Divergence control and yield enhancement in high-harmonic generation in mixed gases

V

RIJE

U

NIVERSITEIT

A

MSTERDAM

B

ACHELOR

T

HESIS

Divergence control and yield

enhancement in high-harmonic generation

in mixed gases

Author:

Roy van der Linden

Supervisor:

Peter Kraus

Bachelor Thesis Physics and Astronomy, performed within the High Harmonic Generation and EUV Science group of Peter Kraus

at the

i

Abstract

Divergence control and yield enhancement in high-harmonic generation in mixed gases

by Roy van der Linden

This thesis presents data on high harmonic generation in a gas mixture of argon and krypton. The data is found in agreement to simulations based on SFA. Through a similar path as Kanei et al. [6], the difference between the phases of the generated EUV photons of the two gases is accountable for the interference effects found in the spectrum of the mixture compared to the single gases. Additionally, we report evidence of improved divergence control of the signal of the gas mixture above the separate gases. The beam profiles have smaller full width half maxima as well as reduced signal in the outer lower areas of the beam profile, namely the wings.

Acknowledgements

I would like to thank all the people at ARCNL for creating such a friendly and mo-tivating work environment. Specifically, I would like to thank the HHG and EUV Science group of Dr. Peter Kraus. I started my research project in April 2020, during a period when the country was in state of emergency because of the COVID-19 virus and although, in the beginning, I was not able to be physically present at the insti-tute, the group included me quickly and taught me a lot in these first few months. During this period, I had the most contact with my supervisor and group leader, Dr. Peter Kraus. He offered me many informative Zoom sessions and guided me through the wonderfull world of High Harmonic Generation. Eventually, the re-strictions loosened and I experienced first hand the professional, kind and support-ive working conditions in the office as well as in the lab. Sylvianne Roscam Abbing was my daily supervisor and has been very educational in regards to the setup, the research and the workings of an academic career and because of that the world of optical physics has really opened up for me. Aside from my counsel of Sylvianne and Peter, Dr. Fillipo Campi has always had an open ’Whatsapp door’ policy and if not there in person would always answer my cries of help with quick and help-ful solutions and answers over Whatsapp. Moreover, I have been gratehelp-ful to have experienced his out of this world BBQ skills. Additionally, I would like to thank Reinout Jaarsma for setting up the gas mixing system and being the lab ’camping dj’. Many thanks again to the rest of the group and people at ARCNL for showing the workings of the scientific world.

iii

Contents

Abstract i Acknowledgements ii 1 Introduction 1 2 Theory 2 2.1 Three-step model . . . 2 2.1.1 Ionization . . . 3 2.1.2 Propagation . . . 4 2.1.3 Recombination . . . 4 2.2 Lewenstein model. . . 6 2.3 Macroscopic effects . . . 7 2.3.1 Quadratic relation . . . 8 2.4 HHG in mixed gases . . . 9 3 Methods 10 3.1 Experimental setup . . . 10 3.1.1 The pulse . . . 10 3.1.2 Generation. . . 10 3.1.3 Detection. . . 12 3.2 Simulations . . . 12 4 Results 14 4.1 Quadratic relation. . . 15 4.2 Intensity modulation . . . 16 4.2.1 Experimental data . . . 16 4.2.2 Simulations . . . 17 4.3 Divergence . . . 18 5 Conclusion 20 A Optical Auto-Correlation 21 A.1 Chirped pulse amplification (CPA) . . . 21

A.2 Spatial Chirp. . . 21

A.3 Mach-Zehnder-type Interferometer . . . 22

Chapter 1

Introduction

A little more than a century ago Ernest Rutherford published a paper presenting the results of his famous Geiger-Marsden-experiment. The New Zealand-er had ob-served the scattering of alpha radiation from a thin gold foil and hypothesized that, because of the nature of the scattering angles, the foil had to consist of separate low mass particles (electrons) orbiting a high mass nucleus. During the following decades, further studies directed towards matter interactions on nanoscales such as molecular vibration, atomic motion and photon emission and absorption came into play. It immediately became clear that these systems happened at timescales of femtoseconds (10−15 s), beyond the measurement capabilities of the time. Fortu-nately, "Ask and it will be given to you" (Matthew 7:7), Pump-probe Spectroscopy was introduced. Here, an initial laser pulse is used to bring a system in the desired disorder. Thereafter, a consecutive pulse is used to image the disorder. By vary-ing the delay time between these pulses different frames are captured showvary-ing the complete effect of the desired observable. The search for better spectral resolution of atomic mechanisms has brought us to a step beyond, femtosecond science, namely attosecond (10−18s) science. Pump-probe experiments using pulses with duration in the attosecond regime have already given insights towards electron behaviour, like ionization and recombination.

