Part 1 :

1. Nature of light
   During the seventeenth century two emission theories on the
   Nature of light were developed, the wave theory of Hooke and Huygens
   and the corpuscular theory of Newten. Subsequent observations by
   Young, Malus, Euler and ithers lent support to the wave theory. Then in
   1864 Maxwell combined the equation of electromagnetism in a general
   form and showed that they suggest the existance of trensverse
   electromagnetic wave. The speed of propagation in free space of these
   waves was given by
   0 0
   1
   c
   ε µ
   = (1)
   where 0
   ε and µ0
   are the permittivity and permeability of free space,
   respectively. Substitution of the experimentally determined values of 0
   ε
   and µ0
   yielded a vaule for c in very close agreement with the value of
   the speed of light in vacuo measured independently. Maxwell therefore
   proposed that light was an electromagnetic wave having a speed of
   8
   c m s = ×3 10 / , a frequency of some 14 f Hz = ×5 10 and a wavelength of
   about λ = 500nm .
   Maxwell theory suggested the posibilty of producing
   electromagnetic waves with a wide range of frequencies (or
   wavelengths). In 1887 Hertz succeeded in generateing non-visible
   electromagnetic waves, with a wavelength of the order of λ =10.0m , by
   discharging an induction coil across a spark gap thereby setting up
   oscillating electric and magnetic fields. Visible light and Hertzain waves
   are part of the electromagnetic spectrum which, as we can see from Table
   1, extends approximately over the wave length range of λ =1.0pm to
   λ =100.0km . The wave theory thus become the accepted theory of light.
   ٣
   However, while the wave theory, as we shell see below, provides an
   explanation of optical phenomena such as interfererence and diffraction,
   it fails completely when applied to situation where energy is exchanged,
   such as in the emission and absorbtion of light.and the photoelectric
   effect. The photoelectric effect, which is the emission of elwctrons from
   the surfaces of solids when irradiated, was explanied by Einstein in
   1905.He suggested that the energy of a light beam is not spread evenly,
   but is concentrated certain regions, which propagate like particles. He
   called these particles' photons.
   Einstein was led to the concept of photons by work of Planck on
   the emission of light from hot bodies. Planck found that the observation
   indicated that light energy is emitted in muliplies of certain minimum
   enregy unit. The size of the unit, which is called a quantum, depends on
   the wavelength λ of the radiation and is given by
   hc E
   λ
   = (1)
   where h is Planckconstant. Planck's hypothesis did not require that the
   energy should be emitted in loclazied bundles and it could, with
   difficulty, be reconcild with the electromagnetic wave theory. When
   Einstein showed, however, that it seems necessary asssume the
   concentration of energy traveling through space as particles, a wave
   solution was excluded. Thus we have a particle theory also; light
   apparently has a dual nature.
   The two theories of light are not in conlict but rather they are
   complementary. For our purposes it is sufficient to accept that in many
   experiments, especially those invloving the exchange of energy, the
   particle (photon or quantum) nature of light dominates the wave nature.
   On the other hand, for experiments invloving interfererence and
   diffraction, where light interacts with light, the wave nature dominates.
   ٤


2. Source of light
   It is well-known that when an electron in an atom undergoes
   transitions between energy state or levels it either emits or absorbs a
   photon, which can be described in term of a wave of frequency ν where
   ν = ∆E h/ , ∆E being, the energy differrence between the two levels
   concerned. Let us consider the electron transitions which may occur
   between the two energy level s of the hypothetical atomic system shown
   in Fig 1. If the electron is in the lower E1
   then in the presence of photons
   of enegy 2 1 ( ) E E− it may be excited to the upper level E2
   by absorbing
   a photon. Alternatively if the electron in the level E2
   it may return to the
   ground stste with the emission of a photon. The emission process may
   occur in two distinct ways. Theses are : (a) the spontaneous emission
   process in which the electron drops to the lower level in an entirly
   random way and (b) the stimulated emission process in which the
   electron is triggered to undergo the transition by the presence of photons
   of energy 2 1 ( ) E E− . There is nothing mystical in this, as the electron
   would undergo this process sooner or later spontaneously : the transition
   is simply initiated by the presence of stimulating photon.
   Fig.1. Two energy level system
   The absorption and emission processes are illustrated in Fig 2.(a),
   (b) and (c). Under normal circustances we do not observe the stimulated
   emission process because the probabilty of the the spontaneous emission
   ٥
   process occurring is much higher. The average time the electron exists in
   the excited state before making a spontaneous transition is called the
   lifetime 21 τ of the the excited state. The subtitle 12 here indicates the
   energy levels involved. Because spontaneous radiation from any atom is
   emitted at random, the radiation emitted by a large number of atoms will
   clearly be incoherent. In contrast to this, the the stimulated emission
   process results in coherent radiation since the wave associated with the
   stimulating and stimulated photons have identical frequencies, are in
   phase, have the same state of polarization and travel in the same
   direction. This means that with stimulated emission the amplitude of an
   incident wave can grow as it passes through a collection of excited atom
   in what is clearly an amplification process. As the absorption transition,
   in common with stimulated emission, can only occur in the presence of
   photon of appropriate energy, it is often referred to as stimulated
   absorption. These two processes may be regarded as the inverse of one
   another.
   Fig.2. Energy level diagram illustrating (a) absorption, (b) spontaneous emission and (c)
   stimulated emission. The black dot indicates the state of the atom before and after the
   transition.


