Degenerate four-wave mixing in transparent two-component medium considering spatial structure of the pump waves

In this paper we investigate spatial selectivity of the degenerate four-wave radiation converter in transparent liquid containing nanoparticles considering spatial structure of the pump waves. The bandwidth of the most efficiently converted spatial frequencies is associated with the rotation and divergence of the pump waves.


Introduction
Accounting for spatial structure of the pump waves in the study of radiation (image) conversion quality by four-wave mixing is the necessary condition. The spatial structure of the pump waves fully determines the four-wave converters resolution [1,2] or it is one of the main factors leading to the deterioration of the image conversion quality [3].
In recent years the ability to use media containing micro-and nanosized particles (colloidal solutions, suspensions, etc.) for realization of the four-wave mixing is actively discussed [4][5][6][7][8][9][10]. If liquids containing nanoparticles are used as media in four-wave mixing then in such media electrostriction and Dufour effect influence significantly on the characteristics of four-wave radiation converter filtering high spatial frequencies of the object wave [11,12], which may be preferable in correction systems of small-scale phase inhomogeneities [13].
The aim of this work is to study influence of the divergence and rotation of the pump waves on spatial selectivity of the degenerate four-wave radiation converter in transparent liquid containing nanoparticles.

Relation of the interacting waves spatial spectra
Let us consider a plane layer thickness ℓ of transparent two-component medium (liquid in which nanoparticles are located). In liquid in the Z-axis direction two pump waves with complex amplitudes The intensity of radiation which propagates in nonlinear medium can be written as follows

   
Here, Then the temperature variation can be expressed as the sum of the quickly   In view of the foregoing, the Helmholtz equation (1)  ,.
We expand the interacting waves in plane waves and expand the quickly varying component of temperature in harmonic gratings Here, j A is spatial spectrum of jth wave,

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T  is spatial spectrum of the temperature grating, j  and Equations (7) and (8) then the spatial spectrum of the temperature grating and the spatial spectrum of the object wave on the front edge of the nonlinear layer where 0  is a parameter which characterizes the first pump wave divergence, 20  is a vector determining the propagation direction of the second pump wave. In view of (11) in the paraxial approximation the expression for the spatial spectrum of the object wave (10)     (b). Four-wave radiation converter with the rotated second pump wave filters low spatial frequencies along with the high spatial frequencies due to the presence of the non-zero wave mismatch projection included explicitly in the last term of the expression (10). A similar behaviour of the spatial spectrum module form with changing of the wave mismatch is observed for nondegenerate four-wave radiation converter in transparent two-component medium [12].
When the second pump wave is plane and propagates accurately along the Z-axis the spatial spectrum of the object wave is symmetrical about an axis passing through the zero spatial frequency and depends on (2), transparent twocomponent medium (1,2), medium with Kerr nonlinearity (3). Here, G is the parameter that determines the efficiency of four-wave radiation converter.
In constructing graphs we equated the spatial spectra modules of the object waves of two fourwave radiation converters at spatial frequency 40 5   . As seen, the spatial selectivity of four-wave radiation converters in transparent liquid containing nanoparticles and medium with Kerr nonlinearity is the same at high spatial frequencies.
In the area of spatial frequencies cut out by four-wave radiation converter the object wave amplitude is independent on the divergence and pump wave rotation.
To characterize the spatial selectivity of four-wave radiation converter in the pump waves flat we use the bandwidth of the most efficiently converted spatial frequencies defined as difference between spatial frequencies at which the maximum value of spatial spectrum module of the object wave is reduced by half (figure 2) [12].
Without the second pump wave rotation the reduction of the bandwidth of the most efficiently converted spatial frequencies with increasing of the first pump wave divergence (figure 3) is well described by the expression of the form Here, α and β are the parameters which are inversely proportional to the multiplication of the wave number on nonlinear layer thickness. As for the values 3 1.5 10 k ,