Using high harmonic generation, a frequency conversion process, a femtosecond pulse can generate within a medium an attosecond pulse, which can then be used for pump-probe spectroscopy. The medium can be in the form of gases as well as solids. Kanei et al. [6] showed in 2007 that using a gas mixture of helium and neon can decrease intensity loss of the frequency conversion. Here, the effects of the sepa-rate gases are shown to interfere constructively and destructively with each other. In the same year Takahashi et al. [8] suggested that next to mutual interference effects, one gas could act as a booster for the conversion process of the higher ionization gas. This thesis aims to bring a clearer view towards the optimization conditions and ef-fects of high harmonic generation in a gas mixture of argon and krypton. Unless stated otherwise atomic units are used during this research thesis.

2

Chapter 2

Theory

The search for more precise temporal and spectral resolution of ultra-fast electronic processes pushed towards shorter and shorter pulses, beyond the femtosecond re-gion, while at the same time increasing the spectral bandwidth. In 1961, when the rest of us were listening to the newest banger of Elvis Presly, Franken et al. [3] discov-ered second harmonic generation; using laser light to generate higher frequencies of light. The generated frequencies are multiples of the incoming laser light. A decade later Burnett et al. [2] focused a nanosecond pulse onto an Aluminium target, cre-ating a plasma and genercre-ating the first high harmonics, specifically up to harmonic eleven of CO2 laser with 10.6 ∼µm center wavelength of the fundamental, which

was above the continuum background of the plasma. Using High Harmonic Gener-ation (HHG) a femtosecond pulsed laser can be used on a target to generate light in the extreme ultra violet and soft x-ray region. HHG has thus become an accessible table top attosecond radiation source.

2.1

Three-step model

The current theory of High Harmonic Generation can be described classically, quan-tum mechanically or within a combination of both. I will start with the classical description as conceptually, it is the most intuitive. First, an incoming laser field modifies the Coulomb barrier of the atom, the electron is then ionized by the incom-ing laser field as a result of the lowered barrier (step 1: ionization). Thereafter the electron is accelerated in the continuum by the first quarter cycle of the laser field (step 2: propagation). Finally, when roughly a half cycle of the laser field is com-pleted (this can differ dependent on the electron trajectory) the electron has a chance of recombining with the parent ion (step 3: recombination), upon which a photon is emitted, see figure2.1

FIGURE2.1: A schematic overview of the three-step model. First the

the electron is ionized (1), second it propagates through the contin-uum, gaining kinetic energy (2). Lastly, after the sign switch of the laser field, the continuum electron is accelerated back towards the ion and has a chance of recombining with the parent ion (3). During the last step the build up kinetic energy is released in the form of an

EUV or soft X-ray photon. This image is altered from [5].

The energy of a generated photon after HHG is fully contributed by the acceleration of the electron in the free field (step 2). In the coming experiments harmonics up to 39 electronvolt (correspond to the 25th harmonic) are generated. In order to get a grip of the gained energy during the propagation step, that will subsequently be transferred to the generated photon upon recombination, the following expression of the pondermotive energy is calculated, where E is the amplitude of the incoming electric field and ω is the frequency of the laser field,

Up= ( E 2ω)

2_{. (2.1)}

2.1.1 Ionization

When pulses with a central wavelength of 800 nanometer are used, the photons have energy around∼1.5 electronvolt. For Argon, with an ionization potential of 15.6 eV, this would mean, following classical mechanics, that in order to ionize there most likely needs to be multi-photon ionization. However, as just described the potential barrier is first of all lowered by the laser field. In addition, quantum mechanically, the electron has a chance of tunneling through the potential barrier. In order to see in what ionization regime HHG systems fall, either multi-photon ionization or tunnel ionization, the Keldysh parameter is introduced in the following manner, with Up the pondermotive energy and Ipthe ionization potential,

γ=

s Ip 2Up

. (2.2)

If γ < 1 the system mainly follows tunnel ionization and if γ > 1 multi-photon ionization is in turn the dominant process generating the high harmonics. In the case of this study, the Keldish parameter is ∼ 0.35 and therefor tunnel ionization overtakes multi-photon ionization.

