**Ultrafast Time-Resolved Spectroscopy**

László Nánai1, Szabolcs Beke1 and Koji Sugioka2 *1University of Szeged, Department of General and Environmental Physics Laser Technology Laboratory 2RIKEN - The Institute of Physical and Chemical Research, Wako, Saitama, 1Hungary 2Japan* 

## **1. Introduction**

226 Femtosecond–Scale Optics

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In this chapter we will focus on the subject of spectroscopical investigation of matter (gas, liquid, solid, low dimension materials, etc) using ultrashort (ps, fs) laser pulses in different wavelength region (visible, infrared, UV, X-ray). We will point out the rigorous difference in aims of classical spectroscopy using highly narrowed electromagnetic waves with high temporal and spatial coherence compared to the wide narrowed spectral structures of ultrashort light pulses.

We will discuss briefly the main characteristics of light-matter interaction on the base of the linear and nonlinear optics assumption. Some outlook of facilities of classical spectroscopy will be made with attempt of time resolved techniques e.g., time of flight and pump and probe techniques. We will also demonstrate some interesting results of ultrafast time resolved spectroscopy in material science and solid state physics.

## **2. Elements of linear and nonlinear optics of light-matter interaction**

The study of phenomena induced by laser beam (continuous (CW) and pulsed) on solid surfaces is a widely investigated subject of modern solid state physics and chemistry. From the advent of the first lasers (late 60s) a huge number of scientific papers had been devoted to the investigations of different kinds of laser – matter processes such as laser induced damage, plasma formation, phase transition in different circumstances, micro- and macroprocessing, laser-induced chemical reaction at solid – gas and solid – liquid surfaces etc. The laser became a very useful tool for initiating unique chemical reactions to produce advanced materials (ultra hard ones for example). Number of technological applications, for example laser induced deposition of metals on porous materials and semiconductor surfaces are already in service of high-tech industry (micro- and nanoelectronics). Production with tailored properties of different nanostructures and components serves in chemical industry as catalysators, etc. [1]

Some preliminaries of the light-matter interaction are treated in a quasi-classical approximation:

Semi-classical approximation means that the electromagnetic wave is described classically on the base of Maxwell's equations while the material is described on quantum

Ultrafast Time-Resolved Spectroscopy 229

<sup>0</sup> ( ) *<sup>z</sup> Iz Ie*

where 0*I* is the intensity of the light at the incidence plane, *I z*( ) the intensity in the depth

2 2 2 2 () ( ) () ( ) , 2 2

The above-mentioned equations describe the light – matter interaction events for relatively low beam intensities where the material constants do not depend on the intensity, but only

For the laser beam intensities comparable to the electric field strength inside the atoms we have to take into account the higher-order (nonlinear) terms in the field to get the

<sup>0</sup> *Pt E E E* ( )

 

As a result we can get different phenomena e.g., 2nd and higher order harmonic generation,

We note that in all processes the conservation laws for photon energy and momentum

123 123

Non-linear optical effects usually called as N wave mixing processes, where N is the number of photons participating in reactions. The more photons the weaker the probability of the effect. At very high laser intensities e.g., at very short pulses very high order effects might have been realized. The materials usually have different refractive indices for different frequencies and polarizations; therefore the phase matching for them will be satisfied only

A number of special methods have been developed (due to nonlinear crystals to produce effects such as second harmonic generation (SHG), optical parametric generation (OPG), optical parametric oscillation (OPO), optical parametric amplification (OPA), quick

Spectroscopy in general is the study of the interaction between light and matter such as determination of quantum energy levels in substrates (gas, solid, liquid, etc.). In this "traditional" simple form, one measures the spectroscopic components of the reflected beam

(1) (2) (3) 2 3

 

*and k k k k* (11)

 

level z, and

The real (

) and imaginary (

**4. Nonlinear optics approximation** 

(phase matching) should be fulfilled as

**5. The spectroscopy 5.1 General remarks** 

on the frequency.

where ( ) *<sup>n</sup>* 

is the absorption coefficient.

to optical material parameters *n* and *k* through Kramers-Kronig relations.

*n k*

polarization *P* . (For H we got <sup>9</sup> 16 2 *E V cm I W cm* ~ 10 / and ~ 10 / )

is the nth order of the susceptibilities of the medium.

sum- and difference frequency generation, parametric generation, etc.

 

for special directions e.g., second harmonics birefringence.

switching features as electrooptical Kerr and Pockel cells [2-4].

(8)

(10)

) parts of dielectric constants at given frequency are related

 

. (9)

mechanical level (taking into account the electronic, vibration and rotation energy levels of the atoms).

## **3. Linear optics approximation**

From Maxwell's equations we get:

$$
\overline{D} = \varepsilon\_0 \overline{E} + \overline{P}(\overline{E}) \quad \text{and} \quad \overline{B} = \mu\_0 \overline{H} + \mu\_0 \overline{M}(\overline{H}) \tag{1}
$$

where *D* - electric flux density, *E* - electric field vector, *B* - magnetic flux density, *H* magnetic field vector, 0 - dielectric constant of vacuum = 8.85·10-12 C2N-1m-2 [F/m], 0 permeability of free space = 4·10-7 H/m (we will care materials for which 0 *M* 0 *B H* ). The polarization

$$
\overline{P}(\overline{E}) = \varepsilon\_0 \chi \overline{E} \tag{2}
$$

where - electric susceptibility.

The wave equation for electric field:

$$\nabla^2 \overline{\mathbf{E}} = \mu\_0 \varepsilon\_0 \frac{\partial^2 \overline{\mathbf{E}}}{\partial t^2} + \mu\_0 \varepsilon\_0 \varkappa \frac{\partial^2 \overline{\mathbf{E}}}{\partial t^2} = \mu\_0 \varepsilon\_0 \varepsilon\_r \frac{\partial^2 \overline{\mathbf{E}}}{\partial t^2} \tag{3}$$

where (1 ) *<sup>r</sup>* is the relative dielectric constant.

It is well known from optics that the frequency of electromagnetic wave and the material refractive index *n* are related to 0*c* (speed of light in vacuum)

$$\left|\nu\right|^2 = \frac{1}{\mu\_0 \varepsilon\_0} \frac{1}{\varepsilon\_r} = \frac{c\_0^2}{\varepsilon\_r} \quad \text{and} \quad n = \frac{c}{\nu} \tag{4}$$

where 0*c* = 3·108 m/s.

