15 January 2001
Optics Communications 187 (2001) 407±414
www.elsevier.com/locate/optcom
Formation of higher-order Bessel light beams in biaxial
crystals
T.A. King a,1, W. Hogervorst b,2, N.S. Kazak c,3, N.A. Khilo d,4,
A.A. Ryzhevich c,*
a
b
Laser Photonics, Department of Physics and Astronomy, University of Manchester, M13 9PL Manchester, UK
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, Belarus
d
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
*
Corresponding author. Tel.: +375-17-2841751; fax: +37517-2840879.
E-mail addresses: terry.king@man.ac.uk (T.A. King), wh@
nat.vu.nl (W. Hogervorst), tol@dragon.bas-net.by (N.S. Kazak,
A.A. Ryzhevich), nkhilo@optoinform.bas-net.by (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.
(see, for example, Refs. [1±5]). The main properties
of BLBs stem from the conical structure of their
spatial spectrum. The result of mutual interference
of plane-wave components of BLB is its multi-ring
spatial structure. The total number of BLB rings is
usually large. The consequence of this is the fact
that the divergence of an individual ring inside a
beam is much smaller than the divergence of the
whole beam. This property is referred to as diffractionless propagation of the BLB and is pronounced in the central zone of the beam at fairly
large lengths and at BLB excitation by a collimated Gaussian-type beam.
0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V.
PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 1 1 2 4 - X
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T.A. King et al. / Optics Communications 187 (2001) 407±414
The ``nondiverging'' optical beams are of interest for alignment and guiding of atoms. For
this purpose, both zero-order and higher-order
BLBs can be used [6±9]. Besides, higher-order
BLBs Laguerre±Gaussian beams are of interest
for studying the processes of creation and annihilation of wavefront dislocations [10±13].
The study and application of BLBs call for effective methods of their obtaining. At present, the
main method of obtaining zero-order BLBs is
based on the use of an axicon [2,14±16]. This
method is simple and permits to convert Gaussiantype beams with an eciency close to 100%. To
obtain higher-order BLBs, a holographic technique is used now [7,17,18]. Bessel beams of arbitrary order can also be produced by illuminating
an axicon with an appropriate Laguerre±Gaussian
mode [9]. In this paper the holographic technique
is used also to transform a Gaussian beam into a
higher order Laguerre±Gaussian beam. The holographic technique is simple and a 40% eciency
can be typically obtained. From the physical point
of view, in the holographic method a transfer of
dislocation of the transmission function from a
spatially inhomogeneous hologram to the light
®eld is realized.
An alternative physical principle is used in the
method of obtaining ®elds with wavefront dislocation based on the use of media with dislocation
of optical properties depending on the direction of
light propagation. An example of such media is
biaxial crystal. They feature dislocation of the
polarization of normal modes at light propagation
in the vicinity of binormals. Below we show theoretically and experimentally that ®rst-order and
higher-order BLBs can be obtained with use of
biaxial crystals.
formed at internal conical refraction and BLB
having a conical structure of the spectrum of
spatial frequencies. The presence of such a relationship would permit obtaining higher-order
BLBs by passing laser radiation through a homogeneous crystal.
It is known that in the vicinity of binormal, the
section of wave vectors surface represents two coaxial coni (Fig. 1). The geometrical parameters of
the coni are such that the wave vectors kp of two
normal modes propagating at a small angle with
binormal are equal
kp …q† ˆ kx e1 ‡ ky e2 ‡ …k ÿ bkx ÿ pbq ÿ q2 =2k†e3 ;
1†
where
q ˆ …kx2 ‡ ky2 †1=2 ,
1=2
ÿ1
b ˆ arctg‰c2 ……eÿ1
3 ÿ e2 †
ÿ1
=2v2 Š is the parameter of crystal an…eÿ1
2 ÿ e1 ††
isotropy, k ˆ x=v, v is the phase velocity of waves
in the direction of binormals, e1;2;3 are the principal
values of the permittivity tensor. Indices p ˆ 1
enumerate the slow and the fast modes, and e1 , e2 ,
e3 are the unit vectors of the Cartesian coordinate
system with the z-axis parallel to the binormal
(Fig. 1).
The polarization of normal waves in the vicinity
of the binormal is linear, and depending on an
azimuth angle u ˆ arctg…ky =kx † [20]. As the azimuth angle is changed by the value of u, the polarization vectors of the normal waves cp … p ˆ 1†
rotate by angle u=2 (Fig. 2). These polarization
vectors can be expressed in the form
2. Theoretical model
It is known that when the light wave propagates
in the vicinity of binormals, the e€ect of internal
conical refraction shows up (see, for example, Ref.
