Nondiffracting vortex-beams in a birefringent
chiral crystal
Tatyana A.Fadeyeva and Alexander V.Volyar*
Physics Department,Taurida National V.I.Vernadsky University,Vernadsky av.4,Simferopol,Ukraine,95007
*Corresponding author:volyar@crimea.edu
Received August18,2009;revid November3,2009;accepted November3,2009;
posted November3,2009(Doc.ID115802);published December3,2009
A vector-wave analysis of nondiffracting beams propagating along a birefringent chiral crystal for the ca of
tensor character of both the optical activity and linear birefringence is prented.We have written character-
istic equations and found propagation constants and amplitude parameters of the eigenmodes.The character-
istic curves have anomalous zones described by an isotropic point or a gap-point,provided that the elements of an optical activity tensor obey the requirement g11g33Ͻ0,͉g33͉Ͼ͉g11͉.In the anomalous zone,a nondiffracting beam can propagate through a purely chiral crystal as if through an isotropic medium.We have shown that the
field of eigenmodes is nonuniformly polarized in the beam cross ction,while thefield with the initially uni-
form polarization distribution experiences periodic transformations.We have revealed that even a purely chi-
ral crystal without linear birefringence can generate optical vortices in an initially vortex-free Besl beam.
©2009Optical Society of America
OCIS codes:350.5030,260.6042,260.1180,260.0260.
1.INTRODUCTION
It is well known[1]that nondiffracting beams(or diffraction-free beams)form a wide class of wavefields as solutions to the Helmholtz equation,provided that the equation is parable while the beams have transmission symmetry along one of the coordinates(say,the z axis). The solutions refer to different kinds of coordinate sys-tems:Cartesian,circular cylindrical,parabolic cylindri-cal,and elliptical cylindrical coordinates.The parability of the Helmholtz equation impos the condition that the solutions of the transver part not depend on the longi-tudinal coordinate.The simplest solution in Cartesian co-ordinates is a plane wave.The most well-studied nondif-fracting wavefields are Besl beams(e,for example, [2]and references therein)that are the result of the solu-tion to Helmholtz equation in circular cylindrical coordi-nates.The solution to the Helmholtz equation in elliptic cylindrical coordinates entails Mathieu beams[1],while employing parabolic cylindrical coordinates results in parabolic diffraction-free beams[3].Experimental sub-stantiation of the latter two kinds of beams have been demonstrated in the literature[4,5].Note that to create an ideal nondiffracting beam would require infinite en-ergy.However,in an experiment such a problem is solved by creating approximately nondiffracting beams with the help of special pha plates offinite sizes,so that the beam properties do not significantly change over short propagating distances[6,7].Improvement of th
e experi-mental technique has stimulated applications of nondif-fracting beams as optical tweezers in microlithography, metrology,medical imaging,etc.[2,8].
One of the key problems of modern singular optics[9]is the control of the properties of nondiffracting beams,in particular,optical vortices embedded in them.An appro-priate medium for this purpo is anisotropic crystals.However,a crystal-traveling beam has a vector character. The vector properties of nondiffracting beams—in par-ticular,different types of Besl beams as solutions to the Maxwell equations in a homogeneous isotropic medium—have been analyzed before[10,11],where the authors showed the beams to be nonuniformly polarized over their cross ction.The analysis of Besl beams in biaxial crystals spreading along one of the optical axes was pre-nted in[12,13],where a spectral integral technique was employed.Berry and Jeffray[14]have analyzed the pic-ture of a conical refraction in terms of singular optics,fo-cusing their attention on the fact that Besl vortex-beams in biaxial crystals are the beams with eigenpolarization.Notice also that a number of papers have been devoted to properties of paraxial Laguerre–Gaussian and Besl–Gaussian beams in uniaxial crys-tals[15–20]where were revealed unique properties of the crystal-traveling beams raising the possibility of generat-ing and annihilating optical vortices.
