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.Journal of Applied Geophysics 45 2000 111125www.elsevier.nlrlocaterjappgeo
Ground penetrating radar polarization and scatteringfrom cylinders
Stanley J. Radzevicius), Jeffrey J. DanielsDepartment of Geological Sciences, The Ohio State Uniersity, 125 South Oal Mall, Columbus, OH 43210-1308 USA
Received 18 October 1999; accepted 10 July 2000
.Ground penetrating radar GPR polarization is an important consideration when designing a GPR survey and is useful toconstrain the size, shape, orientation, and electrical properties of buried objects. The polarization of the signal measured bythe receive antenna is a function of the polarization of the transmit antenna and scattering properties of subsurface targets.Circular cylinders represent important environmental and engineering targets such as buried pipes, wires, and rebar. Thebackscattered fields from cylinders may be strongly depolarized depending on the orientation of the cylinder relative to theantennas, the electrical properties of the cylinders, and the radius of the cylinder compared to the incident wavelength. Thesepolarization dependent scattering properties have important implications for target detection, survey design, and datainterpretation.
As the radius-to-wavelength ratio of metal and plastic pipes decreases, the backscattering properties become morepolarization dependent. When using linearly polarized dipole antennas, metallic pipes and low impedance dielectric pipes arebest imaged with the long axis of the dipole antennas oriented parallel to the long axis of the pipes. High impedance,dielectric pipes, are best imaged with the long axis of the dipoles oriented orthogonal to the long axis of the pipes. q 2000Elsevier Science B.V. All rights reserved.
Keywords: Ground penetrating radar; Polarization; Cylinders
.Ground penetrating radar GPR is a commongeophysical technique for investigating the shal-
low subsurface Annan and Davis, 1989; Ol-hoeft, 1992; Peters et al., 1994; Daniels et al.,
.1998 . The vector nature of the GPR electro-
) Corresponding author.E-mail address: [email protected]
.S.J. Radzevicius .
magnetic field, commonly referred to as polar-ization, is described Beckman, 1968; Born and
.Wolf, 1980; Mott, 1986; Balanis, 1989 and islargely ignored by interpreters of GPR data.
.Investigations by Roberts 1994 and Roberts .and Daniels 1996, 1997 have demonstrated the
potential of using the polarization characteristicsof GPR for defining the size, shape, orientation,and material properties of buried objects.
This paper describes polarization and cylin-der scattering theory and concepts relevant forthe GPR practitioner. Analytic solutions and
0926-9851r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. .PII: S0926-9851 00 00023-9
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125112
GPR data examples over buried pipes of vary-ing radii are used to illustrate polarization con-cepts and verify the applicability of theory tocommercial dipole antennas and physical mod-els. This manuscript also describes which dipoleantenna configurations will result in optimalsurvey design, depending on whether the targetsof interest are metallic or plastic pipes.
The electromagnetic field at a given point inspace, at a given time, has both a magnitude anda direction, and thus is described by vectors. Asthe electromagnetic wave propagates, the orien-tation and magnitude of these vectors change asa function of time. Polarization describes themagnitude and direction of the electromagneticfield as a function of time and space. When the
time varying EM fields vary sinusoidally time.harmonic , polarization may be classified as
linear, circular, or elliptical. If the vector thatdescribes the electric field as a function of timeis always directed along a straight line, the fieldis said to be linearly polarized. If the vectorsweeps out a circle, it is referred to as circularpolarization. Both are special cases of ellipticalpolarization, in which the electric field tracesout an ellipse. An arbitrary electromagnetic fieldcan be described by three orthogonal basis vec-tors. Since the electric and magnetic fields areorthogonal to the direction of propagation, if wechoose one of the basis vectors in the directionof propagation, the electric field can be decom-posed into two orthogonal basis vectors. Theelectric field of a wave traveling in the z direc-tion can be described by two orthogonal compo-
.nents as given by Balanis 1989 :
E z ,t sE eya zcos v tyb zyf . .x x0 xand
E z ,t sE eya zcos v tyb zyf 1 . . .y y0 ywhere a represents the attenuation constant, bthe phase constant, v the angular frequency, f
the phase, and E and E are the maximumx0 y0amplitudes of the E and E components, re-x yspectively.
