Radar Cross Section of Modified Target using Gaussian Beam Methods: Experimental Validation

Transcription

1 Radar Cross Secton of Modfed Target sng Gassan Beam Metho: Expermental Valdaton H. Ghanm, A. Khenchaf Lab-STICC UMR CNRS 685, ENSTA Bretagne 986, Brest, France & P. Polgen French General Drectorate for Armament (DGA) 7559, Pars, France Abstract The am of ths paper s to stdy the Radar Cross Secton (RCS) of modfed radar targets (plate wth notch) sng Gassan Beam technqes. The Gassan metho sed n ths work are Gassan Beam Smmaton (GBS) and Gassan Beam Lanchng (GBL). We establsh the theoretcal formlaton of the GBS and GBL technqes and analyze the nflence of the man Gassan beam parameters on the varaton of the scattered feld. Then, we present the smlatons of RCS. The nmercal reslts are compared wth PO, MoM metho, and also wth expermental measrements performed n the anechoc chamber at Lab-STICC (ENSTA Bretagne). Keywor Radar Cross Secton (RCS); Gassan Beam Smmaton (GBS); Gassan Beam Lanchng(GBL); Physcal Optc (PO); Method of Moment (MoM); Physcal Theory of Dffracton (PTD) I. INTRODUCTION The RCS can be smlated sng rgoros metho sch as Method of Moment (MoM), Fnte Dfference Tme Doman (FDTD) or asymptotc models ncldng Geometrcal Optcs (GO), Physcal Optc (PO), Physcal Theory of Dffracton (PTD), and Geometrcal Theory of Dffracton (GTD). The rgoros metho lke MoM are based on an ntegral formlaton, and they are served to valdate the new asymptotc approach. The asymptotc approach redces the operaton nmber of solvng of hgh-freqency eqatons for large obects [], [9], []. The asymptotc metho sng the hypothess of locally plane wave and hgh-freqency approxmaton are based on the prncple of rays. The applcaton of the type of these metho n a complex propagaton scenaro s reglarly lmted by the transacton between hghlghted and shadowed regon and the castc problem (except the PO method). The vsblty of these lmtatons depen on the complexty of the radar target and the geometrc observaton confgraton. For these reasons we chose to apply two asymptotc technqes based on Gassan beams called Gassan Beam smmaton (GBS) and Gassan Beam Lanchng (GBL), and we stdy the RCS varaton of dfferent radar targets. In the GBL technqe, when a radar target s llmnated by a Gassan Beam, the feld radated s decomposed nto a plane wave spectrm then smmng the contrbton of the radatons of all the beams nteractng wth the target []. The Gassan Beam smmaton as an asymptotc approach for comptng hgh-freqency wave fel was developed by V. Cerveny and M.M. Popov []-[5]. The smmaton of Gassan beams allows solvng some crtcal ponts of the asymptotc ray metho sch as the problems related to the evalaton of wave feld n snglar areas. In addton, the Gassan beams smmaton method offers a solton to overcome the castcs problem [3], [4], [3]-[5]. In ths context, the obectve of ths work s to nvestgate the dfferent mechansms of electromagnetc wave scatterng by a modfed radar targets (plate wth notch) sng GBS and GBL metho and valdate the nmercal smlaton reslts by expermental measrements. II. FORMULATION A. GBS method: formlaton The Gassan Beam method sed for the estmaton of a wave feld n hgh-freqency approxmaton has been developed by V. Cerveny [4], [8] and M.M. Popov [6]. The approach s based on the smmaton of Gassan beams. Ths physcal prncple of Gassan Beam approach conssts n a compatble step assembly. Consder a homogeneos and sotropc medm and an electromagnetc wave propagatng n ths medm whch s actng excted by a pont sorce. In the GBS method, the total feld at the recever s a sm of the contrbtons of all the beams passng n the vcnty of the recever (whch s the same case for each observaton pont). For each consdered ray, we determne a Gassan beam propagatng along the ray. Then we sm the contrbton of each Gassan beam to the recever overall rays [8], [9]. Fg.. Geometrc confgraton and coordnate parameters for Gassan Beam Smmaton (GBS) formlaton. Plate wth a rectanglar apertre (notch). Fg. presents the ray-centered coordnate system (s, q, q ) sed to formlate the eqaton () of the Gassan beam

