Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis
We have experimentally considered evolution of the Gaussian beam propagating nearly perpendicular to the uniaxial crystal axis. Also we have analyzed the spin and orbital angular momenta and found oscillations of the angular momentum when the crystal optical axis is rotated.
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| Мова: | English |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2013
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| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Цитувати: | Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis / B.V. Sokolenko, A.F. Rubass, A.V. Volyar // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 4. — С. 344-348. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1178122017-05-27T03:06:28Z Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis Sokolenko, B.V. Rubass, A.F. Volyar, A.V. We have experimentally considered evolution of the Gaussian beam propagating nearly perpendicular to the uniaxial crystal axis. Also we have analyzed the spin and orbital angular momenta and found oscillations of the angular momentum when the crystal optical axis is rotated. 2013 Article Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis / B.V. Sokolenko, A.F. Rubass, A.V. Volyar // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 4. — С. 344-348. — Бібліогр.: 7 назв. — англ. 1560-8034 PACS 42.50.Tx http://dspace.nbuv.gov.ua/handle/123456789/117812 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
We have experimentally considered evolution of the Gaussian beam
propagating nearly perpendicular to the uniaxial crystal axis. Also we have analyzed the
spin and orbital angular momenta and found oscillations of the angular momentum when
the crystal optical axis is rotated. |
| format |
Article |
| author |
Sokolenko, B.V. Rubass, A.F. Volyar, A.V. |
| spellingShingle |
Sokolenko, B.V. Rubass, A.F. Volyar, A.V. Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Sokolenko, B.V. Rubass, A.F. Volyar, A.V. |
| author_sort |
Sokolenko, B.V. |
| title |
Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis |
| title_short |
Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis |
| title_full |
Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis |
| title_fullStr |
Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis |
| title_full_unstemmed |
Generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis |
| title_sort |
generation of optical vortices in rotating uniaxial crystal for light propagation along the perpendicular to its optical axis |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117812 |
| citation_txt |
Generation of optical vortices in rotating uniaxial crystal
for light propagation along the perpendicular to its optical axis / B.V. Sokolenko, A.F. Rubass, A.V. Volyar // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 4. — С. 344-348. — Бібліогр.: 7 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT sokolenkobv generationofopticalvorticesinrotatinguniaxialcrystalforlightpropagationalongtheperpendiculartoitsopticalaxis AT rubassaf generationofopticalvorticesinrotatinguniaxialcrystalforlightpropagationalongtheperpendiculartoitsopticalaxis AT volyarav generationofopticalvorticesinrotatinguniaxialcrystalforlightpropagationalongtheperpendiculartoitsopticalaxis |
| first_indexed |
2025-07-08T12:50:31Z |
| last_indexed |
2025-07-08T12:50:31Z |
| _version_ |
1837083167770542080 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 344-348.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
344
PACS 42.50.Tx
Generation of optical vortices in rotating uniaxial crystal
for light propagation along the perpendicular to its optical axis
B.V. Sokolenko, A.F. Rubass, A.V. Volyar
Taurida National University, Department of General Physics,
4, Vernadsky Ave., 95007 Simferopol, Ukraine,
Phone: +38(066)574-73-02, e-mail: simplexx.87@gmail.com
Abstract. We have experimentally considered evolution of the Gaussian beam
propagating nearly perpendicular to the uniaxial crystal axis. Also we have analyzed the
spin and orbital angular momenta and found oscillations of the angular momentum when
the crystal optical axis is rotated.
Keywords: angular momentum, singular beam, polarization, optical vortex, umbilics.
Manuscript received 10.07.13; revised version received 03.09.13; accepted for
publication 23.10.13; published online 16.12.13.
1. Introduction
It is well known that the Gaussian beam propagates
perpendicular to the crystal optical axis and splits into
the ordinary and extraordinary ones. The ordinary beam
transmits through the crystal as through the isotropic
medium while the extraordinary beam is elliptically
deformed [1]. In case the beam bears the optical vortex
then the crystal changes the singular structure of the
vortex. The slight changes of the crystal parameters can
result in the critical transformations of the phase
singularity [2-4]. However the question of the vortex
generation in this case remains open.