The solution of the wave equation can be expressed:

$$E(z,t) = \text{Re}\{\overline{E}(z,\alpha)\exp(-i\overline{k}\overline{r} + i\alpha t)\}\tag{5}$$

where *z* is the direction of propagation, 2 the circular frequency and *k* is the wave vector ( *k* 2 / , / *c* the wavelength).


*g d v dk* (6)


$$w\_{\rm plt} = \frac{\alpha \rho}{k} = \frac{c}{n} \tag{7}$$

and the Beer-Lambert law

228 Femtosecond–Scale Optics

mechanical level (taking into account the electronic, vibration and rotation energy levels

<sup>0</sup> 0 0 *D E PE B H MH*

where *D* - electric flux density, *E* - electric field vector, *B* - magnetic flux density, *H* -

permeability of free space = 4·10-7 H/m (we will care materials for which 0 *M* 0 *B H*

<sup>0</sup> *PE E* ( ) 

> 

22 2

*tt t*

*c c*

 (4)

 

(3)

00 00 22 2 0 0 *r EE E <sup>E</sup>*

2

1 1 and *r r*

 

*E z t E z ikr i t* ( , ) Re{ ( , )exp( )} 

> 

> > *g d*

*dk* 

*c*

*k n* 

*v*

*ph*

*v*

*n*

 


( ) (1)

(2)

of electromagnetic wave and the material

the circular frequency and *k* is the wave

(6)

(7)

(5)

0 -

).

of the atoms).

**3. Linear optics approximation**  From Maxwell's equations we get:

magnetic field vector, 0

The polarization

where (1 ) *<sup>r</sup>* 

where 0*c* = 3·108 m/s.

vector ( *k* 2 /



and the Beer-Lambert law

 , / *c*

For the materials we have the

where   - electric susceptibility. The wave equation for electric field:

It is well known from optics that the frequency

The solution of the wave equation can be expressed:

where *z* is the direction of propagation, 2

2

is the relative dielectric constant.

2 0 0 0

 

the wavelength).

refractive index *n* are related to 0*c* (speed of light in vacuum)

( ) and

$$I(z) = I\_0 e^{-\alpha z} \tag{8}$$

where 0*I* is the intensity of the light at the incidence plane, *I z*( ) the intensity in the depth level z, and is the absorption coefficient.

The real ( ) and imaginary ( ) parts of dielectric constants at given frequency are related to optical material parameters *n* and *k* through Kramers-Kronig relations.

$$m = \sqrt{\frac{\sqrt{(\varepsilon')^2 + (\varepsilon'')^2} + \varepsilon'}{2}}, \quad k = \sqrt{\frac{\sqrt{(\varepsilon')^2 + (\varepsilon'')^2} - \varepsilon'}{2}}.\tag{9}$$

The above-mentioned equations describe the light – matter interaction events for relatively low beam intensities where the material constants do not depend on the intensity, but only on the frequency.

## **4. Nonlinear optics approximation**

For the laser beam intensities comparable to the electric field strength inside the atoms we have to take into account the higher-order (nonlinear) terms in the field to get the polarization *P* . (For H we got <sup>9</sup> 16 2 *E V cm I W cm* ~ 10 / and ~ 10 / )

$$
\overline{P}(t) = \varepsilon\_0 \left( \mathcal{X}^{(1)} \overline{E} + \mathcal{X}^{(2)} \overline{E}^2 + \mathcal{X}^{(3)} \overline{E}^3 + \dots \right) \tag{10}
$$

where ( ) *<sup>n</sup>* is the nth order of the susceptibilities of the medium.

As a result we can get different phenomena e.g., 2nd and higher order harmonic generation, sum- and difference frequency generation, parametric generation, etc.

We note that in all processes the conservation laws for photon energy and momentum (phase matching) should be fulfilled as

$$
\rho = o\_1 + o\_2 + o\_3 + \dots \quad \text{and} \quad \overline{k} = \overline{k}\_1 + \overline{k}\_2 + \overline{k}\_3 + \dots \tag{11}
$$

Non-linear optical effects usually called as N wave mixing processes, where N is the number of photons participating in reactions. The more photons the weaker the probability of the effect. At very high laser intensities e.g., at very short pulses very high order effects might have been realized. The materials usually have different refractive indices for different frequencies and polarizations; therefore the phase matching for them will be satisfied only for special directions e.g., second harmonics birefringence.

A number of special methods have been developed (due to nonlinear crystals to produce effects such as second harmonic generation (SHG), optical parametric generation (OPG), optical parametric oscillation (OPO), optical parametric amplification (OPA), quick switching features as electrooptical Kerr and Pockel cells [2-4].

#### **5. The spectroscopy**

#### **5.1 General remarks**

Spectroscopy in general is the study of the interaction between light and matter such as determination of quantum energy levels in substrates (gas, solid, liquid, etc.). In this "traditional" simple form, one measures the spectroscopic components of the reflected beam

Ultrafast Time-Resolved Spectroscopy 231

Ultrafast spectroscopy is based on using ultrashort laser pulses with pulse duration of ps-fs time region. This technique involves temporally short, therefore spectrally broad light pulses. These kinds of pulses are used to probe directly the dynamics of the system rather than the energy levels themselves. It is very important that the duration of the laser pulses

Taking into account the quantum mechanical considerations we reach the appearance of uncertainty principle e.g., time and energy resolution are related to each other through the

Elementary excitations in solids show a complex nonequilibrium behavior. The fastest nonequilibrium processes occur on ultrafast time scales and strongly influence both optical properties and carrier transport. Among condensed phases metals and semiconductors represent an interesting class of practically important targets of nano and microelectronics. During these processes the electronic band structure, optical transition energies, carrier concentration, and phonon frequencies vary over a broad range leading to a variety of ultrafast phenomena. Moreover, the quantum confinement of wave functions in low-dimensional

Optical spectroscopy with ultrashort pulses provides direct insight into these processes

High amount of laser light energy can be deposited in a very small volume determined by the laser focal spot and penetration depth at a given wavelength. The electromagnetic incident wave will lead to photo-excitation of the electrons due to the large difference of electron and phonon heat capacities (*c*p >> *c*e). Therefore in the target material, especially in the case of metals and metal-based nanostructures – one creates a non-equilibrium electron distribution leaving the lattice temperature essentially unchanged (*T* ~ 300 K). The rise time

on the full width at the maximum level (FWHM) can be written as

 

semiconductor nanostructures allows a systematic variation of material properties.