[19]) that is the incident ®eld forms a cone of directions of the energy ¯ow inside a crystal. On the
basis of this, a theoretical model may be proposed,
in which there is a relationship between the ®elds
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 ˆ 1 enumerate a slow and a fast
mode.
T.A. King et al. / Optics Communications 187 (2001) 407±414
409
Fig. 2. Azimuth dependence of the polarization of the fast and slow eigenmodes in the vicinity of the biaxial crystal binormal.
c1 …u† ˆ sin …u=2†e1 ‡ cos…u=2†e2 ;
cÿ1 …u† ˆ cos …u=2†e1 ÿ sin …u=2†e2
2†
or in an alternative form in terms of vectors of the
rightpand
left circular polarization e ˆ …e1 
ie2 †= 2:
ÿi
c1 …u† ˆ p ‰e‡ exp…iu=2† ÿ eÿ exp…ÿiu=2†Š;
2
1
cÿ1 …u† ˆ p ‰e‡ exp …iu=2† ‡ eÿ exp …ÿiu=2†Š:
2
3†
Normally, the e€ect of internal conical refraction is considered for the case of linear polarization of the incident light (see, for example, Refs.
[21,22]). It should be noted that due to the azimuth
dependence of the polarization direction of normal
waves inside the crystal the transmitted radiation
has, accordingly, the azimuthally inhomogeneous
intensity distribution. Therefore, in order to solve
the problem of formation of azimuthally homogeneous beams, it is necessary to use circularly
polarized input radiation.
Speci®cally, we shall assume that the incident
®eld has, for example, right-hand circular polarization with an amplitude at the input face of the
crystal equal to
a0 …q; u† ˆ a0 e‡ fin …q; u†:
4†
To ®nd the refracted waves, we represent the
incident ®eld polarization vector in the form of a
linear superposition of eigenvectors (2) as follows:
X
ap …u†cp …u†
5†
e‡ ˆ
p
p
with eigenvalues a1 …u†
pˆ
 i exp …ÿiu=2†= 2 and
aÿ1 …u† ˆ exp …ÿiu=2†= 2.
Then the Fourier spectrum of the ®eld (4) just
before the crystal can be given in the form
X
6†
A0 …q; u† ˆ a0 Fin …q; u†ap …u†cp …u†;
p
where
Fin …q; u† ˆ
Z Z
fin …q; u†
exp ‰iqq cos…u ÿ u1 †Šq dq du1
7†
is the Fourier spectrum of the function fin …q; u†.
In a particular case of the azimuth-independent
input ®eld,
Z
8†
Fin …q† ˆ 2p fin …q†J0 …qq†q dq;
where J0 …qq† is the zero-order Bessel function.
When passing over the crystal boundary, each
Fourier component (6) experiences re¯ection depending on the quantity q. This dependence can be
neglected for paraxial beams. For such beams the
Fourier components of the ®eld inside the crystal,
near its boundary, are described by expression (6).
Consequently, the ®eld in the crystal is
Z Z
a0 X
F …q; u1 ; z†ap …u1 †cp …u1 †
a…q; u; z† ˆ
2p†2 p
exp ‰iqq cos …u ÿ u1 † ÿ ipbqzŠq dq du1 ;
9†
where
410
T.A. King et al. / Optics Communications 187 (2001) 407±414
F …q; u1 ; z† ˆ Fin …q; u1 † exp …ÿiq2 z=2k†;
q
2
q ˆ …x ÿ bz† ‡ y 2 :
Let the input ®eld be azimuthally symmetric.
Then, substituting the polarization vectors from
Eq. (3) into Eq. (9), integrating with respect to u1
and summing over p, we obtain
Z
a0
a…q; u; z† ˆ
F …q; z†
2p†
‰J0 …qq†e‡ cos …bqz† ÿ J1 …qq†
exp …ÿiu†eÿ sin …bqz†Šq dq: …10†
Now let us assume that the input ®eld is lefthand circularly polarized. In this case, in expan…u†
sion (5) the p
eigenvalues
are equal to a1p
 ˆ

ÿi exp …iu=2†= 2 and aÿ1 …u† ˆ exp …iu=2†= 2. A
calculation similar to the previous one gives
Z
a0
F …q; z†
a…q; u; z† ˆ
2p†
‰J0 …qq†eÿ cos …bqz† ÿ J1 …qq†
exp …iu†e‡ sin …bqz†Šq dq:
11†
As follows from Eqs. (10) and (11), the circularly polarized wave excites in the biaxial crystal
the superposition of two waves with orthogonal
polarizations one of which contains wavefront
dislocation.