3. Propagation of light
   Propagation of light refers to the manner which is an
   electromagntic wave trsnsfer it's energy from one point ot another.
   ٦
   Transmission, reflection and refraction are the three main part of
   propagation of light. Principly, the light can travel in three ways from
   source to another location : (1) directly from the source through empty
   space (2) through various media; and (3) after being reflected from a
   mirror. This propagation is in the form of a transverse electromagntic
   wave.
   This work is concerned with the process of light traveling from
   directly from the source through empty space. This maen that, we will
   derive the main equation described the propagation of light in empty
   space. Two approximations; pure plane wave and Gaussian wave will be
   presented. It is known that the first approximation represents the ideal
   situation and proceed directly from Maxwell's equations. Although this
   approximation does not represent the practical cases, such as Gaussian
   wave and so on , it constitutes the theoretical basis for their derivation.
   3-1. Pure plane wave
   A. Summary of Maxwell's equations :
   The results of combing Faraday's law, Ampere's law and Gauss' law are
   referred to as Maxwell's equations. These equations can be written in
   differential form (or point form) as follow:
   ∇ = .D ρ (1)
   ∇ = .B 0 (2)
   dt
   ∂
   ∇ × = − B
   E (3)
   t
   ∂
   ∇ × = +
   ∂
   D
   H J (4)
   where D is electric displacement, ρ is medium free charge density, B is
   magnetic flux (flow) density, E is electric field strength, H is magnetic
   field strength and J is current flow density. These deceptively simple
   equations are the result of many years of research and study of the
   ٧
   phenomena associated with electricity and magnetism. Although these
   equations have not been derived analytically, the are reasonable and no
   experiments have shown them to be invalid. In the absence of any such
   data, we may accept them as a valid characterization of electromagnetic
   phenomena. However, these equations are applicable only where the
   dimensions are large compared to atomic dimensions.
   Associated with Maxwell's equations, we have equation of continity (or
   conservation of charge)
   t
   ∂ρ
   ∇ × = −
   ∂
   J (5)
   The current flow density and the electric field in the region are related by
   Ohm's law J E =σ where σ is the conductivity. For a linear, isotropic
   and homogeneous medium, we have :
   0 D E = ε ε (6)
   B H = µ µ0
   (7)
   where 0
   ε is permittivity of free space, µ0
   is permeability of free space,
   ε is relative dielectric constant and µ is relative permeability. In the case
   of no free charge ( ρ = = 0, 0 J ), thus Maxwell's equations will be :
   0 ∇ = .( ) ε εE 0 (8)
   0 ∇ = .( ) µ µH 0 (9)
   0
   ( )
   dt dt
   µ µ
   ∂ ∂ ∇ × = − = − B H E (10)
   0
   t t
   ε ε
   ∂ ∂ ∇ × = + =
   ∂ ∂
   D E H J (11)
   B. Waves equations :
   First we derive the wave equation that governs the propagation of all
   electromagnetic wave. Consider a linear, isotropic and homogeneous
   medium. We will assume that the net free charge in the region is zero
   (ρ = 0) and that any currents in the region are conduction ( ) J E =σ .
   ٨
   These type of regions are quite general and including the practical cases
   of free space (σ µ ε = = = 0, 1, 1) as well as most conductors and
   dielectrics. Maxwell's equations in point form for the free space become :
   0
   dt
   µ
   ∂
   ∇ × = − H
   E (12)
   0
