Method of laser acceleration of electrons and device for its implementation. Fifty years of laser. A new step - an accelerator on the table Difficulties of accelerator physics

Secondary beams

Modern experiments widely use secondary particle beams, which are created after the interaction of a primary accelerated particle beam with a target. Using electromagnetic separators and collimators, particles can be separated from the huge number of particles formed on the target certain type and a certain impulse. In nuclear physics, this method produces secondary beams of radioactive nuclei, the lifetime of which can be several milliseconds. Similarly, secondary beams of p- and K-mesons can be obtained. Secondary beams of p-mesons can be used to form neutrino beams, which are obtained during the decay of p-mesons:

p->m- +n m, p+>m+ + nm.

A pure neutrino beam can be obtained by filtering the resulting particles through a thick absorber.

Laser acceleration of electrons

The idea of ​​using lasers to accelerate electrons in plasma was put forward in 1979 by American scientists. For short laser pulses, the first analytical studies were published in 1987 and 1988. In fact, laser acceleration of electrons in plasma is very close to the so-called collective electron acceleration method, which was developed over many years at the Kharkov Institute of Physics and Technology under the leadership of Ya.B. Fainberg. You can read about the problems faced by traditional vacuum accelerator technology and about collective methods of acceleration in plasma in an article previously published in the journal Nature.

Rice. 57

The dotted line shows lines of reduced electron density, the solid line shows lines of increased electron density. The arrow shows the direction of propagation of the laser pulse.

As applied to short laser pulses, the acceleration of electrons in plasma can be schematically represented as follows. Propagating in the plasma, the pulse pushes electrons out of the region where it is located. this moment is located (Fig. 3). In addition to the forces from the momentum, the electrons are subject to an electric field from the plasma ions, which can be considered stationary due to their greater mass. After the pulse has left this region, the electrons are only affected by the charge separation field, which tends to return the electrons to their original position. Having accelerated in this field, the electrons overshoot their initial position and begin to oscillate relative to the ions at the so-called plasma frequency. Since the pulse runs through the plasma and constantly pushes out those electrons that meet in its path, it always starts plasma oscillations behind it. Moreover, the initial phase of these oscillations is different at different points along the pulse path. As a result, a charge separation wave is excited, the phase of which propagates through the plasma at the speed of the pulse (the so-called wake wave, Fig. 4). The electric field of this wave in one half of the period is directed in the direction of propagation of the pulse, and in the other half of the period - towards the direction of propagation of the pulse. If an electron with an initial speed equal to the speed of the pulse is placed in that region of the plasma wave where the force acting on it from the electric field is directed in the direction of its movement, then the electron, moving along with the wave, will begin to accelerate. Such an accelerator is called a “wake-wave accelerator.” For relativistic particles whose speed is close to the speed of light, even a small increase in speed corresponds to a large increase in their energy. As a result of acceleration, the energy of the electron can increase significantly.

Rice. 58 - Perturbation of the electron density in a wake wave excited by a laser pulse with a duration of 30 fs and a power of ~30 TW in a plasma with a density of 2.2·1018 cm-3. Along the vertical axis is the radial coordinate measured from the pulse axis. On the horizontal axis - time after the laser pulse passes through a given point

Experiments carried out in France showed that the mechanism of electron acceleration described above is actually implemented. But the resulting increase in electron energy turned out to be insignificant due to the very short length at which this acceleration occurred.

At first it was believed that laser pulses with a duration close to the period of plasma oscillations were best suited for exciting wake waves, while longer pulses were not suitable for this purpose. But numerical calculations and subsequent experiments showed that this is not so. A laser pulse, the length of which significantly exceeds the length of the plasma wave, and the power exceeds a certain value, changes its shape during propagation in the plasma (Fig. 5). First, a modulation of its amplitude occurs, and then it is divided into a sequence of shorter pulses with a repetition period equal to the plasma period. This effect is called pulse self-modulation. A resonance occurs between the sequence of short pulses and plasma oscillations. Each subsequent short pulse increases the amplitude of the wake wave that was excited by the first short pulse. As a result, already inside the laser pulse the field of the plasma wave becomes very large and reaches 109 V/cm. Some of the plasma electrons are captured in the plasma wave. They begin to move with the wave and are accelerated to an energy of the order of 100 MeV over a length of several millimeters.