= 8,8103 Hz (2940 cm–1).

  of the pulse and its temporal

2 (2)/ *n* (12)

must be shorter than the time scale of the dynamics that one wants to observe.

Laser spectroscopy

**6. Ultrafast spectroscopy** 

Fourier transformation [7].

= 5 fs we get

duration

e.g., if 

laser-induced fluorescence spectroscopy (LIF)

Newton activation analysis spectroscopy (NAA)

 extended X-ray absorption fine structure (EXAFS) and numerous other combined type spectroscopic methods [7-8].

resonance ionization spectroscopy (RIS)

Mossbauer spectroscopy (MOSSBAUER)

X-ray fluorescence spectroscopy (XRF)

For Gaussian profile pulses the spectral bandwidth

occurring on a time scale between about 10-14 and 10-10 s.

**7. Ultrafast laser excitation in materials 7.1 Impact of laser beam energy to a matter** 

Raman spectroscopy (RAMAN)

X-ray and -ray spectroscopies

after transmitted through the medium or emitted from it due to the external excitation of the energetic levels. In a classical his performance one uses – normally – spectrally narrow light beam. This beam may be tuned across discrete energetic levels of the studied target. Different regions of the electromagnetic spectrum provide different kinds of information as a result of the interaction.

The spectroscopic instrument represents - as its main part - a dispersion element (prism or grating with high separation capability of beam wavelengths). Usually this consists also of a slit, light collecting optics, and a detector (see Fig. 1.) called monochromator.

Fig. 1. A general scheme of a monochromator

Depending of the target material structure, composition, and the light-matter interaction type such as direct absorption, transmission, emission of scattering (type Rayleigh, Brillouin, Raman) one can summarize the classical spectroscopy methods as follow [5-11].

	- absorption (AA)
	- emission (AES, OES)
	- fluorescence (AFS)
	- Auger electron spectroscopy (AES)
	- X-ray photoelectron spectroscopy (XPS, ESCA)
	- molecular spectroscopy
	- near infrared absorption spectroscopy (NIR)
	- ultraviolet and visible spectroscopy (UV-VIS)
	- electron paramagnetic resonance spectroscopy (EPR, ESR)
	- nuclear magnetic resonance spectroscopy (NMR)

Laser spectroscopy

230 Femtosecond–Scale Optics

after transmitted through the medium or emitted from it due to the external excitation of the energetic levels. In a classical his performance one uses – normally – spectrally narrow light beam. This beam may be tuned across discrete energetic levels of the studied target. Different regions of the electromagnetic spectrum provide different kinds of information as

The spectroscopic instrument represents - as its main part - a dispersion element (prism or grating with high separation capability of beam wavelengths). Usually this consists also of a

Depending of the target material structure, composition, and the light-matter interaction type such as direct absorption, transmission, emission of scattering (type Rayleigh, Brillouin,

Raman) one can summarize the classical spectroscopy methods as follow [5-11].

slit, light collecting optics, and a detector (see Fig. 1.) called monochromator.

a result of the interaction.

Fig. 1. A general scheme of a monochromator

Auger electron spectroscopy (AES)

X-ray photoelectron spectroscopy (XPS, ESCA)

 near infrared absorption spectroscopy (NIR) ultraviolet and visible spectroscopy (UV-VIS)

nuclear magnetic resonance spectroscopy (NMR)

electron paramagnetic resonance spectroscopy (EPR, ESR)

 Atomic spectroscopy: absorption (AA) emission (AES, OES) fluorescence (AFS) Electron spectroscopy:

Infrared spectroscopy

molecular spectroscopy

Nuclear and electron spectroscopy

Fourier transform spectroscopy (FT)


and numerous other combined type spectroscopic methods [7-8].

## **6. Ultrafast spectroscopy**

Ultrafast spectroscopy is based on using ultrashort laser pulses with pulse duration of ps-fs time region. This technique involves temporally short, therefore spectrally broad light pulses. These kinds of pulses are used to probe directly the dynamics of the system rather than the energy levels themselves. It is very important that the duration of the laser pulses must be shorter than the time scale of the dynamics that one wants to observe.

Taking into account the quantum mechanical considerations we reach the appearance of uncertainty principle e.g., time and energy resolution are related to each other through the Fourier transformation [7].

For Gaussian profile pulses the spectral bandwidth of the pulse and its temporal duration on the full width at the maximum level (FWHM) can be written as

$$
\Delta \nu \tau \ge 2 \ln(2) / \pi \tag{12}
$$

e.g., if = 5 fs we get = 8,8103 Hz (2940 cm–1).

## **7. Ultrafast laser excitation in materials**

## **7.1 Impact of laser beam energy to a matter**

Elementary excitations in solids show a complex nonequilibrium behavior. The fastest nonequilibrium processes occur on ultrafast time scales and strongly influence both optical properties and carrier transport. Among condensed phases metals and semiconductors represent an interesting class of practically important targets of nano and microelectronics. During these processes the electronic band structure, optical transition energies, carrier concentration, and phonon frequencies vary over a broad range leading to a variety of ultrafast phenomena. Moreover, the quantum confinement of wave functions in low-dimensional semiconductor nanostructures allows a systematic variation of material properties.

Optical spectroscopy with ultrashort pulses provides direct insight into these processes occurring on a time scale between about 10-14 and 10-10 s.

High amount of laser light energy can be deposited in a very small volume determined by the laser focal spot and penetration depth at a given wavelength. The electromagnetic incident wave will lead to photo-excitation of the electrons due to the large difference of electron and phonon heat capacities (*c*p >> *c*e). Therefore in the target material, especially in the case of metals and metal-based nanostructures – one creates a non-equilibrium electron distribution leaving the lattice temperature essentially unchanged (*T* ~ 300 K). The rise time

Ultrafast Time-Resolved Spectroscopy 233

Resonant interaction of a coherent ultrashort pulse with a particular transition in the semiconductor creates both a coherent optical polarization between the optically coupled states and carriers (electrons or holes) from energetically lower to higher states in the same

With time evaluation this well-defined phase relation is destroyed by a variety of scattering processes which change the relative phase of the wave function between the ground- and excited states. This phase relaxation or so-called dephasing process means a fast decay of the macroscopic polarization and results in a homogeneous broadening of the particular optical

Therefore, the overall excitation-relaxation process could be characterized as it can be seen

As it had been mentioned before the ultrashort laser pulses provide an excellent tools to realize time-resolved experiments with which one can observe transient species in different chemical reactions and follow the dynamical behavior of physical-, chemical- and biological processes. Another important property is that with modest energy, the fs pulses can have huge peak powers. This also makes them suitable for many tasks that we would not normally think of as 'time resolved', including laser ablation of materials, multi-photon absorption (for imaging of biological materials), fragmentation (e.g., DNA into fragments that may be analyzed using mass spectrometry), the conversion to a range of new wavelengths using nonlinear techniques, e.g., infrared light to visible light conversion and 2-photon excited fluorescence, etc**.** Semiconductor processes and collisions in liquid phase

Fig. 2. The scheme of excitation-relaxation processes [12]

materials are also in the range of a few hundreds of fs [18-20].

