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Scenes from a Mueller matrix movie of a brass cone are shown. From the Mueller matrix image, a variety of polarization measures are calculated and displayed. These polarization images reveal considerable details about the geometry of the cone despite the fact that the images are taken from a single vantage point, i. e. they are “flat images”. As the cone rotates, the orientation of the diattenuation and retardance follow the shape of the cone, providing a clear indication of its shape.
©1999 Optical Society of America
1. Background
Mueller matrix imaging [1] is an active imaging technique that measures the Mueller matrix at every pixel of a scene. The scene is illuminated by a sequence of polarization states, and the polarization of light scattered by the scene is measured. The Mueller matrix image provides a complete description of the basic polarization properties [2] such as retardance, diattenuation, and depolarization at each point in the scene for the illumination and collection solid angles used in the setup. It is of particular interest to relate the geometry of an object within a scene to its measured orientations of retardance and diattenuation.
We hypothesize that for homogeneous scattering objects, diattenuation images and retardance images will indicate geometrical properties of the sample. In particular, the orientation of the diattenuation should be aligned with the plane of incidence of the light at the surface, and the diattenuation magnitude should increase as the angle of incidence and the angle of scatter increase. Similarly, the retardance should also be aligned with either the plane of incidence or the orthogonal s-plane, depending on the nature of the scatterer.
We seek to understand what classes of materials yield useful polarization signatures and which materials are either too rough or otherwise unsuitable to yield valuable polarization images [3–5]. Here, polarization images are presented for one of the most interesting samples from our studies. A more complete study of Mueller matrix images of basic shapes and materials is in preparation for publication.
By a homogeneous scattering object, it is meant that the object’s surface properties are uniform in composition and texture; and that the surface is free from any variations that would influence the polarization of the scattered light. Thus if fingerprints, residues, scratches, dents, etc. that altered the object’s polarization properties were present, then that surface would be considered inhomogeneous.
2. Optical configuration
A Mueller matrix imaging polarimeter (Fig. 1) has been constructed at the University of Alabama in Huntsville for the accurate determination of Mueller matrix images of a wide variety of samples, such as polarization elements, waveguides, beamsplitters, and optical systems [6–10]. In this study, the sample is a small brass cone (maximum diameter of 34mm, length 100mm ) with spherical rounded tip. The surface roughness, Ra, of the cone was estimated to be 0.313 μm using a Taylor-Hobson profilometer. The Mueller matrix images were acquired at a set of angles and these data sets have been animated into a Mueller matrix movie. Figure 1 shows the Mueller matrix imaging polarimeter and sample cone. The bistatic angle between the incident beam and the collected beam was 10 degrees, which is smaller than the angle shown in this figure. The cone is surrounded by a black felt cloth to provide a dark background and minimize stray light.
The polarimeter and its calibration are described extensively elsewhere [1,11]. The light source was a HeNe laser at 632.8 nm. Both the polarization generator and the polarization analyzer are constructed of a stationary linear polarizer and a rotating quarter-wave linear retarder. The two retarders are rotated in angular increments with a 5:1 ratio with steps of 6° and 30°. Sixty images are acquired and processed into each 16-element Mueller matrix.
3. Mueller matrix decomposition
The Mueller matrix describes the transformations between incident and (in this case) scattered Stokes vectors. The properties of diattenuation, retardance, and depolarization are obtained from the Mueller matrix, M, by applying the polar decomposition [12] which yields three Mueller matrices for a pure diattenuator, D, a pure retarder, R, and a pure depolarizer, Dep, related to M by matrix multiplication
From these matrices the diattenuation, retardance, and depolarization properties are readily determined. Figure 2 represents the 16-dimensional Mueller matrix space where intensity of flux has one degree of freedom, diattenuation and retardance each have three degrees of freedom, and depolarization has nine degrees of freedom. The structure and properties of the Mueller matrix has been a fertile area for research [13–16]. For Mueller matrix images, the polar decomposition is applied at each pixel yielding diattenuation, retardance, and depolarization images.