   t
   ε
   ∂
   ∇ × =
   ∂
   E
   H (13)
   ∇ = .E 0 (14)
   ∇ = .H 0 (15)
   Taking the curl of equation (12) and substituting equation (13), we obtain
   2
   0 0 2
   ( )
   dt
   µ ε
   ∂
   ∇ × ∇× = − E
   E (16)
   We similarly obtain, by taking the curl of equation (13) and substituting
   equation (12),
   2
   0 0 2
   ( )
   dt
   µ ε
   ∂
   ∇ × ∇× = − H
   H (17)
   In order to interpret these results, we use the vector identity
   2 ∇ × ∇ × = ∇ ∇ − ∇ ( ) ( . ) A A A (18)
   where 2 ∇ A is the vector Laplancian. In rectangular coordinates, the
   vector Laplancian is given by
   2 2 2
   2
   2 2 2
   A A x z
   A
   y
   x y z
   ∂ ∂ ∂
   ∇ = + +
   ∂ ∂ ∂
   A (19)
   Substituting equation (18) into equations (16) and (17), we obtain
   2
   2
   0 0 2
   dt
   µ ε
   ∂
   ∇ = E
   E (20)
   2
   2
   0 0 2
   dt
   µ ε
   ∂
   ∇ = H
   H (21)
   since ∇ = .E 0 and ∇ = .H 0 for the free space. The vector differential
   equations (20) and (21) are called the wave equations (or Helmholtz
   ٩
   equations). Each equation is composed of three scalar differential
   equations in term of the components of the vectors. For example, by
   matching components, we obtain
   2 2
   2
   0 0 2 2 2
   1 x x
   x
   E E E
   dt c dt
   µ ε
   ∂ ∂ ∇ = = (22)
   And similarly for the other two components of E and the three
   components of H.
   The electric and magnetic fields vibrate perpendicularly to one
   another and perpendicularly to the direction of propagation as illustrated
   in Fig.3 that is, light waves are transverse waves. In describing optical
   phenomena we often omit the magnetic field vector. This simplified
   diagrams and mathematical descriptions but we should always remember
   that there is also a magnetic field component which behaves in similar
   way to the electric field component.
   Fig.3 Electromagnetic wave : the electric vector and magnetic vector vibrate in orthogonal
   planes and perpendicular to the direction of propagation. (p4)
   C. Time-Harmonic field
   Maxwell's equations and all the equations derived from them so far in
   this work hold for electromagentic quantities with an arbitrary timedependance. The actual type of time functions that the field quantities
   assume depends on the source function ρ and J . In engineering,
   ١٠
   sinusoidal time functions occupy a unique position. They are simplest
   form and easy to generate; arbitrary periodic time functions can be
   expanded into Fourier series of haramonic sinusoidal components; and
   transient nonperiodic functions can be expreesed as Fourier integeral.
   Field vectors that vary with space coordinates and are sinusoidal
   fuctions of time can be mathematically represeneted by vector phasers
   that depend on space coordinates and time as follow
   0 E( , , , ) exp ( ) x y z t E i kz t = −ω (23)
   0 H( , , , ) exp ( ) x y z t H i kz t = −ω (24)
   where E( , , , ) x y z t and H( , , , ) x y z t are the value of the electric and
   magentic fields at the point r at time t , E0
   and H0
   are the amplitudes of
   the electric and magnetic waves, ω is the angular frequency and k is the
   wave number ( k = 2π λ/ ). The term ( ) kx t −ω is the phase of the the
   electric and magnetic waves. Thus wave equations (20) and (21) become
   2
   0 0 ∇ = E E i µ ε ω (25)
   2
   0 0 ∇ = H H i µ ε ω (26)
   Expanding equations (25) and (26) in terms of components, the wave
   equations for the phasor components of the field vector become
   2 2 2
   2
   2 2 2
   x x x
   x
   E E E k E
   x y z
   ∂ ∂ ∂