**8. Measurements and instrumentations** 

or a different band.

transition.

in Fig. 2.

of non-degenerate electron distribution creation is in the order of a few fs, thus we can say that high temperature non-equilibrium electron distribution has the same rise time as the laser pulse duration. Then over a time scale of a few fs the non-equilibrium electrons redistribute their energy among themselves. It takes place through e.g., electron – electron coulomb interactions resulting in a local equilibrium with temperature *Te*, called the thermalized electron redistribution (with relaxation time *e e* ).

The excited thermalized electron gas then transforms the energy through electron – phonon interactions within a relaxation time *<sup>e</sup> <sup>p</sup>* .

Energy transports between electron and phonon subsystems [12-16].

This mass energy transferred to phonon bath will be redistributed among phonons during the relaxation time *p p* leading to the equilibrium phonon temperature *Tl*.

Therefore we can consider the kinetic evolution of a photo-excited electron – phonon system.


In semiconductors and complex nanostructures, the relaxation processes are multistep ones and include different mechanism such as:

Below band gap excitations:

Transitions between electronic states may have different origins, such as:

Transitions from atoms or vacancies,

Transitions from impurity levels into the valence or conduction band continuum,

Transitions (indirect) between excited intraband levels,

Transitions due to the so-called free carrier absorption,

 Inter-valence band transitions of holes, and intersubband transitions between valence and conduction subbands

Transitions in low-dimensional semiconductor nanostructures, e.g., quantum wells, wires and dots.

Indirect interband excitations: free carrier absorption due to the presence of free charges in both conduction and valence bands. This requires coupling to a third particle, e.g., a phonon or an impurity because of the conservation laws for wave number vectors.

Inter valence band transitions: due to dipole-allowed transitions of free holes from states in one valence band to states of higher energy in another valence band. For bulk semiconductors with a diamond-like, e.g., silicon and germanium, or zinc-blende lattice like most III-V semiconductors, inter-valence band absorption is dominated by transitions between the heavy hole (HH) and light hole (LH and split-off bands).

Intersubband transitions in quasi-two-dimensional nanostructures: They are characteristics for quantum wells or superlattices in which carrier motion is restricted to a quasi-twodimensional semiconductor layer. Quantum confinement occurs in a situation where the length scale of the potential structure, i.e., the well width, is on the order of the de-Broglie wavelength of the carriers.

Dephasing of coherent polarizations

232 Femtosecond–Scale Optics

of non-degenerate electron distribution creation is in the order of a few fs, thus we can say that high temperature non-equilibrium electron distribution has the same rise time as the laser pulse duration. Then over a time scale of a few fs the non-equilibrium electrons redistribute their energy among themselves. It takes place through e.g., electron – electron coulomb interactions resulting in a local equilibrium with temperature *Te*, called the

The excited thermalized electron gas then transforms the energy through electron – phonon

This mass energy transferred to phonon bath will be redistributed among phonons during



In semiconductors and complex nanostructures, the relaxation processes are multistep ones

Transitions from impurity levels into the valence or conduction band continuum,

Inter-valence band transitions of holes, and intersubband transitions between valence

Transitions in low-dimensional semiconductor nanostructures, e.g., quantum wells,

Indirect interband excitations: free carrier absorption due to the presence of free charges in both conduction and valence bands. This requires coupling to a third particle, e.g., a phonon

Inter valence band transitions: due to dipole-allowed transitions of free holes from states in one valence band to states of higher energy in another valence band. For bulk semiconductors with a diamond-like, e.g., silicon and germanium, or zinc-blende lattice like most III-V semiconductors, inter-valence band absorption is dominated by transitions

Intersubband transitions in quasi-two-dimensional nanostructures: They are characteristics for quantum wells or superlattices in which carrier motion is restricted to a quasi-twodimensional semiconductor layer. Quantum confinement occurs in a situation where the length scale of the potential structure, i.e., the well width, is on the order of the de-Broglie

 leading to the equilibrium phonon temperature *Tl*. Therefore we can consider the kinetic evolution of a photo-excited electron – phonon

).

thermalized electron redistribution (with relaxation time *e e*

.

Energy transports between electron and phonon subsystems [12-16].

thermal processes. The average time scale is about 1-500 fs.

Transitions between electronic states may have different origins, such as:

or an impurity because of the conservation laws for wave number vectors.

between the heavy hole (HH) and light hole (LH and split-off bands).

Transitions (indirect) between excited intraband levels, Transitions due to the so-called free carrier absorption,

interactions within a relaxation time *<sup>e</sup> <sup>p</sup>*

case of metals is about 1 ps to a few ns.

and include different mechanism such as:

Transitions from atoms or vacancies,

Below band gap excitations:

and conduction subbands

wires and dots.

wavelength of the carriers.

Dephasing of coherent polarizations

the relaxation time *p p*

system.

Resonant interaction of a coherent ultrashort pulse with a particular transition in the semiconductor creates both a coherent optical polarization between the optically coupled states and carriers (electrons or holes) from energetically lower to higher states in the same or a different band.

With time evaluation this well-defined phase relation is destroyed by a variety of scattering processes which change the relative phase of the wave function between the ground- and excited states. This phase relaxation or so-called dephasing process means a fast decay of the macroscopic polarization and results in a homogeneous broadening of the particular optical transition.

Therefore, the overall excitation-relaxation process could be characterized as it can be seen in Fig. 2.

