Pattern Recognition Group

Infrared polarization measurements and modeling applied to surface-laid antipersonnellandminesFrank Cremer,MEMBER SPIETNO Physics and Electronics Laboratory P.O.Box96864NL-2509JG,The HagueThe NetherlandsandDelft University of TechnologyPattern Recognition GroupSection of Applied GeophysicsDelft,The NetherlandsE-mail:cremer@fel.tno.nlWim de Jong,MEMBER SPIEKlamer SchutteTNO Physics and Electronics Laboratory P.O.Box96864NL-2509JG,The HagueThe Netherlands Abstract.Linear polarization of thermal infrared(TIR)radiation occurs when radiation is reflected or emitted from a smooth surface(such as the top of a landmine)and observed from a grazing angle.The background (soil and vegetation)is generally much rougher and therefore shows less pronounced linearly polarized radiation.This difference in polarization is utilized to enhance detection of landmines using TIR cameras.A setup has been constructed for the acquisition of polarized TIR images.This setup contains a polarizationfilter that rotates synchronously to the frame sync of the camera.Either a long wave infrared(LWIR)or a mid wave infrared(MWIR)camera can be mounted behind the rotating po-larizationfilter.The synchronization allows a sequence of images to be taken with a predefined constant angle of rotation between the images. Using this image sequence,three independent Stokes images are cal-culated,consisting of the unpolarized radiance,the difference between vertically and horizontally polarized radiances,and the difference be-tween the two diagonally polarized radiances.A model has been devel-oped that describes the polarization due to reflection of and emission from a smooth surface.This model predicts the linear polarization for a landmine‘‘illuminated’’by a source that is either hotter or cooler than the surface of the landmine.The measurement setup has been used to vali-date the model.The measurements agree well with the model predic-tions and can be used for estimating the real part of the refractive index of the rubber surface of the landmine.Besides the indoor measurement, outdoor measurements have also been performed.The results of these measurements show that under the given conditions the majority of land-mines can be observed in the polarized radiance,whereas they are not clearly visible in the normal(unpolarized)radiance or the visual image.©2002Society of Photo-Optical Instrumentation Engineers.[DOI:10.1117/1.1467362] Subject terms:infrared polarization model;measurement setup;measurements; landmine detection.Paper PL-012received Sep.14,2001;revised manuscript received Dec.5,2001; accepted for publication Dec.8,2001.1IntroductionOne of the sensors used to detect landmines is a thermal infrared͑TIR͒camera.Current cameras are able to detect small temperature differences͑as low as15mK͒.Land-mines have different heat conductivity and heat capacity compared to natural backgrounds.Due to these differences in thermal properties,temperature differences between a landmine and the background may develop when the soil is heated or cooled down.The TIR radiation from other sources͑like the sun and the sky͒is reflected by the land-mine and the background.Both emission from the surfaces and reflection of the sources onto these surfaces play a role in the formation of the TIR image.TIR images of landmines in natural scenes contain clut-ter,since other͑natural͒objects like trunks,holes,and rocks also may have different thermal properties compared to the background.In the visual spectrum it is well known that unpolarized light reflected from a smooth surface be-comes polarized.1This is also true for TIR radiation.How-ever,for TIR radiation not only the reflection,but also the emission is polarized.Since in general the surfaces of land-mines are smoother than the surfaces found in a natural background,the presence of significantly polarized TIR ra-diation is an extra indication for landmines͑or other non-natural objects͒.This has been shown in previous work.2,3 In Fig.1,the relation between the various aspects of TIR polarization measurements is shown.The camera observes the scene through a polarizationfilter.The measured radi-ance,consisting of a nonlinear polarized and a polarized part,depends on the temperature of the target͑mine or background͒and the reflection of