A thesis submitted to the Department of Physics in partial fulfillment of the requirements for the degree Of MASTER OF SCIENCE In Physics By SAPNA

A thesis submitted to the Department of Physics in partial fulfillment of the requirements
for the degree
Of

MASTER OF SCIENCE

In
Physics
By

SAPNA
(2015IMSBPH016)

Under the Guidance
Of

Dr. B.K.Singh
DEPARTMENT

DEPARTMENT OF PHYSICS
CENTRAL UNIVERSITY OF RAJASTHAN
NH-8, BANDARSINDRI
AJMER-305817

MAY 2018
Certificate
This is to certify that the thesis entitled ‘Targeting steady state in
discrete system with coexisting attractors’ is submitted by
‘SAPNA’, (ID- 2015IMSBPH016) to this University in partial
fulfillment of the requirement for the award of the degree of
Master of Science in Department of Physics. The work
incorporated in the thesis has not been, to the best of our
knowledge, submitted to any other University or Institute for the
award of any degree/diploma.

Dr. Manish Dev Shrimali
Head of Department
Department of physics
Central University of Rajasthan

Place : Curaj
Date :

1
Declaration
This is to certify that the thesis entitled “Targeting steady state
in discrete systemwith coexisting attractors” submitted by me
to the Department of Physics, Central University of Rajasthan,
Bandarsindri, Ajmer for the award of the degree of Master of
Science is a bona-fide record of research work. The contents of
thesis have been not been copied from anywhere. Further, the
thesis partially or completely will not be submitted to any other
Institute or University for the award of any other degree or
diploma.

Signature
SAPNA
2015IMSBPH016
Date:

2
Acknowledgment
I would like to express my special appreciation and thanks to my
supervisor Dr. B.K.SINGH, you have been a tremendous mentor
for me. I would like to thank you for encouraging my research.

Your advice on both research as well as on my career have been
priceless. I would also like to thank my teacher for their
continuous support and encouragement throughout this project.

This whole work is jointly done with my elder brother Pravesh
kumar and my sister Renu kumari who is a source of
unconditional love and affection to me. His continuous support
and guidance to me at each stage of life is priceless.

A special thanks to my family. Words cannot express how grateful
I am to my brother, father, mother and grandmother for all of the
sacrifices that you’ve made on my behalf. Your prayer for me was
what sustained me thus far.

I would also like to thank all of my classmates who supported me
in writing, and incented me to strive towards my goal. Their timely
help and friendship shall always be remembered.

My special regards to my teachers because of whose teaching at
different stages of education has made it possible for me to see
this day. Because of their kindness I feel, was able to reach a
stage where I could write this thesis. I must thank the office staff
Pushpendra sharma ,Rahul Sharma and khemaram for their kind
support Throughout my tenure at CURAJ.

Abstract
List of Figures
List of Figures
Contents
Chapter 1

Introduction
“Imagination is more important than knowledge”
Albert Einstein

Light can be described as an electromagnetic wave which is
obtained by the same theoretical principles of electromagnetic
radiation, such as X-Rays and Radio Waves. This phenomenon of
light is called electromagnetic wave optics. Electromagnetic
radiation propagates in the form of two mutually coupled wave
vector as an electric-field wave and a magnetic-field wave. When
light waves propagate around an through objects whose
dimensions are much greater than the wavelength of the light, the
wave nature of light is not readily viewed.

Light is referred by a scalar function, which is called the wave
function, that known as the wave equation which obeys a second-
order differential equation. We know that the wave function
represents any of the components of the magnetic or electric
fields.

The Wave Equation
Light travels in the form of waves. Light In free space, travel with
speed C0. A homogeneous transparent medium such as glass is
characterized by a constant, for which refractive index is n (; 1).

In a medium of refractive index n, light waves propagates with a
reduced speed
?0
C= ? –

an optical wave is given by a function of position r =(x, y, z) and
time t, denoted u =(r, t) and known as the wave function. This
satisfies a partial differential equation called the wave equation,

2 1 ?2 ? (1.1)
? u- ? 2 ?? 2
=0

Where ?2 is the Laplacian operator.