Analysis of beam propagation through chiral aniso-tropic crystals(anisotropic crystals with optical acti
vity) is bad as a rule on studying the properties of parate plane waves in a beam and extending them to the beam as a whole[21].Berry and Dennis succeeded in analyzing afine structure of polarization singularities in a birefrin-gent dichroic chiral crystal on the basis of the above ap-proach enhanced by the stereoscopic projection technique [22].In[23],Berry and Jeffray analyzed the view of Poggendorff rings and caustic surfaces under the condi-tion of conical refraction in a birefringent chiral crystal. The propagation of a paraxial Gaussian beam in a bire-fringent chiral crystal was also studied in[24]on the ba of approximate methods.However,the description of the vortex properties of nondiffracting beams in anisotropic chiral crystals on the basis of the electromagnetic mode-
1084-7529/10/010013-8/$15.00©2010Optical Society of America
beam technique that provides a depiction of the vortex states in the beam as a whole has not yet been accom-plished.
The aim of the prent paper is to analyze the optical vortex properties of electromagnetic nondiffracting beams traveling through a uniaxial birefringent chiral crystal employing the mode-beam technique.
In Section2,we derive the basic and characteristic equations for nondiffracting electromagnetic beam
s in a birefringent medium with anisotropic optical activity.Be-havior of a nondiffracting beam in a purely chiral crystal is analyzed in Section3.We solve the characteristic equa-tion for the propagation constants and amplitude param-eters of eigenmodes and describe the evolution of a Besl singular beam with uniform circular polarization at the initial plane of the crystal.Section4is devoted to solving the characteristic equation in a birefringent chiral crys-tal.Also,we analyze optical vortex transformations in a linearly polarized Besl beam bearing a singly charged optical vortex.
2012年中考2.BASIC AND CHARACTERISTIC EQUATIONS
We consider propagation of a nondiffracting monochro-matic beam at the frequencyalong an unbounded uniaxial anisotropic medium with optical activity(chiral-ity).The Maxwell equations are
͑aٌ͒ϫE=−ik B,͑bٌ͒ϫH=ik D,
͑cٌ͒·D=0,͑dٌ͒·B=0,͑1͒with a wavenumber k in vacuum while constitutive equa-tions(e,for example,[25]and references therein)are written in the form