sources on the target.Currently,not a single sensor is sufficient for the detec-tion of all landmines with an acceptable false-alarm rate.4 Therefore research focuses on multisensor systems,gener-ally consisting of a ground penetrating radar͑GPR͒,a metal detector͑MD͒,and a TIR camera,for improved detection performance.5,6Using the polarization setup,the perfor-mance of the TIR camera can be improved and thus have a1021Opt.Eng.41(5)1021–1032(May2002)0091-3286/2002/$15.00©2002Society of Photo-Optical Instrumentation Engineerslarger contribution to the multisensor system.First,an introduction is given on the background of measuring TIR polarization.Furthermore,a model of TIR polarization will be presented along with the model as-sumptions.For measurements of TIR polarization,the con-structed measurement setup is described.In the next sec-tion,the indoor measurements are described.These measurements are analyzed and compared to model predic-tions.Some outdoor measurements are presented next.Fi-nally,conclusions are presented.2Polarization ModelTo describe the polarization effects,a basic model will be introduced.This model only describes the MWIR ͑3to 5␮m ͒polarization effects on the landmine target illuminated by a single source ͑see Fig.2͒.The polarization of the natural background is not taken into consideration.This model is based on a number of assumptions to simplify the calculations.These assumptions are:1.The material of the landmine can be described by a single refractive index for the wavelength band used ͑MWIR ͒.2.The material of the landmine is opaque,meaning that there is no transmission of radiation through the land-mine.3.The surface of the landmine is specular for reflection and obeys the Kirchoff law for radiation.4.The temperature of the landmine is constant,since the landmine is assumed to be in thermal equilibrium with its surroundings.5.The single source,which is radiating on the land-mine,is unpolarized.6.The spectral sensitivity of the MWIR camera is con-stant throughout the wavelength band,ranging from 3to 5␮m.7.The polarization filter is ideal for the wavelength band.8.The transmission through air is 100%and thus there is no path radiance.The source in Fig.2is assumed to be a blackbody ͑im-plying that the emission coefficient is unity for the full wavelength band ͒at a constant temperature T bb .The radi-ated electromagnetic wave can be decomposed into two perpendicular components called polarizations.One com-ponent vector E ip is in the plane of incidence;this is called parallel polarization,hence the subscript p .The plane of incidence is the plane that is spanned by the propagation direction of the incident radiation and the surface normal.The other component E is is perpendicular to the plane of incidence and is given the subscript s .These two incidence polarizations are equal in magnitude,as the source is as-sumed to be unpolarized.The thermal emission of a blackbody in the MWIR wavelength band as function of the temperature T is given by Planck’s equation 7:I BB ͑T ͒ϭ͵3␮m5␮m 2hc2␭51e͑hc /␭kT ͒Ϫ1d ␭͓W/m 2/sr ͔,͑1͒where ␭is the wavelength,h is Planck’s constant,k is Boltzmann’s constant,and c is the speed of light.The instantaneous electrical field (E rp ,E rs )of the TIR radiation cannot be measured by the TIR camera.The fre-quencies involved are too high to make this measurement possible.Instead,the TIR camera measures the amount of power ͑per unit area ͒incident onto the detector,which is called radiance.For electromagnetic waves with field strength E ,the radiance I is given by:I ϭ⑀a c ␲T͵TE 2dt ͓W/m 2/sr ͔,͑2͒where ⑀a is the dielectric constant of the medium ͑air ͒,and the integration period T is large enough (T ϾϾ␭/c ).The radiance incident on the landmine can be described by two radiance terms,I ip and I is ,that are related to E ip and E is ,respectively.These two incident radiance terms are equalinFig.1Overview of the various aspects of infrared polarization.The camera observes the scene through a polarization filter.The mea-sured radiance and the polarization of this radiance depends on the temperature of the target (landmine or background)and the reflec-tion of sources on thetarget.Fig.2Schematic overview of the different components in the sim-plified polarization model.1022Optical Engineering,Vol.41No.5,May 2002magnitude͑as the source is not polarized͒,and the sum of these terms is equal to the blackbody radiance as given in Eq.