Let us consider the particular case of incidence
of a zero-order BLB on the crystal. Then
fin …q† ˆ J0 …qin q†. Neglecting the in¯uence of the
®nite transverse extension of the Bessel beam on its
Fourier spectrum we obtain Fin …q† ˆ 2pd…q ÿ qin †=
qin , where d…x† is the delta function. Integrating
Eqs. (10) and (11) we ®nd the ®eld amplitudes at
the output face of a crystal of thickness L:
the de®nition of F …q; u; z† after Eq. (9), which is
immaterial for calculating the intensity pattern, is
omitted.
Thus, the output ®eld is the superposition of the
zero-order and the ®rst-order BLBs with orthogonal polarizations. The amplitude ratio of these
beams depends on the distance covered in the
crystal by the simple harmonic law. When the
condition
bqin L ˆ …2n ‡ 1†p=2;
where n ˆ 0; 1; . . . is ful®lled, the term in Eqs. (12)
and (13) proportional to the Bessel function
J0 …qin q† becomes zero. Consequently, complete
conversion of the circularly polarized zero-order
Bessel beam into the ®rst-order Bessel beam with
orthogonal polarization will take place. Estimation of the oscillation period L0 ˆ p=2bqin at
b 0:016 (KTP crystal), qin ˆ 2pcin =k, k ˆ 0:63
lm, cin ˆ 0:02 gives L0 0:5 mm.
It is important to note that the process of increasing the Bessel function order can be continued. To do this, it is necessary to extract from
output ®eld Eqs. (12) and (13) the ®rst-order
Bessel beam and realize its repeated passage
through the crystal.
In the general case of incidence on the crystal of
a Bessel beam of order m with the phase factor
exp…imu†, integrating Eq. (7), we ®nd
m
Fin …q; u† ˆ 2p…i† exp…imu†d…q ÿ qin †=qin :
for the right-hand polarized incident beam, and
a…q; u; L† ˆ a0 …J0 …qin q† cos…bqin L†eÿ
15†
Substitution of Eq. (15) into Eq. (9) leads to
formulas generalizing Eqs. (12) and (13):
a…q; u; L† ˆ
a…q; u; L† ˆ a0 …J0 …qin q† cos…bqin L†e‡
ÿ J1 …qin q† exp… ÿ iu† sin …bqin L†eÿ †
12†
14†
a0
Jm …qin q† cos…bqin L†e‡
2p
‡ Jmÿ1 …qin q† exp‰i…m ÿ 1†uŠ sin…bqin L†eÿ †;
16†
a…q; u; L† ˆ
a0
Jm …qin q† cos…bqin L†eÿ
2p
ÿ Jm‡1 …qin q† exp‰i…m ‡ 1†uŠ sin…bqin L†e‡ †:
17†
ÿ J1 …qin q† exp…iu† sin …bqin L†e‡ †
13†
for the left-hand polarized one. In Eqs. (12) and
(13) the phase factor exp …ÿq2in z=2k† inferred from
Formulas (12), (13), (16), (17) describing the e€ect
of BLB order transformation hold for paraxial
BLBs within the applicability limits of the original
relation (1). In the case of a KTP crystal, BLBs
T.A. King et al. / Optics Communications 187 (2001) 407±414
with a cone angle not exceeding 10° can be adequately described by formulas (16) and (17).
Thus, theoretical analysis indicates the possibility of formation in biaxial crystals of light
beams with screw dislocations and, in particular,
of higher-order BLBs.
3. Experimental results and discussion
The e€ect of transformation of the order of a
BLB was tested experimentally. The experimental
setup shown in Fig. 3. A collimated, circularly
polarized Gaussian beam of a He±Ne laser had a
waist size of 2.5 mm and was transformed into a
zero-order BLB using an axicon with an internal
angle of 2:2° and refractive index of 1.5. Consequently, the cone angle cin ˆ qin =k0 of the zeroorder BLB was about cin ˆ 1:1°. This Bessel beam
had illuminated a 12 mm thick KTP crystal oriented perpendicularly to the binormal and located
at a distance of 16 cm from the axicon. The radius of the BLB formed by the axicon, within the
limits of the crystal position, was about 1.5 mm. In
the output ®eld, the right and left circularly polarized components were separated and investigated independently. The intensity distribution in
their cross-section was measured and compared to
squared zero- and ®rst-order Bessel functions in
accordance with Eq. (12). Fig. 4(a) and (b) show
the images of the central part of the beams crosssections beyond the polarizer±analyzer. Fig. 5(a)
and (b) give the corresponding intensity distributions in comparison with the squared Bessel
functions J02 …ck0 q† and J12 …ck0 q†. It is clear that
411
good quantitative agreement between theoretical
and experimental data is obtained.
