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