08 / 25 / 2011 - 14:26 — viktoriya

**УДК 535.14**

**OPTICAL WAVES INTRACAVITY INTERACTION IN DOMAIN STRUCTURES WITH QUADRATIC AND CUBIC NONLINEARITY**

**Rena Kasumova, Dr. of Sc., prof., Gulnara Safarova, magistr Baku State University**

*With account of phase changes of all interacting waves a comparative analysis has been made of efficiencies of doubling and tripling processes of laser radiation frequency in regular domain structures. The investigations have been carried out for the case when the constant –intensity approximation of basic radiation is applied not to regular domain structure as a whole but to each domain taken separately, what permits to make more precise analysis of wave interactions in similar structures.*

**Keywords: **intracavity frequency conversion, constant-intensity approximation, regular domain structures.

*Проведен сравнительный анализ эффективностей преобразования процессов удвоения и утроения частоты лазерного излучения в регулярной доменной структуре с учетом изменения фаз всех взаимодействующих волн. Исследования проведены для случая, когда приближение заданной интенсивности основного излучения применено не к регулярной доменной структуре как целой, а для каждого отдельно взятого домена, что позволяет проводить более корректный анализ взаимодействия волн в подобных структурах.*

**Ключевые слова:** внутрирезонаторное преобразование частоты, приближение заданной интенсивности, регулярные доменные структуры.

Quasi - phase matched interactions of light waves are usually put into effect on the account of the nonlinear gratings specially created in solid media. Realization of quasi - phase matched interactions of waves removes in essence one of crystal application restrictions in nonlinear optics – existence of wave synchronism or extends a field of applying already used nonlinear crystals [1]. Thus, for instance, in crystal LiNbO3 the tensor component of nonlinear susceptibility d33, which may be used only in quasi- phase matched interactions, is of the greatest importance. In recent years there has been made a break in elaboration of a number of technologies of making crystals with periodical modulation of nonlinear susceptibility: obtaining the volumetric regular domain structures in the course of crystal growth, chemical diffusion through a periodical mask drawn by lithografhic way, post-growth over polarization of crystals by a way of applying an electric field [2].

Presently, using quasi – phase matched interactions the coherent radiation from IR to UV range has been received [3-4]. Quasi – phase matched was used for creation of the sources of three basic colours (RGB sources) [5-7]. They use quasi – phase matched interaction at second harmonic generation for determining a quality of crystals. Elaboration of the small sources of a coherent radiation presents an actual problem of laser physics which may be solved using the newest achievements in this field and the modern methods in nonlinear optics [8-11].

Analysis of frequency tripling process in regular domain structures at consecutive quasi – phase matched interaction has been in [12-14], where the constant-field approximation is mainly used. Besides the layers with quadratic nonlinearity the direct third harmonic generation with use of the layers of cubic nonlinearity is possible.

Investigation of quasi – phase matched interactions in the constant –intensity approximation [15] permits to find out a series of the new effects absent in the constant –field approximation. The developed method of analyzing the impact of the task parameters on efficiency of proceeding wave processes allows one to set the optimum values of parameters for making an effective frequency converter on the basis of a periodical structure [16-18]. In the work [18] intracavity quasi - phase matched generation of second harmonic has been investigated.

In the present work with account of phases changes of all interacting waves a comparative analysis has been made of efficiencies of laser radiation frequency doubling and tripling processes in regular domain structure created from crystalline layers with quadratic and cubic nonlinearities. On making a theoretical analysis we take into regard the reverse reaction of excited wave to pumping wave. The investigations have been carried out for the case when the constant –intensity approximation of basic radiation is applied not to regular domain structure as a whole but to each domain taken separately, what permits to make more precise analysis of wave interactions in similar structures.

The reduced equations depicting intracavity of third harmonic generation in this case with account of losses in a medium look as follows:

where – complex amplitudes of laser wave and third harmonic wave at corresponding frequencies mains () in direction of axis (the plus sign) and direction opposite axis (the minus sign), -phase mismatch in each domain (domains are formed from the identical nonlinear layers and differ only by the direction of spontaneous polarizations), are absorption coefficients, are nonlinear coefficients for odd number of domains ( - for even number of domains) of interacting waves at the respective frequencies [18]

- refraction indices at frequencies , - wavelength of pump radiation.

The boundary conditions, with this, are as follows

Here corresponds to an entry to the -th domain, and – the complex amplitudes of laser wave and harmonic wave at entry to -th domain, respectively, and the complex amplitudes of laser wave and harmonic at the outlet from ()-th domain, and – phases changes on the boundary between ()-th and -th domains, respectively, at frequencies .

From the expression for complex amplitude of third harmonic [17], obtained from the system (1) with account of the boundary conditions (2), it follows that harmonic amplitude depending on length is periodical function. First, at a distance of optimum length (coherent length) there occurs transfer of basic radiation energy to harmonic energy. Then, the reverse transfer of energy takes place. With this, coherent length of interaction is defined by the expression

For comparison, a coherent length of interaction in a domain at second harmonic generation is defined by the expression [16]

Analysis of the expressions (3) and (4) shows that in case of interaction of powerful laser fields with a regular domain structure there are possible the versions when in a cubic medium harmonic wave reaches conversion maximum earlier than in a quadratic one, i.e. when . For instance, on equal levels of entry intensity of basic radiation in nondissipative media at second harmonic generation the reduced coherent length of the first domain , length of structure after the second domain , after the third domain the reduced length of domain structure is equal to =1.7396, after the fourth domain -2.3694, after the fifth one it makes up 2.991, and after the sixth domain it reaches 3.5991 (for value of reduced phase mismatch = 2.4) (upper curve in Figure).

In case of third harmonic generation the reduced length of the first domain is 0.5307, structure length after the second domain is equal to 1.9851, after third -=1.6923, after the fourth is =2.3463, after the fifth =3.0003, and after the sixth domain the given length of a structure is equal to 3.6074 (lower curve in Figure). From the calculated values of the coherent lengths of domains it follows that with increase of domain number the layer – domains expand.

Owing to fulfillment of quasi – phase matched conditions on a boundary of domains efficiency of conversion to harmonic at the outlet of each domain acquires the maximum value. In addition, an efficiency in a quadratic medium from domain to a domain monotonously increases: (upper curve in Figure). Analogous values of efficiency in a cubic medium are equal to: (lower curve in Figure)

Fig. Dependence of efficiency of intracavity generation of the second and third harmonics on length of a domain structure comprising two periods of “grating” of modulation of quadratic (upper curve 1) and cubic (lower curve 2) of susceptibilities. = 2.4, 1,2,3=0.

At intracavity arrangement of a domain structure after reflection from a mirror of laser resonator the waves running in reverse direction is accompanied by further increase of conversion efficiency (compare efficiencies for domains 5, 6 and those for first four domains). This increase of conversion efficiency to harmonic is provided by fulfillment of optimum phase relation between interacting waves [18].

Using the obtained analytical expressions for the task parameters in the constant –intensity approximation it is possible to calculate for each concrete experiment the optimum values of basic radiation intensity, coherent lengths of domains, phase mismatch between interacting waves of pumping and its harmonic. As was expected, an account of losses in a medium leads to weakening efficiency. The suggested method for analysis of nonlinear wave interaction may be used for studying intracavity parametric interaction of nonlinear optical waves in similar structures.

This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan under grant № EIF-2010-1(1)-40/14-M-9.

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