D=⑀ˆE−i␥ˆH,B=H+i␥ˆE,͑2͒where the tensors⑀ˆ=diag͑⑀,⑀,⑀3͒and␥ˆ=kgˆ,gˆ=diag͑g,g,g3͒,characterize the linear birefringence and the optical activity of a medium,respectively,inherent, for example,in triclinic,
hexagonal,and other practically important crystal systems[26,27].After simple transfor-mations of Eqs.(1)and(2)we come to the wave equations for the electric and magneticfields in the forms
ٌٌ͑E͒−ٌ2E−k2͑⑀ˆ2−␥2Iˆ͒E
=2k␥ٌϫE+k⌬␥ٌ͑ϫe z E z͒
+e z⌬␥͓kٌ͑ϫE͒z−k2͑␥+␥3͒E z͔,͑3͒H=͓iٌ͑ϫE͒/k͔−i␥E−i⌬␥e z E z,͑4͒
where Iˆis a unit matrix,⌬␥=␥3−␥,and e z stands for a unit vector of the z axis.From Eqs.(1c),(1d)(2)we derive ٌ·D=͑⑀−␥2ٌ͒E+͑⌬␥/kٌ͒͑ϫץz E͒z
+͓⌬⑀−⌬␥͑␥+␥3͔͒ץz E z=0.͑5͒Further,using the property of the tensor⑀ˆonefinds
ץz E z=−͓͑⑀−␥2͒/⑀␥ٌ͔ЌEЌ−͑⌬␥/k⑀␥ٌ͒͑ϫץz E͒z,͑6͒where⑀␥=⑀3−␥32,⌬⑀=⑀3−⑀,ٌЌ=e xץx+e yץy,EЌ=e x E x +e y E y,and e x and e y are unit vectors,so that we obtain ٌE=␣ٌЌEЌ−͑⌬␥/k⑀␥ٌ͒͑ϫץz E͒z,͑7͒with␣=͓⌬⑀−⌬␥͑␥+␥3͔͒/⑀␥.Finally,the basic equation for a birefringent chiral crystal is reduced to the form
ٌ2E+k2͑⑀ˆ−Iˆ␥2͒E+2k␥ٌ͑ϫE͒+͑⌬␥/k⑀␥ٌٌ͒͑ϫץz E͒z
=␣ٌٌ͑ЌEЌ͒−k⌬␥ٌ͑ϫe z E z͒+e z k⌬␥͓k͑␥+␥3͒E z
−ٌ͑ϫE͒z͔.͑8͒Longitudinal symmetry of nondiffracting mode-beams assumes that
E͑x,y,z͒=E͑x,y͒exp͑−iz͒,͑9͒wherestands for a propagation constant of the mode beam.As a result,the equations for the transver and longitudinal components of the electric vector are ٌЌ2E x+U¯2E x+2k␥ٌ͑ϫE͒x+⌬␥͓kٌ͑ϫe z E z͒x
−i͑/k⑀␥͒ץxٌ͑ϫE͒z͔
hal=␣ץxٌ͑ЌEЌ͒,͑10ٌ͒Ќ2E y+U¯2E y+2k␥ٌ͑ϫE͒y+⌬␥͓kٌ͑ϫe z E z͒y
−i͑/k⑀␥͒ץyٌ͑ϫE͒z͔adhereto
=␣ץyٌ͑ЌEЌ͒,͑11͒
reforceE z=−iٌЌEЌ͑⑀−␥2͒/͑⑀␥͒−ٌ͑ϫE͒z⌬␥/͑⑀␥k͒,͑12͒where U¯2=k2͑⑀−␥2͒−2.The above equations can be re-duced to a suitable form if one makes u of the new vari-ables
u=x+iy,v=x−iy,
2ץu=ץx−iץy,2ץv=ץx+iץy,͑13͒so that
ٌЌEЌ=ץv E++ץu E−,andٌ2ϵ4ץuv2,
where we also introduce a circularly polarized basis:
E+=E x−iE y,E−=E x+iE y.͑14͒Then the basic equations take the form
ٌЌ2E++͑U¯2−2k␥͒E+−i2͑+͒ץuٌ͑ϫE͒z
=2͑␣−⍀/͒ץuٌ͑ЌEЌ͒,͑15ٌ͒Ќ2E−+͑U¯2+2k␥͒E−−i2͑−͒ץvٌ͑ϫE͒z
=2͑␣+⍀/͒ץvٌ͑ЌEЌ͒,͑16͒where=⌬␥͑␥+␥3͒/⑀␥,=⌬␥/͑k⑀␥͒,⍀=k͑␥+␥3͒͑⑀−␥2͒/⑀␥,ٌ͑ϫE͒z=i͑ץv E+−ץu E−͒.
We willfind a particular solution to the above equa-tions in the form
E=Aͩץu⌿−ץv⌿ͪ+Bͩץu⌿ץv⌿ͪ,͑17͒
where⌿=⌿͑x,y͒is a scalar function,and A,B are ampli-tude parameters.Note that thefirst and the cond terms in Eq.(17)are of a transver electric(TE)and transver magnetic(TM)wavefield,respectively,for the ca of a pure birefringent crystal,where␥=␥3=0[19].
