Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond

Abstract

We describe monochromatic light propagation in uniaxial crystals by means of an exact solution of Maxwell's equations. We subsequently develop a paraxial scheme for describing a beam traveling orthogonal to the optical axis. We show that the Cartesian field components parallel and orthogonal to the optical axis are extraordinary and ordinary, respectively, and hence uncoupled. The ordinary component exhibits a standard Fresnel behavior, whereas the extraordinary one exhibits interesting anisotropic diffraction dynamics. We interpret the anisotropic diffraction as a composition of two spatial geometrical affinities and a single Fresnel propagation step. As an application, we obtain the analytical expression of the extraordinary Gaussian beam. We then derive the first nonparaxial correction to the paraxial beam, thus giving a scheme for describing slightly nonparaxial fields. We find that nonparaxiality couples the Cartesian components of the field and that the resultant longitudinal component is greater than the correction to the transverse component orthogonal to the optical axis. Finally, we derive the analytical expression for the nonparaxial correction to the paraxial Gaussian beam.

Nonparaxial wave beams in a uniaxial birefringent crystal

“…Notice that deformation of the pattern does not vanish even in the paraxial case [14,18,24], being inherent to linearly polarised beams. The above results are consistent with the theoretical analysis [18,25,26] for paraxial beams propagating perpendicular or nearly perpendicular to the optic axis in crystals. At the same time, this deformation is very difficult to detect experimentally for the beams propagating along the optic axis of any real crystals, since it is proportional to the ratio of refractive indices for the ordinary and extraordinary beams.…”

Evolution Properties of a Partially Coherent Flat-topped Vortex Hollow Beam Propagating in Uniaxial Crystals Orthogonal to the Optical Axis

“…In this case, we mainly consider a laser beam propagating in uniaxial crystals orthogonal to the optical axis. Within the framework of paraxial approximation, the components of the laser beam propagating in uniaxial crystals orthogonal to the optical axis can be treated by the following formulas [2][3][4]: …”

“…dx dy dx dy (4) where W(r 1 , r 2 , z) and W(r 10 , r 20 , 0) represent the crossspectral density functions of the laser beam in the output and input planes, respectively. Assume that the flat-topped vortex hollow beam considered in this work is x-polarized and is incident into uniaxial crystals at the plane z=0, then the flat-topped vortex hollow beam at the source plane z=0 can be written as [24]:…”

“…The propagation properties of laser beams in uniaxial crystals have been treated by solving the boundary value problems of Maxwell's equations in uniaxial crystals [2][3][4]. Since the vectorial theory of laser beam propagation in uniaxial crystals was constructed, propagation of various laser beams [5][6][7][8][9][10][11][12][13][14][15][16][17][18], such as Laguerre-Gauss and Bessel-Gauss beams, Hermite-Gauss beams, dark hollow beams, flat-topped beams, elliptical Gaussian beams, elliptical Gaussian vortex beams and beams generated by a Gaussian mirror resonator in uniaxial crystals, partially coherent flat-topped beams, partially polarized and partially coherent beams, LaguerreGaussian correlated Schell model beams, coherent and partially coherent four-petal Gaussian vortex beams, has been widely investigated.…”

Transformation of Finite Olver-Gaussian Beams by an Uniaxial Crystal Orthogonal to the Optical Axis

“…The theoretical approach to describe the paraxial propagation of laser beam along the optical axis of an uniaxial medium is introduced for the first time by Ciattoni et al [8][9][10]. Subsequently, they have introduced the paraxial and non-paraxial approaches to characterize a laser beam propagating an uniaxial crystal orthogonal to the optical axis [11][12][13]. After these, a lot of studies elaborated in the context have treated the propagation of laser beams through an uniaxial crystal orthogonal to the propagation axis.…”