2.1. Linear polarization
For a wave to have linear polarization, thetime-phase difference between the two compo-nents must be
Dfsf yf snp ns0,1,2,3, . . . 2 .y x
2.2. Circular polarization
Circular polarization is achieved only whenthe magnitudes of the components are the sameand the time-phase differences are multiples ofpr2.
E sEx y
Dfsf yf s" q2n p 3 .y x /2 .where q and y refer to clockwise CW or
.counterclockwise CCW rotation. ns0, 1, 2,3, . . .
2.3. Elliptical polarization
Elliptical polarization is achieved only whenthe time-phase difference between the two com-ponents are odd multiples of pr2 and theirmagnitudes are not the same or when the time-phase difference between the components arenot equal to multiples of pr2, regardless oftheir magnitudes.
Case 1:E /Ex y
Dfsf yf s" q2n p 4 .y x /2ns0, 1, 2, 3, . . . ,where q and y refer to CWor CCW rotation.
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125 113
npDfsf yf /" ns0,1,2,3, . . .y x 2
Df)0 for CW Df-0 for CCW.Pipes and other objects scatter energy prefer-
entially, depending on the incident polarization.The polarization and orientation of the transmitantenna is thus important to ensure sufficientenergy is scattered from subsurface targets toallow measurement by the receive antenna.Preferential scattering may result in depolariza-tion of the incident field. Depolarization occurswhen the amplitude or phase of the incident
..field components Eq. 1 are modified suchthat the scattered field results in a differentpolarization. The ability of the receive antennato measure these scattered fields is determinednot only by the power of the scattered fields,but also by the polarization match between thescattered fields and the receive antenna. Thepolarization of the field incident on the receiveantenna is determined by the polarization of thefield radiated by the transmit antenna and thedegree of depolarization experienced by scatter-ing from subsurface objects. It is thus importantto understand the polarization properties of GPRantennas and scattering from subsurface objects.
Most commercial GPR antennas are dipole orbow-tie antennas that radiate linearly polarizedenergy with the majority of the radiated electricfield oriented along the long axis of the dipoleor bow-tie. For a description of dipole fields
.over a half-space, consult Annan 1973 , Annan . .et al. 1975 , Arcone 1995 , Engheta et al.
. .1982 , Smith 1984 and Radzevicius et al. .2000b . A complete polarization mismatch us-ing dipole antennas results when the scatteredfield and polarization of the receive antenna are
..both linearly polarized Eq. 2 and oriented atright angles to each other. For example, rotatingideal dipole antennas orthogonal to each other .crossed-dipoles results in a complete polariza-tion mismatch. Spiral, or other circularly polar-
ized antennas, are also used for pipe detection.A complete polarization mismatch using circu-larly polarized antennas results when the scat-tered field and receive antenna are both circu-
..larly polarized Eq. 3 , but have electric fieldswith opposite rotation directions left and right
3. Normal incidence plane wave scattering by(circular cylinders fundamental theory and
)conclusions from analytical solutions
Cylinders represent an important class of ob-jects for GPR since they represent importantenvironmental and engineering targets and alsobecause their scattering properties are stronglypolarization dependent. A brief discussion ofthe importance of polarization on the scatteringof plane waves normally incident on both di-electric and conductive circular cylinders is nowdescribed. The reader is referred to Balanis . .1989 and Ruck et al. 1970 for a more de-tailed explanation of equations and for obliqueincidence.
Two linearly independent basis vectors . ..polarizations Eq. 1 are necessary to de-scribe scattering from both dielectric and per-fectly conducting cylinders. It is convenient tochoose these polarization vectors such that onevector is oriented along the long axis of the
cylinder E parallel or transverse-magnetic ..TM and the other vector oriented orthogonal
to the long axis of the cylinder E perpendicular .. .or transverse electric TE Fig. 1 . TM polar-
ization is achieved when the long axis of thetransmit and receive dipole antennas are ori-ented parallel to the long axis of the cylinderand the survey direction is orthogonal to the
.long axis of the cylinder Fig. 2a . TE polariza-tion is achieved when both antenna axes areoriented orthogonal to the long axis of the cylin-der and the survey direction is orthogonal to the
.long axis of the cylinder Fig. 2b .The scattered field is a function of the electri-
cal properties of the cylinder and surrounding
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125114
.Fig. 1. Definitions of E parallel TM and E perpendicular .TE polarizations relative to a cylinder, as defined in
.Balanis 1989 . TM polarization occurs when the electricfield is parallel to the long axis of the cylinder. TEpolarization occurs when the electric field is perpendicularto the long axis of the cylinder. E and H represent theelectric and magnetic fields respectively, while a repre-sents the cylinder radius, b represents the propagationvector, f represents the scattering angle, r represents theradial distance from the cylinder, and x, y, z represent thecoordinates of a cartesian coordinate system.