4. Experimental results
Figure 3 contains four separate Mueller matrix images (3a, 3b, 3c, and 3d) of the brass cone. Each Mueller matrix is a four-by-four array of images, representing the elements of the Mueller matrix. These element images range from -1 (red) to +1 (blue), as expected in a normalized Mueller matrix. Figure (3.a.) contains the full, the normalized Mueller matrix image. . From this Mueller matrix, the pure diattenuation matrix (3b), the pure retardance matrix (3c), and the pure depolarization matrix (3d) are derived via the polar decomposition.
The diagonal elements (m11, m22, m33) and some off-diagonal elements (m01, m10, m12, m12) of the Mueller matrix show interesting stripes related to the shape of the object. When this Mueller matrix image M(r) multiplies a Stokes vector, S, such as {1,1,0,0} for horizontal linearly polarized light, the result, M(r).S, is the Stokes vector image of the scattered light when S is incident. Thus adding the first two columns yields the Stokes vector image for horizontal linearly polarized light incident from this bistatic angle.
Figure 3b is the diattenuation Mueller matrix image, a hermetian matrix at each pixel. The top row is equal to the top row of 3a, and the other twelve elements are determined to yield a pure diattenuator with all retardance and depolarization removed. The top row, pixel by pixel, indicates the Stokes vector with the maximum scattering efficiency. The last element, m03, is nearly zero indicating that right and left circular polarization scatters with nearly equal efficiency.
Figure 3c is the retardance Mueller matrix image, an orthogonal matrix (real unitary) at each pixel, which must be zero in the first row and column except for the m00 element. Figure 3d is a pure depolarization Mueller matrix image where the last three elements on the first row are zero (all diattenuation is removed) and the lower three-by-three submatrix is even (all retardance is removed).
The procedure above has been applied to a series of Mueller images of the cone as it rotates in the sample compartment of the Mueller matrix imaging polarimeter, yielding movies of the polarization parameters. Figure 4 shows the magnitude (first column) and orientation (second column) of the diattenuation with the cone axis positioned at angles of 0° (measured from the incident beam), 30°, 60°, 90°, 120°, and 150°. Since this angle is measured from the incident beam, which is 10° from the camera axis, in the 90° view, the cone axis is not coincident with the camera axis. Note that a vertical mounting rod is visible on the bottom of the cone. The most interesting aspect of these images is the pattern of the diattenuation orientation in the 60°, 90°, and 120° views. Particularly in the 90° view, the orientation of the diattenuation closely follows the shape of the plane of incidence, horizontal along the middle and rotating steadily clockwise around the right side to vertical near the top and bottom. The magnitude of the diattenuation is small, less than 0.1 in the 0°, 30°, and 150° cone orientations. The speckled variations in the data are due to the noise in measuring this small magnitude. The orientation images show how the diattenuation is controlled by the shape of the object, yielding stripes in the diattenuation Mueller matrix image.
Figure 5 shows the corresponding retardance magnitude images (first column) and retardance orientation images (second column). The retardance magnitude is small, measuring less than 18° (a twentieth of a wave) nearly everywhere across the object. The orientation is vertical (red) across the middle of all the figures and rotates steadily counterclockwise toward horizontal at the top. The retardance rotates clockwise toward horizontal toward the bottom in each image. So for scattering from this brass cone the retardance fast axis is closely aligned with the s-plane for this surface providing a clear indication of the shape of the object. This s-plane orientation contrasts with the approximate p-plane orientation of the diattenuation. The magnitude of the retardance is largest in the 90° and 120° figures where the axis of the cone is close to the direction of the incident light and the angles of incidence and scatter are the largest.
