Self-action of a Gaussian beam in a nematic liquid crystal cell
We present a theoretical study of the possibility of optical singularity birth in a wave front of a laser beam passing through a homeotropically aligned nematic liquid crystal (LC) cell. At intensities below some threshold value there is no any distortion in the initial homogeneous homeotropic LC...
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Інститут фізики конденсованих систем НАН України
2001
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Self-action of a Gaussian beam in a nematic liquid crystal cell / V. Reshetnyak, S. Subota // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 307-313. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1204352017-06-13T03:04:20Z Self-action of a Gaussian beam in a nematic liquid crystal cell Reshetnyak, V. Subota, S. We present a theoretical study of the possibility of optical singularity birth in a wave front of a laser beam passing through a homeotropically aligned nematic liquid crystal (LC) cell. At intensities below some threshold value there is no any distortion in the initial homogeneous homeotropic LC alignment. When light intensity becomes higher than this threshold value, LC director deviates from its initial orientation. Strong director anchoring at the cell walls is assumed. Inhomogeneous director profile leads to the modulation of refractive index which in its turn may be considered as a Gaussian lens formed in the LC cell. This lens gives birth to the phase singularities in coherent beam with initially smooth wave front. At small values of director deviation in geometrical optics approximation one can find the light field just after the LC cell, and utilizing the Huygens-Fresnel principle calculate spatial position of the phase singularity at near field as a function of intensity of the input beam. В статті представлене теоретичне дослідження можливості виникнення оптичної сингулярності у хвильовому фронті лазерного пучка, який проходить через комірку нематичного рідкого кристала (РК). При інтенсивностях нижче певного порогового значення немає викривлення в початково однорідному гомеотропному впорядкуванні РК. Коли інтенсивність світла стає більше цього порогового значення, директор РК відхиляється від своєї початкової орієнтації. Вважається, що зчеплення зі стінками комірки жорстке. Неоднорідний профіль директора приводить до модуляції показника заломлення, який, в свою чергу, можна розглядати як гаусову лінзу, утворену в комірці РК. Ця лінза спричиняє виникнення фазової сингулярності в когерентному пучку з початково гладким хвильовим фронтом. При малих відхиленнях директора в наближенні геометричної оптики можна знайти світлове поле відразу після комірки, і, використовуючи принцип Гюйгенса-Френеля вирахувати знаходження в просторі фазової сингулярності в ближній зоні як функцію від інтенсивності падаючого пучка. 2001 Article Self-action of a Gaussian beam in a nematic liquid crystal cell / V. Reshetnyak, S. Subota // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 307-313. — Бібліогр.: 8 назв. — англ. 1607-324X PACS: 61.30.Gd, 42.65.-k, 42.70.Df DOI:10.5488/CMP.4.2.307 http://dspace.nbuv.gov.ua/handle/123456789/120435 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
We present a theoretical study of the possibility of optical singularity birth
in a wave front of a laser beam passing through a homeotropically aligned
nematic liquid crystal (LC) cell. At intensities below some threshold value
there is no any distortion in the initial homogeneous homeotropic LC alignment.
When light intensity becomes higher than this threshold value, LC
director deviates from its initial orientation. Strong director anchoring at the
cell walls is assumed. Inhomogeneous director profile leads to the modulation
of refractive index which in its turn may be considered as a Gaussian
lens formed in the LC cell. This lens gives birth to the phase singularities
in coherent beam with initially smooth wave front. At small values of director
deviation in geometrical optics approximation one can find the light
field just after the LC cell, and utilizing the Huygens-Fresnel principle calculate
spatial position of the phase singularity at near field as a function of
intensity of the input beam. |
| format |
Article |
| author |
Reshetnyak, V. Subota, S. |
| spellingShingle |
Reshetnyak, V. Subota, S. Self-action of a Gaussian beam in a nematic liquid crystal cell Condensed Matter Physics |
| author_facet |
Reshetnyak, V. Subota, S. |
| author_sort |
Reshetnyak, V. |
| title |
Self-action of a Gaussian beam in a nematic liquid crystal cell |
| title_short |
Self-action of a Gaussian beam in a nematic liquid crystal cell |
| title_full |
Self-action of a Gaussian beam in a nematic liquid crystal cell |
| title_fullStr |
Self-action of a Gaussian beam in a nematic liquid crystal cell |
| title_full_unstemmed |
Self-action of a Gaussian beam in a nematic liquid crystal cell |
