LENS ANALYSIS METHODS FOR QUASIOPTICAL SYSTEMS

Transcription

1 LENS ANALYSIS METHODS FOR QUASIOPTICAL SYSTEMS S.B. Sørensen*, K. Pontoppidn* *TICRA, Copenhgen Denmrk, Keywords: qusioptics, dielectric lenses. Astrct The Method of Moments (MoM), Physicl Optics () nd Geometricl Optics () re compred for nlysis of dielectric lenses. It is found tht MoM cn e used (with norml computers) for dimeters of the lens up to 5, eing the wvelength, wheres the pproimte methods nd must e used t higher frequencies. Good greement etween the different methods is found nd efficient clcultion of nd is discussed. Introduction Lenses hve ecome importnt s focusing or phsecorrecting elements in the qusiopticl frequency rnge from pproimtely 300 GH to TH. Bem wveguides in this frequency rnge re normlly designed y mens of Gussin ems, ut often the dimensions of the components re too smll (in wvelengths) to rely solely on opticl methods or Gussin em nlysis. In this pper the Method of Moments (MoM) is compred to Physicl Optics () nd Geometricl Optics () for nlysis of dielectric lenses. 2 Lens nlysis theory Anlysis of dielectric lenses using MoM, nd will e eplined in the following susections. The methods re ll sed on the equivlence principle which llows the scttered field to e represented y the rdition of equivlent currents flowing on the surfces of the lens. Rottionl symmetry is not ssumed for ny of the methods. Incident field: E i, H i 2 Region Totl field: E, H J, M Region 2 Totl field: E, H Figure : Equivlent currents on dielectric oject. The usul choice of the currents J nd M is such tht if ll spce is filled y medium of dielectric constnt, the currents will rdite the field E-E i, H-H i in region nd the field -E i, -H i in region 2. If, on the other hnd, ll spce hs dielectric constnt of 2 the sme currents will rdite ero field in region nd the field -E, -H in region 2. If the equivlent currents re known, it is thus possile to clculte the totl field everywhere, oth outside nd inside the lens. It cn e shown tht the currents re relted to the totl field on the oundry etween region nd 2 y J nˆ H, M nˆ E, () where ˆn is the outwrds pointing unit norml vector. The purpose of ll the following nlysis procedures is to clculte the equivlent currents, such tht the totl field cn e found everywhere. A lens locted in free spce is now considered, s shown in Figure Equivlence principle for lens nlysis J, M J, M In Figure horn rdites field E i, H i in region in medium chrcteried y the dielectric constnt. This field is scttered y homogeneous dielectric oject tht occupies region 2 nd hs different dielectric constnt 2. The resulting totl field in the two regions is denoted E, H. In the MoM theory it is customry to represent the scttered field y set of equivlent electric (J) nd mgnetic (M) currents flowing on the interfce etween region nd 2, []. Figure 2: Equivlent currents on lens. Ais of rdited em

2 The equivlent currents re seprted in the currents J, M on the left hnd surfce nd the currents J, M on the right hnd surfce of the lens. The totl field is found y integrting the currents on oth surfces nd dding the incident field from the horn. It often hppens, however, tht the support structure tht keeps the lens in plce is lrge nd locks ny spill-over round the lens. A lens mounted in n infinite conducting screen will then e etter model, s shown in Figure 3. Infinite conducting screen J, M wve, which llows n pproimte clcultion of the equivlent currents. The first step in the procedure is to determine the currents J, M on the surfce illuminted y the source. In ny point on this surfce the equivlent currents cn e computed using the Fresnel reflection nd trnsmission coefficients for plne-wve incidence on plnr dielectric interfce. The direction of incidence is determined y Poynting s vector such tht the locl reflected nd trnsmitted field cn e computed. When these fields re known the currents J, M follows directly from eq. (). By integrtion of J, M in the dielectric lens mteril it is possile to compute the incident field on the right hnd surfce of the lens so tht J, M cn e found. Agin Poynting s vector nd the Fresnel reflection nd trnsmission coefficients re used. The procedure is illustrted in Figure 4. A suitle integrtion grid is set up on oth surfces nd ech current element on the left hnd surfce rdites onto ech grid point on the right hnd surfce nd contriutes to the equivlent currents J, M. The integrtion procedure uses polr grid comined with the Guss-Legendre integrtion rule, see [2] for further detils. Ais of rdited em J, M J, M Figure 3: Lens mounted in n infinite conducting screen. If the edge illumintion of the lens is low the currents J, M will not e significntly ffected y the screen, nd the field on the right hnd surfce of the conducting screen will e nerly ero. The equivlence principle then shows tht the totl field in the hlf-spce to the right of the screen cn e found y integrting only the currents J, M, ecluding the rdition from J, M nd the incident field from the horn. In this wy good pproimtion to the rdition of lens in conducting screen cn e found. In the following susections numer of methods for determining the equivlent currents will e descried. 2.2 Method of Moments (MoM) The Method of Moments is ttrctive ecuse it is n ect method nd ecuse it is very fleile. It llows metllic support structures to e included together with the dielectrics without pproimtions. The drwck is tht the memory requirement nd computtion time grow rpidly with the sie of the lens. In the present study very efficient MoM formultion is used [] which llows nlysis of lenses of up to 5 on modern PC worksttions. 2.3 Physicl Optics () Ais of rdited em Figure 4: lens clcultion. All currents in the integrtion points on the left hnd surfce rdite through the lens mteril to the integrtion points on the right hnd surfce. As illustrted in Figure 4, the method includes interction etween ll elements, ut multiple interctions etween the two surfces re neglected. If the incident field hs n irregulr ehviour, the locl plne-wve ssumption my not e sufficiently ccurte. This cn e.g. hppen if the lens is locted close to wist in em wveguide. It my then e necessry to epnd the incident field in series of plne wves s descried in [3]. This will ensure n ccurte clcultion of the currents J, M, from which the field inside the lens cn e computed. Also the field inside the lens will e irregulr such tht second plne wve epnsion is needed to compute the currents J, M. Although the method is much fster thn MoM it cn e time consuming to nlyse lenses with dimeters lrger thn 00. This is due to the smll distnce etween the current lyers which requires dense integrtion grid in order to clculte the rdition from one set of currents onto the other. As shown in [4] the convergence cn e improved y inserting n uiliry plne inside the lens mteril. In the Physicl Optics method it is ssumed tht the field on the lens surfces (see Figure 2) ehves loclly s plne

