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Home / Issues / № 2, 2014

Phisics and Mathematics

Umbetov A.U.
An analysis and synthesis of difficult optical elements from anisotropic crystals are of interest for the construction of laser measuring devices. Thus there is a task of development of methodology of calculation of such elements, that more precisely would describe their properties on passing and interference of hertzian waves. The co-version method of Ph.I (is known). Fyodor for the calculation of distribution of electro-¬magnetic waves in anisotropic environments. However this method results in difficult general expressions, and his use for ДКЭ is difficult. On the whole a task is not accessible to the strict analytical decision, thus basic difficulty is ; in the necessity to take into account out-of-parallelism of wave vector to , describing transfer of phase of wave, and radial vector   describing transfer of energy of wave (λ -the length of wave, Е, Н -vectors of interesting of elliptic and magnetic fields.

The structural  features of worked  out DKE are below  given:   DP 1(pic.1), BL (pic. 2). The prisms of type of DPPUD differ in that at normal incidence-fission   of wave   on   the exit   of prism   is absent,   but   here   maximal sensitiveness of "doubling" between о- and е-waves is achieved in relation to angle of incidence. The first   variant   of prism    (DPPUD- 1)    is presented   on  picture 1.   The plane of optical axes is perpendicular an entrance and output to the verges of prism and parallel to the rib of prism.


Fig.1. Construction of двупремляющих prisms of variable angle of doubling of type of DPPUD - 1(а) and DPPUD (b)

For a wave, falling inplane, containing optical axes, at any angles of incidence on the exit of DPPUD formed two о- and е- waves with the ortogonal state of пpolarization . Basic descriptions of DPPUD, measureable on experience, are  doubling angle y  between о- and е- rays on the exit of prism and angle of rejection of х came from the prism of е-ray  from the plane of incidence. Calculation sizes y and х for DPPUD Д - 1 determined on formulas:


  ;                       (1)


 ,                                                   (2)


where parameter of  trifle ((δ)/;,,-main  indexes of refraction of о- and е- waves; -deflectable angle  of wedge of prism; azimuth angle, characterizing the turn of DPPUD about axis of , the flat spreads along thatwave; angle of incidence on the entrance verge of prism, equal to the angle of turn of DPPUD around axis x.

The second variant of prism (DPPUD - 2) is presented on a picture 1,6. The plane of optical axes is perpendicular to  an entrance and output to the verges, and also rib of prism. Here in this case  we get:

  ;            (3)



If a falling wave lies inplane optical axes at any angles of incidence, that takes place at (β=0, then breaking (up "doubling") on an exit takes place in plane of incidence, i.e. х2 = 0. If a wave falls in of,  planes of optical axes, then breaking up is absent, as well as in case of DPPUD of Dependence of kind (1) - (4) are in a good consent with experimental facts  and can be used for drafting of algorithms for COMPUTER in the systems of operation of a laser ray.

Properties of other  element  (BL)  are described  in  works.  Calculation  of DKE  of type  of B.L., conducted on methodology, and explained on picture  2, c.









             а)                                                                       b)













Fig. 2. Constructions of bifocal lenses of type

of  BL- 1(а) and BL - 2(b) and chart of motion of rays in them(c)


Let in the circular polarized  wave spreads in the direction of axis z. At   such   choice   of the state   of polarization   of falling   wave   "attachment"   of vector of polarization to the optical axis of crystal on the entrance of BL appears unimportant, that allows in future to unite the construction of theory for the lenses of type of BL- 1 and BL- 2. Let z=0 жэне z=ℓ are left and right to the verge of BL  accordingly, and spherical border is set with equation where –distance from the beginning of the system of coordinates to the center of spherical surface of БЛ. Directions of optical axes in the areas of I and II are set by the unit vectors of а1 -(1,0,0) and oh, =  x2+у2+(z-δ)2= R2,  where δ - corner between BL and axis  2(fig. 2, c). Let (on the left verge of БЛ in the arbitrary point of М1 the narrow parallel beam of light falls along an axis. The point of М1 has coordinates of  dcosφ, dsinφ,0,  where φ-angle  between the axis of х and radius-vector, conducted from beginning of coordinates z to the point M. Will suppose d<< R  To, where R - radius of curvature of spherical border of division of BL., size (d/R)2  is scorned small. The index of refraction for о- wave in the areas of I and II is identical and equal h0, and for a е- wave depends on  (i=1,2)- single wavy vector in  the areas of I and II accordingly. In area of K1,  (1) coincides with direction of light ray. There is mutual transformation of о- and е-волн on the spherical border of division of БЛ. Wave vector  in area of. II lies in plane, passing through an axis z and determined by an angle  φ:  = (sinα2cosφ, sinα2sinφ, cosα2 ), where аг is an angle  between  and by an axis z. On a spherical border in general case four waves must be considered and according to them four border conditions.

Thus the vector of     and angle of α2 must add indexes(оо),(ое),(ео) ,  meaning:   (оо) and (ee) –keeping of type of polarization of wave;  (ое) and   (ео) transformation of falling о- wave in the refracted е-wave and vice versa. Thus ,   and   angle   can   be   found   from   the law  of refraction   on  a spherical surface:



where I–single vector of normality. Intersection

shining with the spherical border of division has coordinates    Putting in (5) values of vectors of  and аг we will get the angles of .

