Fields in Dispersive Media

Abstract

It is not just the case that matter affects the propagation of light—or more specifically electromagnetic (EM) radiation—it is also the case that light affects the matter through which it propagates. Conversely, this affects the propagation of light through the medium, but in a much more specific way; this effect is a function of the properties of both the material and the incident EM radiation. We will additionally discuss the effects of dispersion in confined (bounded) media, i.e., where the dispersion is a function of the arrangement of certain materials and unbounded media where EM radiation is free to propagate undisturbed. This will be important when we discuss the propagation electric field signals of such media as well.

1. Introduction

Incident electromagnetic (EM) radiation excites the molecules in a material, and these molecules become polarized; they respond according to the direction of the electric field. Figure 1 shows the initial step of the process of material polarization.

Figure 1.

An EM traveling wave is incident on a molecule of a particular material. This induces a unique response in molecules of the material.

It should be adequately noted that there are several ways that material response to light-matter interaction can be analyzed. For the purposes of relevance, this chapter will analyze the mechanical response of the atom—as illustrated in Figure 2—in response to incident EM radiation waves.

Figure 2.

The mechanical response of the material to incoming monochromatic EM radiation. The vibrational response of the molecule is a function of its properties. This same principle is also related to the color of the flame that a molecule produces [1].

In this chapter we will discuss polarization at the level of the interacting material, the effects of dispersion on a train of pulses, various types of polarization, and methods for reducing polarization to maintain the integrity of optical signals. But first, some words on non-dispersion in unbounded media, a more general term for what would normally be called “free space.”

Unbounded media are the baseline of understanding EM wave propagation. In unbounded media, waves are free to propagate unperturbed. Examples of unbounded media include the ocean, the air, and outer space. An unbounded medium would be the ideal location for an isotropic antenna as the radiation would be free to propagate in all directions, only weakening in accordance with the inverse square law. Such is ideal for radio towers which produce low-frequency EM waves which can propagate for kilometers and reach many people.

Additionally, unbounded media are generally non-dispersive. This is to say that the speed of energy propagation is orthogonal to the frequency of the said propagating energy. In the previous paragraph, it was said the waves in unbounded media are free to propagate unperturbed. This is not completely true. Natural disturbances within the air or sea can interfere with the propagation of energy therein. However, these natural impediments are not necessarily treated as features of the media itself. It is for this reason that direct or line-of-sight propagation is key in facilitating communication between antennas and cell towers.

2. Understanding dispersion

Dispersion can be difficult to understand. There are picture balls on two parallel infinite conveyor belts running at different speeds, Belt A at the top and Belt B at the bottom. The conveyor belts in Figure 3 carry balls at 2.5 cm/s and 3.0 cm/s, respectively. Additionally, the centers of the balls on each belt are separated by 10 cm and 12.5 cm, respectively. At time $t=0$ , the centers of Red Ball 0 (RB₀) and Blue Ball 0 (BB₀) are aligned. However, the center of BB₋₁ trails that of RB₋₁ by a distance of 2.5 cm.

Figure 3.

A mechanical kinematic illustration of the principle of the dispersion.

Since Belt B is traveling faster than Belt A, the distance between BB₋₁ and RB₋₁ will decrease. From the perspective of Belt A, Belt B is moving with a velocity of 0.5 cm/s. At time $t=5$ , the center of BB₋₁ is aligned with that of RB₋₁; however, in that time the balls on Belt B have moved a net 15 cm, while those on Belt A have moved a net 12.5 cm. However, if one were to see the conveyor belts as a group, the belts would appear to have moved 2.5 cm (measuring from location of alignment) within that time, giving the conveyor belts a “group velocity” of 0.5 m/s even though the individual components are moving faster.

The situation described is analogous to the behavior of a wave propagating in free space. As the individual waves travel with their respective phase velocities along the guide, the two peaks become disaligned for a period [2]. Because the center of RB₋₁ will have traveled a distance ${\lambda }_{\mathrm{RB}}+\mathrm{\Delta }z$ with a phase velocity $v_{z\mathrm{RB}}$ and the center of BB₋₁ will have traveled a distance ${\lambda }_{\mathrm{BB}}+\mathrm{\Delta }z$ with phase velocity $v_{z\mathrm{BB}}$ , one can then write a set of equations as

$${\lambda }_{\mathit{RB}}+\Delta z=v_{z,\mathit{RB}}\Delta t$$ E1

$${\lambda }_{\mathit{BB}}+\Delta z=v_{z,\mathit{BB}}\Delta t$$ E2

We can then write $\mathrm{\Delta }t$ as

$$\Delta t=\frac{{\lambda }_{\mathit{RB}}-{\lambda }_{\mathit{BB}}}{v_{z,\mathit{RB}}-v_{z,\mathit{BB}}}$$ E3

In the same way, one can express $\mathrm{\Delta }z$ as

$$\mathrm{\Delta }z=\frac{{\lambda }_{\mathrm{RB}}{v}_{z,\mathrm{BB}}-{\lambda }_{\mathrm{BB}}{v}_{z,\mathrm{RB}}}{v_{z,\mathrm{RB}}-v_{z,\mathrm{BB}}}$$ E4