Figure 6 shows the polarization crosstalk between pairs of orthogonal states. Column 1 is the fraction of scattered light that is vertically polarized when horizontal linearly polarized light is incident; note that it increases steadily from the middle to the edges. The coupling from left circular into right circular polarized light, shown in column 2, is noticeably greater. We defined these crosstalks because they use the most commonly encountered polarized states such as linear horizontal and vertical polarizations and circular polarizations. But there is no limitation to define other crosstalks as long as the coupling is calculated between two orthogonal states. Crosstalks are calculated from the normalized Mueller matrix images. Their values range from 0 to +1. The value varies from 0 when the state of polarization of the reflected light remains unchanged, to 0.5 when the reflected light is unpolarized, to +1 when the reflected light is fully coupled into the orthogonal polarized state. Crosstalk values of the brass cone range from 0 to 0.5, meaning that the polarized light incident on the target remains either in the same state of polarization (or close to the polarization incident) or becomes completely depolarized. The horizontal-to-vertical crosstalk remains slightly greater than zero on the middle and on the top and bottom depending on the orientation of the cone. No coupling appears on these regions of the cone. Figures 6d and 6e (cone respectively oriented at 90° and 120°) show very interesting patterns. The horizontal-to-vertical crosstalk slowly increases from 0 in the middle to 0.4 when the surface is oriented at 45°. The magnitude of the crosstalk increases regularly following the shape of the cone. Left-to-right crosstalk images present similar patterns. No coupling occurs when the surface is oriented at 45° and unpolarized light is scattered when the surface is oriented either horizontally or vertically.
Figure 7 shows the polarizance image (first column) and depolarization index image (second column) as the cone rotates. Unpolarized incident light generally becomes partially polarized after scattering from a surface and the polarizance is the degree of polarization of that scattered light. The polarizance is generally very similar to the diattenuation magnitude, as it is for this sample. In contrast, the depolarization index characterizes how completely polarized light incident on a scattering surface scatters with a reduced degree of polarization. In general, different incident polarization states will experience a different reduction of the degree of polarization so an average measure is useful. The depolarization index is defined in terms of the Mueller matrix elements as [2,17]
This index measures how close M is to the set of non-depolarizing Mueller matrices, and it is closely related to the average depolarization of the exiting light. The depolarization index is low along the middle of the cone and increases toward the edges of the cone. As the cone rotates, the depolarization remains small (orange) at the small spot on the rounded tip of the cone corresponding to the specular reflected beam.
Fig. 4. Magnitude and orientation images of the diattenuation for different orientations of the brass cone (0°, 30°, 60°, 90°, 120°, and 150°)
Fig. 5. Magnitude and orientation images of the retardance for different orientations of the brass cone (0°, 30°, 60°, 90°, 120°, and 150°)
Fig.6. Polarization crosstalk images for different orientations of the brass cone (0°, 30°, 60°, 90°, 120°, and 150°)
Fig. 7. Polarizance and depolarization index images for different orientations of the brass cone (0°, 30°, 60°, 90°, 120°, and 150°)
5. Conclusion
Conventional passive polarimetric systems only measure the polarization state of the light reflected or scattered by a scene since there is no control of the illuminating polarization state. In contrast, active imaging systems such as the Mueller matrix imaging polarimeter allow characterization of objects in terms of basic polarization properties of the light. It is a powerful investigation tool that extends our understanding of polarization in scattering.
We have illustrated the capability of such a technique by presenting a data set showing the polarization signature of a simple shape object described in terms of retardance, diattenuation, and depolarization. From these results, it appears that polarization could provide relevant information for determining orientations of surfaces in space. We showed that retardance orientation and diattenuation orientation indicate geometrical properties of the target. The curvature and orientation of the reflection surface give distinct patterns to the polarization orientation.
Mueller matrix images allow the polarization characteristics of scattering objects to be classified and quantified. Small phase differences between different incident states are present (retardance) and for this metal object, the orientation of the retardance provides a clear indication of the shape of the surface. Similarly, the orientation of the diattenuation provides a second measure that may correlate closely with geometric orientation of the object. Further studies can quantify these effects for a range of material to determine the applicability and utility of these observations.
Measurements described in this paper were performed in a laboratory under favorable experimental conditions. Polarization signatures of other objects are not always as representative of their shape and orientation. Several parameters such as roughness, texture, and surface cleanness can affect the polarization properties of the sample. A more complete study will be presented in another publication including a variety of shapes and materials.
Acknowledgment
This work was supported by Air Force Office of Scientific Research under grant F4962-96-0136.
References and links
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