Electromagnetic scattering from large aspect ratio lossy dielectric solids in a conducting medium

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1 Eletroageti satterig fro large aspet ratio lossy dieletri solids i a odutig ediu Gary teve aela, Coastal ystes tatio Code R 6703 West Highway 98 Paaa City, Florida Abstrat The spheroidal T-atrix foralis developed by Haa -3 ad aela 4-6 for aousti satterig is exteded to eletroageti satterig fro lossy dieletri solids i a odutig ediu. The spheroidal T-atrix foralis exhibits superior perforae with respet to the spherial T-atrix foralis for objets that deviate appreiably fro a spherial shape. Both aousti (elasti) ad eletroageti satterig are solutios of the vetor Helholtz equatio. I the ase of elasti wave satterig, the displaeet field has 3 degrees of freedo orrespodig to the polarizatio states of the shear wave ad the logitudial ode. I the ase of eletroageti satterig, the eletri (ageti) field has polarizatio states orrespodig to left ad right-haded photos, but las a logitudial ode. The T-atrix desriptio of eletroagetis iis the T-atrix desriptio of elasti wave satterig i the absee of a logitudial ode. Ideed, the stress tesor of the displaeet is replaed by the exterior derivative of the eletri field i Betti s idetity i the derivatio of the T-atrix foralis of satterig fro a lossy dieletri solid. Cotiuity of the displaeet ad surfae tratio is replaed by otiuity of the tagetial opoets of the eletri ad ageti fields i the boudary oditios. I the ase of a tie haroi field, the presee of a fiite odutivity i the ediu is represeted by the isertio of a iagiary opoet of the waveuber that is proportioal to the odutivity i the ediu. I the ase of oplex waveuber, the Helholtz equatio is o loger a self-adjoit operator, ad the -atrix is o loger uitary. This artile desribes soe of the features uique to satterig i seawater due to the large odutivity of the ediu..0 ITRODUCTIO This artile desribes the appliatio of the spheroidal T-atrix foralis developed by Haa -3 ad aela 4-6 to eletroageti satterig fro isotropi lossy dieletri solids i a odutig ediu. etio.0 begis with a itrodutio to the salar ad vetor spheroidal wavefutios, followed by a derivatio of the geeralizatio of Betti s third idetity for eletroageti satterig, ad oludig with a derivatio of the T-atrix for a large aspet ratio dieletri solid ad a perfetly odutig solid based upo the geeralizatio of Betti s idetity. etio 3 oludes..0 PHEROIDAL T-MATRIX FORMALIM The spheroidal T-atrix desriptio of eletroageti satterig fro a odutig lossy dieletri solid i a odutig ediu is derived i this setio. The sybols ε,, σ, ad ε,, σ are used to deote the eletri suseptibility, ageti pereability, ad odutivity of the edia exterior ad iterior to the satterer, where the sybols ω ε iωσ ω ε iωσ (. are the waveuber i the exterior ad iterior regios, respetively. The otatio of Morse ad Feshbah 7 is used to deote the regular ad irregular spheroidal radial futios of the third id by the sybols je ( ξ ) ad he ( ξ ), respetively. These futios orrespod to the radial futios of the first ad third ids deoted () (3) by R ( ξ ) ad R ( ξ ) by Flaer 8 ad Abraowitz ad tegu 9. The sybols ( η ) ad () ( η ) are used to deote the agular futio of the first ad seod id. Here, f is the diesioless wave uber, where f is the sei-foal distae of the spheroidal oordiate syste. The oralizatio ovetio of Flaer 8 for the expasio oeffiiets d ( ) of the agular futios of the first id is adopted. The regular ad irregular basis futios of the salar Helholtz equatio i the exterior ad iterior of the satterer are defied as follows

