Download Experimental observation of the far field diffraction patterns of

XXIX ENFMC
- Annals of Optics
2006
Experimental observation of the far field diffraction patterns of divergent
and convergent Gaussian beams in a self-defocusing medium
Sabino Chávez-Cerda1, Cesar M. Nascimento2, Márcio A. R. C. Alencar2, Monique G. A. da Silva3,
Mario R. Meneghetti3 and Jandir M. Hickmann2
1
Instituto Nacional de Astrofísica, Optica y Electronica, Apdo Postal 51/216, Puebla, Pue., 72000
Mexico.
2
Departamento de Física, Universidade Federal de Alagoas, 57072 -970, Maceió, AL, Brazil.
3
Departamento de Química, Universidade Federal de Alagoas, 57072-970, Maceió, AL, Brazil.
[email protected]
Abstract
We report on the experimental observation of the far field patterns formation obtained when a
divergent or convergent beam interacts with a thin self-defocusing material. The measurements
were performed using a CW laser beam propagated through a cell containing a colloidal system
that presents a large nonlinear refractive index. The observed patterns reveal the different
behavior on the far field from a convergent or a divergent beam owing to the negative signal of
the medium nonlinearity. Our results are in good agreement with a model for a thin selfdefocusing medium based on the Fresnel-Kirchhoff diffraction theory.
Introduction
A large number of spatial effects can be observed when an intense light beam interacts with a nonlinear medium
[1-7]. In particular, concentric ring intensity distribution pattern can be induced in the far field of a beam after
propagation through a nonlinear material. This effect of spatial phase-modulation (SPM) is due to the intensitydependent complex refractive index and it has been observed in several systems, such as atomic vapors, liquid
crystals, polymers, and nanostructured materials [2-5]. In a large number of cases reported, conical emissions
are due to processes involving electronic transitions but thermal effects may also contribute to SPM generating
spatial redistribution of the laser energy across the beam transverse profile [6].
Although several successful efforts have been made in order to explain the conical effects based on
phenomenological models, the important contribution of the beam wave front curvature connected with the
medium nonlinearity is neglected in most cases [7].
In this work, we report the observation of SPM from convergent and divergent Gaussian beams due to their
propagation in a thin self-defocusing media. Using this approach it has been possible to accurately predict the
generated patterns observed in experiments due to the interaction of a convergent or divergent Gaussian beam
with both self-focusing and self-defocusing media. Two different behaviors for the conical emission were
observed and they are attributed to the interplay of the strong SPM caused by the refractive index change of the
medium and in the sign of the curvature of the wave front of the incident beam on the sample. Our study is a
systematic experimental verification on existence of the two different behaviors of the conical emission
described earlier.
Theoretical Model
Consider that a Gaussian beam, of wavelength λ , propagating through a nonlinear medium, of thickness L and
linear absorption coefficient α , along the direction of the Z-axis. Assuming that the beam-waist position is
taken as the origin of the coordinate system, the complex amplitude of the light electric field at the entrance of
the medium can be written as
E (r , z 0 ) = E (0, z 0 ) exp −
ik 0 n0 r 2
r2
exp
−
2R
w 2p
(1)
XXIX ENFMC
where r is the radial coordinate,
- Annals of Optics
2006
z 0 is the coordinate position of he medium entrance plane, k 0 is the free
w
space wave number, n0 is the refractive index of the air surrounding the medium, p is the beam radius at the
medium entrance plane, R is the radius of the wave front curvature in the corresponding position. Assuming a
Kerr nonlinearity, after the propagation of the Gaussian beam through the medium, the total phase-shift that is
added to the Gaussian beam can expressed as
φ (r ) = k 0
n0 r 2
+ ∆φ (r )
2R
.
(2)
In this equation, the first term corresponds to the contribution of the beam curvature to the phase. The nonlinear
phase shift contribution is related with the intensity-dependent refractive index of the
( )
( )
material, ∆n(z , r ) = n2 I (z , r ) , and can be written as
∆φ (r ) = k 0
z0 + L
z0
∆n(z , r ) dz ≈ ∆φ 0 exp −
(
2r 2
w 2p
,
(3)
)
where ∆φ 0 = k 0 ∆n(z 0 ,0 )L is the peak nonlinear phase-shift induced in the beam. Hence, the complex electric
field of the optical wave, after its propagation along the nonlinear material is
E (r , z 0 + L ) = E (0, z 0 ) exp −
r2
αL
exp −
exp [− iφ (r )]
2
w 2p
.