Thus the aim of the presented paper is the
experimental study of the birth and death of phase
singularities processes when the beam propagates nearly
perpendicular the crystal optical axis.
2. The basic representations
Let us consider the total representations of the properties
of the monochromatic optical fields following the
paper [5].
In the paraxial approximation, the electromagnetic
field can be written in the form:
E = Ex + Ey, H = Hx + Hy (1)
or
,
/
/
)exp(
jzj
y
x
j
j
u
y
x
e
k
i
u
e
e
ikz
H
E
(2)
where j = x, y and k stand for the wavenumber.
The complex amplitude takes the form
)],(exp[),( zrikzrAu jjj (3)
Then the Poynting vector is
.
Re
2
**
**
YYXX
XrrX
HEHE
HEHE
c
S
(4)
After substitution of Eq. (3) into Eq. (4), we can
select the energy flux associated with the spin-orbital
momentum [5]
,)(
16
**
YXYXzc uuuue
k
ic
S
(5)
while the spin angular momentum (SAM) is
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 344-348.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
345
y
Ls
Bs1
P
L1 Cr L2 CCD
x
4
С
0 Bs2
4
Mr1 Mr2
z
Fig. 1. The schematic representation of experimental set-up.
.][
1 2
2
rdSr
c
L cc (6)
At the same time, the volume density of the SAM
is written as YXYX uuuu
i
8
. Thereof, the local
SAM can be expressed by means of the Stokes
parameters:
0
3
22 ||||
)(
S
S
uu
uuuui
A
YX
YXYX
c
, (7)
where
)(and|| **
3
2
0 EEEEiSS YXYXE .
The local SAM equal to the ratio 3 0S S cannot
exceed “1”. In the case of the circular polarized beam,
we have Ac = 1 or per photon.
The second part in Eq. (4) presents the orbital
angular momentum (OAM) written in the form
o z X Ye �S S S S (8)
with
ze �S being the longitudinal flux, while the sum
YX SS is the transverse flux:
2 2 * *( ) , ( )
8 16X Y j j j j j
c ic
A A u u u u
k �S S . (9)
Thus, the total OAM of the beam is
2
2
)]([
1
dSSr
c
L YXo . (10)
OAM is associated with the spatial beam structure.
In contrast to the SAM, the OAM depends on the
coordinate transformations.
Thus, the angular momentum can contain some
parts related to different degrees of freedom of the light
wave: SAM is charged with the polarization state,
whereas OAM characterizes the beam as a whole, its
geometry and space movement [5].
In the case of the anisotropic medium, the changes
in the ordinary and extraordinary beams are also related
with SAM. However, different parts of the beam
undergo different transformations. The beam is spatially
depolarized. SAM decreases. At the same time, the
polarized beam components are transformed, too. These
processes partially compensate each other. (When the
beam propagates along the crystal optical axis the
transformations of SAM compensate each other
precisely.)
3. Experimental study of the spin-orbital coupling
It is known that conversion of the elliptical singular
beams has periodical features taking place along the
beam axis z propagating along the perpendicular to the
crystal optical axis [2]. The most interesting case springs
up when the beam propagates nearly perpendicular to the
optical axis with the deviation angle 7º so that
00sin . Besides, the transformation dynamic of the
beam cross-section is very complex process, because the
ordinary and extraordinary beams lay partially upon
each other for the small deviation angle 0. As the result,
we obtain the complex polarization pattern containing
the chain of the polarization singularities from the
lemon, star and monster [7].
The experimental set-up for studying the Gaussian
beam evolution is shown in Fig. 1.
