I thank my juniors Parvendra Kumar and Samit for their valuable help and support in my research work. I would like to express my sincere thanks to my departmental friends Abu, Rahul, Supriyo, Satchi, Himangshu, Arindam for their company during my research period. I would also like to express my gratitude to my friends Samiran, Jharnali, Mousumi, Arpita, Rohit, Kousik for their moral support and companionship.
My special thanks go to my elder sisters Saswati (Di) & Manideepa (Chhordi) for their constant support, care and love during my research period.
Modulation Instability of coupled nonlinear field equations for pulse propagation
Modulation Instability and solitary wave solutions of the Nonlinear Schr ӧ dinger equation in the context of a Non-Kerr medium
Modulation Instability in Metamaterials embedded into a non-Kerr medium 43
Soliton propagation and soliton-soliton interaction in a silicon waveguide 67
Introduction
- Background
- The relevance and the aim of the topic of research
- Negative refractive index metamaterials
- Thesis Overview
- In this chapter, we have focused on the study of exact solitary wave solutions and MI of the generalized nonlinear Schrödinger equation in the context of a non-Kerr medium
- We give a brief summary of the results and analysis of the problems considered in the thesis. Also, we give the future directions of research in this ever expanding area of
Bright solitons are the solution of the NLS equation in the anomalous GVD regime (β2 < 0). The physical origin of this effect is related to the delayed nature of the Raman reaction (vibration). On the other hand, the TOD results due to the frequency dependence of the GVD parameter.
One of the most fascinating examples of such structures is the so-called negative-index metamaterials (NIM).
Modulation Instability of coupled nonlinear field equations for pulse propagation in Metamaterial
- Introduction
- Generalized Theoretical Model
- Modulation Instability analysis
- Solitary wave solutions for the coupled nonlinear field equations
- Chapter Summary
On the other hand, Lazarides and Tsironis[141] derived a system of coupled nonlinear Schrodinger equations (NLSE) for the shelves of the propagating electric and magnetic fields in an isotropic and homogeneous nonlinear left-handed material taking into account both nonlinear polarization and magnetization. H r ω are the envelope amplitude of the electric and magnetic fields in the frequency domain, respectively, the set of coupled equations for the electric and magnetic fields in the frequency domain could be expressed as follows. Our goal is to obtain a set of coupled equations for the shelves of the electric and magnetic fields in a more general form.
Normalized variables are defined as:. 2.11), where T0 is the pulse width, LD =T02/ β2 the dispersion length, and A0 and B0 the initial amplitude of the electric and magnetic fields. But unlike them, we consider both the electric and magnetic effects of self-gaze. Since SE ≠SH, the initial normalized amplitudes of the electric and magnetic fields should always be different, i.e.
Under these MI conditions, we obtain the gain spectrum ()g Ω of MI as. 2.17) It is clear that the MI gain depends on the initial amplitude of electric and magnetic fields, the perturbation frequency, and the electric and magnetic self-steepening parameter traversals. The so-called reduced self-steepening parameter s could be controlled simply by setting the initial electric or magnetic field amplitudes to a given operating frequency of MM. Now we choose P1= A Sech B T1 [ ( −Z V0 )] and P2 = A Sech B T2 [ ( −Z V0 )], where A1 and A2 are the complex amplitudes of the electric and magnetic fields, B and V0 are respectively temporal width and soliton inverse velocity.
We obtained several physical parameters related to the solitary wave solutions in terms of the medium parameters as follows. We found that one can control the MI gain in a MM by tuning the initial electric or magnetic field amplitudes of the pulse.
Modulation Instability and solitary wave solutions of the Nonlinear Schrödinger Equation in the context
- Introduction
- Theoretical Model
- Solitary wave solutions
- Modulation Instability analysis in cubic-quintic medium
- Chapter Summary
By means of the combined amplitude–phase formalism, we can study the bright and dark single wave solutions individually. In Fig.3.1(a), we have shown the intensity of the bright soliton for different normalized quintic parameter σat s=0.02. In Fig. 3.2(b) we have studied the effect of the SS parameter on the intensity profile of the bright soliton.
We observe that the intensity of the soliton remains constant, while the width of the soliton decreases. We notice that with the increase of the quintic parameter, the amplitude of the soliton decreases and the width of the soliton decreases. We observe that the intensity of the soliton remains constant, but the width of the soliton decreases with the increase of the SS parameter.
Now we investigate the impact of the higher order dispersion effect and the nonlinear effects on the MI gain. Finally, in Figure 3.4(a) we plot the variation of the MI gain as a function of Ω and σ. It is observed that the intensity of the MI gain increases with the increase of the cubic quintic.
In this chapter, we have studied solitary wave solutions and a higher-order MI NLSE analysis describing the propagation of an ultrashort femtosecond pulse in a non-Kerr medium. The role of cubic quintic parameter and self-staring parameter in MI amplification was also discussed.
Modulation instability in Metamaterials embedded into a non-Kerr medium*
- Introduction
- Theoretical Model
- Modulation Instability analysis
- Influence of loss on Modulations Instability
- Chapter Summary
We considered loss in our analysis and found that loss distorts the sidebands of the MI gain spectrum. Fig.4.1: Plot of the refractive index (n), group velocity dispersion (β2), the third order dispersion parameter (β3) and the fourth order dispersion parameter (β4) against the normalized frequency. The behavior of S3 is similar to that of the self-steepening parameterS1, as can be seen from Eq.(4.2).