(4)
The far-field distribution pattern is obtained considering the free propagation of the optical wave through space,
by means of the Fraunhofer approximation of the Fresnel-Kirchhoff diffraction integral as
I (ρ ) = I 0
r2
J 0 (k 0θ r ) exp − 2 − iφ (r ) r dr
0
wp
∞
2
,
(5)
where J 0 ( x ) is the zero-order Bessel function of the first kind, θ is the far field diffraction angle, ρ is the
radial coordinate in the far field observation plane. In the paraxial approximation, the distance from the exit
plane of the medium and the far-field observation plane, D , is related to the radial coordinate and the diffraction
angle in the far-field are related by ρ = Dθ . The parameter I 0 is written as
I 0 = 4π 2
E (0, z 0 ) exp (− α L 2 )
iλ D
2
.
(6)
When the light beam is convergent, the radius of the wave front curvature is negative; while if the beam is
divergent the radius is positive. Moreover, when the beam is transmitted through the self-defocusing (selffocusing) medium, the nonlinear phase-shift is negative (positive). The two different behaviors can be observed
for the far-field patterns in each case.
The first case occurs for a divergent beam in self-defocusing medium ( R > 0 , ∆φ < 0 ), or for a convergent
beam in self-focusing material ( R < 0 , ∆φ > 0 ). In this limit, the number of bright rings ( N ) increases
linearly with the increase of the nonlinear phase shift,
N = ∆φ 0 2π
. The second behavior is obtained for a
convergent beam propagating through a self-defocusing medium ( R < 0 , ∆φ < 0 ), or alternatively for a
divergent beam in self-focusing medium ( R > 0 , ∆φ > 0 ). In this case, the greater the distance to the pattern
XXIX ENFMC
- Annals of Optics
2006
center the smaller the peak intensity of the diffraction rings. The number of bright rings does not follow the
same relation observed in the first case however it increases as the absolute value of the phase shift raises.
Experimental Setup
We have used the experimental setup depicted in figure 1. A Gaussian light beam was obtained from a CW
Argon-ion laser tuned at 514 nm. The beam was focused by a lens of 25 cm of focal length. The laser
propagated through a 1 mm cell containing the self-defocusing material. The sample position could be varied
along the beam path and about the focal plane of the lens making in this way a convergent or divergent beam
impinging on the sample. With this set up, the length of the nonlinear medium was about 10 times smaller than
the Rayleigh length. The far-field diffraction patterns were collected by a CCD camera, and the results were
recorded and analyzed by a computer.
A colloidal solution of castor oil and gold nanoparticles, which presents a large negative nonlinear refractive
index, was used as the self-defocusing medium. The average diameter of the gold particles within this colloid
was about 10 nm. The material has an absorption band centered at 553 nm due to surface plasmons resonance.
Sample
1 mm
D
CCD
camera
Laser
f = 25 cm
z =0
z = 0 is the position of the lens focal plane. For
z < 0 ( z > 0 ) the beam is convergent (divergent).
Figure 1 – Experimental setup.
Results and Discussions
The formation of the diffraction patterns was analyzed in order to study the influence of the beam curvature on
the observed profile. In this case, the far-field pattern was recorded for two different cell positions, which were
symmetric with respect to the lens focal plane. These experimental conditions correspond to the two different
cases studied in Ref. [7]. Figure 2 presents the observed profile and the equivalent numerical result for the case
which R > 0 and ∆φ < 0 .
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2 – Observed far field patterns for divergent beam: (a) P = 35 mW, (b) P = 40
mW and (c) P = 50 mW. Theoretical results: (d) P = 35 mW, (e) P = 40 mW and (f)
XXIX ENFMC
- Annals of Optics
2006
The sample was positioned at z = 32 mm, which corresponds to a curvature radius equal to R = 33.8 mm.
One can observe that the innermost part of the profile is diffuse and that a brighter external ring is generated for
this case. The figure also shows the profile evolution with the increase of the nonlinear phase-shift. The number
of generated rings varies linearly with the laser power. The experimental and theoretical results for the second
case, in which R < 0 and ∆φ < 0 , are presented in figure 3. The sample was in a symmetrical position with
respect to the focal plane. For this condition, two different kinds of rings can be observed with the increase of
the power. The first observed kind consists of thin rings around the beam propagation axis. For higher laser
power, thicker rings can be observed, superimposed to the rings of the first kind. The width of the second kind
of rings is larger for the outer rings than it is for the inner ones. The results observed for the two cases are
essentially different [7].
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3 – Observed far field patterns for convergent beam: (a) P = 35 mW, (b) P = 40 mW
and (c) P = 50 mW. Theoretical results: (d) P = 35 mW, (e) P = 40 mW and (f) P = 50 mW.
Conclusions
In summary, we have studied systematically the generation and evolution of the far-filed diffraction of Gaussian
beams after its propagation through a self-defocusing medium. We verified experimentally the formation of two
different behaviors for divergent and convergent beams. The difference between the diffraction patterns is
attributed to the changes in sign of the wave front curvature of the incident beam and to the interplay of the
strong SPM. The experimental results are in very good agreement with the proposed model.
Acknowledgements
The authors acknowledge the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado de Alagoas (FAPEAL), Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for the financial support.
References
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