We study numerically the effects of finite curvature and ellipticity of the Gaussian beam on propagation through a saturating nonlinear medium. We demonstrate generation of different types of pattern arising from the input phase structure as well as the phase structure imparted by the nonlinear medium

Agarwal, G. S.

Kapoor, Rakesh

English

Publications of the IAS Fellows

Finite beam curvature related patterns in a
saturable medium
Rakesh Kapoor
Center for Advanced Technology, Indore 452 013, India
rkapoor@cat.ernet.in
G.S. Agarwal
Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India
gsa@prl.ernet.in
Abstract: We study numerically the eects of nite curvature and
ellipticity of the Gaussian beam on propagation through a saturating
nonlinear medium. We demonstrate generation of dierent types of
pattern arising from the input phase structure as well as the phase
structure imparted by the nonlinear medium.
c©1998 Optical Society of America
OCIS codes: (190.0190) Nonlinear Optics, (330.5000) Pattern
References
1. S. Camacho-Lopez, R. Ramos-Garcia and M.J. Damzen, ”Experimental study of propagation of
an apertured high intensity laser beam in Kerr active CS2,” J. Mod. Opt. 44, 1671-1681 (1997).
2. G.A. Swartzlander Jr. and C.T. Law, ”Optical vortex solitons observed in Kerr nonlinear medium,”
Phys. Rev. Lett. 69, 2503-2506 (1992).
3. G.A. Swartzlander Jr., D.R. Anderson, J.J. Regan, H. Yin and A.E. Kaplan, ”Spatial dark-soliton
stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583-1586 (1991).
4. V. Tikhonenko, J. Christou and B. Luther-Davies, ”Three dimensional bright spatial soliton
collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698-2701 (1996).
5. G. Grynberg, A. Maˆitre and A. Petrossian, ”Flowerlike patterns generated by a laser beam
transmitted through rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72, 2379-2382
(1994).
6. W.J. Firth and D.V. Skryabin, ”Optical solitons carrying orbital angular momentum,” Phys.
Rev. Lett. 79, 2450-2453 (1997).
7. The transmission of a Gaussian beam through a Gaussian lens has been shown to yield optical
vortices: L.V. Kreminskaya, M.S. Soskin and A.I. Khizhriyak, ”The Gaussian lenses give birth
to optical vortices in laser beams,” Opt. Commun. 145, 377-384 (1998).
8. T. Ackeman, E. Kriege and W. Lange, ”Phase singularities via nonlinear beam propagation in
sodium vapor,” Opt. Commun. 115, 339-346 (1995).
9. J. Courtial, K. Dholakia, L. Allen and M.J. Padgett, ”Gaussian beams with very high orbital
angular momentum,” Opt. Commun. 144, 210-213 (1997)
10. A.E. Siegman, Lasers (University Science Books, Mill Valley, California); Chapter 17. Note that
we adopt the convention exp(ikz− iωt) rather than the one used by engineers exp(−ikz+ iωt).
11. R.W. Boyd, Nonlinear Optics (Academic Press, New York, 1992) p203.
12. V.Tikhonenko, Y. S. Kivshar, V. V. Steblina and A. A. Zozulya, ” Vortex solitons in a saturable
optical medium,” J. Opt. Soc. Am. B 15, 79-86 (1998).
In recent times, there has been considerable interest[1, 2, 3, 4, 5, 6, 7, 8] in the study
of propagation of a beam with certain phase structure, say a vortex structure through
a nonlinear medium. The phase structure can give rise to an angular momentum [9] for
the beam. It is also known that the phase of the Gaussian beam changes sign [10] as one
crosses the focus. It should further be noted that the nonlinear wave equation couples
amplitudes and phases, as the nonlinear polarization is in, general complex. Thus, one
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would expect that the point at which a beam enters the nonlinear medium would be
very important. Besides the sign of the nonlinearly is also important. Keeping this in
view, we have carried out a numerical study of the propagation of a complex Gaussian
beam[10]
E = E0 exp
(
−
ikx2
2qx
−
iky2
2qy
)
eikz−iωt (1)
through a nonlinear medium with saturable nonlinearity. The complex parameter q in
Eq.1 can be written as qx,y = izRx,y − z + z0 where zRx,y = w20x,y= is the Rayleigh
range and the beam waist is located at z = z0. We will throughout assume that the entry
face of the nonlinear medium is at z = 0. Thus for positive (negative) z0 the beam’s
waist is inside (outside before the entry face) the medium and we have a converging
