = κ*A* exp (κ*x*), *x* < 0 (108)

*A* = *B* sin *α* (111) κ*A* = *kB* cos *α* (112)

*B* sin (*ka* + *α*) = *C* (113) *kB* cos (*ka* + *α*) = −κ*C* (114)

*k*

<sup>κ</sup> <sup>+</sup> *<sup>n</sup><sup>π</sup>* (115)

<sup>κ</sup> <sup>+</sup> *<sup>n</sup>*1*<sup>π</sup>* (116)

<sup>2</sup> (*x*) = *kB* cos (*kx* + *α*), 0 < *x* < *a* (109)

<sup>3</sup> (*x*) = −κ*D* exp (κ (*a* − *x*)), *a* < *x* (110)

*k*

*u*<sup>2</sup> (*x*) = *B* sin (*kx* + *α*), 0 < *x* < *a* (106) *u*<sup>3</sup> (*x*) = *D* exp (κ (*a* − *x*)), *a* < *x* (107)

We shall analyze now the same problem, starting from the quantum mechanical side.

**9.2 The quantum mechanical problem: the particle in a rectangular potential well**

*dx*<sup>2</sup> <sup>+</sup> *<sup>V</sup>* (*x*)

; *<sup>U</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*k*<sup>2</sup>

*<sup>k</sup>* tan *<sup>α</sup>*, *<sup>α</sup>* <sup>=</sup> arctan

<sup>κ</sup> , *ka* <sup>+</sup> *<sup>α</sup>* <sup>=</sup> <sup>−</sup> arctan

*d*2

*V* (*x*) =

= − tan Γ<sup>2</sup> (101)

*ψ* (*x*) = E*ψ* (*x*) (102)

<sup>0</sup> <sup>−</sup> <sup>κ</sup><sup>2</sup> (104)

0, *elsewere* (103)

 Γ2 *<sup>K</sup>*<sup>2</sup> <sup>−</sup> <sup>Γ</sup><sup>2</sup> 2

just after (93), has nothing to do with the parity of states).

v.1, pr.25), one of the simplest problems of quantum mechanics:

<sup>E</sup> <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>κ<sup>2</sup> 2*m*

*u*�

*u*�

1 <sup>κ</sup> <sup>=</sup> <sup>1</sup>

*<sup>k</sup>* tan (*ka* <sup>+</sup> *<sup>α</sup>*) <sup>=</sup> <sup>−</sup> <sup>1</sup>

1

*u*�

We are looking for bound states inside the well:

The wavefunction and its derivatives:

must be continuous in *x* = 0:

and in *x* = *a*:

Dividing (111), (112):

and (113), (114):

 − *h*¯ 2 2*m*

Let be:

$$\frac{k}{\varkappa} = -\tan\frac{ka}{2}, \qquad \arctan\frac{k}{\varkappa} = -\frac{ka}{2} \tag{118}$$

and for *n*<sup>2</sup> odd:

$$\frac{k}{\varkappa} = \tan\left(-\frac{ka}{2} + \frac{\pi}{2}\right), \qquad \arctan\frac{k}{\varkappa} = -\frac{ka}{2} + \frac{\pi}{2} \tag{119}$$

Putting

$$\text{C} = \frac{ka}{2}, \text{ C} = \frac{k\_0 a}{2} \tag{120}$$

the conditions (118), (119) become respectively:

$$\tan \xi = -\frac{\xi}{\sqrt{\mathcal{C}^2 - \xi^2}}\tag{121}$$

$$\cot \xi = \frac{\xi}{\sqrt{\mathcal{C}^2 - \xi^2}} \tag{122}$$

Let us write now the wavefunction (106) using the expression (115) for *α*:

$$\mu\_2\left(\mathbf{x}\right) = B\sin\left(k\mathbf{x} + \arctan\frac{k}{\varkappa} + n\pi\right), \quad 0 < \mathbf{x} < a \tag{123}$$

With (118) and (119), the equation (123) splits in two equations:

$$u\_2^\*\left(\mathbf{x}\right) = B\sin\left(k\mathbf{x} - \frac{ka}{2} + n\pi\right) \tag{124}$$

$$u\_2^{\*\*}\left(\mathbf{x}\right) = B\sin\left(k\mathbf{x} - \frac{ka}{2} + \frac{\pi}{2} + n\pi\right) \tag{125}$$

