formation of higher-order bessel light beams in biaxial crystals

Formation of higher-order Bessel light beams in biaxialcrystals

T.A. King a,1, W. Hogervorst b,2, N.S. Kazak c,3, N.A. Khilo d,4,A.A. Ryzhevich c,*

a Laser Photonics, Department of Physics and Astronomy, University of Manchester, M13 9PL Manchester, UKb Laser Centre, Department of Physics and Astronomy, Free University, De Boelelaan 1081, 1081 HV Amsterdam, Netherlands

c Institute of Physics, NAS of Belarus, 68 F. Skaryna ave., 220072 Minsk, Belarusd Division for optical problems in information technologies, NAS of Belarus, 1-2 Kuprevich str., 220141 Minsk, Belarus

Received 5 September 2000; accepted 15 November 2000

Abstract

A transformation of the order of a Bessel light beam (BLB) from zeroth to ®rst and from ®rst to second when

propagating light through a biaxial crystal has been studied theoretically and experimentally. The possibility of the

formation of higher-order BLBs by means of this e�ect is con®rmed. The transformation of beam order is observed

when propagating circularly polarized light along the optical axis of the crystal under conditions of internal conical

refraction. It is shown that a proper choice of the crystal length or conicity angle of the incident beam permits complete

transformation of a zero-order Bessel input beam into a ®rst-order Bessel beam. Ó 2001 Published by Elsevier Science

B.V.

PACS: 42.25.Bs; 42.25.Ja; 42.25.Lc

Keywords: Bessel light beam; Internal conical refraction; Biaxial crystal

1. Introduction

The linear properties of Bessel light beams(BLBs) have been studied in fair detail by now

(see, for example, Refs. [1±5]). The main propertiesof BLBs stem from the conical structure of theirspatial spectrum. The result of mutual interferenceof plane-wave components of BLB is its multi-ringspatial structure. The total number of BLB rings isusually large. The consequence of this is the factthat the divergence of an individual ring inside abeam is much smaller than the divergence of thewhole beam. This property is referred to as dif-fractionless propagation of the BLB and is pro-nounced in the central zone of the beam at fairlylarge lengths and at BLB excitation by a colli-mated Gaussian-type beam.

15 January 2001

Optics Communications 187 (2001) 407±414

www.elsevier.com/locate/optcom

* Corresponding author. Tel.: +375-17-2841751; fax: +375-

17-2840879.

E-mail addresses: [email protected] (T.A. King), wh@

nat.vu.nl (W. Hogervorst), [email protected] (N.S. Kazak,

A.A. Ryzhevich), [email protected] (N.A. Khilo).1 Tel.: +44-1612754181; fax: +44-1612755509.2 Tel.: +31-20-4447947; fax: +31-20-4447899.3 Tel.: +375-17-2841751; fax: +375-17-2840879.4 Tel.: +375-17-2637735.

0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V.

PII: S00 3 0-4 0 18 (0 0 )0 11 2 4- X

The ``nondiverging'' optical beams are of in-terest for alignment and guiding of atoms. Forthis purpose, both zero-order and higher-orderBLBs can be used [6±9]. Besides, higher-orderBLBs Laguerre±Gaussian beams are of interestfor studying the processes of creation and annihi-lation of wavefront dislocations [10±13].

The study and application of BLBs call for ef-fective methods of their obtaining. At present, themain method of obtaining zero-order BLBs isbased on the use of an axicon [2,14±16]. Thismethod is simple and permits to convert Gaussian-type beams with an e�ciency close to 100%. Toobtain higher-order BLBs, a holographic tech-nique is used now [7,17,18]. Bessel beams of arbi-trary order can also be produced by illuminatingan axicon with an appropriate Laguerre±Gaussianmode [9]. In this paper the holographic techniqueis used also to transform a Gaussian beam into ahigher order Laguerre±Gaussian beam. The holo-graphic technique is simple and a 40% e�ciencycan be typically obtained. From the physical pointof view, in the holographic method a transfer ofdislocation of the transmission function from aspatially inhomogeneous hologram to the light®eld is realized.