• 
  
  
  
  • = −
    ∂ ∂ ∂
    (25)
    2 2 2
    2
    2 2 2
    y y y
    y
    E E E
    k E
    x y z
    ∂ ∂ ∂
  
  
  
  


• 
  
  
  
  • = −
    ∂ ∂ ∂
    (26)
    2 2 2
    2
    2 2 2
    z z z
    z
    E E E k E
    x y z
    ∂ ∂ ∂
  
  
  
  


• 
  
  
  
  • = −
    ∂ ∂ ∂ (27)
    2 2 2
    2
    2 2 2
    x x x
    x
    H H H k H
    x y z
    ∂ ∂ ∂
  
  
  
  


• 
  
  
  
  • = −
    ∂ ∂ ∂ (28)
    ١١
    2 2 2
    2
    2 2 2
    y y y
    y
    H H H
    k H
    x y z
    ∂ ∂ ∂
  
  
  
  


• 
  
  
  
  • = −
    ∂ ∂ ∂
    (29)
    2 2 2
    2
    2 2 2
    z z z
    z
    H H H k H
    x y z
    ∂ ∂ ∂
  
  
  
  


• 
  
  
  
  • = −
    ∂ ∂ ∂ (30)
    D. Uniform plane wave
    A uniform plane wave is a particular solution of Maxwell's
    equations with E (and H) assumeing the same direction, same
    magnitude, and same phase in infinite planes perendiduclar to the
    direction of propgation. Stricly speaking a uniform plane wave does nor
    exist in practical because a sources infinite in extent would be required to
    create it, and practical wave sources are alawyes finite in extent. But if
    we are far enough away from a source, the wavefront beacomes almost
    sperical; and very small portion of the surface of a giant sphere is very
    nearly a plane. The characteristics uniform plane waves are particular
    simple, and their study is of fundamental theoretical as well as practical
    importance.
    We may arbitrarity assume the direction of E to be in the postive
    x direction; that is
    ( ) ˆ E = E z a x x (31)
    This x component of E is a function of only z since the field is to be
    uniform over the xy plane x and y is thus independent of and
    coordinates. Therefore , we have
    0
    E E x x
    x y
    ∂ ∂
    = =
    ∂ ∂ (32)
    From equation (12), we have
    0 ∇ × = − E H i µ ω (33)
    and curl of E can be fined as
    ١٢
    ˆ ˆ ˆ
    z x z x y y
    x y z
    E E E E E E
    a a a
    y z z x x y
        ∂ ∂ ∂ ∂ ∂ ∂   ∇ × = − + − + −           ∂ ∂ ∂ ∂ ∂ ∂   E (34)
    ˆ
    x
    y
    E
    a
    z
      ∂ ∴ ∇ × =     ∂
    E (35)
    Thus
    ˆ
    x
    y
    E
    a
    z
      ∂ ∴ ∇ × =     ∂
    E (36)
    Therefore, by matching components of equation (33), we see that the only
    non-zero component of H is the y component, as shown by equation
    (36); that is
    ( ) ˆ H = H z a y y (37)
    0
    1
    ˆ
    x
    y
    E
    a
    i z µ ω
    ∂
    =
    ∂
    H (38)
    This y component of H is also shown independent of x and y . This
    follows directly from equation (38) since Ex
    is independent of x and y .
    Therefore, E and H for a uniform plane wave are orthogonal as shown
    in Fig.3. The resulating components of the wave equation given in
    equations (25)-(30) become particularly simple for this case. Since
    0 E E y z = = and 0 H H x z = = , equations (26), (27), (28), (30) are zero .
    the remaining two equations (25) and (29) become
    2
    2
    2
    0
    x
    x
    d E k E
    dz
  
  
  
  


• = (39)
  2
  2
  2
  0
  y
  y
  d H
  k H
  dz


• = (40)
  where partial derivative have been replaced by ordinary derivative since
  H y
  and Ex
  are functions of only one variable, z . The solution of
  equation (39) is readily seen to be
  ١٣
  ( ) ( ) ( ) E z E z E z x x x


• − = + (41)
  0 0 ( ) exp( ) exp( ) E z E ikz E ikz x


• − = − + − (42)
  where E0

and E0
−
are arbitrary constants that must be dertemined by
boundary condition. Let us now examine the frist phasor term on the right
side of equation (42) and write
0
( ) ( ) exp( ) ˆ ˆ
x x x
z a E z a E ikz + + E = = − (43)
For a cosine reference, the instantaneous expression for E in equation
(43) is
( , ) ( , ) Re ( )exp( ) ˆ ˆ
x x x x
z t a E z t a E z i t ω

= =     E (44)
0 0 ( , ) Re exp ( ) cos( ) ˆ ˆ
x x
z t a E i t kz a E t kz ω ω

= − = −     E (45)
It is clear that quation (45) represent a traveling wave and descibes a
perfectly monochromatic plane wave of infinite extent propgation in the
postive z direction. Equation (45) has been plotted in Figs.4 for several
values of t . At t = 0, 0
( ,0) cos( ) E z E kz x

= is a cosine curve with an
amplitude E0
+
. At successive time the curve effectively travels in the
postive z direction. We have, then, a traveling wave.
١٤
Fig.4. Wave traveling in in postive z direction 0
( , ) cos( ) E z t E t kz x ω

= − for several
values of t .
١٥
Here we should mentioned to that, equation (45) can also be
expressed using a sine rather that a cosine function, or alterntively using
complex expoentials. However, in the plane wave described above and in
other forms of wave there are surfaces or constant phase, which are
referred to as wavesurface or wavefront. As time elapses the wavefront
move through space with a velocity ϑ . Thus if we fix our attantion on a
particular point (a point of particular phase) on the wave, we set
cos( ) tan ωt kz a cons t − = or ωt kz A cons t phase − = tan , from which
we obtain
0 0
1
c
k
ω
ϑ
ε µ
= = = (46)
Equation (46) assures us that the the velocity of propagation of a
equiphase fron (the phase velocity) is equal to the velocity of light. Wave
number k bears a difinite relation to the wavelength
0 0
2
k
π
ω ε µ
λ
= = (47)
We can see, the second phasor term on the right side of equation (42)
0 E ikz exp( ) −
− , represents a cosinusoidal wave traveling in the (−z )
direction with the same velocity ϑ . if we are concerned only with the
wave traveling in (+z ) direction, we set 0 E 0
−
= . The associated
magnetic field Hcan be found from equation (33)
0
0 0 ( )
( ) 0 0
x y z
x x y y z z
x
a a a
i a H a H a H
z
E z
µ ω

 
  ∂
∇ × = = − + +  
  ∂
   
E (48)
which leads to
0 H x
+
= (49)
١٦
0
1 ( )
x
y
E z H
i z µ ω
+

• ∂
  =
  ∂
  0 H z

=
Thus H y
+
is the only nonzero component of corresponding H to the E in
equation (43), and since
0
( ) ( exp( ) ( ) x
x
E z E ikz ikE z
z z
+
∂ ∂ + +
= − = −
∂ ∂
(50)
Equation (48) yields
0 0
1
ˆ ˆ ˆ ( ) ( ) ( ) y y y x y x
k
a H z a E z a E z
µ ω η

