CHAPTER 10. LIGHT WAVES

What is the actual nature of light? Is light a wave phenomenon or a particle phenomenon? The essential feature of a particle is its localization and the essential feature of a wave is its non-localization. Light is a stream of particles (known as phonons). It is also an electromagnetic wave propagating in a medium.

10.1. One-Dimensional WavesThe most familiar waves and the easiest to visualize are the mechanical waves

(Fig10.1).

Wave profile is a function of both position and time,(10.1)

Here we limit ourselves to a wave that does not change its shape as it progresses through space. Suppose the wave pulse moves along the x-axis at a constant speed v. After a time t the pulse has moved along the x-axis a distance vt, but its shape remains unchanged. In the coordinate system S (Fig.1) that travels along with the wave pulse at a speed v, the pulse can be thought as stationary, so that

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Fig.10.1. One-dimensional wave generated in a spring. Longitudinal wave: the medium is displaced in a direction of motion of the wave. Transverse wave: the medium is displaced in a direction of motion of the wave. Feature: the disturbance advances, not the material medium.

Fig.10.2. A moving reference frame where the wave profile is unchanged with time.

Since x =x-vt, it follows that, (10.2)

Eq.(10.2) represents the most general form of the one-dimensional wavefunction. It describes a wave having the desired profile, moving in the positive x-direction with a speed v. Similarly, if the wave travels in the negative x-direction, the wavefunction becomes,

, with v>0 (10.3)

10.2. Harmonic Waves (Sinusoidal Waves)Harmonic wave is the simplest wave form with a profile of a sine or cosine curve.

Any wave shape can be synthesized by a superposition of harmonic waves. (10.4)

where k is a positive constant known as the propagation number, A is the amplitude of the wave. The wave is periodic in both space and time. The spatial period is known as the wavelength .

(10.5)For harmonic wave, .

If the wave is viewed from a fixed position, it is periodic in time with a repetitive unit temporal called temporal period .

The inverse of the temporal period is the frequency f.

Therefore, . The other two quantities are angular frequency and wave number or spatial frequency which is important in spectroscopy. The wave number is the number of whole wavelengths in one meter.

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Advanced Study:Using partial differential equations, the one-dimensional wave equation can be written as,

This is the wave equation for undamped systems that do not contain sources in the region under consideration. It can be shown that both and are solutions of the wave equation.

Fig.10.3. The spatial variation of a harmonic wave at a fixed time.

The harmonic wave functions can be rewritten as

(10.6)

(10.7)(10.8)

(10.9)

All these waves extend from to and each wave is monochromatic.

Example: Given the traveling wave function , find the frequency, the wavelength, the temporal period, the amplitude, and the direction of motion.

(Ans: 6Hz; 10m; 1/6s; 2m; negative x)

Example: The speed of electromagnetic waves in vacuum is 3108 m/s. Find the frequency of yellow-green light of wavelength 555 nm. Overhead power lines radiate electromagnetic waves at a frequency of 50 Hz. Compare the wavelength with yellow-green light. (Ans: 5.411014 Hz; 6106 m)

10.3. Phase and Phase VelocityMore generally, instead of Eq.(10.4), the harmonic wave function can be written as,

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Fig.10.4. The variation of a harmonic wave with time at a fixed position.

(10.10)The argument of the sine function is called phase: . 0 is called the initial phase which is the phase at t=0 and x=0. The rate-of-change of phase at any fixed location is the angular frequency of the wave,

Similarly, the rate-of-change of phase with distance, holding t constant is,

The speed at which the profile moves under the condition of constant phase is called phase velocity of the wave,

(10.11)

The phase velocity is the wave velocity at which the wave profile propagates. The phase velocity is accompanied by a positive sign when the wave moves in the direction of increasing x and a negative one in the direction of decreasing x.

10.4. Properties of WavesWhen a wave propagates in space, the line that links all the points of the same phase

is called wavefront. The propagation of a wave can be regarded as the advancing of a wavefront at the wave (or phase) velocity. The straight line in Fig.10.5 is perpendicular to all wavefronts in the direction of propagation and is called a ray.

Light in nature is an electromagnetic wave. It is a transverse wave in which the electric and magnetic fields oscillate in a direction perpendicular to the direction of wave propagation.

At a given time, when all the surfaces on which a disturbance has constant phase form a set of planes, each generally perpendicular to the propagation direction, the waves are called planes waves. If the wavefronts form concentric spheres that increase in diameter as they expand out into the surrounding space, the wave is a spherical wave.

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Fig.10.5. A sound wave travels from a point source through a three-dimensional medium. The sound is a longitudinal wave.

10.5. Amplitude and IntensityThe wave carries an energy that is proportional to the square of its amplitude. The

amount of energy passing through unit area per second is defined as the intensity of the wave. Therefore the intensity is also proportional to the speed of light in the medium.

