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1 Nght Vson and Electronc Sensors Drectorate RDER-NV-TR-67 A Note on the rewster Angle n Lossy Delectrc Meda Approved for Publc Release: Dstrbuton Unlmted Fo elvor, Vrgna

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3 Nght Vson and Electronc Sensors Drectorate RDER-NV-TR-67 A Note on the rewster Angle n Lossy Delectrc Meda y Ian McMchael October 010 Approved for Publc Release: Dstrbuton Unlmted. Scence and Technology Dvson FORT ELVOIR, VIRGINIA

4 A Note on the rewster Angle n Lossy Delectrc Meda In electromagnetsm, the rewster angle, θ, s the angle of ncdence at whch there s no reflecton of the parallel (.e. vecal) polarzed wave from a planar nterface between materals wth dfferng permttvty. That s, the reflecton coeffcent goes to zero at ths angle and all of the ncdent power s transmtted nto the delectrc medum. The reflecton coeffcent s a functon of the consttutve parameters permttvty,, and permeablty, µ, of the two meda formng the nterface as well as the angle of ncdence. For meda wth µ =, the rewster angle s commonly expressed as θ µ r = tan 1, (1) r where the r subscrpt denotes the relatve consttutve parameter, the subscrpt denotes the parameter of the medum n whch the ncdent wave s propagatng, and the t subscrpt denotes parameter of the medum nto whch the wave s transmtted. We assume the ncdent medum s free space and the transmtted medum s sol for the purposes of ground penetratng radar (GPR). For perpendcularly polarzed waves, no real ncdent angle exsts that wll reduce the reflecton coeffcent to zero. [1] For clarty, we now defne some basc defntons assocated wth plane waves ncdent at a planar nterface: 1. Plane of ncdence the plane contanng the propagaton vector of the ncdent wave k and the unt normal to the nterface.. Parallel polarzaton the electrc feld of the ncdent wave les n the plane of ncdence. The ohogonal magnetc feld s n the drecton transverse to plane of ncdence, so we call ths a transverse magnetc (TM) wave. Ths s referred to as vecal polarzaton n the GPR communty. 3. Perpendcular polarzaton the electrc feld of the ncdent wave les normal to the plane of ncdence. Snce the electrc feld s n the drecton transverse to the plane of ncdence, we call ths a transverse electrc (TE) wave. Ths s referred to as horzontal polarzaton n the GPR communty. In realstc envronments nvolvng GPR, one must consder the lossy component of the sol s permttvty. Sols become lossy as mosture s added, especally n mneralzed sols wth hgh salnty [], [3]. The sol mosture strongly affects the real pa of the permttvty. Whle the salnty of sols does not have a dramatc nfluence on the real pa of the permttvty, the A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 1

5 magnary pa s strongly affected by both the salnty and mosture of the sol, especally at low frequences. Expressng the complex permttvty for the ncdent wave medum and the transmtted wave medum as ~ r = r j r, () ~ = j, (3) we can attempt to fnd the rewster angle usng (1) as follows. We change the complex permttvty nto phasor form to make the mathematcs assocated wth the square root slghtly easer. That s, we can wrte θ = tan 1 r j j r = tan 1 ~ ~ r exp exp ( jφ t ) ( jφ ) = tan 1 ~ ~ r 1 exp j ( φ φ ) t. (4) where we use φ to represent the phase angle as opposed to the azmuthal angle of ncdence. Now, changng back from phasor form n order to take the nverse tangent, the argument becomes ~ ~ 1 1 θ = tan 1 ~ cos [ φ t φ ] + j [ φ t φ ] ~ sn. (5) r r a b We then attempt to solve for the rewster angle usng the defnton of the nverse tangent of a complex number as ( a + jb) ( ) a + jb j 1 j θ = tan 1 ( a + jb) = ln, (not a realzable angle!) (6) 1+ j where the constants a and b are defned n (5) and ln s the natural logarthm. It can be seen from (6) that the rewster angle for a complex valued permttvty s a complex value. Snce complex valued physcal angles do not exst, we can conclude that the reflecton coeffcent cannot truly go to zero n a sol wth complex valued permttvty. In other words, n a lossy sol there s no angle for whch all of the ncdent TM wave power wll be transmtted nto the sol. A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD

6 Though the reflecton coeffcent does not exactly go to zero n a lossy medum, there stll exsts a mnmum value at some angle, whch s often referred to as the pseudo-rewster angle. We defne the reflecton coeffcent for a TM wave ncdent upon a sol from free space as n~ snθ n~ cos θ TM Γ =, (7) n~ snθ + n~ cos θ where θ s the ncdent angle and n ~ s the complex ndex of refracton of the sol n ~ = µ. (8) ( j )( ) If we assume a nonmagnetc medum, the relatve permeablty n (8) wll be unty,.e. µ = 1. We can observe how the magnary component of the permttvty affects the wave propagaton at a hypothetcal nterface by plottng the reflecton coeffcent for several values of magnary permttvty whle keepng the real pa constant. Fgure 1 shows the magntude of the reflecton coeffcent plotted versus angle of ncdence for four dfferent magnary permttvty components. The real component of the permttvty was kept constant at unty. For the case of no loss and a very low (unty) real permttvty, we observe no reflecton. For the lossy case, we can see that the magnary component of the permttvty affects the angle at whch we fnd the dp n the reflecton coeffcent. We can also observe that ths dp becomes sharper for hgher losses. We can draw two mpoant conclusons from these observatons: 1. One cannot smply use the real pa of the permttvty to determne the angle for mnmum reflecton (whch we may call the pseudo-rewster angle ) gven a lossy sol.. In real envronments the sol surface (delectrc nterface) s not perfectly flat. If there s a large amount of loss n the sol, small devatons n angle around the pseudo-rewster angle wll sgnfcantly ncrease the reflecton coeffcent. As an example demonstratng the second pont stated above, the change n the reflecton coeffcent due to angle devatons around the pseudo-rewster angle can be observed n Fgure 1 for the hghest loss case. The mnmum value of the reflecton coeffcent for the case where ~ = 1 j100 occurs when Γ = at θ = 84.5 o. The reflecton coeffcent doubles f the angle of ncdence s reduced by o or f t s ncreased by only 3.5 o. Ths change n A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 3

7 reflecton due to small devatons n angle would preclude the explotaton of the rewster angle for GPR applcatons n lossy sols wth realstc undulatng surfaces. However, f we examne the case of a medum wth low loss, say ~ = 1 j, the reflecton coeffcent doubles from ts mnmum f the angle s reduced by o or f t s ncreased by o. In the low loss case, there s slghtly more leeway for angle devatons whle keepng the reflecton coeffcent relatvely low. Fgure 1. Reflecton coeffcent for meda of varyng levels of loss. When mosture s added to sol, the real pa of the permttvty s affected as well as the magnary component. Usng data collected from the U.S. Army RDECOM CERDEC Nght Vson and Electronc Sensors Drectorate s mne lanes faclty and repoed n [4], the reflecton coeffcent can be determned for a sol wth varyng levels of mosture. The sol under test n the present analyss s called bluestone gravel. The complex permttvty was derved usng S- parameter measurements of a sol sample n a coaxal holder from 1 MHz GHz. Varyng levels of mosture were added to the sample and the volumetrc mosture percentage was recorded. Table 1 shows the permttvty values repoed from the measurements at 00 MHz. Fgure shows the reflecton coeffcent for the bluestone gravel versus ncdent angle as calculated usng equaton (7) for a frequency of 00 MHz. A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 4

8 Table 1. Permttvty values of bluestone gravel sol from the NVESD mne lanes faclty at 00 MHz. Measured Permttvty for luestone Gravel at 00 MHz Volumetrc Mosture (%) Permttvty ( ~ = j ) j j j j j17.0 Fgure. Reflecton coeffcent for sol from the NVESD mne lanes faclty for varyng levels of mosture at 00 MHz. A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 5

9 Table shows the permttvty values repoed from measurements at GHz. Fgure 3 shows the reflecton coeffcent for the bluestone gravel versus ncdent angle as calculated usng equaton (7) for a frequency of GHz. Whle the reflecton coeffcent reaches a mnmum very close to zero n ths case, t should be noted that ths mnmum changes drastcally wth changes n volumetrc mosture. Table. Permttvty values of bluestone gravel sol from the NVESD mne lanes faclty at GHz. Measured Permttvty for luestone Gravel at GHz Volumetrc Mosture (%) Permttvty ( ~ = j ) j j j j j3.3 A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 6

10 Fgure 3. Reflecton coeffcent for sol from the NVESD mne lanes faclty for varyng levels of mosture at GHz. The rewster angle expresson gven n (6) merely shows that there s no physcal angle at whch the reflectvty goes to zero for a lossy medum and should not be used for actual calculatons. One may be tempted to take the real pa or the absolute value of the complex angle, but nether of these corresponds to the mnmum value of the reflecton coeffcent. It can be shown that the reflecton coeffcent reaches a mnmum when the real pa of Γ = 0. As stated n [5] wthout dervaton, the pseudo-rewster angle, whch s the angle for whch the absolute magntude of the complex reflecton coeffcent reaches ts mnmum, ~ θ, s gven as [ 1] ( 1) ( ) ( 1) ( ) ( + ) 1 ~ sn θ =, (9) for r = 1and r = 0 as n the case where the ncdent feld s n free space. The angle gven n (9) s the angle from grazng. Equaton (9) can be verfed by examnng the reflecton coeffcents plotted n Fgure. For example, the calculated mnmum reflecton coeffcent at 00 MHz for 30.3% volumetrc mosture ( ~ o = 19 j17 ) s Γ = 78.74, whch s n mn accordance wth the plotted value. The calculated mnmum reflecton coeffcent at GHz for 30.3% volumetrc mosture ( ~ o = 17.5 j3. 3 ) s Γ = Ths example not only shows mn the valdty of equaton (9), t also shows that dsperson plays a role n determnng the pseudo- rewster angle. Snce the complex permttvty s a functon of frequency, the reflecton coeffcent must be a functon of frequency too. Most GPR systems are wdeband, whch means that the pseudo-rewster angle wll not be constant over the entre radated band. It has been shown n ths note that the rewster angle does not exst for delectrc nterfaces nvolvng complex permttvty. A pseudo-rewster angle does exst, whch s the angle at whch the magntude of the reflecton coeffcent reaches a mnmum. However, the pseudo-rewster angle cannot be calculated usng the standard rewster angle formula. Fuhermore, the reflecton coeffcent can rapdly rse for small angle devatons away from the pseudo-rewster angle when the sol s very lossy. Fnally, the dspersve nature of lossy sols means that the pseudo-rewster angle wll not be constant over a broad band. A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 7

