Light. (Material taken from: Optics, by E. Hecht, 4th Ed., Ch: 1,2, 3, 8)

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1 (Material taken from: Optics, by E. Hecht, 4th Ed., Ch: 1,2, 3, 8) is an Electromagnetic (EM) field arising from the non-uniform motion of charged particles. It is also a form of EM energy that originates from the motion of charged particles between various energy states. The primary characteristics of light are its velocity denoted by c, frequency ν, wavelength λ, Energy E and momentum p. These quantities are related as follows: velocity of light: c(in m/s) = νλ; ν is in Hz (s 1 ); and λ is in m Energy of Photon: E(in J) = hν where h is Planck s constant in Js Also, since ν = λ c, the frequency and wavelength are closely related exhibits wave-like or particle-like behavior depending on conditions. For instance, in the photoelectric effect light behaves like particles (referred to as photons) while in interference conditions (e.g. Young s double slit experiment) it manifests wave-like properties. Wave-particle duality can be expressed conveniently as a relation between wavelength (a wave characteristic) and momentum ( a particle characteristic) using de Broglie s relation: p = h λ 1 Mathematical representation of light (EM waves) is a transverse EM oscillation travelling at speed c. A monochromatic (single frequency) light wave can be expressed as: E( r, t) = E o ( r, t)e i( k r ωt+φ) (1) where, E is the electric field vector with peak value E o and has units of volts/m r is position vector identifying a location in space in Cartesian coordinates k is the wave-vector with magnitude k = 2π λ and has units of m 1 t is time in s 1 ω is the period of oscillation in radians and has magnitude ω = 2πν 1

2 φ is a starting phase angle in radians Alternately, light can also be expressed in terms of an oscillating magnetic field as: B( r, t) = B o ( r, t)e i( k r ωt+φ) where B( r, t) is the space and time-dependent magnetic field in T esla = N/amp m with peak value of B o 1.1 Wave and phase velocity Eq. 1 is a general representation of a plane wave propagating in the positive x-direction including both the real and imaginary components of the wave. Often we are interested in the real part of the wave, the part that actually carries the energy. A much simplified form of the wave can be obtained by first making use of Euler s complex notation through which the plane wave of eq. 1 can also be expressed as: E( r, t) = E o ( r, t)[cos( k r wt + φ) + isin( k r wt + φ)] and as will be often see, the real part of the wave can simply be denoted as: E( r, t) = E o ( r, t)cos( k r wt + φ) (2) Using this form we can express some important features of wave motion: 1. The phase of the wave, θ: The argument of the cosine term is the phase of the travelling wave, i.e: θ = kr wt + φ (3) 2. Rate of change of phase with time or distance gives us the frequency and wavenumber respectively as follows: (a) θ t = ω (b) θ r = k In the above partial derivatives it was assumed that the initial phase angle φ is independent of time and/or position. This need not always be the situation From the rate of change of the phase with time and position and important quantity known as the phase velocity can be extracted as: v = r t = θ t θ r = ω k = 2πν 2π λ = νλ = c So we see that the phase velocity is equivalent to the wave velocity, i.e. the speed of light. 2 August 30, 2006

3 1.2 Relation between E and B Faraday s law which states that a changing magnetic field generates an electric current (i.e. an electric field) relates E and B through the relation: E = B t where, = xi + xj + xk. The above relation gives: B = E c and the condition that B E, i.e. the magnetic and electric fields are always perpendicular to each other. 1.3 Plane monochromatic linearly polarized light wave Referring to Fig. 1 a monochromatic light wave travelling in a single direction (i.e. x direction) that has an electric (or magnetic) field oscillating in a single direction (linearly polarized) say y can be expressed as: E(x, t) = E o (x, t)e i(k xx ωt+φ) j and from Faraday s law, the corresponding magnetic field can be expressed as: B(x, t) = B o(x, t) e i(kxx ωt+φ) ( i j) c where i is the unit vector in the direction of propagation of the wave and j is the polarization vector denoting the direction of oscillation of the electric field. Figure 1: Plane monochromatic linearly polarized wave travelling along the x-direction with the electric field oscillating in the y-direction. 1.4 Energy and Pressure in light EM waves carry energy simultaneously through their electric and magnetic components with the instantaneous energy density, i.e. energy per unit volume U in J/m 3 expressed for each part as: U E = ɛ 2 E2 U B = 1 2µ B2 3 August 30, 2006

