effect of surface roughness and complex indices of refraction on polarized thermal emission

$: Effect of surface roughness and complex indices of refraction on polarized thermal emission$
Effect of surface roughness and complex indicesof refraction on polarized thermal emission

Kristan P. Gurton and Rachid Dahmani

We present a series of measurements characterizing the dependence of polarized thermal emission onsurface roughness. In particular, we measure the spectrally resolved degree of linear polarization (DOLP)for a series of roughened borosilicate (Pyrex) glass substrates as a function of the roughness parameterRa, the root-mean-square slope distribution, and observation angle �. Also measured are a series ofsmooth glass substrates coated with two particular polymers of interest, i.e., a common commerciallyavailable Krylon paint and a chemical-agent-resistant coating paint. The DOLP is measured over a4–13 �m wave band by using a modified Fourier transform IR spectrometer in which a wire-grid polarizerand a quarter-wave Fresnel rhomb are used in conjunction to measure all four Stokes parameters. Inaddition, we show an enhanced DOLP due to anomalous dispersion exhibited by the surface material. ©2005 Optical Society of America

OCIS codes: 120.2130, 120.5410, 110.3080, 240.5770.

1. Introduction

It is well-known that unpolarized light can becomepartially linearly polarized upon reflection from aplanar smooth surface. A similar process occurs forblackbody radiation emitted from such surfaces. As aresult, both types of polarization are often present inthe recorded radiation emitted from a thermal objectand are functionally dependent on from which direc-tion the object is viewed.1,2 The induced polarizedemission originates from differences in orthogonalcomponents of the directional spectral emissivity,��, T, �, ��.3 Here � represents the wavelength, T isthe temperature, � is the azimuthal angle, and � isthe angle between the surface normal n̂ and the de-tector’s line of sight at a given point on the surface.

Because many man-made objects tend to exhibitsurfaces that are smooth when compared with com-mon naturally occurring materials, e.g., soil, vegeta-tion, trees, etc., differences in the polarized thermalemission are present. These differences in partially

polarized emission can be exploited by polarimetricthermal imaging.4–6 By examining the polarizationstate of the image forming thermal radiation, one canimprove discrimination of objects that are normallydifficult to resolve because of the naturally occurringbackground clutter.

Consider a planar surface bound by air with direc-tional spectral emissivity ��, T, �, ��. By consideringthe steady-state form of Kirchhoff’s law, we can ex-press directional spectral emissivity in terms of thedirectional spectral absorptivity ��, T, �, �� and re-flectivity r��, T, �, ��, i.e.,

��, T, �, �� , T, �, �� 1 � r��, T, �, ��.(1)

The total emissivity is expressed in terms of two or-thogonal components, �� and ��, thus restricting theazimuthal dependence, �:

��, T, �� , T, �� , T, ��

2 , (2)

where the symbols � and �� indicate the directionsperpendicular and parallel to the plane of emission,i.e., the plane defined by the surface normal and theline of sight. Similarly, by expressing the absorptivityand the reflectivity in terms of their orthogonal com-ponents, we find that �� and �� can be expressed as

Kristan Gurton ([email protected]) and Rachid Dahmani([email protected]) are with the U.S. Army Research Labo-ratory, 2800 Powder Mill Road, Adelphi, Maryland 20783-1145.

Received 13 August 2004; revised manuscript received 10 March2005; accepted 13 March 2005.

0003-6935/05/265361-07$15.00/0© 2005 Optical Society of America

10 September 2005 � Vol. 44, No. 26 � APPLIED OPTICS 5361

��, T, �� , T, �� 1 � r��, T, ��, (3a)

��, T, �� , T, �� 1 � r��, T, ��, (3b)

Let us now define a degree of linear polarization,DOLP��, ��, for emission polarization as the differ-ence between the orthogonal emissivities divided bythe total emittance. We find that

DOLP��, �� , �� , ��, �� , ��

, (4)

where we have omitted the temperature dependencesfor brevity. By representing the reflectivity in termsof Fresnel coefficients, we find two representationsfor the orthogonal components of the emissivity, de-pending on magnitude of the optical absorption forthe emitting surface.3

By defining the complex refractive index of the emit-ting surface as n�� ik��, we find for surfaces withnegligible absorption, i.e., k�� 0, that the orthog-onal components of the directional emissivity are

��, �� 1 ��n��2 � sin2 ��1�2 � cos �

�n��2 � sin2 ��1�2 cos ��2

, (5a)

��, �� 1 ��n��2 cos � � �n��2 � sin2 ��1�2

n��2 cos � �n��2 � sin2 ��1�2�2

.

