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This chapter discusses light waves. Light can be thought of as a transverse electromagnetic wave propagating through space. As the electric and magnetic fields are linked to each other and propagate together, it is usually sufficient to consider only the electric field at any point; this field can be treated as a time-varying vector perpendicular to the direction of propagation of the wave. If the field vector always lies in the same plane, the light wave is said to be linearly polarized in that plane. The chapter describes intensity in an interference pattern. When two light waves are superposed, the resultant intensity at any point depends on whether they reinforce or cancel each other. This is the well-known phenomenon of interference. This chapter discusses the localization of fringes. When an extended quasi-monochromatic source, such as a mercury vapor lamp with a monochromatic filter, is used instead of a monochromatic point source, interference fringes are often observed with good contrast only in a particular region. This phenomenon is known as the localization of fringes and is related to the lack of coherence of illumination.

2.1 Light Waves

Light can be thought of as a transverse electromagnetic wave propagating through space. Because the electric and magnetic fields are linked to each other and propagate together, it is usually sufficient to consider only the electric field at any point; this field can be treated as a time-varying vector perpendicular to the direction of propagation of the wave. If the field vector always lies in the same plane, the light wave is said to be linearly polarized in that plane. We can then describe the electric field at any point due to a light wave propagating in a vacuum along the z direction by the scalar equation

(2.1)

where a is the amplitude of the light wave, v is its frequency, and λ is its wavelength. Visible light comprises wavelengths from 0.4 μm (violet) to 0.75 μm (red), corresponding to frequencies of around 7.5 × 1014 Hz and 4.0 × 1014 Hz, respectively. Shorter wavelengths lie in the ultra violet (UV) region, while longer wavelengths lie in the infrared (IR) region.

The term within the square brackets, called the phase of the wave, varies with time as well as with the distance along the z axis from the origin. With the passage of time, any surface of constant phase (a wavefront) specified by Eq. 2.1 moves along the z axis with a speed

(2.2)

(approximately 3 × 108 metres per second in a vacuum). In a medium with a refractive index n, the speed of a light wave is

(2.3)

and, since its frequency remains unchanged, its wavelength is

(2.4)

If a light wave traverses a distance d in such a medium, the equivalent optical path is

(2.5)

Equation 2.1 can also be written in the compact form

(2.6)

where ω = 2πv is the circular frequency, and k = 2π/λ is the propagation constant.

Equation 2.6 is a description of a plane wave propagating through space. However, with a point source of light radiating uniformly in all directions a wavefront would be an expanding spherical shell. This leads us to the concept of a spherical wave which can be described by the equation

(2.7)

At a very large distance from the source, such a spherical wave can obviously be approximated over a limited area by a plane wave.

While the representation of a light wave in terms of a cosine function that we have used so far is easy to visualize, it is not well adapted to mathematical manipulation. It is often more convenient to use a complex exponential representation and write Eq. 2.6 in the form (see Appendix A)

(2.8)

where φ = 2πz/λ, and A = a exp(−iφ) is known as the complex amplitude.

2.2 Intensity in an Interference Pattern

When two light waves are superposed, the resultant intensity at any point depends on whether they reinforce or cancel each other. This is the well-known phenomenon of interference. We will assume that the two waves are propagating in the same direction and are polarized with their field vectors in the same plane. We will also assume that they have the same frequency.

The complex amplitude at any point in the interference pattern is then the sum of the complex amplitudes of the two waves, so that we can write

(2.9)

where A1 = a1 exp(−iφ1) and A2 = a2 exp(−iφ2) are the complex amplitudes of the two waves. The resultant intensity is, therefore,

(2.10)

where I1 and I2 are the intensities due to the two waves acting separately, and Δφ = φ1 − φ2 is the phase difference between them.

If the two waves are derived from a common source, so that they have the same phase at the origin, the phase difference Δφ corresponds to an optical path difference

(2.11)

or a time delay

(2.12)

The order of interference is

(2.13)

If Δφ, the phase difference between the beams, varies linearly across the field of view, the intensity varies cosinusoidally, giving rise to alternating light and dark bands or fringes. These interference fringes correspond to loci of constant phase difference (or, in other words, constant optical path difference).

2.3 Visibility of Interference Fringes

The intensity in an interference pattern has its maximum value

(2.14)

when Δφ = 2mπ, or Δp = mλ, where m is an integer, and its minimum value

(2.15)

when Δφ = (2m + 1)π, Δp = (2m + 1)λ/2.

The visibility V of the interference fringes is then defined by the relation

(2.16)

where 0 ≤ v ≤ 1. In the present case, from Eqs. 2.14 and 2.15,

(2.17)

2.4 Interference with a Point Source

Consider, as shown in Fig. 2.1, a transparent plate illuminated by a point source of monochromatic light such as a laser beam brought to a focus. Interference takes place between the waves reflected from the front and back surfaces of the plate. These waves can be visualized as coming from the virtual sources S1 and S2, which are mirror images of the original source S. Interference fringes are seen on a screen placed anywhere in the region in which the reflected waves overlap.

With a plane-parallel plate (thickness d, refractive index n), a ray incident at an angle θ1, as shown in Fig. 2.2, gives rise to two parallel rays. The optical path difference between these two rays is

(2.18)

(note the additional optical path difference of λ/2 introduced by reflection at one surface; see Appendix B). Since the optical path difference depends only on the angle of incidence θ1, the interference fringes are, as shown in Fig. 2.3(a), circles centered on the normal to the plate (fringes of equal inclination, or Haidinger fringes).

With a wedged plate and a collimated beam, the angles θ1 and θ2 are constant over the whole field, and the interference fringes are, as shown in Fig. 2.3(b), contours of equal thickness (Fizeau fringes).

2.5 Localization of Fringes

When an extended quasi-monochromatic source (such as a mercury vapor lamp with a monochromatic filter) is used instead of a monochromatic point source, interference fringes are often observed with good contrast only in a particular region. This phenomenon is known as localization of the fringes and is related to the lack of coherence of the illumination.

We will...