Absorption Cross Section of Moderately Conducting Disks at 35 GHz

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Absorption cross sectionof moderately conducting disks at 35 GHz

Marty J. Venner and Charles W. Bruce

Absorption cross-sectional measurements and model predictions of 25 disks with moderate electricalconductivity were made. The disk radii span part of the resonance region, and their thicknesses are anorder of magnitude less than the skin depth. A photoacoustic apparatus, with a 35-GHz microwavesource, was used to measure the absorption cross sections. The disks tend to be more absorbing whenthey are edge-on rather than broadside to the incident microwaves. Also they absorb most efficientlynear a size parameter of 1. Model predictions generally agree with the measured magnitudes and trendsin the data. © 1998 Optical Society of America

OCIS codes: 290.0290, 290.2200, 290.3030, 300.6430, 010.1110.

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1. Introduction

Absorption and scattering of electromagnetic wavesby small particles of various shapes continue to be anarea of great interest to the scientific community.One shape that has not received as thorough an in-vestigation as spheres or cylinders is the disk. Pre-vious research on disks has focused primarily on theirscattering properties with little or no investigation oftheir absorbing properties.1–9 The motivation forhis previous research stemmed from interest inaking the disk a standard radar scatterer and forodeling radar extinction by ice crystals.5,8 The

prime motivation of this particular research is theestablishment of a basis for the development of in-tentional obscurants. An aerosol cloud of highly ab-sorbing disks could be used to block or partiallyobscure infrared or optical signatures. Even thoughthe measurements and calculations were made atmillimeter wavelengths, the results should be appli-cable to other wavelength regions as long as the sizeparameter and indices of refraction are held constant.The size parameter is specified by ka, where k is theree-space wave number and a is the disk radius.ecause of the emphasis on absorption properties,

M. J. Venner is with the Propulsion Sciences Division, U.S. AirForce Research Laboratory, 10 E. Saturn Boulevard, Edwards AirForce Base, California 93524-7680. C. W. Bruce is with the De-partment of Physics, New Mexico State University, Las Cruces,New Mexico 88003.

Received 20 April 1998.0003-6935y98y307143-08$15.00y0© 1998 Optical Society of America

he disks used in this study were made from a mod-rately conducting material that is nearly 5 orders ofagnitude lower in electrical conductivity than met-

ls such as silver or copper. With significant absorp-ion, the thickness or thickness parameter kt alsoecomes important.

2. Disk Characteristics

The disks are composed of conductive carbon blackcombined with a latex binder. Both the binder andthe black were obtained from Advanced ElectroMag-netics, Inc., Santee, Calif. The black in this partic-ular mixture is particles of graphitic carbonaveraging 30 nm in diameter.10 The mixture con-tains approximately 40% black and 60% latex andother solids. At this density most of the blackshould be touching throughout the polymer matrix,and continuous current densities are possible. Thedisks are cut from thin films of the material. Theirdiameters range from approximately 1 to 9 mm or0.3 , ka , 3. The procedure devised for making thedisks was not well refined, and thicknesses variedconsiderably from disk to disk. As a result, the av-erage thicknesses of those chosen for measurementranged from approximately 8 to 30 mm.

The average material dc electrical conductivity waseasured at 400 V21 m21.11 From this a complex

index of refraction can be calculated assuming thatthe free-electron or Drude model applies. The con-nection between the conductivity and the index ofrefraction is given by

εyε0 5 εyε0 1isvε0

5 n2 2 k2 1 i2nk, (1)

20 October 1998 y Vol. 37, No. 30 y APPLIED OPTICS 7143

ts

m

w

dpvmanidiaeate

cct

7

where ε is the complex dielectric constant, s is thecomplex conductivity, n and k are the real and imag-inary parts of the index of refraction, and v is theangular frequency.12 The Drude model was appliedhere because the conductivity of the black is due pri-marily to quasi-free electrons that flow throughout anetwork of connected particles held in place by thelatex binder.13 The relative dielectric constant ofhe binder, a polyvinyl-chloride-based compound, ismall ~εyε0 ; 2–4! relative to that contributed by the

conductive black at these frequencies.14 Bound elec-trons in the graphitic carbon black form strong copla-nar bonds and are not expected to contributesignificantly to the real part of the dielectric constanteither. Because ε is small relative to the imaginarycontribution but somewhat uncertain, it was set tothe vacuum value in Eq. ~1!. Also, for metals atlonger wavelengths ~i.e., microwaves! the inducedcurrent density is generally in phase with the inci-dent field, and to a good approximation the conduc-tivity is real and independent of frequency.15