11 References [1] C. alans, Advanced Engneerng Electromagnetcs, New York: John Wley & Sons, [] Y. Shao, Q. Hu, H Gao, Y. Lu, Q. Dong, C. Han, Effect of Delectrc Propees of Most Salnzed Sols on ackscatterng Coeffcents Extracted From RADARSAT Image, IEEE Trans. Geosc. Remote Sensng, vol. 41, no. 8, pp , Aug [3] J. Hpp, Sol Electromagnetc Parameters as Functons of Frequency, Sol Densty, and Sol Mosture, Proc. of the IEEE, vol. 6, pp , Jan [4] J. Cus, D. Leavell, C. Wess, R. Noh, E. Smth, J. Coes, R. Castellane, M. Felds, Characterzaton of Sols from the Nght Vson and Electronc Sensors Drectorate Mne Lane Faclty, Fo elvor, VA, U.S. Army Corps of Engneers, Engneer Research and Development Center, July 003. [5] E. Shotland, Three Angles Sgnfcant n Rado Propagaton, IEEE Trans. Antennas Propag., vol. 0, no. 6, pp , Nov A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 8

12 Appendx Dervaton for the pseudo-rewster angle for meda wth complex valued permttvty: The reflecton coeffcent for a parallel polarzed wave ncdent from free space onto a nonmagnetc medum s gven as n snθ n cos θ TM Γ =. (A1) n snθ + n cos θ where n = j for nonmagnetc meda. Wth complex valued permttvty and nonmagnetc meda, we can rewrte (A1) as ( j ) snθ ( j ) cos θ ( j ) snθ + ( j ) cos θ TM Γ =. (A) Each square root can be separated nto a complex number usng the followng dentty: x + y + x x + y x x + jy = ± j, (A3) where the sgn should be chosen to be the same as the sgn of y. After separatng the square roots, we can combne real and magnary terms and then multply the top and bottom by the conjugate of the denomnator to get the followng Γ TM = ( + ) sn θ ( cos θ ) + + j snθ ( A A ) ( + ) sn θ + snθ ( A + A ) + ( cos θ ) + [ ], (A4) ( cos θ ) + ( cos θ ) A = (A5) ( cos θ ) + + ( cos θ ) A = (A6) A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 9

13 The pseudo-rewster angle for lossy meda ~ θ, whch we are defnng as the angle for whch the magntude of the complex reflecton coeffcent reaches ts mnmum, occurs when the real pa of the complex reflecton coeffcent equals zero. Therefore, to solve for the rewster angle, we smply set the real pa of (A4) equal to zero. The denomnator of the real pa of (A4) vanshes when solvng the algebrac equaton so that we only need to solve the followng equaton for θ ( + ) sn θ ( cos θ ) + = 0. (A7) Usng the double angle formula and combnng lke cosne terms, (A7) takes the form cos [( + ) 1] + cos 4 ( + θ ) θ a We then use the quadratc equaton [ ] + [( ) 4( + ) ] = 0 b c (A8) b ± b 4ac cos θ =, (A9) a where the constants are defned as shown n equaton (A8). We choose the root wth the negatve sgn n the numerator to assure that the quotent s between -1 and 1 and corresponds to real ncdent angles. The fnal step s to take the nverse cosne and solve for θ, whch satsfes the condton for ~ θ, as follows The expresson gven n (A10) can be condensed by assgnng whch smplfes the pseudo-rewster angle to ~ 1 1 b b 4ac θ = cos. (A10) a E = +, (A11) r r ~ θ = 1 cos 1 ( 1+ E ) ( 1) ( 1)( ) r r E E E E r ( E 1). (A1) A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 10

14 The soluton (A11) can be shown to be equvalent to that of (9), whch was gven n the lterature. A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD 11

15 Dstrbuton for Repo: RDER-NV-TR-67 Defense Techncal Informaton Center 875 John J. Kngman Hghway Sute 0944 Fo elvor, VA Drector US Army RDECOM CERDEC Nght Vson and Electronc Sensors Drectorate ATTN: RDER-NVS-C Fo elvor, VA 060 D-1 A Note on the rewster Angle n Lossy Delectrc Meda; Ian McMchael, US Army RDECOM CERDEC NVESD