Phase-matching of high harmonics – general considerations | Lecture 23

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Summary

In this lecture, the final installment of the course on strong field laser physics, phase matching in high harmonic generation is explored. The lecture delves into the complexities of harmonics generation on both macroscopic and microscopic levels, emphasizing the interesting nature of microscopic effects. Key concepts discussed include phase matching, voxel definition, constructive interference, and the geometric phase. The session also covers the influence of gas density, refractive index, and coherence length on phase matching, while highlighting the importance of the polarization phase in harmonics emission. By understanding the harmonic q's accumulated phase, polarization phase, and atomic phase, students can comprehend how to achieve perfect phase matching.

Highlights

Exploring the final lecture on strong field laser physics, focusing on phase matching in high harmonic generation. 📚
Understanding the role of microscopic effects in the emission of high harmonics. 🔬
Discussion on phase matching, voxel concept, and constructive interference in harmonics generation. 🎯
Introduction to geometric phase and its importance in calculating the polarization phase. 📈
Exploring refractive index and coherence length, and their contributions to phase matching. 🔍

Key Takeaways

The lecture unravels the complexities of phase matching in high harmonic generation and its reliance on both macroscopic and microscopic effects. 🚀
Geometric phase and polarization phase are crucial in understanding harmonics emission and achieving phase matching. 🔄
Refractive index and coherence length play fundamental roles in the phase matching of harmonics. 🔬

Overview

Welcome to the grand finale of the strong field laser physics series! In this session, we take a deeper dive into high harmonic generation, dissecting the intricate dance of phase matching on both micro and macro levels. It turns out these 'tiny' microscopic effects hold the keys to intriguing harmonic mysteries!

Let's phase it, understanding phase matching is no small feat. Our journey kicked off with voxel definition—a fancy term for 'volume pixels.' The lecture then weaved through the complexities of constructive interference, leading us into the realm of geometric phase, where wavelengths and refractive viabilities guide us!

Precision is key! By the end of this engaging session, we decoded the atomic phase, and uncovered how refractive indices and coherence lengths can make or break the harmonic show. With our newfound insights, we're ready to tackle phase mismatches and lead harmonics to their blazing glory!

Chapters

00:00 - 03:00: Introduction to High Harmonic Generation The chapter introduces the topic of high harmonic generation within the broader theme of strong field laser physics. It emphasizes the need to examine both macroscopic and microscopic effects, highlighting the particular interest in the microscopic effects.
03:00 - 20:00: Phase Matching of High Harmonics The chapter begins with a focus on understanding the phase matching of high harmonics, illustrated through the use of a laser beam. The beam generates high harmonics, and the discussion suggests that the target gas used is likely a common scenario in these experiments. This sets the stage for further exploration of phase matching in high harmonic generation.
20:00 - 35:00: Geometric and Excitation Phases The chapter titled 'Geometric and Excitation Phases' discusses the production of high harmonics in a gas. It explains that the extension of the gas is larger than the focus area, which is not confined to a small part of the focus. This spatial extension allows for the generation of harmonics throughout the focus area. The harmonics are produced at various points within the focus, with a probable increase in generation at the center of the focus.
35:00 - 47:00: Factors Contributing to Phase Matching The chapter titled 'Factors Contributing to Phase Matching' discusses the concept of phase matching in the context of elementary emitters or rays. It raises the question of whether these elements, when generated, arrive in phase in the far field. This is compared to problems encountered in nonlinear optics, such as in a crystal, where the alignment of phases is similarly questioned.

Phase-matching of high harmonics – general considerations | Lecture 23 Transcription