Chapter 2. Theory 4

2.1.2 Propagation

As mentioned before, the energy loss of the returning electron during recombination is given to the HHG photon and originates from the gained energy of the electron in the free field. During this step, the Coulomb force is small compared to the acceler-ation by the laser field and thus can be neglected. In this approximacceler-ation the starting velocity right after tunneling is taken to be zero, which is found to be an accurate assumption. Using the driving laserfield, E = E0cos(ω·t), and Newton’s second

law, we can derive the electron displacement formula’s to be,

¨x=E0cos(ω·t), (2.3) ˙x= E0 ω · (sin(ω·t) −sin(ω·t0)), (2.4) x= E0 ω2cos(ω ·t) − E0 ω · (t−t0) ·sin(ω·t0)) + E0 ω2 cos(ω ·t0), (2.5)

where ω·t0is the ionization phase. The electron displacement set to zero indicates the ionization, as well as the recombination phases and enables us to map every ionization phase to a recombination phase following the relation, see figure2.2

ω·tr= π 2 −3 arcsin  2 πω·t0−1  . (2.6)

FIGURE2.2: The mapping of ionization to recombination phase, see

equation2.6.

2.1.3 Recombination

After the electron recombines with the parent ion it emits a photon with an energy equal to the ionization potential plus the kinetic energy gained by the propagation

step. Using equation2.1,2.4and2.6the HHG photon energy can be derived as, ¯hω = Ip+Ekin (2.7) = Ip+2Up· (sin(ω·tr) −sin(ω·t0))2 (2.8) = Ip+2Up· (cos  3 arcsin 2 πω·t0−1  −sin(ω·t0))2 (2.9) ¯hωmax = Ip+3.17Up. (2.10)

By plotting the HHG photon energy against the recombination time, see figure2.3

it can be seen that for every energy (with exception of the peak value) there are two ionization phases which correspond to two distinct recombination phases. Al-ternatively, an HHG photon with energy ¯hω could have been emitted through two electron trajectories, starting and ending at different phases. Namely, the short tra-jectories, which essentially ionize after the peak in figure2.3aand recombine before the peak of figure2.3b and the long trajectories, which ionize before the peak in figure2.3a. and recombine after the peak of figure2.3b. The long trajectories are ac-celerated longer, but in turn are also, due to the flipping of the laser field, see figure

2.1, more decelerated. The short trajectories spend shorter time in the continuum but are accelerated more intensely.

(A) (B)

FIGURE2.3: The kinetic energy acquired during the propagation step as formulated in the kinetic term of equation2.7, plotted against the

ionization phase in (A) and recombination phase in (B).

In summary, the classical three-step model, describes High Harmonic Generation on a microscopic level. It starts with the ionization of an electron, which can happen via two mechanisms (tunnel ionization and multi-photon ionization). The dominant process depends on the ratio of the incoming laser field and the ionization potential of the medium, given by Keldysh parameter. The second step describes the prop-agation of the ionized electron through the free field, independent of the Coulomb potential. Here the electron builds up kinetic energy through the half cycle of the laser field. Finally, the electron can recombine and emit an HHG photon. The en-ergy contributed to this emitted photon is dependent on the propagation path, for every HHG photon energy there are two paths the electron could have propagated by, which are named the short and long trajectories.

Chapter 2. Theory 6

2.2

Lewenstein model

In 1994 Lewenstein et al. [7] published a paper showing an analytical quantum me-chanical theory of High Harmonic Generation within low frequency laser fields. I will shortly go through the underlying principles of the theory, these concepts will later be used in simulations.

To find a quantum mechanical model describing the process of HHG, the time de-pendent Schrodinger equation,

i¯h ∂

∂t |ψ

(x, t)i = Hˆ |ψ(x, t)i (2.11)

must be solved analytically, taking into account certain conditions given by the pro-cess. The system we are analyzing is made up of a single electron atom in the poten-tial of the incoming laser field and the potenpoten-tial of the nucleus. The Hamiltonian for this system thus becomes,

ˆ

H = −1

2

∂2

∂ x2 +V(x) +E0·cos(t) ·x. (2.12)

