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Diploma Laser in Medicine
Prof. K. I. Hajim
Optics
Chapter One
Nature of Light
1.
Systems in the Universe may be classified into three types:
1- Particle systems (cars, stars, any object having a mass).
2- Wave systems (water waves, sound waves, spring waves, light
waves or anything in which the values of one of its description
parameters is repeated as time goes by and distances are crossed
temporally and spatially).
3- Particle wave systems (for bodies moving with a very high
velocity, having tiny or no mass like electrons, photons
respectively).
* The first step in understanding any system is to:
1- Put forward the proper equation of motion of that system.
2- Solve the equation of motion of the system.
*
The solution of the equation of motion of the system under study
contains all the information about that system from which the history and
full description for that system may be derived.
1.1 The particle system:The equation of motion of particle system is Newton’s Second Law.
This law states that "The force acting on the system (F) is equal to the
mass of the system (m), multiplied by the variation of its velocity (  ),
with respect to variation in time (  t ).
 t 
F= m
t
where,
F is the force,
M is the mass
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Since the velocity is the variation of the distance ( x ) to the variation in
time (  t ).
 xt  


 t  m  2 x(t)

=
 F= m
t
 t2
where,
xt  is the solution of equation of motion of the system:
 xt  


2

t

 = m  xt 
 F= m
t
 t2
By finding the solution of the above equation, we can find any another
quantity wanted like finding the position t  . Also acceleration and speed
can be found.
1.2 The Wave System:
The equation of motion of a wave system is called the wave equation.
In this equation two successive variations of the representative of the
wave with respect to position  2 x1t  is equal to the resprocal of the
speed of the wave multiplied by two successive variation of the
representative of the wave with respect to time.

1  2 x, t 
 2 x, t 
= 2
 t2
x 2

where,
 is the wave representative
 is the velocity of the wave

Sound waves
(Pressure)
Water waves
(Height)
Love waves
(No. of fights
per day)
Spring
Light
(No. of rings per (Electric field ,E)
unit length)
The solution of the wave equation, ( x, t) which is the wave
representative at a point in space, x , at a specific instant of time, t , is
equal to the wave representative at x 0 , t 0 , x 0 , t 0  multiplied by the
2
sinusoidal function with an argument equal to the angular temporal
frequency,  , multiplied by t , minus the angular spatial frequency, k ,
multiplied by x i.e. cos t  k x.
( x, t)   0 x 0 , t 0 cos t  k x 
* If we have a charged particle at a point in space
 The charge is moved to another point  electric field ,E, is varied
 creation of magnetic field ,B,  variation in this magnetic field 
creation of electric field which is also varying  and so on….

The variation of electric field with position is proportional to the
variation of magnetic flux density with time.



………………….... (1)
x
t
And the variation of magnetic field with time is proportion to the
variation of electric field with position.



………………….... (2)
t
x
Maxwell had joined (1) and (2) together, he found that:
Twice the variation of the wave representative with position is equal to
the reciprocal of the speed of light in free space multiplied by twice the
variation of the wave representative with time, i.e,
 2
1  2
 2
………………..(3)
x 2
c  t2
where,
c= speed of light in free space  3  108 m/s
The magnetic flux density, B, is equal to the permeability,  , multiplied
by the magnetic field intensity , H ,
B  H
 Response of matter to magnetism
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and the electric flux density,D, equals to the permittivity,  , multiplied by
the electric field,  ,
D  

 Response of matter to electricity
Equation (1) and (2) can be re- written as:


 
t
x
………………….... (4)
and



………………….... (5)
t
x
2. Maxwell Equations:
* Field: A field represents a set of values assumed by a physical quantity
at various points in a region of space at various instants of time.
* Maxwell Equations  These equations relate field vectors at arbitrary
points in space and time. In short, they cover the behaviors of time
varying electric field and magnetic fields.
1- The curl of the electric field,    i.e. the variation of the electric
field with respect to x, y, z equals to the permeability of free space
,  0 , multiplied by the variation of the magnetic field intensity with
time
i.e.

 
     0

t
t
2- The curl of the magnetic field,    , i.e. the variation of the
magnetic field with x, y, z, equals to the permittivity of free space
multiplied by the variation of the electric field with time plus the
current density.

D
    0
J 
J
t
t
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3- The divergence (variation w.r.t space) of the electric flux density
  D equals to the charge density.
D 
4- The divergence of the magnetic flux density equals to zero.
  0
3. Wave Parameters:
1) t  0

   0 cos t  kx
   0 coskx
Q: Why there is k?
Ans:
To convert the distance traveled by the wave, x, into angle units.
Where the distance traveled, x, divided by the spatial period,  , is

x
 No. of wavelength scanned by the wave within distance x.

x
2  The angle scanned by the wave within distance x.