Rice. 59

On the initial pulse with intensity smoothly varying in space (left figure), amplitude modulation first appears (middle figure), and then it is divided into a chain of pulses of short length (right figure), the distance between which is equal to the plasma wavelength lp.

Experiments carried out in France, the USA, Japan, and England showed that in the self-modulation mode the maximum energy of accelerated electrons is quite high, but the energy spectrum is very wide, which is a disadvantage from the point of view of possible applications.

In 2004, almost simultaneously, three experimental groups discovered a new mode of electron acceleration, in which the energy reached 250 MeV and the energy spectrum was quite narrow. In this mode, the laser radiation intensity exceeded 1019 W/cm2, and the pulse length was close to the plasma wavelength. The high-frequency pressure forces acting on the plasma electrons were so great that immediately behind the pulse there appeared an almost spherical region in which there were practically no electrons. This area began to be called the bubble, and the acceleration mode itself became the bubble mode (Fig. 6). A certain amount of plasma electrons were captured from the plasma into this region, which were accelerated.

At present, significant experimental and theoretical material has already been accumulated, sufficient for the design and construction of a laser accelerator with an electron energy of more than 1000 MeV. Now several such projects are close to implementation.

Rice. 60 - Laser pulse propagation in bubble mode. Immediately behind the pulse, a region is formed in which there are no electrons (electron bubble). A small electron bunch is captured from the plasma and accelerated

proton particle detector acceleration

In 2000, when thin foils were irradiated with high-intensity (more than 1018 W/cm2) laser pulses, protons with energies up to 10 MeV were discovered, escaping mainly from the back wall of the foil in the direction of pulse propagation. This result aroused great interest. The experiments were repeated in many laboratories. The maximum measured proton energy in some of them reached 60 MeV, and their number reached 1012 per laser pulse.

How do protons with such high energy arise? Analysis of experimental data and numerical calculations showed that under the action of a laser pulse, fast electrons appear in the foil, which pass through the foil and fly out from its opposite side. But they cannot fly far. They are stopped by the electric field of the ions remaining in the foil. A negatively charged layer consisting of electrons is formed near the rear surface of the target. The electric field created by these electrons is directed perpendicular to the surface and reaches a value sufficient to ionize the atoms located on the surface. Then, under the influence of the same electric field, the ions begin to accelerate. A double layer appears, consisting of layers of electrons and ions separated in space, which flies out of the target. During the acceleration process, energy is transferred from electrons to ions. Light ions (protons) formed from hydrogen atoms adsorbed on the surface of the foil are most efficiently accelerated (Fig. 61).

Rice. 61 - Acceleration of ions (protons) when irradiating thin foil with a short laser pulse. The laser pulse falls on the left boundary of the foil, fast electrons fly out through the right boundary of the foil and accelerate the ions with their electric field

Such sources of energetic ions are already used in proton radiography, when an image of an object is obtained by shining a beam of protons through it. This method makes it possible, in particular, to determine the structure of electric fields inside the object under study. But laser sources of fast ions have the greatest prospects in medicine (oncology). The fact is that it is more expedient to use protons to influence cancerous tumors. Currently, the sources of such protons are various vacuum accelerators, which are very bulky and expensive. There are hopes that laser sources will be more compact and cheaper.

Charged particle accelerators have long ceased to be exclusively scientific instruments. Today in the world there are more than 30 thousand accelerators, and most of them are used for cancer radiation therapy, sterilization and the production of semiconductor materials. For example, to turn a wafer of pure silicon into a microchip, you need to introduce impurity atoms into strictly designated places, and an accelerator is ideal for this. The more compact, reliable and cheaper accelerators are, the more profitable it is to use them and the more new tasks can be found for them.