2 Report Douetatio Page For Approved OMB o Publi reportig burde for the olletio of iforatio is estiated to average hour per respose, iludig the tie for reviewig istrutios, searhig existig data soures, gatherig ad aitaiig the data eeded, ad opletig ad reviewig the olletio of iforatio. ed oets regardig this burde estiate or ay other aspet of this olletio of iforatio, iludig suggestios for reduig this burde, to Washigto Headquarters ervies, Diretorate for Iforatio Operatios ad Reports, 5 Jefferso Davis Highway, uite 04, Arligto A Respodets should be aware that otwithstadig ay other provisio of law, o perso shall be subjet to a pealty for failig to oply with a olletio of iforatio if it does ot display a urretly valid OMB otrol uber.. REPORT DATE 0 EP 003. REPORT TYPE /A 3. DATE COERED - 4. TITLE AD UBTITLE Eletroageti satterig fro large aspet ratio lossy dieletri solids i a odutig ediu 5a. COTRACT UMBER 5b. GRAT UMBER 5. PROGRAM ELEMET UMBER 6. AUTHOR() 5d. PROJECT UMBER 5e. TAK UMBER 5f. WORK UIT UMBER 7. PERFORMIG ORGAIZATIO AME() AD ADDRE(E) Coastal ystes tatio Code R 6703 West Highway 98 Paaa City, Florida PERFORMIG ORGAIZATIO REPORT UMBER 9. POORIG/MOITORIG AGECY AME() AD ADDRE(E) 0. POOR/MOITOR ACROYM(). DITRIBUTIO/AAILABILITY TATEMET Approved for publi release, distributio uliited. POOR/MOITOR REPORT UMBER() 3. UPPLEMETARY OTE ee also ADM0046. Oeas 003 MT/IEEE Coferee, Held i a Diego, Califoria o epteber -6, 003. U.. Goveret or Federal Purpose Rights Liese 4. ABTRACT 5. UBJECT TERM 6. ECURITY CLAIFICATIO OF: 7. LIMITATIO OF ABTRACT UU a. REPORT ulassified b. ABTRACT ulassified. THI PAGE ulassified 8. UMBER OF PAGE 6 9a. AME OF REPOIBLE PERO tadard For 98 (Rev. 8-98) Presribed by AI td Z39-8

3 ( ) ( ) Re ψ je ( ξ) ( η, ϕ) σ σ he ( ) (, ϕ) ( ) ( ) ψ ξ η σ σ The agular futios (. The atrix U is a syetri, uitary atrix ll, ' * ( ( U ) ( U ) ), whih redues to the idetity atrix ll, ' l', l i the liit the iagiary opoet of the waveuber, vaishes, that is, I( ) 0. ( ) σ ( ηϕ, ) ε ( ) πλ ( ) dη ( η) ( η) δ Λ l l' *( ) ( ) ' ( ) os( ϕ), σ e ( η) si( ϕ), σ o (.3 are the spheroidal equivalet of the spherial harois. ote that i the above defiitio of the or of the agular futios the itegral of the agular futio ad its oplex ojugate is oputed. For oplex waveubers, the above itegral without the oplex ojugate is equal to a uitary atrix, sie the agular wave equatio is o-self-adjoit i the ase of oplex waveubers, ad the orrespodig eigevalues are i geeral oplex. This is to be otrasted with the ase of the spherial wave futios where the agular wave equatio is idepedet of frequey ad hee self-adjoit eve for oplex waveubers. The orrespodig radial wave equatio is superfiially o-self-adjoit, but it is oforally equivalet to a wave equatio that is idepedet of frequey ad self-adjoit by a siple resalig of the radial oordiate. Hee the eigevalues for the spherial radial wave equatio are also real. This differee betwee the behavior of the spheroidal ad spherial wave futios for oplex frequeies has resulted i a uber of errors i the literature, ad subtle differees i various idetities ad expasios for real ad oplex frequeies i the spheroidal ase. For exaple, i the ase of oplex