Finally, one can express the group velocity as

$$\frac{\Delta z}{\Delta t}=v_z=\frac{{\lambda }_{\mathit{RB}}{v}_{z,\mathit{BB}}-{\lambda }_{\mathit{BB}}{v}_{z,\mathit{RB}}}{{\lambda }_{\mathit{RB}}-{\lambda }_{\mathit{BB}}}=\frac{{\lambda }_{\mathit{RB}}{\lambda }_{\mathit{BB}}{f}_{\mathit{BB}}-{\lambda }_{\mathit{BB}}{\lambda }_{\mathit{RB}}{f}_{\mathit{RB}}}{{\lambda }_{\mathit{RB}}-{\lambda }_{\mathit{BB}}}$$ E5

$$=\frac{{\lambda }_{\mathit{BB}}{\lambda }_{\mathit{RB}}\left(f_{\mathit{BB}}-f_{\mathit{RB}}\right)}{{\lambda }_{\mathit{RB}}-{\lambda }_{\mathit{BB}}}=\frac{f_{\mathit{BB}}-f_{\mathit{RB}}}{\frac{1}{{\lambda }_{\mathit{BB}}}-\frac{1}{{\lambda }_{\mathit{RB}}}}\to \frac{{\omega }_{\mathit{BB}}-{\omega }_{\mathit{RB}}}{2\pi \left(\frac{1}{{\lambda }_{\mathit{BB}}}-\frac{1}{{\lambda }_{\mathit{RB}}}\right)}$$

We know that the wavevector $k=2\pi /\lambda$ , so one express the group velocity as

$$v_g=\frac{\mathit{d\omega }}{\mathit{dk}}$$ E6

This is to say the group velocity is the degree change in the temporal frequency for every change in the spatial frequency. What this means is that the value of the frequency is on some level a function of the k-vector and ultimately a function of the wavelength. As illustrated in Figure 4, this has implications for the propagation of light through various media as different wavelengths will travel at different velocities and take various pathways therein [3].

Figure 4.

The separation of white light into different colors by prismatic dispersion.

3. Plasmonic dispersion

The Lorentz force is the force on a point charge due to EM fields. As discussed in the Introduction, EM radiation causes motion among the particles in a material as shown in Figure 2. The direction (vector) of this force is called the polarization. Assuming a sea of free particles in vacuum [4], the strength of the Lorentz force on a point charge is expressed as

$$\overrightarrow{\mathit{F}}=q\left(\overrightarrow{\mathit{E}}+\overrightarrow{\mathit{v}}\times \overrightarrow{\mathit{B}}\right),$$ E7

where $\overrightarrow{\mathit{F}}$ , q, $\overrightarrow{\mathbf{E}}$ , $\overrightarrow{\mathit{v}}$ , and $\overrightarrow{\mathbf{B}}$ are the force, the charge, the electric field strength, the particle velocity, and the magnetic flux density, respectively. For our purposes, we consider an isotropic material where the electric permittivity is a simple scalar $ϵ$ . This means that the Lorentz force vector will be unidirectional. In birefringent and anisotropic, the permittivity would be expressed as a tensor, and the Lorentz force vector would be multidirectional.

We additionally assume a dielectric unmagnetized material such that the value of $\overrightarrow{\mathbf{B}}$ is assumed to be zero. This makes the response of the atoms into a simple harmonic oscillator. In this case we can express the Lorentz force as simply

$$\overrightarrow{\mathit{F}}=q\overrightarrow{\mathit{E}}\mathbf{=}m\overrightarrow{a}=m\left(\frac{d^2\overrightarrow{\mathit{d}}}{d{t}^2}\right)=m{\left(\mathit{j\omega }\right)}^2\overrightarrow{\mathit{d}}=-m{\omega }^2\overrightarrow{\mathit{d}},$$ E8

where $m$ , $\omega$ , and $\overrightarrow{\mathit{d}}$ are the mass, the angular frequency, and dipole vector, respectively. We can simplify to get the dipole vector as

$$\overrightarrow{\mathit{d}}=-\frac{q\overrightarrow{\mathit{E}}}{m{\omega }^2}\mathbf{\to }\overrightarrow{\mathit{P}}\mathbf{=}\mathit{Nq}\overrightarrow{\mathit{d}}\mathbf{=}\mathbf{-}\frac{N{q}^2\overrightarrow{\mathit{E}}}{m{\omega }^2}\mathbf{,}$$ E9

where $\overrightarrow{\mathbf{P}}$ is the polarization vector and $N$ is the dipole density. The electric displacement field $\overrightarrow{\mathbf{D}}$ is the permittivity $ϵ$ multiplied by the electric field $\overrightarrow{\mathbf{E}}$ , but in dielectric materials, the electric field induces a response in the material known as the polarization. This effect is added to the original displacement field to make it so that

$$\overrightarrow{\mathit{D}}\mathbf{=}{ϵ}_0\overrightarrow{\mathit{E}}\mathbf{+}\overrightarrow{\mathit{P}}\mathbf{=}\left({ϵ}_0-\frac{N{q}^2}{m{\omega }^2}\right)\overrightarrow{\mathit{E}}$$ E10

As a result, the effective permittivity is expressed in (11) as

$$ϵ={ϵ}_0\left(1-\frac{N{q}^2}{{ϵ}_{0}m{\omega }^2}\right)={ϵ}_0\left(1-\frac{{\omega }_p^2}{{\omega }^2}\right),$$ E11

where the plasma frequency ${\omega }_p=q\sqrt{N/{ϵ}_{0}m}$ .