In the first accelerators using high voltage, a powerful electrostatic field was created, which picks up and accelerates charged particles. But a generator capable of delivering over a million volts is a complex, expensive and difficult device to use. This voltage can create an electrical discharge towards other objects at a distance of more than a meter. Today, instead of a constant voltage, particles are accelerated by applying an alternating electric field to them many times.

Description

All modern accelerators work this way, but this method has already reached its limit. To develop such devices further, many physicists are studying the possibility of accelerating charged particles by the field that arises when laser radiation interacts with matter. Laser pulses can concentrate energy into very short bursts and thus provide extreme high power without building complex installations.

To accelerate particles (protons, for example) using a laser, physicists in recent decades have been directing laser pulses at thin foil. In this case, the electromagnetic field accelerates some of the electrons inside the light wave, as a result of which they fly through the foil, creating two differently charged areas in the material. And where there are two regions charged with the opposite sign, there is also an electric field, which can then pick up the particles and give them the necessary momentum.

Read also:

Previous experiments with laser beams and foil showed that protons with energies up to 8 MeV could be knocked out of the material. This corresponds to the energy that an electron would receive if it flew between points with a potential difference of 8 million volts. This is already enough for many cases of using accelerators in practice, but not enough for medical accelerators. For example, proton therapy for cancer typically uses particles with energies in excess of hundreds of MeV.

And it is precisely for the targeted burning of tumors that it is important to create the cheapest possible accelerators. The fact is that now for these purposes it is necessary to build complex synchrotrons weighing hundreds of tons, so there are only about fifty places in the world where proton therapy can be carried out (several of them are in Russia). Naturally, the cheaper such a device is, the more of them can be built and the more lives can be saved.

Description

New publication in Communications Physics describes a slightly modified approach: instead of one powerful pulse with an energy of 1.1 joules, Swedish physicists proposed taking two pulses of 0.55 joules each. In practice, this means that one pulse is divided into two using a translucent mirror. Two beams fall on the foil at a certain angle, and, as simulations have shown, this significantly increases the energy of the emitted protons without additional costs. According to scientists' calculations, the two-beam design increases the maximum proton energy to 14 MeV.

But for medicine this is still not enough. The protons produced by the interaction of laser radiation also have too wide a spectrum, the energy of the particles is distributed unevenly between them, and some move noticeably slower than others. But the proportion of particles receiving energies above 1 MeV increased fivefold. This allows us to count on further progress in this area. The researchers emphasize that the laser beam can be divided into a larger number of parts. Maybe, correct selection beam configurations will allow you to achieve even better results.

This method has experimentally produced electron beams with energies exceeding 8 GeV.

Direct acceleration by laser field

Direct acceleration by a laser field is ineffective, since in a strictly one-dimensional problem, an electron entering the field of a laser pulse, after leaving it, has the same energy as at the beginning, that is, it is necessary to carry out acceleration in highly focused fields in which the longitudinal component of the electric field is significant. but in such fields the phase velocity of the wave along the propagation axis is greater than the speed of light, so the electrons quickly lag behind the accelerating field. To compensate for the latter effect, it was proposed to carry out acceleration in a gas, where the relative dielectric constant is above unity and the phase velocity decreases. However, in this case, a significant limitation is that already at radiation intensities of the order of 10 14 W/cm² the gas is ionized, forming plasma, which leads to defocusing of the laser beam. Experimentally, this method demonstrated the modulation of a 3.7 MeV electron beam with an energy of 40 MeV.

Acceleration in a plasma wave

When a sufficiently intense laser pulse propagates in a gas, it ionizes with the formation of a nonequilibrium plasma, in which, due to the ponderomotive effect of laser radiation, it is possible to excite the so-called wake wave - a Langmuir wave, traveling after the pulse. This wave has phases in which the longitudinal electric field is accelerating for electrons traveling along with the wave. Since the phase velocity of the longitudinal wave is equal to the group velocity of the laser pulse in the plasma, which is only slightly less than the speed of light, relativistic electrons can remain in the accelerating phase for quite a long time, acquiring significant energy. This method of accelerating electrons was first proposed in 1979.