2 ampltde (s, q, q, t). Ths coordnate system s connected to any selected Ω ra. In addton to the geometrc parameters, assmptons are also sed to establsh the theoretcal eqaton of GBS. In fact, we start by consderng a homogeneos and sotropc medm an electromagnetc wave propagatng (wth a propagaton velocty v) n ths medm whch s beng excted by a pont sorce. Then, we sppose that some wave process s descrbed by the Helmholtz s wave eqaton and the pont sorce s postoned n the orgn. After, we solve the Helmholtz s eqaton n the neghborhood of rays. Usng the local coordnates and at the recever pont, the solton of the Helmholtz eqaton as a soltary Gassan beam s gven by () [], [3]: v T ( s, q, q, t) exp ω t τ( s) ( q P Q q) det τ ( s) s v [ Q] In (), v represents the propagaton velocty; qt s the transpose of the vector; the travel tme from the sorce along the desgnated ray s represented by τ(s), and the other parameters are cted n the prevos paragraph. The parameters Q and P (whch depend on s) are two-by-two matrx named dynamc qanttes [3], [] whch are related by the Dynamc Ray Tracng (DRT) (). dq dp c. P; and () The system of dfferental eqatons (), s solved by ntrodcng ntal condtons specfed at an arbtrary pont (s s ) on the central ray. The ntal condtons are also related to a three others condtons along the whole rays [6] whch are: even thogh P and Q are not symmetrcal the (P Q - ) mst be symmetrc matrx; Im(P Q - ) s a postve-defnte matrx and (det[q] ). The ntal vales of Q and P can be establshed by sng Hll s ntal data for the Green s fncton [6], [7]: ω. ω r Q. I ; P. I and ; s s (3) c c In (3) ω denotes the begnnng half beam wdth at a freqency f ω r/π. I s the dentty matrx ( ). Usng the ntal condtons n (3), we can fnd the general solton of (), and can be wrtten as follows: ω. ω r Q. +. c c ( s s ). I ; and P. I In the case of homogeneos meda, by sng (4) n (), we retrn to the representaton of the ampltde of the Gassan beam n 3D: ( s, q, q ) c ω. ω + 3. ( s s ).exp ωτ () s + ω ω c. c ( q ) + q.. + ( s s ) Usng the geometrcal confgraton llstrated n Fg., and ntrodcng the sphercal coordnaton system (r, θ, φ), we () (4) (5) can dedce the followng factor (n 6) as fncton of the tance (r) from the transmtter to the recever: ( ϕ) ; and s s. cos( ϕ) q + q r. sn r (6) Fnally, to calclate the fll ampltde ( GBS ) at the recever we mst se an ntegral formlaton as shown n eqaton (7). Ths ntegral wll be calclated on all Gassan beams descrbed by ther characterstc angle (called takeoff angle φ) from the sorce: ( q q ) GBS Φ. s,,. d ϕ ϕ γ (7) γ In (7) the ntegral fncton s the prodct of three terms. The frst term denoted Φϕ s the complex weght fncton whch may dffer from ray to ray, however, remans constant n each consdered ray. The second term s the fncton φ(s, q, q ) s the Gassan beam related to the ray gven by (5). Fnally, the thrd term dγ s expressed accordng to angle φ by the eqaton (π.sn(φ).dφ). The choce of the ntegraton doman γ s related to the fncton of the Gassan beam φ(s, q, q ) and to the central ray. In fact, the doman γ s fxed on the central ray and delmts the Gassan beams propagatng n the neghbor of the central ray. On the other hand, the contrbton of the Gassan beams φ(s, q, q ) s condtoned by the fact that otsde the γ doman do not contrbte effectvely to the wave feld. For a homogeneos medm, the ray asymptotc solton of the Helmholtz eqaton s gven by the followng eqaton: ( r ) exp 4 r ω π c r (8) The GBS ntegral, n (7), may be evalated asymptotcally sng the saddle-pont method. Therefore, ths reslt mst match wth the ray asymptotc solton (8) n the reglar area. The calclaton of the complex weght fnctonφϕ s performed by matchng the asymptotc solton of (7) and (8). Ths ntegral of GBS s evalated and smlated nmercally and qadratcally by a reglar ncrement denoted Δφ k. GBS N k k ϕk ( ϕ ). ϕ k k. π. Φ..sn Δ ϕ The formlaton n (9) wll be sed to compte the scattered feld, then the RCS sng GBS method. After the formlaton of the scattered fled sng GBS method ((7) and (9)), we examne the effect of the man parameters of the Gassan beam on the varaton on the feld ampltde. Then, we compare the solton based on Gassan beam wth the analytcal solton gven by (8). The prncpal nflencng parameters n GBS method n (7) are the nmber (N) of beams and ther wdth vale (ω ). To verfy the valdty of the proposed method, we analyze the behavor of GBS solton for dfferent vale of beam wdth and beam nmber. Fg. compares the ampltde of feld calclated ray asymptotc solton of the Helmholtz eqaton and GBS method, for a freqency eqal to GHz, a beam wdth (ω ) eqal to 8λ (where λ s the wavelength) and for varos beams nmber (N){33,, 4 and 6}. Ths smlaton (Fg. ) has been realzed as a fncton of the tance (r n km) from (9)

3 the sorce to the recever. We can observe, when the nmber of beams s more than, the GBS and the ray asymptotc are n are almost dentcal. Ths, a hgh beam nmber s necessary for hgh precson, whch s the same case of the classcal ray tracng metho. Fg.. Comparson between the ray asymptotc solton and the Gassan beam smmaton method for N33,, 4, 6 the beam wdth s 8λ. The electrc fel scattered from the target srface (Σ) llmnated by the ncdent beam s gven by the ntegraton of the ncdent Gassan beam on the reflector srface (PO ntegral). Ths ntegral s wrtten by (): ( k R) k. Z exp.. E ( r). R R ( ez H( r) ). 4. π Σ ε R r () Fnally, by sng () n () and solvng the ntegral, we can compte the scattered feld applyng GBL formlaton as n []. In both GBS and GBL method, t s reqred to take nto accont the far feld approxmaton parameters (tance r, target sze D and wave length λ). In order that the phases and ampltdes of the waves arrvng from dfferent regons of the target do not vary consderably wth the tance (r), the farfeld regon mst be far enogh away from the sorce. Ths regon of the far feld starts at a tance "r" gven by the followng eqaton (Franhofer crteron) [6] : r ((.D )/λ). Fg. 3. Varaton of percentage error between GBS method and ray asymptotc solton as fncton (r) and for several beam nmber. Fg. 3 llstrates the varaton of percentage error between GBS method and ray solton for dfferent beam densty. We can see that for sffcent beam densty, N6, the relatve error remans below 3% even at 5 km from the sorce. In addton, one shold note that n r the smlaton based on GBS method do not have snglartes, whch s contrary to the ray asymptotc method. Ths reslt confrms that by sng the GBS method we can overcome some lmtaton of the ray asymptotc models. B. GBL method: formlaton The Gassan Beam Lanchng (GBL) technqe has been ntrodced and appled n the research pblshed by H. T. Cho []. Consder a target (plate, c, cylnder, ) llmnated by a Gassan beam, the GBL method s appled to calclate the radaton ntegral of the target scattered fel. For the consdered Gassan beam, the ncdent magnetc feld s gven by the followng form [], [4]: ρ + b x + y H ( ρ ) H ( ) exp k z + z + ρ + b z + ρ + b () In (), the tance between a pont on the llmnated srface and the wast the ncdent Gassan beam s denoted by ρ, the poston vector n the Gassan beam s defned by and b k.ω /, and k, ω are the wave nmber and the half beamwdth respectvely. Fg. 4. Target sze as the fncton of tance for far-feld approxmaton: the Far-feld crteron of Franhofer. Fg. 4 shows the mnmm far-feld range as a fncton of the target dmensons D, and for dfferent freqences vales. In ths work, all smlatons and measrements are realzed n the freqency eqal toghz. III. NUMERICAL AND EXPERIMENTAL RESULTS: VALIDATION AND EVALUATION A. Measrng devces and consdered targets The valdaton of the nmercal smlaton reslts have been done n the monostatc confgraton (where the transmtter and the recever are n n the same poston), whch s located n an anechoc chamber (8m 5m 5m) at ENSTA Bretagne (see Fg. 5).