• 
  
  
  
  • 
    
    
    
    • H = = = (51)
      where η0
      is called the intrinsic impedance of the free space and it is given
      by
      0 0 0
      0
      0 0 0
      120 377
      k
      µ ω µ µ
      η π
      ε µ ε
      = = = = = Ω (52)
      ( ) H z y
    
    
    
    
  
  
  
  

is in phase with ( ) E z x
+
, and we can write the instantaneous
expression for H as
( , ) ( , ) Re ( )exp( ) ˆ ˆ
y y y y
z t a H z t a H z i t ω

= =     H (53)
0
0
( , ) cos( ) ˆ
y
E
z t a t kz ω
η
+
H = − (54)
The above analysis for the phase velocity is true for the case of
monochromatic waves. As we know, it is impossible in practice to
produce perfectly monochromatic waves, we often have the situation
where a group of wave of closely similar wavelength is moving such that
their resultant forms a packet. This packet moves with the group velocity
ϑg
. The group velocity is given by
g
k
ω
ϑ
∂

∂
(55)
١٧
E. Two dimensional plane wave
The mathematical description of the one dimensional plane wave
can be generalized to include plane wave moving in two directions. Such
wave be characterized by a wavevector k , thus equation (23) becomes :
0 E r E k r ( , ) ( , , , ) exp ( . ) t x y z t E i t = = −ω (56)
where r is a vector from the origin to the point ( x y z , , ). Thus, for
example have plane wave propagating in direction yz to the z axis with
its wavefront normal to the yz plane, we can write
ˆ ˆ
y y z z k = + k a k a (57)
ˆ ˆ
y z r = + ya za (58)
.
y z k r = + k y k z (59)
By follow the same analysis of one dimensional plane wave, hence we
can write equation (56) in this case as
0
( , , , ) cos( ) y z E x y z t E t k y k z = − − ω (60)
Equation (60) has been plotted in Fig.5 for t = 0.
An equally important concept is that of spherical waves which, we
can imagine, are generated by point source of light. If such a source is
located in an isotropic medium (such as free space) it will radiate
uniformly in all directions, the wavefront ate thus a series of concentric
spherical shells. We can describe this situation by
( , ) cos( . ) t t
r
ζ
E r k r = − ω (60)
where the constant ζ is known as the source sterngth. The factor 1/ r in
the amplitude term accounts for decrease in amplitude of the wave as it
propagates further and further from the source. As the irradiance (or
int / ( . ) ensity energy time area = is proportional to the square of the
amplitude, there is an inverse-square-law decrease in irradiance. If the
medium in which the source is located is an isotropic, then the wave
١٨
surfaces are no longer spheres; their shapes depend on the speed of
propagation in diferent directions.
Fig.5. Two dimensional plane wave traveling in yz plane for t = 0 and its counter.
١٩
Part 2 :

1. Laser line shapes
   In deriving the expression for the propagation of plane wave
   acuually represents the ideal case. It is implicity assumes that all the
   atoms in either the upper or lower levels would be able to interact with
   the perfectly monchromatic wave with lineshape 0
   f ( ) ν . Although, the
   spectral width of a laser output can be much less than that of ordinary
   light due to the spontaneous emission process, it cannot really be
   considered monchromatic wave. Thus, the laser lineshape will have a
   finite wavelength (or frquency) spread i.e. thay have a spectral width
   f d ( ) ν ν which means the frquency of the result spectral lines lies
   between ν and ν ν +d .
   Thus we see that a photon of energy hν may not necessarily
   stimulate another photon of energy hν . We then take f d ( ) ν ν as the
   probability that the stimualted photon will have an energy between hν
   and h d ( ) ν ν + . This can be simply seen practically in both emission and
   absorption processes and if, for example, we were to measure the
   transmission (or emission) as a function of frquency for transition
   between the energy states E1
   and E2
   , we would obtain a probabilty
   distribution. The probability distributions usually belong to one of two
   classes :
   (1) A discrete probability distribution.
   (2) A continuous probability distribution.
   It is clear that the distribution of the spectral lineshape of laser light will
   be of the type of continuous probability distribution. It has been
   experimentally found that it just takes one of the two kinds, depending on
   the effects leading to the broadening processes :
   a) The normal distribution.
   ٢٠
   b) The Cauchy distribution.
   A. Normal distribution
   The normal distribution is a continuous probabilty distribution. It is
   considered the most important continous distribution because in
   applications many randam variables are normal randam variables, (that is,
   they have a normal distribution) or they are approximately normal or can
   be transformed into normal randam variables in relatively simple fashion.
   Furthermore, the normal distribution is a useful approximation of more
   complicated distributions, and it also occurs in the proofs of various
   statistical tests.
   The normal distribution, also known in physics studies as
   Gaussian distribution, is defined as the distribution with density (or the
   probabilty density function)
   2
   1 1 ( ) exp , ( 0)
   2 2
   x
   f x µ
   σ
   σ π σ
       −
   = − >          
   (60)
   The above formula represents the standared mathematical relation of the
   normal distribution. In this formula, µ is the mean or expectation of the
   distribution while the parameter σ is its standard deviation. The curve
   f x( ) is symmetric with respect to x = µ because the exponent contains
   2
   ( ) x = µ . Changing µ correponds to moveing the curve to another
   postion (translating it), and for µ = 0 it is symmetric with respect to the
   ordinate (i.e. verticle direction), as shown in Fig.6. The curve in Fig.6 are
   called bell-shape curves. They have peak at x = 0 (or x = µ when
   translated). 2 σ is the variance of distribution, and we see that for small
   2 σ we get a high peak and steep slopes, whereas with incrasing 2 σ the
   curve gets flatter and flatter, the denisty is spread out out more and more,
   in agreement with the fact that the variance measures the spread.
   ٢١
   ٢٢
   ٢٣
   Fig.6. Density of the normal distribution for various values of µ and σ .
   The simplest case of a normal distribution is known as the standard
   normal distribution (or unit distribution). This is a special case when
   µ = 0 and σ =1, and desxribed by this probabilty density function
   2
   1
   ( ) exp
   2 2
   x
   f x
   π
    