As the laser pulse intensity increases, the amplitude of the excited plasma wave increases and, as a consequence, the acceleration rate increases. At sufficiently high intensities, the plasma wave becomes nonlinear and eventually collapses. In this case, a highly nonlinear regime of laser pulse propagation in plasma is possible - the so-called bubble (or bubble) regime, in which a cavity similar to a bubble is formed behind the laser pulse, almost completely devoid of electrons. This cavity also contains a longitudinal electric field that can effectively accelerate electrons.

Experimentally, in the linear interaction mode, an electron beam was obtained, accelerated to energies of the order of 1 GeV on a path 3 cm long. To compensate for the diffraction divergence of the laser pulse, in this case, a waveguide in the form of a thin capillary was additionally used. Increasing the power of the laser pulse to the petawatt level made it possible to increase the electron energy to 2 GeV. A further increase in the electron energy was achieved by separating the processes of their injection into the accelerating plasma wave and the acceleration process itself. Using this method, electrons with an energy of about 0.5 GeV were obtained in 2011, and in 2013 the level of 3 GeV was exceeded, and total length the accelerating channel was only 1.4 cm (4 mm - injection stage, 1 cm - accelerating stage). In 2014, the first experimental results on the acceleration of electrons in a 9 cm long capillary using the BELLA laser were obtained at the Lawrence Berkeley National Laboratory. These experiments demonstrated acceleration to energies exceeding 4 GeV with a 0.3 PW laser pulse, a new record. In 2019, a new record was set there - with a peak laser pulse power of 0.85 PW, electrons with an energy of about 7.8 GeV were obtained in a capillary 20 cm long.

In the nonlinear interaction mode, the maximum energy achieved was 1.45 GeV on a 1.3 cm long path. A laser pulse with a power of 110 TW was used in the experiment.

see also

Notes

1. R. Joel England et al. Dielectric laser accelerators (English) // Rev. Mod. Phys. . - 2014. - Vol. 86. - P. 1337. - DOI:10.1103/RevModPhys.86.1337.
2. E. Esarey, P. Sprangle, J. Krall. Laser acceleration of electrons in vacuum (English) // Phys. Rev. E. - 1995. - Vol. 52. - P. 5443.
3. T. Tajima, J. M. Dawson. Laser Electron Accelerator (English) // Phys. Rev. Lett. . - 1979. - Vol. 43. - P. 267.
4. W. P. Leemans et al. GeV electron beams from a centimetre-scale accelerator (English) // Nature Physics. - 2006. - Vol. 2. - P. 696-699.
5. Xiaoming Wang et al. Quasi-monoenergetic laser-plasma acceleration of electrons to 2 GeV (English) // Nature Communications. - 2013. - Vol. 4 . - P. 1988.
6. B. B. Pollock et al. Demonstration of a Narrow Energy Spread, ∼0.5 GeV Electron Beam from a Two-Stage Laser Wakefield Accelerator (English) // Phys. Rev. Lett. . - 2011. - Vol. 107. - P. 045001.
7. Hyung Taek Kim et al. Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses // Phys. Rev. Lett. . - 2013. - Vol. 111. - P. 165002. - DOI:10.1103/PhysRevLett.111.165002. - arXiv:1307.4159.
8. W. P. Leemans et al. Multi-GeV Electron Beams from Capillary-Discharge-Guided Subpetawatt Laser Pulses in the Self-Trapping Regime // Phys. Rev. Lett. . - 2014. - Vol. 113. - P. 245002. - DOI:10.1103/PhysRevLett.113.245002.
9. A. J. Gonsalves et al. Petawatt Laser Guiding and Electron Beam Acceleration to 8 GeV in a Laser-Heated Capillary Discharge Waveguide // Phys. Rev. Lett. . - 2019. - Vol. 122. - P. 084801. - DOI:10.1103/PhysRevLett.122.084801.
10. C. E. Clayton et al. Self-Guided Laser Wakefield Acceleration beyond 1 GeV Using Ionization-Induced Injection // Phys. Rev. Lett. . - 2010. - Vol. 105. - P. 105003.