| title_sort |
self-action of a gaussian beam in a nematic liquid crystal cell |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120435 |
| citation_txt |
Self-action of a Gaussian beam in a
nematic liquid crystal cell / V. Reshetnyak, S. Subota // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 307-313. — Бібліогр.: 8 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT reshetnyakv selfactionofagaussianbeaminanematicliquidcrystalcell AT subotas selfactionofagaussianbeaminanematicliquidcrystalcell |
| first_indexed |
2025-07-08T17:52:44Z |
| last_indexed |
2025-07-08T17:52:44Z |
| _version_ |
1837102181001461760 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 2(26), pp. 307–313
Self-action of a Gaussian beam in a
nematic liquid crystal cell
V.Reshetnyak∗, S.Subota
Physics Faculty, Kyiv Taras Shevchenko University,
6 Glushkova Ave., 03680 Kyiv, Ukraine
Received August 14, 2000
We present a theoretical study of the possibility of optical singularity birth
in a wave front of a laser beam passing through a homeotropically aligned
nematic liquid crystal (LC) cell. At intensities below some threshold value
there is no any distortion in the initial homogeneous homeotropic LC align-
ment. When light intensity becomes higher than this threshold value, LC
director deviates from its initial orientation. Strong director anchoring at the
cell walls is assumed. Inhomogeneous director profile leads to the modula-
tion of refractive index which in its turn may be considered as a Gaussian
lens formed in the LC cell. This lens gives birth to the phase singularities
in coherent beam with initially smooth wave front. At small values of di-
rector deviation in geometrical optics approximation one can find the light
field just after the LC cell, and utilizing the Huygens-Fresnel principle cal-
culate spatial position of the phase singularity at near field as a function of
intensity of the input beam.
Key words: liquid crystal, phase dislocations, phase singularities, light
induced Frederiks transition
PACS: 61.30.Gd, 42.65.-k, 42.70.Df
1. Introduction
In early 70’s, the structure of wavefronts in monochromatic wave was analysed
in detail by Nye and Berry [1] and by Wright [2]. It was shown that imperfections
of regularity in optical wavefront can occur even in a pure monochromatic wave,
namely screw and edge dislocations. Optical singularities are intriguing topological
structures in modern optics. In [3] it was reported that the experimental study of
the nucleation of wave front phase dislocations in a Gaussian beam experienced
the self-action in a nematic liquid crystal. They investigated cases of normal and
oblique incidence of the input beam. In the first case the appearance of circular edge
dislocation was observed. In the case of oblique incidence, the situation resembles
∗E-mail: reshet@marion.iop.kiev.ua
c© V.Reshetnyak, S.Subota 307
V.Reshetnyak, S.Subota
the Gaussian beam passing the astigmatic lens. To describe this process theoretically
it is necessary to solve the Maxwell’s equations for light propagation simultaneously
with equations for LC reorientation. We study this problem approximately. First we
find the liquid crystal director profile in a nematic cell illuminated with Gaussian
light beam, neglecting the feedback. After that the light diffraction caused by the
director inhomogeneity is considered.
2. Director profile
In this paper we consider a homeotropically aligned nematic liquid crystal cell.
Linearly polarised light with intensity I = I0e
−ρ2/R2
illuminates the cell. Oz axis
of the Cartesian frame is along a non-perturbed direction of the director, and Ox
axis is along the polarization of light. Strong director anchoring at the cell walls is
assumed.
L
E
z
y
x
θ
Figure 1. LC cell geometry, electrical field ~E and director orientation.
The total free energy to be minimized consists of two parts: elastic energy as-
sociated with the director deformation in the cell volume and interaction between
liquid crystal and light beam. For simplicity we employ one elastic constant approx-
imation, and in that case obviously director reorientation occurs in XOZ plane. It
is convenient to describe the director field by the angle θ between director and Oz
axis:
n = (sin θ, 0, cos θ) . (2.1)
The total free energy in a polar coordinate system now has the form:
F =
K
2
∫
V
(
(θ′x)
2 + (θ′y)
2 + (θ′z)
2
)
dV −
ǫa
16π
∫
V
sin2 θ|E|2dV, (2.2)
where K is elastic constant, ǫa is the anisotropy of the LC dielectric tensor, V –
volume of the cell. To find the director profile under the action of Gaussian light
beam we use the Ritz’s variational method, namely we seek solution in the form:
θ = θ0 exp
(
−
x2 + y2
a2
)
sin
(
πz
L
)
, (2.3)
308
Self-action of a Gaussian beam in a nematic. . .
1 2 3 4 5 6 7 8 9 10
1,2
1,4
1,6
1,8
2,0
I/I
0
a/R
1 2 3 4 5 6 7 8 9 10
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8θ
0
I/I
0
Figure 2. a dependency on the intensity
of light beam.
