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Self-focusing and defocusing of twisted
light in non-linear media
Anita Thakur1,2and Jamal Berakdar2,∗
1Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany
2Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle-Wittenberg, Heinrich-Damerow-Str. 4,
06120 Halle (Saale), Germany
3athakur@mpi-halle.mpg.de
4jamal.berakdar@physik.uni-halle.de
http://qft0.physik.uni-halle.de
Abstract:
rying angular momentum (called twisted light) propagating in a nonlinear
medium. We derive a differential equation for the beam width parameter
f as a function of the propagation distance, angular frequency, beam waist
and intensity of the beam. The method is based on the Wentzel-Kramers-
Brillouin and the paraxial approximations. Analytical expressions for f are
obtained, analyzed and illustrated for typical experimental situations.
We study the self-focusing and defocusing of a light beam car-
© 2010 Optical Society of America
OCIS codes: (260.5950) Self-focusing; (190.4350) Nonlinear optics at surfaces.
References and links
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1. Introduction
The self-focusing of light beams is a basic phenomena in nonlinear optics [1] with a variety
of important applications [2] that rely on the manipulation and control of the photon beam [3].
Generally, the theory of self focusing is well established with the propagation characteristics
found to be closely related to the properties of the medium [1–4] and to the pulse width of laser
beams [5]. Self-focusing and de-focusing of electromagnetic beams in nonlinear media was
reviewed by Akhmanov et. al [6]. Recently, several investigations were conducted to study the
propagation properties of Cosh Gaussian and Hermite Gaussian beams in different media [7,
8]. Here, we investigate the self focusing of Laguerre Gaussian beams [9–12] in a nonlinear
medium using the the Wentzel-Kramers-Brillouin (WKB) and the paraxial approximations [6,
13]. Laguerre Gaussian (LG) beams with a central hole singularity have been shown to play
an important role in several areas of optics [14,15]. In particular, light carrying orbital angular
momentum l, also called twisted light, is described by LG modes with a term describing the
on-axis phase singularity of strength l; hence the name optical vortices [16–19] for this type of
intensity distribution. In addition to the winding number l, LG modes are characterized by their
radial index p and their waist size w0. Here we use LG modes with p = 0; for l?= 0. In this case,
the intensity cross-sections perpendicular to the propagation direction consists of one bright
ring with no on-axis intensity. This feature makes them ideal for applications in optical trapping
and optical tweezers. Furthermore, as LG beams can transfer orbital momentum to the trapped
particle, it can also act with a torque on the trapped particle [16–19]. LG tweezers can also trap
metallic particles with a refractive index higher than that of the surrounding medium [20,21].
All of these applications rely on the light scattering and hence they are related to the strength
and the distribution of the intensity. The focusing and defocusing are thus important, e.g. in
the above context they allow the manipulation of the trapping spot size and the strength of the
tweezers.
The paper is structured as follows: starting from the amplitude distribution of LG beams
propagating in a nonlinear dielectric medium in section 2, we derive a general differential equa-
tion for the beam width parameter. Utilizing the WKB and paraxial approximations in section
3 we derive analytical expression for the intensity distribution as a function of the beam’s pa-
rameters. In section 4 results are presented graphically and discussed. A brief conclusion and
future prospectives are given in section 5.
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2.Theoretical background
2.1.
The amplitude distribution of the LG beam uLG
being along the beam propagation direction, is [24]
Laguerre Gaussian beams
lp(r,φ,z) in a cylindrical coordinate with z axis
uLG
lp(r,φ,z)=
Clp
w(z)
?√2r
w(z)
?
?l
exp
?−r2
R)−ilφ +i(2p+l+1)arctan(z
w2(z)
?
Ll
p(2r2
w2(z))
×exp
−i
kr2z
2(z2+z2
zR)
?
,
(1)
where r is the radial coordinate and φ is the azimuthal angle. w(z) = w0
radius of the beam at z, and zRis the Rayleigh range. w0is the beam waist at z = 0 . Llp(x) is the
associated Laguerre Polynomial, Clpis the normalization constant, and (2p+l +1)arctan(z
is the Gouy phase. At the beam waist, z = 0, the amplitude of a Laguerre- Gaussian beam
simplifies to
?√2r
w0
?
1+(z2/z2
R) is the
zR)
uLG
lp(r,φ,z = 0) =Cl
p
?l
exp
?−r2
w2
0
?
Ll
p(2r2
w2
0
)exp(−ilφ).
(2)
2.2.
We consider a nonlinear medium characterized by the dielectric function ε = ε0+F(EE∗),
i.e. ε(r,z) depends upon the beam irradiance; the functional dependence of F is determined
by the physical situation/mechanism under study. In turn |E|2depends on z in a manner yet to
be determined. In the spirit of the paraxial approximation, we expand F in a Taylor series in
powers of r2and retain terms up to r2. This leads to
Self-focusing and defocusing in a nonlinear medium
ε(r,z) ≈ ε0(z)−r2ε2(z).
(3)
In the wave equation governing the propagation of the laser beam
∇2E +ω2
c2εE +∇
?E∇ε
ε
?
= 0(4)
the third term can be neglected if k−2∇2(lnε)?1, where k is the wave vector. This inequality is
satisfied in almost all cases of practical interest. For a cylindrically symmetric beam a solution
for
∇2E +ω2
c2εE = 0
we obtain using WKB and the paraxial approximation Refs. [6,13] as
(5)
E(r,φ,z) = A(r,φ,z)exp[i(ωt −kz)]
√ε0and A(r,φ,z) is the complex amplitude of the electric field. Substituting for
E(r,φ,z) and neglecting∂2A
acteristic distance of intensity variation is much greater than wavelength. We obtain
(6)
where k =ω
c
∂z2on the basis of WKB approximation which implies that the char-
2ik∂A
∂z=∂2A
∂r2+1
r
∂A
∂r+1
r2
∂2A
∂φ2−ω2
c2ε2r2A.
