Review of Optics

In this chapter, we will review those aspects of optics that are most relevant to the study of photonics. It is natural to begin with the fundamental question, What is light? Historically, light has at times been considered to be in the form of particles, or corpuscles, a point of view favored by Isaac Newton. The view of light as a wave was promoted in the 17th century by Christiaan Huygens, among others, and came to dominance after the experiments of Thomas Young on light interference in the early 19th century. Our modern view of light arose during the early part of the 20th century with the advent of quantum mechanics. In this view, light must be considered to be both a particle and a wave, in the same way that material particles such as electrons have both a particle and wave nature. Generally, the classical, or wave nature of light is appropriate when light is propagating from one point to another, whereas the quantum, or particle nature of light manifests itself when light is absorbed or emitted by atoms. During absorption or emission, light acts like a stream of particles or packets of energy called photons. Each photon contains energy equal to

where h = 6.63 x 10^-34J s is Planck's constant, v and are the frequency and wavelength of the light wave, respectively, and c is the speed of light in a vacuum. In most situations other than absorption and emission, light can be treated as a wave, consisting of oscillating electric and magnetic fields. The variation of these two fields in space and time is governed by Maxwell's equations, the treatment of which are outside the scope of this book. We will, however, quote certain results from Maxwell's equations from time to time and use these results to explain various phenomena relevant to photonics. The interested reader is referred to the bibliography for more advanced treatments that show how these results follow from Maxwell's equations.

One simple solution to Maxwell's equations in a uniform medium is that of a plane wave, in which the electric field is constant everywhere along a plane (at a particular instant in time), and varies sinusoidally in a direction perpendicular to that plane. For example, if the electric field varies in the x direction, then

where k 2π/ is the wave vector magnitude or wavenumber, 2πv is the angular frequency (measured in radians per second), and the quantity ø=(kx - t) is the phase of the wave. Here and throughout the book, it will often be convenient to use the complex exponential notation for waves and oscillations, with the understanding that the real part of the expression corresponds to the physical oscillation. Using Euler's identity eⁱ = cos( ) + i sin( ), the wave in Eq. (2-2) is then equivalent to

The electric field amplitude E₀ is a vector in the y-z plane. If E₀ = E_o , the wave is said to be polarized in the y direction, whereas if E₀ = E₀,it is polarized in the z direction. Any other direction for E₀ can be described by a linear combination of polarizations in the y and z directions, so we say in general that there are two distinct polarizations for a given plane wave. Figure 2-1 shows the variation of E_ywith x and t for y-polarized light. The value of E_ydepends on the phase øof the wave at a particular x and t. When ø = 0, E_y is at a positive maximum, and when ø= /2, E_y = 0. A phase ø= gives a negative maximum in E_y , and ø = 2πgives again a positive maximum. The wave is therefore periodic in phase with period 2π. It is periodic in space with wavelength and periodic in time with period T.

The light wave contains not only an electric field, but also an oscillating magnetic field. As indicated in Fig. 2-2, the magnetic field has the same dependence on time and space as the electric field, but is perpendicular to both the electric field and the direction of propagation. The relative orientation of E and B is always such that the cross product E x B is in the direction of wave propagation. For an arbitrary wave direction, Eq. (2-2) can be generalized to

In this case, the planes of constant phase are perpendicular to the wave vector k, which specifies the direction of wave propagation. The wavelength is related to the wave vector by k = k = 2 / .

The wave in Eq. (2-2) is characterized by planes of constant phase at x = t/k where the amplitude is a maximum. As time advances, these planes of constant amplitude propagate in the +x direction with a speed

which is referred to as the phase velocity of the wave. For electromagnetic waves in a vacuum, this phase velocity is v_p = c, where c = 3 10⁸ m/s is the speed of light. In a material

Figure 2-1 Electric field oscillation in time and space.

Figure 2-2 Transverse electromagnetic wave.

medium, the atoms interact with the light, and the phase velocity of the wave is changed to

where n is the index of refraction and we have defined the free-space wavenumber k₀ = 2π/ ₀ in terms of the free-space wavelength ₀ = c/v. The effect of a higher refractive index is to slow the wave down and to decrease the wavelength to = ₀/n. Table 2-1 gives the index of refraction for a few materials.