But it is observed that the maximum entropy of the system increases continuously when calculated in the case of the regime beyond DA. But in the case of DA beyond, the multipole will merge all the states and gives one.
High Harmonic Generation (HHG)
In other words, high harmonic generation (HHG) is a process in which usually noble gas atoms excited by an intense laser field of frequency ω emit radiation of higher frequencies that are usually integer multiples of ω. Directed by an infrared laser, the high harmonic radiation can extend from the optical frequency to the extreme ultraviolet (XUV) [1, 2] frequency range.
Literature survey
Also, at higher gas pressures, the laser energy at the output of the fiber is reduced, either from ionization or from other loss mechanisms in the waveguide. This is because the extension of the interaction system is generally much smaller than the wavelength of the laser ie.
Three Step Model (Ionization, Propagation And Recombination)
In the first step (ionization) of the model, the high electric field of the laser ionizes the electron by tunneling. The laser is turned on and the electron moves away from the nuclei in the first half cycle of the laser pulse.
High Harmonic Generation Beyond Dipole Approximation
H =H0+H′ where H0 is the field-free Hamiltonian and H′ is an interaction term given by H′ =µ. E0cos(ωt−kz). the dipole interaction term remains non-zero over the order of atomic size, but the wavelength is of the order of thousands of ˚A. But at large intensities this term should not be overlooked and is considered beyond the dipole approximation.
Optimization of harmonics using genetic algorithm
The fitness function is the function value for the optimization parameter corresponding to that chromosome being optimized. Based on the fitness function, other chromosomes (next generation) are prepared by mutation and crossover etc.
Application
Second reason for this low conversion is that different phase velocities of the fundamental and of the harmonic beam result in a phase mismatch. Again the fitness function is calculated for this population and generation of population and calculation of fitness is repeated until the best chromosome is found. The Floquet theory states that the solution of periodic time-dependent Schrodinger equation can be written as, .
The above equation (2.5) is an eigenvalue equation in two variables, (x,t), with time-dependent eigenvalues given by the Floquet energies, ǫλ also called quasi-energy. The key result of Floquet theory is that the Floquet eigenvalues φλ(x, t), are periodic in time with the same period T as the Hamiltonian.
Dynamical Symmetry Rules
2.15) the equality holds only for odd n, so only x polarized odd harmonics will be generated and no even harmonics will be generated. The rules of DS can be extended to the case of DS of arbitrary order N. In this case, non-zero harmonics will be obtained if and only if n=N m±1, where m is an arbitrary integer.
Applying this PN operator to the Hamiltonian, we get, . solving this simple trigonometry we get, By applying PN we get, . 2.21) the equality only holds for mN ±1, therefore only n=mN ±1 harmonics are produced (where m is a positive integer).
Tight binding method
Matrix elements for the H0 operator in the bond-dependent coordinate system using the semi-empirical method listed in the literature [11]. The bond-dependent coordinate system has the same definition as in the case of C2H2, the global and local z directions are the same. The value of the effective nuclear charge is z2p = 1.78 for 2ptype orbitals, where we choose to describe the atomic orbitals in the local coordinate system.
In the second category, the matrix elements are calculated when the basis set is described in a bond-dependent coordinate system as shown in fig. And the matrix elements calculated in bond-dependent coordinate system are rotated to global coordinate system.
Formulation
Usually, the wavelength of the applied field is much larger than the dimension of the system. Using Floquet's theorem, the solution of the time-dependent Schrodinger equation can be written as, The probability of emitting the next harmonic (In) is obtained from the Fourier transform of the system's time-dependent momentum.
Within the dipole approximation, due to the dynamic symmetry of the system, only odd-order harmonics will arise, but in the case outside the dipole approximation, all harmonics (odd and even order) are expected. We examine the validity of the dipole approximation as a function of the number of monomeric units in the system, which can be quantified by comparing the intensities of even and odd harmonics.
Introduction
It is noted that even if the electron oscillation is confined to a small region, the non-dipole term cannot be ignored.
Results and discussion
As the intensity increases, even-order harmonics become more pronounced and the difference between intensity of even-order harmonics and odd-order harmonics continues to increase (here the dynamics are extremely complicated and we have not yet studied its reasoning). In later part of the plateau near the cutoff, the even order harmonics are much more order intense than odd order harmonics. Near the cutoff point, more intense even-order harmonics can be generated by increasing number of monomer units of ethylene even if a relatively moderate intensity laser is used.
Even-order harmonics are found to be more intense even if only two ethylene monomer units are included in the model system. As the laser intensity increases, the DA becomes worse and near the limit, the even harmonics are much higher in intensity than the odd order harmonics.
![Figure 3.1: HHG spectra of model polyethylene [H(C 2 H 2 ) n 0 H] system where π elec- elec-tron is interacting with linearly polarized laser light of electric field strength 0.9 a.u.](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10473858.0/49.892.178.703.269.872/figure-spectra-polyethylene-interacting-linearly-polarized-electric-strength.webp)
Abstract
Introduction
Results and discussion
It has been observed that the maximum spectral entropy does not change in the case of within DA, but it increases continuously in the case of BDA. This explains the decrease in the intensity of harmonic generation and the increase in the plateau cutoff. Population of dipole forbidden MO is small but significant enough to contribute to the entropy and higher intensity of equal order harmonics.