material, the distance from the cylinder, and thescattering angle f, as defined in Fig. 1. Ascattering angle of 1808 represents backscatter-ing. Since circular cylinders have a cylindricalshape, their scattering properties are conve-niently described using Hankel and Bessel func-tions because they represent cylindrical waves.Incident and scattered fields for conductive anddielectric cylinders are described in terms ofcylindrical coordinates in Appendix A.
.The radar cross-section RCS represents aconvenient way to describe the strength of scat-tered fields observed in the far-field. The RCSis defined as the the area intercepting the amountof power that when scattered isotropically, pro-duces at the receiver a density that is equal to
the density scattered by the actual target Bal-.anis, 1989 .
< s < 2E2RCSs lim 4p r 6 .2ir` <
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125 115
alternatively the RCS per unit length Balanis,.1989 .
< s < 2ESWs lim 2p r 7 .2ir` <
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.Fig. 3. Scattering widths for metallic cylinders normalized by the wavelength l of the incident field. As the radius of thecylinder becomes small compared to wavelength, TM scattering widths become nearly constant as a function of scatteringangle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TM polarizationbackscattering widths are greater than the TE polarization scattering widths for most cylinders.
when they scatter electric field components or-thogonal to the field components radiated by thetransmit antenna. Scattered cross-componentsare produced by scattering from rough planes or
most small objects. Cross-components compo-.nents not present in incident field are not intro-
duced by scattering of plane waves normallyincident on infinite circular cylinders, as seen in
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.Fig. 4. Scattering widths for high impedance dielectric cylinders, normalized by the wavelength l of the incident field. Asthe radius of the cylinder becomes small compared to wavelength, TM scattering widths become nearly constant as afunction of scattering angle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TEpolarization backscattering widths are greater than the TM polarization scattering widths for small diameter cylinders.
. . .Eqs. 12 23 . Balanis 1989 describes thecase of a plane wave obliquely incident on bothdielectric and conducting cylinders. Balanis ob-served that scattering from a perfectly conduct-ing infinite cylinder does not introduce addi-
tional components in the scattered field that arenot present in the incident field. This is not thecase for dielectric cylinders, which introduceorthogonally polarized components underoblique wave incidences. Cylinders having a
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125118
.Fig. 5. Scattering widths for low impedance dielectric cylinders, normalized by the wavelength l of the incident field. Asthe radius of the cylinder becomes small compared to wavelength, TM scattering widths become nearly constant as afunction of scattering angle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TMpolarization backscattering widths are greater than the TE polarization scattering widths for small diameter cylinders.
finite length also produce depolarization fromedge scattering.
It is not necessary for a target to produce ascattered cross-component to be visible with
cross-pole antennas. A long metallic pipe, while .a strong depolarizer Fig. 3 does not produce
field components that were not originally pre-sent in the incident field. The linear geometry of
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125 119
pipes results in field components that are ori-ented both parallel and orthogonal to the longaxis of the pipe. The reflected and transmitted
fields for thin pipes are related by the following .relationships Daniels et al., 1988 :
yi k r S Se x x x yE sE ysinu cosus t / S Sr y x y y
=cosu 9 .sinu
E eyi k rss S yS . y y x x /E 2 rt
=sin2uyS sin2uqS cos2u 10 ./x y y xwhere E and and E are the scattered ands ttransmitted fields, respectively, and u is theangle between the long axis of the transmitdipole and the long axis of the cylinder. Forlinear targets S and S are small comparedx y y xto other components and thus
E eyi k rss S yS sin2u 11 . .y y x x /E 2 rt
The ratio E rE is maximized when us458 fors tboth dielectric and conductive pipes and thuscrossed-dipole antennas at 458 with respect tocylinders represent the best antenna geometry to
.image cylinders Fig. 2c .