   = −   
   (61)
   The variable has a mean of 0 and a variance and standard normal
   deviation of 1. The density ( ) unit f x is has its peak 1/ 2π at x = 0 and
   inflection points at x = +1and x = −1. Although the above density is
   most commonly known as the standard normal distribution, some authors
   have used that term to describe other versions of the normal distribution.
   For example, once defined standard normal as
   1 2
   f x x ( ) exp
   π
   = −    (62)
   which has a variance of 1/ 2 but other authors defined standard normal as
   2
   f x x ( ) exp = −  π   (63)
   ٢٤
   which has a simple functional form and a variance of 2 σ π =1/ (2 ) . In
   general, there are many mathemtical details related to this distribution
   and its properties, but what has been presented here is sufficient to cleaify
   the spectral lineshape of laser light output.
   B. Cauchy distribution
   The Cauchy distribution is a continuous probabilty distribution. It
   is also considered an important continous distribution. In particular, the
   Cauchy distribution is important as an example of a pathological case. It
   is alos a useful approximation of more complicated distributions, and it
   also occurs in the proofs of various statistical tests.
   The Cauchy distribution, also known in physics studies as
   Lorentzian distribution. The general formula for the probability density
   function of the Cauchy distribution is
   2
   0
   1
   ( )
   ( ) 1
   f x
   x x
   πγ
   γ
   =
       −
     +      
   (64)
   ( ) 2 2
   0
   ( )
   ( )
   f x
   x x
   γ
   π γ
   =

• − (65)
  where 0
  x is the location parameter and γ is the scale parameter. Fig.7.
  shows The Cauchy distribution and it is clear that the curve f x( ) is also
  symmetric. The curve in Fig.7 are also called bell-shape curves.
  However, the case where 0
  x = 0 and γ =1 is called the standard
  Cauchy distribution. The equation for the standard Cauchy distribution
  reduces to
  ( ) 2
  1
  ( )
  1
  f x
  π x
  =

(66)
٢٥
٢٦
٢٧
Fig.7. Density of the Cauchy distribution.
Finally we should mentioned to that Cauchy distributions look
similar to a normal distribution. However, they have much heavier tails.
When studying hypothesis tests that assume normality, seeing how the
tests perform on data from a Cauchy distribution is a good indicator of
how sensitive the tests are to heavy-tail departures from normality.
Likewise, it is a good check for robust techniques that are designed to
work well under a wide variety of distributional assumptions. Finally,
The above line profile f x( ) for ; normal and Cauchy distributions are
given in their normalized forms so that
0
f x dx ( ) 1
∞
= ∫
(67)
٢٨
Part 3 :

1. Types of laser spectral lineshape broadenings
   The spectral lineshape broadening is actually due to a number of
   external factors and internal atomic processes. They can be divided into
   two categories; homogenous and in homogenous broadening. In fact, we
   will not expand on that, because their details out of our interest here, but
   it can be mentioned in summary as follow.
   A. Homogenous broadening
   Here all atoms in the medium experience the same perturbations.
   The processes involved may be : (1) collision or (2) electromagnetic or
   (3) just the uncertainty broadening associated with the spontaneous
   lifetime. Practically, the homogenous broadening mechanisms lead to a
   Lorentzian lineshape which may be written in terms of frequency as
   ( )
   2
   2
   0
   ( / 2) ( )
   2
   L
   f
   ν
   ν
   ν
   π ν ν
   ∆
   =
       ∆
     − +      
   (68)
   where ∆ν is the linewidth (full-width half maximum), that is the
   separation between the two points on the (frquency) curve where the
   function falls to half of its peak value which occurs at frquency ν 0
   (the
   central line frequency).
   It is completely identical to the general form of the probability
   density function of the Cauchy distribution in which one was replaced
   γ ν = ∆ / 2 , x =ν and 0 0 x =ν . In order to comply with what is required
   in laser light because the broadening in it occurs to the frequency (or
   wavelength). Putting ν ν= 0

