Aspects of Rough Surface Scattering and Related Phenomena, Yountville, June 26 - 28, 2006

Higher-Order 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 non-Gaussian 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. (Springer-Verlag, 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), 491-496, (2005).

3

J. Ellis and A. Dogariu, "Complex degree of mutual polarization", Opt. Lett. 29(6), 536-538

(2004)

S

3

S

2

S

1

S

3

S

2

S

1

Joint Distributions p(s1, s2, s3)

accessible through intensity measurements for spatially-resolved speckle fields

S

3

S

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S

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S

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Joint Distributions p(s1, s2, s3)

accessible through intensity measurements for spatially-resolved speckle fields

S

3

S

2

S

1

S

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S

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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 field-field correlations (intensities), they can not be

distinguished one from another because the intensities obey negative exponential

distributions leading to so-called “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, higher-order 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

non-absorbing 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 higher-order 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 multiple-scattering phenomena.

However, both the numerical simulations and the experimental results indicate the

presence of non-Gaussian 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 higher-order correlators to determine

additional field properties does not require more measurements. Once conventional

Stokes polarimetry is performed (spatially- or temporally-resolved), no additional

measurement is necessary. The analysis is based on the already available data, i.e. the

spatially- or temporally-resolved 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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