(diverging) beam. The saturable nonlinearity will be the one produced by a medium
modelled as a collection of two level atoms. Hence, the induced polarization is taken as
[11]
P = nd
(
+ i
2 + 1 + 2jGj2
)
G
 = (!0 − !)=γ and G = ~d  ~E=hγ; (2)
where d is the dipole matrix element for the transition with frequency !0. All frequencies
have been scaled with respect to the half width γ of the transition.The sign of the
detuning determines whether the nonlinearity is of focussing ( < 0) or defocussing
( > 0) type. The parameter 2G is the scaled Rabi frequency and n is the density of
atoms. On scaling all frequencies with respect to γ and all lengths with respect to l (the
length of the medium), the wave equation in slowly varying envelope approximation can
be written as
@G
@
=
i
2kl
r2?G+
il
2
(
+ i
2 + 1 + 2jGj2
)
G; (3)
r2? =
(
@2
@x20
+
@2
@y20
)
;
 =
z
l
; x0 =
x
l
; y0 =
y
l
;
where  is the absorption coecient at line centre
 =
4nd2!
hcγ
: (4)
We will solve Eq.3 subject to the initial condition
G = G0 exp
(
−
ikx20l
2
2qx
−
iky20l
2
2qy
)
; (5)
with complex q.
The pattern formation is very sensitive to the focussing or defocusing nature of
the medium as well as to the converging or diverging nature of the Gaussian beam. The
results also depend on the ellipticity of the beam. Simulations were done for propagation
of a converging elliptic Gaussian beam through a focussing nonlinear medium with  =
−18 and  = 300. The medium thickness was taken to be 7:5cm: The Gaussian beam of
 = 780nm: and complex radius of curvatures qx = :12 + 2:5cm: and qy = :21 + 2:5cm:
was propagated through the medium. To nd out the proper aperture size and correct
number of iterations the simulations were done with following parameters.
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1. The simulations were done rst on a 256x256 mesh. The iteration number in
each case was decided by observing the convergence of the pattern for dierent number
of iterations. Therefore, the number of iteration for dierent cases varies from 60 to 100.
For converging beams, it was found that the entrance aperture, of four times that of
the beam size along both the axes was sucient for the propagation of 99 percent of
the total beam. In case of diverging beams, the aperture size was adjusted to allow 99
percent of the beam through it at the exit plane.
2. In all cases of converging beams, the simulations were repeated with a 512x512
mesh and the aperture was ve times of the beam size at the entrance plane. It was
found that the results were more or less same as obtained with 256x256 mesh. Therefore,
it was decided to use a 256x256 mesh with four times beam size aperture along both
the axes.
In successive gures, we present patterns for dierent cases. All the gures are
drawn in pseudo colors. The color shades change from red, yellow, green, blue and
magenta for the intensities changing from zero to one. In Figs. 1-6, we present results
for the focussing ( < 0) medium. Some results for the defocusing medium ( > 0)
are shown in Fig.7 and Fig. 8. The gures 1-3 (4-6) show the formation of patterns
for convergent (divergent) beams. Fig.9 shows the intensity prole and zero lines of the
real and imaginary part of the beam at dierent planes inside the nonlinear medium.
This gure shows several crossings of the contours of Re E = 0; Im E = 0 suggesting
the generation of vortices [7, 12]. It must be added that the propagation of an elliptic
beam through a Kerr medium has been studied previously [1]. For weak ellipticity, the
propagation through a saturating medium has also been studied [8]. The flower-like
patterns were observed by Grynberg et al [5] who used a feedback mirror. In their case
the distance between neighbouring flower petals is roughly determined by the distance
of the feedback mirror to the medium. We have presented results in altogether dierent
regimes where the interplay of diraction and the self-induced phase shift in beam is
causing the pattern formation. We also note that for z0 >> zR, the incoming beam
is almost a plane wave. However the strong nonlinearity is quite sensitive to the small
curvature of the wavefront.
In conclusion, we have shown numerically how the nite curvature of the in-
put beam can generate very dierent kind of patterns which depend on the conver-
gent/divergent nature of the beam and on the focussing or defocusing characteristics of