Substituting Eq.(17)into Eqs.(15)and(16)wefind the expressions
ٌЌ2⌿+U12⌿=0,ٌЌ2⌿+U22⌿=0,͑18͒
U12=
本科文凭查询͑U¯2−2k␥͒͑A+B͒
A͑1++͒+B͓1+͑⍀/͒−␣͔
,
U22=
͑U¯2+2k␥͒͑A−B͒
A͑1−+͒−B͓1−͑⍀/͒−␣͔
.͑19͒
The function⌿will obey the equation
ٌЌ2⌿+U2⌿=0,͑20͒only provided that U1=U2=U,whence we come to the ex-pressions
͓U¯2−U2͑1+͔͒A−͓2k␥+͑⍀/͒U2͔B=0,͑21͒
−͑2k␥+U2͒A+͓U¯2−U2͑1−␣͔͒B=0.͑22͒This t of equations has a nontrivial solution relative to amplitude parameters A and B if
ͫk2͑⑀−␥2͒−2−U2͑1+͒−͓2k␥+͑⍀/͒U2͔−͑2k␥+U2͒k2͑⑀−␥2͒−2−U2͑1−␣͒ͬ=0.͑23͒The above expression is of a characteristic equation for a propagation constantof nondiffracting beams transiting through a birefringent chiral crystal.Its solution is ±2=k2͑⑀+␥2͒−͓͑⑀3+⑀−2␥␥3͒/͑2⑀␥͒U2͔±ͱD,͑24͒where D=4k4⑀␥2+2k2␥͑⑀␥3−⑀3␥+⑀⌬␥͒/⑀␥U2+͓͑⌬⑀/2͒2 +⌬␥͑⑀␥3−⑀3␥͔͒/⑀␥2U4,together with the condition A=1,de-fines two propagation constants of eigenmodes and their amplitude parameters:
B±=͑±/k͓͒k2͑⑀−␥2͒−±2−U2͑⑀3−␥2͒/⑀␥͔
ϫ͓2±2␥+U2͑⑀−␥2͒͑␥+␥3͒/⑀␥͔−1.͑25͒
3.PURELY CHIRAL CRYSTAL
A.Characteristic Equations
First we consider the ca of a purely chiral crystal with-out a linear birefringence⑀=⑀3,⌬⑀=0.One distinguishes two ranges of beam parameters:(1)when the parameter U is much less than the wave vector,UӶkͱ⑀,and(2)U is comparable with the wave vector Uϳkͱ⑀.Thefirst condi-tion characterizes a paraxial region of the beam propaga-tion,while the cond one is a nonparaxial region.When U=0,we deal with a z-propagating plane wave.Then,Eq.
(25)gives two values of the propagation constants
±=k͑n±␥͒,͑26͒while the beam parameters B±→ϯ1.The propagation constant−corresponds to a right-hand polarized(RHP) component E+of the wave with B+=−1,whereas the value +describes the left-hand polarized(LHP)component E−with B−=1.In the paraxial ca of beam propagation,con-tribution of the value⌬␥to the propagation process is negligibly small,and we obtain from Eqs.(24)and(25) the beam parameters in the form
±Ϸk͑n±␥͓͒1−͑U/kͱ⑀͒2/2͔,B±Ϸϯ͓1−͑U/kͱ⑀͒2/2͔.
͑27͒The tensor character of the optical activity gˆmanifests itlf most brightly in the nonparaxial region Uϳkͱ⑀.In-deed,let us require that D=0in Eq.(24).Then wefind the characteristic parameter U=U is to be
U is2=−2k2͑␥/⌬␥͒⑀␥,+͑U is͒=−͑U is͒.͑28͒Equations(28)describe anomalies in a nondiffracting beam–crystal system with isotropic point U=U is.Since⑀ӷ␥2,the wave parameter U is retains a real value only if ␥/⌬␥=g/͑g3−g͒Ͻ0.On the other hand,the wave param-eter U is cannot exceed the critical value U isജU critϷkͱ⑀, so that0ϽU is2ഛk2⑀or0Ͻ−g/͑g3−g͒ഛ1.The last require-ment can be reduced to the simple form
g3gϽ0,͉g3͉Ͼ͉g͉.͑29͒For the remaining cas(in particular,for media with iso-tropic optical activity⌬␥=0),there is no isotropic point in the beam–crystal system.