Figure 3. θ0 dependency on the inten-
sity of light beam.
here L is the cell thickness. Substituting this expression into the free energy func-
tional and integrating over x, y, z we get free energy in a simpler functional form.
Now free energy function depends on variables a, θ0 and intensity I0 appears in it as
the parameter. The necessary condition for minimum of the free energy function is
that partial derivatives of F with respect to a, θ0 are equal to zero simultaneously.
This condition gives us a system of non-linear equations. Solving this system, we
found that the director reorientation has a threshold behaviour. At intensities below
some threshold value there is no distortion in the initial homogeneous homeotropic
LC alignment. The director deviates from its initial direction at intensities higher
than some threshold value. This threshold value of intensity is given by:
Ith =
8πK
ǫa
(
π
L
)2 (
1 +
L
πR
)2
. (2.4)
It is seen that at L/R ≪ 1 the threshold intensity tends to a constant value, which
coincides with a threshold of reorientation of the director in a field of a plane wave
[4] and at L/R ≫ 1 it increases very sharply. Similar behaviour of the threshold
value of intensity upon ratio L/R results was also obtained by Khoo et al. [5]. In
figures 2, 3 we plot the numerically found a, θ0 dependencies on the intensity of light
beam.
Figure 4 presents the free energy for different values of intensity. It is seen that
for values of intensity below threshold the free energy has minimal value only at a
and θ0 both equal to zero. For values of intensity higher than threshold, nontrivial
minimum appears.
3. Phase singularity birth
Now we shall investigate Gaussian light beam diffraction caused by the inhomo-
geneous director profile, the last being found in the previous section. The appearance
309
V.Reshetnyak, S.Subota
0
0.5
1
1.5
a/R
0
0.2
0.4
0.6
0
0.5
F
0.5
1
1.5
a/R
0
0.2
0.4
0.6
0.8
0
0.1
F
(a) (b)
Figure 4. Free energy function F (θ0, a) at the intensity (a) below threshold value,
I < Ith and (b) above threshold value, I > Ith.
of wavefront singularities, called optical vortices (OV), was observed experimentally
in LC by Pishnyak et. al. [3]. A necessary condition of their existence is equality to
zero of wave amplitude on some line being axis of vortex [6]. As a result in a light
wave there appears a dark spot and the phase becomes undetermined. The line of ze-
ro amplitude can coincide with an axis of a light bundle, or to have a more composite
behaviour. A special interest presents a case, at which the axis V is perpendicular
to a direction of distribution of a bundle. Thus the edge dislocation of a wavefront
is born. To find a light field behind LC cell we used the Huygens-Fresnel princi-
ple. Each point of a wavefront can be considered as a source of secondary spherical
wave and these waves interfere. Mathematically, in Fresnel’s approximation, which
is correct in a short-range zone, the principle looks like [7]:
U (x1, y1, z1) =
eikz
iλz
∫ ∫
Σ
U (x, y) exp
{
i
k
2z
[
(x1 − x)2 + (y1 − y)2
]
}
dxdy, (3.1)
where U (x1, y1, z1) is complex wave amplitude in a plane, which is at distance z
apart from the initial plane, U (x, y) is the beam complex amplitude in the initial
plane z = z0, k wave vector, λ – wavelength. To calculate the behaviour of the light
field in our case it is necessary to know the light field at the exit of the cell. In a ray
(geometrical optics) approximation this field has the form [8]:
U (x, y) ∼ |E|2 exp
(
i
ω
c
ψ1 (L)
)
, (3.2)
where
ψ1 (L) =
∫ L
0
none
(
n2
o sin
2 θ + n2
e cos
2 θ
)1/2
dz′ (3.3)
is the light phase retardation after passing through the cell, θ is the director deviation
angle, which we have found in the previous section. We calculated numerically the
changes of amplitude and phase of the light wave with distance from the cell for
particular value of intensity, I = 1.045Ith. Figures 5, 6 present light wave amplitude
and phase as the function of radius for a different value of the distance z from LC
cell. When the distance z increases, a local minimum appears in amplitude, while, at
310
Self-action of a Gaussian beam in a nematic. . .
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
0,0
0,5
1,0
1,5
2,0
2,5
r
1
A
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
5
6
7
8
9
10
r
1
F
(a) (a)
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
0,0
0,5
1,0
1,5
2,0
2,5
3,0
r
1
A
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
5
6
7
8
9
r
1
F
(b) (b)
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
0,0
0,5
1,0
1,5
2,0
2,5
3,0
r
1
A
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
6
8
10
12
14
r
1
F
(c) (c)
Figure 5. Wave amplitude at: (a) z =
1.1zR, (b) z = 1.299172zR , (c) z =
1.4zR.