The Gaussian beam radiated from the He-Ne laser
with the wavelength λ = 0.6328 m splits with the plate
Bs1 forming together with the plate Bs2 an
interferometer. After transmitting the λ/4 plate, the beam
gets the circular polarization in one interferometer arm.
Then, the beam is focused by the lens L1 at the input
crystal Cr face with beam waist w0 ≈ 0.02 mm. The
crystal (SiO2) is mounted in the specially designed
optical stand and turned at the angle 0 relatively to the
beam axis. Besides, the crystal optical axis can turn at
the angle ψ. Then, the beam is collimated by the lens L2,
passes through the circular polarizer P 4/ being
recorded by the CCD camera after mixing with the
reference beam.
In order to understand the obtained results, let us
consider beam propagation nearly perpendicular to the
uniaxial crystal optical axis following the paper [1]. The
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 344-348.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
346
complex amplitudes of the ordinary xE
~
and
extraordinary yE
~
components, in the coordinates related
with the crystallographic axes, have the form
oooooox fwYXE //exp
~ 222 , (11)
e
ye
e
xe
e
yx
y f
w
Y
w
Xi
E
2
2
2
2
exp
~
, (12)
where: 2/2
1 oo wkz , 2/2
2 ex wkz , 2
2
2
1
2
2 2/ nnwkz ey ,
oo zzi /1 , xx zzi /1 , yy zzi /1 ,
cosooo zixX , sinooo ziyY , cosxxe zixX ,
sinyye ziyY , 2/1
2
ooo zkaf ,
2/sin/2/cos 2
211
22
2
2 yyxxe znnkazkaf .
In the laboratory referent frame, coordinate
transformation is as follows:
sincos 11 yxx , cossin 11 yxy (13)
while basic transformation has the form
sincos1
yxx EEE , cossin1
yxy EEE . (14)
The theoretical and experimental patterns shown in
Fig. 2 correspond to the intensity distributions for the
right-hand polarized component.
Fig. 2 illustrates the intensity distribution in the
nearly standard conoscopic pattern. A small asymmetry
of the pattern is the result of a slight inclination (2º)
of the crystal and the beam axes. At the same time, the
asymmetry type and shape of the beam cross-section can
be controlled by the 0 and ψ angles.
a
b
Fig. 2. The conoscopic patterns of the beam field after the
crystal with z = 2 cm: theory (a) and experiment (b).
Evidently, the expression (7) enables us to calculate
SAM provided that we can define the Stokes parameters
S3 and S0 experimentally. These measurements permit to
implement our experimental set-up. This process is
demonstrated in Fig. 3. The shown curves are SAM as a
function of the ψ angle, when the angle 0 is a
parameter. Rotation of the optical axis (the angle ψ
comes to the oscillations of SAM. The angles 0 are
chosen in such a way that the ordinary and extraordinary
beams do not split along the crystal length for account of
the crystal birefringence. Within a rather large range of
the 0 angles, we observe large amplitude oscillations.
However, there are some angles where the
oscillation amplitude of SAM is very small. In our
opinion, this process is related with the conversion of the
spin and orbital angular momenta. Indeed, for the
relatively large angles 0 the ordinary and extraordinary
beams are partially separated. The shape of the beam
cross-section is strongly distorted increasing OAM. In its
turn, SAM decreases for account of the spin-orbital
coupling (see the next Section). OAM can be experi-
mentally estimated as a ratio of the average small and
large ellipse axes. Such a process is illustrated in Fig. 4.
rotation angle ψ, deg
Fig. 3. The SAM dynamics plotted for 0 = 7º (red line ●),
0 = 17º (blue line ▲) and 0 = 23º (black line ■). Rotational
angle ψ = 0…360º.
rotation angle ψ, deg
Fig. 4. The OAM dynamics plotted for 0 = 7º (red line ●),
0 = 17º (blue line ▲) and 0 = 23º (black line ■). Rotational
angle ψ = 0…360º.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 344-348.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
347
The comparison of the curves in Figs 3 and 4
shows that the growth of SAM is accompanied by
decreasing the average OAM and vice versa.