(5b)

For moderate to highly absorbing surfaces wherek�� 0, e.g., representative of most metals in the IR,we find that

��, �� 4n�� cos �

cos2 � 2n��cos � n2�� k2��, (6a)

��, �� 4n�� cos �

�n2�� k2��cos2 � 2n�� cos � 1.

(6b)

In both cases, we have assumed air to be the bound-ing medium with the surface whose complex refrac-tive index is taken to be approximately 1.

Equations (4)–(6) are valid only for smooth homo-geneous surfaces that are contamination free. As aresult, the predicted DOLP expressed by Eq. (4) is anideal case. Minor surface contaminations and defectscan result in great differences between the measuredand the calculated DOLP.

Various models have been proposed for the par-tially polarized emission when the surface is rough-ened or contaminated by small particles. Earlymethods were based on surface-scatter models inwhich results from bidirectional reflection measure-ments were modified in an attempt to describe polar-

ized emittance as a function of surface-roughnessparameters.7,8 Jordan and Lewis first reported bothmeasured and calculated partially polarized thermalemission over the wave band of 10–11 �m for a seriesof roughened glass and aluminum substrates as afunction of surface orientation and a measured root-mean-square (rms) slope distribution.9,10 They pre-sented a modified Fresnel model that effectivelyweighted the predicted emissivity relative to the nor-malized rms slope distribution. They reported goodagreement between the measured and the calculatedDOLP for light to moderately roughened surfaces,but agreement was less certain as the roughenedfacets became larger.

Others have further developed the slope-weightedFresnel approach by including facet shadowing andmultiple scattering.11–13 In a manner similar to Jor-dan, they treat the roughened surface as a series ofinfinitesimally small planar facets and integrate overall possible orientations described by the rms slopedistribution.

Recently we measured and modeled the effects ofsurface contaminates on the degree of polarized emit-tance.14,15 Regardless of the type of surface anomaly,most geometric-based models describe the attenua-tion of polarized emission as a reduction in the effec-tive projected area when compared with the idealcase described by Fresnel equations.

To better understand how polarized thermal emis-sion is affected by various surface conditions, a seriesof spectral IR polarimetric measurements have beenconducted on a variety of substrates. In particular,the DOLP was measured at various angles ofobservation for a variety of well-characterized boro-silicate glass substrates that were roughened to vary-ing degrees by sandblasting.

To spectrally resolve the measured DOLP over thewavelength region of 4–13 �m, we used a polari-metrically modified Fourier transform IR interferom-eter16 (FTIR). This allowed for direct measure of thespectrally resolved four Stokes parameters (i.e.,S0, S1, S2, S3) that totally describe the polarizationstate of the thermal emission, as well as the resultantDOLP. All measurements were conducted as a func-tion of surface orientation and two-measured rough-ness metrics, i.e., the surface roughness parameter,Ra, and rms slope distribution.

2. Experiment

Borosilicate glass (Pyrex) plates were chosen as thesample material because of their initial smoothness,readily available indices of refraction, and high emis-sivity value of 0.94. Figure 1 shows the complex in-dices of refraction used in Fresnel calculationsconducted for comparison with measured DOLPvalues.17 Three-inch-square (1 in. � 2.54 cm) glassplates were sandblasted using various types of abra-sives and pressures. The surface-roughness parame-ter Ra and the rms slope m0 were measured for eachplate produced, at a number of different locations, byuse of a Tencor surface profilometer equipped with a5 �m radius diamond tip stylus. Profilometer param-

5362 APPLIED OPTICS � Vol. 44, No. 26 � 10 September 2005

eters used during the measurement include a stylusloading force and speed of 2 mg and 5 �m�s, respec-tively, for a sample length of 100 �m. Because of thestructure of the glass, a large range of Ra values werenot achieved. Although over a dozen plates were pro-duced, only four glass plates were studied, since theyspanned the limited range of Ra values produced, i.e.,0.007 (smooth), 3.98, 4.80, and 6.68 �m (very rough).The corresponding rms slopes were 9.76°, 8.60°, and11.90°, respectively. Figure 2 shows the respectivenormalized rms slope distributions where a Gaussianform was assumed for each.