4. Linear polarization and scattering from( )pipes physical model examples
To test the applicability of the analytical . . .solutions Eqs. 12 23 and Fig. 6 , data were
.Fig. 6. Backscattering widths as a function of radius a , .normalized by the wavelength l of the incident field.
Solid lines represent TM polarization and dashed lines .represent TE polarization. a TM backscattering widths
are greater than TE backscattering widths for most metallic .cylinders. b TE backscattering widths are greater than
TM backscattering widths for small diameter, high .impedance, dielectric cylinders. c TM backscattering
widths are greater than TE backscattering widths for smalldiameter, low impedance, dielectric cylinders.
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125120
Fig. 7. Geometry of bow-tie antenna elements for 500 .MHz air multi-component antenna. The antenna consists
.of four transmitting elements T1, T2, T3, T4 and two .receiving elements R1, R2 . The figure is drawn to scale.
recorded in a sand test pit using a Geophysical .Survey Systems GSSI bow-tie, multi-compo-
nent antenna having a center frequency of 500 . .MHz in air Fig. 7 . The antenna consisted of
two receiving elements and four transmittingelements eight different transmittingreceiving
.combinations that were used to record the data.Fig. 8 is a plot of the amplitude spectrum
obtained by taking the Fourier transform of bothco-pole and cross-pole traces with no pipespresent. The soil interface influences currentdistribution and impedance of GPR antennas byan amount determined by the antenna designand the electromagnetic properties of the ground.A soil interface causes the antenna to radiate at
.lower frequencies 270 MHz peak on interface .than in an air whole-space 500 MHz peak .
The amplitude spectrum varies smoothly as afunction of frequency no nulls at a given fre-
.quency and is approximately Gaussian in shape.A large polarization mismatch with cross-pole
antennas is observed in Fig. 8 co-pole larger.amplitude than cross-pole for all frequencies
because each frequency is linearly polarized.Constant polarization for all frequencies is a
desirable feature of dipole antennas de Jongh et.al., 1998 . While time-domain impulse radar
. .Fig. 8. Amplitude spectrum for co-pole solid and cross-pole dashed traces for an antenna located on the soil interfacewith no buried pipes. The amplitude spectrum varies smoothly as a function of frequency and is approximately Gaussian inshape with a peak frequency of 270 MHz. The strong polarization mismatch for cross-pole antennas results in larger co-poleamplitudes for all frequencies because each frequency is linearly polarized.
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125 121
radiates a pulse composed of many frequencies .Fig. 8 , the frequencies centered about 270MHz have the largest amplitudes and are themost significant for determining the responseobserved in field data.
To illustrate the above concepts, data wererecorded over plastic and metal pipes with sur-vey directions normal to the long axis of thepipes and at 158, 308, and 458 angles to the longaxis of the pipes. Long pipes were used to avoidresonance and edge effects. Pipes of varyingradii were used to demonstrate the most signifi-cant features observed in the analytic solutions.Fig. 9 shows the results of data recorded over a0.0032 m radius copper pipe buried at a depthof 0.46 m and having a length of 3.05 m. A soilpermittivity probe, described by Caldecott et al. .1985 and operating at 40 and 60 MHz fre-quencies gave relative permittivity values of 4near the surface and graded to a permittivity of7 at a depth of 0.6 m. The peak 270 MHzfrequency has a nominal wavelength of 0.42 mand thus the copper pipe with a radius of 0.0032m yields a radius-to-wavelength ratio of 0.008.While this radius-to-wavelength ratio is only fora single frequency, most of the energy radiatedfrom the antenna is composed of frequenciesthat have TM polarization scattering widthsgreater than TE polarization scattering widths .Figs. 3 and 6a . Fig. 9 demonstrates that this is
.the case with the TM component T2R1 much .greater than the TE component T1R2 . Cross-
.pole components T2R2 and T3R1 yield maxi-mum values when the dipole antennas are ori-ented at 458 to the long axis of the cylinder, as
. .described by Eqs. 9 11 . No difference be-tween cross-pole configurations would be ex-pected with ideal antennas over homogeneoussoils or at a 458 survey angle for co-pole config-urations. Coupling between the four transmit-ting elements and two receiving elements, inaddition to small construction and alignmentdifferences, results in a slightly different an-tenna response between the two cross-pole . .T2R2 and T3R1 and co-pole T2R1 and T1R2configurations. Heterogeneities in the soil also
Fig. 9. GPR survey normal and at 158, 308, 458 to the longaxis of a copper pipe buried at a depth of 0.46 m, having aradius of 0.0032 m, and a length of 3.05 m. A 270 MHzcenter frequency antenna on soil with a relative permittiv-ity of 7 has a nominal wavelength of 0.42 m and thus thecopper pipe yields a radius-to-wavelength of 0.008. Figs. 3and 6a suggest that this small radius-to-wavelength ratio
.results in larger TM scattering widths T2R1 Fig. 7 than .TE scattering widths T1R2 Fig. 7 . Cross-pole compo-
.nents T2R2 and T3R1 yield maximum values when thedipole antennas are oriented at 458 to the long axis of the
. .pipe as described by Eqs. 9 11 .
