Nonlinear optical investigation of Gaussian laser beam propagating ...

Nonlinear optical investigation of Gaussian laser beam propagating in a dye-doped
nematic liquid crystal
S.H. Mousavi a,d, ⁎, E. Koushki b,c , H. Haratizadeh a
a Physics Department, Shahrood University of Technology, Shahrood, Iran
b Photonic Laboratory, Physics Department, Tarbiat Moallem University, Tehran, Iran
c Physics Department, Tarbiat Moallem University of Sabzevar, Sabzevar, Iran
d Physics Department, Semnan University Semnan, Iran
article info
Article history:
Received 13 September 2009
Received in revised form 22 January 2010
Accepted 25 January 2010
Available online 1 February 2010
Keywords:
Curvature radius
Beam radius
Nonlinear optical material
Liquid crystal
1. Introduction
abstract
Liquid crystals are mesophases between crystalline solids and
isotropic liquids. Recently, the nonlinear effects of azo-dye-doped
liquid crystals (ADDLCs), such as photorefractive effect [1–3] and
degenerate four-wave mixing [4], have attracted much interest due to
their applications in optical devices such as liquid crystal displays
which are used in electronic watches, calculators, laptop and desktop
computers.
In linear optical processes the physical properties of the liquid
crystal, such as molecular structure, individual or collective molecular
orientation, temperature, density, and population of electronic levels
are not affected by the optical fields. However the direction,
amplitude, intensity, and phase of the optical fields are affected in a
unidirectional way. Liquid crystals are optically highly nonlinear
materials and their physical properties i.e. temperature, molecular
orientation, density and electronic structure can be easily perturbed
by applied optical field [5–9]. Since liquid crystalline molecules are
anisotropic hence a polarized light from a laser source can induce an
alignment or order in the isotropic phase, or realign the molecules in
the ordered phase.
⁎ Corresponding author. Physics Department, Shahrood University of Technology,
Shahrood 3619995161, Iran. Tel.: +98 912 1909450; fax: +98 273 3335270.
E-mail addresses: hadi_mousavi@yahoo.com, mousavi@phys.tus.ac.ir
(S.H. Mousavi).
0167-7322/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.molliq.2010.01.008
Journal of Molecular Liquids 153 (2010) 124–128
Contents lists available at ScienceDirect
Journal of Molecular Liquids
journal homepage: www.elsevier.com/locate/molliq
The laser beam shape and variation of the curvature radius of the wavefront have been simulated when the
Gaussian laser beam passes through a dye-doped nematics liquid crystal. The effect of different dyes is
investigated in the wavefront distortion as well as the beam shape due to its quality factor. We have reported
the dependency of the curvature radius of the wavefront to the nonlinearity of the sample. Also we have
obtained the beam shape of three dyes doped nematic liquid crystal as a nonlinear optical material compared
with the experimental results.
© 2010 Elsevier B.V. All rights reserved.
The optical properties of liquid crystals can be controlled by
external applied DC or low-frequency fields; which gives rise to a
variety of electro-optical effects which are widely used in many
electro-optical display and image-processing applications.
There are several methods for nonlinear characterization of
materials such as z-scan which is a powerful technique to obtain both
the sign and magnitude of the complex nonlinear refractive index of
some optical materials. This technique is based on the principle that
spatial variations of the incident intensity distribution can photoinduce
a lens in the nonlinear material which affects the posterior propagation
of the beam and intensity changes at the far-field.
During developing the z-scan technique [10,11], manyimprovements
and modifications have been suggested due to its sensitivity and
simplicity such as eclipsing [12],tophatbeams[13], two colours [14] and
reflection [15]. Detailed analysis of the parameters affecting the z-scan
measurements was reported by Chapple et al. [16]. In this investigation, a
Gaussian beam incident to a thin Kerr nonlinear sample was considered
to obtain simple analytical formulas relating the z-scan curve obtained
from the on-axis intensity at the far-field. Gaussian decomposition
method has been used to analyze the characteristics of the z-scan curves
for thin samples with small or large nonlinear phase shifts.
In this paper, we focus on nonlinear properties of dye-doped liquid
crystal. Also, we have studied the laser beam propagation in a thin
nonlinear material (dye-doped liquid crystal) and compared with
experimental data. Furthermore, the radius of curvature of the
wavefront was calculated and it is found that convergent (divergent)
Gaussian beam passing through a self-focusing sample, the radius of
curvature of the wavefront becomes larger (smaller).