The plasma frequency is the natural oscillation frequency of the displaced electrons in a neutral plasma [4, 5] of free electrons where it is assumed that collisions are rare. Altogether we can express the dispersion relation thusly

$$k^2={\omega }^2\mathit{ϵ\mu }={ϵ}_0{\mu }_0\left({\omega }^2-{\omega }_p^2\right)\to \omega \left(k\right)=\sqrt{k^2{c}^2+{\omega }_p^2}$$ E12

This means that the group velocity is expressed as

$$v_g=\frac{k{c}^2}{\sqrt{k^2{c}^2+{\omega }_p^2}}=c\sqrt{1-\left(\frac{{\omega }_p^2}{{\omega }^2}\right)}$$ E13

In a material like germanium (Ge) the intrinsic carrier density $N_{\mathrm{Ge}}$ is $2.5\times {10}^{19}{\mathrm{m}}^{-3}$ [6]; the rest of the variables are known constants. This gives germanium a plasma frequency of $f_{p,\mathrm{Ge}}=44.8\mathrm{GHz}$ . We apply the relation in (13) to germanium to measure the frequency-dependent group velocity of light therein as demonstrated in Figure 5.

Figure 5.

The group velocity of light through germanium as a function of frequency.

Below the plasma frequency, the value of the k-vector becomes purely imaginary, meaning that the electric field is purely evanescent—and non-propagating—at that point. The group velocity is the slope of the dispersion curve in Figure 6. For both curves, the lowest allowable frequency is the plasma frequency. Below this frequency, the permittivity is negative, the k-vector is imaginary, and, consequently, the field simply evanesces into the material and propagation completely ceases.

Figure 6.

The dispersion relation for germanium where $\omega$ is function of $\mathit{k}$ .

4. Dispersion in conductive media

In the previous section, we derived plasmonic dispersion from understanding the Lorentz force law as it pertains to electrons in a dielectric material. Dispersion can also occur in a conducting material where the charges therein are unbound. When we apply our knowledge of electromagnetics and electrostatics to a conductor, we discover a few things:

1. There is zero net charge within a conductor, although charges can still randomly move around therein as seen in Figure 7.

2. All the charge rests on the surface of the conductor, perpendicular to the surface.

3. The presence of an electric field induces the motion of electrons, creating a conduction current [7].

Figure 7.

The distribution of freely moving protons (red) and electrons (blue) in a conductive material in electrostatic equilibrium where ${\mathit{Q}}_{\mathbf{\Sigma }}\mathit{=}\mathbf{0}$ .

Additionally, in a conductor the free charges are always pushed to the surface so that electrostatic equilibrium is maintained. This means that the gradient of the electric field $\overrightarrow{\mathbf{\nabla }}·\overrightarrow{\mathbf{E}}$ equals zero. Things are different in the presence of an electric field, e.g., Figure 8, where the electrons are pulled in the opposite direction of the electric field.

Figure 8.

The kinetic behavior of electrons in the presence of an electric field produces a steady-state current known as the conduction current.

With regard to conductive media, we can do the same thing through an understanding and application of Faraday’s law which states that a time-varying magnetic flux density always accompanies a nonconservative electric field, i.e., a field $\overrightarrow{\mathbf{E}}$ for which $\overrightarrow{\mathbf{\nabla }}\mathit{\times }\overrightarrow{\mathbf{E}}\mathit{=}\mathbf{0}$ . This is demonstrated by the equation

$$\overrightarrow{\nabla }\mathbf{\times }\overrightarrow{\mathit{E}}=-\frac{\partial \overrightarrow{\mathit{B}}}{\partial t}=-{\mu }_0\frac{\partial \overrightarrow{\mathit{H}}}{\partial t}$$ E14

We can combine this with Ampère’s law where the curl of the magnetic field generates a time-varying electric field such that

$$\overrightarrow{\nabla }\mathbf{\times }\overrightarrow{\mathit{H}}={\overrightarrow{\mathit{J}}}_f+ϵ\frac{\partial \overrightarrow{\mathit{E}}}{\partial t}=\sigma \overrightarrow{\mathit{E}}+\frac{\partial \overrightarrow{\mathit{E}}}{\partial t}=\left(\sigma +\mathit{j\omega ϵ}\right)\overrightarrow{\mathit{E}}\mathbf{=}\mathit{j\omega }{ϵ}_{\mathit{eff}}\overrightarrow{\mathit{E}},$$ E15

where ${\overrightarrow{\mathbf{J}}}_f$ is the free current density vector produced in the presence of an electric field as exemplified in Figure 8 and $\sigma$ is the conductivity. We can calculate the effective permittivity this way such that ${ϵ}_{\mathit{eff}}=ϵ\left(1-\frac{\mathit{j\sigma }}{\mathit{\omega ϵ}}\right)$ . In this way we can write the dispersion relation.

