19.1.1 Characteristics of light

Light is described as an oscillation of ‘something’ at right-angles to the direction of propagation – a transverse wave. The ‘something’ that is oscillating is the amplitude of electric and magnetic fields, as shown in Fig. 19.1. They oscillate with a frequency ν and a wavelength λ. Sitting at one point in space and counting how many complete oscillations the wave makes in a given length of time is a measure of the frequency, the units of which are per unit time, s⁻¹ or hertz (Hz). The distance between two successive maxima (or two successive minima) is equal to the wavelength, which is measured in units of distance. Maxima and minima collectively are known as extrema and also as antinodes. The sign of the oscillating field changes from (+) to (–) and passes through zero at a node. Another commonly encountered descriptor of electromagnetic waves is the wavenumber , which is measured in inverse length, m⁻¹ in SI. However, because it is so ingrained in the literature, the more commonly encountered unit is cm⁻¹ and this will be used here.

The speed of light was the first physical constant to be assigned an exact value. This is because it is used as the basis for defining many other fundamental constants. The speed of light in vacuum is represented by c, and is

(19.1)

If we need to differentiate it from the speed of light in a medium, we will denote the speed of light in vacuum as c₀ and the speed of light in a medium as c. The refractive index is the ratio between the speed of light in vacuum and the speed of light in a medium

(19.2)

The product of wavelength λ and frequency ν is equal to the speed of light

(19.3)

The wavenumber is the inverse of the wavelength thus,

(19.4)

Implicit in our discussion below, and throughout this book, is that we are considering propagation of light in vacuo – that is, in a medium with refractive index n = 1. This is a simplifying approximation so that we do not have to keep track of this factor of n and label everything with a subscript 0. When light enters a dense medium, its speed decreases in proportion to the refractive index, as shown in Eq. (19.2). The electromagnetic field of light establishes a polarization in the medium that oscillates at the same frequency as the incident light. Therefore, the frequency of light in the medium remains constant at ν. However, in accord with Eq. (19.3) the wavelength of light in the medium λ must be reduced from its value in vacuo λ₀ such that λ = λ₀/n. The refractive index is a slowly varying function of frequency in a transparent material, but changes rapidly near absorption features. The refractive index of air is n_air = 1.0003 at 589 nm (a standard wavelength associated with the familiar yellow D line emission of Na). When performing high-resolution spectroscopy it is important to correct for this factor that would change the D₁ line from λ₀ = 589.76 nm to λ = 589.58 nm for light propagating in air at atmospheric pressure. The refractive index is proportional to density. Therefore, it depends not only on the composition of the medium but also on temperature and pressure. For example, water at atmospheric pressure and 20 °C has n = 1.3330.

James Clerk Maxwell developed the mathematical theory of magnetic fields and showed that electricity and magnetism cannot exist in isolation. He suggested that light was composed of oscillating electric E and magnetic B fields, which are perpendicular to each other, and that these field can be represented by sine or cosine waves,

(19.5)

(19.6)

where ω = 2πν is the angular frequency (units of radians per second represented by s⁻¹ and never Hz), and k = 2π/λ is the wavevector. The maximum values of the amplitudes of the electric and magnetic fields are E₀ and B₀, respectively. The argument of the cosine function is known as the phase of the wave φ₀

(19.7)

Light propagating in a uniform medium has a constant phase. Thus, differentiating this equation with respect to time and setting it equal to zero, we obtain the phase velocity of the wave

(19.8)

What is commonly called the ‘speed of light’ is the speed at which a point of constant phase (an antinode, for example) propagates past a fixed observer. An electromagnetic wave is two-dimensional and c is the propagation velocity of a plane of constant phase along the x-axis.

The intensity of a classical wave is proportional to |A|², where A is the amplitude of the wave. Thus, the intensity of an electric field (what is commonly measured in any light detector since the interaction of matter with the magnetic field is generally much weaker) is proportional to E²₀. φ describes the relative phase (phase difference) of the waves. The term ‘relative phase’ describes the displacement of one wave with respect to another. Waves that have constant displacement are coherent waves – that is, they are in phase with each other. Waves that are in phase add constructively, while waves that are out of phase add destructively. A shift in phase by 90° (= π/2 radians) changes a sine wave into a cosine wave

(19.9)

Recall that a cosine function can also be represented by a complex exponential function

(19.10)

and that i² = −1. Therefore, an equivalent way to write the wave equation of the electric field is

(19.11)

where the relative phase φ has been included explicitly. The real part of E represents the physical wave. Using this complex notation, the intensity is proportional to the absolute square of the field, that is by the product of the complex conjugate of the field E^* with the field E

(19.12)