4 approaches. The measrements are performed at a freqency of GHz, the nmercal smlaton parameters are fxed as follows: beam wdth of ω λ, beam nmber eqal to, azmth angle ϕ and an ncdent angle 8 < θ < 8. (c) Fg. 5. General descrpton of the expermental setp, flat plate, (c) plate wth an apertre. The characterstcs of varos components of measrements system are: All walls are covered wth absorbent materal. A compter controls the Vectoral Network Analyzer (Anrts 37347D) whch operates n the freqency range from 4MHz to GHz and the postonng system. An elevaton motor for atng the heght of the target. The NEWPORT postonng system wth an anglar resolton eqal to. and an angle vary between -9 and 9. The dmensons of the metallc plate and t notch are shown n Fg. 6, ths target wll be sed for expermental measrements. The sze of the plate and ts rectanglar notch are (3cm 3cm) and (5cm 5cm) respectvely. The geometry of the consdered radar target (plate wth notch) meshed wth a tranglar patch (sng CATIA software) s present n Fg. 6. In ths nmercal model of the radar target, each facet s represented by a trangle node (n ble), t centered n red color (brght ponts) and black cross n the mddle of the external edge and also n the notch edge. The RCS of the plate wth an apertre ncldes the srface scattered feld and the edge dffracted feld (by the notch and the external of the plate). In the GBS method, to consder the edge dffracton contrbton, we have chosen to se the physcal theory of dffracton (PTD) method whch s detaled n []. In fact, PTD method s sed to compte a dffracton part for each ray that hts the target srface n the vcnty of an edge and s calclated n the complex weght fncton n the ntegral (7). 3cm Fg. 7. Comparson between GBS+PTD, GBL, PO, MoM metho and expermental measrements n vv polarzaton: PEC flat plate, f GHz. In Fg. 7, we compare the GBS+PTD and GBL+PTD wth the expermental measrements and the nmercal models (PO, MoM n FEKO software). It s observed that, when the dffracton contrbtons are acconted, the combned GBS+PTD and GBL+PTD metho gve an accrate qaltatve representaton of the RCS varaton for all observaton angles. We can also remark that PO s nsffcent n comptng of edge plate dffracton contrbtons, althogh the agreement of GBS+PTD and GBL+PTD wth the rgoros MoM solton and measred data are good n the most of scatterng angles. These reslts wll serve as a benchmark for analyss of the nflence of the presence of the notch n the RCS varatons. 5cm 3 cm 5 cm Fg. 6. Dmensons and mesh of PEC plate wth an apertre. The next part presents the comparson between nmercal and expermental RCS reslts. Frstly the stdy the RCS of a PEC flat plate s exposed. Then the nflences of the presence of the rectanglar apertre on the RCS varatons are played. B. RCS estmaton: nmercal and expermental reslts To valdate the smlaton reslts obtaned sng the Gassan technqes (GBS, GBL), the varaton of the RCS of a sqared PEC flat plate (sze λ λ) n the monostatc case s measred n an anechoc chamber frstly. Then a comparson s also performed wth others nmercal Fg. 8. Comparson between GBS+PTD, GBL+PTD, MoM, PO metho and expermental measrements n vv polarzaton: PEC plate wth notch, f GHz, ϕ. Fg. 8 shows the RCS of a PEC plate wth rectanglar notch obtaned sng (GBS+PTD, GBL+PTD, MoM, PO) metho and expermental measrement (same parameters n Fg. 7). It s observed that for the ncdent between and, the nmercal models and expermental reslts are n good