Therefore s is assumed real and set equal to theeasured dc conductivity. Solving Eq. ~1! for n and

k in terms of s leads to

n2 512

112 F1 1 S s

vε0D2G1y2

, (2)

k 5s

2vε0 n. (3)

For a conductivity of 400 V21 m21, n and k are bothapproximately 10. The skin depth of this material isthen approximately 135 mm.

3. Absorption Measurements

A. Measurement Method and Procedure

Disk absorption was measured with a photoacousticdetector.16 The device consists of two identicalsealed chambers placed in thermal contact with eachother. The disk to be measured is placed in one ofthe chambers. Coherent microwaves irradiate bothchambers simultaneously so the only pressure differ-ence between the chambers results from the heateddisk. The change in pressure is recorded in time andthe slope is converted to a cross section. The absorp-tion cross section is defined as the power ~watts! ab-sorbed by the object of interest divided by theirradiance ~watts per square meter! incident upon it.If we assume that all the absorbed power is convertedto heat, the absorption cross section sa is

sa 51

Irad

dQdt

, (4)

here dQydt is the rate of heat produced within thedisk and Irad is the incident irradiance. From basic

144 APPLIED OPTICS y Vol. 37, No. 30 y 20 October 1998

thermodynamic relations the cross section, as mea-sured by this apparatus, can be written as

sa 5Vcv

RIradSdDp

dt D , (5)

where dDpydt is the volume-averaged rate of pres-sure difference between the two chambers, V is thetotal volume of one chamber, R is the gas constant forair, and cv is the molar-specific heat of air.16 The

ifferential pressure rate of change, then, is directlyroportional to the particle cross section. This de-ice can easily measure pressure changes of a fewillitorr. The disks were attached to sharpened

crylic rods and then mounted on a rotating mecha-ism inside the chamber. Each disk measured was

ncrementally rotated through a full 360 deg withata obtained at 6- or 12-deg increments. At eachncremental position, the disks were irradiated forpproximately 2–3 s eight times and the results av-raged. All the measurements were then reduced toverages over 0–90 deg. To test repeatability, mul-iple disks were measured at or near the same diam-ter.

B. Calibration and Measurement Errors

Because the pressure change is proportional to thecross section, a particle with a known cross sectioncan be used to calibrate the photoacoustic detector.Carbon fibers were chosen because they were avail-able and had been recently characterized using thesame apparatus.16 Calibration with a 13.9-mm-longfiber was performed at least once each day that diskabsorption measurements were made. An averageof all the 13.9-mm fiber calibrations was used to con-vert the disk measurements to cross sections. Thisapproach results in larger statistical errors but main-tains better consistency between the measured rawdata and the derived cross sections. Standard sta-tistical methods are used to estimate the magnitudeof the error. The major source of random error is theaveraged calibration constant derived from the car-bon fiber. It contributes approximately two thirds ofthe total standard deviation. Also contributing isthe averaged absorption from multiple measure-ments at each disk orientation. Standard devia-tions from both of these averages are combined toestimate the total error. They are typically 12–15%of the cross-sectional value. It is important to notethat this is the error in the overall magnitude. Er-rors in the trends are typically less than 5%.

It was necessary to apply a thermal correction tosome of the thicker disks. The theory presentedabove assumes that the rate of heat generated in theparticle is equal to the rate of heat absorbed into thesurrounding air. There can be a significant differ-ence in the cooling rate between disks of the samediameter but with thicknesses that differ by morethan 3–5 mm. This thermal lag reduces the derivedross section from the actual value. When the mi-rowave power is turned off, the resulting exponen-ial differential pressure decay rate provides a

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measure of the energy still stored on the disk. Acorrection was applied to the disks to account for thisstored energy.17 Corrections ranged from less than10% for thicknesses less than 15 mm and as high as41% for those greater than 25 mm. Once convertedto cross sections, this data should then be directlycomparable to the model calculations. Residual er-rors from the thermal correction are not included inthe above statistical uncertainties and may push theactual error to 20% in some cases.