(7)
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To solve Eq. (7) we express A(r,φ,z) as
A(r,φ,z) = A0(r,z)exp{i[−kS(r,z)−lφ]}
(8)
where A0and S are real functions of r, φ and z and the Eikonal S is
S =r2
2β(z)+Θ(z).
(9)
Θ(z) is an additive function where
β(z) =1
f
d f
dz.
(10)
The parameter β(z) is the curvature of the wavefront. Substituting for A(r,φ,z) and S from
Eq. (9) and Eq. (10) in Eq. (8) one obtains
?∂S
∂A2
0
∂z+∂A2
∂r
2∂S
∂z+
∂r
?2
=
1
k2A0
?∂2A0
0
∂S
∂r+A2
∂r2+1
r
∂A0
∂r-l2
?∂2S
r2A0
?
∂S
∂r
−ε2
?
ε0r2
(11)
0
∂r2+1
r
= 0(12)
The solution of Eq. (11) for LG beam can be written as
A0(r,z) =E0
f(
√2r
w0f)lexp(−r2
w2
0f2)Ll
P(2r2
w2
0
).
(13)
For l = 1 and p = 0, substituting for S and A0from Eqs. (9), (13), (11) yields
1
f
d2f
dz2=
4c2
ω2ε0(z)w4
0f4−ε2(z)
ε0(z).
(14)
Equation (14) can always be solved by considering the conditions
f = 1 and
d f
dz= 0 at z = 0.
(15)
It is, however, convenient to reduce Eq. (14) to a dimensionless form by transforming the coor-
dinate z to the dimensionless distance of propagation
zc
w2
ξ =
0ω
(16)
and the beam width w0to the dimensionless beam width
ρ =w0ω
c
(17)
Substituting Eq. (16) and (17) in Eq. (14) yields
ε0(z)
f
d2f
dξ2=4
f4−ρ2w2
0ε2(z).
(18)
In case of a parabolic nonlinearity, that is when the nonlinear term is proportional to E2we
have the r dependent term
ε2(f)r2=αE2
0
f2
r2
0f2
w2
(19)
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where α is a constant. Substitution of ε2(f) in Eq. (18) yields
1
f
d2f
dξ2=
1
ε0f4(4−ρ2αE2
0).
(20)
The analytical solution of Eq. (20) under the conditions (15) is
f =
?
ε0+4ξ2−E2
0αξ2ρ2
ε0
.
(21)
For further analysis it is useful to write Eq. (20) in the form
d2f
dz2=
c2
0ω2
ε0w4
?
4−αw2
0ω2
c2
E2
0
?
f−3.
(22)
3.Results and discussions
Equation (20), or (22) is the fundamental second order differential equation governing self-
focusing/defocusing of LG beams in a parabolic medium. Essentially, the effect of the non
linearity is dictated by the second term on the right of Eq. (20), or (22). In the absence of
this termd2f
leading to a steady divergence. This effect is the natural diffraction divergence. The second term
containing the nonlinear effect is negative and acts in the opposite direction tending to converge
the beam. The convergence (focusing) or divergence (defocusing) of the beam depends on
which of the two terms predominates. Equation (22) makes also clear that for the focusing
the product (w0ωE0)2is relevant, i.e. for a focused beam the frequency has to be increased
when the intensity (or the waist) is lowered to maintain focusing.
Figures 1(a) and 1(b) illustrate the focusing effects for typical experimentally feasible pa-
rameters.
dξ2remains positive causing the beam width parameter (f) to increase continuously
? ?
r [cm]
intensity
0.1
(a)
r [cm]
intensity
0
_
_
0.08
(b)
Fig. 1. (color online) (a) Intensity in CGS units of LG beam verses the radial distance
from the propagation direction (in cm) for the l = 1, and p = 0. The angular frequency is
ω =2 ×1014rad/sec,w0=1[cm],ε0=1,α =1,E0=0.3[StatV/cm].ρ =0.66 ×104.(b)
Initialintensityprofile(dottedcurve)comparedtothepropagatedintensityatξ =4 ×10−4.
As clear from this figure, focusing and intensity increase occur until at certain value of the
normalized distance of propagation (ξ) = 0.00062, after that de-focusing sets in. This is due
to the fact that at higher intensity nonlinear refractive term dominates over the diffractive term
for some initial distance of propagation after that the diffractive term strongly overcomes the
nonlinear refractive term and therefore the beam defocuses.
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From Fig. 1 it is also obvious that the focusing effect can be utilized, e.g. for creating tighter
and stronger three-dimensional optical traps by crossing two LG beams at the focused distance.
The predicted focusing effect can also be used for the realization of more versatile optical
tweezers. From Eq. (22) we infer that to achieve results similar to those in Fig. 1 for a smaller
staring waist one has either to increase the intensity (or the frequency) by the roughly the same
amount.
4. Conclusions
We studied the self focusing of a twisted light beam in a nonlinear dielectric medium by using
the paraxial approximation. The differential equation for the beam waist is solved analytically.
The occurrence of the focussing is pointed out and its dependence on the beam’s parameters is
worked out analytically and illustrated by numerical calculations. Some practical applications
of the predicted effect are pointed out.
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