Continuing we make use of the Strong Field Approximation (SFA). In this approxima-tion, the ionization potential of the atom is relatively small compared to the pon-deromotive energy. Therefor the Keldish parameter is smaller than one and we have entered the tunnel ionization regime explained in section2.1.1. It should be noted that the ponderomotive energy is not high enough to saturate the ionization of the system, the case of ionization of all electrons. Further, ionization is assumed to only effect ground state of the cation, no excited states of the neutral atom or ion are ex-cited. Moreover, we assume that post-ionization the electron enters the continuum, just like in the three step model, with zero velocity. Lastly, because the pondermo-tive energy is large, the kinetic energy build up by the electron, in the continuum, outmatches the atomic potential of the parent ion. The SFA is summed up as follows, i. Ionized electrons entering the continuum were all bound and ground state

electrons, excited states can be neglected.

ii. There is no depletion by ionization of these ground state levels.

iii. The influence of the Coulomb potential on the continuum electron can be ne-glected.

Next lies the task to specify the wave function of the electron in the SFA system. The wave function will be analyzed as a composite of the bound ground state wave function,|0i, and the free continuum wave function, often donated as|vi,

|ψ(x, t)i =e−i·Ip·t(a(t)|0i + ( Z

d3v b(v, t))|vi). (2.13)

Here the only missing values are the constants a(t)and b(t). Yet, because we assume that the intensity is below the saturation intensity, the second item of the SFA, we can take a(t) ' 1. The wave function, together with the Hamiltonian in equation2.11, can be injected in the TDSE to find b(t). This gives a differential equation, which when solved, results in,

b(~v, t) =i Z t −∞dt 0_~ E(t0) ·dx(~v + ~A(t) − ~A(t0)) ·ei ~ S(~(p,t0,t) (2.14)

with A being the vector potential of the laser field, d~ x the dipole moment in the x direction and~S(p, t0, t)representing the classical action of the electron in the contin-uum. Additionally, t0 can be see as t0, the ionization time in the three step model. Here, using the substitution,~p= ~v + ~A(t), the classical action is now given by,

~_{S(p, t}0 , t) = Z t t0 dt 00_(~ p − ~A(t00))/2 + Ip. (2.15)

Now we use the principle of least action to find the true trajectory from the initial state of ionization to the final state of recombination. Taking the derivative of the stationary action to all three variables gives the following three statements,

∇_~p~_{S(p, t}0_{, t) =x(t) −x(t}0_{) =0 (2.16)} ∂~S(p, t0, t) ∂t0 = [~p− ~A(t0_)]2 2 +Ip (2.17) ∂~S(p, t0, t) ∂t = [~p− ~A(t)]2_{− [~p− ~A(t}0_)]2 2 = ωXUV (2.18)

Substituting the second equation into the third shows that energy is conserved and that the kinetic energy is given to the emitted photon,

∂~S(p, t0, t) ∂t = [~p− ~A(t)]2 2 +Ip, (2.19) =E_kin+Ip, (2.20) =ωXUV. (2.21)

In order to simulate a high harmonic spectrum later on, we need to Fourier trans-form the time dependent dipole, defined as,

d(t) = hψ(~x, t)|x|ψ(~x, t)i. (2.22)

Using the previous lined up equations for the wavefunction and it’s constants we arrive at our final expression for the dipole,

d(t) =i Z t −∞dt 0Z_d3_~pE~_(t0_{) ·d} x(~p− ~A(t0)) | {z } ionization ei~S(~(p,t0,t) | {z } propagation ·d∗_x(~p− ~A(t0)) | {z } recombination +c.c. . (2.23)

The first term represents the ionization of the electron into the continuum due to the laserfield. The second term shows the phase build up during the propagation step. The third term ombodies the recolision with the parent ion.

2.3

Macroscopic effects

The three step model describes HHG on the microscopic scale, however, in order to optimize the efficiency macroscopic effects need to be taken into account as well. Phase matching plays the most important role. The generated EUV pulses can prop-agate in the medium with alternate phases, which can either build up coherently or interfere destructively. To completely optimize the phase matching effect the phase velocity of the incoming IR beam must equal the phase velocity of the generated EUV. Phase changes arise from differences in refractive indexes for IR and EUV

Chapter 2. Theory 8 light, both in the normal medium and the plasma (if there is any). Moreover, po-tential plasma strongly absorbs. To account for phase matching proper absorption length,intensity and pressure must be chosen for certain ionization levels.