Fig. 2. The scheme of excitation-relaxation processes [12]

## **8. Measurements and instrumentations**

As it had been mentioned before the ultrashort laser pulses provide an excellent tools to realize time-resolved experiments with which one can observe transient species in different chemical reactions and follow the dynamical behavior of physical-, chemical- and biological processes. Another important property is that with modest energy, the fs pulses can have huge peak powers. This also makes them suitable for many tasks that we would not normally think of as 'time resolved', including laser ablation of materials, multi-photon absorption (for imaging of biological materials), fragmentation (e.g., DNA into fragments that may be analyzed using mass spectrometry), the conversion to a range of new wavelengths using nonlinear techniques, e.g., infrared light to visible light conversion and 2-photon excited fluorescence, etc**.** Semiconductor processes and collisions in liquid phase materials are also in the range of a few hundreds of fs [18-20].

Ultrafast Time-Resolved Spectroscopy 235

The delay in the probe arm is usually realized with an optical path enhancement done by a

As we can see the main laser beam is split with a mirror into 2 parts: pump beam with intensity of about 90% of the original and a probe beam of about 10 % of original. Both pulses are focused upon the target with their spatial overlapping. The delay is realized with variation of a beam path length compared to probe one. The weaker pulse in some of his characteristic (e.g., intensity, polarization, temporal duration) will be modified varying the delay *(Δt).* This is the results of excitation in target material by the pump beam. Repeating the measurements by varying the time delay one determines the temporal dynamics of the

In some of more sophisticated measurements, one tries to use a focusing object as it can be

Fig. 5. Schematic of the transient grating experiment. Two excitation pulses are crossed in time and space in the sample. The resultant spatially periodic material excitation is probed

Different variations of this technique can be used to determine dynamics of events in different fields, such as electron transport in solids, hetero- and nanostructures, induced spin dynamics by magnetic influences etc. Numerous applications had already been developed for chemistry, biology, and life sciences. The resolution achieved by pump and

by diffraction of a third, variably delayed beam [32].

probe method nowadays reaches as hundreds of attoseconds.

mirror system (Fig 4.).

excitation.

seen in Fig 5.

Direct measurements in fs region are not possible using electrical methods and other nonoptical techniques. The use of specialized photodetectors such as streak cameras or avalanche photodiodes that can resolve picosecond or even 100s of femtoseconds transients in real-time, but are not able to resolve a necessary few fs events, therefore alternative detection techniques are required.

The techniques that are used most frequently are based on the auto- or cross-correlation of two beams of femtosecond pulses. If the target is a nonlinear crystal used for sum-frequency generation, this technique can be used to determine the shape and relative arrival time of two short pulses. If the sample consists absorbing materials normally one uses pump-probe experiments for temporal registrations of events [1,21-30]. Therefore, if we want to measure the dynamics of a fast event, we have to apply a faster tool to do it. Moreover, the use of a not as short as possible laser pulse can induce the shortening transient behavior [31]. The most commonly used scheme of a general pump and probe equipment is sketched in Fig 3. [24]

Fig. 3. Schematic of a general pump and probe equipment [24]

Fig. 4. Sketch of pump and probe for different sources

234 Femtosecond–Scale Optics

Direct measurements in fs region are not possible using electrical methods and other nonoptical techniques. The use of specialized photodetectors such as streak cameras or avalanche photodiodes that can resolve picosecond or even 100s of femtoseconds transients in real-time, but are not able to resolve a necessary few fs events, therefore alternative

The techniques that are used most frequently are based on the auto- or cross-correlation of two beams of femtosecond pulses. If the target is a nonlinear crystal used for sum-frequency generation, this technique can be used to determine the shape and relative arrival time of two short pulses. If the sample consists absorbing materials normally one uses pump-probe experiments for temporal registrations of events [1,21-30]. Therefore, if we want to measure the dynamics of a fast event, we have to apply a faster tool to do it. Moreover, the use of a not as short as possible laser pulse can induce the shortening transient behavior [31]. The most commonly used scheme of a general pump and probe equipment is sketched in Fig 3. [24]

detection techniques are required.

Fig. 3. Schematic of a general pump and probe equipment [24]

Fig. 4. Sketch of pump and probe for different sources

The delay in the probe arm is usually realized with an optical path enhancement done by a mirror system (Fig 4.).

As we can see the main laser beam is split with a mirror into 2 parts: pump beam with intensity of about 90% of the original and a probe beam of about 10 % of original. Both pulses are focused upon the target with their spatial overlapping. The delay is realized with variation of a beam path length compared to probe one. The weaker pulse in some of his characteristic (e.g., intensity, polarization, temporal duration) will be modified varying the delay *(Δt).* This is the results of excitation in target material by the pump beam. Repeating the measurements by varying the time delay one determines the temporal dynamics of the excitation.

In some of more sophisticated measurements, one tries to use a focusing object as it can be seen in Fig 5.

Fig. 5. Schematic of the transient grating experiment. Two excitation pulses are crossed in time and space in the sample. The resultant spatially periodic material excitation is probed by diffraction of a third, variably delayed beam [32].

Different variations of this technique can be used to determine dynamics of events in different fields, such as electron transport in solids, hetero- and nanostructures, induced spin dynamics by magnetic influences etc. Numerous applications had already been developed for chemistry, biology, and life sciences. The resolution achieved by pump and probe method nowadays reaches as hundreds of attoseconds.

Ultrafast Time-Resolved Spectroscopy 237

spacing. Under these conditions X-ray diffraction would be strongly dominated by the bulk

In such a way, different X-ray spectroscopical techniques have been developed as X-ray absorption spectroscopy (XAS), extended X-ray absorption fine structure spectroscopy (EXAFS) absorption near edge spectroscopy (XANES) inelastic X-ray Raman scattering

Fig. 7. Left: Concept of XRS. The energy transfer from an inelastically scattered photon results in the excitation of a core electron into an empty state. Right: Complete scattering spectrum from graphite. Intensity versus incident energy E0 is plotted, analyzer energy E' is

Concerning the instrumentation different kinds of wavelength dispersive devices are in utilizations for spectroscopical applications e.g., cylindrically curved analyzers and position

Fig. 8. Schematic setup four arrays of cylindrically curved crystals in sagital focusing mode. Scattering of a point source beam is analyzed at different energies (see vertical cut) resulting

in a spectrum on the PSDs. For XRS the setup is rotated by 90o for scattering in the

As a sample of nice characteristic results of XANES/EXAFS we turn to Fig. 9.

crystal ignoring the damaged or melted subsurface layers.

As an example, we demonstrate the concept of XRS (Fig. 7.).

(XRS), and X-ray emission spectroscopy (XES).

fixed at 6460 eV [41].

sensitive detectors (PSD) (see Fig. 8)

predominantly vertical plane [41].