͑1͒:I ip͑T bb͒ϭI is͑T bb͒ϭ12I BB͑T bb͒.͑3͒The surface of the landmine in Fig.2is assumed to be specular for reflection.For specular reflection,the angle of incidence␪i is equal to the angle of reflection␪r.To de-termine the reflection coefficients,it is necessary to define the angle of transmission␪t,even though the transmission is assumed to be zero.The relationship between the angle of incidence␪i and the angle of transmission␪t is given by Snell’s law1,8:n0sin͑␪i͒ϭn1sin͑␪t͒,͑4͒where n0is the refractive index of the air and n1ϭnϩik is the refractive index of the landmine͑in complex notation͒. The refractive index of air is assumed to be1,so n0ϭ1. The refractive index of the landmine is unknown,but will be calculated in Sec.5.Part of the radiance from the source is reflected by the landmine’s surface.The amount that is reflected is given by the reflection coefficients,which differ for the two polar-ization orientations.The reflection coefficients,in terms of radiance,are given by1:␳p͑␪i͒ϭI rpI ipϭͯE rp E ipͯ2ϭͯtan͑␪iϪ␪t͒tan͑␪iϩ␪t͒ͯ2,͑5͒␳s͑␪i͒ϭI rsI isϭͯE rs E isͯ2ϭͯϪsin͑␪iϪ␪t͒sin͑iϩ␪tͯ2.͑6͒The remainder of the radiance from the source is either transmitted through or absorbed by the landmine.The transmission through the landmine is assumed to be zero. The theoretical situation in Fig.3is considered for obtain-ing a relationship between the absorption,reflection,and emission coefficients.In this situation a landmine with a specular surface is in thermal equilibrium with a blackbody. The incidence radiance I i(␪I)for angle␪iϭ␪r is given by: I i͑␪i͒ϭI BB͑T bb͒.͑7͒This radiance is partly absorbed and partly reflected by the landmine͑transmission through the landmine is assumed to be zero͒:I i͑␪i͒ϭ␣͑␪i͒I i͑␪i͒ϩ␳͑␪i͒I i͑␪i͒.͑8͒The existence radiance I o(␪i)for the same part of the land-mine consists of the emitted radiation and the reflected ra-diation:I o͑␪i͒ϭ⑀͑␪i͒I BB͑T m͒ϩ␳͑␪i͒I i͑␪i͒.͑9͒Due to thermal equilibrium,the two temperatures T bb and T m must be the same,as do the incoming and outgoing radiances I i(␪i)and I o(␪r).This reduces Eqs.͑8͒and͑9͒to:␣͑␪i͒ϭ⑀͑␪i͒ϭ1Ϫ␳͑␪i͒.͑10͒This relationship is also known as the Kirchoff law of radiation.9,10Without giving proof,these relationships are true for both polarization directions independently.So the emission coefficients for both polarization orientations are given by:⑀p͑␪i͒ϭ1Ϫ␳p͑␪i͒,͑11͒⑀s͑␪i͒ϭ1Ϫ␳s͑␪i͒.͑12͒The radiance originating from the landmine consists of the reflected radiance from the source and the emitted radi-ance due to the temperature T m of the surface of the land-mine:I p͑T m,T bb,␪i͒ϭ12I BB͑T m͒⑀p͑␪i͒ϩ12I BB͑T bb͒␳p͑␪i͒ϭ12I BB͑T m͒ϩ12␳p͑␪i͓͒I BB͑T bb͒ϪI BB͑T m͔͒,͑13͒I s͑T m,T bb,␪i͒ϭ12I BB͑T m͒⑀s͑␪i͒ϩ12I BB͑T bb͒␳s͑␪i͒ϭ12I BB͑T m͒ϩ12␳s͑␪i͓͒I BB͑T bb͒ϪI BB͑T m͔͒.͑14͒Both polarized radiances depend on the difference in radiance between the blackbody and the landmine.This dif-ference can be seen as an effective illumination radiance I e(T bb,T m)incident on the landmine:I e͑T bb,T m͒ϭI BB͑T bb͒ϪI BB͑T m͒.͑15͒This radiance from Eqs.͑13͒and͑14͒passes through an ͑assumed to be͒ideal polarizer and falls subsequently onto the detector array of the camera.The detector of the camera returns a value linearly related to the incoming radiance. This value depends on the orientation of thefilter.For a given angle␸between the principal axis of the polarization filter and the horizontal axis,the radiance I c after calibra-tion,as measured by one detector element of the camera,is givenby:Fig.3The landmine in thermal equilibrium with a blackbody source.1023Optical Engineering,Vol.41No.5,May2002I c ͑␸͒ϭ12͓I ϩQ cos ͑2␸͒ϩU sin ͑2␸͔͒,͑16͒where ␸ϭ0deg represents the situation that horizontally polarized radiance passes through the linear polarizer unat-tenuated.The parameters I ,Q ,and U are three of the four Stokes parameters.1The fourth parameter V defines the cir-cular polarization,is not considered in the model,and can-not be measured by only a linear polarization filter.If we define the horizontal and vertical polarization to be s-and p-polarization,respectively,then the Stokes param-eters relate to the total radiance as given in Eqs.