Computation, visualization, and animation ofinfrared Mueller matrix elements byscattering from surfaces that are absorbingand randomly rough

Arthur H. Carrieri, Charles Jeffrey Schmitt, Craig M. Herzinger, and James 0. Jensen

Computation of Mueller matrix elements by infrared scattering from randomly rough two-dimensionalsurfaces and results of a method for graphic display of the data are presented. A full waveelectromagnetic scattering model first generates raw data elements of the 4 x 4 Mueller matrix.7(0, nx, k, s2 , (h2 )) in beam backscattering angle (0) ranging from normal to oblique incidence, inrefractive index of the beam scatterer (nX - ike) spanning the 9 < X < 12.5 Elm midinfrared band, and inmean-squared slope ((aS2 ) and mean-squared height ((h2)) of the scattering surface. These data are nextcompressed into a graphics format file occupying considerably less computer storage space and mappedinto color images of the Mueller elements as viewed on a high-resolution graphics terminal. Thediagonal and two off-diagonal elements are animated in the X-0 plane according to variations in Us2 and(h2). Predicted elements for polarized IR beam energies on vibrational resonance of the surfacemolecules, and particularly the off-diagonal elements, show subtle properties of the scatterer as viewed inthe animation sequences.

Key words: Computer visualization, Mueller matrix, infrared scattering, remote detection.

1. Introduction

Here at the U.S. Army Chemical and BiologicalDefense Agency we embarked on a remote detectionprogram based on scattering Mueller matrix spectros-copy. In this technique energy-select infrared (IR)laser beams that are modulated in polarization irradi-ate and stimulate vibrational activity in moleculesthat form a surface targeted for detection (the chemi-cal analyte). The scattered beam radiance carries apolarization optical encoding that, when modulated asecond time, can be electronically transformed to theMueller matrix of the scatterer.' This resultantMueller matrix is a complete optical description of the(elastic) scattering surface. Detection is sought froma susceptible field of Mueller elements measuredon-then-off absorption by the analyte. The incident

A. H. Carrieri and J. 0. Jensen are with the Edgewood Research,Development, and Engineering Center, U.S. Army Chemical andBiological Defense Agency, Aberdeen Proving Ground, Maryland21010-5423. C. J. Schmitt is with the Department of Computerand Information Sciences, Towson State University, Baltimore,Maryland 21204. C. M. Herzinger is with the University ofNebraska, Lincoln, Nebraska 68588-0511.

Received 12 June 1992.

beam tuned to absorption or extinction by the analyte(vibrational resonance) yields uncommon elements ofthe analyte, while the detuned (nonresonance beam)provides a set of reference elements. Identification,in situ, is determined by an algorithm that operateson a vector whose M < 16 components are theMueller elements measured at the analyte-suscep-tible beam energy-angle parameter pairs.2

In performing the detection task it is first useful topredict Mueller elements of scattering over a continu-ous range of parameters that are sensitive to theanalyte. We selected the full wave scattering theoryreported by Bahar3-5 and Bahar and Fitzwater6 ,7 forour model and analyzed its results in a multidimen-sional graphics format that succinctly presents a fieldof Mueller elements, with scattering topography andpolarized beam parameters, that best contrast targetfrom background. The information generated in atypical animation of six elements by this model overgeometric variations of the scatterer (specular todiffuse) over the 9.0-12.5-pum midinfrared beam wave-length band (i.e., the optical bandwidth of a phase-sensitive detection instrument built here) and overoblique-to-normal backscattering angles can exceedgigabytes of computer storage memory. Significanteffects embedded in such large databases can be

6264 APPLIED OPTICS / Vol. 32, No. 31 / 1 November 1993

visualized by computer algorithm. We set out todevelop such codes and better comprehend scatteringby an absorbing and geometrically random surface.