Wave vector      of ray      going      out from      BL,      we will write      down in      a kind

     Obviously, that              and law of refraction    (eo) of wave on the border of will look like :



where is a normal to the plane of z = е . From(6) will we get angles .

For being of е-ray in area of II it is required to consider single vector of group     speed    on     correlation     of  where   μ1,μ2-coefficients. It is necessary to angle between in an anisotropic environment. It is possible to show that expression for    describing the trajectory of е-ray in BL. looks like




Case of =0  suits to  element of type of  BL- 1 (рicture .2, а), and - element of type BL- 2 (picture2, b). For the Icelandic spar n0>ne   and  - we have   . This means that о-ray going to the environment of II on leaving from BL- 2 walks away from the axis of Z; consequently, falling on BL - 2 a parallel bunch with this polarization will be going (picture.2, c) away. For(ео) - and(ee) - rays at  (in BL- 1) we have:  and

Consequently,(ео) - and(ее) -rays will cross the axis of  z  in two different points corresponding to two focuses of . Thus, by means of  BL- 1 the interesting case of SDM of flat wave will be realized on two spherical waves with divided along an axis by 2 focuses into a size , depending on double-refracting properties of crystal and thickness of  h of plane-convex lens(picture  2, а). This property of BL- 1 can be used for a holographing in polarized light  in subsequent bunches with the managed intensity.

The calculations given above allowed in theory to predict, and in and experimentally to find out a spatial uninvariance( to  irreversibility of passing of hertzian wave in relation to the axis of z) at the analysis of distribution of laser bunch through BL- 1 in directions( + Z) (- Z). The invariance of the polarization linear optical systems is unobvious. On the contrary, there is a necessity to examine the location of elements of optical chart in a polarization optics. By the methods of matrix optics easilyto prove, for example, that a result of passing of hertzian wave through a double-base polarizing element(polarization +λ/4 ) will be different on the state of  polarization depending on that, from what part a wave falls on difficult element 2. DKE of type of BL- 1 demonstrates an unique case in this sense, when not only the state of polarization but also amount of waves on an exit and picture of interferencepolarized waves are different  for opposite directions of distribution of light. At falling of flat wave on BL- 1 outside plane-convene  lenses(picture 2, а) there is breaking up on four waves, from which waves that(оо) - and(ое) - I are: parallel to axes z, and(ee) and(ео) –waves  are spherical waves with carrying (along an axis 2) focuses. In case of falling of flat wave on БЛ- 1 from the side of  plane – convex  lens(area of III on a fig. 2, а) on an exit formed one parallel and one converging   astigmatic bunches with the ortogonal states of polarization.  Unlike BL- 1  DKE  of type of BL- 2  property  of uninvariance does not possess. For BL- 2(fig. 2,b) forming is characteristic converging(ео) and going(ое) away waves as a result of transformation of е-waves  in о- waves and vice versa.

BP(fig. 3, а) is two prisms from a monaxonic crystal as equal-side trapezoids (prisms of Dove), agglutinate with  large grounds by the layer of glue, having an intermediate index of refraction of . Optical axes in making prisms located in a plane perpendicular to the grounds of prisms parallel between then selves  and form a angle 45° with the plane of gluing(fig. 3,6) together. the choice of orientation of optical axes is comfortable during work with a laser source;-, to the vertical  orientation  of vector of Е of laser  radiation,   (in  parallel to the rib of falling normally on an entrance verge A1  C1  a laser ray is divided into о- and е-rays. thus о-ray tests a complete internal reflection from the layer of glue. At falling on a verge A 1 B1 of the    second laser ray (from an independent source or first ray) formed by an optical division е-ray passes without rejections along an axis Z In subsequent о- and е-buckles there is interference on the exit of BP.

At the turn of BP on a small angle  and about axis parallel to the rib of  BP, о- and е-burkes on the exit of  BP  have      relative     movement 3.      The condition      of existence   of interference(фотосмешения) is переналожение   (cross-correlation)   о-  and  е-burcles at bringing their vibrations over to one plane by means of analyzer.


Thus top limit of measureable angles of turn where r-radius of the mixed up bunches; 2а, е, 0 are parameters of BP : length of the general founding, length of lateral side, corner at founding(fig. 3, в). For making  from the Icelandic spar of BP  with and=12mm, е=11mm, θ =65° from (8) we get α≤3º  at the r  of 3 mm.


1. Barsukov К. А., Оsipov U.V., Popov V.N. Optics and spectroscopy, 1980, p. 605-61, 1981, p. 191-196.

2. Оsipov U.V., Optico-mechanic industry, 1973, № 5. p. 5; 1972, № 12, p. 1822.

З. Barsukov К. А., Оsipov U.V., Umbetov A.U. Distribution electromagnetic bullock two-component дlenses from monaxonic crystals. (Theory and experiment). 4260-83 of 01.08.83

Bibliographic reference

Umbetov A.U. PARTICULARITIES DVUHKOMPONENTNYH KRISTALLOOPTICHESKIH SYSTEMS. International Journal Of Applied And Fundamental Research. – 2014. – № 2 –
URL: www.science-sd.com/457-24646 (25.01.2020).