## **9. Ultrafast X-ray spectroscopy**

X-rays are very useful tools of modern science as well as solid state physics. The determination of the atomic structures became possible with achievement of coherent X-ray applications. In that frame one uses the static X-ray diffraction technique based on Bragg reflections.

However, the appearance of new pulsed coherent X-ray sources with extremely short pulse duration had opened a way for time dependent investigations. Femtosecond X-ray pulses enable atomic spatial (~0,1 mm) and high enough temporal resolution to observe the evolution of atomic configurations. In such a way one gets a direct dynamic structural picture [31-35].

Until now, a variety of methods have been developed to generate fs X-ray beams. For example, during the interaction of very high intensity laser pulses with material due to results of electron-atom interaction processes one yields to characteristic brehmstrahlung and line emission. The time duration of X-ray beam generated like as generating fs laser pulse duration, and the energies are in range of 10 eV ~ 1 MeV. Also high intensity coherent X-ray beams may be emerged from laser-produced plasma sources or laser-driven electron X-ray sources and synchrotron radiation induced sources [31, 33, 35].

To perform time-resolved measurements in the X-ray regime one can use suitable variants of pump and probe techniques like in optical region (Fig. 6.).

Fig. 6. Schematic of an optical pump- X-ray probe experiment [42].

One of the advantages of using X-ray beams for spectroscopic aims is the deeper penetration of coherent X-ray beam into the material if the wavelength is less or in the order of lattice 236 Femtosecond–Scale Optics

X-rays are very useful tools of modern science as well as solid state physics. The determination of the atomic structures became possible with achievement of coherent X-ray applications. In that frame one uses the static X-ray diffraction technique based on Bragg

However, the appearance of new pulsed coherent X-ray sources with extremely short pulse duration had opened a way for time dependent investigations. Femtosecond X-ray pulses enable atomic spatial (~0,1 mm) and high enough temporal resolution to observe the evolution of atomic configurations. In such a way one gets a direct dynamic structural

Until now, a variety of methods have been developed to generate fs X-ray beams. For example, during the interaction of very high intensity laser pulses with material due to results of electron-atom interaction processes one yields to characteristic brehmstrahlung and line emission. The time duration of X-ray beam generated like as generating fs laser pulse duration, and the energies are in range of 10 eV ~ 1 MeV. Also high intensity coherent X-ray beams may be emerged from laser-produced plasma sources or laser-driven electron

To perform time-resolved measurements in the X-ray regime one can use suitable variants

X-ray sources and synchrotron radiation induced sources [31, 33, 35].

Fig. 6. Schematic of an optical pump- X-ray probe experiment [42].

One of the advantages of using X-ray beams for spectroscopic aims is the deeper penetration of coherent X-ray beam into the material if the wavelength is less or in the order of lattice

of pump and probe techniques like in optical region (Fig. 6.).

**9. Ultrafast X-ray spectroscopy** 

reflections.

picture [31-35].

spacing. Under these conditions X-ray diffraction would be strongly dominated by the bulk crystal ignoring the damaged or melted subsurface layers.

In such a way, different X-ray spectroscopical techniques have been developed as X-ray absorption spectroscopy (XAS), extended X-ray absorption fine structure spectroscopy (EXAFS) absorption near edge spectroscopy (XANES) inelastic X-ray Raman scattering (XRS), and X-ray emission spectroscopy (XES).

As an example, we demonstrate the concept of XRS (Fig. 7.).

Fig. 7. Left: Concept of XRS. The energy transfer from an inelastically scattered photon results in the excitation of a core electron into an empty state. Right: Complete scattering spectrum from graphite. Intensity versus incident energy E0 is plotted, analyzer energy E' is fixed at 6460 eV [41].

Concerning the instrumentation different kinds of wavelength dispersive devices are in utilizations for spectroscopical applications e.g., cylindrically curved analyzers and position sensitive detectors (PSD) (see Fig. 8)

Fig. 8. Schematic setup four arrays of cylindrically curved crystals in sagital focusing mode. Scattering of a point source beam is analyzed at different energies (see vertical cut) resulting in a spectrum on the PSDs. For XRS the setup is rotated by 90o for scattering in the predominantly vertical plane [41].

As a sample of nice characteristic results of XANES/EXAFS we turn to Fig. 9.

Ultrafast Time-Resolved Spectroscopy 239

Fig. 10. Setup for the measuring of polarization change

Fig. 11. A typical setup for transmission THz spectroscopy

A typical setup for transmission THz spectroscopy is shown in Fig. 11.

Fig. 9. XAS spectrum of a molecule (PtPOP) in solution illustrating the two regions: the lowenergy XANES region up to ~50 eV above the IP and the high-energy EXAFS region >50 eV. The spectrum has been normalized.

## **10. Time resolved THz spectroscopy**

In the optical wavelength scale the THz region makes a bridge between microwaves and infrared domains. This is located at about 1012 Hz, so called terahertz region. Because of the quite low phonon energies in this region, the terahertz spectroscopy mainly is devoted to carry investigations in the exploration of infraband/subband excitations (transition). The Trays are harmless for the human body; therefore, one can find applications in basic medical research and security [34].

The materials used for generation of terahertz radiation by optical rectification can also be used for its detection by using the Pockels effect where certain crystalline materials become birefringent in the presence of an electric field. The birefringence caused by the electric field of a terahertz pulse leads to a change in the optical polarization of the detection pulse, proportional to the terahertz electric-field strength. With the help of polarizers and photodiodes, this polarization change can be measured.

238 Femtosecond–Scale Optics

Fig. 9. XAS spectrum of a molecule (PtPOP) in solution illustrating the two regions: the lowenergy XANES region up to ~50 eV above the IP and the high-energy EXAFS region >50 eV.

In the optical wavelength scale the THz region makes a bridge between microwaves and infrared domains. This is located at about 1012 Hz, so called terahertz region. Because of the quite low phonon energies in this region, the terahertz spectroscopy mainly is devoted to carry investigations in the exploration of infraband/subband excitations (transition). The Trays are harmless for the human body; therefore, one can find applications in basic medical

The materials used for generation of terahertz radiation by optical rectification can also be used for its detection by using the Pockels effect where certain crystalline materials become birefringent in the presence of an electric field. The birefringence caused by the electric field of a terahertz pulse leads to a change in the optical polarization of the detection pulse, proportional to the terahertz electric-field strength. With the help of polarizers and

The spectrum has been normalized.

research and security [34].