In addition to these radial distributions of intensity, an important attribute of ®rst-order BLBs
is their wavefront dislocation. To display this, an
interference of the BLB with plane and spherical
reference waves has been studied. The numerically
calculated interference pictures are shown in Fig. 6.
It is visible that in the case of interference with a
plane reference wave, a characteristic indication of
dislocation is the bifurcation of the maximum at the
center of the BLB and in case of interference with a
spherical wave ± a spiral structure (see, also Ref.
[23]). The above properties have been observed in
experimental interference patterns. Images of these
patterns are given in Fig. 7(a) and (b) for plane and
spherical reference waves respectively.
Note that the direction of spiralling as well as
the orientation of the bifurcated maximum reverse
when we change the sign of input beam circular
polarization.
After crystal 6 the output ®rst- and zero-order
Bessel beams by virtue of the small cone angle and
relatively small longitudinal dimensions of the
crystal, exist practically in the same region in
which the zero-order initial beam would exist if the
crystal was removed from the system. This region
approximately corresponds to the ®gure formed by
rotating the shaded rhomb (if the refraction in the
crystal is ignored) and has ®nite dimensions (see
Fig. 3). The maximum distance from axicon 5, on
which the ®rst-order output BLB still exists,
zmax ˆ Rd =cin where Rd is the radius of the diaphragm opening limiting the Gaussian beam illuminating 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.
412
T.A. King et al. / Optics Communications 187 (2001) 407±414
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).
Fig. 4. Images of the ®eld beyond the polarizer±analyzer for
radiation with circular polarization (a) coinciding with the input one (zero-order BLB) and (b) orthogonal to it (®rst-order
BLB).
and interference patterns given by us correspond
to one of the cross-section of this region. However,
in all the cross-sections of the above region, after
the polarizer±analyzer 8 the intensity distribution
is analogous to that given in the ®gure. At distance
from axicon 5 larger then zmax ®rst-order BLB is
transformed into a ring ®eld which preserves,
nevertheless, the azimuthal phase modulation inherent in the ®rst-order BLB. Using an additional
axicon 11, this ring ®eld can be transformed into a
®rst-order BLB again.
We also checked the theoretical result of Section 2, indicating the possibility of stepwise
increase of the order of Bessel function under repeated passage of light through a crystal. For this
purpose we used an experimental set-up with two
crystals in series. In the ®rst stage a zero-order
BLB, incident on the KTP crystal, is transformed
into a ®rst-order BLB. Next, the ®rst-order beam
polarization is converted from left-circular to
right-circular polarization and the beam is directed
T.A. King et al. / Optics Communications 187 (2001) 407±414
413
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.
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.
towards the second biaxial crystal a-HIO3 (iodic
acid) along the direction of its optical axis. According to Eq. (16) the output ®eld then involves a
superposition of ®rst- and second-order BLBs.
Fig. 8 shows the radial intensity distribution of the
output ®eld component with polarization orthogonal to the input polarization. It is seen that this
distribution is reasonably well approximated by
the squared second-order Bessel function.
When incident on the crystal is Gaussian beam
instead of Bessel one, it can be transformed into a
Laguerre±Gaussian mode LG01 where radial index
is 0 and azimuthal index is 1. Then we generated a
®rst-order BLB by illuminating an axicon with this
mode with total eciency of about 60%.
It can be concluded that the e€ect of transformation of Bessel beams when propagating along
the biaxial crystal binormal as expressed by Eqs.
(12) and (16) is con®rmed experimentally.
414
T.A. King et al. / Optics Communications 187 (2001) 407±414
biaxial crystals permits to transform several light
beams simultaneously.
Acknowledgements
Financial support from INTAS (INTAS-BELARUS 97-0533) is gratefully acknowledged.
References
Fig. 8. Experimental intensity distribution (curve 1) of the
output ®eld behind the second crystal with polarization orthogonal to the input ®eld polarization as compared to designtheoretical second-order Bessel beam intensity distribution
(curve 2).
4. Conclusions
In this work an optical e€ect of transformation
of transversal structure of light ®eld propagating
along a binormal of biaxial crystal is investigated
theoretically and experimentally. It is shown that
circularly polarized input beam excites two circularly polarized beams in the crystal. The beam
polarized orthogonally to the input beam has a
screw wavefront dislocation if the input beam had
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based on biaxial crystals is that transformation
eciency is close to 100%. Besides, because of their
high radiation damage threshold, biaxial crystals
can be used as intracavity elements. Let us note
that transversal invariance of the scheme based on
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