4. Absorption Results

A. Model Description

Willis and Weil have developed a model that calcu-lates absorption and scattering cross sections fordisks in the resonance region.18 It is a steady-statesolution that combines an integral equation and mo-ment method to solve for the current density inducedby an incident plane wave. The current density isexpressed as a summed series of basis functions withunknown coefficients. The functions are specificallychosen to give a good approximation for circular diskswith thickness kt ,, 1 and radius ka ; 1. Thecoefficients are determined by use of the method ofmoments. Absorption and scattering cross sectionsare then calculated from the derived current density.The disks are assumed to be isotropic and homoge-neous with a known complex index of refraction.Because no assumption is made with respect to thematerial properties, calculations can be made forproperties ranging from dielectric to highly conduc-tive. The model requires as input the real and imag-inary parts of the index of refraction, the diskthickness parameter kt, and size parameter ka. Theoordinate system used by the model is shown in Fig.. The incident-wave direction is specified in spher-cal polar coordinates where ui is the angle formed by

the z axis and the propagation direction. The azi-muthal angle wi is in the plane of the disk.

B. Data and Model Comparisons

Data were obtained for all three possible disk rota-tions relative to the incident wave. For reference,the three orientations are depicted in Fig. 2. Figure3 shows a representative sample of the absorptionmeasured and calculated for all three disk rotations.

Fig. 1. Disk coordinate system.

In these and future plots, all the values are presentedas efficiencies, that is, the cross section divided by thedisk area. The functional dependence for all threerotations in both the data and the model values isquite smooth and generally the magnitudes comparewell. The increasing absorption with E-axis rota-tion from 0 ~broadside! to 90 deg ~edge-on! incidences designated as an edge effect. It can be seen in allhe data and nearly all calculations. This increaserom broadside to edge-on is generally the largestespecially in the data! near a size parameter of 1.verall, the data values and model predictions com-are most favorably for broadside incidence. Theest comparison in the trend, however, is with themaller disks. In Fig. 3~e! the E-axis rotation trendn both the data and the model show some additionaltructure not seen in the smaller sizes. The dataalues in Fig. 3~e! are low, partially because of anncomplete thermal correction. Not enough dataere obtained beyond power shut-off to sufficientlyccount for all the residual heat remaining in thisarge disk. Also, because of the large size and be-ause the field is not truly a plane wave, the intensityncident at the edges is slightly lower than that at theenter. This would reduce the absorption, especiallyt broadside incidence.The H-axis rotation and the k-axis rotation data

enerally match up with the E-axis rotation data atroadside and edge-on incidence as they should.hen the disk plane becomes nearly perpendicular to

he incident electric field, however, the data divergeubstantially above that predicted by the model. Al-hough not obvious in Figs. 3~a!–3~e!, it can be 2rders of magnitude higher. At this orientation, theignal-to-noise ratio is low ~,5!. Small angle errorsnd shape imperfections can lead to a large increasen the signal relative to that from an ideal disk. Thengle uncertainty is approximately 1 deg, and alight curvature across some of the disks is notice-ble. These errors would act to bias the data high.he photoacoustic device, as designed, could not de-ect pressure changes corresponding to the modelredictions for this orientation. The large discrep-ncy could indicate that the model does not properlyalculate all the induced currents as well. Compar-

Fig. 2. Possible disk orientations relative to the incident wave:~a! E-axis rotation, ~b! H-axis rotation, and ~c! k-axis rotation.


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isons with scattering data, presented in Ref. 9, showgood agreement with both highly conductive and di-electric disks at these large angles.