Glass substrates were mounted to a brass heatingelement that rotates through the angular range 0°� � � 90°, which varies the angle of emission col-lected by the polarimeter, where � represents theangle between the surface normal and the line ofsight as shown in Fig 3. Thermal contact between theglass plate and the brass heating element was main-tained by a thin layer of a heat-sink compound.

The spectropolarimetric measurement was con-ducted with a modified FTIR spectrometer.18,19 Asimple diagram showing the various components ofthe experiment is seen in Fig. 3. An approximate1 cm2 region of the heated substrate was subtendedby an f�10 BaF2 lens as shown. The resultant radia-tion was then collimated and polarimetrically ana-lyzed by using an IR wire-grid polarizer (with an

extinction ratio of approximately 170) in conjunctionwith a modified ZnSe Fresnel quarter-wave rhombpurchased from II-VI Inc. Since IR retarders are no-torious for being highly dispersive and may vary sig-nificantly from the vender’s specifications, the rhombwas carefully calibrated by a rotating sample methodbetween parallel polarizers at IR laser wavelengths,i.e., 3.49, 5.20, and 10.60 �m.20 Since the dispersionof ZnSe is small in the IR region of interest, thevariation in the retardance is small. Because of this,rhombs fabricated from low-dispersive materialshave a distinct advantage over other IR retarder de-signs, e.g., reflective phase retarders and zero-multiple-order cadmium sulfide (CdS) wave plates.

Polarized radiation from the heated plates was cou-pled into a Bomem FTIR spectrometer by using anf�7 BaF2 lens that matched the entrance pupil of thespectrometer. This polarimetric configuration allowsfor analysis of linear and circular polarized compo-nents necessary to calculate all four Stokes vectors aswell as the DOLP.

Aberrations that would affect the polarization statewere kept to a minimum by ensuring that all anglesbetween the refracted rays and optical elements werekept to less than 20°. This condition is easily achievedfor the polarimetric collection optics but is less cer-tain for the internal beam path of the FTIR. It wasconcluded that because of the ideal nature of theanalyzer mounted prior to the FTIR, i.e., extinctionratio 170, polarimetric effects induced by the de-vice were minor. Nevertheless we found a slightasymmetry in the polarimetric response of the FTIRand the 5–6 �m region when unpolarized radiancewas measured. We attribute this relatively small po-larimetric aberration to dispersion in the BaF2 wire-grid polarizer, which was most evident in the 5–6 �mregion. As a result, a correction was produced andapplied to all spectropolarimetric data measured.

The measurement was well shielded from ambientthermal radiation generated within the laboratory byplacing numerous cold shields around the experi-ment, as shown in Fig. 3.

Target plates were heated to approximately 65 °C,where surface temperatures were monitored with anembedded thermocouple. This temperature wasfound sufficient to produce a good signal-to-noise ra-tio. Each sample plate was oriented at a particularangle within a 0° � � � 80° range, positioned at 10°increments. Each sample plate was set at a given

Fig. 1. Complex indices of refraction used for the borosilicate(Pyrex) glass plates.

Fig. 2. Normalized rms slope distributions measured for thethree roughened borosilicate (Pyrex) glass substrates (Gaussiandistribution assumed).

Fig. 3. Simple schematic showing the experimental setup usedduring the surface-roughness spectropolarimetric measurement.


angle while the Fresnel rhomb and wire-grid polar-izer were oriented in positions to analyze in sequencefour of the linear states, 0°, 90°, and �45° (relative tothe vertical), and either right- or left-handed circularstates. Because no fine structure was expected in thepolarimetric spectra, the FTIR was operated at therelatively modest spectral resolution of 8 cm�1, and aBartlett apodization function was applied to each in-terferogram. The resultant interferograms were fi-nally converted to an emission spectrum by applyinga conventional Fourier transform. Six resultant po-larimetric spectra were then generated, one repre-senting each of the polarization states sampled, i.e.,I�0°�, I�90°�, I�45°�, I�135°�, I�R�, and I�L�. Once acomplete set of polarimetric spectra was measured,the sample orientation angle was rotated to the nextangle, and the Stokes spectra measurement wasrepeated.