Literature

Scientific

• E. Esarey, C. B. Schroeder, W. P. Leemans.

Hello, my name is Alexander and I am a physicist. From the outside, this may sound like a sentence, but in fact it is. It turns out that I am engaged in fundamental research in physics, namely, I am studying accelerated charged particles: protons and all those larger ones - positive ions, that is. In my research, I don’t use large accelerators like the LHC, but I shoot a laser at the foil, and a pulse of protons comes out of the foil.

Now a few words about me. I graduated from the Faculty of Photonics and Optical Informatics at ITMO in St. Petersburg, then went to a master’s program at Aalto University (in Finland) in micro- and nanotechnology, and then gave up on all these little things, microscopes, and especially the clean room. And I went into fundamental science with large lasers. Now I am working in graduate school in the southwest of Sweden in the city of Lund at the university of the same name. This is about a cannon shot from Copenhagen.

As soon as I sped it up, it flew

Let's compare these two examples of gloomy physical genius in more detail. Look at the left accelerator and the right one, then again at the left one and again at the right one: yes, mine is on a horse (joke - author's note). In fact, mine is only a meter in diameter, and the protons themselves are accelerated from a piece of foil. Its holder is located exactly in the middle of the circle, with a beautiful copper skirt on it. This is much simpler and more compact than the left sample, which is the size of a bus and, in addition, is filled with asphyxiating gas. So, having asserted yourself enough (in physics it often happens that the less the better), you can turn to the physics of the acceleration process.

Since we accelerate charged particles, it is most logical to do this with an electric field. We will characterize the field by tension. For those who after school went to the front and back end, let me remind you: electric field strength - vector physical quantity, characterizing the electric field at a given point and numerically equal to the ratio of the force acting on a stationary point charge placed at a given point in the field to the magnitude of this charge(dirty copy-paste from Wikipedia). Has the dimension V/m. Returning to the comparison, the accelerator on the left accelerates protons to 4 MeV (Megaelectronvolt), that is, 2.77 * 10 7 m/s or 9.2% of the speed of light. Since the proton charge is 1, and the length of the accelerator is two meters, the field strength will be 2 MV/m. Here we assumed that in all places the field is directed in one direction and, in general, we were very close to the truth. The stylish accelerator has a field strength of the order of several TV/m, that is, about a million times greater. Still, it is worth recognizing that its length is only a few microns.

So, by now we have found out whose field is steeper. It's time to turn to the physical and engineering mechanisms that create this field. In the case of a conventional accelerator, there are two metal sheets, one of which is negatively charged, and the second is not charged at all. Remember the school experiment about rubbing an ebonite stick with a piece of wool. Here the principle is absolutely the same, but the execution is much more complicated. If you accelerate protons from foil, then the field is created by electrons, electrons fly out of the hot plasma, the plasma is produced and heated by a laser, and the rest of the post is about all this.

Do you want me to hit him and he will turn speckled purple?

In my case, I shoot the foil with a laser. I will describe it in more detail after explaining the non-trivial physics of the processes of obtaining warm dense matter, which is the scientific name for plasma, which is the culprit of the triumph of the acceleration of my protons. Now let's talk about everything in order.

The laser generates pulses with a wavelength of 800 nm and 35 fs with a duration of (10 -15 s), that is, the actual pulse length in a vacuum is approximately 10 microns. This impulse contains approximately 2 J of energy, which is a lot. If you take this pulse and focus it on the foil into a neat round spot 5 microns in diameter, then the intensity will be about 10 20 W/cm 2. This is already an indecent amount. Again, a little comparison: steel can be safely cut at an intensity of 10 8 W/cm 2 (or so).

In fact, the laser pulse, due to the design of the amplifier, has a preceding pedestal with a duration of approximately 500 ps, ​​and this very pedestal greatly helps to accelerate protons well.