4. Generation of the polarization singularities
and the chains of optical vortices
in the vector beam components
It is interesting to note that the strong downfall of the
beam polarization ellipticity for some angles 0 and ψ is
related with not only the strong beam deformation but
also the complex polarization distribution at the beam
cross-section. But complex polarization distribution is
always associated with the polarization singularities
where the field in one of the circularly polarized
components is equal to zero, i.e., the corresponding
components carry over the optical vortices.
Fig. 5. The polarization distributions against the background of
the right hand polarized component: 0 = 2º, ψ = 210º, w0 =
5 m, z = 2 cm.
ψ = 215º
o
ψ = 218º
ψ = 223º ψ = 228º
Fig. 6. The intensity distributions and interference patterns.
In order to estimate appearance of the polarization
singularities and optical vortices, let us make use of Eqs
(11) and (12) for the relatively small 0 angles within the
range ψ = 210º…300º.
The polarization pattern shown in Fig. 5 illustrates
the chain of the polarization singularities in the form of
the point with circular polarization enclosed by the
standard polarization tracery corresponding to the
singularities of the lemon (green lines) and star (yellow
lines) types separated by the line with linear polarization
(L-line). Circular polarizations are positioned at the
places with the zero intensity corresponding to the
optical vortices in the other circular polarized
components.
We have experimentally verified this effect, as it is
shown in Fig. 6. The interference patterns in the form of
the complex spirals show definitely that the
corresponding circularly polarized components carry
over the single-charged optical vortices. The vortices are
not motionless: during rotation of SiO2 crystal around z-
axis, the vortex becomes to move and changes its
geometry. The intensity minimum spreads and, finely, is
separated into two vortices with opposite signs. Both
vortices move separately, while the first one annihilates
with another singularity at the periphery, and the latter
one, with the opposite sign, still remains in the beam.
The whole process has periodic recurrence and repeats
every 90 degrees of crystal rotation.
5. Conclusion
Beam propagation perpendicular to the crystal optical
axis breaks the circular symmetry. Naturally, the vortex
conversion in this case results in transforming the orbital
angular momentum. But now, three processes take part
in the phenomenon: spatial depolarization, related with
SAM; ellipticity of beam cross-section caused by OAM;
generation of polarization singularities and optical
vortices. The spin and orbital angular momentum are
supplemented by the response of the crystal medium. It
is the sum of these three processes that must reduce to
conservation of the total angular momentum.
Recently, it was revealed that the extraordinary
paraxial beam, when the initial beam has a circular
symmetry, is subjected to the complex elliptical
deformation. Such a geometrical transformation is
conditioned by different scales along the x- and y-axes
for the extraordinary beams in the crystals. Naturally,
deformation distorts polarization distribution at the beam
cross-section and restricts application of the
conservation law for the angular momentum [6].
References
1. T.A. Fadeyeva, A.F. Rubass, B.V. Sokolenko,
A.V. Volyar, The precession of vortex-beams in a
rotating uniaxial crystal // J. Opt. A: Pure Appl.
Opt. 11(9), p. 53-55 (2009).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 344-348.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
348
2. T.A. Fadeyeva, C.N. Alexeyev, B.V. Sokolenko
M.S. Kudryavtseva, A.V. Volyar, Non-canonical
propagation of high-order elliptic vortex beams in a
uniaxially anisotropic medium // Ukrainian Journal
of Physical Optics, 12, p. 62-82 (2011).
3. A. Ciattoni, G. Cincotti, and C. Palma, Angular
momentum dynamics of a paraxial beam in a
uniaxial crystal // Phys. Rev. E, 67, 036618 (2003).
4. K.Y. Bliokh, E.A. Ostrovskaya, M.A. Alonso,
O.G. Rodriguez-Herrera, D. Lara, C. Dainty, Spin-
to-orbital angular momentum conversion in
focusing, scattering, and imaging systems // Opt.