waveuber, the salar Gree s futio of the Helholtz equatios is o loger give by the siple biliear suatio i ters of the regular ad irregular spheroidal wavefutios. Grr (, ') exp[ i rr' ]/4 π rr' i ψ Reψ σ σ σ Rather, it is give by a biliear expasio of the followig for. Grr (, ') exp[ i rr' ]/4 π rr' i ψ U Reψ σ l, l ' σ ' σ, l ' (.4 (.5 ( ) Defie the followig basis { } for the vetor τ, σ Helholtz equatio i spheroidal oordiates i the exterior ad iterior regios of the satterer. ( aψ ) ( ) ( ), σ σ ( ) ( ) ( ), σ, σ ψ ( ) ( ) 3, σ σ a r f ξ f η ( a a, a ) ξ, ϕ η ( f ξ,0, f η), oblate (, 0, ), prolate (.6 The first pair of vetor basis futios is a pair of solutios of the trasverse vetor Helholtz equatio, ad the third basis futio is a solutio of the logitudial vetor Helholtz equatio. ie there is o logitudial opoet of the eletri ad ageti field, the vetor idex τ is restrited to the values ad orrespodig to the trasverse odes. Defie the followig agular futios, as the geeralizatio of the vetor spherial harois. ( ) ( ) A ( ˆ ϕ η ˆ η ), σ η ϕ σ η A A ( ˆ η η ˆ ϕ ) η ˆ ξ ( ) ( ), σ η ϕ σ ( ) ( ) 3, σ σ (.7 The above futios are ot solutios of the vetor Helholtz equatio. They satisfy the followig orthogoality oditio, where the upper sig is used for prolate oordiates ad the lower sig for oblate oordiates. 0

4 ,, Ω Ω ll, ' ll, ' A δ δ Ω σ ' ',, σ, σ ' ' l ' σ l, l ' A δ δ Ω σ ' ',, σ, σ ' ' l ' σ l, l ' A δ δ Ω σ ' ' 3, 3, σ 3, σ ' ' l ' σ l, l ' A 0, σ 3, σ ' ' l ' A 0, σ 3, σ ' ' l ' A 0, σ, σ ' ' l ' ' dη( η ) (, η) (, η) ' η η ' dη (, η) (, η) ' ( η ) (.8 The geeralizatio of Betti s idetity for aoustis satterig to eletroageti satterig follows. Theore: Let u ad v be a pair of arbitrary solutios of the soure-free, trasverse, vetor Helholtz equatio i the regio. u u 0 Defie the exterior derivative T( u) αβ ad its oral opoet t( u) as follows. T( u) u u αβ α β β σ t( u) T( u) α σα α (. (. 3, l, l ' ' ' Ω dη (, η) (, η) τ, The atrix Ω is a syetri uitary atrix i the ll, ' ase of oplex waveuber. I the ase of real waveuber they redue to the followig atries.,, l' Ω Ω λ δ ll, ' ll, ' l Ω δ dηη η ' ' 3, l' ll, ' l (, ) (, η) (.9 Here, λ are the eigevalues of the spheroidal agular (radial) wave equatio. At this poit it is worth otig that the vetor ( ) basis futios { } have the followig asyptoti τ, σ behavior for large radial oordiate ξ. he (, ξ ) A, σ, σ he (, ξ ) A he (, ξ ) A, σ ξ, σ 3, σ ξ 3, σ (.0 This liit oupled with Betti s idetity is useful i provig various orthogoality relatios betwee the vetor basis futios. Here ˆ is the outward goig oral to the boudary of. The the followig surfae itegral over the boudary of vaishes. da{ u t ( v) t ( u) v} (.3 The above theore follows fro Gree s theore ad the fat that the trasverse wave equatio a be reast i the followig for i ters of the exterior derivative. 0 σ ( u u) T( u) u α σα α (.4 Here, the exterior derivative T( u) αβ plays the role of the stress tesor i elasti wave satterig, ad the iterior derivative t( u) is aalogous to the surfae tratio of the stress tesor. The followig is a useful orollary of the above theore. Corollary: Let 0 be a arbitrary losed, opat surfae about the origi, the sphere at ifiity, ad the volue whose boudary osists of the uio of these two surfaes. The give a arbitrary pair of solutios of the soure-free, trasverse, vetor Helholtz equatio i the regio, the followig itegrals over the surfaes 0 ad are equal. da{ u t( v) t( u) v} da{ u t( v) t( u) v} 0 (.5 The previous theore ad its orollary allows oe to write the followig orthogoality relatios for the 03