Ionized means armed

So, we have a pedestal, which we also focus perfectly on a piece of foil, and it is perfectly absorbed there. In order not to go into the complexities of solid state physics, let us consider the absorption of a free-standing atom. From quantum mechanics We know that only a photon can be absorbed, the energy of which is exactly equal to the energy of the transition of an electron from one state to another. If the photon energy is greater than the ionization energy (that is, sending an electron from the parent nest on a free journey), then the excess will turn into the kinetic energy of the electron, everything is simple. In our case, photons with a wavelength of 800 nm do not have enough energy (this is the energy of one photon, not the entire pulse!) to ionize the target, but here physics comes to our aid. Remember I mentioned the high radiation intensity? If, in addition, we remember that light can be represented as a stream of photons, and the intensity is directly proportional to it, then it turns out that the stream of photons is very large. And if the flux is so large, then there is a high probability that several photons will arrive in one place and at the same time, and when their energies are absorbed, they will add up, and ionization will still occur. This phenomenon, oddly enough, is called multiphoton ionization, and we use it regularly.

At the moment, we have that the electrons have been successfully torn off, which means that the main pulse arrives at the ready-made plasma and begins to heat it.

Basics of plasma physics (didn't come up with a joke, ah)

Our main characteristic of plasma will be its density (the number of electrons per unit volume), which we will further denote as $n_e$ (not to be confused with the refractive index!), and the temperature of these same electrons, that is, their average speed of movement. This is described by the Boltzmann distribution in the same way as in the school physics course:

$$display$$\frac(m_e v^2)(2) = \frac(1)(2) k_B T_e,$$display$$

$$display$$\langle v \rangle = \sqrt(k_B T_e/m_e),$$display$$

Now we will apply an electric field to the plasma already described. Let me remind you that plasma consists of charged particles, which means that at a given density at some distance from the place where we applied the field, electrons will obscure (screen) the source (such a crowd of little Matrosovs - author's note). The distance required for this is called the Debye length and is given by the equation

$$display$$ \lambda_D = \sqrt(\frac(\epsilon_0 k_B T_e)(q^2_e n_e)). $$display$$

Due to the screening effect and a very specific (depending on temperature) average speed of electron motion, the plasma does not react instantly to a suddenly arriving field. It is logical to assume that the response time is related to the Debye length and the speed of electron motion. A good analogy is throwing a stone into a lake. Compared to the whole lake, the stone affects the surface of the water pointwise. Part of the water changes immediately (this is where it splashed), and then the waves begin to spread across the water surface. In the case of plasma, the suddenly appearing electric field is a rock. The size of the splash is determined by the length of the shielding (the field does not act beyond it), and the propagation of the waves depends on how close the electrons are to each other. We can introduce such a characteristic as the plasma response time:

$inline$ t_D = \lambda_D / v $inline$ . By and large, it shows us the time during which information about a change in the applied field will reach those electrons that, as it were, did not see this field.

Since we are physicists, we don't really like time. It is much more convenient to work with frequencies, so we will introduce the concept of the natural frequency of the plasma. This value will show us how often we can change the field so that the entire accumulation of electrons, which we proudly call plasma, has time to respond to these changes. Well, what could be simpler? Let's divide one by the response time, and here it is - frequency:

$$display$$ \omega_p = \frac(1)(t_D) = \sqrt(\frac(q^2_e n_e)(\epsilon_0 m_e)). $$display$$

The natural frequency of a plasma also determines its refractive index. If we honestly write the wave equation for the collective motion of electrons in a plasma, and then assume small changes in the electron density (we won’t do this here, because it’s boring), then the refractive index is set as follows:

$$display$$ \eta = \sqrt(1-\frac(\omega^2_p)(\omega^2_0)). $$display$$

Let's look closely at this expression. As an experimental physicist, I don’t like real numbers, but I try to ignore complex numbers, especially the complex refractive index. Well, how can light, after all, propagate through matter? i times slower than in a vacuum? This is some kind of nonsense! Actually no, but more on that another time. If $inline$\omega_0 > \omega_p$inline$ , then the expression is valid, and the alternating electric field propagates inside our plasma. Everyone is happy, and we will call such a plasma insufficiently dense. However, if $inline$\omega_0< \omega_p$inline$ , то показатель преломления становится не то что комплексным, а целиком мнимым. В этом случае (и не просто потому что я так захотел) волна вообще не будет там распространяться, а сразу отразится без потерь. Это слишком плотная плазма. Очень классное явление, кстати. Называется плазменным зеркалом.

