Theoretical simulation of High Harmonic Generation by timedependent Schrödinger equation
Abstract
Abbreviations
The abbreviations that will be used in this project are as follows:
HHG  High Harmonic Generation 
VUV  Vacuum Ultraviolet 
IP  Ionization Potential 
ESP  Electrostatic Potential 
INTRODUCTION
Background/Rationale
The generation of the attosecond pulse using HHG can be explained using a threestep semiclassical model. The model was proposed by Corkum and Kulander independently in the year 1993 ^{[2]}^{[3]}. It consists of three steps  ionization, acceleration, and recombination ^{[4]} ^{[5]}.
Statement of the Problem
The purpose of this project is to simulate the HHG spectrum for a number of atoms and molecules and compare it with the experiments. From the HHG spectrum, we can find out the cutoff frequency for each atom and hence the maximum kinetic energy the electron can have. The cutoff frequency determines the maximum harmonic order upto which the HHG can be achieved. The cutoff frequency depends on the laser parameters used in the experiment, and the type of atoms or molecules (i.e. the potential experienced by the electron).
First, the electrostatic potential of the atom or molecule is to be determined. Then the effects of the highintensity Laser pulse is studied by simulation using the timedependent Schrödinger equation and the spectrum is obtained. By matching the theoretically obtained spectrum with the experimental one we can verify if we guessed the correct potential of the atom. Elements or compounds with different ionization energies have different cutoff frequencies. Higher the ionization energy, higher will be the cutoff. The threestep model can be used to determine the electron trajectories for different harmonic orders.
Objectives of the Research
The objectives of this project are:
 To determine the potential of the atoms and molecules.
 To simulate the HHG for different atoms and molecules.
Scope
High Harmonic Generation is widely used to produce attosecond pulses which is a prerequisite for attosecond physics. It can be used to produce a wide range of harmonics. The harmonic cutoff varies linearly with increasing laser intensity up until the saturation intensity I_{sat }where harmonic generation stops. The cutoff can be increased by changing the atomic species to lighter noble gases but these have a lower conversion efficiency, so there is a balance to be found out. High harmonics are emitted colinearly with the driving laser and can have a very tight angular confinement, sometimes with less divergence than that of the fundamental field and near Gaussian beam profiles ^{[6]}. It can also be used as an alternative xray light source.
LITERATURE REVIEW
Information
The first experiment on HHG was done by Rhodes and his colleagues showed strong fluorescence for noble gas ions like Ar, Kr, and Xe along with high order harmonics for Ne using a 248nm excimer laser ^{[1]}. Shortly afterward, higher order harmonics were observed in Xe, Kr and Ar too using a 30ps Nd:YAG laser (1064nm) ^{[7]}. From the various experiments it was revealed that the HHG spectrum is characterized by a rapid intensity decrease for the first orders, a plateau from the seventh harmonic up to a very high order (e.g., the 29th in Ar) and an abrupt cutoff. Theorists tried to reproduce the experimental results by numerically solving the timedependent Schrödinger equation for a single active electron, considering only the single atom response and succeeded. However, the early single atom results did not show a locked HHG phase which is a necessary condition for generation of attosecond pulse. In 1993, a simple physical picture of the HHG process was proposed by Corkum and Kulander independently ^{[2]}^{[3]}. According to this model, the electron tunnels through the Coulomb energy barrier, which is suppressed by the presence of the linearly polarised laser electric field. The electron then undergoes classical oscillations in the laser field. The influence of the Coulomb force from the nucleus is practically negligible at this time. If the electron comes back to the vicinity of the nucleus, it may recombine back to the ground state, thus producing a photon with energy IP (the ionization potential) plus the kinetic energy acquired during the oscillatory motion. The trajectories are obtained by solving the Newton's equation:
assuming that the electrons are born with zero velocity at t=0. The energy of emitted HHG photons is then simply
where IP is the ionization energy. This simple classical calculation allows us to understand the cutoff behavior at
where $U_P=\frac{e^2E^2}{4m\omega^2}$is the ponderomotive energy.