2.3.1 Quadratic relation

Analyzing yield enhancement becomes a tricky task when phase matching effects are in play. In order to fully compare yield enhancement between single gases and gas mixtures, the signal should not be hindered by de-phasing effects. This can hap-pen at certain pressures and intensities. In order to best compare the generation from the different gases, the pressure must be chosen in such a way that that the gener-ation originates from coherent build up of single atom responses. The intensity of the fundamental laser pulses must be chosen such that there is no saturation of these single atom responses. In this research the observable is the intensity of the HHG. The intensity goes by the electric field squared. The electric field scales linearly with pressure for perfect phase matching. So, when the intensity scales quadratically with pressure, we can assume single atom responses.

2.4

HHG in mixed gases

Phase matching happens between emitters of the same kind, with equal ionization potentials. Yet, phase matching is also affected if there is more than a single kind of atom. The conditions for perfect phase matching are then changed. In case of a mixture of two separate gases, there are two emitters with differing ionization potential. These two classes of atoms can both emit photons which either phase match, interfere constructively, or be out of phase, interfere destructively. Lifeng Wang et al. [9] formulated this as follows,

I1+2 = I1+I2+2· p

I1I2·cos(∆φ). (2.25) Here Iirepresents the intensity of the generation and∆φ is the phase difference from the emitted HHG photon of the two gases. Kanei et al.[6] showed that if just the short trajectories are selected, the phase difference of the two emitters can be related to the difference in ionization potential,∆Ip, and excursion time, τ, of the continuum electrons (which only dependents on the laserfield and not the ionization potential), ∆φ∼ ∆Ip·τ. (2.26)

Hereby they found a method of extracting excursion times from the phases of the emitters. Note that from equation2.15we can see that the difference of ionization potential causes altered phase build up between the generation processes of the two media during the propagation step. This is phase is then given to the emitted pho-ton, creating a phase difference between the emitted photon of the two gases. The recombination term, in equation2.23, also contributes to the phase, yet is not depen-dent on the ionization potential and so will not contribute to the phase difference. In 2013, Brizuela et al. [1] used a mixture of argon and neon to generate and for-mulated that emission of argon alters the electric field of the incoming IR pulse in such way that it boosts generation of neon. Specifically, the low order harmonincs generated from argon alters the fundamental pulse so that the short trajectories of the neon are increased. This additional effect is different from the optimized phase matching Kanei et al. [6] described and was not taken into account in their research. There only after generation, emission from both gases influence each other construc-tively and destrucconstruc-tively. Gas mixtures open up new possibilities of optimizing phase matching and so efficiency, also it offers extensions of applications of HHG, like in the fields of imaging and spectroscopy.

FIGURE2.4: A visual representation of the phase difference originat-ing from the difference in ionization potentials, taken from Kanei et

10

Chapter 3

Methods

This section will give an overview of the method and setup used to measure the modulations of the spectra of HHG in the gas mixture described in previous section. First, the experimental setup is shown with corresponding laser quantities. There-after, I will give a brief description of the code used to obtain the theoretical values which will later be compared to the experiments.

3.1

Experimental setup

The setup is divided intro three main regions; the laser, generation and detection of the signal.

3.1.1 The pulse

The laser used is the Astrella Titanium:Sapphire Amplifier from Coherent. It has a pulse repition rate of 1000 Hz, each pulse has a central wavelength of 800 nm and op-timized duration of 40 femtoseconds. The laser makes use of Chirped Pulse Ampli-fication in order to generate the femtosecond pulses. During this research problems were encountered within the compression part of the CPA, causing spatially chirped pulses. For more information about the origin and detection of spatial chirp, seeA.

3.1.2 Generation

After the laser the femtosecond pulses are then shot into a vacuum chamber, see part (2) of figure3.2. Attached to the topside of the chamber is the gas system, which consists of three separate tanks, see part (1) of figure3.2. A schematic view of the gas system is shown in figure3.1.

FIGURE3.1: A schematic view of the gas mixing system used.

The first tank is filled with argon, the second tank with krypton and the final tank is used to compose a mixture of both. To acquire a gas ratio of argon and krypton of 2:1, used in the coming experiments, with a total pressure of 60 bar, one can first fill the gas mixing tank by releasing 40 bar pressure argon from the argon tank into the gas mixing tank. This is done by opening the argon tank valve and setting the direction regulator downwards. The gas mixture is filled from below. Now the argon valve is closed and the pressure of the gas mixing tank is raised to 60 bar by releasing 60 bar krypton from the corresponding tank after opening the krypton valve. The krypton valve is closed, the direction regulator is set upwards, in direction of the gas chamber and the mixing tank valve is opened to release the 2:1 mixture into the gas chamber. The backing pressure from the mixing tank is set around 5 bar.