$$k=\omega \sqrt{{\mu }_0{ϵ}_{\mathit{ff}}\left(\omega \right)}=\omega \sqrt{\frac{{\mu }_0ϵ}{{ϵ}_0}}\sqrt{1-\frac{\mathit{j\sigma }}{\mathit{\omega ϵ}}}=\sqrt{\frac{{\mu }_0ϵ}{{ϵ}_0}}\sqrt{{\omega }^2-\frac{\mathit{j\sigma \omega }}{ϵ}}$$

$$=\sqrt{\frac{{\mu }_0ϵ}{{ϵ}_0}}\sqrt[4]{{\omega }^4+\left(\frac{{\sigma }^2{\omega }^2}{{ϵ}^2}\right)}{e}^{-i\left(\frac{{\mathit{tan}}^{-1}\left(\frac{\sigma }{\mathit{ϵ\omega }}\right)}{2}\right)}$$ E16

This wavenumber $k$ is also complex where $k=k^{\prime }-j{k}^{\prime \prime }$ , where

$$\begin{array}{cc}{k}^{\prime }& =\sqrt{\frac{{\mu }_0ϵ}{{ϵ}_0}}\sqrt[4]{{\omega }^4+\left(\frac{{\sigma }^2{\omega }^2}{{ϵ}^2}\right)}cos\left(\frac{{tan}^{-1}\frac{\sigma }{\mathit{ϵ\omega }}}{2}\right)\leftrightarrow \\ k^{\prime \prime }& =\sqrt{\frac{{\mu }_0ϵ}{{ϵ}_0}}\sqrt[4]{{\omega }^4+\left(\frac{{\sigma }^2{\omega }^2}{{ϵ}^2}\right)}sin\left(\frac{{tan}^{-1}\frac{\sigma }{\mathit{ϵ\omega }}}{2}\right).\end{array}$$ E17

The real part of the wavenumber $k^{\prime }$ represents wavenumber inside the material and the imaginary part $k^{\prime \prime }$ represents the attenuation coefficient inside the material.

$$v_g\mathbf{=}{\left(\frac{d{k}^{\prime }}{\mathit{d\omega }}\right)}^{-1}=\sqrt{\frac{{ϵ}_0}{{\mu }_0ϵ}}{\left\{\frac{\left(\frac{{\sigma }^2}{{ϵ}^2}+2{\omega }^2\right)cos\left(\frac{{tan}^{-1}\frac{\sigma }{\mathit{ϵ\omega }}}{2}\right)}{2\omega \left(\frac{{\sigma }^2}{{ϵ}_0^2}+{\omega }^2\right)}+\frac{\frac{\sigma }{ϵ}\sqrt[4]{{\omega }^4+\left(\frac{{\sigma }^2{\omega }^2}{{ϵ}^2}\right)}}{\left({\omega }^2+\frac{{\sigma }^2}{{ϵ}^2}\right)}sin\left(\frac{{tan}^{-1}\frac{\sigma }{\mathit{ϵ\omega }}}{2}\right)\right\}}^{-1}$$

$$=\frac{2\sqrt{\frac{{ϵ}_0}{{\mu }_0ϵ}}\left({\omega }^2+\frac{{\sigma }^2}{{ϵ}^2}\right)}{\left(\frac{{\sigma }^2}{{ϵ}^2\omega }+2\omega \right)\mathit{cos}\left(\frac{{\mathit{tan}}^{-1}\frac{\sigma }{\mathit{ϵ\omega }}}{2}\right)+\frac{\sigma }{ϵ}\sqrt[4]{{\omega }^4+\left(\frac{{\sigma }^2{\omega }^2}{{ϵ}^2}\right)}\mathit{sin}\left(\frac{{\mathit{tan}}^{-1}\frac{\sigma }{\mathit{ϵ\omega }}}{2}\right)}$$ E18

Of course, there are various conditions that could simplify Eq. (16). The $\sigma /\mathit{\omega ϵ}$ component in (15) is known as the dielectric loss tangent $tan\delta$ [8]. This figure $\delta$ describes a materials inherent ability to dissipate electromagnetic energy within it. For example, a material with a high conductivity will absorb—dissipate—EM radiation more quickly. Such a material has would have a high loss tangent compared to a good dielectric.

We can express the dielectric relaxation time ${\tau }_d$ as $ϵ/\sigma$ . In cases that $\delta \ll 1$ or ${\tau }_d^{-1}\ll \omega$ —as would be the case in a lossy dielectric—we can approximate the permittivity as ${ϵ}_{\mathit{eff}}\approx 1-\frac{\mathit{j\sigma }}{2\mathit{\omega ϵ}}$ using the binomial expansion theorem. The material behaves like a standard dielectric as $v_g=c/n$ , where $n$ is the index of refraction, and $k^{\prime \prime }=\mathit{\sigma \eta }/2$ , where $\eta$ is the impedance of the material $\sqrt{{\mu }_0/ϵ}$ .