In 1999, the quantum theory of the threestep model was presented by M. Yu. Kuchiev and V. N. Ostrovsky ^{[8]} and was accepted. Later in 2007, XueShen Liu and Nana Li theoretically studied the interaction of a He+ ion with fundamental laser pulses and different combined laser pulses ^{[9]}. They found out that for low laser intensity the cutoff energy is extended; however, for higher laser intensity the cutoff of the enhanced harmonics is extended under the combination of two fundamental laser pulses. These are some of the works which are relevant to this project.
METHODOLOGY
Concepts
To study the interaction of atoms and molecules with highintensity laser pulses, we have to solve the onedimensional timedependent Schrödinger equation, which can be expressed as (atomic units are used otherwise stated):
where x is the electronic coordinate, ε(t) is the electric field of the laser pulses which is linearly polarised. Here we have used two types of potential functionsone with four parameters and another with seven parameters.
These parameters can be adjusted in order to get the same ground state binding energy of the respective atoms, ions or molecules. The electric field of the simulated HHG is given by
where f(t) is the pulse envelope function for the fundamental Laser pulse given as
where n is the number of cycles in the envelope. The intensity of the Electric field used is 5.0×10^{14} W/cm^{2}. The number of pulses in the enevelope is 46. However, for atoms or molecules with low IP, n is lowered to 7. The value of ω_{0} is taken to be 0.056 (for a wavelength of 750nm).
The harmonicgeneration spectrum is proportional to the modulus squared of the Fourier transform of d (t). The harmonic order of the cutoff is given by
Methods
The timedependent Schrödinger equation is solved and the harmonic spectra is obtained theoretically using a python program for the noble atoms (He, Ne, Ar, Kr), Helium ion, different types of molecules( H_{2}O, and N_{2}). At first, the potential at various points along the required direction is determined using the Gaussian ^{[10]} and plotted as a function of distance from the center of the nuclei. Then the points are fitted using the two potential functions and the different parameters are obtained. Then we put the values of the parameters in the Python program. The python program returns the first four energy eigenvalues and also plots the harmonic spectra as a function of the harmonic order. The calculated value of the cutoff is then compared to that obtained from the simulation.
RESULTS AND DISCUSSION
 Helium Ion (He^{+}): Xueshen Liu and Nana Li in their paper ^{[9]} simulated the HHG of Helium ion using a 'soft Coulomb potential', $V(x)=\frac Z{\sqrt{x^2+a}}$. But the ESP obtained from Gaussian differs from this 'soft Coulomb potential'. Our aim is to simulate the spectrum using the two potential functions in Equation 5a & 5b.
The two potential functions were used to fit the obtained ESP and then the HHG spectrum was obtained.
The values of different parameters obtained after curvefitting were used to simulate the HHG spectrum. Here, we present a comparison of the spectra obtained by using the three types of potential functions.
The IPs obtained from the Python program using the three different potentials are given below:
Type of Potential Function  Ionization Potential Values(in eV) 
Experimental  54.4 
Soft Coulomb Potential  54.4 
One Fitted Potential  54.98 
Two Fitted Potential  54.28 
We also used the two potential functions to simulate the HHG spectra for He, Ne, Ar and Kr.
The IP of Helium atom obtained from the onefitted function is 25.88 eV and that from the twofitted function is 24.97 eV. The experimental value of the IP is 24.6 eV.
Atom  Experimental IP (eV)  Calculated IP using OneFitted Function (eV)  Calculated IP using TwoFitted Function (eV) 
He  24.6  25.88  24.97 
Ne  21.6  20.05  21.63 
Ar  15.7  16.72  15.68 
Kr  14.1  15.79  14.03 
Atom  OneFitted Function  TwoFitted Function 
He  $y=\frac{0.46}{\sqrt{(x0.13)^2+0.03}}0.01$  $y=\frac{0.046}{\sqrt{(x0.41)^2+0.01}}\frac{0.48}{\sqrt{(x(0.03))^2+0.05}}0.0001$ 
Ne  $y=\frac{0.34}{\sqrt{(x0.34)^2+0.02}}0.03$  $y=\frac{0.033}{\sqrt{(x0.53)^2+0.001}}\frac{0.41}{\sqrt{(x0.13)^2+0.05}}0.01$ 
Ar  $y=\frac{0.24}{\sqrt{(x0.5)^2+0.005}}0.03$  $y=\frac{0.06}{\sqrt{(x0.74)^2+0.002}}\frac{0.37}{\sqrt{(x0.11)^2+0.17}}0.01$ 
Kr  $y=\frac{0.22}{\sqrt{(x0.53)^2+0.002}}0.01$  $y=\frac{0.02}{\sqrt{(x2.39)^2+0.2}}\frac{0.24}{\sqrt{(x0.67))^2+0.01}}0.03$ 
 Nitrogen Molecule: The two potential functions were used to generate the HHG spectrum of the Nitrogen molecule by using ESP determined along three directions along the bond, perpendicular to the Natom and perpendicular to the bond.