Exp. 19, 26132 (2011).
5. A.Y. Bekshaev, M.S. Soskin, M. Vasnetsov,
Paraxial Light Beams with Angular Momentum.
Nova Science Publishers, New York, 2008, p. 112.
6. B.V. Sokolenko, M.S. Kudryavtseva,
A.V. Zinovyev, V.O. Konovalenko, A.F. Rubass,
Optical vortex conversion in the elliptic vortex-
beam propagating orthogonally to the crystal
optical axis: The experiment // Proc. SPIE, 8338
(D-8), p. 83380D-83380 (2011).
7. J.F. Nye, Natural Focusing and Fine Structure of
Light: Caustics and Wave Dislocations. London,
Institute of Physics Pub. Science, 1999, p. 328.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 344-348.
PACS 42.50.Tx
Generation of optical vortices in rotating uniaxial crystal
for light propagation along the perpendicular to its optical axis
B.V. Sokolenko, A.F. Rubass, A.V. Volyar
Taurida National University, Department of General Physics,
4, Vernadsky Ave., 95007 Simferopol, Ukraine,
Phone: +38(066)574-73-02, e-mail: simplexx.87@gmail.com
Abstract. We have experimentally considered evolution of the Gaussian beam propagating nearly perpendicular to the uniaxial crystal axis. Also we have analyzed the spin and orbital angular momenta and found oscillations of the angular momentum when the crystal optical axis is rotated.
Keywords: angular momentum, singular beam, polarization, optical vortex, umbilics.
Manuscript received 10.07.13; revised version received 03.09.13; accepted for publication 23.10.13; published online 16.12.13.
1. Introduction
It is well known that the Gaussian beam propagates perpendicular to the crystal optical axis and splits into the ordinary and extraordinary ones. The ordinary beam transmits through the crystal as through the isotropic medium while the extraordinary beam is elliptically deformed [1]. In case the beam bears the optical vortex then the crystal changes the singular structure of the vortex. The slight changes of the crystal parameters can result in the critical transformations of the phase singularity [2-4]. However the question of the vortex generation in this case remains open.
Thus the aim of the presented paper is the experimental study of the birth and death of phase singularities processes when the beam propagates nearly perpendicular the crystal optical axis.
2. The basic representations
Let us consider the total representations of the properties of the monochromatic optical fields following the paper [5].
In the paraxial approximation, the electromagnetic field can be written in the form:
E = Ex + Ey, H = Hx + Hy
(1)
or
,
/
/
)
exp(
÷
÷
ø
ö
ç
ç
è
æ
ú
û
ù
ê
ë
é
¶
¶
¶
¶
+
þ
ý
ü
î
í
ì
´
=
þ
ý
ü
î
í
ì
j
z
j
y
x
j
j
u
y
x
e
k
i
u
e
e
ikz
H
E
(2)
where j = x, y and k stand for the wavenumber.
The complex amplitude takes the form
)]
,
(
exp[
)
,
(
z
r
ik
z
r
A
u
j
j
j
j
=
(3)
Then the Poynting vector is
[
]
[
]
(
[
]
[
]
)
.
Re
2
*
*
*
*
Y
Y
X
X
X
r
r
X
H
E
H
E
H
E
H
E
c
S
´
+
´
+
´
+
´
p
=
(4)
After substitution of Eq. (3) into Eq. (4), we can select the energy flux associated with the spin-orbital momentum [5]
[
]
,
)
(
16
*
*
Y
X
Y
X
z
c
u
u
u
u
e
k
ic
S
-
Ñ
´
p
-
=
(5)
while the spin angular momentum (SAM) is
.
