Download Light Waves

The Hong Kong Polytechnic University
Light Waves
Nature of Light:
Light can be viewed as both stream of particles and electromagnetic waves.
One-Dimensional Wave:
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.
Optics II----by Dr.H.Huang, Department of Applied Physics
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Wave Profile:
In positive x-direction:  ( x, t )  f ( x  vt)
Light Waves
 ( x, t )  f ( x, t )
In negative x-direction:  ( x, t )  f ( x  vt)
 ( x, t )  f ( x)
x  x  vt
Optics II----by Dr.H.Huang, Department of Applied Physics
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Light Waves
The Hong Kong Polytechnic University
Harmonic Waves (Sine Waves) :
 x, t   Asin k x  vt
Propagation number: k
Amplitude: A
Wavelength:  
2
k
Temporal period:  
Frequency:
k
2


v
f 1 
Angular frequency:
  2   2f
v  f
Wave number:   1 
x
v
x t
 



  x, t   A sin 2 
Various forms:  x, t   A sin kx  t 


 x, t   A sin 2f   t 
 x, t   A sin 2 x  ft 
Optics II----by Dr.H.Huang, Department of Applied Physics
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Light Waves
The Hong Kong Polytechnic University
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. The
wave function and x are in meters, t is in seconds.
 1  Asin k x  vt  2 sin 0.2 x  60t 
A2m

2
 10 m
k
k  0.2 m1


v

1
s
6
v  60 m/s
in negative x - direction
f  1   6 Hz
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.
8
f 1   v  
3 10
14

5
.
41

10
Hz
9
555 10
v 3 108
 
 6 106 m
f
50
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Light Waves
Phase and Phase Velocity:
 x, t   A sin kx  t  0 
Phase:  x, t   kx  t   0
Phase velocity:
vp 

k
Initial Phase: 0
v
Properties of Waves:
Wavefront: line that links all the points of the same phase
Ray: line perpendicular to all wavefronts
Plane waves: wavefronts are planes
Spherical waves: wavefronts are spheres
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Light Waves
Polarization:
If all the oscillations are confined within a plane, the wave is called a plane polarized
wave.
If the wave oscillations are random in all directions, it is an unpolarized wave.
Unpolarized wave consists of randomly oriented plane polarized waves.
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.
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Light Waves
Wavelength and Frequency in a medium:
v
c
n
m 

n
The frequency of light in the medium is unchanged.
Wave Packets or Groups:
Mathematically, 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
N
Optics II----by Dr.H.Huang, Department of Applied Physics
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Light Waves
The Hong Kong Polytechnic University
Reflection and Transmission:
Er  n cos i  n cos i 
r 

Ei  n cos i  n cos i 
t 
Et 
2n cos i

Ei  n cos i  n cos i 
r// 
Er // n cos i  n cos i 

Ei // n cos i   n cos i
t // 
Et //
2n cos i

Ei // n cos i   n cos i
sin i  i 
r  
sin i  i 
2 sin i cos i
t 
sin i  i
tan i  i
r// 
tan i  i
2 sin i cos i
t // 
sin i  i cosi  i
For normal incidence:
r//  r
t //  t 
Optics II----by Dr.H.Huang, Department of Applied Physics
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Light Waves
The Hong Kong Polytechnic University
Reflectance and Transmittance (normal incidence):
S  c 2 0 E  B
Poynting vector:
I S
T

c 0 2
E
2
I r vEr2 Er2 Er2  Er2// r2 Ei2  r//2 Ei2// r2 Ei2   r  Ei2//
2
R  2  2  2



r

I i vEi
Ei
Ei   Ei2//
Ei2  Ei2//
Ei2  Ei2//
2
I t vEt2 n Et2 n Et2  Et2// n t2 Ei2  t //2 Ei2// n t2 Ei2  t2 Ei2// n 2
T 




 t
2
2
2
2
2
2
2
2
I i  0vEi
n Ei
n Ei   Ei // n Ei   Ei //
n Ei   Ei //
n
 n  n   n  n 
Rr r 
 



n

n
n

n

 

2
2

2
2
//
T
n 2 n 2
4nn
t   t // 
n
n
n  n2
R T 1
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Light Waves
Phase Change:
For n>n, r<0 while r// change sign at the incidence
angle ip, where i+ip=90. The reflected ray is
perpendicular to the transmitted one.
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).
n
tan i p 
n
In the internal reflection case, there will be no
transmission when the incidence angle is larger than
a critical angle ic,
n
sin iC 
n
For near normal incidence or the glancing incidence,
there is a phase shift  for external reflection and
zero phase shift for internal reflection.
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.
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Light Waves
Stokes Relation:
Using the principle of reversibility for optical rays:
r 2 Ei  t tEi  Ei
tt   1  r 2
r tEi  trEi  0
r   r
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Light Waves
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?
60
1v
8
d



8
.
3

10
m

360
6 f
  t  2ft  3.77 1012 radian
N  ft  61011
Example:
Write an expression for the infrared wave shown in
the right figure.
A  75 unit ;   800 nm;   2 1015 s
k
2



4
107 m 1 ;   2f 
2

  1015 ;


 x, t   75 sin  107 x   1015 t 
4

Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Light Waves
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= ?
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?
2
 2 
energy  IAt  20   
 120  9600 J
 
Example:
The vacuum wavelength of a light wave is 750 nm. What is its propagation number
in a medium of refractive index 1.5?
m 
k

n
2
m

750
 500 nm
1.5

2
7

1
.
26

10
m
9
500 10
Optics II----by Dr.H.Huang, Department of Applied Physics
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Light Waves
The Hong Kong Polytechnic University
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.
t1 
t2 
L
L
30
7



1
.
33

10
s
v1 c n1 3 108 1.33
t2  t1  3 108 s
L
L
30
7



1
.
63

10
s
v2 c n2 3 108 1.63
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.
 n  n   1.50  1 
R
 
  4.0%

 n  n   1.50  1 
 n  n   1.45  1 
R
 
  3.4%
 n  n   1.45  1 
2
2
 n  n   1.55  1 
R
 
  4.7%

 n  n   1.55  1 
2
2
2
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Light Waves
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?
f 
t

nf

555
 424 nm
1.31
3
 f  318 nm
4
Homework: 10.1; 10.3; 10.5;
10.7; 10.9; 10.14
Optics II----by Dr.H.Huang, Department of Applied Physics
15