Electromagnetic radiation can possess a near infinite range of wavelengths. The span of wavelengths is termed the electromagnetic spectrum; this is shown in Fig. 19.2. The electromagnetic spectrum ranges from radio waves at low energy up to cosmic rays at the high end. The length unit of choice depends on the spectral range. Different parts of the spectrum are associated not only with different names, but also with different characteristic molecular/atomic excitations as noted in Table 19.1. This means that different regions are also associated with different spectroscopic techniques, which also tend to have different preferred units. Hence, pure rotational spectroscopy often uses wavelength in μm, frequency in MHz, or wavenumber in cm⁻¹; vibrational spectroscopy (also called infrared spectroscopy) is most commonly conducted in cm⁻¹ but sometimes also in μm. Visible light is commonly referenced by wavelength in nm, though wavenumber in cm⁻¹ is also used. This is true for electronic spectroscopy throughout the UV/Visible range. X-ray spectroscopy and X-ray diffraction often reference the wavelength in Å (now sometimes in pm) or photon energy in eV. Gamma spectroscopy is often performed in energy units of keV or MeV.

Several other terms are commonly associated with light and electromagnetic waves. White light is comprised of waves of many frequencies. White light can be characterized by a central frequency and a broad bandwidth of frequencies of random phase on either side of this central value. This is in contrast to monochromatic light (light of a single frequency) or narrow bandwidth light (light of a small frequency range). Plane-polarized light has its E field in a well-defined plane. The plane of the E field is termed the plane of polarization. In circularly polarized light, the E field rotates in a plane orthogonal to the propagation direction.

19.1.2 Superposition of waves

In addition to oscillating with a definite frequency and wavelength, one of the essential defining characteristics of a wave is its ability to exhibit interference phenomena as a result of the relative phase behavior of combined waves. In Figs 19.3–19.5 we see the results of the superposition of two waves of equal amplitude propagating collinearly. Three different cases are shown: Fig. 19.3 shows equal frequency and in-phase; Fig. 19.4 shows equal frequency and out-of-phase; and Fig. 19.5 shows initially in-phase but unequal wavelength.

All electromagnetic waves of the same wavelength travel at the same speed in the same medium. Therefore, waves of equal wavelength that are in-phase at the origin remain in phase everywhere. Note that waves with a well-defined phase relationship need to be added coherently. That is, the electric field of the combined wave is obtained by adding the electric fields of the component waves. The intensity of the combined wave is then obtained by squaring the summed wave. In a coherent sum, add waves then square to obtain the intensity.

Consider two waves that they have the same frequency and amplitude but a phase difference φ. They may be written

(19.13)

(19.14)

The electric field of the superposed wave is obtained by addition of the two waves

(19.15)

Factoring out a term of e^{− iφ/2} from the term in parenthesis and expanding the result using Eq. (19.10), we rewrite the electric field of the superposed waved as

(19.16)

This allows us to write E as a product of real functions and complex exponential functions only. This simplifies the result because in taking the product involving the complex conjugate,

(19.17)

Therefore, the intensity (irradiance) is

(19.18)

19.1.2.1 Directed practice

Complete the algebraic steps necessary to derive Eq. (19.18).

If the waves did not have a well-defined phase relationship, an incoherent sum would be performed. In an incoherent sum, the intensity of the each wave is first obtained by squaring the field strength, then the component intensities are added to obtain the resultant intensity of the combined wave. In a coherent sum of two waves with equal amplitude E₀, the maximum intensity is 4E²₀. In an incoherent sum of two waves with equal amplitude E₀, the maximum intensity is 2E²₀. The output of two conventional light bulbs adds incoherently: two 100 W bulbs have the same intensity as one 200 W bulb (avoiding for the moment any consideration of distance dependences and shadowing from the construction of the bulbs).

However, if we have coherent sources – most easily constructed from lasers but also possible with classical sources under special conditions – then we need to consider the relative phases of the constituent waves. Constructive interference is perfect for waves of the same wavelength and amplitude that are in phase φ = 0 or, more generally, differ in phase by φ = 2nπ, where n is an integer. Constructive interference becomes progressively weaker as the phase difference increases. Time invariant interference between two sources of light can only take place if they are coherent sources, that is, they must have the same frequency and have a constant phase difference.

When the phase difference increases to π radians, the interference becomes perfectly destructive for waves of the same wavelength and amplitude. This is shown in Fig. 19.4. Interference is a periodic phenomenon and destructive interference becomes increasingly less effective as the phase difference increases beyond π radians. Destructive interference is a maximum at φ = (2n + 1)π, with n again an integer.