produce differences due to the antenna elementoffsets.
Data were also recorded over metal and plas-tic pipes, having a radius of 0.0381 m, a lengthof 3.66 m, and buried at a depth of 0.61 m. A270 MHz antenna yields a nominal radius-to-wavelength ratio of approximately 0.09. Figs. 3and 6a suggest similar backscattering widths forTM and TE polarizations for metallic pipes
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125122
having this radius-to-wavelength ratio. In con-trast to the thin metal pipes, Fig. 10 demon-
.strates that the TM polarization T2R1 is only .slightly larger than the TE polarization T1R2 ,
as predicted from analytic solutions. Cross-pole .components T2R2 and T3R1 still yield maxi-
mum values when the dipole antennas are ori-ented at 458 to the long axis of the metal pipe,as in the thin metallic pipe case. In contrast to
Fig. 10. GPR survey normal and at 158, 308, 458 to thelong axis of a steel pipe buried at a depth of 0.61 m,having a radius of 0.0381 m, and a length of 3.66 m. A270 MHz center frequency antenna yields a nominal radiusto wavelength ratio of approximately 0.09. Unlike the thinpipe, Figs. 3 and 6a suggest similar TM and TE scatteringwidths for metallic pipes having this radius-to-wavelength
.ratio. In this figure, the TM T2R1 polarization is only .slightly brighter than the TE polarization T1R2 , as pre-
.dicted. Cross-pole components T2R2 and T3R1 still yieldmaximum values when the dipole antennas are oriented at458 to the long axis of the pipe, as in the thin pipe case.
Fig. 11. GPR survey normal and at 158, 308, 458 to thelong axis of a PVC pipe buried at a depth of 0.61 m,having a radius of 0.0381 m, and a length of 3.66 m. A270 MHz center frequency antenna yields a nominal radiusto wavelength ratio of approximately 0.09. Figs. 4 and 6bsuggest larger backscattering widths for TE compared toTM, given this radius-to-wavelength ratio. This figureverifies that the TE polarization is greater than the TM
.polarization. Cross-pole components T2R2 and T3R1still yield maximum values, as in the metallic pipe case,when the crossed-dipole antennas are oriented at 458 to thelong axis of the pipe.
metal pipes, Fig. 11 demonstrates that the TE .polarization T1R2 is larger than the TM polar-
.ization T2R1 , as predicted from analytic solu- .tions. Cross-pole components T2R2 and T3R1
still yield maximum values when the dipoleantennas are oriented at 458 to the long axis ofthe plastic pipe, as in the metallic pipe case.
( )S.J. Radzeicius, J.J. DanielsrJournal of Applied Geophysics 45 2000 111125 123
The scattering properties of cylinders arestrongly polarization dependent. It is thus im-portant to understand the scattering properties ofcylinders and the polarization properties of an-tennas used in the GPR survey. Knowledge ofthe electrical properties of the buried pipe andsurrounding soil allows one to constrain thediameter of the pipe. The analytical solutionsand field data demonstrate the importance ofconducting GPR surveys, with two axially ro-tated measurements to avoid nulls over thinmetallic pipes, when using linearly polarizedco-pole or cross-pole antennas. An alternative isto use circularly polarized antennas that auto-matically rotate the polarized vector in spaceand thus removes the direction of signal nulls.As the radius of the metallic pipes increased,the TM and TE polarization backscatteringwidths become more similar and the need foraxially rotated measurments diminished.