In Section 2 we discuss the assumptions and condi-tions imposed by full wave theory on the scatteringmodel and how the Mueller elements are computed.In Section 3 a sample of model data is graphicallypresented, and in Section 4 we conclude with adiscussion of future direction for theory and visualiza-tion software.

2. Computing Scattering by Arbitrary Surfaces with aFull Wave Model

The phrase full wave is used in the literature byBahar3-5 and Bahar and Fitzwater6 ,7 and relates to atheory of scattering of electromagnetic plane wavesfrom statistically modeled rough surfaces of relativepermittivity Er(X), where is the free-space wave-length of the incident radiation. A claim is made bythis theory that physical optics and perturbationmethods are combined in such a way that duality andreciprocity relationships are satisfied. This resultprovides electromagnetic field solutions expressed interms of vertically and horizontally polarized wavesand evanescent waves. It differs from other effective-medium and multiple-scale surface roughness modelsin how initial conditions are imposed. The theoryreportedly predicts cross-polarized and like-polarizedelectromagnetic fields from both specular point scat-tering and diffuse Bragg-type scattering in a self-consistent manner. Furthermore, the theory claimsthat a priori selection of multiple roughness scaleparameters to define the scattering surface geometryis not required in the solution of Maxwell's equations.

The first task was to write a numerical implementa-tion program of full wave theory. This program,which we named RETRO, computes six elements of theMueller matrix that are nonzero from field relationsderived by the model and arranges their values in adata file containing a description of the scatterer, itscomplex refractive index N _ n(X) - ik(X) = Er"(X)mean-squared slope (US2 ) and height ((h2)) (in Gauss-ian and other distributions), incident beam wave-length (X), and backscattering angle (0).

The following clarifies some notational differencesbetween what is published in the literatures and whatwas programmed in the model.

(a) The surface boundary is defined asy - h(x, z),where (h) = 0.

(b) The reference plane is defined asy = 0.(c) Only backscattering is considered, namely, the

incident and final field directions lie on the same lineso that Oi - - Of = 0 (the plane of incidence is assumedto be the x-y plane and Oi is the angle between thedirection of the wave and the normal to the referenceplane).

(d) The mean-squared height of the surface is(h2 ).

(e) The mean-squared slope of the surface is US2 =

(h, 2 + hz2), hx = ah/Ox, hz = ah/Oz.

(f) The correlation length of the surface, lc isdefined by l2 4(h 2 )/rS 2 .

(g) The autocorrelation function of the surfaceheights is rhh'(Xd, Yd) = (hh')/(h2 ), where h = h(x, z),h = h(x', z'), Xd = X - X', Zd = Z - z'.

The Mueller matrix AF' reported by Bahar3 wasderived from modified Stokes vectors. Our modelRETRO(3) code transforms this into the more desiredMueller matrix SF representing the standard notationof a Stokes vector. The notational difference is asfollows. Given IM as the Stokes vector in the modi-fied notation (Bahar) and Is in the standard notation:

IM = S3

S4

then IM =same light:

So So + 1SIS m =

S2 S2

S3 S 3

RFIM and Is' = FIs must represent the

SF= 'W,

where

1

1

j = 0

0

1 0 0

-1 0 0

0 00 ,0 0 1

/2

0

0

1/2 0 0

-1/2 0 00 1 0

0 0 1

A. Assumptions Built into the Model Code

This RETRO(3) version of full wave theory has two setsof assumptions imposed: process and surface. Pro-cess assumptions address how the light is scattered,for example, how many times it strikes the surface,how the emitter and receiver are oriented, and howdiffuse the scattered light is. The model code assertssingle-scattering processes only; light measured atthe detector struck the rough surface exactly onceand multiple scattering is not considered. This al-lows a second-order iterative full wave solution thatplaces a limit on how rough the surface can be.Other process assumptions imply that the light sourceand detector are on the same optical path with thesame orientation (namely, we consider only backscat-tering), the Mueller elements are calculated on a persolid unit angle basis, and the scattered radiation istotally diffuse. This latest statement means that thecoherent portion of the scattered light return (thereturn without the surface roughness, times a con-stant) is null. That limits how smooth the surfacecan be.