And as dessert $inline$\omega_0 = \omega_p$inline$ . This is plasma of critical density. In this case, it begins to resonate with the forcing (supplied by us) alternating electric field. For such a special case, you can even introduce the concept of critical density and define it like this:

$$display$$ n_c = \frac(\epsilon_0 m_e \omega^2_0)(q^2_e). $$display$$

SHOCK! Plasma heating! To do this you only need...

To begin with, let the plasma that we formed as a pedestal have a smooth density gradient, in this case we have heating through resonant absorption. An illustration of this is in the picture below.

Illustration of the resonant absorption process: a) electron density distribution near the front side of the target; b) refraction of a laser beam in a plasma with a density gradient; c) electric field in plasma

So, the laser shines on our plasma at an angle, maybe 45 degrees, and at the same time it is polarized in the plane of incidence. Polarization is indicated by red arrows in the figure. Our plasma has a density gradient, which means its refractive index is continuously changing (here, growing). At some point, it will happen that a certain layer of plasma for our laser will become “rotary” and it will be reflected, that is, it will propagate parallel to the critical layer for some time. It is important to note that it will turn before it reaches the layer with the critical density, since we launched it at an angle to the normal. The plasma density at which the laser beam will turn is given by the following equation:

$$display$$ n_t = n_c \cos^2 \alpha,$$display$$

Now the fun begins. Let us remember that light is not only a stream of photons, but also an electromagnetic wave, that is, our pulse has an electric field that oscillates harmoniously with a large amplitude. When light propagates parallel to the critical layer, a standing wave, which does not change over time (of course, as long as the laser pulse is in place). The field of this wave, in fact, penetrates further than the plasma layer where the light turned and reaches the critical layer. Let me remind you that the frequency of plasma oscillations in the critical layer is the same as the frequency of laser radiation, which means resonance occurs. When the laser stops shining, the energy it imparted to the electrons in the critical layer is distributed through impacts to the remaining electrons, which means that the plasma has heated up.

So where, exactly, is the acceleration?

	Plasma formation by the pedestal, that is, ionization of the front side of the target. Foil with a thickness of 0.4 - 12 microns is usually used as a target.
	Here the main part of the pulse interacts with the created plasma and heats it. Some electrons are so well heated that they fly out from the back of the target.
	When enough electrons have flown out, the remaining positive charge in the foil pulls them back. In the plasma they heat up again and fly out. For some time, dynamic equilibrium is established. The electric field is directed perpendicular to the target
	This same electric field lifts protons and other ions (depending on what was there at all) from the back surface of the target, and then accelerates them. By the time the ions accelerate, the electron cloud has already collapsed, and all the particles begin to fly further together. And then we begin to believe that they no longer interact.

Divide and rule

In addition, we can look at the profile of a beam of flying protons. It's round, by the way. And if we can also measure the energy of the protons in each part of the beam, then we will be able to unambiguously restore the shape of the electron cloud that accelerated our protons.

Instead of a conclusion

For those especially curious about the laser itself and its structure

I won’t go into much detail about laser theory. The basics of where laser radiation comes from are perfectly outlined in the Wikipedia article on stimulated emission. To be very brief, laser radiation requires three components: an active medium (from which photons are emitted), a pump (it maintains the active medium in a state in which there are more excited atoms that can emit), and a resonator ( it ensures that photons copy each other during repeated passages through the active medium). If you put all the components together and pray, the laser will begin to shine, but continuously. If you try a little more, you can make it generate pulses, including such short ones as on my installation. For the more curious, the method of generating femtosecond pulses is called passive mode locking. And now a small feature of very short pulses. It is often believed that a laser shines at a single wavelength, and in continuous mode, as well as with long pulses, this can even be called true. In fact, due to a number of complex physical processes, which we certainly will not discuss here, the temporal shape of the pulse and its spectrum are related by the Fourier transform. That is, the shorter the pulse, the wider its spectrum.

