Chapter 2

Included in this chapter:

2.1 Introduction

2.2 Physical Optics

2.2.1 Amplitude, Wavelength, Frequency, and Velocity

2.2.2 Phase and Coherence

2.2.3 Wave Phenomena

2.3 Quantum Optics

2.4 Geometrical Optics

2.4.1 Reflection and the Law of Reflection

2.4.2 Refraction and Snell's Law

The theories and models of light are an integral part of the study of fiber optics. Very often, students of optical fiber technology come an electronics background and are unfamiliar with the terms and concepts used by physicists and optical engineers who study the nature of light. One of the goals of this chapter is to help such students achieve a better understanding of light.

Traditionally, the study and applications of light are divided into three broad categories: physical optics, geometrical optics, quantum optics. In each category, light is studied with a different model. Physical optics use the wave model (similar to radio and TV waves). Quantum optics uses a particle, or photon, model; geometrical optics considers light as a straight line ray. Each model is valid within limitations. The complete model of light involves all three. We will discuss only those elements necessary for understanding the topics in this book.

Wave number and angular frequency are useful for writing the equation in its simplest form, but the wavelength and the freq represent actual, physical properties of the wave and are used often in wave calculations. Sometimes it is necessary to deri wavelength and frequency from a wave equation using the shown in Example 2. 1.

2.2.1 Amplitude, Wavelength, Frequency, and Velocity

The basic wave properties are summarized in Table 2.1, which also indicates the symbols commonly used to represent them and the base unit used in their measurement. Each wave property is described in detail in the following sections.

The wavelength (l) is the distance between two like points along the wave, and is usually measured between two peaks. The amplitude (A or E) is the height of the wave and is similar to the peak voltage of an AC signal. Both of these properties are illustrated in Figure 2.2.

Table 2.1 Basic Wave Properties

Figure 2.2 Amplitude (A) and Wavelength (A)

The frequency and velocity of the wave arise from the motion of the wave. Since the wave is moving, it has a speed or velocity associated with it. The number of wavelengths that pass a fixed point in a second is known as the frequency. The concepts of velocity, frequency, and wavelength can be envisioned by considering a train passing by an intersection. The cars of the train have a certain length (corresponding to wavelength), and the train is traveling at a certain velocity. If you count the number of train cars that pass you in one second, you are measuring the train's frequency.

The train analogy also illustrates the relationship between velocity, wavelength, and frequency. The frequency (number of train cars that pass) is affected by the velocity. If the train moves faster, more cars pass in one second; if the train moves slower, fewer cars pass. In other words frequency and velocity are directly proportional.

Frequency is also affected by wavelength. If each train car is shorter, more cars are able to pass in one second; if each car were longer, fewer cars could pass. The wavelength and the frequency are indirectly proportional. The relationships between wavelength, frequency, and velocity are expressed mathematically as:

v = f l (2.4)

For most computations, the velocity of light is assumed to be a constant corresponding to the velocity of light in a vacuum, 3 x 10⁸ m/s and referred to with the letter c.

  EXAMPLE 2.2

Given electromagnetic waves in a vacuum, determine the wavelength for the following frequencies: 150 GHz, 225 MHz, 14 THz.

In a vacuum, the velocity of the waves is 3 x 10⁸ m/s. Using Equation 2.4,

v = c = 3 x 10⁸ m/s = f / l

Rearranging Equation 2.4 to solve for wavelength (A) gives

l = (3 X 10⁸ m/s) / f

The wavelengths for the given frequencies are then

f = 150 GHz

l = (3 x 10⁸ m/s) / 150 GHz 2.00 mm

f = 225 MHz

l = (3 X 10⁸ m/s) / 225 MHz 1.33 m

f = 14THz

l = (3 x 10⁸ m/s) / 14 THz = 21.4 mm

As Example 2.2 shows, waves with high frequencies have short wavelengths, whereas waves with low frequencies have long wavelengths. The difference between various types of electromagnetic waves lies in the difference in their wavelength (or frequency). Light waves are high frequency (short wavelength), and radio and TV waves are low frequency (long wavelength). A list of all types of electromagnetic waves in order by frequency or wavelength is known as the electromagnetic spectrum (illustrated in Figure 2.3)

Notice that the electromagnetic spectrum is arranged by both wavelength and frequency. For most areas of optics, the wavelength is used in describing the wave.

