How big must an aperture in a spatial filter be if we created it from a lens with a diameter of $40 \mathrm{cm}$ and its focal length is $4 \mathrm{m}$? Our Gaussian laser beam has an input diameter $30 \mathrm{cm}$ and a wavelength $1~053 nm$. The radius of the focus (parameter $\sigma $) of the Gaussian beam can be obtained using

\[\begin{equation*}
r = \frac {2}{\pi }\lambda \frac {f}{D}
\end {equation*}\]
where $D$ is the diameter of the beam, $f$ is the focal length of the lens and $\lambda $ is the wavelength of the laser.

The laser beam is focused on a surface of a nuclear fuel pellet of a $1 \mathrm{mm}$ diameter. What energy should it have in order for the intensity in its focus to reach $10^{14} W.cm^{-2}$? The radius of the focus is $25 \mathrm{\micro m}$ and a pulse lasts $10 \mathrm{ns}$. How many beams do we need to equally cover the surface of a pellet? What is their total energy?
What energy must the laser beam have if it is not focused on a surface of a nuclear fuel pellet, but the beam diameter matches exactly the diameter of the pellet and the density is its focus reaches $10^{14} W.cm^{-2}$? Assume that we have one such beam and it shines homogenously on the pellet „from all directions“.

What intensity must a laser with a wavelength of $351 \mathrm{nm}$ have in order to stabilize a Rayleigh-Taylor (RT) instability using the surface ablation of a fuel pellet? Suppose the boundary between the ablator and DT ice is corrugated with a wavelength of

$0,2 \mathrm{\micro m}$,

$5 \mathrm{\micro m}$.

How will the intensity of the laser change if we also apply a magnetic field with magnitude $5 \mathrm{T}$?

What else can help us minimize the RT instability?

How far from the surface of the target (suppose it is made of carbon and the laser has wavelength of $351 \mathrm{nm}$) is critical surface situated and how far does two-plasmon decay occur, if the characteristic length of plasma^{1)}

^{1)}

The density of plasma $n_e$ is typically expressed as a funciton $n_e = f\(\frac {x}{x_c}\)$, where $x$ is the distance from the target and $x_c$ is so called characteristic length of plasma, which represents scale parameter for the distance from the target.))is~$50 \mathrm{\micro m}$? Next assume

that the density of the plasma decreases exponentially with distance from the target,

that the density of the plasma decreases linearly with distance from the target.

What energy must electorns have in order to go through the critical surface to the real surface of the target? To calculate the distance electron travels in carbon plasma use an empirical relationship $R = 0{,}933~4 E^{1{,}756~7}$, where $E$ has units of \jd {MeV} and $R$ has units of \jd {g.cm^{-2}}.

What is the distance that an electron has to travel in the electric field of the plasma wave in order to reach the energies determined in second exercise?

Which wavelengths of scattered light are present in the case of stimulated Raman scaterring for laser with wavelength of $351 \mathrm{nm}$?

What energy must a laser impulse lasting $10 \mathrm{ns}$ have in order for the shock wave generated by it to be able to heat the plasma to a temperature at which a thermonuclear fusion reaction can occur? What will be the density of the compressed fuel? Note: Assume that the initial plasma is a monatomic ideal gas.

Assume a charged chord with linear density $\rho $, uniformly charged with linear charge density $\lambda $. The tension in the chord is $T$. It is placed in a magnetic field of constant magnitude $B$ pointing in the direction of the chord in equilibrium. Your task is to describe several aspects of the chord's oscillations. First, we want to write the appropriate wave equation. Neglect the effects of electromagnetic induction (assume the chord to be a perfect insulator; that also means the charge density does not change) and find the Lorentz force acting on an unit length of the chord for small oscilations in both directions perpendicular to the equilibrium position. Use this force to write the wave equation (which will also include the effects of the tension). Apply the Fourier substitution and determine the disperse relation in the approximation of a weak field $B$; more specifically, neglect the terms that are of higher than linear order in $\beta = \frac {\lambda B}{k \sqrt {\rho T}} \ll 1$, where $k$ is the wavenumber. Find two polarization vectors, this time neglect even the linear order of $\beta $. Now suppose that in a particular spot on the chord, we create a wave oscilating only in one specific direction. How far from the original spot will be the wave rotated by ninety degrees from the original direction?

Štěpán was nostalgically remembering the third serial task.

(10 points)5. Series 34. Year - S. resonance and damped oscillations

On a tense rope, waves can exist with the deflection $\f {u}{x, t}$ from the equilibrium, that satisfy the wave equation with damping

\[\begin{equation*}
\ppder {u}{t} = v^2 \ppder {u}{x} + \Gamma \pder {u}{x} ,
\end {equation*}\]
where $v$ is the phase velocity and $\Gamma $ is the coefficient of damping. Do a fourier substitution and find the dispersion relation. Solve it for the wavenumber $k$. What condition, in terms of frequency $\omega $, phase velocity $v$ and the coefficient $\Gamma $, must the waves meet in order to create nodes on the rope (i.e. points in which the rope stays in equilibrium position, but around which the rope is moving)?

Consider a jump rope attached firmly at one end to a fixed wall. At the distance $L$ from the wall, we start moving the rope up and down to create waves. The jump rope has a linear density $\lambda $ and the constant tension $T$ in the direction away from the wall. The deflection then satisfies the equation

\[\begin{equation*}
\ppder {u}{t} = \frac {T}{\lambda } \ppder {u}{x} .
\end {equation*}\]
For the deflection of the end of the rope that is moving satisfies $\f {u_0}{t} = A \f {\cos }{\omega _0 t}$. Assume the solution can be written in the form of two planar waves moving in the opposite direction to each other. Find the solution using only the parameters given in this problem statement, that is $T$, $\lambda $, $L$, $A$ and $\omega _0$. For certain frequencies, the solution has a diverging amplitude (i.e. growing beyond any limits). Find their values and the respective wavelenghts.

Let's have a point source of light and a planar glass panel with a refractive index $n = 1,50$. In the foot of the perpendicular from the source to the panel there are wavefronts with a radius of curvature $R = 5,00 \mathrm{m}$ inside the glass. What is the real distance between the source and the panel?

A solar sail with the surface area of $S = 500 \mathrm{m^2}$ and area density $\sigma =1,4 \mathrm{kg\cdot m^{-2}}$ is located at the distance of $0,8 \mathrm{au}$ from the Sun. What force does the solar radiation act on the sail at the beginning of the sail's motion? What is the acceleration of the sail at that moment? The luminosity of the Sun is $L_{\odot } =3,826 \cdot 10^{26} \mathrm{W}$. Assume that the radiation approaches the sail from a perpendicular direction and scatters elastically. Hint: We recommend you find the acceleration for small initial velocity $v_0$ and then let $v_0 = 0$.

(9 points)5. Series 33. Year - 5. optically relativistic

Calculate the phase shift $\Delta \Phi $ when an optical beam with a wavelength $\lambda _0$ goes through a glass plate with thickness $h$ and the index of refraction $n$ that is moving along the beam with constant speed $v$ relative to a case when the plate is stationary. We are interested mainly about the first nonzero term of Taylor series of $\Delta \Phi (v)$.