General solutions of the theme “ Light propagation in optical uniaxial crystals ”

In this article, we introduce a new approach to receive general solutions which describe all of the properties of the light propagating across optical uniaxial crystals. In our approach we do not use the conception of refractive index ellipsoid as being done in references. The solutions are given in analytical expressions so we can handly calculate or writing a small program to compute these expressions.


INTRODUCTION
The problem of lights propagation in optical uniaxial crystals, i.e. crystals of trigonal, tetragonal and hexagonal systems, was solved by the application of Maxwell's equations.Solving the Maxwell's equations for a plane wave light propagating in transparent non-magnetic crystals, one can derive two refractive indices of the two propagating modes of light [2,3] In (1), ij  (i, j = 1, 2) are the components of the dielectric impermeability tensor of crystal.In expression (1), the light direction is taken in parallel to axis OX 3 of an arbitrary coordinate axes   in (1) also vary in according to the light direction.Therefore in references, in order to eliminate this difficulty, one can only solve this problem in crystal coordinate axes * i OX with the help of the conception of refractive index ellipsoid, but this approach can only be applied in some limited cases when light propagating in some special symmetric directions of the crystal.The refractive index ellipsoid of optical uniaxial crystals is an ellipsoid of revolution.It has an important property: the central section perpendicular to the light direction

THEORETICAL CALCULATIONS
The transformation cosinus matrix   k i


In an arbitrary coordinate axes i OX (i = 1, 2, 3) we choose the 3 OX axis which is parallel to the light direction m, which has three components (cosinus) in crystal coordinate axes: m 1 ; m 2 ; m 3 .Thus, the unit vector   3 u along axis equals to m Denoting g and h two vectors (not unit vectors) prolonging axes 1 OX , 2 OX respectively.We can write: Where (1 , 0 , 0) is the unit vector along * 1 OX and  is a coefficient derived from the orthogonal condition are the components of g in axes From these orthogonal conditions we find: From expressions (2), ( 3), (4) we can derive the components of h along the axes

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The components of g and h along the OX , but these direction cosinus can be derived by dividing these components by their vector length, i.e. g and h .Finally, we obtain the direction cosinus matrix We can verify the truth of this matrix by these tests: In expression (6) we used Einstein notation, i.e. to take the summation of the repeated indices by running this index from 1 to 3.

12
 and 22  from (7) into the general expression (1) we can solve the proposed problem.
After a long way of calculations we derive the refractive indices for the two propagating modes of light: The corresponding refractive index of ordinary and extra-ordinary rays Now here, we discuss what of the sign (+ or -) in ( 8) of which the refractive index of ordinary ray will be taken.For the convenience of discussion we rewrite expression (8) in the form: * For ordinary ray: The velocity of extra-ordinary ray: The polarization of the two rays from (10) into the equations ( 14) we have: we solve the equations and derive the components as follows:

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Applying the transformation rule of the components of a vector: OX , i.e. the optical axis of crystals.We can verify the truth of ( 18) and ( 19) by the following tests: The lack of the coincidence between the light direction m and the direction of light energy transfer P (the Poynting vector) According to [2], [3] the angle  of the lack of coincidence between the light direction m and the direction of light energy transfer, i.e.Poynting vector P is determined by the following expression: Thus, in order to calculate  we have to determine the electric vector *

E . Because in crystal coordinate axes
For the ordinary ray from (18):

E D E D cos
Thus, for the ordinary ray, there is no lack of coincidence between m and P. * For the extra-ordinary ray: , KDP is a negative optical crystal.Its optical axe is the 4 A axe and in this case is parallel to axis * 3 OX .We apply our above results in three cases: The light direction is along the optical axe of KDP In this case we have m 1 = m 2 = 0 and m 3 = 1 This is the simplest case of light propagation in optical uniaxial crystals and interestingly to be discussed here.In references, we know that in this case we have only one ray propagating along the optical axe of KDP.This is the ordinary mode.Its polarization can be taken in any direction belonging in the plane perpendicular to the optical axe.
Which, for our results: * For the ordinary ray: limitd values, which is not infinity but depends on the light polarization entering the crystal.Imagine a laser beam with any polarization entering along the crystalographic axe of crystal.The polarization of the laser beam can now combine two perpendicular components lying in the plane perpendicular to the optical axe of crystal.Each of the components is the polarization vector for mode n o or n e .Although their lengths are not equal to 1, but as shown in [2] the important thing is not the eigen vector but eigen direction as all vectors of arbitrary lengths provided lying along this direction are also eigen vectors of a second rank tensor.Thus, in references we frequently speak about eigen direction instead of the eigen vector.In our case the laser beam will propagate across the crystal with its original polarization.It is the ordinary ray.The laser beam can be polarized in any direction so the plane perpendicular to optical axe of KDP is an eigen plane.
In conclusion of this discussion, our results are the same already known in the classical approach.
The light direction is along one of the two axes A 2 of KDP (along There is a coincidence between m and P.
In this case, we can say that we have two ordinary rays propagating with different velocities along an A 2 of KDP.Trang 88 The light direction  Trang 89

CONCLUSION
Based on the general expression of refractive index (1), by the transformation cosinus matrix and tensorial calculations, we have completely solved the theme "Light propagation in optical uniaxial crystals".These analytical expressions describe all the properties of light propagating across the crystal.We have some remarks: the polarization of the two propagating modes depends only on the light direction whereas the light velocities and the angle of lack of coincidence between m and P of extraordinary ray depend on the crystal and light direction.
With the exception of the cubic system, which is an isotropic medium in optical aspect, our approach can be applied to orthorhombic and monoclinic systems.Of course the calculations will be more complex and take longer time because of in these cases ** 11 22   .
Acknowledgments: The authors wish to give their thanks to Pr. Lê Khắc Bình for his comments and helps during the preparation of this article.

Fig
Fig 1.The crystal coordinate axes expression (1) we can solve the given problem.

DD
, the unit vectors of polarization of the two rays in coordinate axes i OX .Because the light is transversal, so in we have to solve the equations determined the eigen vectors of a twodimension tensor [ ij  ] having known eigen values n o and n e .can write these above equations in the form: with the normalized condition of   e around its diagonal by an angle π we have: Fig 2. A) Polar projection of point group 42m of KDP

From
means that, in this case we have only one mode propagating along optical axe of KDP.It is the ordinary ray.Light polarization is calculated from (27):

Fig 3 .
Fig 3.The polarization of the ordinary ray and extra-ordinary ray.
Fig 4. The polarization of the ordinary ray and extra-ordinary ray OX * 1 : In this example we have taken into account tensor [ Analogously, we can derive all the components of tensor [ ij  ] in the coordinate axes i OX as follows: In this case, because of the refractive index of ordinary ray n o < n e , the quantity A o must be greater.On the other side, in this case    takes the sign (+).Finally, regardless of positive or negative optical crystals, the refractive index of ordinary and extra-ordinary rays have the expressions: In this case it is difficult to use the refractive index ellipsoid approach to solve the problem.