An alternative physical principle is used in themethod of obtaining ®elds with wavefront dislo-cation based on the use of media with dislocationof optical properties depending on the direction oflight propagation. An example of such media isbiaxial crystal. They feature dislocation of thepolarization of normal modes at light propagationin the vicinity of binormals. Below we show theo-retically and experimentally that ®rst-order andhigher-order BLBs can be obtained with use ofbiaxial crystals.

2. Theoretical model

It is known that when the light wave propagatesin the vicinity of binormals, the e�ect of internalconical refraction shows up (see, for example, Ref.[19]) that is the incident ®eld forms a cone of di-rections of the energy ¯ow inside a crystal. On thebasis of this, a theoretical model may be proposed,in which there is a relationship between the ®elds

formed at internal conical refraction and BLBhaving a conical structure of the spectrum ofspatial frequencies. The presence of such a rela-tionship would permit obtaining higher-orderBLBs by passing laser radiation through a homo-geneous crystal.

It is known that in the vicinity of binormal, thesection of wave vectors surface represents two co-axial coni (Fig. 1). The geometrical parameters ofthe coni are such that the wave vectors kp of twonormal modes propagating at a small angle withbinormal are equal

kp�q� � kxe1 � kye2 � �k ÿ bkx ÿ pbqÿ q2=2k�e3;

�1�where q � �k2

x � k2y �1=2

, b � arctg�c2��eÿ13 ÿ eÿ1

2 ��eÿ1

2 ÿ eÿ11 ��1=2

=2v2� is the parameter of crystal an-isotropy, k � x=v, v is the phase velocity of wavesin the direction of binormals, e1;2;3 are the principalvalues of the permittivity tensor. Indices p � �1enumerate the slow and the fast modes, and e1, e2,e3 are the unit vectors of the Cartesian coordinatesystem with the z-axis parallel to the binormal(Fig. 1).

The polarization of normal waves in the vicinityof the binormal is linear, and depending on anazimuth angle u � arctg�ky=kx� [20]. As the azi-muth angle is changed by the value of u, the po-larization vectors of the normal waves cp �p � �1�rotate by angle u=2 (Fig. 2). These polarizationvectors can be expressed in the form

Fig. 1. Cross-section of the wave vector surface in the vicinity of

the binormal by the crystallographic plane XZ. a is anisotropy

parameter and indices p � �1 enumerate a slow and a fast

mode.

408 T.A. King et al. / Optics Communications 187 (2001) 407±414

c1�u� � sin�u=2�e1 � cos�u=2�e2;

cÿ1�u� � cos�u=2�e1 ÿ sin�u=2�e2

�2�

or in an alternative form in terms of vectors of theright and left circular polarization e� � e1��ie2�=

��2p

:

c1�u� � ÿi��2p �e� exp�iu=2� ÿ eÿ exp�ÿiu=2��;

cÿ1�u� � 1��2p �e� exp�iu=2� � eÿ exp�ÿiu=2��:

�3�

Normally, the e�ect of internal conical refrac-tion is considered for the case of linear polariza-tion of the incident light (see, for example, Refs.[21,22]). It should be noted that due to the azimuthdependence of the polarization direction of normalwaves inside the crystal the transmitted radiationhas, accordingly, the azimuthally inhomogeneousintensity distribution. Therefore, in order to solvethe problem of formation of azimuthally homo-geneous beams, it is necessary to use circularlypolarized input radiation.

Speci®cally, we shall assume that the incident®eld has, for example, right-hand circular polar-ization with an amplitude at the input face of thecrystal equal to

a0�q;u� � a0e�fin�q;u�: �4�To ®nd the refracted waves, we represent the

incident ®eld polarization vector in the form of alinear superposition of eigenvectors (2) as follows:

e� �X

p

ap�u�cp�u� �5�

with eigenvalues a1�u� � i exp�ÿiu=2�= ��2p

andaÿ1�u� � exp�ÿiu=2�= ��

2p

.Then the Fourier spectrum of the ®eld (4) just

before the crystal can be given in the form

A0�q;u� � a0

Xp

Fin�q;u�ap�u�cp�u�; �6�

where

Fin�q;u� �Z Z

fin�q;u�� exp �iqqcos�uÿ u1��qdqdu1 �7�

is the Fourier spectrum of the function fin�q;u�.In a particular case of the azimuth-independent

input ®eld,

Fin�q� � 2pZ

fin�q�J0�qq�qdq; �8�

where J0�qq� is the zero-order Bessel function.When passing over the crystal boundary, each