Each   corresponds to 2 .
The wave representative at x and t  0 is equal to the wave
representative at x  0 , t  0 , multiplied by a sinusoidal function with an
argument equals to:

1- Number of wavelengths scanned by the wave
multiplied by 2
x
during x

x

x,0   0 cos .2 


Or
2- The angular spatial frequency,
distance x.
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2

, multiplied by the traveled
 2 
x,0   0 x,0cos .x 
  
x,0   0 x,0 cosk x 


 = Spatial period
1

2
= Spatial frequency
= Angular spatial frequency =k

2) x  0
   0 cos t 

Q: Why there is  ?
Ans:
To convert the time scanned by the wave, t , into angle units.
Where the time, t , scanned by the wave divided by the temporal period,
 ,is

t

t

 number of temporal periods scanned by the wave within time t.
2  is the angle scanned by the signal within time t.
 Each
  corresponds to 2  .
x
2 is the angle scanned by the signal within the time t.

 The wave representative at time = t and x  0 is equal to the wave
representative at ( x  0, t  0) multiplied by a sinusoidal function with an

argument equals to:
1. Number of temporal periods scanned by the wave,
multiplied by 2 .
t

0, x    0 cos .2 


Or
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t

during x
2. The angular temporal frequency,
2

, multiplied the time, t , scanned
by the wave.
 2 
0, t    0 0, t  cos t 
  

0, t    0 0, t  cos t 
  =Temporal period
1
=Temporal frequency

2
=Temporal angular frequency

3.  0  maximum value of the wave representative x  0, t  0  .
*The concept of phase difference:
Phase difference 
t
.2

x
2.
.2

1.
  t
 k x
4. The refractive index:
In any material, the speed of light is less than the speed of light in
free space.
Each material can be characterized by defining the index of refraction of
that material.
The refractive index, n , is equal to the speed of light in vacuum, c , to the
speed of light in the material, .
n
c

The speed of light in vacuum is equal to the reciprocal of the square root
of the permeability of vacuum,  0 , multiplied by the permittivity,  0 , of
vacuum. While the speed of light in the material is equal to the reciprocal
of the square root of the permeability,  , of the material multiplied by the
permittivity  , of the material.
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n
1
 0 0

1

1
 0 0
 0  r  0 r


 0 0
for most materials,    0


 r
0
n
5. Polarization
1) Linear polarization:
Consider an electromagnetic wave for which the fields E & H are
given by:
E= E0 exp i (k .r-t)
H= H0 exp i (K .r-t)
-If the amplitudes E0& H0 are constants, the wave is said to be linearly
polarized.
* In the case of natural or unpolarized light, the instantaneous
polarization fluctuates rapidly in a random manner.
2) Circular polarization:
Consider a special case of two linearly polarized waves of the same
amplitude E0, polarized orthogonally to each other.
Suppose the wave have a phase difference of



.
2
î 0 coskz  t  and ĵ  0 sinkz  t 
The total electric field, is


 total   0 î coskz   t   ĵ sin kz   t 
From this equation, the electric field vector at a given point is constant in
magnitude but rotates with temporal angular frequency  . this type of
wave is said to be circularly polarized.
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3) Elliptical polarization:
If the components of electric fields are not of the same amplitude,
say:
i  0 coskz  t 
and
j 0 sin kz  t 
 0  0
where
 The resultant electric field vector, at a given point in space, rotates and
also changes in magnitude.
In this case the wave is said to be elliptically polarized.
* Generally:
E= (Ex + i Ey) exp i k x   t 
is Ey =0
 Linear polarization.
is Ex= Ey
 Circular polarization.
is Ex  Ey  Elliptical polarization.
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Chapter (1) – Nature of light
(Exercises)
1- A radio wave has a wavelength (spatial period) of 30 m. Find the
temporal frequency and the temporal period of the wave.
2- Ordinary a current has a temporal frequency of 60 Hz. find the
wavelength and the temporal period of the wave.
3- The light produced by a He-Ne laser has a temporal period of
2.11  10-15sec. Find the angular temporal frequency.
4- Light travels through a ruby laser rod at a speed of
1.74  108m/sec.Find the index of refraction of ruby.
5- A He-Ne laser beam (λ= 633 nm) travels through a glass window
with an index of refraction of 1.65. Find the speed and temporal
period inside the glass.
6- Given
x
 1 


y
 2 t
 sin2
100
 20 


Find :
Spatial period, temporal period, angular spatial frequency,
Maximum amplitude.
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The electromagnetic spectrum
Region
Gamma Rays
Spatial period
(λ)/in meter
(0.0≈0.1)nm
Temporal freq.
(f)/in Hertz
31018≈∞
Energy
/in (joules)
1.910-15~1.910-16
Energy of a
photon/in (eV)
12403.8~1240
X-Rays
(0.1~20)nm
1.51016~31018
1.910-16~9.910-16
1240~62
Ultraviolet
(20~400)nm
7.51014~1.51016
9.910-18~2.710-19
62~3.1
Light
(400~700)nm
4.21014~7.51014
4.910-19~2.710-19
3.1~1.77
Infrared
(700)nm~
(1.0)cm
(1.0)cm~
(1.0)m
(1.0) m~
(1000)m
31010~4.21014
2.710-19~1.910-23
1.77~1.110-3
3108~31010
1.910-23~1.910-25
1.110-3~1.110-6
3102~3108
1.910-25~1.910-31
1.110-6~1.110-9
Microwave
Radiofrequency
waves
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