Aspects of Rough Surface Scattering and Related Phenomena, Yountville, June 26  28, 2006
HigherOrder Correlations in Electromagnetic Fields
and Applications for Imaging and Sensing
Aristide Dogariu
CREOL,
The College of Optics and Photonics
University of Central Florida
Correlations in Electromagnetic Fields and Applications
Random electromagnetic fields are determined by propagation of EM radiation and/or its
interaction with material objects. The complex characteristics of these fields including
intensity, phase distribution, state of coherence and polarization carry information about
the properties of the objects such as, for instance, refractive index, structure or shape.
The statistics of random complex fields give rise to various measurable
distributions. Random complex fields arise in many practical situations, such as speckle
phenomena resulting from the coherent illumination of diffusive random media and
scattering from rough surfaces.
1
In fact, each speckle can be viewed as a realization of an
underlying complex field distribution. Frequent assumptions imply a circular Gaussian
distribution for the complex fields but they are not always valid in applications.
Furthermore, as Gaussian statistics are completely determined by their second order
moments, they provide little information about the underlying scattering system.
A number of measurable distributions, including intensity, phase difference,
ellipticity, and states of polarization can be derived from the statistics of random complex
fields. One useful tool for visualizing the polarization statistics is the Poincare sphere,
where specific distributions of pure states of polarization are mapped onto the surface of
a sphere of unit radius while partially polarized and unpolarized states lie somewhere
inside the sphere.
Every point in a globally depolarized field, whether this is a point in space, time,
or a member of an ensemble, can be considered as resulting from the summation of a
large number of underlying fields. The probability distribution of these underlying fields
determines the state of polarization at that particular point.
Under certain conditions, it is possible that the underlying fields are locally
unpolarized, in which case no information about them can be retrieved beyond the
intensity statistics. It is also possible that the underlying fields are locally polarized, in
which case the resulting distribution of polarization states can be studied on the surface of
1
E. Jakeman, ‘‘Polarization characteristics of nonGaussian scattering by small particles,’’
Waves Random Media 5, 427–442, (1995).
J. W. Goodman, ‘‘Statistical properties of laser speckle patterns,’’ in Laser Speckle and Related
Phenomena, J. C. Dainty, ed. (SpringerVerlag, New York, 1975), pp. 9–75.
J. Uozumi, K. Uno, and T. Asakura, ‘‘Statistics of Gaussian speckles with enhanced
fluctuations,’’ Opt. Rev. 2, 174–180, (1995).
Aspects of Rough Surface Scattering and Related Phenomena, Yountville, June 26  28, 2006
the Poincare sphere. In the case when a globally depolarized field is composed of locally
polarized states, one can differentiate between various types of globally unpolarized light,
and, therefore, infer additional information about the underlying random fields. This
situation is usually encountered when coherent laser radiation interacts with randomly
inhomogeneous media.
By considering the Poincare sphere, it is easy to visualize a virtually infinite
number of distributions that result in globally unpolarized light. Any distribution that has
an equal probability of polar pairs on the sphere will average to a completely unpolarized
field in the global sense. For instance, one can consider various geometrical structures on
the surface of the Poincare sphere, as long as they have a necessary polar symmetry. It is
worth pointing out that it is not even necessary for the points to lie on the surface of the
sphere or in other words to represent pure states of polarization.
Different globally unpolarized fields composed of locally pure states of
polarization, can be classified based on their invariance to (i) choice of reference frame,
(ii) choice of right handed circular polarization, and (iii) an arbitrarily introduced
retardation. In terms of polarization states distributed on the surface of the Poincare
sphere, these conditions are equivalent to distributions being (i) invariant under rotation
about the
V
axis, (ii) symmetric about the
Q U
,
plane, and (iii) invariant under an
arbitrary rotation of the three axes.
2
As mentioned before, the globally unpolarized distributions of the electric field
are of particular interest in the case of speckle patterns arising from coherent scattering
by a diffusive medium. Different random fields can be regarded as being globally
unpolarized
3
. A practical situation can be envisioned where fully polarized speckles are
averaged over the entire field of view of an optical system to produce a globally
unpolarized response. An example is illustrated in Figure 1 where various distributions of
polarization states are depicted on the Poincare sphere. Only the distribution on the left,
however, corresponds to a field that obeys Gaussian statistics.