5 vetor basis futio for a arbitrary opat, losed surfae. da{ t(re ) Re Re t(re )} 0 τ, σ τ ', σ ' ' l ' τ, σ τ ', σ ' ' l ' dat {( ) t ( )} 0 τ, σ τ ', σ ' ' l ' τ, σ τ ', σ ' ' l ' da{(re t ) Re t( )} τ, σ τ ', σ ' ' l ' τ, σ τ ', σ ' ' l ' i δ δ δ τ Ω 0, τ 3 or τ ' 3 ll' σ ' ' τ ' τ, / ( i), 3 σ τ l, l' (.6 ( f ) f, ' ' ' ( ) ( ) {Re t( E ) t(re ) E } T ( a) a ', ' ' ( ) ( ) { t( E ) t( ) E } i δ δ δ ( i) Ω σ ' ' τ ' ll' τ, τσ, τ ' σ ' ' l ' σ τ l, l ' T (.8a (.8b The above orthogoality relatioships follow Here, E deotes the liit as the exterior eletri field fro the previous orollary, where the asyptoti liit approahes the surfae, ad E deotes the liit as the of the vetor basis futios { τ } i the liit, σ ξ, iterior eletri field approahes the surfae. ad the orthogoality oditios of the vetor spheroidal harois { A τ are used to evaluate the surfae, σ } ext, oe applies Betti s idetity to the iterior itegral over the sphere at ifiity. surfae fields. Here the assuptio that the iterior is soure free is ade, ad the iterior eletri field a At this poit the ahiery eessary to derive be expaded i ters of the regular basis futios. I the spheroidal T-atrix desriptio of eletroageti this ase the followig surfae itegral vaishes, sie satterig fro a lossy dieletri solid has bee Betti s idetity applied to a pair of regular vetor basis itrodued. I the followig, the idies ad are futios vaishes. used to olletively desigate the idies τ, σ ad ( ) ( ) τ ', σ ' ' ', respetively. I geeral, the idex τ is {Re ψ te ( ) t(re ψ ) E} 0 (.9 restrited to the values of or, sie the exterior derivative of the third vetor basis futio vaishes idetially. There are 3 equatios for the iterior ad The iidet ad sattered eletri fields have the followig expasios i ters of the regular ad irregular vetor basis futios i the regio exterior to the satterer. ( ) E Re i a ψ (.7 ( ) E fψ s exterior surfae fields. The followig boudary oditios are utilized to rewrite these 3 equatios i a ore tratable for. ˆ E ˆ E (.0 ˆ H ˆ H Here, the soure-free Maxwell s equatios are utilized to relate the ageti field to the url of the eletri field. Usig the above orthogoality oditios for the vetor basis futios ad the above expasios for the iidet ad sattered fields, oe arrives at the followig liear equatios for the iidet ad sattered fields. H E iω Usig the above relatioship betwee the eletri ad ageti field, the origial boudary oditios a be rewritte i the followig for. (. 04

6 ˆ E ˆ E (. ˆ ( E ) ˆ ( E ) Usig the above boudary oditios, oupled with the followig vetor idetities, t( E) ˆ ( E) E t( u) E ( ˆ ( u) ( E ˆ) ( u) (.3 oe a re-express equatios.8 ad.9 i ters of the surfae fields E, ad t( E ). ( f ) f, ' ' ' ( ) ( ) {Re t( E ) t(re ) E } T ( a) a ', ' ' ( ) ( ) { t( E ) t( ) E } ψ ψ ( ) ( ) {Re te ( ) t(re ) E} 0 (.4 σ ' ' τ ' ll' τ, τσ, τ ' σ ' ' l ' σ τ l, l ' ( ) ( ), ' ' ( ) ( ), ' ' ( ) ( ), ' ' ( ) ( ), ' ' ( ) ( ), ' ', ' i δ δ δ ( i) Ω ( ) Re Q t(re ) A Q t( ) A Re M Re A M A R t(re ) A P ( ) ( ) Re A ' (.7 The solutio of equatios.6 for the sattered field is of the followig for. f Ta (.8 T is the spheroidal T-atrix give below. The surfae fields E, ad t( E ) a be expressed by the followig expasio i ters of the vetor spheroidal harois. E te ( ) α A ( ) β