]
[
1
2
2
r
d
S
r
c
L
c
c
ò
´
=
(6)
At the same time, the volume density of the SAM is written as
(
)
Y
X
Y
X
u
u
u
u
i
*
*
-
pw
8
. Thereof, the local SAM can be expressed by means of the Stokes parameters:
0
3
2
2
|
|
|
|
)
(
S
S
u
u
u
u
u
u
i
A
Y
X
Y
X
Y
X
c
=
-
=
*
*
,
(7)
where
)
(
and
|
|
*
*
3
2
0
E
E
E
E
i
S
S
Y
X
Y
X
E
¶
¶
-
¶
¶
=
Ñ
=
^
.
The local SAM equal to the ratio
30
SS
cannot exceed “1”. In the case of the circular polarized beam, we have Ac = (1 or
h
±
per photon.
The second part in Eq. (4) presents the orbital angular momentum (OAM) written in the form
ozXY
e
^^
=++
P
SSSS
(8)
with
z
e
P
S
being the longitudinal flux, while the sum
Y
X
S
S
^
^
+
is the transverse flux:
22**
(),()
816
XYjjjjj
cic
AAuuuu
k
pp
^
=+=Ñ-Ñ
P
SS
.
(9)
Thus, the total OAM of the beam is
2
2
)]
(
[
1
d
S
S
r
c
L
Y
X
o
ò
^
^
+
´
=
.
(10)
OAM is associated with the spatial beam structure. In contrast to the SAM, the OAM depends on the coordinate transformations.
Thus, the angular momentum can contain some parts related to different degrees of freedom of the light wave: SAM is charged with the polarization state, whereas OAM characterizes the beam as a whole, its geometry and space movement [5].
y
Ls
Bs
1
P
L
1
Cr
L
2
CCD
x
4
С
0
Bs
2
4
Mr
1
Mr
2
z
In the case of the anisotropic medium, the changes in the ordinary and extraordinary beams are also related with SAM. However, different parts of the beam undergo different transformations. The beam is spatially depolarized. SAM decreases. At the same time, the polarized beam components are transformed, too. These processes partially compensate each other. (When the beam propagates along the crystal optical axis the transformations of SAM compensate each other precisely.)
3. Experimental study of the spin-orbital coupling
It is known that conversion of the elliptical singular beams has periodical features taking place along the beam axis z propagating along the perpendicular to the crystal optical axis [2]. The most interesting case springs up when the beam propagates nearly perpendicular to the optical axis with the deviation angle ((7º so that
0
0
sin
a
»
a
. Besides, the transformation dynamic of the beam cross-section is very complex process, because the ordinary and extraordinary beams lay partially upon each other for the small deviation angle (0. As the result, we obtain the complex polarization pattern containing the chain of the polarization singularities from the lemon, star and monster [7].
The experimental set-up for studying the Gaussian beam evolution is shown in Fig. 1.
The Gaussian beam radiated from the He-Ne laser with the wavelength λ = 0.6328 (m splits with the plate Bs1 forming together with the plate Bs2 an interferometer. After transmitting the λ/4 plate, the beam gets the circular polarization in one interferometer arm. Then, the beam is focused by the lens L1 at the input crystal Cr face with beam waist w0 ≈ 0.02 mm. The crystal (SiO2) is mounted in the specially designed optical stand and turned at the angle (0 relatively to the beam axis. Besides, the crystal optical axis can turn at the angle ψ. Then, the beam is collimated by the lens L2, passes through the circular polarizer
P
-
l
4
/
being recorded by the CCD camera after mixing with the reference beam.