The light waves that are visible to the human eye are roughly in the 400 nm to 700 nm range, but there are two other ranges that are not visible but are usually referred to as light. From roughly 3 nm to 400 nm lies ultraviolet light, and from roughly 700 nm to 10,000 nm is infrared light. The wavelength (or frequency) of visible light corresponds to the color of light. Green light is around 500 nm, red, 600 nm, and violet, 400 nm.

Figure 2.3 Electromagnetic Spectrum

Light occurs naturally as a combination of several different waves each with a separate wavelength. Even light that we think of as one color (the red light on a police car, for example) is actually several waves whose wavelengths are close in value. When light is traveling, each wavelength travels at a slightly different velocity. As a result, the velocity of light in a material can be described in one of two ways. Group velocity refers to the velocity of all the waves traveling together. Phase velocity is the velocity of a single point of constant phase in the wave.

Light sources generally emit multiple waves, and the relationship among these waves is described in terms of phase. Phase indicates the position of one wave relative to another. The difference in position between two waves is measured as the phase difference, expressed in degrees or radians. Two waves that are aligned, as shown in Figure 2.4(a), are commonly known as in-phase or are said to have a phase difference of zero degrees (0 radians). The two waves shown in Figure 2.4b are out of phase by 180 degrees (p radians).

The phase relationships of waves from a source of light are often described by the term coherence. Light that has coherence (or coherent light) has a fixed phase among the waves. In other words, all of the waves of a coherent light source are aligned the same way. Incoherent light waves have a phase that is constantly changing.

Coherence is also generally intended to mean that all of the waves are of the same wavelength (also known as monochromatic). To clarify the terms, light is sometimes described as spatially coherent (waves have a fixed phase) and temporally coherent (waves are monochromatic, i.e., have the same wavelength).

Figure 2.4 (a) In-Phase Waves (b) Out-of-Phase Waves

Light waves exhibit some distinct phenomena which are an important part of many applications. Foremost among these phenomena are interference, diffraction, and polarization. Understanding each of these is important in studying light.

Interference is an effect observed under special conditions. When coherent waves are combined, they exhibit one of two effects. If the light waves are 180 degrees out of phase, the sum of the two waves will be a third wave with an amplitude equal to the difference between the amplitudes of the individual waves. If the two waves that are combined have the same amplitude, the result will be no wave at all. This effect is known as destructive interference.

Constructive interference is when light waves are in phase with each other, resulting in a third wave whose amplitude is equal to the sum of the amplitudes of the interfering waves. Constructive and destructive interference are illustrated in Figure 2.5.

Interference is closely related to another wave effect known as diffraction. When light passes through a small opening, it spreads out to fill areas that would be expected to be in a shadow. As shown in Figure 2.6, diffraction causes a change in the shape of the incoming light. The exact effect of diffraction depends largely on the size of the opening, the distance between the light source and the opening, and the wavelength.

Diffraction is classified as either Fraunhofer or Fresnel. Fraunhofer (or far field) diffraction occurs when the light source, the opening, and the observation point are large distances away. Fresnel (or

---------------------------------------------------------------------------------- (B) -------------------------------------------------------------

Figure 2.5 (a) Constructive Interference (b) Destructive Interference

near field) diffraction occurs when a light source and observation point are near the opening.

The concept of diffraction has been used to construct useful optical devices such as diffraction grating which can be used to separate light into its component wavelengths. Diffraction is also the principle used to describe the adverse effects of emitting light through a s opening. Light sources used in fiber optics are generally of the semiconductor type (see Chapter 7 for a complete discussion). These sources emit light from a small opening at the junction between two semiconductor materials, and, as a result, the emitted light is diffracts spread out.

Because light spreads out from a semiconductor source, it is difficult to inject it into the small core of a fiber. The effects of diffraction must be limited to make launching light into the fiber feasible. Diffraction effects are partially tempered by the construction of the light source and they can be further limited by placing optical components between source and the fiber.

Polarization is the last major concept in wave optics. The polarization of a wave is really just an expression of its orientation in space. polarization should not be confused with phase which describes orientation of a wave relative to another wave. For a typical source of light, the waves emitted are randomly oriented. Some waves oscillate vertically while others oscillate horizontally, and still others oscillate at some angle in between. This haphazard distribution of orientations is normally known as random polarization (or unpolarized light).