Another important feature of MI in MM is that it can be activated in both normal and anomalous dispersion regimes by reasonable choice of parameters. In Fig.4.3 we show the variation of MI gain with normalized perturbation frequency for different values of normalized frequency ωpe withγe =0.01. We note that the amplitude of the MI gain as well as the gain bandwidth decreases with the increase of the normalized frequency ω ω.
We also investigated the role of the nonlinear parameterp, which results from the combined effect of cubic-quintic nonlinearities, on the MI gain spectrum. From Figure 4.5 (a), we see that both the MI gain and the gain bandwidth increase with the increase of the nonlinear parameter p. But if loss is included, as the value of loss coefficient increases, the peak of MI gain remains almost the same, but the sidebands become distorted.
35], the peaks of the sidebands are distinct and distortions in the sideband could be fairly neglected. Self-stare3 parameters arising from χ(5)non-linear polarization are shown to have similar MI characteristics to self-stare1 parameters.
Solitary wave solutions of Nonlinear Schrödinger equation with evolution parameter dependent
- Introduction
- Evolution parameter dependent nonlinear Schrodinger equation
- Dark solitary wave solution of evolution parameter dependent NLSE
- Bright and Dark solitary wave solution with constant co-efficients of generalized NLSE
- Exact Bright soliton solutions
- Exact Dark soliton solutions
- Chapter Summary
As a special case, we also studied the solitary wave solutions of the NLSE with constant higher order dispersion and nonlinear coefficients. In Eq.(5.1), the complex value function q(x, t) represents the wave profile where the independent variables or evolution parameters are the spatial x and time t. Eq.(5.1) with α =0 is known as the generalized Radhakrishnan-Kundu-Lakshmanan (RKL) equation[174,181]. The bright soliton solution of the RKL equation has been reported[174].
In section (5.4), we would report the solution of the bright and dark solitary wave NLSE with constant coefficients. Because of the time-dependent coefficients in equation (5.1), the soliton frequency κ, wavenumber ω, and phase constant θ generally also depend on time. Now, setting the coefficients of the other two linearly independent functions in equation (5.8) to zero yields
Finally, equating the two values of the soliton velocity υ given by eqn (5.17) and eqn (5.18) gives the relation (5.19) and this closes the case and also shows that the method is consistent. This happens due to the reduction of the speed of the soliton as evidenced by equation (5.18) through its dependence on A based on equation (5.15). This unknown exponent will be determined during the derivation of the soliton solution in equation (5.26).
Thus, the bright 1-soliton solution of the NLSE with power law nonlinearity is given by We have also reported the exact bright and dark one solitary wave solutions of the generalized NLSE with constant coefficients.
Soliton propagation and soliton-soliton interaction in a silicon waveguide*
- Introduction
- TPA effect on soliton propagation in silicon waveguide
- Effect of TPA on N=2 Soliton
- Soliton-soliton interaction
- Chapter Summary
In fact, the presence of a TPA parameter in the nonlinear Schrödinger equation (NLSE) makes the equation non-integrable and we must use numerical methods to investigate solitons in a silicon-based waveguide. The transmission can give us an estimate of the effect of TPA on the energy or power of the pulse propagating through the silicon waveguide. normalized TPA parameter for N=1, 2 and 3 soliton. It is highly expected that the transmission of the waveguide decreases as the TPA coefficient increases due to nonlinear absorption occurring in the waveguide.
The higher the order of the soliton, the greater the effect of TPA on transmission. In Fig.6.2 we plot the intensity of a second-order soliton in the output of the given silicon wave for different values of the TPA parameter. The spatio-temporal evolution of the N=2 soliton with r=0.006, through the silicon waveguide and its transformation into an N=1 soliton at the output of the waveguide can be clearly seen in Fig.6.3.
But if the TPA parameter 'r' is greater than 0.1, it evolves into a single soliton at the output end of the silicon wave. We find that when the TPA parameter is small, as the TPA parameter increases the soliton pair is evolving into a number of smaller pulses with decreasing amplitude. In Fig.6.6 we draw the spatio-temporal evolution of a soliton pair N=2 with the TPA parameter r=0.5.
We have shown the formation of a fundamental soliton from a higher-order soliton if the TPA parameter is judiciously chosen. In addition, the influence of the TPA parameter on the soliton-soliton interaction in a silicon waveguide is discussed.
Summary and Future Aspects
- Summary of the thesis work
- Future Direction
- International Journals
- Manirupa Saha and Amarendra K. Sarma, “Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion
- Amarendra K. Sarma and Manirupa Saha, “Modulational instability of coupled nonlinear field equations for pulse propagation in a negative index material embedded
- Conferences (International/National)
Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank”, Phys. Li, “New types of solitary wave solutions with the higher-order nonlinear Schrodinger equations,” Phys. Sridhar, "Refraction of electromagnetic energy for wave packets incident on a negative index medium is always negative", Phys.
Tsironis, "Nonlinear Coupled Schrodinger Field Equations for Propagation of Electromagnetic Waves in Left-Handed Nonlinear Materials," Phys. Kinsler, “Unidirectional optical pulse propagation equation for electrically and magnetically responsive materials,” Phys. Stockman, “Criteria for negative refraction with low optical losses from a fundamental principle of causality,” Phys.
Mihalache, "Generation, compression and propagation of pulse trains in the nonlinear Schrodinger equation with distributed coefficients", Phys. Luther-Davies, "Dark-as-black soliton: Dark soliton with total phase shift greater than π", Phys. Wieman, "Formation of bright matter-wave solitons during the collapse of attractive Bose-Einstein condensates", Phys.