2. Theoretical approach
A plane wave and a spherical wave represent two opposite
extremes of angular and spatial confinement. Waves including
wavefront normals which make small angles with the z-axis are
called paraxial waves and they must satisfy the paraxial Helmholtz
equation [17].
An important solution of the mentioned equation which exhibits
the characteristics of an optical beam is a wave called the Gaussian
beam. The electric field of a TEM00 Gaussian beam with beam waist
radius w0 travelling in the z direction can be written as
W 0
Einðr; zÞ
= A0 Wz ðÞ
Wz ðÞ= W0 1+ z
z0 Rz ðÞ= z 1+ z0 z
ρ2
exp −
W2 !
ðÞ z
2
2 1= 2
exp −ikz−ik ρ2
+ iζ z
2RðÞ z
ðÞ
!
where I 0, W(z), R(z) and z 0 are respectively the irradiance at beam
waist of the Gaussian beam, the beam radius at z, the radius of
curvature of the wavefront at z and the diffraction length of the beam
respectively and ζðÞ= z tan −1 z
z ;W0 =
0
λz0 π
1 = 2
.
In this study, first of all some of Gaussian beam parameters such
as intensity, beam radius and curvature radius are introduced.
Within any transverse plane, the beam intensity assumes its peak
value on the beam axis, and drops by the factor l/e 2 at the radial
distance ρ=W(z). Since 86% of the power is carried within a circle of
radius W(z), we regard W(z) as the beam radius. The phase of the
Gaussian beam could be calculated from Eq. (1). It is given by
φρ;z ð Þ = kz−ζðÞ+ z
kρ2
: ð4Þ
2RðÞ z
On the beam axis (ρ=0) the phase
φρ;z ð Þ = kz−ζðÞ: z
ð5Þ
The first term of Eq. (4), kz, is the phase of a plane wave and the
second represents a phase retardation E(z) given which ranges from
−π/2atz=−∞ to +π/2atz=∞. This phase retardation corresponds
to an excess delay of the wavefront in comparison with a plane wave
or a spherical wave. The total accumulated excess retardation as the
wave travels from z=−∞ to z=∞ is π. The last term is responsible for
wavefront bending. It represents the deviation of the phase at off-axis
points in a given transverse plane from that at the axial point. Because
the variations of tan −1 (z/z0) and R(z) are slow, the constant phase
surfaces satisfy the following equation:
z +
2
r
2RðÞ z
ζ z
= m + ðÞ
2π
ð1Þ
ð2Þ
ð3Þ
× λ≈constant: ð6Þ
This is the equation of a paraboloidal surface of curvature radius
R;whereR(z) is the radius of curvature of the wavefront at position z
on the beam axis. The radius of curvature R(z) isinfinite at z =0,
corresponding to plane wavefronts. It decreases to a minimum value
of 2z0 at z = z0. The radius of curvature subsequently increases with
further increase of z until R(z)=z for z ≫ z 0. The wavefront is then
approximately the same as that of a spherical wave. For negative z
the wavefronts follow an identical pattern, except for a change in
sign.
S.H. Mousavi et al. / Journal of Molecular Liquids 153 (2010) 124–128
If the Gaussian beam passes through a thin third-order nonlinear
optical sample with nonlinear refraction c (MKS) located at z, the
phase shift of the Gaussian beam is given by:
Δφ0 Δφ0ðz;rÞ =
1+ z
z 0
2
2 r
2
e−w 2 ðÞ z
ð7Þ
where Δφ0=kγI0Leff=kΔnI0 and Leff = 1−e−αL ð Þ .
. The electric field
α
at the exit plane of the sample is obtained [10], and the far-field at a
vertical plane in the far-field is given by:
Er;z ð Þ = Ein z;r
ð Þe −αL= ∞
2 ∑
m =0
½iΔφ0ðÞ z Š m
wm0 exp −
m! wm r 2
w 2 m
− ikr2
!
+ iθ m
2R m
where the term E in(r=0, z) is the input electric field of the incident
Gaussian beam at the sample plane. Defining d as the propagation
distance in free space from the sample to the plane of the far-field and
g=1+d/R(z), the remaining parameters in Eq. (8) are expressed as:
w 2
m 0 = w 2 ðÞ z
2m +1
dm = kw2 m 0
2
w 2 2
m = wm 0 g 2 + d2
d2 " #
m
R m = d½1−
g
g 2 2
d
+
θm = tan −1 d " #
dm
g
:
d 2
−1
mŠ
The electric field of the beam is now a summation of some
subbeams having individually own wavefront curvature radius at
position z. The parameter W(z) guides us to stimulate the beam profile
before and after transmitting the beam from the sample.