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Fig.10.6. A point emits a spherical wave. At distance far away from the source, the wavefronts flattens into a plane wave.

Fig.10.7. (Left) If all the oscillations are confined within a plane, the wave is called a plane polarized wave. (Right) If the wave oscillations are random in all directions, it is an unpolarized wave. Unpolarized wave consists of randomly oriented plane polarized waves.

Advanced Study:The wave function of a plane wave can be written as,

where k is the propagation vector or the wave vector. The wavefront is a plane that is perpendicular to the wave vector k.

The wave function of a harmonic spherical wave can be expressed as,

The plane wave has a constant amplitude, while the spherical wave decreases its amplitude with increasing distance from the source. Both of the two wave functions satisfy the three-dimensional wave equation,

Advanced Study:The energy streaming through space in the form of an electromagnetic wave (such as light) is

shared equally between the constituent electric and magnetic fields. The energy flow in space can be represented by a vector (known as Poynting vector),

where E and B are the electric and magnetic field vectors, respectively.

10.6. Wavelength and FrequencySuppose c and are the speed and wavelength of light in vacuum, respectively. In a

medium of refractive index n, the speed and wavelength are reduced, respectively, and (10.12)

The frequency of light in the medium is unchanged.

Radio Waves: (frequency) a few Hz up to about 109 Hz. Microwaves: (frequency) 109 Hz to 31011 Hz.Infrared: (wavelength) 1 mm to about 760 nm. Visible light: (wavelength) 760 nm to about 390 nm. Ultraviolet: (wavelength) 390 nm to 1 nm.

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10-14

X-RAYS

MICROWAVES

Fig.10.8. The electromagnetic spectrum.

X-Rays: (wavelength) 1 nm to 6.010-12 m.

10.7. Wave Packets or GroupsA sinusoidal wave implies that the source is vibrating infinitely. In reality, the

source of light can only produce interrupted waves with finite length. Such an interrupted wave is known as a wave packet, wave group, or wave train. Mathematically (from Fourier analysis), a wave packet with a finite length consists of a group of frequencies. The smaller the number of frequencies in a wave packet then the greater will be the length of the wave packet. Different frequencies in a group mean different wavelengths. It can be shown that the spread of wavelength about the mean wavelength is,

(10.13)where N is the number of waves in the wave packet.

10.8. Reflection and TransmissionFresnel’s equations (10.14-17) are derived to calculate the amount of light reflected

and transmitted at an interface between two optical media.

(10.14)

(10.15)

(10.16)

(10.17)

where r is the amplitude coefficient of reflection perpendicular to the plane of incidence, whilst t is the amplitude coefficient of transmission perpendicular to the plane of incidence. Similarly, r// is the amplitude coefficient of reflection parallel to the plane of incidence, whilst t// is the amplitude coefficient of transmission parallel to the plane of incidence.

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Fig.10.9. A wave packet containing N waves, being the mean wavelength.

Advanced Study:A periodic or bounded function f(x) with a period can be expressed as a Fourier series,

The coefficients A and B can be calculated by,

and

Using the Snell’s law, the above equations can be simplified to,

(10.18)

(10.19)

(10.20)

(10.21)

When particular values are obtained from the above Eqs.(10.14-21), a minus sign in the result for r, t, r// and t// simply means the electric vector direction is opposite to that indicated in Fig.10.10.

For nearly normal incidence, i and i are very small so that the sine is approximately equal to the tangent and the cosine is equal to unity. From the Eqs.(10.14-17), we have,

and (10.22)

The reflectance is defined as the fraction of the incident intensity reflected. Recall that the intensity is proportional to the square of the amplitude and also proportional to the speed of light in the medium, we have,

The reflectance for near normal incidence is then,

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Fig.10.10. A schematic showing the electric vector E incident, reflected and transmitted at the interface of two optical media. The electric vector E is resolved into two perpendicular components: one parallel to the plane of incidence (subscript //) and one perpendicular (subscript ). The subscripts i, r, and t stand for incident, reflected and transmitted, respectively. The incident and reflected rays are in the medium of refractive index n, whilst the transmitted (refracted) ray is in the medium of refractive index n.

(10.23)

The transmittance is defined as the fraction of the incident intensity transmitted.

The transmittance for near normal incidence is then,

(10.24)

From Eqs.(10.23) and (10.24) we can get, (10.25)

This is an expected result which means that if there is no absorption, the sum of the reflected and transmitted intensities should be equal to the incident intensity.