The edge effect seen in the data is likely due, inpart, to field nonuniformities in the volume of spacethe disk is rotated through. The distance from thehorn to the disk center ~position at which the calibra-tion fiber is placed! is 34 mm. At this distance, thedisks are not far enough into the Fraunhofer regionwhere the E-field phase and amplitude approximatethat of a plane wave. Because the model assumes auniform irradiance at all points on the surface re-gardless of disk orientation, field nonuniformitiescould explain the larger discrepancy between thedata and the model for disks at edge-on. But it doesnot explain the increase at edge-on predicted by themodel. A crude measurement of the volumetric in-tensity profile was made over the volume of space thelargest disk could occupy. The measured intensityalong the z axis was, in fact, roughly inverse rsquared in form. Using this volumetric data, aspace-integrated average of the intensity was deter-mined for the largest disk at edge-on. The value isapproximately 7% larger than that at the center ofthe disk and only partially explains the edge effect inthe data.

An explanation that applies to the model and thedata is that, at edge-on incidence, the total surface

Fig. 3. Data and model comparison for allthree orientations. The dashed curves aremodel predictions. F, E-axis rotation; ■,H-axis rotation; }, k-axis rotation.


area is exposed to twice the irradiance. Grazing-incidence waves propagate along both surfaces, a por-tion of each being refracted into the medium. Theenergy per unit area entering the disk may actuallybe less than at broadside incidence. Regardless ofthe incident angle, any waves ~both phase and am-plitude! refracted into the medium will propagatewith an angle nearly normal to the surface. Theangle of refraction for a thin film of this conductivity,calculated using a complex form of Snell’s law, is lessthan 6 deg from normal with grazing incidence.19

The interference that arises from multiple reflectionscan occur in a thin film of this material and presum-ably inside an edge-on disk as well.

Calculations of the field structure at the surface ofa disk were carried out with the model to furtherinvestigate possible diffraction and interference ef-fects. Figures 4~a! and 4~b! show the total field am-plitude contours for broadside and edge-on incidence,respectively. In Fig. 4~a! the induced field appearso have a dipole structure. In Fig. 4~b! a much moreomplicated structure is apparent, indicating effectsypically associated with diffraction and interference.learly this induced field structure is responsible for

he predicted increase in absorption at edge-on. It isrobable that the increased intensity near the lead-ng edge relative to the calibration point at the diskenter enhances this structure in the data. This

would explain the larger discrepancy between thedata and the prediction at edge-on compared withbroadside. From Fig. 3 the enhancement is minimalfor the smallest size where the field nonuniformityover the disk is small. The departure between dataand model then becomes substantial to ka ; 2.

The data and model for the largest diameter disksdiffer much less at edge-on than expected given thefield nonuniformity. Most likely, because of the largesize, any induced currents at the far half of the disksare weak and do not add significantly to the overallcurrent density. The currents induced in the nearhalf, on the other hand, are increased substantially bythe inverse r-squared intensity. Some averaging hastaken place, keeping the measured values near thosepredicted but reducing enhancement of the edge effectby the intensity variation. Also, at this large size, thephase change along the incident-wave surface normalto the propagation direction becomes significant. Byuse of a Fresnel approximation, the calculated phasechange from the disk center to the edge is approxi-mately 15 deg.20 This could also be a significant fac-tor in the overall comparison because the datagenerally fall below the model predictions @Fig. 3~e!#.The intensity variation across the disk would explainwhy the data curve in Fig. 3~e! increases to approxi-

Fig. 4. Total field amplitude ~arbitrary units! on the disk surface:~a! broadside incidence and ~b! edge-on incidence.

mately 30 deg, then stays relatively flat beyond that.Figure 5 repeats that result and includes the angulardependence calculated by a thin-film model.21 Inter-estingly, the data trend matches the thin-film trendnicely to 40 deg instead of following the disk modeltrend. This would be expected if averaging has takenplace with the disk data as explained above. Becauseof their large size, these disks tend to absorb with anangular dependence similar to the thin film, at leastuntil the inverse r-squared intensity variation be-comes significant ~i.e., .40 deg!. The broad mini-mum near 40 deg, predicted by the model, probablyresults from interference as well. Calculations withincreased material conductivity removes this mini-mum, supporting the assertion.