Figure 1 Three different distributions of polarization states corresponding
to globally unpolarized radiation.
2
J. Ellis and A. Dogariu, ‘‘Differentiation of globally unpolarized complex random fields,’’ J.
Opt. Soc. Am. A 21, 988–993, (2004).
J. Ellis and A. Dogariu, "Discrimination of globally unpolarized fields through Stokes vector
element correlations", JOSA A22(5), 491496, (2005).
3
J. Ellis and A. Dogariu, "Complex degree of mutual polarization", Opt. Lett. 29(6), 536538
(2004)
S
3
S
2
S
1
S
3
S
2
S
1
Joint Distributions p(s1, s2, s3)
accessible through intensity measurements for spatiallyresolved speckle fields
S
3
S
2
S
1
S
3
S
2
S
1
S
3
S
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S
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S
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S
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1
Joint Distributions p(s1, s2, s3)
accessible through intensity measurements for spatiallyresolved speckle fields
S
3
S
2
S
1
S
3
S
2
S
1
Aspects of Rough Surface Scattering and Related Phenomena, Yountville, June 26  28, 2006
To illustrate the possible relationships between the detailed structure of a random
medium and the complex statistical properties of the scattered fields, Figure 2 presents
speckle patterns produced by different media which can all be considered as being highly
diffusive. These fields are all globally unpolarized as seen in the measured Stokes
vectors. On the basis of conventional fieldfield correlations (intensities), they can not be
distinguished one from another because the intensities obey negative exponential
distributions leading to socalled “fully developed” speckle fields. However, the
underleying random fields are different and this can be established by analyzing higher
order correlations of the electromagnetic field properties. This requires, for instance,
examining the values of the Stokes vectors, higherorder correlations between different
Stokes vector elements can be calculated such as:
<s
1
2
>,< s
2
2
>,< s
3
2
>,< s
1
s
2
>,< s
1
s
3
>,< s
2
s
3
>.
a)
b)
Figure 2
a) speckle fields generated by the different media shown as insets in b). They are different types of
nonabsorbing porous materials.
b) values of Stokes vector correlators corresponding to the media producing the speckle patterns
shown in a).
We calculated these correlations for the random fields in Figure 2 and, as can be seen, the
higher order correlators are quite different. This proves that the higherorder correlations
can provide additional structural information about the scattering media. As can be seen
in Figure 2, the higher order correlators can be quite different proving that additional
structural information about the media can be extracted. This approach permits
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Aspects of Rough Surface Scattering and Related Phenomena, Yountville, June 26  28, 2006
distinguishing between different field distributions and, therefore, between different
random media which determine those scattered fields. It is worth pointing out that, in the
case of highly diffusing media, the differences in the values of these correlators are
caused by the specific properties of the medium’s interface. To illustrate these surface
effects we will present several experiments and also results of numerical calculations
based on the coupled dipole approximation.
Gaussian statistics is usually invoked in studying multiplescattering phenomena.
However, both the numerical simulations and the experimental results indicate the
presence of nonGaussian distributions, as seen in the deviation of the moments of the
joint probability distribution of the Stokes vector elements. It is worth noting that the
absolute value of this deviation of Stokes vector element autocorrelations is
experimentally detectable, even when there are no measurable differences in the first
order moments or the intensity distribution.
Finally, it is worth noting that, the use of higherorder correlators to determine
additional field properties does not require more measurements. Once conventional
Stokes polarimetry is performed (spatially or temporallyresolved), no additional
measurement is necessary. The analysis is based on the already available data, i.e. the
spatially or temporallyresolved Stokes parameters.
These results illustrate the dependence between the distribution of underlying
fields and the morphological characteristics of the scattering system that gave rise to the
distribution of polarization states. The use of correlations between different elements of
the Stokes vector can become a powerful tool for differentiating between different field
distributions which are globally unpolarized.
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