A two-color interferometer for high-order harmonic generation

Abstract

High-order harmonic generation (HHG) is a highly nonlinear optical process in which extreme ultraviolet (XUV) radiation is generated due to the interaction of an intense laser field with atoms. This process enables the production of high harmonics, which would not be achieved with standard nonlinear media. The utilization of two-color laser fields (formed by combining a fundamental field with one of its harmonics) has revealed that the electron trajectories involved in the process can be manipulated, resulting in the enhancement of High Harmonic Generation yield and overall efficiency. In particular, combinations involving second or third harmonics with the fundamental field, have been shown to significantly enhance the High Harmonic Generation efficiency or selectively enhance the High Harmonic Generation flux of specific harmonics, by fine tuning of the relative phase and intensity ratio of the two fields.

This thesis presents the design and construction of a setup based on an Ytterbium laser system, in which the generation of third harmonic was accomplished, with a 15% conversion efficiency. Initially, theoretical comparisons between direct third-harmonic generation (THG) and a cascaded approach, which is second-harmonic generation (SHG) followed by sum-frequency generation (SFG), indicated that the latter results in a higher conversion efficiency for third harmonic. Based on this result, an interferometric setup was implemented, and further limitations that could decrease the efficiency of Third Harmonic Generation were investigated. For instance, the impact of spatial and temporal walk-off, common limitations following the cascaded process, were analyzed to further optimize the system. Power scans were performed to identify the optimal intensity ratios between the second harmonic and residual IR fields, leading to maximized Third Harmonic Generation efficiency. The presented setup is an early version of a final system in which all three beams (fundamental, second harmonic, and third harmonic) will be an output with controllable relative powers. This is important for two-color High Harmonic Generation experiments where the flexibility to adjust the intensity ratio between the fields is needed.

To study how waveform shaping, of the incident laser field, impacts High Harmonic Generation, simulations were performed based on the three-step model, using a combination of the fundamental and its third harmonic field. These simulations investigated the impact that the relative phase and intensity ratio of the two fields have upon the electron trajectories and resulting harmonic yield. The focus of the simulations was on identifying waveform configurations that lead to flatter return energy curves, which suggests that a large number of electrons are ‘forced’ to release the same kinetic energy, thus enhancing the yield at specific harmonic orders.

Popular Abstract

Light is an electromagnetic wave, and has some properties that remain unchanged in the linear regime. One of those properties is the frequency, a parameter which determines how fast the electric field oscillates, and ultimately even the color of the light. So, when light travels through a piece of glass, maybe it will come out less intense (because some of it may be absorbed), but in most cases, we do not expect it to have a different color. However, if we have strong enough light, then we can enter the nonlinear regime, where a material will not respond ‘linearly’, and we can even produce light with different frequencies.

To study the nonlinear regime, we can use ultrafast optics, where we have access to laser pulses that are not only incredibly short (on the order of femtoseconds 10-15 second), but also extremely intense. The shorter the pulse, the higher its peak intensity for the same energy. When a weak light hits a material, it causes the electrons to jiggle gently, and the response of the material is linear. However, if the light is strong enough, then it shakes the electrons of the material so violently that the electrons will emit light with a new frequency. We can think of the medium like a box, where on one end we shoot light, for instance with a red color, and at the output the light will be blue! This box generates new frequencies of light, integer multiples of the original one, which is also called harmonic generation. The medium can emit light twice the frequency of the input wave (Second Harmonic Generation), three times the frequency of the input (Third Harmonic Generation) and so on. In principle, all materials can behave in such a way with a strong enough light. But, in practice, to observe those new harmonics, we need ‘non-ordinary’ materials, that will make the process more efficient, also known as nonlinear media. However, nonlinear materials become less effective as we push toward higher harmonic orders. For example, it becomes increasingly difficult to observe the third harmonic compared to the second. But scientists found a clever workaround. Since the usual rules of linear optics don’t apply in the nonlinear regime, they asked: what if we mix light with light? And that’s exactly what they did. By overlapping different light frequencies in a nonlinear material, they were able to generate new frequencies through a process called wave mixing. In fact, Second Harmonic Generation is one of the simplest forms of this (where two photons of the same frequency combine to produce light at twice the frequency). For Third Harmonic Generation we can combine three photons of the same frequency, but this process is inefficient. Thus, the combination of two beams with different frequencies will boost the efficiency of Third Harmonic Generation and make otherwise hard-to-detect harmonics more accessible. This process is described as \omega\ +\ 2\omega\ \rightarrow\ 3\omega, and since the end product is a higher frequency produced by the sum of two lower frequencies, then it is known as Sum Frequency Generation. However, reaching even higher harmonics by using those techniques becomes impossible, because the intensity of harmonics drops as the harmonic order increases. This is described by this scaling law I_q\ =\ I^q, where q is the harmonic order, and I is the intensity. We are limited in low-order harmonics, unless we change the conditions once again. This is what scientists did, instead of a bulk nonlinear medium, they used gases, and focused extremely intense lasers onto them. This is when we enter the highly nonlinear regime.