4 where ɛ and µ are the permittivity and permeability of the medium in which light is travelling. For the case of vacuum or air, ɛ = ɛ o = C 2 s 2 m 3 kg 1 and µ = µ o = 4π 10 7 m Kg C 2. The total energy density carried by light is then: U T otal = U E + U B = ɛ 2 E µ B2 further, since E = cb and c = 1 ɛµ the above can be expressed as: U T otal = ɛe 2 = 1 µ B2 Pressure (P): Since the unit of pressure is N/m 2 = J/m 3 which is the same as U so the energy density carried by the EM field is also the pressure of the EM field, i.e. P = ɛe 2 Rate of energy flow S: The flow of energy in EM fields is often expressed by estimating the instantaneous energy flowing across a unit are per unit time. Since the energy is being transported in the direction of the light beam, it is useful to express the flow as a vector quantity known as the Poynting vector S which has units if W/m 2 given by: S = 1 µ E B = c 2 ɛ E B (4) Intensity of light I: Often it is useful to evaluate the time averaged light energy incident on a unit area per unit time (as compared to the instantaneous energy transported by the Poynting vector). Using eq. 4 the time averaged intensity can be evaluated as: I =< S > t = c 2 ɛ E B Using the fact that the intensity is a real quantity the above equation can be evaluated using the real components of E and B to give: I = c 2 ɛ E o B o < Cos 2 (kr ωt + φ) >= 1 2 c2 ɛe 2 2 Polarization In the previous section, linearly polarized light was introduced by stating that the electric (or magnetic) field oscillated in a single direction. In general a beam of light can have the E and B fields oscillating in any arbitrary direction with the only constraint that the oscillation is perpendicular to the wave propagation direction. The polarization state of a beam is a function of two quantities: (i) the oscillation direction of the E field and (ii) the behaviour of this oscillation with time. Based on this, and referring to Fig. various types of polarization states may be defined: 1. Random polarization: here the oscillation of the E field is equally likely to be in any direction at a 4 August 30, 2006

5 given instant of time (but always perpendicular to the direction of propagation). In other words, if snapshots of the E-field vector were to be captured, the vectors would lie in random directions. 2. Circular polarization: here the E-field is observed to rotate about the direction of propagation maintaining equal magnitude as a function of time. 3. Elliptical polarization: the electric field again rotates about the direction of propagation nut its magnitude varies with position. The various polarization types may be visualized in Fig. 2. To visualize the different polarizations one may make use of an oscilloscope and combine two sinusoidal signals in perpendicular directions to create Lissajous figures. This web-link allows to to do this through java applets: Figure 2: Different polarization states viewed along the direction of propagation. The direction of the electric field represents a time average over sufficiently long times. 3 Huygens-Fresnel principle While a ray of light travels in a straight line, a collection of rays originating from a light source varies spatially according to the nature of the source and other obstacles in the path of light. Huygens came up with an approach to determine the distribution of light intensity for an arbitrary situation given its value at some earlier instant. To understand the Huygens principle, we must first understand the concept of a wavefront. The wavefront is the surface or locus of all point on the travelling wave having identical phase. From eq. s 2 and 3 the mathematical condition describing a wavefront is that all points have the same θ value. Fig. 3(a) denotes a light source and its wavefront. Huygens theorem is especially important because a simple geometrical construction can be used to determine the wavefront at any later time. The theorem goes as follows: 5 August 30, 2006