C. Diameter and Thickness

The cross-sectional dependence on diameter is shownby the two plots in Fig. 6. Because the disks vary inthickness, calculations were performed for threethicknesses that bound most of those measured.The data itself was subdivided into thickness rangesfor better comparison. Cross sections for the small-est disks are roughly a factor of 2 higher than theircalculated values. Otherwise, at broadside the dif-ferences are no larger than 30% and at edge-on theyare no larger than 52%.

From the plots some trends with diameter becomeapparent. At ka , 0.7 the model predicts a crosssection roughly proportional to the square of the diskarea for both broadside and edge-on incidence. Thedata set at ka ; 0.65 matches well, but data for thesmaller disks do not and imply a different trend.The signal-to-noise ratios for those disks are againlow ~;5!. Because of their small size, however,shape and orientation uncertainties are not as likelyto be major factors in the difference. Residual scat-tering off the chamber walls could be another source,but it is not known if that could account for the dis-crepancies. At ka ; 1 the wave interacts most effi-ciently with the disk, and diffraction can have thegreatest effect on the overall induced current density.The result is a predicted maximum in the absorption

Fig. 5. Comparison of disk absorption data to a one-dimensionalthin-film model and a disk model prediction versus angle of inci-dence: —, film model; ---, disk model; ■, data.


b

7

efficiency just beyond ka 5 1 for both angles of inci-dence. Peak values in the data also occur for thosesize parameters nearest the predicted maximum.Interestingly, both data and model absorption effi-ciencies are greater than unity in that region. Themeasured values, however, are somewhat high be-cause of the inverse r-squared irradiance variationmentioned above. Removal of this effect may actu-ally improve the comparison, but measurement un-certainties would still limit confirmation of the modelvalues above unity. For values of ka much greaterthan 1, diffraction effects on the overall current den-sity at broadside incidence become small, and theabsorption efficiency drops. The disk should beginto absorb similar to a semi-infinite thin film ~i.e.,independent of size! of the same thickness. Both thedata and the model predictions appear to be asymp-totically approaching a constant value for broadsideincidence. In fact, the value in Fig. 5 at 0-deg inci-dence is only slightly less than the calculated value inFig. 6~a! for ka . 3.

As a further check on the overall characteristics ofthese disks, the total scattering and extinction werealso calculated versus diameter. Figure 7 shows cal-culated curves of absorption, total scattering, andextinction for disks at broadside and edge-on. Be-yond ka ; 2, the extinction appears to asymptoticallyapproach unity for broadside incidence. In the limitas ka3 `, the extinction efficiency for an opaque disk

Fig. 6. Diameter dependence of the absorption cross section: ~a!roadside incidence and ~b! edge-on incidence. Curves depict the

calculations and symbols depict the data: ■, ,12; Œ, 12–17; �,17–22; }, 22–27; F, .27 mm.


is 2.22 Because these disks are not opaque, it is rea-sonable that the limiting value is something less than2. Calculations with the thin-film model indicatethat approximately half of the incident energy is ab-sorbed, one quarter is transmitted, and the remain-ing quarter is reflected. Because the transmittedportion is not energy removed from the beam, diffrac-tion must account for the other quarter calculated bythe disk model. These disks in the large-size limitappear to be removing energy from the beam by dif-fraction equivalent to one quarter of the area ratherthan equal to the area as in the case of an opaquedisk. The other feature of interest in Fig. 7 is thedominance of absorption for intermediate values ofka. Normally absorption is expected to dominatethe total scattering for ka ,, 1. Here absorption isslightly greater than the total scatter at ka 5 2 anddominates for smaller values. This might be ex-pected as kt is much less than 1.

The plots in Fig. 6 also reveal an additional com-plexity that is due to disk thickness. For ka , 0.7the model actually predicts more absorption for athinner disk, then reverses that trend for larger di-ameters. Calculations versus thickness at four spe-cific diameters also reveal that the absorption is notnecessarily a maximum for the thickest disk. Fig-ure 8 shows these results along with data matchingthose diameters. The largest three diameters ex-hibit a broad maximum, then only a slight decrease inabsorption. Most apparent is the peak for the small-est diameter disk near kt ; 0.003. The exact datavalues corroborate the trends shown but the uncer-tainties again limit confirmation of the model.