But in the case of BDA, despite the above interaction, π1 interacts with π2 and π∗3 also via non-dipole effects. As the intensity increases, even harmonics become more prominent and the difference between the intensity of even harmonics and odd harmonics increases, at least in the plateau region of the HHG spectra.
![Figure 4.1: Entropy of model polyacetylene [H(C 2 H 2 ) n 0 H] system where π electron is interacting with linearly polarized laser light of electric field strength of 0.9 a.u.](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10473858.0/57.892.185.747.368.785/entropy-polyacetylene-electron-interacting-linearly-polarized-electric-strength.webp)
Summary
This comparison shows that although the laser intensity increases the HHG plateau, it has no effect on the entropy. And in the case of an approximation outside the dipole, one must obtain all harmonics regardless of the polarization direction of the emitted light.
![Figure 4.4: Comparison of HHG spectra and Entropy of only monomer unit of model polyacetylene [H(C 2 H 2 ) n 0 H] system as a function of laser intensity in the case of beyond dipole approximation.](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10473858.0/61.892.183.747.387.836/figure-comparison-spectra-entropy-polyacetylene-function-intensity-approximation.webp)
Introduction
Classification of CNT
Non dipole effects
That is why HHG frequencies cannot be increased by a simple increase in laser intensity. An interesting possibility to counteract the relativistic drift in the weakly relativistic regime, based on the use of antisymmetric molecular orbitals, is shown in [11]. Due to symmetry, the momentum distribution of the antisymmetric molecular orbital has two peaks at the nonzero momentum component along the molecule's axis.
Therefore, in a strong laser field, the electron tunnels out of the molecule with non-zero momentum along the molecular axis. If the axis is oriented in the direction of propagation of the laser, the initial momentum of the ionized electron will counteract the relativistic drift, allowing rescattering and thus significantly increasing the harmonic signal.
Results and discussion
The odd harmonic excitation is expected due to symmetry requirements and in the case of off-dipole approximation the symmetry is lost, therefore inclusion of off-dipole approximation is important. 5.1, left column of the Fig 5.2, contains HHG spectra inside for DA regime and the right column for BDA regime. As it can be seen that only odd order harmonics are generated in the case of inside DA.
5.3, the HHG spectra are plotted for a CNT containing different numbers of unit cells irradiated by a circularly polarized laser with an electric field strength of 0.9 a.u. 760 nm), regardless of the polarization of the emitted light, in the case of circularly polarized laser light, only 5n±1 harmonics are allowed. The same applies to the spectra in the case of DA also for circularly polarized laser light.
Summary
5.2, x polarized HHG spectra are plotted for the same system and the same parameters in Fig. We have presented here a study of the optimization of higher order harmonics generated in two different cases of (a) within dipole approximation and (b) beyond dipole ap - proximity. We have considered model system of ethylene irradiated with intense, linearly polarized (perpendicular to the system axis) laser field propagating along the system axis.
It is found that although the harmonic intensity increases in both cases, the optimization is always higher in the case of the beyond-dipole approximation. The study of the optimization of harmonics of even order as well as harmonics of odd order has been done.
Introduction
However, given the initially selected population, there may not be enough diversity in the sets to ensure that the GA covers the entire problem space. The mechanism of the process is qualitatively described by a semi-classical three-step model [19]. Here, we optimized the desired 51st (i.e., odd-order) harmonic in both the inside-DA and outside-DA cases, and surprisingly, it was found that the consideration of dipole effects [20–27] significantly improves the harmonic intensity compared to the case without dipole consideration.
It has been shown that with two-color laser optimization, harmonic intensities are amplified by orders of magnitude. A few experimental studies have been carried out, where the optimization is achieved up to a maximum of 17 times [6,9,10].
Results and discussion
The better optimization in the case of BDA can be understood as follows: Instead of optimizing the laser parameter, we let the writing wave function at t= 0 as,. Clearly, as the number of terms in the sum increases, better optimization will be achieved. In the case of DA as shown by entropy plots, only two non-zero terms contribute.
Moreover, in the case of BDA, non-dipole mechanism contributes to high harmonic generation optimization. It is found that increment in the intensity of the optimized harmonic is up to 10 times in the case of DA, and up to 458 times in the case of beyond DA, so We can say that outside dipole approximation plays an important role in the optimization of selected harmonic.
Summary
As shown both the HHG and entropy spectra change radically in the case beyond DA. We observe the significant increase of entropy in the case of the regime beyond DA. Then we have shown that in the case of the intra-dipole approximation regime and for linearly polarized laser light, due to dynamical symmetry, polarized even-order x harmonics and odd-order polarized y harmonics are produced.
And the difference between the intensity of even and odd order harmonics increases to the order of 6 in the case of optimization of even order harmonics outside DA. This difference in intensity between even and odd order harmonics is limited to a magnitude of order 3.5 only in the case of odd order harmonics optimization.
![Figure 4.2: HHG spectra of model polyacetylene [H(C 2 H 2 ) n 0 H] system where π elec- elec-tron is interacting with linearly polarized laser light of electric field strength of 0.9 a.u](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10473858.0/59.892.174.699.289.885/figure-spectra-polyacetylene-interacting-linearly-polarized-electric-strength.webp)