When using linearly polarized dipole anten-nas, metallic pipes are best imaged with thelong axes of the dipoles oriented parallel to thelong axis of the pipe. Small diameter, highimpedance, dielectric pipes are best imaged withthe dipole axes oriented orthogonal to the longaxis of the pipe. Crossed-dipole antennas can beused to reduce clutter and improve antennaisolation when stratigraphy is considered clutterand only pipes or other depolarizing targets areof interest. Maximum amplitudes are observedover pipes when the crossed-dipoles are ori-ented at 458 to the pipe. The best choice ofantennas and polarizations for a particular sur-vey depends on the targets of interest and thefield conditions.
( )Appendix A. Cylinder scattering equations
Below are equations, in cylindrical coordi-nates, that describe normally incident plane wavescattering from infinitely long conductive and
dielectric circular cylinders. The scattered fieldis a function of the electrical properties of thecylinder and surrounding material, the distancefrom the cylinder, and the scattering angle. Han-kel and Bessel functions represent cylindricalwaves and are useful for describing scattering
. .from cylindrical cylinders. Eqs. 12 23demonstrate that cross-components components
.not present in the incident field are not intro-duced by scattering of plane waves normallyincident on infinite cylinders. The reader is
. .referred to Balanis 1989 and Ruck et al. 1970for a more detailed explaination of equations.TM polarization normally incident on a per-fectly conducting cylinder case:
E i sE eyib0 x 12 .z 0` J b a .n 0s yn 2. i nfE syE i H b r e .z 0 n 02.H b a .n 0nsy`
Es sEs s0 14 .r fTE polarization normally incident on a perfectlyconducting cylinder case:
H i sH eyib0 x 15 .z 0` XJ b a .n 0s yn 2. i nfH syH i H b r eX .z 0 n 02.H b a .n 0nsy`
H s sH s s0 17 .r fTM polarization normally incident on a dielec-tric cylinder case:
E i sE eyib0 x 18 .z 0`
s ynE sE iz 0nsy`
=X X . . . .J b a J b a y e rm J b a J b a'n 0 n 1 r r n 0 n 1
XX 2. 2. . . . .e rm J b a H b a yJ b a H b a' r r n 1 n 0 n 1 n 0= 2. . i nfH b r e 19 .n 0
Es sEs s0 20 .r f
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TE polarization normally incident on a dielec-tric cylinder case:
H i sH eyib0 x 21 .z 0`
s ynH sH iz 0nsy`
=X X . . . .J b a J b a y m re J b a J b a'n 0 n 1 r r n 0 n 1
XX 2. 2. . . . .m re J b a H b a yJ b a H b a' r r n 1 n 0 n 1 n 0= 2. . i nfH b r e 22 .n 0
H s sH s s0 23 .r fThe TE polarization case is expressed in termsof magnetic fields for convenience and one mayconvert between electric and magnetic fieldsusing Amperes and Faradays laws. E i is theincident field, Es is the scattered electric field,H i is the incident magnetic field, H s is thescattered magnetic field, r, f, z are the stan-dard coordinates in a cylindrical coordinate sys-tem, r is the radial distance of the observationpoint, a is the cylinder radius,
xsrcosf 24 . .J b r is the Bessel function of the first kind orn
X .order n, J b r is the derivative of the Besselnfunction of the first kind with respect to theentire argument of the Bessel function,
2. .H b r is the Hankel function of the secondn2.X .kind of order n, H b r is the derivative ofn
the Hankel function of the second kind withrespect to the entire argument of the Besselfunction, b is the phase constant of the mate-0rial surrounding the cylinder, b is the phase1constant of the cylinder, is the ratio of therpermittivity of the cylinder to the surroundingmaterial, and m is the ratio of the permeabilityrof the cylinder to the surrounding material. Thee iwt time convention is used in this manuscript.
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