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 and the amplitude of the second (negative) oscillation is smaller than (Figure 4). On the attosecond time scale, the femptosecond pulse can be considered quasistatic.

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 (Figure 5).

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 the second potential well disappears and a catastrophe occurs: the charged particle jumps from the high second minimum _{ }to the low first minimum and radiates an attosecond

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.

Figure 4. The femptosecond laser pulse.

Figure 5. The four-stroke cycle of the hypothetical attosecond generator.

Stroke 4 According to the principle of maximum delay, in the moment when the negative amplitude of the leading edge equals the first potential well disappears and a catastrophe occurs: the charged particle jumps from the high first minimum _{ }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.

3. The real attosecond electromagnetic pulses 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 [15] - [17] . Note that the work of a real generator of attosecond pulses can be explained by using the concepts of the hypothetical generator of attosecond pulses and the so-called semi classical approximation of quantum mechanics.

According to de Broglie, electrons have wave properties. An electron is described by a wave function. The wave function has a wave length. In semi classical Bohr model of the atom (1913), valence electrons rotate in circular stationary orbits around the atom nucleus [18] . The stationary orbit satisfies the standing wave condition: the whole number of the electron wavelengths l must fit along the circumference of the orbit [18] :

, (12)

where is an integer, is the radius of the -orbit. In this case the energy level of the electron in the - orbit (the so-called ionization energy [19] ) is

, (13)

where is the effective nuclear charge [20] . In contrast to the classical particle with an elementary charge, an electron doesn’t lie at the bottom of the electrostatic potential well, but lies at the n-th energy level.

Let the atom be illuminated by a focused femtosecond powerful laser pulse. If the strength of the electric field of the laser pulse is close the strength of the Coulomb field of the atom nucleus, the resulting potential well for the valence electron becomes a superposition of the Coulomb potential well and the linear function (in volts) [21] :

, (14)

where is the charge of the electron, is Coulomb’s constant. Note that the resulting potential, equation (14), has the potential barrier with the height (see Figure 6). The width of the barrier is determined by the distance between the turning points and, where the potential is equal to the basic energy level ?:

. (15)

The quadratic equation with respect to

(16)

gives two solutions:

(17a)

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

(17b)

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) allow us to formulate a simple criterion: the electron tunnels through the barrier if the barrier width equals the diameter of the stationary orbit divided by (points “−2” and “2” in Figure 7). Note that this criterion can be written in the arithmetic form as

Figure 6. The resulting potential. Value corresponds to the basic energy level of the hydrogen atom.

Figure 7. The tunneling curve on the control plane.

. (18)

. (19)

The generation of ultrashort pulses is a process when is constant and is a function of time, as we

can see in Figure 7. Let, then the tunneling curve degenerates into two points:

(point “2” in Figure 7) and (point “−2” in Figure 7). If,

then tunneling is not impossible (points “−1”, “0”, “1” in Figure 7). If , then the left potential barrier can be tunneled by the electron (point “−2” in Figure 7) and if, the right potential barrier can be tunneled by the electron (point “2” in Figure 7).

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 and the amplitude of the second (negative) oscillation is smaller than (Figure 4). On an attosecond time scale, the femtosecond pulse can be considered as quasistatic.

According to Keldysh [22] [23] , the process of tunneling ionization of the valence electron is “quasistatic” too, if the carrier frequency of the laser pulse is significantly less than the frequency of an electron tunneling through the potential barrier

,

where, and are the charge, mass and energy of the electron, and is the maximum amplitude of the laser pulse.

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 (Figure 8).

Stroke 2 When the electric strength reduces from to 0, the electron reserves the energy

.

Stroke 3 When the electric strength equals 0, the potential barrier disappears and, according to the principle of maximum delay, a catastrophe occurs: the electron jumps from the zero energy level to the basic energy

level and radiates an attosecond pulse with a carrier frequency of.

Figure 8. The six-stroke cycle of the real attosecond generator.

Stroke 5 When the electric strength increases from to 0, the electron reserves the energy

Stroke 6 When the electric strength equals 0, the potential barrier disappears and, according to the principle of maximum delay, a catastrophe occurs: the electron jumps from the zero energy level to the basic energy

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 [19] : each period begins at a minimum for the alkali metals, and ends at a maximum for the noble gases. So, to generate attosecond pulses the hydrogen or the noble gases are used. As the H ionization energy [19] is 13.59 eV, He―24.58 eV, Ne―21.56 eV, Ar―15.76 eV, Kr―13.99 eV, Xe―12.13 eV, Hg―10.43 eV, Rn―10.74 eV, the corresponding radiation refers to the soft EUV region of the spectrum. The duration of the ultra short pulse in a photon energy range of 10 eV to 25 eV cannot be less than 100 as.

In the catastrophe theory the principle of maximum delay is widely used [13] . In this article we have used this principle too. However, it does not allow taking into account the kinetic energy of an electron oscillating in an external laser field the so-called ponder motive energy. To produce the hard EUV-rays or even X-rays ponder motive energy must be taken into account. In this case, the principle of maximum deceleration should be replaced by a different, more suitable principle.

4. Conclusion

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.

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