In cases where $1\ll \delta$ or $\omega \ll {\tau }_d^{-1}$ , the case of a good conductor, $k\approx \sqrt{{\mathit{\sigma \mu }}_0\omega /2}\left(1-j\right)$ and $v_g\approx \sqrt{8\omega /\sigma {\mu }_0}$ . In the latter case, the field penetrates only slightly into the material as $\mathcal{O}\left(k^{\prime }\right)=\mathcal{O}\left(k^{\prime \prime }\right)$ .

$$\frac{2\sqrt{\frac{{ϵ}_0}{{\mu }_0ϵ}}\left({\omega }^2+\frac{{\sigma }^2}{{ϵ}^2}\right)}{\left(\frac{{\sigma }^2}{{ϵ}^2\omega }+2\omega \right)\mathit{cos}\left(\frac{{\mathit{tan}}^{-1}\frac{\sigma }{\mathit{ϵ\omega }}}{2}\right)+\frac{\sigma }{ϵ}\sqrt[4]{{\omega }^4+\left(\frac{{\sigma }^2{\omega }^2}{{ϵ}^2}\right)}\mathit{sin}\left(\frac{{\mathit{tan}}^{-1}\frac{\sigma }{\mathit{ϵ\omega }}}{2}\right)}$$ E19

Additionally, the attenuation is a new phenomenon resulting from the fact that the k-vector is complex. When we take the reciprocal of this coefficient get another quantity known as the skin depth or penetration depth ${\delta }_p$ , the extent to which the field penetrates the material. In a good conductor, ${\delta }_p$ takes a value $\sqrt{2/\sigma {\mu }_0\omega }$ . The values of $ϵ$ and $\sigma$ are listed in Table 1 for some good conductors [9].

Material	Conductivity (S/m) @ $T=293\mathrm{K}$
Silver	$6.1\times {10}^7$
Copper	$5.8\times {10}^7$
Gold	$4.1\times {10}^7$
Aluminum	$3.5\times {10}^7$
Brass	$1.5\times {10}^7$

Table 1.

Conductivities of some good conductors.

Good conductors are characterized by an abundance of free electrons. Good conductors (good dielectrics), on the other hand, are characterized by a paucity of such electrons. Examples of good dielectrics are enumerated in Table 2 [10].

Material	Permittivity (F/m)
Calcium	$2.685\times {10}^{-11}$
Marble	$7.08\times {10}^{-11}$
Silicon dioxide (SiO2)	$3.452\times {10}^{-11}$
Slate	$3.54\times {10}^{-11}$
Polytetrafluoroethylene (PTFE)	$1.77\times {10}^{-11}$

Table 2.

Conductivities of some good dielectrics.

Figure 9 shows the interaction between an electric field and an good conductor (e.g., gold) at a wavelength of 1.55 $\mathrm{\mu m}$ . The field decays very rapidly as gold has a skin depth of 5.66 nm and a dielectric relaxation time of 1.49 attoseconds (10⁻¹⁸ s) at this wavelength. This contrasts with a lossy dielectric where the skin depth is much larger, and there is thus more field penetration as displayed in Figure 10 with germanium where the skin depth is roughly a centimeter and the dielectric relaxation time is 66.1 picoseconds.

Figure 9.

The rapid decay of an electric field upon impact interaction with gold.

Figure 10.

The comparatively slow decay of an electric field upon impact interaction with germanium.

Conductive media have more free electrons that move in the presence of an electric field than those that simply vibrate. Conductive media produce a loss (attenuation) component in propagation of incoming EM radiation. Additionally, the drift movement of electrons in conductive media causes heat to which the electrical energy from the incoming field is transferred. This energy conversion causes the dissipation of the field. Nevertheless, in the context of the phenomenon of dispersion, the degree of this attenuation is a function of the frequency of the incoming EM radiation.

5. Modal dispersion

So far, this chapter has covered the form that dispersion takes in media (i.e., material dispersion). However, there are other types of dispersion that exist which ultimately stem from the properties of the material. One of these other ways is modal dispersion [11]. This is when the velocity of various modes changes as a function of the input wavelength. As displayed in Figure 11, the different modes that propagate at different group velocities and angles, which themselves are functions of an input wavelength. These modes that arise in a waveguide are the different forms that the electric field can take in a geometrically confined structure. These forms are functions of satisfying the Helmholtz equation for the geometry of the confining structure. Only a finite number of fields do this and produce the appropriate wavevector in the z-direction, $\mathit{\beta }$ . Consequently, only a finite number of modes will be allowed in the structure.

Figure 11.

The various modes within a step-index waveguide—where h, ${\mathit{n}}_{\mathit{c}}$ , and ${\mathit{n}}_{\mathit{cl}}$ are the height and the core and cladding indices, respectively.

