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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 Waves
The most familiar waves and the easiest to visualize are the mechanical waves
(Fig10.1).
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.
Wave profile is a function of both position and time,
 ( x, t )  f ( x, t )
(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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 ( x, t )  f ( x)
Since x =x-vt, it follows that,
 ( x, t )  f ( x  vt)
(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,
 ( x, t )  f ( x  vt) , with v>0
(10.3)
Advanced Study:
Using partial differential equations, the one-dimensional wave equation can be written as,
 2
1  2

x 2 v 2 t 2
This is the wave equation for undamped systems that do not contain sources in the region under
consideration. It can be shown that both  x, t   f x  vt and  x, t   f x  vt are solutions of the
wave equation.
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.
 x, t   Asin k x  vt
(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 .
 x, t    x  , t 
(10.5)
For harmonic wave, k  2  .
Fig.10.3. The spatial variation of a
harmonic wave at a fixed time.
If the wave is viewed from a fixed position, it is periodic in time with a repetitive
unit temporal called temporal period .
  v
The inverse of the temporal period is the frequency f.
f 1 
Therefore, v  f . The other two quantities are angular frequency   2   2f and
wave number or spatial frequency   1  which is important in spectroscopy. The wave
number is the number of whole wavelengths in one meter.
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Fig.10.4. The variation of a harmonic wave with
time at a fixed position.
The harmonic wave functions can be rewritten as
x t
 x, t   A sin 2   
(10.6)
  
 x, t   A sin 2 x  vt
(10.7)
 x, t   A sin kx  t 
(10.8)
x 
 x, t   A sin 2f   t 
(10.9)
v 
All these waves extend from x   to x   and each wave is monochromatic.
Example: Given the traveling wave function  1  2 sin 2 0.1x  6t  , 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 3108 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: 5.411014 Hz; 6106 m)
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10.3. Phase and Phase Velocity
More generally, instead of Eq.(10.4), the harmonic wave function can be written as,
 x, t   A sin kx  t  0 
(10.10)
The argument of the sine function is called phase:  x, t   kx  t   0 . 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,

 
t x
Similarly, the rate-of-change of phase with distance, holding t constant is,

k
x t
The speed at which the profile moves under the condition of constant phase is called
phase velocity of the wave,

t x 
x

 v
(10.11)

t 
k
x t
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 Waves
When 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.
Fig.10.5. A sound wave travels from a point
source through a three-dimensional medium.
The sound is a longitudinal wave.
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
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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.
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,
 r   Ae i ( kr t )
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,
A
 r , t   e ik r vt 
r
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,
1  2
 2  2 2
v t
10.5. Amplitude and Intensity
The 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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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),
S  c 2 0 E  B
where E and B are the electric and magnetic field vectors, respectively.
10.6. Wavelength and Frequency
Suppose 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,
c

v
m 
and
(10.12)
n
m
The frequency of light in the medium is unchanged.
10-14
X-RAYS
MICROWAVES
Fig.10.8. The electromagnetic spectrum.
Radio Waves: (frequency) a few Hz up to about 109 Hz.
Microwaves: (frequency) 109 Hz to 31011 Hz.
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Infrared: (wavelength) 1 mm to about 760 nm.
Visible light: (wavelength) 760 nm to about 390 nm.
Ultraviolet: (wavelength) 390 nm to 1 nm.
X-Rays: (wavelength) 1 nm to 6.010-12 m.
10.7. Wave Packets or Groups
A 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,
 1

(10.13)

N
where N is the number of waves in the wave packet.
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,


A
f  x   0   Am cos mkx   Bm sin mkx
2 m 1
m 1
The coefficients A and B can be calculated by,
2 
2 
Am   f x  cos mkxdx
Bm   f x sin mkxdx
and
 0
 0
10.8. Reflection and Transmission
Fresnel’s equations (10.14-17) are derived to calculate the amount of light reflected
and transmitted at an interface between two optical media.
E
n cos i  n cos i 
r  r  
(10.14)
Ei  n cos i  n cos i 
E
2n cos i
t  t 
(10.15)
Ei  n cos i  n cos i 
E
n cos i  n cos i 
r//  r // 
(10.16)
Ei // n cos i   n cos i
Et //
2n cos i

(10.17)
Ei // n cos i   n cos i
where r is the amplitude coefficient of reflection perpendicular to the plane of incidence,
t // 
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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.
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.
Using the Snell’s law, the above equations can be simplified to,
sin i  i 
r  
(10.18)
sin i  i 
2 sin i cos i
t 
(10.19)
sin i  i
tan i  i
r// 
(10.20)
tan i  i
2 sin i cos i
t // 
(10.21)
sin i  i cosi  i
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,
r//  r
t //  t 
and
(10.22)
The reflectance is defined as the fraction of the incident intensity reflected. Recall
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that the intensity is proportional to the square of the amplitude and also proportional to the
speed of light in the medium, we have,
2
vIr Er2 Er2  Er2// r2 Ei2  r//2 Ei2// r2 Ei2   r  Ei2//
R
 2  2


 r2
2
2
2
2
2
vIi Ei
Ei   Ei //
Ei   Ei //
Ei   Ei //
The reflectance for near normal incidence is then,
 n  n   n  n 
Rr r 
 

 n  n   n  n 
2
2

2
2
//
(10.23)
The transmittance is defined as the fraction of the incident intensity transmitted.
vI t n Et2 n Et2  Et2// n t 2 Ei2  t //2 Ei2// n t 2 Ei2  t 2 Ei2// n 2
T




 t
vIi
n Ei2 n Ei2  Ei2// n Ei2  Ei2//
n Ei2  Ei2//
n
The transmittance for near normal incidence is then,
n
n
4nn
T  t 2  t //2 
(10.24)
n
n
n  n2
From Eqs.(10.23) and (10.24) we can get,
R T 1
(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.
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).
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,
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

n sin i p  n sin i   n sin 90  i p  n cos i p
This gives the method the determine the Brewster’s angle,
n
tan i p 
n
(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 than a
critical angle ic, there will be no transmission and all the light will be reflected. This
critical angle can be calculated by,
n
sin iC 
(10.27)
n
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,
r 2 Ei  t tEi  Ei
r tEi  trEi  0
and
The stokes’ relation is then,
tt   1  r 2
r   r
and
(10.28)
Fig.10.12. Graphs used for the derivation of the Stokes’ relation.
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 3108 m/s and the
frequency is 61014 Hz? What phase shift occurs at a given point in 10 -3 seconds and
how many waves pass by in that time?
(Ans: 8.310-8m; 3.7681012radian)
22
Example:
Write an expression for the
infrared wave shown in the right figure.
Example:
Given
the
function
 x, t   3sin kx  t    , 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 4  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: 1.26107 rad/m)
23
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=3108 m/s.
(Ans: 310-8s)
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
  10 sin 2 100 x  3  1010 t . The amplitude is in meters, x in meters and t in
seconds.
 

10.2 A wave propagating with =30010-9m at a speed of 3108m/s has what period?
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.
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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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