0
2 1 ( ) L
f ν
π ν ν
= ≈
∆ ∆
(69)
٢٩
B. Inhomogeneous broadening
Here each atom in the medium experiences a different line profile.
The addition of all the line profiles from all the atoms give the total line
profile. An example is Doppler broadening due to the thermal motion of
emitting atomic in a gas with Doppler broadening. Practically, the
inhomogeneous broadening mechanisms lead to a Gaussian lineshape
which may be written in terms of frequency as
2
0
1 1 ( ) exp
2 2
G
f
ν ν
ν
ν π ν
    −
= −    ∆     ∆  
(70)
where ∆ν and ν 0
are as defined above. In some lasers books, this
lineshape is called Doppler frequency distribution because it is source of
inhomogeneous broadening.
It is completely identical to the general form of the probability
density function of the normal distribution in which one was replaced
σ ν = ∆ , µ ν= 0
and x =ν . In order to comply with what is required in
laser light because the broadening in it occurs to the frequency (or
wavelength). and Putting ν ν= 0
, gives
1 1 ( )
2
G
f ν
ν π ν
= ≈
∆ ∆
(71)
As we mentioned before, the above line profile L
f and G
f are given in
their normalized forms so that
0
f d ( ) ν ν 1
∞
= ∫
(72)
2. Laser modes
Examination of the laser output with a spectrometer of very high
resolving power, such as the scanning Fabry-perot interferometer, reveals
that it consists of a number of discrete frequency components (or very
narrow spectral lines). To appreciate how these discrete lines arise and
٣٠
how they are related to the laser transition lineshape we need to examine
the effects of the mirror on the light within the laser cavity. In fact, we
have two types of laser modes; longitudinal (axial) and transverse modes.
The probability distribution that were previously put forward are not
related to both. In next section, we will outline these two class of modes
in order to determine to which of them are related to those probability
distribution.
A. Longitudinal modes
The two mirror of the laser form a resonant cavity and standing
wave patterns are set up between the mirror in exactly the same way that
standing waves develop on the string. The standing waves satisfy the
condition
&
2 2
nc L n
L
λ
= = ν (73)
where L is the optical path length between the mirror, in which case λ
would be the vacuum wavelength and n is an integer number of halfwavelength able to ‘fit’ between two mirrors. It is clear the frequency
separation δν between adjacent modes is given by
1
( )
2
n n
c
L
δν ν ν = − = +
(74)
As equation (74) is independent of n , the frequency separation of
adjacent modes is the same irrespective of their actual frequencies. The
modes of oscillation of the laser cavity will consist, therefore of a large
number of frequencies, each given by equation (73) and separated by
c L / 2 , as shown in Fig.8. It should be appreiated, however, that while all
the integers n give possible axial cavity modes only those which lie
within the gain curve or laser transition line will actually oscillate.
This is exactly the inverse of the time for a photon to do a round
trip within the cavity. Although the longitudinal modes show the state of
٣١
discrete frequency components, they are not concerned with laser
propagation where the longitudinal modes all contribute to a single 'spot'
of light in the laser output, whereas the transverse modes discussed below
may give rise to a pattern of spot in the output.
Fig.8. Broadened laser transition line ( or irradiance against frequency) (a) and (b) cavity
modes (c) axial modes in the laser output.
B. Transverse modes
Longitudinal modes are formed by the plane waves traveiling
axaially along the laser cavity on a line joining the centers of the mirror.
For any real laser cavity there will probably be waves traveiling just offaxis that are able to replicate themseelve after covering a colsed path such
as Fig.9. These will also give rise to resonant modes, but because they
have components of their electromagnetic fields which are transverse to
the direction of propagation they are termed transverse electromagnetic
(or TEM) modes.
In fact, on this class of laser output, the above probability
distribution apply. Experimentally found that it just takes one of the two
٣٢
kinds; the Lorentzian or Gaussian distributions. This depends on the type
of effects that lead to the broadening processes of laser lineshape. The
TEM modes are characterized by two integer p and ℓ so that we have
TEM
pℓ
(such as TEM 00 ,TEM 01 ,TEM 11 ,etc) . Here p gives the number
of minimum as the beam is scanned horizontally and ℓ the number of
minimum as it scanned vertically.
Fig.9. Example of a nonaxail self-replicating ray that gives rise to transverse modes.
3. The fundamental transverse electromagnetic mode (TEM 00 )
In the case of a fundamental transverse electromagnetic TEM 00
mode, the irradiance distribution across the beam is Gaussian, and so may
write the electric field variaition as
2 2
0 2
( , ) exp( ) x y E x y
w
ζ
−
= − (75)
where x and y are measured in directions perpendicular to the laser axis
which is taken to be a long z direction. The sideways spread of the beam
is determined by the value of the parameter w , which is a function of
the distance z . When 2 2 2
x y w − > , the field falls off rapidly with
distance away from the laser axis. The value of w is determined by the
٣٣
locus of point where the field amplitude has fallen to 1/ e of its
maximum value (that is, where 2 2 2
x y w − = ) .
Fig.10 shows the typical variation of w , with position, within a
cavity formed by two concave mirrors of radius of curvature 1
r and 2
r
separated by L . Such a cavity can be shown to be stable when L r r ≤ +1 2 .
The surfaces of constant phase are not in general plane, but are
perpendicular to the contour of constant field strength. It can be seen that
the mirrors themselves are surfaces of constant phase. This is not accident
but merely a direct consequence of the requirement that the mode be selfreplication as the light energy flows backwards and forwards between
mirrors . At one position within the cavity, the wavefront become plane
and, in fact a plane mirror replace at this point would give rise to a
hemispherical cavity. At this point also w has its smallest value, w 0
. The
variation of w with z is given by
0 2
0
( ) 1 z
w z w
w
λ
π
   