Fourier component (6) experiences re¯ection de-pending on the quantity q. This dependence can beneglected for paraxial beams. For such beams theFourier components of the ®eld inside the crystal,near its boundary, are described by expression (6).Consequently, the ®eld in the crystal is

a�q;u; z� � a0

�2p�2X

p

Z ZF �q;u1; z�ap�u1�cp�u1�

� exp �iqqcos�uÿ u1� ÿ ipbqz�qdqdu1;

�9�where

Fig. 2. Azimuth dependence of the polarization of the fast and slow eigenmodes in the vicinity of the biaxial crystal binormal.

T.A. King et al. / Optics Communications 187 (2001) 407±414 409

F �q;u1; z� � Fin�q;u1�exp�ÿiq2z=2k�;q �

��xÿ bz�2 � y2

q:

Let the input ®eld be azimuthally symmetric.Then, substituting the polarization vectors fromEq. (3) into Eq. (9), integrating with respect to u1

and summing over p, we obtain

a�q;u; z� � a0

�2p�Z

F �q; z�

� �J0�qq�e� cos�bqz� ÿ J1�qq�� exp�ÿiu�eÿ sin�bqz��qdq: �10�

Now let us assume that the input ®eld is left-hand circularly polarized. In this case, in expan-sion (5) the eigenvalues are equal to a1�u� �ÿi exp�iu=2�= ��

2p

and aÿ1�u� � exp�iu=2�= ��2p

. Acalculation similar to the previous one gives

a�q;u; z� � a0

�2p�Z

F �q; z�

� �J0�qq�eÿ cos�bqz� ÿ J1�qq�� exp�iu�e� sin�bqz��qdq: �11�

As follows from Eqs. (10) and (11), the circu-larly polarized wave excites in the biaxial crystalthe superposition of two waves with orthogonalpolarizations one of which contains wavefrontdislocation.

Let us consider the particular case of incidenceof a zero-order BLB on the crystal. Thenfin�q� � J0�qinq�. Neglecting the in¯uence of the®nite transverse extension of the Bessel beam on itsFourier spectrum we obtain Fin�q� � 2pd�qÿ qin�=qin, where d�x� is the delta function. IntegratingEqs. (10) and (11) we ®nd the ®eld amplitudes atthe output face of a crystal of thickness L:

a�q;u; L� � a0 J0�qinq�cos�bqinL�e��ÿ J1�qinq�exp�ÿ iu� sin�bqinL�eÿ�

�12�for the right-hand polarized incident beam, and

a�q;u; L� � a0 J0�qinq�cos�bqinL�eÿ�ÿ J1�qinq�exp�iu� sin�bqinL�e��

�13�for the left-hand polarized one. In Eqs. (12) and(13) the phase factor exp�ÿq2

inz=2k� inferred from

the de®nition of F �q;u; z� after Eq. (9), which isimmaterial for calculating the intensity pattern, isomitted.

Thus, the output ®eld is the superposition of thezero-order and the ®rst-order BLBs with orthog-onal polarizations. The amplitude ratio of thesebeams depends on the distance covered in thecrystal by the simple harmonic law. When thecondition

bqinL � �2n� 1�p=2; �14�

where n � 0; 1; . . . is ful®lled, the term in Eqs. (12)and (13) proportional to the Bessel functionJ0�qinq� becomes zero. Consequently, completeconversion of the circularly polarized zero-orderBessel beam into the ®rst-order Bessel beam withorthogonal polarization will take place. Estima-tion of the oscillation period L0 � p=2bqin atb � 0:016 (KTP crystal), qin � 2pcin=k, k � 0:63lm, cin � 0:02 gives L0 � 0:5 mm.

It is important to note that the process of in-creasing the Bessel function order can be contin-ued. To do this, it is necessary to extract fromoutput ®eld Eqs. (12) and (13) the ®rst-orderBessel beam and realize its repeated passagethrough the crystal.

