A the Department of Physics, Central University

A thesis submitted to the Department of Physics in partial fulfillment of the requirementsfor 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 CertificateThis is to certify that the thesis entitled ‘Targeting steady state indiscrete system with coexisting attractors’ is submitted by’SAPNA’, (ID- 2015IMSBPH016) to this University in partialfulfillment of the requirement for the award of the degree ofMaster of Science in Department of Physics. The workincorporated in the thesis has not been, to the best of ourknowledge, submitted to any other University or Institute for theaward of any degree/diploma.Dr. Manish Dev ShrimaliHead of DepartmentDepartment of physicsCentral University of RajasthanPlace : CurajDate : 1 DeclarationThis is to certify that the thesis entitled “Targeting steady statein discrete systemwith coexisting attractors” submitted by meto the Department of Physics, Central University of Rajasthan,Bandarsindri, Ajmer for the award of the degree of Master ofScience is a bona-fide record of research work.

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The contents ofthesis have been not been copied from anywhere. Further, thethesis partially or completely will not be submitted to any otherInstitute or University for the award of any other degree ordiploma.SignatureSAPNA2015IMSBPH016Date: 2AcknowledgmentI would like to express my special appreciation and thanks to mysupervisor Dr. B.K.SINGH, you have been a tremendous mentorfor me.

I would like to thank you for encouraging my research.Your advice on both research as well as on my career have beenpriceless. I would also like to thank my teacher for theircontinuous support and encouragement throughout this project.

This whole work is jointly done with my elder brother Praveshkumar and my sister Renu kumari who is a source ofunconditional love and affection to me. His continuous supportand guidance to me at each stage of life is priceless.A special thanks to my family. Words cannot express how gratefulI am to my brother, father, mother and grandmother for all of thesacrifices that you’ve made on my behalf.

Your prayer for me waswhat sustained me thus far.I would also like to thank all of my classmates who supported mein writing, and incented me to strive towards my goal. Their timelyhelp and friendship shall always be remembered.

My special regards to my teachers because of whose teaching atdifferent stages of education has made it possible for me to seethis day. Because of their kindness I feel, was able to reach astage where I could write this thesis. I must thank the office staffPushpendra sharma ,Rahul Sharma and khemaram for their kindsupport Throughout my tenure at CURAJ.AbstractList of FiguresList of FiguresContentsChapter 1Introduction “Imagination is more important than knowledge” Albert EinsteinLight can be described as an electromagnetic wave which isobtained by the same theoretical principles of electromagneticradiation, such as X-Rays and Radio Waves. This phenomenon oflight is called electromagnetic wave optics.

Electromagneticradiation propagates in the form of two mutually coupled wavevector as an electric-field wave and a magnetic-field wave. Whenlight waves propagate around an through objects whosedimensions are much greater than the wavelength of the light, thewave nature of light is not readily viewed.Light is referred by a scalar function, which is called the wavefunction, that known as the wave equation which obeys a second-order differential equation.

We know that the wave functionrepresents any of the components of the magnetic or electricfields.The Wave EquationLight travels in the form of waves. Light In free space, travel withspeed C0.

A homogeneous transparent medium such as glass ischaracterized by a constant, for which refractive index is n (; 1).In a medium of refractive index n, light waves propagates with areduced speed ?0 C= ? -an optical wave is given by a function of position r =(x, y, z) andtime t, denoted u =(r, t) and known as the wave function. Thissatisfies a partial differential equation called the wave equation, 2 1 ?2 ? (1.

1) ? u- ? 2 ?? 2 =0Where ?2 is the Laplacian operator.Intensity, Power, and EnergyIntensityThe optical intensity I = (r, t), defined as the optical power per unitarea (units of watts / cm2 ), The optical intensity is proportional tothe average of the squared wave function: I(r ,t)=2 Optical IntensityPowerThe optical power P (t) ,(unit of power is watt) is flowing into anarea A, it is normal to the direction of propagation of light is theintegrated intensity P(t) = ?A I(r ,t) ?AEnergyThe optical energy (units of energy -joules) collected in a giventime interval is the integral of the optical power over the timeinterval.MONOCHROMATIC WAVESMonochromatic waves are radiation of a single wavelength or of avery small range of wavelengths. A monochromatic wave isdescribed 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) = phasev =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 acomplex 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 alsosatisfy the wave equation.The Helmholtz EquationEquation (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 waveequation (1.1). It leads to a differential equation for the complexamplitude 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 opticalaxis and lies near to the axis throughout the system. A wave is called tobe paraxial if the wave front normal of this wave are paraxial rays.

Theway of constructing a paraxial wave is to begin with a plane waveA.exp (-jkz), like a “carrier” wave, and modulate its complex envelopeA, and making it a slowly varying function of position, A(r), so thecomplex 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 withposition z must be slow within the distance of a wavelength ? =2 ?/? so that the wave approximately maintains its plane-wavenature.

Figure1.1: The wave front normals and wave fronts of a paraxialwave in the x-z plane.The Paraxial Helmholtz Equation:The paraxial wave (1.6) to satisfy the Helmholtz equation (1.5), thecomplex envelope A(r) must satisfy another partial differential equationthat is obtained by substituting (1.

6) into Helmholtz equation. It isassume 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 ?2Where ?2? = ?? 2 + ?? 2 is the transverse Laplacian operator. Equation(1.7) is the slowly varying complex envelope approximation of theHelmholtz equation. We simply call it the paraxial Helmholtz equation.The solution of the paraxial Helmholtz equation is the paraboloidalwave which is the paraxial approximation of a spherical wave.