͑13͒and ͑14͒as follows:I c ͑90deg ͒ϭ12I Ϫ12Q ϭI p ͑T m ,T bb ,␪i ͒,͑17͒I c ͑0deg ͒ϭ12I ϩ12Q ϭI s ͑T m ,T bb ,␪i ͒.͑18͒This only defines two Stokes parameters.The third param-eter U ͑diagonal polarization ͒must be zero for this situa-tion with a horizontally placed landmine and an unpolar-ized source.From Eqs.͑17͒and ͑18͒,the radiance I and the Stokes parameter Q can be calculated:I ϭI c ͑0deg ͒ϩI c ͑90deg ͒ϭ12͓␳s ͑␪i ͒ϩ␳p ͑␪i ͔͒I e ͑T bb ,T m ͒ϩI BB ͑T m ͒,͑19͒Q ϭI c ͑0deg ͒ϪI c ͑90deg ͒ϭ12͓␳s ͑␪i ͒Ϫ␳p ͑␪i ͔͒ϫI e ͑T bb ,T m ͒.͑20͒Often different representations are used for the polarization:LP ϭ͑Q 2ϩU 2͒1/2,͑21͒DoLP ϭLPI,͑22͒␺ϭ12arctan ͑U /Q ͒,͑23͒with LP being the amount of linear polarization,DoLP thedegree of polarization,and ␺the angle of polarization ͑ori-entation of the polarization ellipse ͒.In the specific situation of Eqs.͑19͒and ͑20͒,the angle ␺is either 0or 90deg.As is shown later on,due to thermal emission,DoLP never reaches 1for TIR.In Fig.4the Stokes parameters I and Q are plotted for a range of incidence angles ␪i and differences between the radiance of the blackbody and the landmine.For the refrac-tive index of the landmine surface,the value 1.445ϩ0.05i is chosen.This figure clearly shows a linear rela-tion between the radiance difference and the two Stokes parameters.This model is only valid for specular reflection.In real-ity,the surface of the landmine may not be absolutely smooth.To accurately model this,it is necessary to include surface roughness using either bidirectional reflectance functions ͑BRDFs ͒or a slope approximation.11However,the model described here provides a first order approxima-tion.3Measurement SetupGenerally there are two different approaches used for themeasurement of ͑infrared ͒polarization.Either time or spa-tial division is necessary to measure up to four elements of the Stokes vector using only one focal plane.However,there are approaches that claim to measure infrared polar-ization without reducing either spatial or temporal resolu-tion;these approaches are discussed at the end of this sec-tion.With time division,different polarization images are measured sequentially.This is usually performed by mount-ing a polarization filter in front of the camera and taking a sequence of images with different polarization directions.For measurements of the full ͑four elements ͒Stokes vector,a retarder ͑for instance a quarterwave plate ͒is rotated fol-lowed by a fixed linear polarization filter.12,13This common approach of either rotating a polarizer or a retarder is re-ported by the majority of the literature.14Fig.4Model calculations for different angles of incidence ␪i for the Stokes parameters (a)I and (b)Q as function of the radiance difference I e (T bb ,T m )between the blackbody and a landmine with a refractive index of 1.445ϩ0.05i .1024Optical Engineering,Vol.41No.5,May 2002Alternatively,the different polarizations are measured simultaneously at the cost of reduced spatial resolution.For example,every four adjacent pixels of a focal plane array ͑FPA ͒are grouped.In front of each of these four pixels a different polarization filter is mounted,each with a different orientation.This approach is followed by two parties.First,there is Nichols Research in cooperation with DERA ͑UK ͒15and the University of Alabama.16Second,there is Physics Innovations 17in cooperation with Lockheed Martin.18Because of the large development costs,this ap-proach is less common.However,due to the fact that a full set of four Stokes images can be acquired in a single frame,this approach has advantages for applications in rapidly changing environments.There are at least two other approaches that claim not to suffer from either time or spatial division of the FPA.The first one is a hyperspectral imager as made by Aerodyne.19The other is developed by FOI ͑Swedish patent FOA-R-99-01090-408-SE ͒and is a construction of a stack of detector elements,where each element only detects one polarization direction and passes the others.3.1Measurement Setup ConstructionOur approach for the measurement setup is the use of time division and a rotating polarization filter.An overview of this setup is given in Fig.5͑a ͒.The setup consists of a polarization filter ͑a wire-grid polarizer ͒,a motor to rotate the polarizer,and controller electronics ͑see Fig.6͒.In this setup different infrared