Thus,a nondiffracting beam with the U parameter de-scribed by Eq.(28)does not exhibit optical activity.The beam propagates through the chiral crystal as if through an isotropic medium.However,in the vicinity of the iso-tropic point(28),the beam parameters B+and B−change their signs into opposite ones,while the spectral curves for propagation constants+and−experience a sharp bend.This means that a purely chiral crystal changes handedness of the optical activity to the opposite one with respect to such nondiffracting beams.
Figures1(a)and1(c)illustrate the ca without anoma-lous behavior of nondiffracting beams.The curves in Fig. 1(c)for the propagation constants±=͑U͒have a smooth form in a broad range of the
wave parameters U up to the value U=U crit.In the region UϾU crit the propagation con-stant becomes a pure imaginary one,and we are dealing with evanescent nondiffracting beams[28].The spectral curves±=͑U͒with the same values of␥but different values of␥3practically coincide with each other in a paraxial region but have different behavior near critical values UϽU crit.The anomalous ca,where the crystal changes handedness of its optical activity,is shown in Figs.1(b)and1(d).The characteristic curves B±͑U͒expe-rience sharp changes in the vicinity of the isotropic point U=U is that look like a step-transformation both in the large scale of Fig.1(b)and the small scale of Fig.2(a). Such a step-transformation is caud by the fact that the
beam parameters are uncertain at the isotropic point B ±͑U =U is ͒.They can have arbitrary values in an isotropic medium.It is worthy of note that the spectral curve ±͑U ͒has a sharp bend in this range (e Fig.2(c)).The unique ability of the crystal to change its properties depending on the beam parameters forces us to consider behavior of a nondiffracting beam–crystal system rather than of the beam and the crystal parately.
Thus,in a purely chiral crystal,the wave field of the simplest nondiffracting beam is characterized by two eigenmode-beams [e Eq.(17)]:
E ͑+͒=
ͩ͑1+B +͒ץu ⌿−͑1−B +͒ץv ⌿ͪexp ͑−i +z ͒,
E ͑−͒=
ͩ
͑1+B −͒ץu ⌿−͑1−B −͒ץv ⌿
ͪ
exp ͑−i −z ͒,
͑30͒
where the upper indices (Ϯ)in the components of the elec-tric field E refer to the indices in the propagation constant
±and the mode parameters B ±.The beam field is non-uniformly polarized at the beam cross ction.In a paraxial region,we can approximately consider that B +Ϸ−1and B −Ϸ1,so that the E ͑+͒field has a dominant LHP component while the E ͑−͒field has a dominant RHP com-ponent.Note also that the fields E ͑+͒and E ͑−͒have differ-ent cutoff parameters U =U ±͑crit ͒
in a nonparaxial region when transforming the propagating eigenmodes into eva-nescent ones at ±͑U crit ͒=0.
The absolutely opposite situation occurs if we require the field of a nondiffracting beam to be uniformly polar-ized over the whole beam cross ction at the plane z =0in a nonparaxial region.In this ca,we have to compensate for one of the circular polarizations (E +or E −)in the ini-tial superposition of eigenmodes in Eq.(30).Such a new nondiffracting beam los its structural stability when
transmitted along the crystal.Consider the process in de-tail with the example of Besl beams.
B.Besl Beams
The Helmholtz Eq.(20)has a solution in circular cylindri-cal coordinates in the form
⌿l =J l ͑Ur ͒exp ͑±il ͒.
͑31͒
Inrting Eqs.(31)with l =0into Eq.(30)and taking into account that the operators ץu and ץv in polar cylindrical coordinates have the form
ץu ϵe −i ͓ץr −͑i /r ͒ץ͔,ץv ϵe i ͓ץr +͑i /r ͒ץ͔,
we obtain the transver circularly polarized components E Ќ,1͑±͒of the electric field in the form
2020全国高考语文E +,1
͑±͒
=͑1+B ±͒e −i ±z e −i ͓ץr −͑i /r ͒ץ͔⌿0,E −,1͑±͒=−͑1−B ±͒e −i ±z e i ͓ץr +͑i /r ͒ץ͔⌿0,
͑32͒
where r 2=x 2+y 2,and stands for an azimuthal angle.
