Fourier processing of the speckle structure of the optical field formed in the process
of multiphonon Bragg diffraction
V. M. Kotov,
a
G. N. Shkerdin, D. G. Shkerdin, A. I. Voronko, and S. A. Tikhomirov
Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino, Moscow Oblast
J. Stiens, R. Vounckx, and W. Vandermeiren
Vrije Universiteit Brussel, Belgium
Submitted November 1, 2005
Opticheski Zhurnal 74, 12–15 February 2007
The quantitative characteristics of the speckle structure of an optical field are obtained by Fourier
processing the optical signal, using a transparency that differentiates the speckle pattern. It is
shown in terms of the Gauss–Schell model that the maximum values of the differential of the
processed pattern are proportional to the degree of spatial coherence of the corresponding section
of the optical field. The results are found to be in good agreement with the data obtained when
other methods are used in particular, a method based on the angular selectivity of multiphonon
acoustooptic Bragg scattering.© 2007 Optical Society of America.
Acoustooptic AO interaction is a widely used method
of filtering partially coherent optical radiation according to
the degree of its spatial coherence.
1–4
Among all possible
forms of AO diffraction that make it possible to efficiently
solve this problem, the most attractive is multiphonon Bragg
diffraction, whose angular selectivity is significantly greater
than that of ordinary Bragg scattering.
5,6
It is shown in Ref. 5 that the character of the graininess
of the speckle structure of an optical field substantially
changes as one goes from one diffraction order to another.
However, that paper gives only the qualitative characteristics
of the degree of graininess. This paper proposes a fairly
simple method of quantitatively describing the speckle struc-
ture of a field obtained in experiment. The method is based
on the Fourier processing of images using filters—in particu-
lar, amplitude transparencies see, for example, Refs. 7 and
8. The essence of the method is elucidated by Fig. 1, which
shows an optical system consisting of two lenses with focal
length f . The initial image described by function h , is
formed in the front focal plane P
1
of lens L
1
, where and
are the coordinates of the field in plane P
1
. A field will then
be formed in the back focal plane P
2
with the form
7–9
gx, y =
1
i f
h, exp
2i
f
x + y
dd
=
1
i f
Fu, v . 1
Here is the wavelength of the light, Fu , v is the Fourier
transform of the function h , , and u =
x
f
and v =
y
f
are the
spatial frequencies. After the radiation passes through lens
L
2
, the reverse transformation occurs, i.e., the field formed in
the back focal plane P
3
of lens L
2
coincides with the initial
field h , . The characteristics of the output light field can
be substantially affected by placing various transparencies in
plane P
2
the spatial-frequency plane.
Let a filter whose transmission varies according to the
law Tu = u be used as a transparency
7
the one-
dimensional case is considered here for simplicity: u is the
coordinate, and is some constant; i.e., the filter’s transmit-
tance increases as u increases. If the initial field is described
by function f x, the field in plane P
2
will be the Fourier
transform of f x—i.e., Vu = J f x. Taking into account
the influence of filter Tu, the resultant field Gu in plane
P
2
has the form Gu = VuTu = uVu. The image at the
output of the optical system will be the Fourier transform of
function Gu:
gx =
uVuexp2iuxdu
=
2i
d
dx
Vuexp2iuxdu
=
2i
df - x
dx
,
2
i.e., a field that is the differential of the initial field f x
appears in image plane P
3
. Because the differential of a func-
tion reaches an extremum in the region where the field am-
plitude varies most rapidly, such filtering accomplishes “con-
touring” of the image. In our case, the image is a speckle
pattern, and differentiation of the image therefore results in
FIG. 1. Optical layout of the system that carries out Fourier processing of an
image.
84 84 J. Opt. Technol. 74 2, February 2007 1070-9762/2007/020084-03$15.00 © 2007 Optical Society of America