We translate now the coordinate *x*, so that the origin of the new axis is placed in the center of the well:

$$x = y + \frac{a}{2} \tag{126}$$

and we get:

$$\, \, \, \, U\_2^\* \left( y \right) = u\_2^\* \left( y + \frac{a}{2} \right) = B \sin \left( ky + n\pi \right) = (-1)^n B \sin ky \tag{127}$$

$$\mu U\_2^{\*\*}\left(y\right) = \mu\_2^{\*\*}\left(y + \frac{a}{2}\right) = B\sin\left(ky + \frac{\pi}{2} + n\pi\right) = (-1)^n B\cos ky\tag{128}$$

So, the wavefunctions corresponding to the eigenvalues obtained from (121), (122) have well defined parity.

Let us stress once more that the core of the optical-quantum analogy consists in eqs. (74), (75), which can be formulated as follows: the refractive index for the propagation of light plays a similar role to the potential, for the propagation of a quantum non-relativistic particle, and both the electric (or magnetic) field and the wave function are the solution of essentially

the 2DESs formed in semiconductor heterostructures, but is inappropriate for metallic thin films, where the electron density is much higher, and even at nanoscopic scale, there are tens of occupied bands; so, the system is merely 3D. Consequently, the dimensionality of a system depends not only on its geometry, but also on its electron concentration. Let us remind that the conductive / dielectric properties of a sample depends on frequency of electromagnetic waves: so even basic classification of materials is not necessarely intrinsec, but it might

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 107

We shall discuss now the concept of transverse modes or subbands, which are analogous to the transverse modes of electromagnetic waveguides [11]. In narrow conductors, the different transverse modes are well separated in energy, and such conductors are often called electron

We consider a rectangular conductor that is uniform in the *x*−direction and has some transverse confining potential *U*(*y*). The motion of electrons in such a conductor is described

+ *U* (*y*)

We assume a constant magnetic field *B* in the *z*−direction, perpendicular to the plane of the

+ *U* (*y*)

+ *U* (*y*)

Ψ (*x*, *y*) = EΨ (*x*, *y*) (132)

Ψ (*x*, *y*) = EΨ (*x*, *y*) (134)

*χ* (*y*) = E*χ* (*y*) (136)

<sup>0</sup>*y*<sup>2</sup> (137)

*Ax* = −*By*, *Ay* = 0 (133)

<sup>√</sup>*<sup>L</sup>* exp (*ikx*) *<sup>χ</sup>* (*y*) (135)

2

depend of the value of some parameters.

waveguides.

Writing

by the effective mass eq (131):

so that the effective-mass equationcan be rewritten as:

<sup>E</sup>*<sup>s</sup>* <sup>+</sup> (*px* <sup>+</sup> *eBy*)

2*m*

<sup>E</sup>*<sup>s</sup>* <sup>+</sup> (*hk*¯ <sup>+</sup> *eBy*)

2*m*

we get for the transverse function the equation: 

**11. Transverse modes (or magneto-electric subbands)**

<sup>E</sup>*<sup>s</sup>* <sup>+</sup> (*ih*¯ <sup>∇</sup> <sup>+</sup> *<sup>e</sup>***A**)

2*m*

conductor, which can be represented by a vector potential defined by:

2

<sup>Ψ</sup> (*x*, *<sup>y</sup>*) <sup>=</sup> <sup>1</sup>

2

<sup>+</sup> *<sup>p</sup>*<sup>2</sup> *y* 2*m*

*<sup>U</sup>* (*y*) <sup>=</sup> <sup>1</sup>

is often a good description of the actual potential in many electron wave guides.

We are interested in the nature of the transverse eigenfunctions and eigenenergies for different combinations of the confining potential *U*and the magnetic field *B*. A parabolic potential

> 2 *mω*<sup>2</sup>

<sup>+</sup> *<sup>p</sup>*<sup>2</sup> *y* 2*m*

the same (Helmholtz) equation. So, if the dynamics of a particle, given by the Schrodinger equation, can be considered as the central aspect of quantum mechanics, the scattering of light by a medium with refractive index *n* −→*r* can be considered as the central aspect of optics, at least when the Maxwell equations can be reduced to a Helmholtz equation. Remembering Goethe's opinion, that the "Urphänomenon" of light science is the scattering of light on a "turbid" medium, one could remark that his theory of colours is not always as unrealistic as it was generally considered. [18]