In order to understand the obtained results, let us consider beam propagation nearly perpendicular to the uniaxial crystal optical axis following the paper [1]. The complex amplitudes of the ordinary
x
E
~
and extraordinary
y
E
~
components, in the coordinates related with the crystallographic axes, have the form
(
)
[
]
o
o
o
o
o
o
x
f
w
Y
X
E
s
-
s
+
-
=
/
/
exp
~
2
2
2
,
(11)
ú
ú
û
ù
ê
ê
ë
é
-
s
-
s
-
s
s
=
e
y
e
e
x
e
e
y
x
y
f
w
Y
w
X
i
E
2
2
2
2
exp
~
,
(12)
where:
2
/
2
1
o
o
w
k
z
=
,
2
/
2
2
e
x
w
k
z
=
,
2
2
2
1
2
2
2
/
n
n
w
k
z
e
y
=
,
o
o
z
z
i
/
1
-
=
s
,
x
x
z
z
i
/
1
-
=
s
,
y
y
z
z
i
/
1
-
=
s
,
y
a
+
=
cos
o
o
o
z
i
x
X
,
y
a
+
=
sin
o
o
o
z
i
y
Y
,
y
a
+
=
cos
x
x
e
z
i
x
X
,
y
a
+
=
sin
y
y
e
z
i
y
Y
,
2
/
1
2
o
o
o
z
k
a
f
=
,
2
/
sin
/
2
/
cos
2
2
1
1
2
2
2
2
y
+
y
=
y
y
x
x
e
z
n
n
k
a
z
k
a
f
.
In the laboratory referent frame, coordinate transformation is as follows:
y
+
y
=
sin
cos
1
1
y
x
x
,
y
+
y
-
=
cos
sin
1
1
y
x
y
(13)
while basic transformation has the form
y
-
y
=
sin
cos
1
y
x
x
E
E
E
,
y
+
y
=
cos
sin
1
y
x
y
E
E
E
.
(14)
The theoretical and experimental patterns shown in Fig. 2 correspond to the intensity distributions for the right-hand polarized component.
Fig. 2 illustrates the intensity distribution in the nearly standard conoscopic pattern. A small asymmetry of the pattern is the result of a slight inclination (((2º) of the crystal and the beam axes. At the same time, the asymmetry type and shape of the beam cross-section can be controlled by the (0 and ψ angles.
a
b
Fig. 2. The conoscopic patterns of the beam field after the crystal with z = 2 cm: theory (a) and experiment (b).
Evidently, the expression (7) enables us to calculate SAM provided that we can define the Stokes parameters S3 and S0 experimentally. These measurements permit to implement our experimental set-up. This process is demonstrated in Fig. 3. The shown curves are SAM as a function of the ψ angle, when the angle (0 is a parameter. Rotation of the optical axis (the angle ψ comes to the oscillations of SAM. The angles (0 are chosen in such a way that the ordinary and extraordinary beams do not split along the crystal length for account of the crystal birefringence. Within a rather large range of the (0 angles, we observe large amplitude oscillations.
However, there are some angles where the oscillation amplitude of SAM is very small. In our opinion, this process is related with the conversion of the spin and orbital angular momenta. Indeed, for the relatively large angles (0 the ordinary and extraordinary beams are partially separated. The shape of the beam cross-section is strongly distorted increasing OAM. In its turn, SAM decreases for account of the spin-orbital coupling (see the next Section). OAM can be experimentally estimated as a ratio of the average small and large ellipse axes. Such a process is illustrated in Fig. 4.
rotation angle ψ, deg
Fig. 3. The SAM dynamics plotted for (0 = 7º (red line ●), (0 = 17º (blue line ▲) and (0 = 23º (black line ■). Rotational angle ψ = 0…360º.
rotation angle ψ, deg
Fig. 4. The OAM dynamics plotted for (0 = 7º (red line ●), (0 = 17º (blue line ▲) and (0 = 23º (black line ■). Rotational angle ψ = 0…360º.
The comparison of the curves in Figs 3 and 4 shows that the growth of SAM is accompanied by decreasing the average OAM and vice versa.
4. Generation of the polarization singularities
and the chains of optical vortices
in the vector beam components
It is interesting to note that the strong downfall of the beam polarization ellipticity for some angles (0 and ψ is related with not only the strong beam deformation but also the complex polarization distribution at the beam cross-section. But complex polarization distribution is always associated with the polarization singularities where the field in one of the circularly polarized components is equal to zero, i.e., the corresponding components carry over the optical vortices.