Surface assumptions deal with the statistical repre-sentation of the surface heights and slopes. Thismodel assumes an isotropic scattering surface, i.e.,scattering is independent of the rotational or transla-tional position of the rough surface (invariant to arotation or translation of the x and z axes). In

1 November 1993 / Vol. 32, No. 31 / APPLIED OPTICS 6265

Bahar's formalism this means rhh'(Xd, Zd) -> rhh'(rd),rd2 = Xd2 + Zd2. The surface must also be uniformand that implies that the probability density of theheights and slopes are independent, p(h, ho, hz) >Ph(h)ps(hX, h5). Moreover, Ph and Ps are, respec-tively, Gaussian and jointly Gaussian probabilitydensity functions.

To meet the single-scattering restriction US2 < 2and to satisfy the condition of purely diffuse scatter-ing 4(h2 )ko2 >> 1, where ko 2( rr/X).

B. Consequences of Assumptions

Under these conditions and assumptions the scatter-ing Mueller matrix SF' from Bahar's expressions3-7 isreduced to the following:

UV

hvIhv

SF' =0

0

vh

hh

hh

0

0

hh h

0 R1+1vu vh

0

0

h{

Im IUV

hh hh u0 -Im(I} Re I- z/v v {V Vh

where

k1l Di jDkl Q 2k P2 psdhxdh_,mi( -- Y

Q = 2ko 27 {exp[v' 2(h 2)(1 - rhh')]2

- exp(-v2 (h ))}

X Jo(vrd)rddrd. (3)

The normalized scattering cross sections are givenby Eqs. (2) and (3) from which the polarization-dependent scattering coefficients DLi and Dk1* (theasterisk denotes complex conjugation), shadow func-tion P2, scattering surface slope density function p,and geometric parameters vy, v., rd, rhh', h,,Y, ay, and nare derived elsewhere.2 3 Note that the process andsurface assumptions imposed by this model reduceSF' to eight nonzero.

We finally note Bahar's use of three statistics3-7 forcomputing an autocorrelation function rhh' when arandomly rough surface is characterized. In thismodel they are written as Gaussian, N = 8, andN= 6.

For the Gaussian autocorrelation function

( = rd2)rhh'(rd) = exp -2 (4)Jci

For the N = 8 autocorrelation functionrhh'(rd) -8+32 + 3072] )

t2 4 96_4Ko6

0.4Us2

4 = rdK8, K8 (h2) (5)

For the N = 6 autocorrelation function

rhh'(rd) = 1 - -_CK(C) + 1+

(TS2

C rdK6, K6 (h 2) (6)

C. Visualization and Animation of the Computed MuellerElements

We begin our analysis by compressing data from the(1) full wave model and rearranging it into a binary file.

The binary file is then programmed to map theMueller elements in color code to the viewing cathoderay tube in which a scroll option allows interactivedata presentation (animate). This latter programmanages the display through a code developed by oneof us (Schmitt).

A menu is used to select the type of data presenta-tion. The following lists a meaning of the menuchoices shown in Fig. 1:

SigmaThis is a slider bar that permits selection of any of

the available US 2 parameters.Prev Sigma

This is a pushbutton that changes the sigma param-eter to the next lower available value.Next Sigma

This is a pushbutton that changes the sigma param-eter to the next higher available value.Msht

This is a slider bar that allows selection of any ofthe available (h2) parameters.Prev Msht

This is a pushbutton that changes the Msht param-eter to the next lower available value.Next Msht

This is a pushbutton that changes the Msht param-eter to the next higher available value.Expand

Small variations in ranges of the US2 and (h2)parameters are sometimes imperceptible in the colormappings. The expand slider bar allows the choiceof a scale factor for the Mueller matrix data. Afterthe values are multiplied by the scale factor, the colormap range is amplified.Movie Sigma