cameras can be mounted.The po-larization filter has a large spectral range and can be used for either LWIR or MWIR cameras.The controller electronics have the task to synchronize the rotation of the polarization filter with the frame sync of the camera.In Fig.5͑b ͒,the motor controller is shown in more detail.For each rotation of the filter,between 6and 60frame syncs of the camera occur,depending on the set-ting of the multiplier.For the motor speed control there aretwo options:either the angle readout is divided by a num-ber or the frame sync is multiplied by a number.If the angle readout is divided,then the frequency of the phase lock loop ͑PLL ͒matches the frame sync frequency.As the frame sync frequency is usually lower than the angle read-out frequency,the controller dynamics have a tendency to become unstable.For the present setup,the choice has been made for multiplying the frame sync rather than dividing the angle readout.Because of this multiplication,the PLL works at a higher frequency and can make faster speed corrections.With this multiplication,the PLL is in lock even for the lowest rotation speed of one rotation per sec-ond ͑for a camera with a frame rate of 60Hz ͒.The angle readout is also counted ͑and reset for each rotation ͒so that the angle can be recorded for each frame sync.Once the PLL is locked in,the upper bound on the error in the angle is 3deg.One problem with this setup is that the polarization filter reflects radiation from the camera housing,the lens,and through the lens the cooled detector array.This reflected image seems to be just out of focus.So,instead of just acquiring the image of the scene,the reflected image of the camera is added to it.This effect is called narcism.There are two ways of correcting this effect.The first one is to tilt the polarizer so that it does not look at the camera,but at a uniform source.20For a small field of view ͑FOV ͒,this tilting seems possible.However,when observing mines at relatively close distance ͑a few meters ͒,a wide FOV lens is necessary and the filter has to be tilted much more ͑even up to 11deg 20͒.It is expected that this tilting may give rise to problems ͑the transmission through the filter may change ͒and seems therefore not feasible.The second approach for correcting narcism is to mea-sure and subtract the reflected image by the filter.This ap-proach is an integral part of our measurement calibration procedure and is described in Sec.4.1.4Measurements4.1System CalibrationThe camera measures the radiance over some integration period ͑fixed for these measurements at 1ms ͒.This value is digitized with a resolution of 12bits.The measured value depends on the instantaneous field of view,the aperture ͑lens diameter ͒,the efficiency of each detector element,as well as the conversion offset in the analogue to digital con-version.The fabrication process for construction offocalFig.5(a)Infrared polarization setup,consisting of a rotating polar-ization filter (a wire-grid polarizer),a motor,an infrared camera,and custom-made controller electronics.(b)The motor controller multi-plies the frame sync by a number specified by the processing com-puter.This is input to the PLL,which drives the motor such that the angle pickup has the same frequency as the multiplied frame sync.The angle pickup produces a pulse for every 3deg ofrotation.Fig.6Photo of the infrared polarization setup.1025Optical Engineering,Vol.41No.5,May 2002plane arrays͑FPAs͒sensitive to TIR radiation is not as well defined as for silicon-based FPAs.The variations in this process lead to differences in sensitivity of detector ele-ments,also called nonuniformity.When all these camera parameters are known,the mea-sured bit value can be converted to radiance in W/m2/sr. However,as already mentioned in Sec.3.1,due to the setup of camera andfilter,narcism occurs.This means that the measured radiance is the sum of͑a fraction of͒the radiance of the scene and͑a fraction of͒the reflected radiance of the camera itself.Thus,it is impossible to directly calculate the scene radiance based on only the camera parameters.We have developed a two-point calibration procedure to correct for narcism and at the same time perform a calibra-tion of the radiance.This calibration is performed sepa-rately for each detector element and thus