= +        
(76)
where z is measured from the position of minimum beam diameter. The
precise value of w 0
depend on the type of cavity.
Fig.10. TEM 00 mode within the cavity. The mode adjusts itself so that the mirror surface
are surfaces of constant phase. The value of w is determined by the locus of points where the
field amplitude has fallen to 1/e of its maximum value..
٣٤
4. High order TEM modes or Hermite–Gaussian modes :
We know the usual starting point for the derivation of laser beam
propagation modes is solving the scalar Helmholtz equation (wave
equation) within the paraxial approximation. Under some conditions (not
shown here) it gives what is known as Hermite–Gaussian modes. In this
case electric field distributions are essentially given by the product of a
Gaussian function and a Hermite polynomial, apart from the phase term
as follows :
( )
2
0
0 2
2
2
2 2
( , , ) . 2 exp
( ) ( ) ( )
. 2 exp .exp [ (1 ) .
( ) ( )
( )
arctan
2 ( )
nm n
m
R
w x x E x y z E H
w z w z w z
y y H i kz n m
w z w z
z x y k
z R z
   
= −        
   
    − − − + +
   
+
+
(1)
where H n (.....) is the Hermite polynomial with the non-negative integer
index n . The table shows the first six cases of Hermite polynomials.
( ) n H n ϒ
0
0 H ( ) ϒ =1
1
1 H ( ) ϒ = ϒ2
2
2
2 H ( ) ϒ = ϒ − 4 1
3
3
3 H ( ) ϒ = ϒ − ϒ 8 12
4 2
4
4 H ( ) ϒ = ϒ − ϒ + 16 48 12
4 2
4
5 H ( ) ϒ = ϒ − ϒ + 16 48 12
The indices n and m determine the shape of the profile in the x
and y direction, respectively. The quantities w and R evolve in the z
direction. The intensity distribution of such a mode has n nodes in the
horizontal direction and m nodes in the vertical direction. For n m= = 0,
٣٥
a Gaussian beam is obtained. This mode is called the fundamental mode
or axial mode, and it has the highest beam quality. Fig.11 shows the
square of real part of the amplitudes within the plane z = 0 for different
values of n m, (with arbitrary parameters).
TEM 00
TEM 01
TEM 10
٣٦
TEM 11
TEM 02
TEM 02
Fig.11. Square real part of the amplitude of Hermite-Gauss modes, within the plane z = 0 ,
for laser beams with beam waist radius 0 w = 4λ .
٣٧
References
Electromagnetics theory
[1] Cheng D. ' Fundamental Engineering Electromagnetics' 1st edn.
(Addison-Wesley Publushing Company 1993).
[2] Paul C and Naser S. ' Introduction to Electromagnetic Field' 2nd edn.
(McGRAW-Hill International edition 1987).
Mathematics
[4] Erwin K. ' Advaced Eneginering Mathematics' 7TH edn. (John Wiley
& Sons ).
Lasers light
[5] Besssely M. J. 'Lasers and their applications' Taylor-Francis , London
[6] Oshea D. C. et al 'Introduction to lasers and their
application'(Addison-Wesley Reading , Mass. 1977
[7] Gower J., ‘Optical Communications Systems’ 2nd edn. (Prentice Hall
International, Hemel Hempstead 1995).
[8] Senior J. M., ‘Optical Fiber Communications Principles and Practices’
2
nd edn. (Prentice Hall International, Hemel Hempstead 1992).
[9] Wilson J and Hawkes J. ‘Optelectronics An interoduction’ 2nd edn.
(Prentice Hall International, Hemel Hempstead 1989).