In the general case of incidence on the crystal ofa Bessel beam of order m with the phase factorexp�imu�, integrating Eq. (7), we ®nd

Fin�q;u� � 2p�i�m exp�imu�d�qÿ qin�=qin: �15�Substitution of Eq. (15) into Eq. (9) leads to

formulas generalizing Eqs. (12) and (13):

a�q;u;L� � a0

2pJm�qinq�cos�bqinL�e�� Jmÿ1�qinq�exp �i�mÿ 1�u� sin�bqinL�eÿ�;

�16�a�q;u;L� � a0

2pJm�qinq�cos�bqinL�eÿ�ÿ Jm�1�qinq�exp �i�m� 1�u� sin�bqinL�e��:

�17�Formulas (12), (13), (16), (17) describing the e�ectof BLB order transformation hold for paraxialBLBs within the applicability limits of the originalrelation (1). In the case of a KTP crystal, BLBs


with a cone angle not exceeding 10° can be ade-quately described by formulas (16) and (17).

Thus, theoretical analysis indicates the possi-bility of formation in biaxial crystals of lightbeams with screw dislocations and, in particular,of higher-order BLBs.

3. Experimental results and discussion

The e�ect of transformation of the order of aBLB was tested experimentally. The experimentalsetup shown in Fig. 3. A collimated, circularlypolarized Gaussian beam of a He±Ne laser had awaist size of 2.5 mm and was transformed into azero-order BLB using an axicon with an internalangle of �2:2° and refractive index of 1.5. Con-sequently, the cone angle cin � qin=k0 of the zero-order BLB was about cin � 1:1°. This Bessel beamhad illuminated a 12 mm thick KTP crystal ori-ented perpendicularly to the binormal and locatedat a distance of �16 cm from the axicon. The ra-dius of the BLB formed by the axicon, within thelimits of the crystal position, was about 1.5 mm. Inthe output ®eld, the right and left circularly po-larized components were separated and investi-gated independently. The intensity distribution intheir cross-section was measured and compared tosquared zero- and ®rst-order Bessel functions inaccordance with Eq. (12). Fig. 4(a) and (b) showthe images of the central part of the beams cross-sections beyond the polarizer±analyzer. Fig. 5(a)and (b) give the corresponding intensity distribu-tions in comparison with the squared Besselfunctions J 2

0 �ck0q� and J 21 �ck0q�. It is clear that

good quantitative agreement between theoreticaland experimental data is obtained.

In addition to these radial distributions of in-tensity, an important attribute of ®rst-order BLBsis their wavefront dislocation. To display this, aninterference of the BLB with plane and sphericalreference waves has been studied. The numericallycalculated interference pictures are shown in Fig. 6.It is visible that in the case of interference with aplane reference wave, a characteristic indication ofdislocation is the bifurcation of the maximum at thecenter of the BLB and in case of interference with aspherical wave ± a spiral structure (see, also Ref.[23]). The above properties have been observed inexperimental interference patterns. Images of thesepatterns are given in Fig. 7(a) and (b) for plane andspherical reference waves respectively.

Note that the direction of spiralling as well asthe orientation of the bifurcated maximum reversewhen we change the sign of input beam circularpolarization.

After crystal 6 the output ®rst- and zero-orderBessel beams by virtue of the small cone angle andrelatively small longitudinal dimensions of thecrystal, exist practically in the same region inwhich the zero-order initial beam would exist if thecrystal was removed from the system. This regionapproximately corresponds to the ®gure formed byrotating the shaded rhomb (if the refraction in thecrystal is ignored) and has ®nite dimensions (seeFig. 3). The maximum distance from axicon 5, onwhich the ®rst-order output BLB still exists,zmax � Rd=cin where Rd is the radius of the dia-phragm opening limiting the Gaussian beam illu-minating the axicon. The intensity distributions

Fig. 3. Optical system for transformation of the order of the Bessel beam: 1 ± 20� telescope; 2 ± polarizer; 3 and 7 ± quarter-wave

plates, 4 ± diaphragm, 5 ± axicon; 6 ± KTP crystal; 8 ± polarizer±analyzer; 9 ± microscope; 10 ± recording system; 11 ± additional

axicon.


and interference patterns given by us correspondto one of the cross-section of this region. However,in all the cross-sections of the above region, afterthe polarizer±analyzer 8 the intensity distributionis analogous to that given in the ®gure. At distancefrom axicon 5 larger then zmax ®rst-order BLB istransformed into a ring ®eld which preserves,

nevertheless, the azimuthal phase modulation in-herent in the ®rst-order BLB. Using an additionalaxicon 11, this ring ®eld can be transformed into a®rst-order BLB again.