Let's say that we have launched a master oscillator, but the energy of its pulses is several nJ. Remember at the beginning I said that the energy in the pulse that arrives at the target is about 2 J? So, this is a billion times more. This means that the impulse needs to be strengthened, and we will talk about this in more detail.

Short pulses are generally characterized by very large peak powers (remember, divide energy by time?), and this has a number of complications. If you shine radiation into the medium with high intensity (power per unit area), it will burn, but if the active medium has burned, then nothing will be amplified. That's why we choose a repetition rate of 10 Hz and only amplify them. Since there is a lot of equipment and it all operates at exactly this frequency, we have a special box that distributes these 10 Hz to all hardware, and for each device you can select the delay in receiving the signal with an accuracy of several picoseconds.

There are two ways to deal with high intensity. As you can easily guess from its definition, you need to either increase the area or reduce the power. With the first, everything is very clear, but the second method was a breakthrough in laser technology in the twentieth century. If the impulse is initially very short, it can be stretched, strengthened, and then compressed again.

To understand how to do this, let's look at the basics of optics. For different lengths waves, the refractive indices in the medium are different, which means (by the definition of the refractive index, by the way) that as the refractive index increases, the speed of light propagation in the medium decreases. And so we launched our pulse into the environment, and its red part passed through the material faster than the blue one, that is, the pulse became longer, and its peak power decreased. Hurray, now nothing is on fire! For deeper knowledge in this area, I recommend Googling and reading about chirped pulse amplification (also known as Chirped Pulse Amplification or CPA).

All we have to do is strengthen the impulse, compress, focus and send it to make a hole in the foil!

And now some pictures with captions.

Actually a photo of the laboratory. The cylindrical thing in the middle is a vacuum chamber, because protons fly very lousy in the air and constantly bump into air molecules. Well, in general, everything looks cooler with a vacuum. The blue thing on the right is a lead wall, so that you don’t accidentally get superpowers and radiation sickness. The laser itself is located behind the door on the left with the yellow Achtung sign

And here is the wall itself in profile. Yes, it's stuffed with lead inside, like Winnie the Pooh.

Our command post is located behind the wall; when we shoot, for safety reasons we are supposed to sit behind it. Of course, we won’t die from radiation, but we can easily go blind. There are five monitors for two computers, it’s very easy to get confused in all this junk. One of the computers has speakers, so while working in the dungeon you can listen to Loboda and the Big Russian Boss; for inexplicable reasons, my colleagues also like them. Only half of them are Swedes, by the way.

We still have a lead sliding door. It is hydraulically driven.

Here we are inside the room with the laser. This is a photograph of the first table on which the laser pulse is generated. Here it is pre-amplified (approximately 1000 times) and stretched. On the shelf above there is a bunch of very important and necessary electronics, without which the laser will not work.

This is the second table in which radiation is amplified after the first. This amplifier is our main workhorse - it increases energy by forty thousand times. In fact, it contains two amplifiers of different design: multi-pass and regenerative. In the first, the pulse simply passes through the active medium several times. The second has its own resonator. Using electro-optical gates (Pockels cells), a pulse is launched inside, it passes there several times until the gain is saturated, and then it is released further. This is where the speed and accuracy of opening and closing shutters is so important.

This is the third table, there is a gain of about 15 times. The tower in the middle, which sticks out above the lid, is a cryostat. It contains a huge crystal in a vacuum, which is cooled with liquid helium to a temperature of -190 degrees Celsius.

This is a separate room that contains the third table pumping power supplies and the main vacuum pumps. The efficiency from the outlet of the system is so-so, approximately 0.1%. I somehow calculated that the electrical power consumed was approximately 160 kW. This is approximately 960 video cards that can be powered and mine, mine, mine. That's how much electricity is consumed when amplifying at a repetition rate of 10 Hz. If we tried to boost 80 MHz, the consumption would increase 8 million times.

Thank you for your attention!