The fields E Ќ,1͑±͒
are the generatrix vector functions for two major groups of fields.
The first group E ͉m −1͉͑±͒
is formed by using a simple rela-tion
E ͉m −1͉͑±͒
=ץv m
E Ќ,1͑±͒=͕e i ͓ץr +͑i /r ͒ץ͔͖͑m ͒E Ќ,1͑±͒.͑33͒
By making u of the Besl function properties [29]
J m −1͑x ͒=J m
Ј͑x ͒+͑m /x ͒J m ͑x ͒,J m +1͑x ͒=−J m
Ј͑x ͒+͑m /x ͒J m ͑x ͒,
we obtain the first t of Besl beams
E +,͉m −1͉͑±͒
=͑1+B ±͒e −i ±z e i ͑m −1͒J m −1͑Ur ͒
,
Fig.2.(Color online)Dispersive curves (a),(b)B ±͑U ͒and (c),(d)␦±2͑U ͒of nondiffracting beams in an optically active medium perturbed by a weak linear birefringence ⌬⑀,g =3ϫ10−10,g 3=−8ϫ10−10
.
Fig.1.(Color online)Dispersive curves for (a),(b)the mode pa-rameters B ±͑1,2,3͒and (b),(c)the propagation constants ±
͑1,2,3͒
/k in a purely chiral medium with n o =n 3=1.55:(1)g =3ϫ10−10
,g 3=8ϫ10−10;(2)g =g 3=3ϫ10−10;(3)g =3ϫ10−10,g 3=−8ϫ10−10;(d)͓g ͔=m .
E −,͉m −1͉͑±͒
=͑1−B ±͒e −i ±z e i ͑m +1͒J m +1͑Ur ͒,
͑34͒
where we omit the factor ͑−U ͒m +1in both components and where m =0,1,2,3,....Each circularly polarized compo-nent of the beam carries over the optical vortex.The vor-tex in the RHP component has a topological charge l +=m −1,while the vortex in the LHP component has a to-pological charge l −=m +1.
The cond group of fields E Ќ,m +1͑±͒
can be obtained as
E Ќ,͉m +1͉͑±͒=ץu m E Ќ.1͑±͒=͕e
roadtrip−i ͓ץr −͑i /r ͒ץ͔͖͑m ͒E Ќ,1͑±͒,͑35͒
so that the components of the mode field E ͑±͒take the form
E ˜+,͉m +1͉͑±͒=͑1+B ±͒e
−i ±z e −i ͑m +1͒J m +1͑Ur ͒,E ˜−,͉m +1͉͑±͒=͑1−B ±͒e
−i ±z e −i ͑m −1͒J m −1͑Ur ͒.͑36͒
The vortices embedded in the RHP and LHP components
of the cond group of fields have topological charges l +=−͑m +1͒and l −=−͑m −1͒,respectively.This situation has much to do with paraxial Laguerre–Gaussian beams in purely birefringent crystals [16,19].
In the simplest cas,for example,for beams with a uniformly distributed circular polarization state over the beam cross ction at the z =0plane,the field can be writ-ten as a superposition:
E m =a E ͉m −1͉͑+͒
blamed+b E ͉m −1͉͑−͒
.