FIGURE3.2: A schematic overview of the setup, including gas mix-ing system (1), the generation chamber where the pulse enters and the signal is generated (2) and the detection chamber where a XUV grading diffracts the generated signal, a MCP measures the intensity

and the camera captures the signal (3).

From the gas system a gas line leads the gas into the vacuum chamber through a gas cell. Before the gas cell, a pressure flow meter is connected to the line indicating the final backing pressure into the chamber. This backing pressure can then be regulated

Chapter 3. Methods 12 by a manual valve above the meter. The gas cell has an opening in the order of a hundred micrometer through which the incoming IR pulses travel and generate the EUV pulses. A gas cell is chosen instead of a gas jet because the backing pressure given by the valve is more accurate for the cell. The pressure below a gas jet is harder to regulate as it drops rapidly.

3.1.3 Detection

The generated EUV light from the gas cell is then directed onto a EUV grating, split-ting the the photons according to their energy and wavelength. Following the grat-ing, a MCP picks up the light creating a signal, which is then captured by the camera behind it. The MCP is set to voltage difference of 3 volts between 1.55 and 4.55. To capture the signal, a total exposure time of two minutes is used.

3.2

Simulations

The simulations given in the next chapter are based on the Saddle point Strong Field Approximation within the Lewenstein model, see section2.2. The electric field is di-vided into half-cycles reflecting the characteristics of the Astrella laser described in section3.1.1. For each halfcycle the XUV emission is calculated. Using a filter, long and short trajectories can be selected, this is done by finding the peak of momen-tum distribution. In the scope of this research only the short trajectories are chosen in order to use the phase difference and ionization potential relation formulated by Kanei et al. [6], see equation2.26. For each half-cycle the code follows the method described by Vladislav et al. [10]. It finds the real part of the dipole moment using,

d(t) =Re[e−iπ/4

∑

trajectories

aionization(t)apropagation(t)arecombination(t)], (3.1)

where astep are the respective probability amplitudes. Because not all ionization phases recombine, we don’t have to sum over all ionization phases and we can just take the sum of all trajectories. The amplitudes are given by,

aionization=

 dn(tb) dt

1₂

, (3.2)

here n(tb) is the chance that an atom is ionized at time tb, it is dependent on the ionization rate of the process,

apropagation =  2π t−tb  ₍ 2Ip) 1 4 |EL(tb)| exp[−i(t−tb)Ip−iS(t)], (3.3) arecombination= q n(tr) AL(tb) −AL(t) [2Ip+{AL(tb−AL(t)}2]3 . (3.4)

The term is ALgoes by the vector potential of the electric field ELof the fundamen-tal pulse. Additionally, n(tr)is the probability that an electron recombines with the parent ion at time tr. This is done for both separate gases, each with given recombi-nation phases taken from literature. The emission for a single half-cycle, which is the product of the three amplitudes multiplied by an exponent, like given in equation

FIGURE3.3: The emission for a single half-cycle calculated using sim-ulations. The emission is equal to the product of the three amplitudes

multiplied by an exponential pre-factor

Next, all half-cycle contributions are added to find the dipole, shown in equation

3.1. A Fourier transform gives the corresponding spectrum for the separate gases. In case of the mixture the two dipoles are normalized and added to find the total dipole moment of the gas mixture. To receive the spectrum we then once again Fourier transform the total dipole moment of the mixture gas.

14

Chapter 4

Results

In this chapter the results obtained in this research project are displayed. First, to make sure we compare the experimental data correctly to the simulations, the quadratic relation of pressure against intensity is shown. Thereafter the spectrum of the generated XUV emission of the separate gases and mixture obtained in the experiments are analyzed against those of the simulations. Next, the phase differ-ences are extracted from both the spectra of the experiments and the simulations and again compared to one and other. Finally, we show the total beam profile from the experimental results and the full width half maximum per harmonic to investi-gate the overall trend of divergence control. For all experimental data two pressure regimes are shown, namely the high pressure regime; argon of 3.70 mbar, krypton of 1.85 mbar and mixture of 5.55 mbar and the low pressure regime; argon of 3.20 mbar, krypton of 1.60 mbar and mixture of 4.80 mbar. The naming does not corre-spond to different effects by the pressure but is just as characterization of the two regimes. The two pressure regimes are investigated to see if the amount interference is influenced by the pressure.