6 Every point of a wavefront may be considered a source of small secondary wavelets, which spread out in all directions from their centers with a velocity equal to the velocity of the propagating wave. Tye new wavefront it then found by constructing a surface tangent to the secondary wavelets, thus giving the locus of all points with the same phase. Such a construction is demonstrated in Fig. 3(b). The original wavefront, S-S, is traveling as indicated by the arrows. The shape after a time t can be obtained by constructing a number of circles centred on S-S with radius r = vt, where v is the propagation velocity. The common tangent drawn between these circles will give the new wavefront. One of the problems with the Huygens construction is that according to the form, a wave must be travelling in the backward direction - but this is not the case. This problem was solved by Fresnel when he modified the construction to account the magnitude and phase of the local electric field, i.e. the math that gives us the interference phenomenon (to be discussed later). Now the modified theorem is called the Huygens-Fresnel principle. Figure 3: Example of an expanding wavefront and the construction of the Huygens wavefronts. 4 Geometrical forms of sources The spatial variation of light intensity differentiates some common types of propagating waves. These propagating waves can also be distinguished on the basis of the shape of their wavefronts and with reference to Fig. 4 the three common forms are: 1. Plane waves: Here the wavefront is a planar surface perpendicular to the propagation direction. The plane EM wave can be expressed as: E(x, t) = E o e i(kx ωt+φ) i The intensity of a plane wave as a function of space and time is I E. E E 2 o and is independent of r i.e. the distance from the source. 2. Spherical waves: Here the wavefronts are concentric spheres about the source of the waves and the electric field can be expressed as: E(r, t) = E o r ei(kr ωt+φ) p 6 August 30, 2006

7 where p is the polarization direction. The intensity of a spherical wave as a function of its distance from the source will be I E 2 E2 o r 2, which falls of as the square of the distance from the source. 3. Cylindrical waves: Here the wavefronts are concentric cylinders about the source of the waves and the electric field can be expressed as: E(r, t) = E o r e i(kr ωt+φ) p from which the intensity will change as I E2 o r 7 August 30, 2006

8 (a) (b) (c) Figure 4: Types of wavefronts and light sources. (a) Plane wave, (b) spherical waves; (c) cylindrical waves. Source of figures: Hecht, Ch 2. 5 sources and the EM spectrum The most important natural light source is the sun, which as Newton discovered, consists of white light. Unlike monochromatic light, white light is composed of numerous frequencies (or wavelengths). The reason the sun emits white light is primarily because it is a very dense and hot body of gases comprising various neutral and charged species, including ions and electrons. The random collision of the charged particles leading ti acceleration/deceleration, i.e. non-uniform motion, results in various frequencies being emitted. 8 August 30, 2006

9 In comparison to the sun, artificial light sources can be made to emit in specific frequencies. These light sources are known as spectral lamps and typically make use of the well defined energy levels in atoms to emit light of well defined frequencies, as indicated by the relation between the energy and frequency of light. The principle of a spectral lamp is as follows: A lamp consists of gas of a specific element or compound, such as Na, Hg, Ne, etc. An electric discharge is generated in the lamp by using a high-voltage source to produce excitation of the electrons in the gas atoms to higher energy levels. When these electrons fall back to their original energy levels, they radiate light of frequency proportional to the difference in energy levels. Consider a Na atom with equilibrium electron configuration given by 1s 2 2s 2 2p 6 3s 1. By the high-voltage, lets say the 3s 1 electron is excited to the 3p level. On falling back to the ground state the frequency of light emitted will be: ν = E 3p E 3s h The web-link introduces you to various sources of light and color. The study involving the observation of frequencies emitted/absorbed by various objects is called spectroscopy while the measurement of the emitted/absorbed frequencies is called spectrometry. The range of frequencies commonly encountered by us has been categorized into a spectrum (ref. p 74, Hecht) based on wavelength, energy, color etc. Some examples are given in table 1. name Radio frequency waves (RF) frequency ν in Hz wavelength λ Energy E in ev Uses 1kHz-1GHz km s to m s 10 6 Radio communication; power lines Microwaves cm 1 mm 10 6 E 10 3 vibration frequencies in water microwaves resonate with and is used in a microwave over Infrared (IR) Visible light ultraviolet (UV) x-rays Gamma rays mm 780 nm 10 3 E 1 Most solids absorb and emit IR and so these are most efficient at delivering energy to solids nm 380 nm 1 E 3 visible colors in this range nm 400 nm 3.2 E Å 100 Å 100 E > < 0.01 Å > Table 1: EM spectrum. Source: Hecht, Ch 3 Rays harmful to the skin. Also absorbed by the cornea of the eye, which is the primary cause of snow blindness. Wavelength are comparable/smaller then inter-atomic distances in solids and so x-ray diffraction is a powerful materials science tool 9 August 30, 2006