**10. Time resolved THz spectroscopy** 

photodiodes, this polarization change can be measured.

Fig. 10. Setup for the measuring of polarization change A typical setup for transmission THz spectroscopy is shown in Fig. 11.

Fig. 11. A typical setup for transmission THz spectroscopy

Ultrafast Time-Resolved Spectroscopy 241

As an example of a possible IR ultrashort arrangement, we show a scheme of experimental

equipment devoted to examine charge-carrier dynamics in a polymer (PoV) (Fig. 13).

Fig. 13. Diagrams of the beam and sample geometries used for (A) 2D IR and (B) visible pump IR probe experiments. These geometries eliminate non-resonant signals that

array detector [33].

frequently interfere with ultrafast vibrational spectroscopic studies. The symbols represent: etalon—tunable Fabry-Perot interference filter, T—optical delay between the pump and probe pulses, P—polarizer, MCT Array—mercury cadmium telluride multichannel infrared

As a nice example of THz transient conductivity spectrum we do example of Si measured and fitted by Drude model (Fig. 12) [37].

Fig. 12. Variation of conductivity in Si [37]

## **11. Ultrafast infrared spectroscopy**

In the infrared spectral region (1-25 m wavelengths) the consequences of Heisenberg uncertainty principle are especially significant, e.g., for a 200 fs pulse duration we get spectral bandwidth about 75 cm–1.

For investigations of large and complex molecules the use of visible spectrum is not so convenient because of the overlapping features due to broad spectrum one can get only a few structural information. Therefore, to get useful information with high temporal resolution and sensitivity with proper reliability, we usually use combined (visible and IR) pump and probe variations [38].

240 Femtosecond–Scale Optics

As a nice example of THz transient conductivity spectrum we do example of Si measured

In the infrared spectral region (1-25 m wavelengths) the consequences of Heisenberg uncertainty principle are especially significant, e.g., for a 200 fs pulse duration we get

For investigations of large and complex molecules the use of visible spectrum is not so convenient because of the overlapping features due to broad spectrum one can get only a few structural information. Therefore, to get useful information with high temporal resolution and sensitivity with proper reliability, we usually use combined (visible and IR)

and fitted by Drude model (Fig. 12) [37].

Fig. 12. Variation of conductivity in Si [37]

**11. Ultrafast infrared spectroscopy** 

spectral bandwidth about 75 cm–1.

pump and probe variations [38].

As an example of a possible IR ultrashort arrangement, we show a scheme of experimental equipment devoted to examine charge-carrier dynamics in a polymer (PoV) (Fig. 13).

Fig. 13. Diagrams of the beam and sample geometries used for (A) 2D IR and (B) visible pump IR probe experiments. These geometries eliminate non-resonant signals that frequently interfere with ultrafast vibrational spectroscopic studies. The symbols represent: etalon—tunable Fabry-Perot interference filter, T—optical delay between the pump and probe pulses, P—polarizer, MCT Array—mercury cadmium telluride multichannel infrared array detector [33].

Ultrafast Time-Resolved Spectroscopy 243

It is well known that VO2 at moderate temperature T > 67 °C goes through a phase

The conductivity changes dramatically (~ about 105 fold) while the crystal structure changes

Fig. 15. The "structural bottleneck" in the transition arises from the time needed for the vanadium atoms to change from the monoclinic structure associated with the insulator to

Fig. 16. The reflectivity of the metallic state is higher than that of the insulating state; use of an ultrafast probe pulse allows the measurement of the transition time with 15 fs resolution

The difference in a reflectivity of the structures is also dominant.

transformation from an insulating to a metallic phase [41-43].

from monoclinic to rutile.

the rutile structure of the metal. [44].

(1 fs = 10-15 second) [44].

With the help of continuous wave IR source is used as a probe beam. Passing through the sample this probe beam is overlapped with fs pump beam. The time resolution is determined by applying a second fs visible pulse to gate the probe beam by up conversion. As a result of interaction we get a pulse with a sum frequency of IR and visible beams while the intensity will be related to the IR absorption.

The temporal delay changing of up conversion one can swap the dynamics of the event. Moreover due to such a up conversion linear detectors in visible are used with high sensitivity and resolution. The CW IR source usually have a very narrow line width, therefore the resolution depends only on homogeneous and inhomogeneous broadening caused by the sample itself.

There is another method avoiding the consequences of Heisenberg principle based on using a short IR pulse to be passed through the sample. After that this beam is dispersed in a monochromator to raise the frequency resolution [40].

In such a realization, the short pulse causes polarization field in the sample. As a polarization a coherent radiation emerges with the probe beam. Therefore, the resolution will be limited with the interaction time of the field with the sample (optical dephasing).

The ultrafast infrared spectroscopy proved to be an excellent tool to carry out structural and dynamical investigations in different areas of chemistry, biophysics, and organic chemistry, especially in the diagnostic of transient states. As an example of the dynamics of dissociation of ICN molecule is shown in Fig. 14.

Fig. 14. Dissociation of ICN molecule

## **12. Some results of ultrashort spectroscopy methods' applications in different fields of material science**

Observations of ultrafast phase transitions

242 Femtosecond–Scale Optics

With the help of continuous wave IR source is used as a probe beam. Passing through the sample this probe beam is overlapped with fs pump beam. The time resolution is determined by applying a second fs visible pulse to gate the probe beam by up conversion. As a result of interaction we get a pulse with a sum frequency of IR and visible beams while

The temporal delay changing of up conversion one can swap the dynamics of the event. Moreover due to such a up conversion linear detectors in visible are used with high sensitivity and resolution. The CW IR source usually have a very narrow line width, therefore the resolution depends only on homogeneous and inhomogeneous broadening

There is another method avoiding the consequences of Heisenberg principle based on using a short IR pulse to be passed through the sample. After that this beam is dispersed in a

In such a realization, the short pulse causes polarization field in the sample. As a polarization a coherent radiation emerges with the probe beam. Therefore, the resolution will be limited with the interaction time of the field with the sample (optical dephasing). The ultrafast infrared spectroscopy proved to be an excellent tool to carry out structural and dynamical investigations in different areas of chemistry, biophysics, and organic chemistry, especially in the diagnostic of transient states. As an example of the dynamics of dissociation

**12. Some results of ultrashort spectroscopy methods' applications in** 

the intensity will be related to the IR absorption.

monochromator to raise the frequency resolution [40].

caused by the sample itself.

of ICN molecule is shown in Fig. 14.