The Stokes vectors that totally describe the polar-ization state of the thermal emission were calculatedby using the following equations:

S0 � I�0°� I�90°�, (7)

where S1, S2, and S3 are defined as

S1 � I�0°� � I�90°�, (8)

S2 � I�45°� � I�135°�, (9)

S3 � I�R� � I�L�, (10)

and I�0°�, I�90°�, I�45°�, I�135°�, I�R�, and I�L� repre-sent the aforementioned polarimetric spectra.

Additionally, we calculated the DOLP and degreeof circular polarization (DOCP) by using the followingrelations:

DOLP �S1

2 S22

S0, (11)

and

DOCP � S3�S0. (12)

However, we should note that, for all samples mea-sured, both S2 and S3 were found to be zero within thenoise inherent in the measurement, and, as a result,the DOCP was also considered to be zero.

3. Results

We first measured a smooth borosilicate glass sub-strate. Figure 4 shows the measured spectral DOLPas a function of angle of emission � for the sampleheld at a temperature of 65 °C. Also shown in Fig. 4is a series of Fresnel-based calculations using Eqs. (4)and (6) for 9.28 �m (see triangular data points). Theagreement between the measured DOLP and theFresnel model was quite good. Fresnel calculations atother wavelengths showed similar agreement. The

negative values in Fig. 4 are a result of noise inherentin the measurement. This is especially apparent atlow-signal levels, and, as a result, values for theDOLP were taken to be effectively zero. Figures 5–7show a series of DOLP measurements for the threeroughened plates. The measured DOLP steadily de-clines as the glass substrates become progressivelyrougher.

We explore the strong spectral dependence of theDOLP shown in Figs. 4–7 by examining what hap-pens when the same glass substrates are paintedwith a thin dielectric layer. We measured two types ofcoating: (1) a common commercially available Krylonblack spray paint and (2) a chemical-agent-resistantcoating (CARC) paint that is produced from a multi-component epoxy resin. In applying the paints, wecoated the surfaces with relatively thick layers toreproduce typical field conditions.

The painted plates were placed in the brass heatingelement and warmed to a temperature of 65 °C, andthe same procedure as described above was repeated.

Fig. 4. Measured DOLP as a function of wavelength and detec-tion angle for a smooth borosilicate glass substrate heated to 65 °C.Triangles are the corresponding calculated DOLP for an emissionwavelength of 9.28 �m based on the Fresnel relations of Eqs. (4)–(6).

Fig. 5. Measured DOLP for a lightly roughened borosilicate platewith Ra � 3.98 �m and rms slope � 9.67° as a function of wave-length and detection angle. The plate temperature was held at65 °C.


Figures 8 and 9 show the measured DOLP for the twosmooth painted plates. Both figures show that thestrong spectral characteristics exhibited by the un-coated glass plates seen in Figs. 4–7 were replaced bya nearly featureless spectral polarized response forboth paints. As a result, the spectral polarimetriccharacteristics of the paint–substrate combinationseen in Figs. 8 and 9 are primarily a function of thepigment only. This obvious result highlights the factthat polarized thermal emission is solely a surface-generated phenomenon.

4. Discussion

We intended to create a set of roughened glass sur-faces that spanned as large of a range as possible.However, the profilometer measurements indicated alimited range of Ra and rms slope, i.e., Ra from 3.98to 6.68 �m and rms slope from 8.60° to 11.90°. Weattribute this inability to generate a more diverse set oftest surfaces to microfracture characteristics of amor-phous glass plates. Jordan and Lewis experienced sim-ilar limitations in their glass and aluminum samples.