Waves with different frequencies and a well-defined initial phase relationship display variable interference because the relative phase difference changes with time. This is shown in Fig. 19.5 and is known as a beat pattern. Note that energy is conserved. Interference leads to changes in the distribution of radiant energy. Destructive interference at the node redistributes energy into the antinodes. Interference phenomena can only occur where beams overlap and the occurrence of destructive interference at one point in space and time must be accompanied by constructive interference elsewhere to balance it. For complete destructive interference to occur at all points in time and space it would mean that the two beams would have to interfere at their sources such that no energy is radiated at all.² We will see in Chapter 27 that fitting cycles of light inside an optical cavity such that they interfere constructively is important in the design of lasers.

If the two waves have slightly different values of wavelength described by k + Δk and ω + Δω, the total electric field that results from the two waves propagating collinearly is

(19.19)

This equation has the same form as Eq. (19.15) but with the relative phase difference replaced by

(19.20)

The superposed intensity is then described by

(19.21)

The cosine factor describes an envelope function that modulates the traveling waves. This traveling wave propagates with a group velocity of

(19.22)

This represents the velocity at which a plane of constant phase of the envelope travels. It need not be equal to the phase velocity of the individual waves.

19.1.3 Double slit interference

Pick a plane of constant phase, say a maximum, and follow it as the wave propagates. We refer to this plane as the wavefront (we could pick any plane of constant phase but the maximum is the most commonly chosen). The wavefront is normal to the direction of propagation. A plane wave is a collimated beam with plane wavefronts.

Consider the situation depicted in Fig. 19.6 in which a monochromatic plane wave is incident on two infinitesimal slits a distance d apart. Each slit acts as a point source and we observe the light transmitted from these slits on a screen at a distance L away. The electric field at point P is the sum of the fields emanating from our two sources,

(19.23)

where A is the amplitude and r₁ and r₂ are the distances of the two slits from the point P. When L is large compared to d, the paths from the two slits are effectively parallel and r₁ and r₂ differ by only d = sin θ, a quantity that is called the optical path difference (OPD). Therefore,

(19.24)

which corresponds to a phase difference between the two waves of

(19.25)

The intensity is obtained from the absolute square of the electric field,

(19.26)

For large L, the angle θ is small and

(19.27)

The light incident on the screen exhibits an interference pattern that has a cos² variation with the distance x from the origin along the screen. Maxima (instances of constructive interference) occur whenever the OPD is equal to an integral multiple of π,

(19.28)

and minima (instances of destructive interference) occur when

(19.29)

The resulting intensity pattern is shown in Fig. 19.7. The fringes do not extend to infinity in any real case because the source cannot be perfectly monochromatic and collimated (though lasers can get extremely close to this) and the slits cannot be infinitesimally narrow, lest no light pass. This second effect causes the light passing through the slits to be emitted in cones rather than hemispheres. Because of this, the intensity is damped as the distance from the origin increases.

19.1.4 Multiple-slit interference

If we generalize the geometry of Fig. 19.6 from one slit to N slits, we find that the OPD between rays emanating from adjacent slits is again dsin θ. The OPD between the first and the jth slit is then (j − 1)dsin θ. Summing the electric fields of the individual slits and squaring, we find the intensity of the resulting interference pattern to be given by

(19.30)

The large principal maximum occurs at

(19.31)

where m = 0, ±1, ±2, …. This is more commonly presented as the grating equation,

(19.32)

where m is known as the order. The grating equation predicts the angle at which different wavelengths will leave a diffraction grating. As is obvious from this derivation, a diffraction grating does not actually diffract the light. The light is subject to multiple slit interference in reflection. Interference is the result of several beams of light interacting with one another. Diffraction is more accurately defined as the interaction of light with a single aperture.

19.1.5 ‘Diffraction’ in crystals

The lattice structure of a crystal resembles a diffraction grating where the ‘slits’ are the spaces between the ordered atoms of the crystal, as shown in Fig. 19.8. If d = 2 Å (a distance on the order of a bond length or the spacing of planes in a crystal) and sinθ = 0.3534 (which from the above example seems like a reasonable sort of angle), then the wavelength of light that would diffract into this angle is

This is in the X-ray region.

Because of this, X-ray diffraction is used to determine the structures of crystals. Diffraction patterns are collected from crystals irradiated with X-rays of a known wavelength and spacings are found by solving the Bragg equation,

(19.33)

We recognize this as the grating equation, with the order now represented by n and d_hkl is the spacing between planes designated by the Miller indices hkl. In Fig. 19.8, the sum of the red line segments PB and BQ represents the pathlength difference between the ray that reflects from the lattice plane at point A and the ray that reflects from the lattice plane below it at point B (similarly and successively for the planes below it). When the pathlength difference corresponds to integer values, constructive interference is observed. When it corresponds to half-integer values, the waves associated with the two rays are out-of-phase and destructively interfere with one another.