We also checked the theoretical result of Sec-tion 2, indicating the possibility of stepwiseincrease of the order of Bessel function under re-peated passage of light through a crystal. For thispurpose we used an experimental set-up with twocrystals in series. In the ®rst stage a zero-orderBLB, incident on the KTP crystal, is transformedinto a ®rst-order BLB. Next, the ®rst-order beampolarization is converted from left-circular toright-circular polarization and the beam is directed

Fig. 4. Images of the ®eld beyond the polarizer±analyzer for

radiation with circular polarization (a) coinciding with the in-

put one (zero-order BLB) and (b) orthogonal to it (®rst-order

BLB).

Fig. 5. Experimental intensity distribution (curve 1) of the

output ®eld with polarization (a) equal to the input one and (b)

orthogonal to it as compared to design-theoretical zero- and

®rst-order Bessel beam intensity distributions (curve 2).


towards the second biaxial crystal a-HIO3 (iodicacid) along the direction of its optical axis. Ac-cording to Eq. (16) the output ®eld then involves asuperposition of ®rst- and second-order BLBs.Fig. 8 shows the radial intensity distribution of theoutput ®eld component with polarization orthog-onal to the input polarization. It is seen that this

distribution is reasonably well approximated bythe squared second-order Bessel function.

When incident on the crystal is Gaussian beaminstead of Bessel one, it can be transformed into aLaguerre±Gaussian mode LG01 where radial indexis 0 and azimuthal index is 1. Then we generated a®rst-order BLB by illuminating an axicon with thismode with total e�ciency of about 60%.

It can be concluded that the e�ect of transfor-mation of Bessel beams when propagating alongthe biaxial crystal binormal as expressed by Eqs.(12) and (16) is con®rmed experimentally.

Fig. 6. Distribution of interference maxima, calculated for the

case of interference of the ®eld �exp�iu� with (a) a plane wave

and (b) a spherical wave.

Fig. 7. Images generated by interference of a ®rst-order Bessel

beam (a) with a reference plane wave and (b) with a reference

spherical wave.


4. Conclusions

In this work an optical e�ect of transformationof transversal structure of light ®eld propagatingalong a binormal of biaxial crystal is investigatedtheoretically and experimentally. It is shown thatcircularly polarized input beam excites two circu-larly polarized beams in the crystal. The beampolarized orthogonally to the input beam has ascrew wavefront dislocation if the input beam hadno dislocation. When the input beam contains adislocation of order m, the above mentioned beamhas a dislocation of m� 1 or mÿ 1 order. Thephysical essence of such wavefront transformationis a transfer of singularity of optical properties ofbiaxial crystal in the vicinity of its binormal to thelight ®eld wavefront.

Using this method it is possible to obtain aBessel beam of ®rst and higher orders from aninput zero-order BLB. The advantage of themethod of ®rst-order Bessel beams productionbased on biaxial crystals is that transformatione�ciency is close to 100%. Besides, because of theirhigh radiation damage threshold, biaxial crystalscan be used as intracavity elements. Let us notethat transversal invariance of the scheme based on

biaxial crystals permits to transform several lightbeams simultaneously.

Acknowledgements

Financial support from INTAS (INTAS-BEL-ARUS 97-0533) is gratefully acknowledged.

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Fig. 8. Experimental intensity distribution (curve 1) of the

output ®eld behind the second crystal with polarization or-

thogonal to the input ®eld polarization as compared to design-

theoretical second-order Bessel beam intensity distribution

(curve 2).


formation of higher-order bessel light beams in biaxial crystals

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