This is a pushbutton that begins a movie sequence,varying US2 from the lowest value to the highest

6266 APPLIED OPTICS / Vol. 32, No. 31 / 1 November 1993

Fig. 1. Backscattering (horizontal axis in degrees) infrared (vertical axis in micrometers) predictions by full theory of diagonal and twooff-diagonal Mueller matrix elements of an absorbing DMMP liquid layer on a randomly rough soil substrate. The coating is optically thickwith 10-pm2 mean-squared height and 0.20 mean-squared slope in the top row of elements, 10 pLm2 and 0.70 in the bottom row. Wecomputed these data by using the Gaussian autocorrelation function of Eq. (4). (Using N 6 or N = 8 statistics does not affect the resultsin any appreciable way.) Molecular absorption in DMMP maximizes at wavelengths listed in Table 1 and is shown in k(X) of Fig. 2.

value. For each US2 value both the upper and lowerscreen displays will be updated. The upper andlower screen displays will usually have different (h2)values for comparison.Movie Msht

This is a pushbutton that begins a movie sequence,varying (h2) from the lowest value to the highestvalue. For each (h2) value both the upper and lowerscreen displays will be updated. The upper andlower screen displays will usually have different US2

values for comparison.Copy Down

This is a pushbutton that copies the informationappearing on the top of the display to the bottom ofthe display. The use of two display areas allowsdirect comparison of elements for different parametersets.Double Buffer

This turns on or off the double buffering option.On some display terminals this can significantlyspeed up the presentation of data.Quit

The quit button is used to exit from the program.Control is returned to the Unix shell.

3. Results

We present here a typical result from execution of themodel program with parameters representing scatter-ing by a slab soil surface coated by a well-knownsimulant of liquid chemical G-agent: dimethylmeth-ylphosphonate (DMMP). Experimental values of thischemical's real and imaginary index of refraction arepassed from the parameters file to the model code.These IR index measurements were performed byQuerry8 and, with his permission, we reproduce thespectra in Fig. 2. We assume that the DMMP coat-ing on the soil substrate is optically thick, i.e., scatter-ing only by the top liquid layer taking on the soilsurface profile. Other input parameters used ingenerating the output RETRO(3) data file of Muellerelements for this surface structure include surfacemean-squared slopes ranging from 0.05 to 2.0, mean-squared heights from 10 to 1000 jim2, a beam wave-length interval of 9.0 < X < 12.5 jim, and backscatter-ing angle varying from oblique incidence (88) tonormal incidence (00). One complete computer runof the program produced a data file with 156,000Mueller matrix records. (A SUN Microsystems com-

1 November 1993 / Vol. 32, No. 31 / APPLIED OPTICS 6267

Table 1. Infrared Wavelengths of Relative Intense Absorption and TheirVibrational Mode Assignments for the Chemical Analyte DMMP

Major Vibrational Bands

Contaminant Wavelength (pm) Assignment

CH3 PO(OCH3 )2 9.43 (C-O-P-O-C) stretchDMMP 9.70 (C-O-P-O-C) stretch

10.94 (P-CH3 ) rock12.21 (P-C) stretch

puter system required more than a month of continu-ous computation at 95% CPU capacity to produce asingle output file.)

