Part 1 : 
1.References 
Electromagnetics theory 
[1] Cheng D. ' Fundamental Engineering Electromagnetics' 1st edn.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).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.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.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.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.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 = -?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.= = (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.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.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 ?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- .Let us now examine the frist phasor term on the right 
side of equation (42) and write 
0
( ) ( ) exp( ) ^ ^
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( ) ^ ^
x x x x
z t a E z t a E z i t ?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.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.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
?Square real part of the amplitude of Hermite-Gauss modes, within the plane z = 0 , 
for laser beams with beam waist radius 0 w = 4?=10.0m , by 
discharging an induction coil across a spark gap thereby setting up 
oscillating electric and magnetic fields.Fig.3 Electromagnetic wave : the electric vector and magnetic vector vibrate in orthogonal 
planes and perpendicular to the direction of propagation.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 .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
+
?Thus, the laser lineshape will have a 
finite wavelength (or frquency) spread i.e. thay have a spectral width 
f d ( ) ?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.Types of laser spectral lineshape broadenings 
 The spectral lineshape broadening is actually due to a number of 
external factors and internal atomic processes.Broadened laser transition line ( or irradiance against frequency) (a) and (b) cavity 
modes (c) axial modes in the laser output.For any real laser cavity there will probably be waves traveiling just off axis 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.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 ?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.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.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.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.(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 time dependance.(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.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.= = (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 ?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.Changing u correponds to moveing the curve to another 
postion (translating it), and for u = 0 it is symmetric with respect to the 
ordinate (i.e. verticle direction), as shown in Fig.6.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 
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.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 .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.Then in 
1864 Maxwell combined the equation of electromagnetism in a general 
form and showed that they suggest the existance of trensverse 
electromagnetic wave. 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 ?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.D E H J (11)
B. Waves equations : 
First we derive the wave equation that governs the propagation of all 
electromagnetic wave.Each equation is composed of three scalar differential 
equations in term of the components of the vectors.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.We may arbitrarity assume the direction of E to be in the postive 
x direction; that is 
( ) ^ 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.( ) ( ) ( ) 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.Thus if we fix our attantion on a 
particular point (a point of particular phase) on the wave, we set 
cos( ) tan ?t kz a cons t - = or ?t kz A cons t phase - = tan , from which 
we obtain 
0 0
1
c
k
?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 
^ ^
y y z z k = + k a k a (57)
^ ^
y z r = + ya za (58)
.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 ( ) ?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.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 u
?It 
is alos a useful approximation of more complicated distributions, and it 
also occurs in the proofs of various statistical tests.The processes involved may be : (1) collision or (2) electromagnetic or 
(3) just the uncertainty broadening associated with the spontaneous 
lifetime.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 ?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
?= - = +
(74)
As equation (74) is independent of n , the frequency separation of 
adjacent modes is the same irrespective of their actual frequencies.This is not accident 
but merely a direct consequence of the requirement that the mode be self replication as the light energy flows backwards and forwards between 
mirrors .Subsequent observations by 
Young, Malus, Euler and ithers lent support to the wave theory.On the other hand, for experiments invloving interfererence and 
diffraction, where light interacts with light, the wave nature dominates.Let us consider the electron transitions which may occur 
between the two energy level s of the hypothetical atomic system shown 
in Fig 1.Under normal circustances we do not observe the stimulated 
emission process because the probabilty of the the spontaneous emission 
?Because spontaneous radiation from any atom is 
emitted at random, the radiation emitted by a large number of atoms will 
clearly be incoherent.Energy level diagram illustrating (a) absorption, (b) spontaneous emission and (c) 
stimulated emission. Associated with Maxwell's equations, we have equation of continity (or 
conservation of charge) 
t
??x ?x = - E
E (16)
We similarly obtain, by taking the curl of equation (13) and substituting 
equation (12), 
2
0 0 2
( )
dt
u ?In describing optical 
phenomena we often omit the magnetic field vector.(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, ?(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
?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.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.u
= = = (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.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.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 = - - ?) ensity energy time area = is proportional to the square of the 
amplitude, there is an inverse-square-law decrease in irradiance.Likewise, it is a good check for robust techniques that are designed to 
work well under a wide variety of distributional assumptions.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
?In order to comply with what is required 
in laser light because the broadening in it occurs to the frequency (or 
wavelength).In some lasers books, this 
lineshape is called Doppler frequency distribution because it is source of
inhomogeneous broadening.In order to comply with what is required in 
laser light because the broadening in it occurs to the frequency (or 
wavelength).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
( , , ) .and u0
 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 = x3 10 / , a frequency of some 14 f Hz = x5 10 and a wavelength of 
about ?Planck found that the observation 
indicated that light energy is emitted in muliplies of certain minimum 
enregy unit.When 
Einstein showed, however, that it seems necessary asssume the 
concentration of energy traveling through space as particles, a wave 
solution was excluded.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.Two energy level system 
 The absorption and emission processes are illustrated in Fig 2.(a), 
(b) and (c).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.Propagation of light 
 Propagation of light refers to the manner which is an 
electromagntic wave trsnsfer it's energy from one point ot another.This maen that, we will 
derive the main equation described the propagation of light in empty 
space.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.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.= .H 0 (15)
Taking the curl of equation (12) and substituting equation (13), we obtain 
2
0 0 2
( )
dt
u ?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
u ?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.At successive time the curve effectively travels in the 
postive z direction.= 500nm .=100.0km .??2.3.?(1)
?????= .( ) ????????????????- ?( ) ( .???= + +
?????????? ?/ ).?(25)
2
0 0 ??????????????????(28)
???????????= =
????????????????????????????????????????????????=
?+ +
= = ????????????.????.+ + +
+
??????????????+
+ ?=
??????= = = = = ?+ +
= = ???????=
?(55)
????.??.??+ .??????????????????2 ???????.????????????(63)
???=1/ (2 ) .?=
????-
??+ ????????????= ?(67)
????????=
???????- + ???????= ?.???= ???(69)
???????????????????????= ?.???= ????1
?= ?(72)
2.= = ???3.?????= + ??????????4.?????????????????????= ?= ?- ?= ?- ?= ?- ?.??