00:00 - 00:30 so welcome to the final lecture of this course strong field laser physics and other second laser physics what we want to do today is to look at high harmonic generation again but this time on the macroscopic effect well actually it will turn out that we also need to deal with the microscopic microscopic effects and as you might expect these are the particularly interesting ones
00:30 - 01:00 um so we will look at phase matching of high harmonics um well let's start with a general description of the problem so what i have here already is the focus of a laser beam say of the laser beam that is that generates the high harmonics right and then you could assume um and this is actually usually the case that our target gas so the gas that
01:00 - 01:30 produces the high harmonics that this that the extension of that gas is larger than the focus at least it's not confined to a small fraction of the focus regularly so this means that we have we generate at least in principle throughout this focus we generate harmonics right so we generate harmonics here we generate a little bit of harmonics here probably more there in the center of the focus where the
01:30 - 02:00 intensity is highest and so on and so forth and the question is of course whether all these let's say elementary emitters or elementary rays that are generated here whether they arrive in phase in the far field right so this is pretty much the same thing um so the same kind of problems that we would have in nonlinear optics there it would be a crystal and we would ask whether all the
02:00 - 02:30 emission in the different regions of the crystal whether they interfere constructively this is phase matching right here this thing is a little bit well it's a little bit special um okay so what we do in order to address that would be that we say that we define here well a small volume i call it a voxel a volume pixel so a voxel
02:30 - 03:00 okay so let's write down a few remarks and we consider here so the emission of this voxel for for a certain harmonic say the q is harmonic let's write that down we consider a specific harmonic let's say the q symmonic
03:00 - 03:30 and now in each voxel some queues some well some cute harmonic radiation will be generated so in each voxel
03:30 - 04:00 some cues harmonic radiation and i would call it deep psi q yeah so q for the q's harmonic and similar to diffraction theory so i would write here r prime so the voxel has infinite decimal volume and therefore i
04:00 - 04:30 write an infinite decimal a wave here so this will be generated will be generated yeah and um the question now is with the interfere whether all the to a dq no d psi q whether all of them interfere yeah
04:30 - 05:00 so the question is well they interfere but the questions whether they interfere constructively so the question is is whether all d psi q interfere
05:00 - 05:30 constructively for all r prime yeah so for all positions in um in the focus okay well there is fortunately one thing that we know for sure we know that that this elementary wave well is a wave and therefore its phase develops as the
05:30 - 06:00 phase of a wave develops yeah so [Music] we know the phase development the phase development of d q a deep psi q [Music]
06:00 - 06:30 as it is a wave and so what we know is that this phase phi of this high harmonic q at some point at some distant point say or screen or something like that right that is given by omega by q times omega times t right so omega this is the fundamental
06:30 - 07:00 um here now so this here is the fundamental of the laser right and now we have to write down also the other part so i split off the refractive index at the q's harmonic so the refractive index
07:00 - 07:30 in the xuv i call this nq and then we have the wave vector this is just 2 pi over lambda q and then we would have r minus r prime right so something like this yeah so this here would be the point of observation of observation this here would be the position of the
07:30 - 08:00 voxel this here would be the refractive index yeah and this here is 2 pi over lambda q so which means lambda of the laser divided by q right okay so that's equation one
08:00 - 08:30 to make things simple we confine ourselves to a position on the screen which is on the optical axis yeah so somewhere downstream here so if the laser would come from this direction we would look well a meter behind or something like
08:30 - 09:00 that right and um just see what what we would would get yeah and on this screen we would say that the phase for each of these um yeah so for each of these elementary waves should be the same then we have constructive interference okay let's write that down
09:00 - 09:30 so now we could look at a point r on the screen so for simplicity
09:30 - 10:00 a point on the optical axis so in the center of the screen for simply simplicity a point on the optical axis that generalization to any other point is trivial we'll briefly come to that um another for this point um all
10:00 - 10:30 d psi q of yeah should arrive as i could write here of r prime should arrive with the same phase this is what perfect phase matching would demand
10:30 - 11:00 well and any deviation would be of course a mismatch and then from that point we could follow back each ray right so each ray back to where the voxel is and then say well which is the phase with which i need to generate actually
11:00 - 11:30 the harmonic such that after this propagation they arrive all with the same phase right well just to motivate this a little bit more so what you have to consider is that we have well that we might have different gas density in this volume in particular we will definitely also have three electrons that produce a refractive index so we have an inhomogeneous medium here right and um
