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The generation of an attosecond pulse in the ultraviolet range is described in the terms of the catastrophe theory. A simple criterion of tunneling is proposed. The criterion allows constructing the quasiclassical model of the generator of attosecond laser pulses based on the interaction of an electric field of extremely powerful femtosecond pulse with the valence electron in the potential well of the gas atom.

Since the advent of the laser in 1960, there has been a sustained interest in the quest of generating laser pulses of the shortest duration and of the maximum power. A pulse is the packet of monochromatic waves and the central frequency of the packet is the so-called carrier frequency of the pulse. Thus, there is a fundamental physical limit of duration of a pulse. It is the period of its carrier frequency. The pulse whose duration is of the order of the period of its carrier frequency is called the ultrashort pulse.

In the visible range of the electro-magnetic spectrum, the ultra short laser pulse can have femtosecond durations (1 fs = 10^{−}^{15} s). Such laser pulses can be directly produced by modern mode-locked lasers [^{15}W). Focusing with parabolic mirrors [^{22} W/cm^{2}, which corresponds to the electric field with the strength well above the interatomic electric field (about several volts per angstrom, 10^{9} V/cm).

The so-called attosecond (1 as = 10^{−}^{18} s) pulse can be created only in the EUV regions of the spectrum. However, in these spectral regions the mode-locking method and the chirped pulse amplification method are no longer applicable.

Fortunately, there are the so-called “catastrophe machines” which transform smooth changes of the input signal into a quick change of their states [

The heavy ball in the gravitational potential well can be replaced by an electron in the electrostatic potential well and the external influence by an electric field of a femtosecond laser pulse. In the same way we can create an “electrostatic catastrophe machine” in which the electron jumps from one local minimum with high energy to another one with lower energy. If the difference of the energy levels is of the order of tens of electron volt, the electron jump is accompanied by emission of attosecond electromagnetic pulse in the ultraviolet range of spectrum.

The aim of the article is to explain the work of the hypothetical electrostatic generator of attosecond pulses from the point of view of the catastrophe theory and classical mechanics, and to use the obtained concepts for a quantum description of the real (quantum) electrostatic generator of attosecond pulses.

Let’s consider a classical particle with an elementary charge in a potential well

where

point

and a maximum at the point

has a W-shaped profile with two minima at points “

Since the distance between the minima (i.e. the width of the potential barrier)

is proportional to the square root of the parameter

If we add the potential of the electric field of the laser pulse with the maximum strength

and its potential minima will be redistributed.

The state of the system (a classical particle with elementary charge in a potential well) is described by the in-

ternal variable

tem is quasistatic or adiabatic. When the control variables

an equilibrium state where the internal variable

where

Combining (6) and (7) one gets the equation of equilibrium states

Equation (8) gives a surface

Let the

Minima of the function

Thus, if the control variables

The discriminant set

we obtain

The generation of ultrashort pulses is a process when

local minimum (point “−3” in

Let a femtosecond laser produce an ultra short laser pulse in the form of two big oscillations and the amplitude of the first (positive) oscillation is bigger than

If the quasistatic laser pulse falls on the system “the charged particle in the second potential well”, we have a generation of ultrashort pulses which are described by a four-stroke cycle (

Stroke 1 The leading edge of the positive oscillation raises the charged particle in the second potential minimum

until this potential well disappears. During this time interval, the particle reserves the energy

Stroke 2 According to the principle of maximum delay, in the moment when the positive amplitude of the leading edge equals _{ }to the low first minimum

pulse with a carrier frequency of

Stroke 3 The leading edge of the negative oscillation raises the charged particle in the first potential mini-

mum until the potential well disappears. During this time interval the particle reserves the energy

Stroke 4 According to the principle of maximum delay, in the moment when the negative amplitude of the leading edge equals _{ }to the low second minimum _{ }and radiates an attosecond pulse with a carrier frequency of

At the end of the fourth stroke the system returns to its initial state and the cycle can be repeated. Thus, if this catastrophe machine could exist in nature, it would be a perfect attosecond pulse generator.

There is a real generation of attosecond pulses in which an electric field of a focused extremely powerful femtosecond pulse interacts with a valence electron in the potential well of the noble gas atom [

According to de Broglie, electrons have wave properties. An electron is described by a wave function. The wave function has a wave length

where

where

Let the atom be illuminated by a focused femtosecond powerful laser pulse. If the strength

where

The quadratic equation with respect to

gives two solutions:

is the left turning point (a particle from the region I falls into the region II),

is the right turning point (a particle from the region II falls into the region III).

For further calculations it is necessary to choose a simple criterion of the barrier width at which the electron tunnels through the barrier. The condition of the stationary orbit, equation (12), and the condition for tunneling (the width of the barrier is comparable to the wavelength of the electron

The generation of ultrashort pulses is a process when

can see in

then tunneling is not impossible (points “−1”, “0”, “1” in

Let a femtosecond laser produce an ultrashort laser pulse in the form of two oscillations where the amplitude of the first (positive) oscillation is bigger than

According to Keldysh [

where

If a quasistatic laser pulse falls on a quasistatic system “electron in the potential well of the atom nucleus”, we have the generation of ultrashort pulses which is described by a six-stroke cycle (

Stroke 2 When the electric strength

Stroke 3 When the electric strength

level and radiates an attosecond pulse with a carrier frequency of

Stroke 5 When the electric strength

Stroke 6 When the electric strength

level and radiates an attosecond pulse with a carrier frequency

At the end of the sixth stroke the system returns to its initial state and the cycle can be repeated.

In the table of the elements there is a periodic trend for ionization energy [

In the catastrophe theory the principle of maximum delay is widely used [

The transformation from the input femtosecond pulse in the visible spectrum to the output attosecond pulse in the ultraviolet spectrum is a transformation of a smooth changing input signal to a quickly changing output signal, so it is a field of interest of the catastrophe theory. We propose a criterion for tunneling (18) and a quasiclassical model of the transformation of femtosecond laser pulses into attosecond pulses described as an electrostatic catastrophe machine.