In the highly nonlinear regime, electrons no longer get excited through typical ionization, where they have enough energy to surpass the energy dictated by the Coulomb barrier. You can think of the Coulomb barrier as the invisible wall that holds the electron close to the nucleus. It exists because the positively charged nucleus pulls on the negatively charged electron, and it takes a certain amount of energy for the electron to break free. But in this strong field regime, the strongly focused laser bends that wall, and it allows for the electron to tunnel through and break free. Afterwards, the electron starts moving, and follows the strong electric field. Then, at just the right moment, the electron may find itself near the nucleus again. If the electric field points in the right direction, the electron may recombine with its parent ion. But since it was accelerated while it was free, it gained energy, and that energy needs to go somewhere. So, when the electron recombines, it releases that energy in the form of light. That process happens incredibly fast, specifically it happens every half of the laser cycle. But the laser pulse is not just one cycle long, it actually contains several cycles, so those three steps will happen multiple times. Each time, the electron may be freed and recombine at different points of the laser cycle, which means that a different amount of kinetic energy will be released in the form of light. The total light that we detect is the desired higher-order harmonics. The higher the kinetic energy of the recombining electrons, the higher the frequency of the emitted light, and thus the higher the harmonic order, giving us the possibility of detecting light, which falls into the extreme ultraviolet range (XUV). This process highly depends on the laser field and the gas that we are using. It is known as High Harmonic Generation. However, this process is not very efficient on its own. Only a small fraction of electrons follow the right path and recombine. So, to improve the efficiency of the process, we can customize the laser field. By changing the shape or properties of the incident laser field, we can manipulate how the electrons move, guiding more of them to recombine at the right time. One way to shape the total field is by combining two different laser fields, the main (with the fundamental frequency) and one of its low harmonics, such as the second or the third harmonic. We can make different shapes of the total field, by adjusting their relative intensities, how much of the fundamental and third harmonic is contributing to the resultant total field. But another key parameter is the relative phase between the two fields, which means how ‘in sync’ the two waves are relative to each other. Careful adjustments of those parameters, lead to specific field shapes that can lets us selectively enhance the generation of certain harmonics. And why do we care so much about these harmonics? Because if those harmonics are in phase, they can give rise to attosecond pulses, flashes of light that last only in the range of a billionth of a billionth of a second. These pulses are so incredibly short that they can actually catch the motion of electrons as it happens. What once seemed impossible (observing electron dynamics in real time) is now within reach thanks to these ultra-short bursts of light.
In this study, we built an experimental setup that allows the generation of a multi-color, femtosecond laser field for high-order harmonic generation. Third harmonic light in the UV was generated with state-of-the-art efficiency. More importantly, the setup serves as an initial platform that outputs fundamental, second harmonic and third harmonic pulses at the same time. With this combination, we can start exploring different shapes of the electric field to better manipulate the electron’s motion. The ultimate goal is to use this control to force more electrons to emit selected high-order harmonics, and in turn, boost the amount of extreme ultraviolet light we can generate.

• author: Pefani, Eleni
• supervisor: Anne-Lise Viotti; Melvin Redon
• organization: Atomic Physics; Department of Physics
• course: PHYM03 20251
• year: 2025
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• keywords: Laser, Nonlinear Optics, High Harmonic Generation, Third Harmonic Generation, Interferometer, two-color HHG, Femtosecond
• publication/series: Lund Reports on Atomic Physics (LRAP)
• report number: LRAP 609
• language: English
• id: 9188018
• date added to LUP: 2025-05-06 10:28:14
• date last changed: 2025-05-06 10:28:14