11:30 - 12:00 therefore the different rays that we propagate back yeah so for a voxel say here and another box there and there they pick up um different faces now and the question is um so to put it the other way around at which phase do i need to produce the harmonic at the different places such that at a
12:00 - 12:30 distant point on the screen say they arrive with the same face yeah um okay um so then we can follow all rays back
12:30 - 13:00 to the voxel where they were created and determine the phase
13:00 - 13:30 accumulated on this path and therefore this gives us the information um with which phase
13:30 - 14:00 the harmonic the q harmonic in this case should be created such that all rays arrive on the screen with the same face
14:00 - 14:30 the same phase and if we want to if we want to say this in mathematical form then we would say that that this phase with which they should be graded we call it the polarization phase so it's a polarization that
14:30 - 15:00 causes the emission of the harmonic um yeah so that this that the difference between this accumulated phase on the during the propagation that this shouldn't vary with with position yeah okay so this is the phase
15:00 - 15:30 um with which the q's harmonic well oh yeah so i wrote that actually already so this face yeah so let's write it like this so this phase
15:30 - 16:00 is called phi polarization so and phase matching demands phase matching demands simply that the derivative and if we look just on the optical axis then it's just the derivative with respect to z um of phi polarization minus phi high harmonic generation therefore the q's harmonic i could add
16:00 - 16:30 but sometimes i just leave that out and so this is the condition yeah and um now we can generalize this to to off axis positions and also usually actually we we so most of the time we just consider points that are close to the axis but of course it can be generalized quite trivially
16:30 - 17:00 and also for off axis positions for off axis directions i should say directions we expect of course that that the phases run out
17:00 - 17:30 run out of phase more quickly so if you that's the reason why a beam propagates on straight lines yeah because the molecules it excites there they their radiation is is out of phase for all directions different than the beam direction yeah so quite obviously so for up for off axis directions we [Music]
17:30 - 18:00 expect um for harmonies it's not always fulfilled i have to say but we may come to that later we expect that the race from different voxels run out of phase more quickly
18:00 - 18:30 no same as in usual optics even linear optics although in general the phase matching conditions condition would be like this you know it's a gradient the phase mismatch is the gradient of this difference of these faces of these two phases
18:30 - 19:00 and this delta k is referred to as the phase mismatch has mismatch no just the same things yeah in the same spirit as in non-linear optics and um you may we may also introduce the coherence length as it's the coherence length of nonlinear optics you know that
19:00 - 19:30 in regular optics there's also a thing a quantity called coherence length which is something completely different another coherence length so when we get out of phase then the elementary waves start to interfere destructively so they lose their well in some sense their coherence this is why it's called coherence links i don't like this name very much
19:30 - 20:00 but it's called like that so what can i do okay so well this means that with this kind of analysis we are almost pretty much done in principle yeah so um if we assume that we know um the the refractive index of the medium yeah then actually pretty much as
20:00 - 20:30 everything is said the only thing that remains to be known is the the polarization phase yeah okay yeah so assuming that we know the refractive index of the medium as a of the gas
20:30 - 21:00 the only problem well this is of course that this only problem sounds [Music] sounds sounds harmless but of course the only problem is usually the the thing that makes or causes the work so the only problem is to compute
21:00 - 21:30 the polarization phase yeah and once this polarization phase is known the macroscopic field can easily well perhaps it's not easy but it's at least straightforward so can
21:30 - 22:00 [Music] be computed in a straightforward way now we just use the superposition principle now we add them according to the amplitude and to the face
22:00 - 22:30 okay so now we need to look at the phase of the of the excitation yeah so um you know um the problem is um that we have a focused beam right and if we look at the focused beam then um the it's different from a plane wave now
22:30 - 23:00 this is the essential point a focus beam is different from a from a plane wave and we call this the geometric phase so this is 8.2 chapter 8.2 the geometric phase
23:00 - 23:30 no the polarization phase is the phase of the polarization yeah so the laser excites the polarization yeah and uh excites the the atoms right and uh in this way some polarization is created right but of course um the if you change the phase of the laser also that the the the phase of the polarization
23:30 - 24:00 will change right so uh definitely the phase of the laser and thus the geometric phase will have a major impact yeah so we need to add this to the um so this is part of the of the of the phase of the polarization okay the geometric phase
24:00 - 24:30 yeah so first remark goes exactly to your question and also the polarization phase will definitely contain the phase of the exciting wave yes the phase of the exciting wave
24:30 - 25:00 which is of course the laser the fundamental of the exciting wave which means the phase of the fundamental and the fundamental is a focus beam