Fig. 14. Dissociation of ICN molecule

**different fields of material science** 

Observations of ultrafast phase transitions

It is well known that VO2 at moderate temperature T > 67 °C goes through a phase transformation from an insulating to a metallic phase [41-43].

The conductivity changes dramatically (~ about 105 fold) while the crystal structure changes from monoclinic to rutile.

Fig. 15. The "structural bottleneck" in the transition arises from the time needed for the vanadium atoms to change from the monoclinic structure associated with the insulator to the rutile structure of the metal. [44].

The difference in a reflectivity of the structures is also dominant.

Fig. 16. The reflectivity of the metallic state is higher than that of the insulating state; use of an ultrafast probe pulse allows the measurement of the transition time with 15 fs resolution (1 fs = 10-15 second) [44].

Ultrafast Time-Resolved Spectroscopy 245

[3] DeLong, K. W., R. Trebino, J. Hunter, and W. E. White, J. Opt. Soc. Amer. B, v. 11, p.

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p. 27

The dynamics of the changes of electron correlation and so far reflectivities was studied with pump and probe spectroscopic methods; using ultrashort X-ray spectroscopy. The time transition limit for phase transformation from monoclinic to rutile was found about 80 fs [44].

Fig. 17. The observed transition time for the insulator-metal transition (red data points) becomes progressively shorter with pulse duration down to 80 fs, at which point a fundamental limit is imposed by the material properties [44].

Direct band-to-band recombination in silicon nanocrystals and porous silicon

The silicon-bulk-crystal has interband optical transitions only with the participations of phonons (to evaluate the energy and momentum conservation laws). The reduction of size of silicon crystal e.g. nanosized or porous silicon show zero-phonon optical transitions which is promising for fabrication of light emitting devices for Si microelectronics. [44]

To demonstrate the existence of such a transition and to get data for dynamics of the different ultrafast spectroscopic methods had been used such as time resolved photoluminescence, pump and probe phonon echoes and for wave mixing.

## **13. References**


244 Femtosecond–Scale Optics

The dynamics of the changes of electron correlation and so far reflectivities was studied with pump and probe spectroscopic methods; using ultrashort X-ray spectroscopy. The time transition limit for phase transformation from monoclinic to rutile was found about 80 fs

Fig. 17. The observed transition time for the insulator-metal transition (red data points) becomes progressively shorter with pulse duration down to 80 fs, at which point a

The silicon-bulk-crystal has interband optical transitions only with the participations of phonons (to evaluate the energy and momentum conservation laws). The reduction of size of silicon crystal e.g. nanosized or porous silicon show zero-phonon optical transitions which is promising for fabrication of light emitting devices for Si microelectronics. [44] To demonstrate the existence of such a transition and to get data for dynamics of the different ultrafast spectroscopic methods had been used such as time resolved

[1] László Nánai, Zsolt Benkő, Renat R. Letfullin and Thomas F. George, Laser-Matter

Interactions: Nanostructures, Fabrication and Characterization in T. F. George, D. Jelski, R. R. Letfullin and G. P. Zhang, Editors, Computational Studies of New Materials II: From Ultrafast Processes and Nanostructures to Optoelectronics, Energy Storage and Nanomedicine (World Scientific, Singapore, 2011), 540 pages. [2] Trebino R., Frequency-resolved opticalgratings: The measurements of ultrashort pulses,

Direct band-to-band recombination in silicon nanocrystals and porous silicon

photoluminescence, pump and probe phonon echoes and for wave mixing.

fundamental limit is imposed by the material properties [44].

[44].

**13. References** 

Kluwer publisher 2002


A.V. Andreev, S.Yu. Stremoukhov and O.A.Shoutova *Physics Department, M.V. Lomonosov Moscow State University*

**of Intra-Atomic Field Strength** 

**Interaction of Atom with Laser Pulses** 

In spite of the twenty-year history, the effect of the high-order optical harmonic generation (HHG) is still under great interest of both experimentalists and theoreticians (see review (Ganeev, 2009)). The origin of this interest is manifold. From the practical point of view, the HHG is one of the effective mechanisms for producing coherent emission in broad range of electromagnetic wave spectrum. The plateau in the harmonic amplitude distribution in extreme ultraviolet (XUV) region affords grounds for development of subfemtosecond pulse formation methods (Paul, 2001). As a result, the new frontiers are opened up in science by extending the nonlinear optics and time-resolved spectroscopy to the XUV region (Papadogiannis, 2003) and pushing ultrafast science to the attosecond domain, enabling XUV spectroscopy and imaging of molecular orbitals (Itatani, 2004), surface dynamics (Tobey, 2007), and electron motion. Hence, the HHG effect opens up the new perspectives in

At present days, the efficiency of conversion to high-order harmonics turns out to be insufficient for using them as real coherent short-wavelength radiation sources in biology, plasma diagnostics, medicine, microscopy, photolithography, etc. Hence, the search for ways of increasing the cut-off frequency and HHG efficiency in the XUV spectral range is still among

The HHG effect was observed with the large number of periodic table elements having usually small and middle atomic numbers (Ganeev, 2009; 2005; 2007a;b; Redkin, 2010). As a rule, the interaction medium is gas jet, cell or laser plasma which is prepared by irradiation of metal surface by laser pre-pulse. The maximum harmonic order, or cut-off frequency (CF), obtained in plasma media to date varies from the sixties to seventies harmonics of fundamental frequency (Redkin, 2010; Suzuki, 2007). The highest-order harmonics (the 101st harmonic, *λ* =7,9 nm) have been obtained in manganese plasmas (Ganeev, 2007b). The efficiency of conversion in the plateau region amounts to <sup>∼</sup> <sup>10</sup>−<sup>5</sup> (Ganeev, 2007a). Recently, in experiments with silver plasma the CF values approximating seventies orders have been obtained (Ganeev, 2005). This value depends on both atomic levels structure and laser pulse parameters (intensity, energy and duration, envelope time-dependence, carrier-envelope phase). CF extension into higher frequencies band promises new possibilities in creating X-ray coherent sources and so the study of its behavior in dependence on media features and external laser

From the general point of view, it is clear that the emission spectrum is sensitive both to the spectral composition of the laser field and its polarization structure. At the beginning of

pulse parameters is the problem of significant scientific and technical interest.

**1. Introduction**

attosecond science (Corkum, 2007; Sola, 2006).

the most topical problems of nonlinear optics.

*Russia*

**10**