The angle at which certain modes propagate within makes a difference as to the speed of propagation because shallower angles have less deviation from the center equilibrium as shown in Figure 12.

Figure 12.

A mode in a rectangular waveguide propagates in the x- and y-directions at various angles.

In a rectangular waveguide, the total distance a mode must travel is expressed as

$$L=\frac{L}{\mathit{cos}{\theta }_x\mathit{cos}{\theta }_y}$$ E20

where ${\mathit{\theta }}_{\mathit{x}}$ and ${\mathit{\theta }}_{\mathit{y}}$ are the angles of deviation in the x- and y-directions, respectively. The higher the order of the mode, the steeper the angle of the travel, and the more distance the mode must traverse through the waveguide. In Figure 13, the same principle holds true in an optical fiber. The rays of the fundamental modes (TM and TE) take a meridional form (a) passing through origin as they propagate and the hybrid modes—HE and EH—spiral in a helical form taking a twisted path through the fiber.

Figure 13.

Rays taking meridional (a) and helical (b) forms in an optical fiber [12].

Additionally, modes that have a shallower angle of interaction with the core-cladding boundary will have higher group velocities. A higher-order mode has a lower $\mathit{\beta }$ value as its $\mathit{k}$ -vectors in other directions are higher, leading to steeper interaction angles with the core-cladding boundary.

The way that the modes propagate through the fiber at different velocities makes the waveguide an effective multipath propagation channel where the received signal is a sum of the various scaled “echoes” of the original signal carried by the various modes that propagate therein.

As a result, this dispersion distorts the pulse and corrupts the information. As shown in Figures 14 and 15, the pulses deform from their respective Gaussian and rectangular forms as a result of the different path delays, a phenomenon known as delay distortion [13]. Calculation of modal delays is used in order to set various bit rates for optical pulses.

Figure 14.

The intermodally dispersive time evolution of a Gaussian optical pulse in a multimode rectangular waveguide.

Figure 15.

The intermodally dispersive time evolution of the more commonly used rectangular pulse in a multimode rectangular waveguide.

One way to ameliorate modal (intermodal) dispersion is to use a single-mode waveguide where the only one mode is carried through. Another way is to make use of waveguides with a parabolic index of refraction, producing a quasi-sinusoidal pathway through the waveguide as demonstrated in Figure 16.

Figure 16.

The pathway through the waveguide is sinusoidal [14] as it is bound to its axis.

The ray switches paths as it passes the region of highest index, meaning that it spends less of its time in the region of lowest index. This is to say that the phase velocity of the wave changes within the waveguide. From a ray optics perspective, the crest of the wave has a position-varying velocity, effectively making continuous switches through “different materials.”

The dispersion characteristic for graded-index (GRIN) waveguides is quite different as well. Figure 17 shows that higher wavelengths are correlated with less dispersion for GRIN waveguides. The step-index, on the other hand, maintains a positive wavelength dispersion characteristic.

Figure 17.

The fundamental mode (TE₀) dispersion characteristic of GRIN slab waveguide vs. that of a step-index slab waveguide.

It should be noted that modal dispersion is more of a function of the arrangement of media, not the sole medium itself. Changes in size, cladding material, and wavelength can affect the propagation of light therein.

6. Chromatic dispersion

Changes in wavelength are important in understanding how light will propagate. As shown in Figure 18, the way these $\mathit{\beta }$ values vary with the wavelength produces various group velocities for different modes. The higher-order modes have a lower group velocity as they travel. The index of the fused silica core in the waveguide described in Figure 18 is 1.5, meaning that light will travel roughly $\mathbf{2}\mathit{\times }{\mathbf{10}}^{\mathbf{8}}$ m/s therein. However, the contribution from dispersion increases the effective index ${\mathit{n}}_{\mathit{eff}}$ “experienced” by each mode at a particular wavelength.

Figure 18.

The group velocities of the transverse electric (TE) modes in a fused silica rectangular waveguide as a function of the wavelengths in the visible spectrum.

The plot in Figure 18 can be reconfigured to show that each mode has an effective index of refraction in the core that varies with its wavelength. This means that different “colors” of light will have different refractive indices in the waveguide. Figure 19 shows that as $\mathit{\lambda }$ increases, the group velocity decreases.

Figure 19.

Dispersion creates a higher refractive index for the modes as $\mathit{\lambda }$ increases.

Figure 19 shows that there is an added index of refraction that comes from the dispersion that contributes the delay of the modes; this is also known as the group index ${\mathit{N}}_{\mathit{g}}$ .

7. Intramodal dispersion

Dispersion can occur even in the context of a single-mode waveguide (of whatever geometry). In the previous section, we demonstrated a difference in ${\mathit{n}}_{\mathit{eff}}$ as it pertains to various wavelengths stimulating a particular mode. However, we can observe a similar phenomenon in a single-mode fiber.