also functions as a nonuniformity correction of the detector elements.Thefirst step is to place a blackbody with a low temperature T l in front of the polarizationfilter and acquire a sequence of images͑one for every orientation␸).The next step is to repeat thefirst step with a blackbody with a high tempera-ture T h.The calibrated radiance for orientation␸of the polarizationfilter is given for each pixel(x,y)by:I c͑␸,x,y͒ϭI BB͑T l͒ϩv͑␸,x,y͒Ϫv l͑␸,x,y͒v h␸,x,yϪl␸,x,yϫ͓I BB͑T h͒ϪI BB͑T l͔͒,͑24͒where v(␸,x,y)is the measured value of the scene͑in bits͒,and v l(␸,x,y)and v h(␸,x,y)are the measured cali-bration value of the blackbody with low and high tempera-ture,respectively.The calibration values are averaged over 30measurements to reduce the error due to camera noise. For a good calibration there must be sufficient difference between the two temperatures and they must be in the range of the apparent scene temperatures.This two-point calibration procedure removes the nar-cism from the measured images as well as corrects for transmission through thefilter.The quality of the polarizer ͑the amount of passed radiance in the cross direction͒hasnot been determined and thus will contribute to the unpo-larized radiance I.4.2Indoor Landmine MeasurementsA measurement setup,analogous to the model as shown in Fig.2,has been used to determine the polarization effects of the hot/cold blackbody source reflected by the surface of a landmine.For this experiment a dummy antipersonnel landmine,known as PMN,21is chosen͑see Fig.7͒.This PMN landmine has aflat rubber top.A piece of 10ϫ40cm of the same rubber is used for the measure-ments.Measurements are taken for six different incident angles:48.8;58.5;70.2;75.4;80.4;and85.1deg with an accuracy of around0.5deg.The distance between the rub-ber and the camera is150cm.The two-point calibration procedure͑see Sec.4.1͒is performed for each angle of incidence.The hot/cold source is varied betweenϪ10andϩ20K around ambient temperature.Each temperature of the hot/ cold source as well as the temperature of the rubber is measured with a hand ing Eq.͑1͒the radi-ances I BB(T bb)and I BB(T m)are calculated.The tempera-ture of the rubber varies only with the room temperature.A sequence of images is taken for the following temperatures around the ambient temperature:Ϫ10.0;Ϫ5.0;Ϫ2.5;0.0;5.0;10.0;15.0,and20.0K with an accuracy of0.1K.Each sequence consists of60images,with the polarizer rotated over6deg between two images.Since the camera has a frame rate of60Hz,the acquisition time is1s for each sequence.5AnalysisThe radiance as measured by the camera behind the polar-izer is given in Eq.͑16͒.This radiance is measured for a full rotation of thefilter.An estimate of the Stokes-Mueller polarization parameters I˜,Q˜,and U˜is given by:I˜ϭ2N͚jϭ1NI c͑␸j͒,͑25͒Q˜ϭ4N͚jϭ1NI c͑␸j͒cos͑2␸j͒,͑26͒U˜ϭ4N͚jϭ1NI c͑␸j͒sin͑2␸j͒,͑27͒where Nϭ60is the number of frames,j is the frame num-ber,and␸jϭ2␲j/N is the angle of the linear polarizer for frame j.The exact orientation of the polarization is not calibrated.The orientation is found by minimizing U˜over all the measurements.The resulting offset in angles has been used to correct the results.In Fig.8,the͑estimated͒Stokes parameter images I˜,Q˜, and U˜are shown for the angle of incidence␪iϭ85deg͑a grazing angle͒with the source set at10K below the ambi-ent temperature͑297K͒.The measurement setup is similar to Fig.2.The black square in the top of the image in Fig. 8͑a͒is the blackbody.The smaller black square in the middle is the rubber of the landmine resting on top of an aluminum table.In the radiance image I˜the rubber has a lower value than the surroundings,but is higher thanthe Fig.7The dummy antipersonnel landmine PMN.Only rubber on the top of this dummy landmine was used for the indoor experi-ments.1026Optical Engineering,Vol.41No.5,May2002blackbody source.In the Q˜image the rubber is clearly visible with the lowest values.The blackbody source,which is colder than the rubber of the landmine,does nothave a lower polarized radiance Q˜,since the radiance from the blackbody source is not polarized,but it can be seendue to the edges.The U˜image is almost zero ͑observe the scale ͒and contains only camera noise.This is expected asthere are no diagonally oriented surfaces present in the scene.These measurements