A ( ) (.5 ubstitutig these expasios ito equatios.4, we arrive at the followig 3 liear algebrai equatios. f Re Qα Re M β a Qα M β 0 Rα Pβ (.6 Here the atries are defied as the followig surfae itegrals, where there are o loger ay uows appearig i the itegrals. T (Re QR P Re M )( QR P M ) (.9 This for is aalogous to the for developed by Haa 3 to desribe the aousti satterig fro a elasti solid. I order to obtai a expressio for the iterior field i ters of the iidet field, we ae the followig expasio of the iterior field i ters of the regular basis futios i the iterior of the target. E b Re (.30 it ( ) ext, Betti s idetity is used to obtai the followig relatioship betwee this expasio ad the surfae fields. ' b {( t ) E t( E )} ( ) ( ) ( ) ', ' i δ δ δ ( i) Ω ( ) ( ) σ ' ' τ ' ll' τ, τσ, τ ' σ ' ' l ' σ τ l, l ' (.3 05

7 Apply the boudary oditio, te ( ) te ( ) (.3 to equatio.3 to obtai the followig expressio for this liear equatio i ters of the surfae fields, E ad t( E ). ' b {( t ) E t( E )} (.33 ( ) ( ) ( ) ', ' Isert the expasios give i equatios.5 for the surfae fields ito equatio.33 to obtai the followig liear equatio for the iterior field. ( ) b Bα C B C β t( ) A ( ) ( ), ' ' A ( ) ( ), ' ' (.34 olvig equatios.6 ad.34, we obtai the followig expressio for the iterior field i ters of the iidet field. ( ) b T a T ( ) ( BR PC)( QR PM) ( ) ( ) (.35 I the ase of a perfetly odutig solid, the boudary oditio is that the tagetial opoets of the eletri field vaish at the surfae. ˆ E ˆ E 0 (.36 The disotiuity i the tagetial opoet of the ageti field is proportioal to the idued surfae urret. I this ase the T-atrix for a perfetly odutig solid is give by the followig expressio. the T-atrix for elasti satterig fro a elasti solid, where the stress tesor is replaed by the exterior derivative of the eletri field. Future wor will fous o the effets of boudaries o the satterig fro proud, buried, ad partially buried targets. ACKOWLEDGEMET This wor was fuded by Code 3MIW of the Offie of aval Researh. REFERECE [] Haa, Roger H., (984), The trasitio atrix for aousti ad elasti wave satterig i prolate spheroidal oordiates, J Aoust. o. Aer., 75(), pp [] Haa, Roger H., ad Douglas Todoroff, (985), A appliatio of the spheroidal oordiate based trasitio atrix: The aousti satterig fro high aspet ratio solids, J. Aoust. o. Aer., 78(3), pp [3] Haa, Roger H., Aousti satterig fro elasti solids, i Uderwater atterig ad Radiatio, Physial Aoustis, olue, edited by A. D. Piere ad R.. Thursto (Aadei Press, a Diego, 994), pp [4] aela, Gary., (00), Low Frequey atterig fro Large Aspet-Ratio Elasti Targets, Oeas 00 Coferee Proeedigs, Ot 30, 00. [5] aela, Gary., (00), Aousti atterig fro Large Aspet-Ratio Elasti Targets, Coastal ystes tatio, 6703 West Highway 98, Paaa City, Florida, , Tehial Report, 0-8. [6] aela, Gary., Low Frequey atterig fro Elasti Targets, paper to be subitted to J. Aoust. o. Aer. [7] Morse, Phillip M., ad Hera Feshbah, Methods of Theoretial Physis: olues ad, (MGraw-Hill Boo Copay, ew Yor, 953). [8] Flaer, C., pheroidal Wave Futios, (taford Uiversity Press, taford CA, 957). [9] Abraowitz, Milto, ad Iree tegu, Hadboo of Matheatial Futios, (Dover Publiatios, ew Yor, 97). T M M Re ( ) ( COCLUIO The spheroidal T-atrix for a lossy dieletri solid i a odutig ediu was derived fro a geeralizatio of Betti s idetity to eletroageti satterig. The resultig T-atrix is forally idetial to 06