Although the variance among roughness parameterand the rms slope appear small, the test samples cre-ated showed significant differences based on visualand tactile inspection. The rms slope distribution isthought by many theorists to be the most importantparameter influencing the attenuation of polarizedemission due to surface roughness; in practice, accu-rate determination of this parameter is quite problem-atic. Bennett et al. reported that, even under the bestof conditions, measurement of the rms slope can varyby as much as a factor of 50 from one practitioner toanother, depending on the instrumentation used, cut-off length, profilometer stylus geometry, scan rate,etc.21 Much of the emphasis placed on the use of therms slope is based on the premise that roughened sur-face facets may be treated as a series of infinitesimallysmall planar surfaces oriented at various angles de-scribed by the rms slope distribution. Even if goodreproducible rms slope values exist, researchers mustthen assume a distribution to apply the rms. For ex-ample, Jordan and Lewis reported significant variancein the calculated DOLP for a series of roughened sur-

Fig. 7. Measured DOLP for a very rough borosilicate plate withRa � 6.68 �m and rms slope � 11.90° as a function of wavelengthand detection angle. The plate temperature was held at 65 °C.

Fig. 6. Measured DOLP for a moderately roughened borosilicateplate with Ra � 4.80 �m and rms slope � 8.60° as a function ofwavelength and detection angle. The plate temperature was heldat 65 °C.

Fig. 8. Measured DOLP as a function of wavelength and detec-tion angle for a smooth borosilicate plate coated with a thin layerof black Krylon paint. The plate temperature was held at 65 °C.

Fig. 9. Measured DOLP as a function of wavelength and detec-tion angle for a smooth borosilicate plate coated with a thin layerof a green CARC paint. The plate temperature was held at 65 °C.


faces depending on whether a Gaussian or Cauchydistribution was applied.9 Regardless of the difficultyinherent in measuring an accurate surface profile, it isnot clear that a simple faceted model is valid when thedimensions of the facets are comparable with orsmaller than the emission wavelength, as is the casehere.

Figure 10 shows a comparison of measured DOLPvalues taken at the emission wavelength of 9.90 �mfor several values of surface roughness as a functionof orientation. Also shown are the DOLP results ofJordan and Lewis for their roughened glass sub-strate, Ra � 9.89 �m and rms slope � 24.5°. Oneinteresting feature seen in Fig. 10 is that with theexception of a few data points for the Ra � 3.98 �msample, the measured functional dependence ap-pears very similar for all roughened samples consid-ered regardless of surface roughness or rms slope. Weexpect different polarimetric responses as the sub-strates transition from a specular to a diffuse surfaceas the roughened facets become larger. We speculatethat the insensitivity to various surface features isdue to an insufficiently large range of Ra values. Thetransition from specular to diffuse surfaces in themeasured DOLP should be evident if Ra values be-tween 0.5–2.0 �m could be examined. An informalsurvey of measured Ra values (measured using ahandheld profilometer) taken from numerous civilianand military vehicles shows a range of Ra values, i.e.,0.02–3.65 �m, where the higher values could beloosely described as quite abrasive.

One of the more interesting aspects highlighted bythe spectrally resolved DOLP shown in Figs. 4–7 isthe strong wavelength dependence that polarizedthermal emission exhibits for certain types of mate-rial. In particular, by examining the refractive-indexdata seen in Fig. 1, we see that the resultant DOLPincreases significantly in regions of anomalous dis-persion, i.e., where dn�d� is positive. The spectral

variation of the complex refractive index will sup-press or enhance the degree to which a material emitspolarized thermal emission. It is apparent from Fig. 4that the DOLP will vary as a function of spectralbandwidth, e.g., a thermal polarimetric imaging sys-tem integrating over a 8.0–9.5 �m region. For Krylonand CARC paint, the polarization spectra are fairlyfeatureless and obscure the polarized emission fromthe heated borosilicate glass substrate. As seen inFigs. 5–7, all roughened surfaces have a measurableDOLP for angles of emission in excess of 40°. Similarresults have been reported elsewhere.9,10

We thank Huey Anderson of the U.S. Army Avia-tion and Missile Command for his continued supportand assistance in conducting this research.

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Fig. 10. Comparison of measured DOLP (taken at 9.90 �m) as afunction of substrate orientation for various roughened glassplates. Also shown are the measured and Fresnel-based calcula-tions for smooth glass substrates (top dashed curve) and resultsfrom Jordan and Lewis (bottom dashed curve).9 LOS, line of sight.


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effect of surface roughness and complex indices of refraction on polarized thermal emission

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