Figure 2.6 Diffraction

Some types of light sources, and some optical effects, produce light with a set polarization where all the waves are oriented in the same direction. A good example is reflection, in which reflected light waves are all polarized in a direction parallel to the surface of reflection. In fiber optics, reflected light causes loss in signal strength and can cause noise in the system. The polarizing effect of reflection is used to design devices with minimal reflection effects.

The principles of quantum optics are based on the theories of quantum mechanics and the structure of the atom. The atom, commonly described as the building block of matter, consists of a central structure called the nucleus which is surrounded by orbiting electrons. The system is somewhat similar to our solar system where the sun is orbited by planets. The nucleus of an atom is composed of two types of particles: the proton, which has a negative charge, and the neutron, which has no charge. The electrons orbiting these particles have a negative charge. The energy levels of these electrons is the key to the production of light.

Atoms may contain varying amounts of energy depending on their structure. The amount of energy that an atom may contain is quantitized, or limited to specific and discrete amounts. To illustrate this point, consider the makeup of the simplest atom-the hydrogen atom. As shown in Figure 2.7, the hydrogen atom has a nucleus containing a single proton (no neutrons). Orbiting this nucleus is a single electron. The electron travels around the nucleus, similar to a planet orbiting the sun, but the electron is able to follow one of several possible paths (illustrated by the dotted circles in Figure 2.7). The path (or orbital) the electron chooses depends on the energy state of the atom.

Figure 2.7 The Hydrogen Atom

The important aspect of the electron orbit is that, although the electron has several possible orbitals, it is confined to these orbitals and these alone. The electron may use orbitals n = 2, 3, 4, etc., but it cannot exist in the areas between. The position of the electron is therefore quantitized.

Since the orbital used by an electron depends on the energy state of the atom, it follows that these energy states must also be quantitized. The atom can, therefore, contain certain specific levels of energy which correspond to certain specific electron orbitals. If energy is supplied to the atom, it will be absorbed in these specific amounts; if energy is released from the atom, it will be released in the same specific amounts.

Exchange of energy by an atom is important to optics since this is how light is produced. Consider an atom which contains its minimum amount of energy (called a ground state atom). Its electron(s) is orbiting close to the nucleus, but if energy were absorbed by this atom, its valence (outermost) electron would be moved to an orbital farther from the atom. After absorbing the energy, the atom is said to be in an excited state.

Atoms do not remain in excited states for very long. After a short time, the atom returns to its ground state by releasing the energy it has absorbed. This released energy is in the form of a particle or packet called a photon, which, depending on the atom and energies involved, could be some form of light. The wavelength of light produced in this manner depends on the amount of energy that the atom absorbed and then released. This relationship is governed by the equation

E = h l (2.5)

where E is the energy, h is a constant (known as Planck's Constant), and A is the wavelength.

The relationship between wavelength and energy becomes more significant when it is realized that the energy absorbed and emitted by a particular atom is limited to certain discrete values. Also, the energy values are different for different types of atoms (for example, helium has one set of energies whereas sodium has another). Because of this, we conclude that each type of atom will absorb and emit its own unique group of wavelengths.

Table 2.2 shows some of the wavelengths emitted by common atoms. These emissions can be observed in everyday life by looking at the emissions of known atoms. For example, neon, used in neon signs, emits several bright colors such as red, green, and orange.

The field of geometrical optics, founded on the principle of rectilinear propagation (light travels in a straight line), uses lines (or rays) to illustrate the path that light follows. Two basic concepts are defined in geometrical optics: reflection and refraction. The rules governing the effects of these concepts on the direction that light travels are the fundamental rules of geometrical optics.

Reflection occurs when light traveling in one material reaches the surface of another material and bounces off. The light may either completely reflect or part of it may reflect with part passing into the other material, depending on the materials involved and the wavelength of the light. Reflection is categorized into three types: fresnel, specular, and diffuse.

Table 2.2 Some Emission Wavelengths of Common Atoms

Atom Emission Wavelength (nm) Brightness

Mercury 404.66 (violet) Faint

546.07 (green) Bright

576.96 (yellow) Bright

Helium 447.15 (blue) Bright

501.57 (blue-green) Bright

667.2 (red) Medium

Neon 632.8 (red) Bright

543 (green) Faint

In Fresnel reflection, light is incident with the surface of a transparent or semitransparent surface (such as glass). Most of the light passes into the new material, but a small percentage (usually about 4%)

reflects.