5 agreement. However, n the case of the angle close to 8, a dfference appears between the dfferent crves of RCS. Secondly, we can see that for the scatterng angle larger than 35, the RCS crves sng GBS+PTD and GBL+PTD are near to those obtaned by MoM method and measrement, bt hgher than the PO model whch may be related to the dffracton of the edge of notch and external of the plate. To obtan more nformaton abot the mpact of the notch of the plate, we compare n the RCS of a PEC wth and wthot notch sng GBS+PTD and GBL+PTD metho. ntegral over all Gassan beams passng n the vcnty of the recever. In the GBL method, when a target s llmnated by a Gassan beam, the radated feld s calclated by decomposng t nto a plane wave spectrm and then smmng the contrbtons of the radatons of all beams nteractng wth the target. In addton, to take nto accont the edge dffracton of the apertre and the external part of the plate, GBS and GBL metho have been combned wth the Physcal Theory of Dffracton (PTD). The nmercal and expermental reslts show that the two metho GBS and GBL model well the varaton of the feld radated by a modfed canoncal target (plate wth an apertre). To extend the applcatons of the GBS and GBL, one of the perspectves of the presented work s to stdy the RCS of delectrc targets by sng the GBS/GBL metho. ACKNOWLEDGMENT The athors wsh to thank the DGA (Drecton Générale de l Armement, France)-MRIS for ts spport of the SOFAGEMM proect, where ths work s n Progress. Fg. 9. Comparson between RCS of a flat plate and a plate wth notch: f GHz, ϕ, GBS+PTD, GBL+PTD. Fg. 9 shows a comparson between RCS of a flat plate (n ble) and a plate wth a notch (n red). It s observed, the effect of the presence of notch on the RCS crve s dependng on the vale of ncdent angle. In fact, at θ the ampltde of RCS decrease, nonetheless for the others ncdent angle vales the RCS of a flat plate s reglarly lower than that of the plate wth a notch. IV. CONCLUSION AND FUTURE WORK In the present paper, new technqes n the electromagnetc scatterng from a modfed canoncal radar target (flat pale wth notch) have been appled. The GBS and GBL metho sed n ths work are based on the Gassan beams. In the GBS technqe, the total feld at the recever s eptomzed by the REFERENCES [] F.Wenmann, Ray tracng wth PO/PTD for RCS modelng of large complex obects, IEEE Ant Prop, vol. 54, no. 6, pp , 6. [] V. Červený, Smmaton of paraxal Gassan beams and of paraxal ray approxmatons n nhomogeneos ansotropc layered strctres, In Sesmc waves n Complex 3-D Strctres, no., pp. 59,. [3] M. M. Popov, A new method of comptaton of wave fel sng Gassan beams, Wave Moton., vol. 4, pp , 98. [4] V. Červený, Expanson of a Plane Wave nto Gassan Beams, Stda geoph. And geod., vol. 6, pp. - 3, 98. [5] V. Červený, Sesmc Ray Theory, Cambrdge: Cambrdge Press,. [6] V. Červený, Comptaton of wave feld n homogeneos meda, Geophys. J. R. astr. Soc., vol. 7, pp. 9-8, 98. [7] N. R. Hll, Gassan beam mgraton, Geophyscs, vol. 55, no., pp , 99. [8] B. S. Whte, A. Norrs, A. Baylss and R. Brrdge, Some remarks on the Gassan beam smmaton method, Geophyscal Jornal of the Royal Astronomcal Socety, vol. 89, pp , 987. [9] H. T. Cho and P. H. Pathak, Unform asymptotc solton for electromagnetc reflecton and dffracton of an arbtrary Gassan beam by a smooth srface wth an edge, Rado Scence, vol. 3, no.4, pp , 997. [] P. Polgen, L. Lcas, F. Mller, S. Qete and C. Terret, Calclaton and analyss of electromagnetc scatterng by helcopter rotatng blades, IEEE Trans. Ant. Prop, vol. 5, no., pp ,. [] H. T. Cho, P. Pathak et R. J. Brkholder, Novel Gassan Beam Method for the Rapd Analyss of Large Reflector Antennas, IEEE Trans Antennas and Propagaton, vol. 49, n 6, pp ,. [] B. Blesten, Mathematcs of Modelng, Mgraton and Inverson wth Gassan Beams, Colorado, USA, 8. [3] M. Katsav and E. Heyman, Gassan Beam Smmaton Representaton of Beam Dffracton by an Impedance Wedge: A 3D Electromagnetc Formlaton Wthn the Physcal Optcs Approxmaton, IEEE Trans. Antennas Propagaton, vol. 6, no., pp ,. [4] P.O. Leye, A. Khenchaf, and P. Polgen, The Gassan Beam Smmaton and the Gassan Lanchng Metho n Scatterng Problem, J. Elect. Analyss and Applcatons, vol. 8, pp.9-5, 6. [5] H. Ghanm, A. Khenchaf, P. Polgen and P. O. Leye, Stdy of RCS of complex target: Expermental measrements and Gassan beam smmaton method, IEEE CAMA conference, pp , 7. [6] Warren L. Sttzman, Gary A. Thele. Antenna Theory and Desgn. John Wleys & Sons, Inc., 997.