Wave front dislocations appearance under the laser beam self-action in liquid crystal

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Abstract

We present theoretical study of optical singularity birth and behaviour in an initially smooth wave front of the incident astigmatic Gaussian light beam. Linearly polarised light beam illuminates a homeotropically aligned nematic liquid crystal cell. Strong director anchoring at the cell walls is assumed. Director reorientation profile is found numerically solving Euler-Lagrange equations. We found the threshold intensity for light induced Fredeericksz-type transition. The threshold intensity of light beam is appeared to increase with increasing of beam asymmetry under the constant value of laser beam area and cell thickness. The results are compared with those calculated using Gaussian-like trial function.

Utilizing the Huygens-Fresno principle we calculate the propagation of the distorted light beam after the liquid crystal cell. It is found that with distance increasing we can observe at first the dipole, then the quadrupole and then again the dipole of optical vortices whis unit charge. Thus, the trajectory of zero amplitude resembles a deformed rubber ring symmetrical in the xz-, yz-planes and stretched along z-axis.

Introduction

Wave front dislocations can be observed in some wave fields, even in a pure monochromatic wave. According to Berry [1] the universal mechanism of a phase dislocation birth in some wave fields at the given space-time point is a full interference of different partial contributions of a wave at this point. It was shown theoretically [2] and experimentally [3], [4] that phase singularity appears due to laser beam self-action in non-linear medium such as photorefractive crystals. Two conditions should be fulfilled simultaneously: (i) the non-linear refractive-index modulation creates a lens-like structure with coexisting focusing and defocusing parts in contrast to an usual spherical lens, (ii) the optical strength of the induced lens reaches some definite threshold value.

Liquid crystal is another example of such non-linear medium, in which self-defocusing of Gaussian beam occurs and as a result phase dislocation appears. The experimental study of the nucleation of wave front dislocations in a Gaussian beam experienced the self-action in a nematic liquid crystal was reported at [5]. So, it is important to investigate this task theoretically. It was made for the case of stigmatic Gaussian beam [6], [7]. But experimentally Gaussian beam often possesses some asymmetry. The problem of self-action was considered earlier [8]. The aim of present work is to validate some estimates made in [8] and improve the results. The results of this work can be used for creating optical vortex tweezers [10].

For present theoretical investigation we should calculate changes of beam amplitude and phase with distance after the cell boundary. It is necessary to solve the Maxwell's equations for light propagation simultaneously with equations for LC reorientation. We solved this problem approximately. First, we find the liquid crystal director profile in a nematic cell illuminated with Gaussian light beam, neglecting the feedback. Euler-Lagrange equation is solved numerically for the case. The light diffraction caused by the director inhomogeneity is considered after that.

Section snippets

Profile of director distribution

Linearly polarised light with astigmatic Gaussian profile:E=exI1/2,I=I0e(x2/R12)(y2/R22)where Ri=ϖi2 and R1  R2, illuminates the homeotropically aligned nematic liquid crystal cell. Further in this article we will use R1 and R2 as characteristics of the Gaussian beam.

Oz axis of the Cartesian frame is directed along a non-perturbed direction of the director n0=(0,0,1), and Ox axis is directed along the polarisation of light (Fig. 1). Strong director anchoring at the cell walls is assumed,

Light field after passing nematic liquid crystal cell

To find light field behind LC cell the Huygens-Fresnel principle is used. In Fresnel's approximation, which is correct in a near field, the principle gives [11]:U(x1,y1,z)=eik0ziλzΣU(x,y)expik02z(x1x)2+(y1y)2dxdy,where U(x1, y1, z1) is complex wave amplitude in the plane X1OY1, which position is at distance z from the initial plane XOY, U(x, y); beam complex amplitude in the initial plane z = z0; k, wave vector; λ, wavelength.

To find behaviour of the light field in our case it is necessary

Conclusion

Self-action of astigmatic Gaussian light beam was considered in the nematic liquid crystal cell with strong homeotropic anchoring. The director distribution profile is found solving numerically Euler-Lagrange eaquation. It has the Gaussian-like form but does not correspond it completely. Anyway, a Gaussian-like trial function can be used for simplifying calculation substantially. Director reorientation is threshold like. For typical values of parameters (liquid crystal ZLI-4119)—K = 10−11N, ɛa = 

Acknowledgements

I am pleased to thank Professor V.Yu. Reshetnyak, Professor M.S. Soskin, Dr. V.I. Zadorozhnyy for fruitful discussions.

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