Fig. 5. The polarization distributions against the background of the right hand polarized component: (0 = 2º, ψ = 210º, w0 = 5 (m, z = 2 cm.
ψ = 215º
215o
ψ = 218º
ψ = 223º
ψ = 228º
Fig. 6. The intensity distributions and interference patterns.
In order to estimate appearance of the polarization singularities and optical vortices, let us make use of Eqs (11) and (12) for the relatively small (0 angles within the range ψ = 210º…300º.
The polarization pattern shown in Fig. 5 illustrates the chain of the polarization singularities in the form of the point with circular polarization enclosed by the standard polarization tracery corresponding to the singularities of the lemon (green lines) and star (yellow lines) types separated by the line with linear polarization (L-line). Circular polarizations are positioned at the places with the zero intensity corresponding to the optical vortices in the other circular polarized components.
We have experimentally verified this effect, as it is shown in Fig. 6. The interference patterns in the form of the complex spirals show definitely that the corresponding circularly polarized components carry over the single-charged optical vortices. The vortices are not motionless: during rotation of SiO2 crystal around z-axis, the vortex becomes to move and changes its geometry. The intensity minimum spreads and, finely, is separated into two vortices with opposite signs. Both vortices move separately, while the first one annihilates with another singularity at the periphery, and the latter one, with the opposite sign, still remains in the beam. The whole process has periodic recurrence and repeats every 90 degrees of crystal rotation.
5. Conclusion
Beam propagation perpendicular to the crystal optical axis breaks the circular symmetry. Naturally, the vortex conversion in this case results in transforming the orbital angular momentum. But now, three processes take part in the phenomenon: spatial depolarization, related with SAM; ellipticity of beam cross-section caused by OAM; generation of polarization singularities and optical vortices. The spin and orbital angular momentum are supplemented by the response of the crystal medium. It is the sum of these three processes that must reduce to conservation of the total angular momentum.
Recently, it was revealed that the extraordinary paraxial beam, when the initial beam has a circular symmetry, is subjected to the complex elliptical deformation. Such a geometrical transformation is conditioned by different scales along the x- and y-axes for the extraordinary beams in the crystals. Naturally, deformation distorts polarization distribution at the beam cross-section and restricts application of the conservation law for the angular momentum [6].
References
1. T.A. Fadeyeva, A.F. Rubass, B.V. Sokolenko, A.V. Volyar, The precession of vortex-beams in a rotating uniaxial crystal // J. Opt. A: Pure Appl. Opt. 11(9), p. 53-55 (2009).
2. T.A. Fadeyeva, C.N. Alexeyev, B.V. Sokolenko M.S. Kudryavtseva, A.V. Volyar, Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium // Ukrainian Journal of Physical Optics, 12, p. 62-82 (2011).
3. A. Ciattoni, G. Cincotti, and C. Palma, Angular momentum dynamics of a paraxial beam in a uniaxial crystal // Phys. Rev. E, 67, 036618 (2003).
4. K.Y. Bliokh, E.A. Ostrovskaya, M.A. Alonso, O.G. Rodriguez-Herrera, D. Lara, C. Dainty, Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems // Opt. Exp. 19, 26132 (2011).
5.
A.Y. Bekshaev, M.S. Soskin, M. Vasnetsov, Paraxial Light Beams with Angular Momentum. Nova Science Publishers, New York, 2008, p. 112.
6. B.V. Sokolenko, M.S. Kudryavtseva, A.V. Zinovyev, V.O. Konovalenko, A.F. Rubass, Optical vortex conversion in the elliptic vortex-beam propagating orthogonally to the crystal optical axis: The experiment // Proc. SPIE, 8338 (D-8), p. 83380D-83380 (2011).
7. J.F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations. London, Institute of Physics Pub. Science, 1999, p. 328.
�
Fig. 1. The schematic representation of experimental set-up.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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