In the Introduction, we discussed how one can model the plasmonic impulse response of a material in the presence of an electric field as a mechanical system for which

$$\overrightarrow{\mathit{x}}\left(t\right)\mathbf{=}\frac{-e/m}{\left({\omega }^2-{\omega }_0^2\right)+\mathit{j\gamma \omega }}{E}_0{e}^{\mathit{j\omega t}},$$ E21

where $E_0$ , ${\omega }_0$ , $e$ , $m$ , and $\gamma$ are the electric field amplitude, the resonant frequency, the electric charge, the mass of the electron, and the damping constant, respectively [11]. We can multiply the quantity described in (21) be $e$ to get the dipole moment vector. We additionally multiply by the dipole density $N$ to get the bulk polarization. We see in (9) that the displacement field is equal to the added effect of the field and the polarization response of the material. However, we get a different value because we are not dealing in the plasmonic regime. The relative permittivity or the square of the index $n$ is expressed as

$$n^2\left(\omega \right)\mathbf{=}\frac{ϵ\left(\omega \right)}{{ϵ}_0}\mathbf{=}1\mathbf{+}\frac{N{e}^2/{ϵ}_{0}m}{\left({\omega }^2-{\omega }_0^2\right)+\mathit{j\gamma \omega }}$$ E22

We can further imagine that there are a certain number of mechanical systems operating simultaneously. This means that for the ith system, there will be a certain fraction of atoms $f_i$ in that system; i.e., the effective dipole density for the ith system is $f_{i}N$ .

$$n^2\left(\omega \right)=\sum _i^{Z}1+\frac{f_i\left(\frac{N{e}^2}{m{ϵ}_0}\right)}{\left({\omega }^2-{\omega }_i^2\right)+j{\gamma }_i\omega }=\sum _i^{Z}1+\frac{f_i\left(\frac{N{e}^2}{m{ϵ}_0}\right)}{\left(\frac{4{\pi }^2{c}^2}{{\lambda }^2}-\frac{4{\pi }^2{c}^2}{{\lambda }_i^2}\right)+\frac{j2\mathit{\pi c\gamma }}{\lambda }}$$

$$=\sum _i^{Z}1+\frac{f_i\left(\frac{N{e}^2}{m{ϵ}_0}\right){\lambda }^2}{\left(4{\pi }^2{c}^2-\frac{4{\pi }^2{c}^2{\lambda }^2}{{\lambda }_i^2}\right)+j2\mathit{\text{πcγλ}}}=\sum _i^{Z}1+\frac{f_i{\lambda }_i^2\left(\frac{N{e}^2}{m{ϵ}_0}\right){\lambda }^2}{\left(4{\pi }^2{c}^2{\lambda }_i^2-4{\pi }^2{c}^2{\lambda }^2\right)+j2\mathit{\pi c\gamma }{\lambda }_i^2\lambda }$$ E23

where c is the speed of light.

We then make the assumption that $\gamma$ , the damping constant, goes to zero for the assumption of a simple harmonic oscillator.

$$n^2\left(\lambda \right)=\sum _i^{Z}1+\frac{f_i{\lambda }_i^2\left(\frac{N{e}^2}{4{\pi }^2{c}^{2}m{ϵ}_0}\right){\lambda }^2}{\left({\lambda }_i^2-{\lambda }^2\right)+\frac{\mathit{j\gamma }{\lambda }_i^2}{2\mathit{\pi c}}\lambda }\underset{\gamma \to 0}{\underbrace{\approx }}\sum _i^{Z}1+\frac{f_i{\lambda }_i^2\left(\frac{N{e}^2}{4{\pi }^2{c}^{2}m{ϵ}_0}\right){\lambda }^2}{\left({\lambda }_i^2-{\lambda }^2\right)}$$ E24

We can make further simplifications to say that

$$n^2\left(\lambda \right)=\sum _i^{Z}1+\frac{G_i{\lambda }^2}{\left({\lambda }_i^2-{\lambda }^2\right)},$$ E25

where ${\lambda }_i$ is the ith resonant wavelength and the coefficient $G_i$ is the strength of that resonance. The coefficient $G_i$ is expressed as

$$G_i=\left(\frac{N{e}^2}{4{\pi }^2{c}^{2}m{ϵ}_0}\right)f_i{\lambda }_i^2$$ E26

These coefficients are known as Sellmeier coefficients [15]. These coefficients play a role in determining the group velocity of a particular wavelength in a fiber.

One of the misconceptions brought about by our knowledge of lasers is the “myth of monochromaticity.” The light sources we use—even for optical fibers—are never fully monochromatic. Rather they emit a spectrum of frequencies (and wavelengths) concentrated around a particular center [16]. This spectrum will generate respectively scaled eigenmodes with their own group velocities and delays through the fiber. This causes pulse broadening even when only a single mode is propagating.