4.1

Quadratic relation

As mentioned before, it is desirable to be in the regime of optimized phase matching to make sure the collective phase of all single atom responses build up construc-tively and there is no phase mismatching. In figure 4.1 the mean argon intensity per harmonic is plotted against increasing pressure. A second order quadratic fit, a+bx+cx2, is used from pressure (mbar) 2.8 to 4.20. The bx term is included be-cause there is a threshold pressure for which HHG can be observed. The fitted data have slopes in the same order of magnitude and the quadratic fit is done with a 95% accuracy. Beyond the pressure of 4.20 mbar de-phasing happens and so in order to have generation from single atom responses the gas pressure range of argon is cho-sen below 4.20 mbar. The same was done for krypton and so it can be concluded that the two pressure regimes are within perfect phase matching.

FIGURE4.1: Intensities for given pressures in mbar. The data is fitted with a second order quadratic relation, a+bx+cx2

Chapter 4. Results 16

4.2

Intensity modulation

Below are the experimental results of the spectra of argon, krypton and a 2:1 ra-tio mixture. Using the method explained in secra-tion2.4the phase difference of the emission of the argon and krypton atoms is extracted.

4.2.1 Experimental data

In figure4.2 the experimental data and extracted phase differences are graphed of the high pressure regime. Looking at figure4.2a, harmonic order 13, 15 and 17 show clear signs of destructive interference, while harmonic order 19, 21 and 23 display constructive interference. Figure4.2bconfirms this, destructive interference corre-sponding to a phase difference of∼πand constructive interference to∼0. A

tran-sition is seen between harmonic order 17 and 19. For harmonic orders including and above 21 the extracted phase difference equals zero. This points at further effects beyond phase matching between the two emitters, which can not be explained by just constructive interference. The combined spectrum seems to be too large for just phase matching effects. This could be due to an accidental higher pressure than the relative separate gases. However, it does not seem to effect the harmonics overall as strong destructive interference is still observed.

(A) The spectra of argon, krypton and gas mixture

(ratio of 2:1)

(B) The extracted phase difference per harmonic

order of the spectra

shown on the left. FIGURE4.2: Spectrum and phase difference of the generated

harmon-ics in the high pressure regime.

To see how the modulations are effected by pressure range, the same experiments were done for lower pressures, see figure4.3. The relation between phase difference and interference effects are seen to be the same in this lower pressure region.

(A) The spectra of argon, krypton and gas mixture

(ratio of 2:1)

(B) The extracted phase difference per harmonic

order of the spectra

shown on the left. FIGURE4.3: Spectrum and phase difference of the generated

harmon-ics in the low pressure regime.

4.2.2 Simulations

As explained in more detail in section3.2, simulations were done based on the SFA Lewenstein model, see section2.2. A similar relation between phase difference and modulation is found. Like in the experimental results, the destructive interference appears to be in the plateau harmonics, while the slight constructive interference is seen in the cut off. The simulations only take into account the short trajectories.

(A) The spectra of argon, krypton and gas mixture

(ratio of 2:1)

(B) The extracted phase difference per harmonic

order of the spectra

shown on the left. FIGURE4.4: Simulated data following the method of section3.2. Only

Chapter 4. Results 18

4.3

Divergence

The results for divergence optimization are listed below. This data is taken from the same signal as the spectra from the previous section. Figure4.5shows the total normalized beam profile of the two corresponding pressure regions. In both cases a narrower beam profile is given by the gas mixture. It should be noted that for the normalized spectrum the noise ratio for argon is dominant. Narrower beams point at more contributions of the short trajectories and overlapping foci origins of the emitted long and short trajectories. This effect generally increases divergence control.

(A) Low pressure region. (B) High pressure region.

FIGURE4.5: Beam profile of the total spectrum taken for argon, kryp-ton and the the mixture (2:1) at two different pressure ranges

To investigate further, the full width half maximum (FWHM) of the normalized beam profiles per harmonic are compared for the three generation media, see fig-ure4.6.

(A) Low pressure region. (B) High pressure region.