Intensity, Power, and Energy
Intensity
The optical intensity I = (r, t), defined as the optical power per unit
area (units of watts / cm2 ), The optical intensity is proportional to
the average of the squared wave function:

I(r ,t)=2 Optical Intensity

Power
The optical power P (t) ,(unit of power is watt) is flowing into an
area A, it is normal to the direction of propagation of light is the
integrated intensity

P(t) = ?A I(r ,t) ?A

Energy
The optical energy (units of energy -joules) collected in a given
time interval is the integral of the optical power over the time
interval.

MONOCHROMATIC WAVES
Monochromatic waves are radiation of a single wavelength or of a
very small range of wavelengths. A monochromatic wave is
described by a wave function with harmonic time dependence,
this wave function is-

U (r, t) = a(r). cos 2?vt+?(r) (1.2)

a(r) =amplitude

?(r) = phase

v =frequency (cycles/s or Hz)

w = 2?v =angular frequency (radians / sec )

T =l/v = 2? / W =period (s).

Complex wave function:
It is simple to represent the real wave function u(r, t) in terms of a
complex function

U (r, t) ==a(r). exp j ?(r).exp (j 2?v t) (1.3)

u (r, t) =Re{U(r, t)}

The function U(r, t) is the complex wave function, U(r, t) must also
satisfy the wave equation.

The Helmholtz Equation
Equation (1.3) may be written in the form of complex amplitude as

U (r,t)= U( r ).exp( j2?v t) (1.4)
Substituting U (r,t)= U( r ).exp( j2?v t) from (1.4) into the wave
equation (1.1). It leads to a differential equation for the complex
amplitude U (r) :

?2 U+k2U=0 (1.5)

This equation known as the Helmholtz equation
2?v ?
K= =
? ?

Where k is the wave number.

Paraxial Waves:
A paraxial wave is a wave which makes a small angle (?) to the optical
axis and lies near to the axis throughout the system. A wave is called to
be paraxial if the wave front normal of this wave are paraxial rays. The
way of constructing a paraxial wave is to begin with a plane wave
A.exp (-jkz), like a “carrier” wave, and modulate its complex envelope
A, and making it a slowly varying function of position, A(r), so the
complex amplitude of the modulated wave becomes

U (r)= A( r ).exp(-jkz) (1.6)

The variation of the envelope A(r) with position and its derivative with
position z must be slow within the distance of a wavelength ? =
2 ?/? so that the wave approximately maintains its plane-wave
nature.

Figure1.1: The wave front normals and wave fronts of a paraxial
wave in the x-z plane.

The Paraxial Helmholtz Equation:
The paraxial wave (1.6) to satisfy the Helmholtz equation (1.5), the
complex envelope A(r) must satisfy another partial differential equation
that is obtained by substituting (1.6) into Helmholtz equation. It is
assume that A (r) varies slowly with respect to z. Substituting (1.6) into
?2 ? ??
(1.5), and neglecting in comparison with k or k2A, leads to a
?? 2 ??
partial differential equation for the complex envelope A (r ):

?? (1.7)
?2? ? – j2k
??

This is known as the Paraxial Helmholtz equation.

?2 ?2
Where ?2? =
?? 2
+
?? 2
is the transverse Laplacian operator. Equation
(1.7) is the slowly varying complex envelope approximation of the
Helmholtz equation. We simply call it the paraxial Helmholtz equation.

The solution of the paraxial Helmholtz equation is the paraboloidal
wave which is the paraxial approximation of a spherical wave. One of
the simplest interesting and useful solutions, however, is the Gaussian
beam .Paraxial Waves are the waves whose wave front normals make
small angles with the z axis. Paraxial waves must satisfy the paraxial
Helmholtz equation. The Gaussian beam is an important solution of this
paraxial Helmholtz equation that shows the characteristics of an optical
beam.

Gaussian beam
In optics, a Gaussian beam is a monochromatic electromagnetic
radiation whose transverse electric and magnetic field amplitude
profiles are given by the Gaussian function.