For n>n, according to Snell’s law, we have i<i. From Eqs.(10.18) and (10.20) we see that r is always negative while r// change sign (from positive to negative) at the incidence angle ip. The condition for the change of sign is that i+ip=90. At this incidence angle, r//=0, the reflected electric vector is plane polarized in a direction perpendicular to the plane of incidence. The angle ip is called the polarization angle or Brewster’s angle (iB). Using Snell’s law, we have,

This gives the method the determine the Brewster’s angle,

(10.26)

The case where n>n is commonly called external reflection. When n<n, it is called internal reflection. In the internal reflection case, when the incidence angle is larger

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Fig.10.11. Phase changes for the reflected components of the electric vector E at the interface between two media with refractive indices n=1.5 and n=1. The perpendicular component E has a phase shift for all the incident angles. This can be clearly seen from Eq.(10.18).

than a critical angle ic, there will be no transmission and all the light will be reflected. This critical angle can be calculated by,

(10.27)

One important conclusion from the Fresnel’s equations is that for near normal incidence or the glancing incidence, there is a phase shift for external reflection and zero phase shift for internal reflection.

Another way to derive the reflection and refraction coefficients is to use the Stokes’ treatment. Suppose the amplitude of incident electric vector is Ei, the amplitude coefficients for reflection and transmission are r and t, respectively. Using the principle of reversibility for optical rays (Fig.10.12), we have,

andThe stokes’ relation is then,

and (10.28)

Example: What is the distance along the wave (in the direction of propagation) between two points having a phase difference of 60 if the wave velocity is 3108 m/s and the frequency is 61014 Hz? What phase shift occurs at a given point in 10 -3 seconds and how many waves pass by in that time? (Ans: 8.310-8m; 3.7681012radian)

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Fig.10.12. Graphs used for the derivation of the Stokes’ relation.

Example: Write an expression for the infrared wave shown in the right figure.

Example: Given the function , what is the value

of the function at x=0 when t=0, t=/4, t=/2, t=3/4, and t= ? (Ans: 0, 3, 0, -3, 0)

Example: A parallel circular beam of light is incident normally on a perfectly absorbing plane surface. If the intensity of the beam is 20 W/cm2 and its diameter is cm, how much energy is absorbed by the surface in 2 minutes? (Ans: 9.6 kJ)

Example: The vacuum wavelength of a light wave is 750 nm. What is its propagation number in a medium of refractive index 1.5? (Ans: 1.26107 rad/m)

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Example: An empty tank is 30 m long. Light from a sodium lamp (=589 nm) passes through the tank in time t1 when filled with water of refractive index 1.33. When filled with carbon disulphide of refractive index 1.63 it takes a time t2. Find the difference in the transit times. The speed of light in vacuum is c=3108 m/s.

(Ans: 310-8s)

Example: A beam of white light is incident normally on a plane surface separating air from glass. If the refractive indices for particular ‘red, green, and violet’ light wavelengths are 1.45, 1.50, and 1.55, respectively, find the reflectances for each color.

(Ans: 3.4%, 4.0%, 4.7%)

Example: A film of cryolite (Na3AlF6) of refractive index 1.31 is deposited on a glass substrate. If three-quarters of a wavelength of light with vacuum wavelength 555 nm is to occupy the film, how thick must the film layer be? (Ans: 317.7nm)

Exercises:10.1 State the amplitude, the propagation number, the frequency, the angular frequency,

the wavelength, the period, and the direction of propagation of the wave . The amplitude is in meters, x in meters and t in

seconds.

10.2 A wave propagating with =30010-9m at a speed of 3108m/s has what period?

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10.3 60 water wave crests pass a fixed post in 15 seconds traveling at a speed of 3m/s. Find the frequency and wavelength.

10.4 Red light, =600nm, enters a glass block whereupon its velocity is two-thirds that of its velocity in vacuum. Find the refractive index of the glass for this wavelength and the wavelength in the glass.

10.5 A ray of monochromatic light is incident at 40 on a plane air/glass boundary. If the refractive index of the glass is 1.5 for this wavelength, calculate the amplitude reflection and transmission coefficients, assuming nair=1. At what angle of incidence is r//=0?

10.6 Calculate the reflectance and transmittance for monochromatic light incident normally (a) externally and (b) internally on an air/glass boundary for which ng=1.5.

10.7 A system of N flat plates of glass, separated by air, transmits a normally incident beam of monochromatic light. Show that the total transmittance is (1-R)2N. If ng=1.5, find the total transmittance for (a) 2 plates and (b) 10 plates.

References:1. A.H.Tunnacliffe and J.G.Hirst, Optics, The Association of British Dispensing

Opticians, 1996. 2. F.L.Pedrotti and L.S.Pedrotti, Introduction to Optics, Prentice Hall, New Jersey,

1993.3. E.Hecht, Optics, Addison Wesley, San Francisco, 2002. 4. D.Halliday, R.Resnick and J.Walker, Fundamentals of Physics, 5th edition, John

Willey & Sons, Inc., Singapore, 1997.

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