In specular reflection, light reflects from a highly reflective surface (such as a mirror), and a large percentage (generally close to 100%)

reflects.

Diffuse reflections come from a rough, opaque surface (such as a piece of wood). The amount of light that reflects is variable, but the refl.ected light scatters in several directions, unlike specular and Fresnel

reflection.

All reflections are governed by the law of reflection. This law states that the angle at which the light approaches a reflective surface is equal to the angle at which it reflects. As illustrated in Figure 2.8, the angles involved are measured relative to a line (called the normal to the surface) drawn perpendicular to the reflective surface. Using the quantities defined in Figure 2.8, the law of reflection is expressed mathematically as

q _I = q _r (2.6)

Figure 2.8 Reflection

When light passes from one material to another, it changes the direction in which it is traveling. This bending of light is known as refraction. The amount and direction of the bend can be determined by knowing the index of refraction of the materials involved.

Index of refraction is a measure of the speed that light will travel through a certain material. Recall from Section 2.2.1, that the speed of light in a vacuum is 3 X 108 m/s. When light travels through any material (such as glass or plastic), its speed is less than its speed in a vacuum. The index of refraction (n) of a material is calculated from the ratio of the speed of light in a vacuum to the speed of light in the material.

Table 2.3 lists the index of refraction for some common materials along

with the speed light travels in them.

Table 2.3 Index of Refraction for Common Materials

Material Index Speed (mls)

air 1.000292 2.991 10⁸

water 1.333333 2.250 10⁸

sodium chloride 1.54 1.948 10⁸

diamond 2.42 1.181 10⁸

  EXAMPLE 2.3

Given the speed of light in the following types of glass, calculate the index of refraction for each one.

Using Equation 2.7,

(3 x 10⁸ m/s)

n = ------------------------

(speed in the material)

Barium Flint 3 x 10⁸ m/s

N = ---------------------------- = 1.59

1.89 x 10⁸ m/s

Spectacle Crown 3 x 10⁸ m/s

N = ---------------------------- = 1.52

1.97 x 3 x 10⁸ m/s

Fused Quartz 3 x 10⁸ m/s

N = ---------------------- = 1.4

2.06 x 10⁸ m/s

The index of refraction is used to calculate the angles involved in refraction with the help of an equation known as Snell's Law. Consider the diagram in Figure 2.9. Snell's Law states that the index of r fraction of the two materials and angles involved are related by t equation,

n₁ sin q ₁ = n ₂ sin q 2 (2.8)

where the subscripts I and 2 refer to the angles and the indexes of the two materials.

Snell's Law can be used to calculate the angle at which light w refract, the indexes of the material, or the incident angle required to p duce a certain refracted angle. Some of the problems that can be solv using Snell's Law are given in Example 2.4.

Figure 2.9 Snell's Law

EXAMPLE 2.4

a. Light travels from air (n = 1.00) to water (n = 1.33) at an incident angle of 40'. Calculate the refracted angle.

Using Equation 2.8,

ni sin 01 = n2 sin 02

divide both sides by n2

ni/n2 sin 01 = sin 02

Take the arcsine of both sides

02 = arcsine(nl/n2 sin 01)

= arcsine(l.00/1.33 sin (40)) = 28.90

b. What index should a material have if it were used in place of the air in part (a) to produce a refracted angle of 500?

Using Equation 2.8,

ni sin 01 = n2 sin 02

divide by sin 01

ni = n2 sin 02/Sin 01

= 1.33 sin (50)/sin (40)

= 1.59

Example 2.4 shows that when light travels from a material with a larger index of refraction to a material with a smaller index of refraction, it refracts away from the normal. Light traveling from a material with a smaller index to a material with a larger index refracts toward the normal. In the first case (larger index to smaller index), increasing the angle of incidence increases the angle of refraction to the point where the light actually reflects instead of refracts. The angle of i

dence (0,) that produces this reflection can be calculated by

0, = aresine(n2/ni)

 EXAMPLE 2.5

Calculate the critical angle between two materials with indices of ni

1.45 and n2 =1.40.