## Abstract

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.

© Optical Society of America

## Corrections

Rakesh Kapoor and G. S. Agarwal, "Finite beam curvature related patterns in asaturable medium: errata," Opt. Express**4**, 229-230 (1999)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-4-7-229

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 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]

through a nonlinear medium with saturable nonlinearity. The complex parameter
*q* in Eq.1 can be written as *q*_{x,y}
=
*iz*_{Rx,y}
- *z* +
*z*
_{0} where *z*_{Rx,y}
=
${\pi \omega}_{0x\mathit{,}y}^{2}$/*λ*
is the Rayleigh range and the beam waist is located at *z* =
*z*
_{0}. We will throughout assume that the entry face of
the nonlinear medium is at *z* = 0. Thus for positive (negative)
*z*
_{0} 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]

$$\Delta =\frac{\left({\omega}_{0}-\omega \right)}{\gamma}\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}G=\frac{\overrightarrow{d}\xb7\overrightarrow{E}}{\mathit{\u0127\gamma}},$$

where **d** is the dipole matrix element for the transition with frequency
*ω*
_{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*
(Δ < 0) or *defocussing* (Δ
> 0) type. The parameter 2*G* 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

$${\nabla}_{\perp}^{2}=\left(\frac{{\partial}^{2}}{\partial {x}_{0}^{2}}+\frac{{\partial}^{2}}{\partial {y}_{0}^{2}}\right),\phantom{\rule{8.2em}{0ex}}$$

$$\zeta =\frac{z}{l},\phantom{\rule{.2em}{0ex}}{x}_{0}=\frac{x}{l},\phantom{\rule{.2em}{0ex}}{y}_{0}=\frac{y}{l},\phantom{\rule{7.2em}{0ex}}$$

where *α* is the absorption coefficient at line centre

We will solve Eq.3 subject to the initial condition

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 Δ = - 18 and
*α* = 300. The medium thickness was taken to be
7.5*cm*. The Gaussian beam of *λ* =
780*nm*. and complex radius of curvatures
*q*_{x}
= .12 + 2.5*cm*. and
*q*_{y}
= .21 + 2.5*cm*.
was propagated through the medium. To find out the proper aperture size and correct
number of iterations the simulations were done with following parameters.

- The simulations were done first on a 256×256 mesh. The iteration number in each case was decided by observing the convergence of the pattern for different number of iterations. Therefore, the number of iteration for different 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 sufficient 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.
- In all cases of converging beams, the simulations were repeated with a 512×512 mesh and the aperture was five times of the beam size at the entrance plane. It was found that the results were more or less same as obtained with 256×256 mesh. Therefore, it was decided to use a 256×256 mesh with four times beam size aperture along both the axes.

In successive figures, we present patterns for different cases. All the figures 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
(Δ < 0) medium. Some results for the defocusing medium
(Δ > 0) are shown in Fig.7 and Fig. 8. The figures 1–3 (4–6) show the formation of
patterns for convergent (divergent) beams. Fig.9 shows the intensity profile and zero lines of the real
and imaginary part of the beam at different planes inside the nonlinear medium. This
figure 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 different
regimes where the interplay of diffraction and the self-induced phase shift in beam
is causing the pattern formation. We also note that for
*z*
_{0} ≫ *z*_{R}
, 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 finite curvature of
the input beam can generate very different kind of patterns which depend on the
convergent/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.

## References

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**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). [CrossRef]

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**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.

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