Further calculations with the thin-film model par-tially explain the phenomena. Figure 9 shows thefraction of the incident energy transmitted, reflected,and absorbed versus thickness for films with conduc-tivities similar to the disk material. From Fig. 9 it isapparent that these thin films also have a peak in theabsorption at a finite thickness. The total reflectedfield amplitude for a film is given by

r 5r12 1 r23 exp~ib!

1 1 r12r23 exp~ib!, (6)

Fig. 7. Calculated absorption, total scattering, and extinction fordisks with a thickness of 15 mm.

Tipri

where r12 and r23 are complex reflection coefficientsfor the reflected waves at the front and back surfacesof the film.21 The phase and amplitude of the wavesinside are expressed by the complex coefficient b.

he thickness and material conductivity control thenternal interference pattern because b is directlyroportional to both. In Fig. 9 it can be seen that theate of decrease of the transmissivity and the rate ofncrease of the reflectivity combine ~A 5 1 2 T 2 R!

to produce an absorption peak at a finite thickness.The absorption is quite dependent on the interferenceof the multiply reflected waves inside the film. Asmight be expected, the variation in conductivityshifts the location of the peak. Presumably, this in-terference effect explains the decrease in absorptionproduced by the disk model. In fact, the percentagedecrease from a maximum near kt 5 0.010 ~;13 mm!to kt 5 0.022 ~;30 mm! is nearly the same in bothmodels for the larger disks. The limited data avail-able for comparison appear to verify the trends. Thedramatic thickness dependence predicted for ka 50.348 in Fig. 8 is not understood. The statisticaluncertainties in the data points for that size would

Fig. 8. Thickness dependence of the cross section for selecteddisks at 0-deg incidence. Curves depict the calculations and sym-bols depict the data: ka 5 ■, 0.348; Œ, 0.910; �, 1.889; }, 3.264.

Fig. 9. Thickness dependence on the radiative properties of thinfilms with differing conductivity: ~1! 200, ~2! 400, ~3! 500, and ~4!600 V21 m21.

allow a trend parallel to that shown by the diskmodel.

5. Conclusions

Measurement and model calculations indicate thatthese moderately conducting disks, which are muchthinner than the skin depth, absorb a significant frac-tion of the incident microwaves. Interestingly, theabsorption increases as the disk is rotated frombroadside to edge-on incidence. The amount of in-crease at edge-on, however, is highly dependent onboth diameter and thickness. Regardless of orien-tation, calculations suggest that absorption accountsfor at least half of the extinction even for the largest-diameter disks. Model calculations for size param-eters near 1 predict absorption efficiencies slightlygreater than unity. Measured data values partiallyconfirm the predictions but are hampered somewhatby statistical uncertainties and expected field non-uniformity effects. For large-size parameters ~ka .2!, the disks begin absorbing with magnitudes simi-lar to that calculated for a semi-infinite thin film.Absorption also appears to peak for a finite thickness,although again data uncertainties limit confirmationof this result. A Drude model was used to derive anindex of refraction of approximately 10 for both thereal and the imaginary parts. This approximationseems reasonable because data and model differencesat broadside incidence are no larger than 30% formost of the disks.

The authors thank the U.S. Air Force for its sup-port of this research. Also, thanks go to VictorHarding of Advanced ElectroMagnetics, Inc. forsupplying the raw materials and Thomas Willis ofAT & T Bell Labs for providing a copy of the diskmodel.

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10. Cabot Corporation, Carbon Blacks for Specialty Applications, tion cross section for a moderately conducting thin cylinder,”

7

North American Tech. Rep. S-136 ~Cabot Corporation SpecialBlacks Division, Billerica, Mass., 1995!.

11. M. J. Venner, “The absorption and scattering cross-section ofmoderately conducting disks at 35 GHz,” Ph.D. dissertation~New Mexico State University, Las Cruces, N.M., 1997!, pp.10–12.

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and absorption of electromagnetic radiation by disc shapedaerosols; an improved computational formulation and com-puter code,” Radiation Laboratory Rep. RL023618-1-T ~Elec-trical Engineering and Computer Sciences Department,University of Michigan, Ann Arbor, Mich., 1986!.

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Absorption Cross Section of Moderately Conducting Disks at 35 GHz