One ofthe simplest interesting and useful solutions, however, is the Gaussianbeam .Paraxial Waves are the waves whose wave front normals makesmall angles with the z axis. Paraxial waves must satisfy the paraxialHelmholtz equation. The Gaussian beam is an important solution of thisparaxial Helmholtz equation that shows the characteristics of an opticalbeam.Gaussian beamIn optics, a Gaussian beam is a monochromatic electromagneticradiation whose transverse electric and magnetic field amplitudeprofiles are given by the Gaussian function.The Gaussian beam is an important solution of the paraxial waveequation that exhibits the properties of an optical beam.

The beampower of gaussian beam is principally concentrated within a smallcylinder that surrounds the beam axis. The intensity distribution in anytransverse plane of gaussian beam is a circularly symmetric Gaussianfunction which is centered about the beam axis. The width of thisgaussian function is minimum at the beam waist and gradually becomeslarger as the distance from the waist increases in both directions. Thewave fronts are approximately plane wave near the beam waist,gradually become curve as the distance from the waist increases andbecome approximately spherical far from the waist of beam.Complex AmplitudeThe concept of paraxial waves was introduced in equation 1.6.

Aparaxial wave is a plane wave that traveling along the z direction exp-(jkz) (with wave number k = 2 ?/? and wave- length ?). This planewave is modulated by a complex envelope A(r) that is a slowly varyingfunction 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 thecomplex envelope A(r) must also satisfy the paraxial Helmholtzequation. So a simple solution of the paraxial Helmholtz equation leadsto the Gaussian beam.

So the simple expression for the complexamplitude 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), isrelated to the full width at half maximum (FWHM, where the amplitudebecome half of its maximum value). At beam width the beam intensityassumes 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 + ?? 2Beam waist W0-This is the measurement of the shape of a Gaussian beam for a givenwavelength ? is described by one parameter; this is called the beamwaist w0. This is a measure of the beam size of gaussian beam at thepoint of its focus (z=0 in the above equations) where the beam widthw(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 rangezR, the beam width W is larger than beam waist where it is at the focuswhere w = w0, The Rayleigh range or range zR is determined: ??? ? ?? = ?WhereR(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 zalong the beam.Beam Divergence-The angular divergence of the beam- 4 ? ?= ? 2?0The divergence angle is directly proportional to the wavelength ? andinversely proportional to the spot size 2W0 .Squeezing the spot size 2W0(beam-waist diameter) so that beam divergence increased. So it is clearthat a highly directional beam is made by making use of a smallwavelength and a thick beam waist diameter.Intensity:The optical intensity I( r)= |U(r)| 2 is a function of the radial and axialpositions , ?=?(? 2 + ? 2 ) an Z respectively .

?(?) 2 2?2 I (?, Z) = ?0 exp? (1.10) ?0 ? 2 (?)Where ?0 =|A0 | 2On the z axis The Gaussian function has its peak, at ?= 0, and when ?increase the gaussian function decreases monotonically. The beamwidth 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 ?0distance ? at Z=0 in 2 Dimensional . ?Figure 1.4: The normalized beam intensity as a function of the radial ?0distance ? 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 peakintensity at the location Z=?0-At location Z=?0 the beam width is ?2 greater than the width at thebeam waist.-The phase on the beam axis at Z=?0 is reduced by an angle ? / 4relative 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 paraxialHelmholtz equation. There are other solutions that exhibit non-Gaussian intensity distributions but share the paraboloidal wave frontsof the Gaussian beam. It is possible to break a coherent paraxial beamusing the orthogonal set of so-called Hermite-Gaussian modes, whichare given by the product of x and y factor.

Such solutions are possible toseparate in x and y in the paraxial wave equations written in Cartesiancoordinates. This mode are given in order ( l , m) referring to the x andy directions.1. The phase of hermite gaussian beam is the same as that of theGaussian beam, except for an excess phase Z (z) that is not dependentof x and y. If Z(z) is a slowly varying function of z, both hermite andgaussian beam have paraboloidal wave fronts with the same radius ofcurvature R( z).2.

This hermite distribution is as Gaussian function which is modulatedin the x and y directions by the functions ? 2 (.) and ? 2 (.),respectively.The wave which is modulated in x, y direction, therefore represents abeam of non-Gaussian intensity distribution.

Complex Amplitude: ?0 ?2 ?2??,? (?, ?, ?)=??,? ?? ? ?? ? ?(?) ?(?) ?(?) ? 2 +? 2 ?exp -jkz-jk + ?(? + ? + 1)?(?) 2?(?) Hermite Gaussian Beam (1.11) ?2Where ?? (u) = ?? (u) exp(- ) l=0,1,2,3…… 2?? (u) Is known as the Hermite-Gaussian function of order l, and ??,? isa constant.If ?0 (u) = 1, then the Hermite-Gaussian function of order 0 is simplyknown 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 2These 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 isgiven by- ?0 2 ?2 ?2 ??,? (?, ?, ?)=??,? 2 ?? 2 ? ?? 2 ? ?(?) ?(?) ?(?) (a) (0,0) Mode (b) (1,0) Mode (c) (0,2) Mode (d) (2,2) ModeFigure1.6: 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 theparaxial Helmholtz equation. Any other solution of beams can bewritten as a superposition of these beams.

This complete set ofsolutions known as Laguerre -Gaussian beams. Laguerre gaussianbeams are described by writing the paraxial Helmholtz equation incylindrical 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 theGaussian beam on ? and z, but the phase has an extra term which isproportional to the azimuthal angle ?, and on a Gouy phase that isgreater by the factor (l + 2m + 1).

Because of this linear dependence ofthe phase on ? (for l ? 0) when the wave travels in the z direction, thenwave front tilts helically as show in Fig. Figure1.7: Wave front Figure1.8: Intensity distribution

x

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