3. Discussion and experimental approach
We used three azo-dyes (Sudan black b, Sudan III and Sudan IV)
doped to a nematic liquid crystal (w1680) which their nonlinear
optical properties have been investigated in recent works [18,19].
Sudan dyes were obtained from Merck and used as the guest. A
nematic mixture of w1680 was synthesized in the Institute of
Chemistry of the Military Technical Academy, Warsaw, Poland, and
used as the host. The dyes are dissolved in a nematic liquid crystal
with 1% concentration. The guest–host cell was made by sandwiching
the solutions between two optical glass plates (2×1.2 cm 2 ). The
homeotropic orientation of the guest and host molecules was
achieved by depositing of a thin layer of lecithin on the clean glass
surfaces and mylar film (12.5 μm) was used as the spacer. The plates
were sealed together by epoxy resin glue. The liquid crystal cells were
Table 1
Nonlinear parameters of samples.
Doped material Laser intensity I 0
(W/cm 2 )
Rayleigh range
(cm)
n 2
(cm 2 /W)
Sudan black b 641 0.31 1.01 ×10 −6
Sudan III 794 3.12 2.1×10 −6
Sudan IV 1440 3.12 8.04 ×10 −6
125
ð8Þ
ð9Þ

126 S.H. Mousavi et al. / Journal of Molecular Liquids 153 (2010) 124–128
Fig. 1. Curvature radius of wavefront as function of z when nematic liquid crystal doped
with Sudan black b (solid line for sample at z=z 0=0.31 cm and Δφ 0=6.3×10 −3 ).
checked under crossed polarizers and used in subsequent optical
studies.
The nonlinear refraction indices have been measured using z-scan
and moiré deflectometry methods. The results show positive
nonlinear refractive indices being in order of 10 −6 or 10 −7 cm 2 /W
for the corresponding dyes doped nematic liquid crystal, demonstrating
self-focusing optical nonlinearity. These values are summarized
in Table 1.
From the standpoint of optical properties, the doping of liquid
crystals by appropriately dissolved concentrations and types of dyes
clearly deserves special attention. The most important effect of dye
molecules on liquid crystals is the modification of their nonlinear
optical properties. These molecules are generally elongated in shape
and can be oriented and reoriented by the host nematic liquid crystals.
Light propagation through a medium depends on whether the
light–matter interaction is linear or nonlinear. In linear optics, the
light fields are not intense enough to create appreciable changes in the
optical properties of the medium (e.g., refractive index, absorption or
scattering cross sections) and so its propagation through the medium
is therefore dictated primarily by its properties. In nonlinear optics,
the light–matter interactions are sufficiently intense so that the
optical properties of the medium are affected by the optical fields. For
example, the optical field is intense enough to cause director axis
reorientations. Such interactions give rise to many so-called selfaction
effects such as self-focusing.
Fig. 2. Curvature radius of wavefront as function of z when nematic liquid crystal doped
with Sudan III (solid line for sample at z=z 0=3.1 cm and Δφ 0=2.4×10 −3 ).
Fig. 3. Curvature radius of wavefront as function of z when nematic liquid crystal doped
with Sudan IV (solid line for sample at z=z0=3.1 cm and Δφ0=2.1 ×10 −3 ).
The experiments for pure nematic crystal indicate that these dyes
can enhance the optical nonlinearity. It can be inferred that the n2 for
dye-doped liquid crystal is about two orders of magnitude greater
than that for pure liquid crystals.
In order to obtain the maximum deformation of laser beam we
put the sample at z0. Using our numerical method we revealed that
the transmitted beam is yet a Gaussian beam, it is expected that the
shape of the wavefronts remains a paraboloidal surface and Eq. (6) is
valid. In order to find the radius of the curvature of the wavefront at
position z, the phase magnitude of the beam on the propagation axis
at z was calculated. Figs. 1–3 show the radii of curvature of the
wavefront as function of z for Sudan black b, Sudan III and Sudan IV
respectively (assuming the sample is placed near z 0). The curves show
that in small values of z (near z=0) the curvature radius is very large
while R(z)≈z for z≫z 0 region which is in good agreement with
Eq. (3). Also for different values of nonlinear refractions, the slopes of
lines are unique and don't depend on the position or quality of the
sample (the position of the sample only affects the magnitude of the
refractive nonlinearity according to Eq. (7)). The changes in refractive
nonlinearity cause small shifts of lines in vertical direction while the
slope remains constant. The maximum beam distortion from the
original one appears in Sudan black b which has the greater
Fig. 4. Laser beam shape (a) without sample and (b) with sample “Sudan black b”
placed at z 0. (c) with sample by applying 0.5 (V) ac voltage. Squares and dots are
experimental data for v=0 and v=0.5 (V) respectively.