The Gaussian beam is an important solution of the paraxial wave
equation that exhibits the properties of an optical beam. The beam
power of gaussian beam is principally concentrated within a small
cylinder that surrounds the beam axis. The intensity distribution in any
transverse plane of gaussian beam is a circularly symmetric Gaussian
function which is centered about the beam axis. The width of this
gaussian function is minimum at the beam waist and gradually becomes
larger as the distance from the waist increases in both directions. The
wave fronts are approximately plane wave near the beam waist,
gradually become curve as the distance from the waist increases and
become approximately spherical far from the waist of beam.

Complex Amplitude
The concept of paraxial waves was introduced in equation 1.6. A
paraxial wave is a plane wave that traveling along the z direction exp-(
jkz) (with wave number k = 2 ?/? and wave- length ?). This plane
wave is modulated by a complex envelope A(r) that is a slowly varying
function of position so that its complex amplitude is

U (r)= A( r ).exp(-jkz)

So the complex amplitude U(r) satisfied the Helmholtz equation, so the
complex envelope A(r) must also satisfy the paraxial Helmholtz
equation. So a simple solution of the paraxial Helmholtz equation leads
to the Gaussian beam. So the simple expression for the complex
amplitude U(r) of gaussian beam is :
?0 ?2 ?2
U(r) = ?0 exp (? ) exp (-jkz- jk + ??(?))
?(?) ? 2 (?) 2?(?)
Complex Amplitude (1.8)
Beam width W (z)-

At any position of z along the beam, the radius of the beam w (z), is
related to the full width at half maximum (FWHM, where the amplitude
become half of its maximum value). At beam width the beam intensity
assumes its peak value on the beam axis, and decreases by the factor 1/
e2 = 0.135 of the initial value at the radial distance ?= W (z).

?2
W (z) =W0?1 +
?? 2

Beam waist W0-

This is the measurement of the shape of a Gaussian beam for a given
wavelength ? is described by one parameter; this is called the beam
waist w0. This is a measure of the beam size of gaussian beam at the
point of its focus (z=0 in the above equations) where the beam width
w(z) (as defined above) is the smallest and where the intensity on-axis
(r=0) is the largest.

Rayleigh range?? :

Rayleigh range is a distance from the waist equal to the Rayleigh range
zR, the beam width W is larger than beam waist where it is at the focus
where w = w0, The Rayleigh range or range zR is determined:
??? ?
?? =
?

Where
R(z) is the Radius of the curvature where

?? ?
R (z) = z 1+
??

? (Z) is the Gouy phase at z, an extra phase term beyond to the phase velocity of light.

?
? (Z) =tan-1( )
??

Figure 1.2: Gaussian beam width w(z) as a function of the distance z
along the beam.

Beam Divergence-

The angular divergence of the beam-
4 ?
?=
? 2?0

The divergence angle is directly proportional to the wavelength ? and
inversely proportional to the spot size 2W0 .Squeezing the spot size 2W0
(beam-waist diameter) so that beam divergence increased. So it is clear
that a highly directional beam is made by making use of a small
wavelength and a thick beam waist diameter.

Intensity:

The optical intensity I( r)= |U(r)| 2 is a function of the radial and axial
positions , ?=?(? 2 + ? 2 ) an Z respectively .

?(?) 2 2?2
I (?, Z) = ?0 exp? (1.10)
?0 ? 2 (?)

Where ?0 =|A0 | 2

On the z axis The Gaussian function has its peak, at ?= 0, and when ?
increase the gaussian function decreases monotonically. The beam
width W(z) of the Gaussian beam increases with the axial distance z.

?
Figure 1.3: The normalized beam intensity as a function of the radial
?0
distance ? at Z=0 in 2 Dimensional .

?
Figure 1.4: The normalized beam intensity as a function of the radial
?0
distance ? at Z=0 in 1 Dimensional.

Properties of the Gaussian Beam at some Special Locations:

-The intensity of the gaussian beam on the beam axis is 1/2 the peak
intensity at the location Z=?0

-At location Z=?0 the beam width is ?2 greater than the width at the
beam waist.

-The phase on the beam axis at Z=?0 is reduced by an angle ? / 4
relative to the phase of a plane wave.