The experimental IP of the Nitrogen molecule is 15.58 eV. The calculated IPs and the different parameters obtained from curvefitting is given below.
Direction  Calculated IP (eV)  Parameters of the OneFitted Function 
Along the NN bond  15.36  $y=\frac{0.3}{\sqrt{(x0.43)^2+0.027}}0.02$ 
Perpendicular to the N atom  15.64  $y=\frac{0.3}{\sqrt{(x0.44)^2+0.027}}0.03$ 
Perpendicular to the NN bond  15.71  $y=\frac{0.34}{\sqrt{(x0.27)^2+0.04}}0.03$ 
 Water (H_{2}O): The ionization potential of water is near to the IP of nitrogen molecule. So, we can expect their cutoffs to be near as well. We generated the HHG spectrum of water. Here we present a comparative study of the HHG spectrum of water and nitrogen molecule.
Molecule  Experimental IP (eV)  Calculated IP (eV) 
H2O  12.62  12.66 
N2  15.58  15.49 
The function used for fitting the ESP in case of water is $y=\frac{0.22}{\sqrt{(x0.39)^2+0.01}}0.02$.
CONCLUSION AND RECOMMENDATIONS
To understand the HHG process from the atomic point of view, it is essential to know how it generates and how the spectra are obtained. Here we studied the theoretical simulation of the HHG process. We have used electrostatic potential (ESP) for the study of atoms and molecules. This helps to get the HHG spectrum for any kind of molecule; and obtaining HHG spectrum for molecule is one of the leading scientific problems to the HHG spectroscopy community. After investigation we found following key features:
 The ESP can be used to construct a model potential for atom or for molecule.
 The HHG spectrum generated by using ESP can reproduce the structure and the cut off of the HHG spectrum.
 Cut off comparison has been made with He, Ne, Ar and Kr atom where IP gradually decreases and as a result the cut off also decreases.
 For molecules, N_{2} and H_{2}O have been compared with respect to their IPs and this reflects in its HHG spectrum as well.
Though it has some limitations and we should rectify it by introducing a few terms during the solution of time dependent Schrödinger equation. For example, various people have observed minima on Argon HHG spectrum, which we do not see in our work. To understand more about specific features and introduce the corresponding term in the TDSE solution, detailed quantum mechanical simulation is required.
ACKNOWLEDGEMENTS
To start with, I would like to express my sincere gratitude to my guide, Dr. Atanu Bhattacharya for his excellent guidance, caring, patience and providing me with an excellent atmosphere for doing research. I also extend my gratitude to all the faculty members of the Inorganic and Physical Chemistry department for providing me a wonderful opportunity to get to know the higher end research here in IISc.
Also, I would like to thank my mentor, Mr. Sankhabrata Chandra for guiding me through every step in this project and helping me figure out the various intricacies involved. Without his constant help, this work could not have been completed.
I am extremely grateful to the Indian Academy of Sciences for providing me with the fellowship and such a great platform to do a fruitful summer project.
I would like to thank my friends and teachers at IIT Bhubaneswar who have continuously helped me and supported me since the last year.
Most importantly, I thank my family who have always supported and helped me to pursue my dream. My greatest debt of all to my Mom, the best educators of life lessons, is acknowledged in the dedication.
References

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Zenghu Chang, 2016, Fundamentals of Attosecond Optics, pp. 165172

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Source

Fig 3: Liu, X., and Li, N.(2007) Efficient extension and enhancement of highorder harmonics of He+ by combined laser pulses. J. Phys. B, 41,015602
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