#### 1. Introduction

There is a class of materials in which their optical and spectroscopic properties under irradiation vary depending on the density of the incident energy and the process is accompanied by the formation of a modulated thermal field, leading to a change in the optical characteristics of the medium. The mechanism of changing of the optical parameters responsible for the change in the permittivity of a substance due to the action of a modulated thermal field can be different [1 - 2]. One of them is associated with the expansion of the medium under the action of heat and, accordingly, a change in its density. The change in the density of the medium leads to a change in its refractive index, which can be used to modulate an external light beam. Depending on whether the imaginary, real or both parts of the substance permittivity change, an amplitude, phase or amplitude-phase lattice, respectively, is formed.

The practical application of such a scheme is connected with the finding of a medium having a sufficiently high expansion coefficient. A number of semiconductor materials, for example, chalcogenide vitreous semiconductors (including

In these materials a linear relationship between the amplitude of the refractive index changing

Here

The main disadvantage of these media is the need to heat the irradiated region of the film to a temperature that is close to its melting point to record data. This circumstance limits the possibilities of improving the technical parameters of recording media. Clearly, it is necessary to specify the power of the energy pulse with great accuracy, since a relatively small overheating of the film leads to its irreversible structural destruction, for example, to decay into droplets. Or vice versa, underheating to the desired temperature leads to the fact that the information will not be recorded. Such stringent requirements for the energy pulse parameters greatly complicate the use of recording media. Carrying out theoretical studies that allow to calculate the values of the given power of the energy pulse to obtain optimal temperature conditions for the surface of the recording layer as a function of its thermophysical characteristics and taking into account the parameters of the radiation source is one of the solutions to the problem. Publications on this issue are not enough. In the present work, a theoretical study of the features of the formation of flat holograms in absorbing substances changing their refractive index as a result of heat release is considered. Since the coefficient

#### 2. Heating of photothermal materials in an interference field created by laser

Typically, the recording medium is a kind of a layered dielectric-semiconductor structure that is formed from a recording layer enclosed between two transparent coatings, one of which is a substrate and the other is a thermally insulating film. Let two plane coherent waves of wavelength
*y*, but also on the coordinate *z* and the time *t*.

The distribution of spatially inhomogeneous surface heat sources in a thin semiconductor layer is determined by the nature of this interference pattern formed by two coherent beams with intensities described by the expression for the case of a rectangular pulse

(1)
where
*m* - period and depth of modulation of the light energy density in a semiconductor layer; (

In the general case,
*z*, and the distribution of the temperature field

(2)
Equation (2) is solved under the following conditions: the boundary *z* = 0 will be considered thermally insulated, that is

(3)
and the absorbing semiconductor layer

(4)
The initial condition for the induced value of the temperature field

(5)
where

Here,

Let us consider, depending on the conditions of the problem, various approximate solutions of the two-dimensional heat equation (2).

1. Consider the case when the recording of information is carried out in a pulsed mode and at a high radiation energy density. In this case conditions
*y* and *z* in (2), we easily find the temperature changes in the antinodes of the interference fringes along the y axis on the surface (*z* = 0) of the semiconductor layer.

(6)
The first term describes the temperature of the surface under uniform illumination (*k* = 0), and the second term - the contribution from the modulation of the interference of light waves, which depends on the parameter *k*. In the process of forming the thermal lattice, the spatial distribution of the phase of the temperature wave on the surface does not change with time, and remains proportional to

(7)
The temperature at the antinodes of the interference fringes will be higher by an amount *m* than the background temperature and

As stated above, the temperature change in the phase of the heat wave

(8)
As is known, the values

(9) The diffractive properties of light-induced gratings will be described using the concept of diffraction efficiency (DE). According to (9), the complex transmission of the thermal lattice is defined as

(10)
Taking into account the formula

(11)
Where
*S* is the transmission of a layer of thickness *d* at the wavelength

(12)
It can be seen that the DE depends quadratically on the density of the absorbed energy. As an illustration, Figure 1 shows the dependence

The temperature field on the surface can be represented in the form of two terms:

(13)
The first of them describes the temperature of the surface of the recording layer when it is uniformly illuminated, i.e. when

(14)
where

The second term is the contribution of the spatially modulated component, and it has the form:

(15)
Then, according to (13), the total surface temperature of the recording layer

(16)
where the coefficient (or amplitude) of the modulation
*y* axis is determined from the relation

(17)

As can be seen from this expression, the modulation factor depends on the spatial frequency, on the thermal diffusivity of the material, on the duration of the action of the laser radiation, and on the modulation of the reference and object beams. It does not depend on the power of the heating radiation and on the thermal conductivity of the material. The temperature at the antinodes of the interference fringes, according to (16), is determined by the relation

and it can be considered as a time-dependent Laplace function. Figure 2 shows an example of the distribution of a harmonic temperature field in a medium illuminated by laser radiation. The spatial frequency of the interference pattern is N = 1000 / mm. Depending on the spatial frequency,
*k* the temperature *T* becomes smaller than

Using the properties of the function

(18)
The fulfillment of condition (18) can be achieved in two ways: 1). If the exposure time is large,
*k*. 2). If the exposure time is fixed, i.e.

(19) In both cases, the modulation factor of the thermal field will tend to

(20) And the corresponding value of the temperature at the antinodes of the interference fringes becomes

(21)
For small values of the argument

(22)
The temperature of the interference bands

(23) and its amplitude does not depend on the spatial frequency.

Now, in accordance with (8), writing down the expression for the phase of the thermal wave as

(24) we have for the diffraction efficiency of the thermal phase lattice, by analogy with (10), the following relation:

(25) Taking into account (17), relation (25) can be rewritten in the following form:

(26)
When

(27)
And when

(28)
For small arguments

(29)

(30) Then the relation

(31)
behaves as a square of the Laplace function, depending on the argument

We note that the transition (26) to (28) is possible only under the condition that the function
*k*, there will be a need to increase the pulse energy in accordance with (21), so that the condition
*k* when the function

#### 3. Conclusion

Formulas that make it possible to estimate the required laser radiation power for recording information in the material under study, depending on the spatial frequency and the duration of the radiation exposure are obtained. It is shown that the amplitude of the modulation of the temperature field in the interference bands depends on the value of the given spatial frequency, and the recording of high frequencies requires an increase in the energy of the recording pulse. An analytical expression is found for calculating the diffraction efficiency of thermal gratings in the case of recording information due to the mechanism of thermal expansion of matter in photothermal media. It has been found that in order to increase the recording density in these media, it is necessary to select a material with a low thermal diffusivity and the recording is preferably carried out with short pulses to minimize the length of thermal diffusion.