are performed for six different angles ␪i and eight different temperatures T bb .This gives a total of 48sequences.For every sequence,the calculated Stokes parameters are averaged over the rubber of the land-mine.The results of these calculations are shown in Fig.9Fig.8One set of images for the blackbody set at 10K lower than the rubber of the landmine and ␪i ϭ85deg (a grazing angle).The radiance I ˜is given in (a).The big black square in the upper half of the image is the blackbody and the small dark gray rectangle in the middle is the rubber of thelandmine.The linearly polarized radiance Q˜and U ˜are given in (b)and (c),respectively.Note that the radiance scale of (c)is a factor 5lower than the scale of (b)and that these scales are also different compared to(a).Fig.9The three Stokes parameters I ˜,Q˜,and U ˜and the angle ␺˜of the rubber of the landmine as a function of the radiance difference I e (T bb ,T m )and the reflection angle ␪i .1027Optical Engineering,Vol.41No.5,May 2002as functions of the effective illumination I e (T bb ,T m )͓see Eq.͑15͔͒.Both the radiance I ˜and the polarized radiance Q ˜,re-spectively in Figs.9͑a ͒and 9͑b ͒,show a linear relation with the effective illumination for each incidence angle.The lin-ear relationship was predicted by Eqs.͑19͒and ͑20͒.The radiance I ˜increases continuously with the angle of inci-dence.The polarized radiance Q ˜,however,reaches a maxi-mum for 80deg.It is obvious from Fig.9͑c ͒that the polarized radiance U˜is zero for all effective illumination values and all angles.This can be expected from theory,since there are no diago-nally oriented surfaces and thus the polarization can only be horizontal or vertical.This also is reflected in Fig.9͑d ͒.Except for zero effective illumination,for which the angle is not well defined,the angle of polarization is either 90͑vertical ͒or 0deg ͑horizontal ͒.Using these experimental values for I ˜and Q˜,it is pos-sible to estimate the average refractive index of the rubber over the wavelength band.In fact,these measurements al-low us to make two independent estimates of the refractive index ͑compared to only one estimate in our previous publication 22͒.For each of the curves in Figs.9͑a ͒and 9͑b ͒,a best fit for the slopes is made using the linear relationships of Eqs.͑19͒and ͑20͒.This gives an estimate of both the sum ͑for the I parameter ͒and the difference ͑for the Q parameter ͒of the two reflection coefficients for each angle of incidence ͑and reflection ͒.In Fig.10the sum ͓Fig.10͑a ͔͒and the difference ͓Fig.10͑b ͔͒of the two reflection coefficients are shown.The sum of the reflection coefficients increases con-tinuously with the angle of incidence.However,the differ-ence between the reflection coefficients reaches a maxi-mum at 79deg.Note that the position of this maximum does not occur at the Brewster angle at 55deg ͑for n 1ϭ1.445),where ␳p equals zero.1The measured radiance is fully polarized under the Brewster angle,when there is no thermal emission ͓I BB (T m )ϭ0͔and so Eqs.͑13͒and ͑14͒reduce to:I p ͑T m ,T bb ,␪i ͒ϭ0,͑28͒I s ͑T m ,T bb ,␪i ͒ϭ12␳s ͑␪i ͒I BB ͑T bb ͒.͑29͒However,due to the thermal emission of the landmine,the measured radiance will not be fully polarized:I p ͑T m ,T bb ,␪i ͒ϭ12I BB ͑T m ͒,͑30͒I s ͑T m ,T bb ,␪i ͒ϭ1I BB ͑T m ͒ϩ1␳s ͑␪i ͒I e ͑T bb ,T m ͒,͑31͒with I e (T bb ,T m )as defined in Eq.͑15͒.In Fig.11the re-lationship between the refractive index,the Brewster angle,and the angle of maximum polarization magnitude is shown.The angle of maximum polarization is almost con-stant around 78deg over a wide range of refractive indices.5.1Estimation of the Refractive IndexEquations ͑19͒and ͑20͒give the radiance I and Q as func-tion of the angle ␪i and the effective illumination I e (T bb ,T m ),i.e.,the difference between the blackbody and rubber radiances.These equations depend on the reflection coefficients ␳p and ␳s and via Eq.͑4͒on therefractiveFig.10The sum (a)and the difference (b)of the reflection coefficients ␳s and ␳p for the rubber of the landmine.The refractive index of the model is calculated using the measurements for both the radi-ances I and Q.Fig.11The Brewster angle and the angle for which the magnitude of the polarization has a maximum as a function of the refractive index.1028Optical Engineering,Vol.41No.5,May 2002。