Figure 1 shows the result. The normalized Muel-ler element amplitudes (all elements except [1, 1]) arebounded between -1.0 and +1.0 and displayed aspixels ranging from red (-1.0) to green (0.0) to blue(+ 1.0) contiguous color shades. The color bar gaugesintensity and polarity of the element signatures.Several observations are made from these and otherdata: (1) an expected result; the rougher the surface(greater spatial mean slope US

2) the more diffuse it is a

reflector of the IR beam, (2) variations in mean-squared height of the analyte coating, as high as 10001Im

2 , was insensitive to all elements, and (3) a not sointuitive result; on vibrational resonance of the ana-lyte, off-diagonal elements [1, 2] and [3, 4] have peakextremum intensities at 30° beam incidence. Thissignal propagates through a polarity reversal as thesurface slope parameter US2 increases. In Fig. 1 wecaptured frames of Mueller elements in the top and

2.5-

2-2.5 -

RefractiveIndex

0.5 -A

9 10 11 12Wavelength (>Lm)

Fig. 2. Infrared refractive-index measurements N(X) = n(X) -

ik(A) of liquid analyte DMMP. The real and imaginary parts areshown as the top and bottom curves, respectively, and molecularvibrational modes are identifed in Table 1. These data arereproduced with the permission of M. Querry.

bottom rows that show the diagonal elements inpolarity transition. Through the animation se-quences we also observe a broadening and a slightshift of these off-diagonal Mueller element extremestoward oblique incidence with increasing US2 .

In addition to these data we made Mueller elementcalculations from contaminant coatings of diisopropyl-methylphosphonate, tributylphosphate, dimethylsi-loxane, diethylsulfide, Diesel fuel, fogoil, soot, andthree fatty acids.9 We found distinct spectral struc-ture yet similar patterns in the animated Muellerelements for all coating compounds.

4. Conclusion and Future Directions

With the rapid advances being made in computertechnology, performing lengthy and complicated com-putations on workstations (the size of a filing cabinet)while graphically analyzing the results interactivelyis practical and becoming more affordable. Whenproperly developed and applied, visualization algo-rithms are invaluable aids in comprehending thesignificant informational content in massive theoreti-cal and experimental databases.

In this problem polarized light scattering wasapplied to a remote-sensing problem: detecting IR-active chemicals (analytes) that wet terrestrial andmanufactured surfaces (background or interferentscatterer). We visualized from a full wave scatteringmodel expected behavior, i.e., a backscattered inci-dent beam spatially diffuses as roughness of thescatterer increases. We also observed patterns inthe Mueller scattering elements that are not soobvious. For example, when plotted in the angle-wavelength plane, element signals S 12 and S 34 in allabsorbing media show up on vibrational resonance ofthe scatterer away from normal incidence, broaden,and shift slightly toward oblique backscattering angle(away from normal incidence) as the surface mean-squared slope increases to an upper limit. The mostintriguing result uncovered was a polarity reversal inthese off-diagonal Mueller elements on resonance ofthe scatterer by variation of surface slope.

There are several areas for which revisions intheory and enhancements in graphical analysis code(or new software) can be developed and may proverewarding. The assumptions built into this RETROmodel limits scattering from restrictive surfaces thatare not realistic in remote sensing of the complexcontaminated terrains (the real world). However,this model is progressing to describe scattering fromsurface structures that are anisotropic. (The secondand fourth quadrants of elements of the Muellermatrix are not necessarily zero for many materials ofinterest.) Polarized scattering by stratified, aniso-tropic, and chiral material is now being studied forpurposes of biological detection. One approach formodeling chiral materials is to characterize theirdielectric relative permittivity by a tensor, ratherthan a scalar, at resonance beam energies. (TheMueller elements [1, 4] and [4, 1] transform the hand-edness of circular polarizations and are, when the

6268 APPLIED OPTICS / Vol. 32, No. 31 / 1 November 1993

beam irradiation is tuned to a specific vibrationalenergy, good candidate vibrational circular dichroismfeatures for detecting biological organisms that arechiral.) Also, the question of multiple scatteringmust be addressed.