FIGURE 4.6: Full Width Half Maximum of consecutive harmonics

taken for argon, krypton and the the mixture (2:1) at two different pressure ranges

In all cases the FWHM of the mixed spectrum is lower than that of the single gases. However, the FWHM does not give a full descriptive analysis of divergence con-trol. Divergent beams namely exists as a double gaussian beam, where the lower

amplitude gaussian beam causes a wider beam profile for lower intensities. These extensions on the lower side of the beam profile are called wings. When extracting the FWHM, these wings are not fully considered. Below we see further evidence of Divergence control in mixed. The results in figure4.7show the integrated wings of the total normalized beam profile. We cut off the the beam profile at the correspond-ing percentage value of the x axis and integrate the amount of signal underneath. 50 % corresponds to the amount of signal underneath the FWHM. In all cases the integrated wings from the separate gases are higher than that of the mixture and so points towards more divergence control in the gas mixture.

(A) Low pressure region. (B) High pressure region.

FIGURE 4.7: Integrated amount of signal in the wings of the total beam profile taken for argon, krypton and the the mixture (2:1) at

20

Chapter 5

Conclusion

During this research an analysis is given of High Harmonic Generation in mixed gases. Experimental results were obtained using argon, krypton and a 2:1 mixture of both. Also, results from simulations based on the Strong Field Approximation were gathered to compare the spectra. From previous literature relations for modulations in mixed gases were taken and tested for the given gases. Additionally, divergence control was tested and found for the gas mixture. Experimental results were cap-tured in two different pressure ranges.

After comparing the combined spectrum of the mixture to the single gases of rela-tive equal pressures strong interferences were detected. In the plateau harmonics, namely harmonic order 13 to 17, destructive interference was seen, with phase dif-ferences close or equal to π. In the cutoff harmonics, namely harmonic order 19 to 23, we found constructive interference with extracted phase differences of close or equal to 0. These results were confirmed by the simulations, however, the effect of the constructive interference was lower than in the experimental results. This, to-gether with phase difference values of exactly zero and π could point at additional effects such as seeding. Next, total beam profiles as well as beam profiles of separate harmonics of the same signal as the spectrum were analysed of the single gases and mixture gas. In the mixture lower FWHM were calculated for all harmonics. Signal under the wings were also found to be lower for the mixture indicating an increase in divergence control.

Appendix A

Optical Auto-Correlation

A.1

Chirped pulse amplification (CPA)

The table-top generation of femtosecond pulses is over the past decade revolution-ized by the introduction of Chirped Pulse Amplification (CPA). This technique, driv-ing High Harmonic Generation, is based on compression of a stretched pulse with the aid of two parallel aligned gratings. These diffraction gratings deflect photons with an angle based on the wavelength of the photons and the spacing between the gratings, following Huygens–Fresnel principle. To generate high energy femtosec-ond pulses this is implemented as follows, first an initial short pulse is stretched using two anti parallel gratings, usually with a factor of a thousand. Beam This second order chirped low power pulse is then amplified to form a stretched high energy pulse. Finally, the pulse enters the compressor, which consists of two parallel aligned gratings, and forms high energy compressed femtosecond pulse.

A.2

Spatial Chirp

Due to the spatial and angular manipulation of the wave front and frequency dis-tribution, pulses generated using compressors are often prone to spatial and angu-lar chirp. This chirp is characterized by temporal delay between different spectral components of the pulse. Regular pulses have a constant frequency throughout the whole pulse, while for a chirped pulse this can differ. This manipulation of fre-quency can be linear (second order spatial chirp) or exponential (third order spatial chirp). In the case of second order chirp the frequency change against time correla-tion is linear, the exponential chirp is described similarly.

In the case of CPA, the angular dispersion caused by the first grating is resolved by the anti parallel aligned second grating. However, almost inevitable small misalign-ment’s can leave some spatial chirp in the output beam.

Appendix A. Optical Auto-Correlation 22

A.3

Mach-Zehnder-type Interferometer

To check whether there is spatial chirp you can build a Mach-Zehnder-type Interfer-ometer shown in figureA.1. This setup consists of two mirrors (dark lines) and two beam splitters (grey lines). With no spatial chirp these beams should interfere and create straight fringes as shows in the image. However, when one of the mirrors is replaced by a set of two mirrors with a respective angle of 45 degrees and this set of mirrors is pulled away from the setup, such that the path of the top beam path changes slightly you can start to see spatial chirp effects. If spatial chirp is present in your beam than one side of the fringes start to fade. This was check was done on our setup, yet no clear spatial could be seen.

FIGUREA.1: A Mach-Zehnder-type Interferometer to check spatial chirp, adjusted from [4]

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[3] eg PA Franken et al. “Generation of optical harmonics”. In: Physical Review Letters 7.4 (1961), p. 118.

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