25:00 - 25:30 um and um its phase is of course the phase of a plane wave plus some geometric phase term its phase is the phase of a plane wave plus some geometric phase which is well known
25:30 - 26:00 at least people believe that it's well known um and this phase i call phi geometric and it depends of course on um z prime and r prime right so um r prime is now the radial coordinate and that is the coordinate on the on the
26:00 - 26:30 axis okay yeah so this means that the the excitation polarization this would be written as omega times t now so omega is the frequency of the fundamental yeah so frequency of fundamental as before and then [Music] we have of course
26:30 - 27:00 n of omega so the refractive index at typically visible or infrared wavelengths times now the k vector yeah so given as 2 pi over lambda say for titanium sapphire laser 2 pi over well 800 nanometers and then we have here our prime minus this geometrical first
27:00 - 27:30 phase term yeah so something like that now in the following i will drop these primes because because i'm get tired of writing these primes all the time okay as i said for um at least for monochromatic gaussian beams yeah you would immediately say well when is a beam gaussian perfectly yeah and in
27:30 - 28:00 particular you would say i need short pulses they are never monochromatic but nevertheless let's do this assumption at least as long as the pulse is not too short it's actually a reasonable approximation for a monochromatic gaussian beam
28:00 - 28:30 the geometrical phase is phi geometrical we have the arcus tongans of z over z0 which is the confocal parameter and um yeah so you see an arc tank tangent like behavior of the phase
28:30 - 29:00 so it changes by the phase changes as we go through the focus by 180 degrees right there's actually also something that's relevant for regular nonlinear optics but the phase fronts they are also curved right so we have one plus set zero over z squared um yeah so it looks like that yeah
29:00 - 29:30 equation four yep and the first term this is the well-known known gui phase so where the first term is the well-known
29:30 - 30:00 gooey phase well a little anecdote here at this point as a here we first have gui um a french scientist yeah and this is why he is pronounced that way um anthony siegmann a famous laser physicist who wrote a famous book on lasers yeah so here you
30:00 - 30:30 see a old copy here uh he introduced an alternative writing of of gui with g-u-o-y yeah and well and everybody who read this book and there were many physicists reading this book followed his example and also wrote him the wrong the wrong way so he actually came once when i was still a
30:30 - 31:00 phd student he came to garching and to visit the max planck institute of quantum optics and i had to explain to him my poster he was a very friendly man and he said well i'm delighted that you read my book and i said what and he said yeah you write it the same way then i write it wrote it but it's wrong so this is how i i learned
31:00 - 31:30 that you should write it this way i'm not sure whether a later edition of this book whether it was corrected or not okay a little anecdote um yeah so we can look at uh the phase distribution yeah so this is uh here you see um the contour lines here they are lines of equal face right looks
31:30 - 32:00 quite funny the phase distribution in a gaussian phase but if you just take the cut through this here then you get this curve here yeah so this holds for monochromatic beams um if we have a few cycle laser beam then things get complicated right because each of these colors and if it's a few cycle pulse then it can span an entire optical octave so basically span the entire
32:00 - 32:30 visible range right and each of these frequencies is focused if you use a focusing element and you need to use one each of them is focuses in a different way right so the blue colors they will have a smaller focus than the red colors and now of course it depends on how on how the different colors are distributed in the laser beam
32:30 - 33:00 right so when you generate you you all the time you deal with a with a very complex situation so typically you may use a fiber compressor right then at the exit of the fiber the diameter of all colors is probably pretty much the same right and you would focus them um then all of them have a different divergence there's a really a lot of fun yeah complicated situation um yeah so and once we investigated this
33:00 - 33:30 together with the group of uh peter hommelhoff in in ellen and we took a near field probe here to measure um to measure the coil phase in a certain yeah for a certain fuel cycle laser beam and you see that it doesn't behave it doesn't behave like a gauche like so this here would be the arkansas uh aqueous tangents this uh the dash curve would be the koi face
33:30 - 34:00 right and it deviates significantly right okay well um this is still in the vacuum yeah so this is um so we focus this in actually high ultra high vacuum operators so it has nothing to do with this medium
34:00 - 34:30 yeah well because uh so the question is why is so far away from the phase um because the the bandwidth is so so broad yeah so we use the four or five femtosecond laser pulses and this means that um that the blue that we had almost an entire optical octave right so the the
34:30 - 35:00 the wavelengths at one end is is half the wavelength at the other end and now if you look at um yeah so you know the formulas for for the beamed waist it depends on uh on the wavelengths right so there are strong effects of that well
35:00 - 35:30 now it depends on how uh you could still say that that this is similar to this dashed curve right so actually the situation is is even more complex it depends on which on which distribution of divergences and beam diameters and so on you have in your beam yeah so the koi phase or this this focal phase can also have this behavior then it's really