For a typical laser with a homogenous linewidth undergoing natural line broadening due to atomic decay, we can express the line-shape function as a typical Lorentzian for which

$${\varphi }_N\left(\nu \right)=\frac{1}{\pi }\frac{\Delta {\nu }_N/2}{{\left(\nu -{\nu }_0\right)}^2+{\left(\frac{\Delta {\nu }_N}{2}\right)}^2},$$ E27

where $\mathrm{\Delta }{\nu }_N$ is the FWHM frequency point or the natural linewidth [17]. This is the spectral profile of a typical laser centered around a frequency $\nu ={\nu }_0$ . We can convert this line-shape function to one of the k-vector $k$ where

$${\varphi }_N\left(k\right)=\frac{1}{\pi }\frac{\Delta {\nu }_N/2}{{\left(\frac{\mathit{kc}}{2\pi }-\frac{k_{0}c}{2\pi }\right)}^2+{\left(\frac{\Delta {\nu }_N}{2}\right)}^2}=\frac{2\mathit{\pi \Delta }{\nu }_N/c^2}{{\left(k-k_0\right)}^2+{\left(\frac{\mathit{\pi \Delta }{\nu }_N}{c}\right)}^2}.$$ E28

where $k_0$ is the center spatial frequency.

The spectrum described in (28) shows that there is an uncountable infinity—i.e., a continuum—of k-vectors (and thus wavelengths) that is allowed.

In the case of a gas, the same can be done with the Boltzmann relation that is used to compute the statistical distribution of velocities (and thus k-vectors) under Doppler [18] broadening where the velocity component v is normally distributed $\mathcal{N}\left(0,k_{B}T/m\right)$ while the energy E undergoes a chi-squared distribution ${\chi }_1^2\left(E\right)$ as shown in (28):

$$P\left(v\right)=\sqrt{\frac{m}{2\pi k_{B}T}}{e}^{-\frac{m{v}^2}{2k_{B}T}}\underset{E=m{v}^2/2}{\leftrightarrow }P\left(E\right)=\frac{1}{\sqrt{4\pi k_{B}T}}\frac{e^{-\frac{E}{k_{B}T}}}{\sqrt{E}}$$ E29

where m, $k_B$ , T, and v are the particle’s mass, Boltzmann’s constant, the temperature, and the velocity, respectively.

Because of the Doppler effect, the radiation will be shifted to a higher frequency $\nu$ which can be expressed as

$$\nu ={\nu }_0\left(1+\frac{v}{c}\right)$$ E30

where ${\nu }_0$ is the center frequency.

With (30) in mind, we can rewrite (28) in terms of $\nu$ and convert to the k-domain like so

$$P\left(\nu \right)=\frac{c}{{\nu }_0}\sqrt{\frac{m}{2\pi k_{B}T}}{e}^{-\frac{m{c}^2{\left(\nu -{\nu }_0\right)}^2}{2{\nu }_0^2{k}_{B}T}}\leftrightarrow P\left(k\right)=\frac{c}{k_0}\sqrt{\frac{m}{2\pi k_{B}T}}{e}^{-\frac{m{c}^2{\left(k-k_0\right)}^2}{2k_0^2{k}_{B}T}}$$ E31

What this means is that in the case of a realistic laser, dispersion within a confined medium will result even if the medium allows for only one mode to be activated. This is to say that Lorentzian homogenous broadening and Gaussian Doppler broadening both result in intramodal pulse broadening.

What is interesting is the counterintuitive connection between the frequency broadening and the temporal pulse broadening as their behaviors mirror each other. However, this is because we are dealing with two separate systems that happen to be connected. The first system is that of the laser linewidth and the decaying atoms and spontaneous thermal processes that diminish monochromaticity. These processes spread the spectrum out and a broader range of frequency (and wavelength) contribution to the spectrum. The second system is that of the delays accrued within the fiber-waveguide channel according to the wavelength contributions of the input spectrum. The input laser spectrum allows for a larger contribution to delays that will be “experienced” by the larger range of frequency and wavelength components, which broaden the temporal profile of the signal.

As noise is intrinsic to a lasing system so is dispersion when that system is directed into a medium; this is especially true in the context of confined or bounded media and their applications in modern communication networks.

8. Conclusion

Dispersion has been discussed in this chapter as it relates to electric fields in various media. We first begin with an understanding of electron responses to EM radiation; i.e., the light-matter interaction as understood in classical electromagnetics. When then used this concept to understand how the velocity of a signal through a medium is a function of the frequency of that signal.

Additionally, we explained and developed the concept of a group velocity with a simple analogy and then moved on to discuss that concept at length in various media. We contrasted group velocity with phase velocity and developed dispersion relations which contrasted the two.

We took the discussion of dispersive materials to light-confining structures. Optical waveguides and fibers function as dispersive materials not only because of their material properties but also because these materials are arranged. The arrangement of these materials—i.e., core-cladding indices, size, index distribution, etc.—all plays a role in the phenomenon of dispersion. This also influences the allowable forms of light that propagate in one of these confined media; the properties of the medium affect these modes of propagation. We additionally learned about ways in which this effect can be eliminated for engineering applications.

We finally ended this chapter with a discussion of effect of intramodally dispersive media on the frequency spectrum and time evolution of electric field signals. This was demonstrated by first dismantling the “myth of monochromaticity” and understanding light itself as a spectrum. This helped us better understand how a single mode can undergo dispersion with itself. It most importantly demonstrated the degree to which dispersion in inherent in EM propagation through all media.