-The radius of curvature of the wave front gets its minimum value, R
=2?0 , so the wave front has the greatest curvature.

HERMITE-GAUSSIAN BEAMS:
The Gaussian beam is not the only the solution of the paraxial
Helmholtz equation. There are other solutions that exhibit non-
Gaussian intensity distributions but share the paraboloidal wave fronts
of the Gaussian beam. It is possible to break a coherent paraxial beam
using the orthogonal set of so-called Hermite-Gaussian modes, which
are given by the product of x and y factor. Such solutions are possible to
separate in x and y in the paraxial wave equations written in Cartesian
coordinates. This mode are given in order ( l , m) referring to the x and
y directions.

1. The phase of hermite gaussian beam is the same as that of the
Gaussian beam, except for an excess phase Z (z) that is not dependent
of x and y. If Z(z) is a slowly varying function of z, both hermite and
gaussian beam have paraboloidal wave fronts with the same radius of
curvature R( z).

2. This hermite distribution is as Gaussian function which is modulated
in the x and y directions by the functions ? 2 (.) and ? 2 (.),respectively.

The wave which is modulated in x, y direction, therefore represents a
beam of non-Gaussian intensity distribution.

Complex Amplitude:
?0 ?2 ?2
??,? (?, ?, ?)=??,? ?? ? ?? ?
?(?) ?(?) ?(?)

? 2 +? 2
?exp -jkz-jk + ?(? + ? + 1)?(?)
2?(?)

Hermite Gaussian Beam (1.11)
?2
Where ?? (u) = ?? (u) exp(- ) l=0,1,2,3……
2
?? (u) Is known as the Hermite-Gaussian function of order l, and ??,? is
a constant.

If ?0 (u) = 1, then the Hermite-Gaussian function of order 0 is simply
known as the Gaussian function. Continue next higher order,
?2
?1 (u) =2u.exp (- ) is an odd function.

2

?2
?2 (u) =(4?2 ? 2).exp (- ) is an even function.

2

3 ?2
?3 (u) =(8? ? 12?).exp (- ) is an odd function
2

These all functions are displayed schematically:

(a) ?0 (u) (b) ?1 (u)

Figure1.5: Low-order Hermite-Gaussian functions: (a) ?0 (u) (b) ?1 (u) (c)
?2 (u) (d) ?3 (u).

?2 (u) ?3 (u)

Intensity Distribution:
The optical intensity of the Hermite-Gaussian beam ??,? = |Ul,m | 2 is
given by-
?0 2 ?2 ?2
??,? (?, ?, ?)=??,? 2 ?? 2 ? ?? 2 ?
?(?) ?(?) ?(?)

(a) (0,0) Mode (b) (1,0) Mode
(c) (0,2) Mode (d) (2,2) Mode

Figure1.6: Intensity distributions of low-order Hermite-Gaussian beams,
The order (l,m) is indicated in each case.

LAGUERRE-GAUSSIAN BEAMS-
The Hermite-Gaussian beams are a complete set of solutions to the
paraxial Helmholtz equation. Any other solution of beams can be
written as a superposition of these beams. This complete set of
solutions known as Laguerre -Gaussian beams. Laguerre gaussian
beams are described by writing the paraxial Helmholtz equation in
cylindrical coordinate (?, ?, z). Then using the separation-of-variables in
? and ?, rather than in x and y. The complex amplitude of the Laguerre-
Gaussian beam is
?0 ? ? ?2 ? 2 ??2
??,? (?, ?, z)=??,? ??? exp
?(?) ?(?) ? 2 (?) ? 2 (?)

?2
?exp -jkz – jk2?(?) ? j?? + ?(? + 2? + 1)?(?)

Laguerre gaussian beam (1.12)
In equation of LG beam the phase has the same dependence, as the
Gaussian beam on ? and z, but the phase has an extra term which is
proportional to the azimuthal angle ?, and on a Gouy phase that is
greater by the factor (l + 2m + 1). Because of this linear dependence of
the phase on ? (for l ? 0) when the wave travels in the z direction, then
wave front tilts helically as show in Fig.

Figure1.7: Wave front

Figure1.8: Intensity distribution