In a future scattering model we expect to accessrefractive-index data not from experimental measure-ment but rather from quantum mechanics calcula-tions on molecular structures alone. We can predictthe analyte's absorption frequencies and intensitiesto 10% accuracy or better by optimizing the molecularmodel quantum code.10 The molecular model thatperforms these ab initio calculations uses a Hartree-Fock level of theory with a finite basis of Gaussian-type wave functions. 0 The least-energy geometricconfigurations of the molecule are computed thatyield the absorption intensities. For the analytemedium containing many, many molecules we broadenthese predicted spectra in a Lorentzian distribution.Once absorption (the imaginary index of refraction) ofthe analyte is computed over a wide spectral band,Kramers-Kronig relations compute the real index ofrefraction and then total index values are passed tothe macroscopic scattering code.

This work was supported by the U.S. Army Chemi-cal and Biological Defense Agency, Edgewood Re-search, Development, and Engineering Center(ERDEC), under the auspices of an In-House Labora-tory Independent Research grant. We express ourthanks to M. Querry (University of Missouri) and M.Milham (ERDEC) for use of measurements on therefractive indices of chemical and mineral com-pounds, to E. Bahar (University of Nebraska) fordiscussions and enlightenment on full wave theory, toJ. Bottiger (ERDEC) for many helpful discussions onpolarized scattering, and to Ronald Piffath (ERDEC)for helping us interpret IR spectra.

References and Notes1. A. H. Carrieri, J. R. Bottiger, D. J. Owens, C. E. Henry, C. M.

Herzinger, S. M. Haugland, J. 0. Jensen, K. E. Schmidt, andJ. L. Jensen, "Mid infrared polarized light scattering: applica-tions for the remote detection of chemical and biologicalcontaminations," Internal Report CRDEC-TR-318 (U.S. ArmyChemical Research, Development, and Engineering Center,Aberdeen Proving Ground, Md., 1992).

2. S. M. Haugland, E. Z. Bahar, and A. H. Carrieri, "Identifica-tion of contaminant coatings over rough surfaces using polar-ized IR scattering," Appl. Opt. 31, 3847-3852 (1992).

3. E. Bahar, "Review of the full wave solutions for rough surfacescattering and depolarization: comparisons with geometricand physical optics, perturbation, and two-scale hybrid solu-tions," J. Geophys. Res. 92, 5209-5224 (1987).

4. E. Bahar, "Full-wave solutions for the depolarization of thescattered radiation fields by rough surfaces of arbitrary slope,"IEEE Trans. Antennas Propag AP-29, 443-454 (1981).

5. E. Bahar, "Scattering and depolarization of electromagneticwaves-full wave solutions," Internal Report RADC-TR-83-118 (Rome Air Development Center, Air Force Systems Com-mand, Griffiss Air Force Base, N.Y., 1983).

6. E. Bahar and M. A. Fitzwater, "Scattering cross sections forcomposite rough surfaces using the unified full wave approach,"IEEE Trans. Antennas Propag. AP-32, 730-734 (1984).

7. E. Bahar and M. A. Fitzwater, "Copolarized and cross-polarized incoherent specific intensities for waves at obliqueincidence upon a layer of finitely conducting particles withrough surfaces," J. Opt. Soc. Am. A 4, 41-56 (1987).

8. M. R. Querry, "Optical constants of minerals and othermaterials from the millimeter to the ultraviolet," InternalReport CRDEC-CR-88009 (U.S. Army Chemical Research,Development, and Engineering Center, Aberdeen ProvingGround, Md., 1987); M. R. Querry, Department of Physics,University of Missouri-Kansas City, Kansas City, Mo., andM. E. Milham, U. S. Army Edgewood Research, Development,and Engineering Center, Aberdeen Proving Ground, Md.(personal communication, 1986).

9. R. Piffath, U.S. Army Edgewood Research, Development, andEngineering Center, Aberdeen Proving Ground, Md. (personalcommunication, 1988).

10. H. F. Hameka, A. H. Carrieri, and J. 0. Jensen, "Calculationsof the structure and the vibrational infrared frequencies ofsome methylphosphonates," Phosph. Sulfur Silicon RelatedElem. 66, 1-11 (1992).

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