the medium. The ellipticity of the beam gives rise to optical vortices, which multiply as
the nonlinearity of the medium increases.
Figure 1. Patterns obtained with propagation of a converging circular-Gaussian
beam through a focusing nonlinear medium. The medium was placed at several
positions before the beam waist. ∆ = −18, α = 300, G0 = 30, and q = (z0 + .21i)
cm. The successive frames (from left to right) are for z0 = 3.0, 3.5, 4.0 and 5.0 cm.
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Figure 2. Patterns obtained with propagation of a converging elliptic-Gaussian
beam through a focusing nonlinear medium. The medium was placed at several posi-
tions before the beam waist. ∆ = −18, α = 300,G0 = 30, and q = [z0+.12i, z0+.21i]
cm. The successive frames (from left to right) are for z0 = 2.5, 3.0, 3.5, 4.0 and 5.0
cm.
Figure 3. Patterns obtained with propagation of a converging elliptic-Gaussian
beam through a focusing nonlinear medium. The medium was placed at before the
beam waist. ∆ = −18, q = [z0 + .12i, z0 + .21i] cm. Patterns were generated for
different values of α and G0. For different frames the value of α = 300, 400, and
500 (from top row to bottom row) and the value of G0 = 30, 40, and 50 (from left
column to right column)
Figure 4. Patterns obtained with propagation of a diverging circular-Gaussian
beam through a focusing nonlinear medium. The medium was placed at several
positions after the beam waist. ∆ = −18, α = 300, G0 = 30 and q = (z0+ .21i) cm.
Aperture was 8 times of the input beam size along both the axes. The successive
frames (from left to right) are for z0 = −3.0, -3.5, -4.0 and -5.0 cm.
(C) 1999 OSA 1 March 1999 / Vol. 4,  No. 5 / OPTICS EXPRESS  164
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Figure 5. Patterns obtained with propagation of a diverging elliptic-Gaussian
beam through a focusing nonlinear medium. The medium was placed at several po-
sitions after the beam waist, ∆ = −18, α = 300,G0 = 30, and q = [z0+.12i, z0+.21i]
cm. Aperture was 8 times of the input beam size along both the axes. The successive
frames (from left to right) are for z0 = −2.5, -3.0, -3.5, -4.0 and -5.0 cm.
Figure 6. Patterns obtained with propagation of a diverging elliptic-Gaussian
beam through a focusing nonlinear medium. The medium was placed at 2.5cm
after the beam waist. ∆ = −18, and q = [−2.5 + .12i,−2.5 + .21i] cm. Patterns
were generated for different values of α and G0. Aperture was 8 times of the input
beam size along both the axes. For top row frames α = 300, and for the bottom
row frames α = 400. The value of G0 = 30, 40, 50 (from left to right)
Figure 7. Patterns obtained with propagation of a diverging elliptic-Gaussian
beam through a defocusing nonlinear medium. The medium was placed at sev-
eral positions after the beam waist. ∆ = −18, α = 300, G0 = 30, and
q = [z0 + .12i, z0 + .21i] cm. Aperture was 12 times of the input beam size along
both the axes. The successive frames (from left to right) are for z0 = −3.0, -4.0 and
-5.0 cm.
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Figure 8. Patterns obtained with propagation of a converging elliptic-Gaussian
beam through a defocusing nonlinear medium. The medium was placed at sev-
eral positions before the beam waist. ∆ = −18, α = 300, G0 = 30, and
q = [z0 + .12i, z0 + .21i] cm. Aperture was 12 times of the input beam size along
both the axes. The successive frames (from left to right) are for z0 = 3.0, 4.0 and
5.0 cm.
Figure 9. Intensity profiles (left) and zero lines (right) of the real (red line)
and imaginary (blue line) part of an elliptic beam propagating through a focusing
nonlinear medium. The input beam was a converging beam as the medium was
placed at 2.5 cm before the waist of the beam. ∆ = −18, α = 300, G0 = 30, and
q = [2.5 + .12i, 2.5 + .21i] cm. Aperture was 4 times of the input beam size along
both the axes. The successive frames were recorded at different positions inside the
medium. From top to bottom z = 4.5, 6.75 and 7.5 cm.
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Finite beam curvature related patterns in a saturable medium

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Finite beam curvature related patterns in a saturable medium

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