35:30 - 36:00 different okay but to produce such a distribution wouldn't be so so easy okay well um [Music] yeah so um one important point to make here is that the fundamental wave is not only in homogeneous with respect to the phase
36:00 - 36:30 but of of course also with respect to the intensity right and this causes um and we know that um that the processes inside inside the atoms that go on when high harmonics are produced that they do depend on the intensity um [Music] a specific harmonic yes so if you look
36:30 - 37:00 at a specific harmonic um say if you look at the 25th harmonic right that is produced at different intensities then the trajectories that led to their production are different yeah okay so um let's write that down so the
37:00 - 37:30 fundamental wave is also inhomogeneous in terms of intensity that's a trivial remark but the consequence is a given harmonic
37:30 - 38:00 a given harmonic produced at different intensity will originate
38:00 - 38:30 and from different from different trajectories and so in particular different t0 and t1 so different times at which these trajectories were born and at which they recombine yeah
38:30 - 39:00 so and this gives rise to another phase factor or to another phase term not factor phase term namely um the polarization um due to the emission time right now this means that the polarization phase this is the sum of the excitation
39:00 - 39:30 plus the polarization for the emission and that's often called the atomic phase and the atomic phase is kind of the most interesting ones if we speak about strong feed laser physics
39:30 - 40:00 another everything else is from the viewpoint of of strong field laser physics [Music] trivial more or less but the atomic phase is not one has to say that the atomic phase is often not the dominating phase it's not the dominating contribution to phase match to phase matching but i would claim it's
40:00 - 40:30 the most interesting one okay at this point i would say we do an in the interim review yeah so kind of a interim summary and just list which factors contribute to phase matching and then we'll have a small break and continue afterwards so an interim summary
40:30 - 41:00 um [Music] so there are four contributions there are four contributions to phase matching of high harmonics yeah so the first one would be the refractive index
41:00 - 41:30 of causes of the gas that we are using then the refractive index of the free electrons then the geometric phase
41:30 - 42:00 well the coil phase in the simplest case and the atomic phase and as i said the first three items they are trivial in the sense that they are textbook knowledge and can be looked up in the internet right as of the refractive index for example
42:00 - 42:30 and here i have a resource for you now so these are the so-called hank tables um so if you just google hank tables then you would end up at this uh a webpage right a fairly old style webpage but nevertheless very informative extremely informative and here you find the link index of refraction for a compound material
42:30 - 43:00 it also works for for a uniform material right and if you click on that then you get a mask like this and here i entered already something yeah so argon at a density of 1.6 gram per cubic centimeter that's of course a lot right but actually i tried to enter 1.6 10 to the minus six gram per cubic centimeter and doesn't
43:00 - 43:30 like that but of course i can divide um through ten to the six uh or the results right and then you can you can type in the range in which you want to know it and can request a plot or a text file right and if we look at the plot then you would get something like this yeah so we get the delta which is the refractive index or
43:30 - 44:00 the difference of the refractive index with respect to one and so one minus delta is the index of refraction right so the real part right so this is this blue curve here and beta is proportional or um yeah so it describes the the absorption and we'll come to the absorption into course of this lecture
44:00 - 44:30 okay yeah either we write this to our summary so the first three items are trivial this is not a friendly word but it has a refractive index
44:30 - 45:00 we would go to the hanky tables then [Music] the three electrons we would take a textbook and we would find that the refractive index of the three electrons this is of course basically given by the plasma frequency or in other words by the density of free
45:00 - 45:30 electrons another the refractive index is a function of omega and the formula for the um for the plasma frequency is 4 pi e square density of the electrons divided by the mass of the electrons omega squared still
45:30 - 46:00 so that's here is density of electrons of free electrons geometric phase at least for an approximation the phase i have to pay attention the gooey face i should
46:00 - 46:30 say yeah so he's a frenchman gooey face and the atomic face well see below see later right and we have to balance these four contributions i have a slide for that that shows this balance i've stolen this figure
46:30 - 47:00 from the dissertation of robot class who wrote a extremely beautiful work on that and so we he calls it the intrinsic phase yeah what i call the atomic phase then this is the refractive index on of the atoms as a in general this gives a delta k smaller than one and then we have the phase and the phase due to the three electrons which
47:00 - 47:30 give a positive contribution right and if you balance it in the right way then you can achieve delta k equal to 0 at least in some sub volume of the focal volume yeah and we'll analyze this in the next lecture in a little bit more detail so see you in five minutes or three minutes or whatever thank you