Fig. 5. Laser beam shape (a) without sample and (b) with sample “Sudan III” placed at
z 0. (c) with sample by applying 0.5 (V) ac voltage. Squares and dots are experimental
data for v=0 and v=0.5 (V) respectively.
nonlinearity properties and larger values of Δφ 0 while for Sudan IV
the minimum distortion is observed because the curves in Fig. 3 tend
to each other. These results revealed that the doping of liquid crystals
by appropriately dissolved concentrations and types of dyes clearly
deserves special attention. The most important effect of dye molecules
on liquid crystals is modification of their nonlinear and optical
properties. These molecules are generally elongated in shape and can
be oriented and reoriented by the host nematic liquid crystals.
In the other simulation, we investigated the beam shape
propagating through the mentioned samples. So that, the sample
was put at a fixed point (z=z 0). The beam radius can be obtained by
Eq. (8) at a far vertical plane located at z. This plane has been moved to
the sample place in finite negligible stages and calculations were done
for each stage. A series of continuous beam radii has been obtained
and the asymptotic curve is the beam profile. Experimentally the
beam profiles are obtained by the edge-scan technique which can be
compared with solid line in Figs. 4–6. InFig. 4, the dashed line is a
Fig. 6. Laser beam shape (a) without sample and (b) with sample “Sudan IV” placed at
z 0. (c) with sample by applying 0.5 (V) ac voltage (very similar to ideal Gaussian beam
shape). Squares and dots are experimental data for v=0 and v=0.5 (V) respectively.
S.H. Mousavi et al. / Journal of Molecular Liquids 153 (2010) 124–128
Table 2
Δφ 0 and M 2 -factor of samples.
Doped
material
theoretical curve when Sudan black b is placed at z=z 0. The squares
are experimental data which are in good agreement with the
theoretical curve. Also they are compared with the beam shape in
the case that the sample does not exist. The beam shape before the
sample is obtained from simple paraxial approximation. The results
are compared with the condition that voltage is used. Using 0.5 (V) ac
voltage, the nonlinearity decreases and the theoretical curve (solid
line in Fig. 4) tends to the Gaussian beam shape. Also this curve is in
good agreement with experimental data (dot points). Applied voltage
in the direction of the NLC director on a homeotropic-aligned cell can
affect both τ → em and τ → dye. The former is because of the fact that the
induced polarization of the NLC molecules under interaction with the
external field will fix at the initial alignment direction. By increasing
the applied voltage, the voltage-induced torque finally overcomes the
light-induced torque and the light-induced reorientation effect,
consequently τ → em, will disappear. Therefore, by increasing the applied
voltage the dye molecules can be aligned in the direction of the
external field. In this case, all the dye molecules situated around the
host NLC molecules will have the same excitation probabilities, hence
→
τ dye disappears due to this process so the nonlinearity decreases.
Fig. 4 and Fig. 5 show these analyses for Sudan III and Sudan IV
respectively.
For Sudan IV with the smallest nonlinearity the curves are very
similar with the case that the sample does not exist. The deviation from
an ideal Gaussian beam shape is expressed by M 2 -factor parameter, this
deviation is defined as the distortion of the Gaussian shape causes
changes in the angle between the beam shape and the z-axes
[20]. Table 2 shows the obtained values for Δφ0 and M 2 -factor.
4. Conclusion
The nonlinear properties of the Gaussian laser beam have been
investigated for three different azo-dyes. A theoretical method to
calculate the curvature radius of the wavefront of a Gaussian beam
passing through different dye-doped nematic liquid crystals is
expressed. Also the effect of the nonlinearity on the laser beam
shape and deviation from ideal the beam profile are calculated and
compared with experimental data. The results show larger nonlinearity
decreases curvature radius and distortion from the ideal
Gaussian beam shape.
References
Δφ0 without
voltage
Sudan black b 6.3 ×10 −3
Sudan III 2.4 ×10 −3
Sudan IV 2.1 ×10 −3
M 2 -factor
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0.5 (V) ac voltage
0.9 2.2×10 −3
0.93 1.0×10 −3
0.95 0.8×10 −4
M 2 -factor by
applying 0.5 (V)
ac voltage
0.95
0.975
0.99
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