Figure 6. Wave phase at: (a) z = 1.1zR,
(b) z = 1.299172zR, (c) z = 1.4zR.
311
V.Reshetnyak, S.Subota
once after value z = 1.299172zR, where zR is Rayleigh length, the amplitude becomes
strictly zero. Simultaneously, the phase is curved, there appears a ledge, which at
vanishing amplitude tends to π, and its walls become vertical. When we increment
slightly the distance z, instead of a ledge there is a trough with vertical walls of height
π too. The circular edge dislocation is born. The dislocation disappears immediately
with the smallest increase of distance z. The valley bottom becomes nonzero, the
phase jump transforms to a smooth ledge with slanted walls.
4. Conclusions
In this paper we found the director profile in the cell with strong homeotropic an-
choring under the action of Gaussian light beam. Director reorientation is threshold
like, and the value of threshold intensity depends crucially on the ratio between cell
thickness and beam half-width. Using the obtained expressions for director profile
we investigated the birth of the circular edge phase dislocation in coherent beam
with initially smooth wave front. Also, spatial position of the phase singularity at
near field as a function of intensity of the input beam was calculated.
5. Acknowledgement
We acknowledge useful conversations with M.S.Soskin.
Work was partially supported by INTAS grant 97–635.
References
1. Nye J.F., Berry M.V. Dislocations in wave trains. // Proc. Roy. Soc. Lond. A, 1974,
vol. 336, p. 165–190.
2. Wright F.J. Wavefront dislocations and their analysis using catastrophe theory. – In:
Structural Stability in Physics, Eds. W. Guttinger and H. Eikemeier, Berlin, Springer-
Verlag, 1979.
3. Pishnyak O.P., Reznikov Ya.A., Vasnetsov M.V., Yaroshchuk O.V., Gorshkov V.N.,
Soskin M.S. // Mol. Cryst. Liq. Cryst., 1998, vol. 324, p. 25–30.
4. Zel’dovich B.Ya., Tabiryan N.V., Chilingarjan Yu.S. // ZhETF, 1981, vol. 81, p. 72–83
(in Russian).
5. Khoo I.C., Liu T.H., Yan P.Y. Nonlocal radial dependence of laser-induced molecular
reorientation in a nematic liquid crystal: theory and experiment. // J. Opt. Soc. Am. B,
1987, vol 4, p. 115–120.
6. Vasnetsov M., Staliunas K. Optical Vortices. New-York, Nova Science, 1999.
7. Goodman J.W. Introduction to Fourier Optics. McGraw-Hill Book Co., 1996.
8. Zel’dovich B.Ya., Tabiryan N.V. // ZhETF, 1982, vol 82, p. 1126–1142.
312
Self-action of a Gaussian beam in a nematic. . .
Самодія гаусового пучка світла в комірці
нематичного рідкого кристала
В.Решетняк, С.Субота
Фізичний факультет,
Київський національний університет ім. Тараса Шевченка,
03680 Київ, просп. Глушкова, 6
Отримано 14 серпня 2000 р.
В статті представлене теоретичне дослідження можливості виник-
нення оптичної сингулярності у хвильовому фронті лазерного пуч-
ка, який проходить через комірку нематичного рідкого кристала (РК).
При інтенсивностях нижче певного порогового значення немає ви-
кривлення в початково однорідному гомеотропному впорядкуванні
РК. Коли інтенсивність світла стає більше цього порогового значен-
ня, директор РК відхиляється від своєї початкової орієнтації. Вважа-
ється, що зчеплення зі стінками комірки жорстке. Неоднорідний про-
філь директора приводить до модуляції показника заломлення, який,
в свою чергу, можна розглядати як гаусову лінзу, утворену в комір-
ці РК. Ця лінза спричиняє виникнення фазової сингулярності в коге-
рентному пучку з початково гладким хвильовим фронтом. При ма-
лих відхиленнях директора в наближенні геометричної оптики можна
знайти світлове поле відразу після комірки, і, використовуючи прин-
цип Гюйгенса-Френеля вирахувати знаходження в просторі фазової
сингулярності в ближній зоні як функцію від інтенсивності падаючого
пучка.
Ключові слова: рідкий кристал, фазові дислокації, фазові
сингулярності, світлоіндукований перехід Фредерікса
PACS: 61.30.Gd, 42.65.-k, 42.70.Df
313
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