͑37͒
From this point of view,the Besl beam in a nonparaxial region U ϳk ͱ⑀without an optical vortex m =1(l =0in the initial beam)in the RHP component at the plane z =0transiting through a purely chiral crystal will generate a doubly charged vortex l −=2in the LHP component simi-lar to that for the paraxial Gaussian beam [16,19]in a purely birefringent crystal.Using Eqs.(34)and (37)and the property J −m ͑x ͒=͑−1͒m J m ͑x ͒,we obtain a =1and b =−͑1−B +͒/͑1−B −͒.The LHP component for m =0has the simple form
E −=2i ͑1−B −͒e i e −i ˜
z sin ͑␦z ͒J 1͑Ur ͒,
͑38͒
where ␦=͑+−−͒/2,˜
=͑++−͒/2.Each circularly po-larized component carries over singly charged optical vor-tices with opposite handedness of the helices.When propagating along a purely chiral crystal,both the RHP and LHP components oscillate.At first glance,the oscil-lations em to be strange phenomena.Indeed,a circu-larly polarized plane wave propagates through a purely chiral crystal without any amplitude transformations.However,we deal here with a combined nonparaxial beam consisting of a great number of circularly polarized plane waves.The wave vectors of the waves lie on a cone surface,projections of their polarization states onto the obrvation plane containing both the RHP and LHP com-ponents.It is this property that is reflected in the wave structure of eigenmode-beams in Eqs.(36).The mode state with the only circular polarization at the z =0plane is not inherent to a nonparaxial beam.One of the compo-nents in Eq.(37)is suppresd by the interference effect only at the initial plane.As a result,we obrve ampli-
ooc什么意思
tude oscillations of the beam components along the crys-tal with period ⌳=2/␦.The amplitude oscillation van-ishes in a paraxial region.
第三英文
Such amplitude oscillations perturb the field structure of the beam,a polarization distribution at the beam cross ction being continuously transformed as the beam propagates.Typical maps of the polarization states [19]on the background of the total intensity distribution for a nonparaxial region of the beam propagation are illus-trated by Fig.3.The integral spirallike curves in the fig-ures reprent the lines tangential to the long axis of the polarization ellip at each point of the beam cross c-tion.There are a great number of singular planes located at the distances ⌬z =⌳,where the LHP component van-ishes.As the beam approaches the planes,the form of the spiral integral curves starts to be transformed:the spirals either are straightened (e z =10m z Ϸ65m in Fig.3)or tend to a t of concentric circles (e z Ϸ36m)forming at the planes z n =⌳/2͑2n +1͒,n =0,1,2,3,...,the fields being similar to TE modes with a linear polarization over the beam cross ction.The hand-edness of the spirals is converted to the opposite one when transiting the planes.However,the handedness of the polarization states is not transformed.Note that similar effects in gyrotropic crystals have been described in [30]for the energy conversion between TE and TM modes in a Besl beam on the basis of a spectral integral technique.
4.BIREFRINGENT CHIRAL CRYSTAL
A.Characteristic Equation
Singular properties of Besl beams in a purely birefrin-gent crystal are studied in detail in [11].In this ction,we concentrate on the more general ca of a birefringent chiral crystal.The basic properties of the nondiffracting beams in this ca are described by the characteristic Eqs.(24)and (25).A general analysis shows that the spectral curves for propagation constants ±and beam parameters B ±have similar features both for chiral birefringent and purely chiral crystals in a paraxial region becau the op-tical activity suppress a linear birefringence for the beams traveling along the crystal optical axis (compare Fig.1(a)and Fig.4(a)).Major differences in the properties of the nondiffracting beam–crystal system appear in a nonparaxial region due to a tensor character of the optical
activity ␥
ˆ.Let us consider the key cas.(1)The ca g 3g Ͻ0,͉g 3͉Ͼ͉g ͉.An isotropic point is an unstable one in a beam–crystal system.Even a very small perturbation in the form of a linear birefringence ⌬⑀Ӷ␥removes the isotropic point U =U is from the characteristic curves ±͑U ͒and B ±͑U ͒.Figures 2(b)and 2(d)show that the step-transformation in the dependency B ±͑U ͒is re-placed by smooth curves with a new stable singular point,a so-called gap-point:as U is →U g a sharp bend,a point of crossing branches of the spectral curve ±͑U ͒is smoothed,while different spectral curves are pushed
away.The ap-pearance of the gap-point is evidence of the dominant role of a linear birefringence in the beam–crystal system.In the vicinity of this point,the linear birefringence sup-press optical activity.The form of characteristic curves inside such an anomalous range for the crystals with ⌬⑀ӷ␥is shown in Fig.4(b).The parameter B −vanishes at