3 2.4 Geometricl optics () In the nlysis it is lso the gol to compute the equivlent currents on the surfces of the lens. Here the propgtion of the field inside the lens is sed on such tht the power is conserved in ry tues s illustrted in Figure 5. A similr procedure hs een descried in [5], ut it will here e shown how to rrnge the clcultions in simple nd efficient wy. Like, the nlysis requires tht the field ehves loclly s plne wve. The is more sensitive to irregulr ehviour of the field thn, ut, on the other hnd, it is much fster thn. where (, y) is the -coordinte of the surfce. In order to use simple forwrd ry trcing through the lens, the prmeteristion of surfce is chosen such tht r (, y ) is the intersection point on surfce of the refrcted ry through r (, y ). This cn lso e epressed s r r r ˆ, (4) where ˆr is the direction of the refrcted ry given y Snell s lw. In this wy ckwrds ry trcing is voided which cn e prolemtic when the incident field is not necessrily rdited y point source. Surfce Surfce The cross-section re surfce element ds of the ry tue is relted to the on surfce y ds ( rˆ n ˆ ), (5) ds ds where nˆ is the inwrds pointing norml on surfce. Furthermore, ds is given y the stndrd reltions Figure 5: lens clcultion. The power is conserved in the ry tue inside the lens. If the field E just inside the lens on surfce is known, it is possile to clculte the field E inside the lens on surfce y the stndrd reltion e jk 2 E E, (2) where the squre root fctor is the divergence fctor tht reltes the mplitude of the field on surfce to the mplitude on surfce ( nd re the cross-section res of the ry tue t surfce nd, respectively). The eponentil fctor contins the phse, where k 2 is the wvenumer 2 / 2 inside the dielectrics nd is the length of the refrcted ry from surfce to surfce. When the incident field on surfce from the source is known, its direction of propgtion is given y Poynting s vector such tht the trnsmission coefficient of Fresnel cn e pplied to otin E. Herefter, E cn e found from eq. (2). The Fresnel coefficients re gin used to find the field just outside the lens on surfce, nd finlly this field together with eq. () gives the equivlent currents. A simple method to compute the divergence fctor in (2) will now e descried. It is convenient to prmeterise surfce y the coordintes nd y in the coordinte system shown in Figure 5. The surfce is given y r (, y) ˆ yyˆ (, y) ˆ, (3) ds N ddy, (6) N ˆ ˆ ˆ y y (7) nd N nˆ / n ˆ. In the sme wy it is found for surfce ds ds ( rˆ n ˆ ), (8) N ddy. (9) From this, the divergence fctor simply ecomes rˆ N rˆ N, (0) The remining compliction is tht the norml vector must e computed from the generl reltion r r N, nˆ N / N, () y N where the derivtives re not known nlyticlly, ut must e found y numericl differentition of (4), e.g. r / ( r (, y) r (, y)) /. (2) The sme integrtion grid cn e used on oth surfces y mens of (6) nd (9) which reltes the integrtion element ddy to the corresponding surfce elements.

4 2.5 Multimode Gussin ems (MMGB) The uthors hve investigted the Guss-Lguerre multimode em epnsion in connection with em wveguides with reflectors. The mode mtching on the reflectors ws done y the method descried in [6]. It ws found tht the procedure is very fst nd tht it ccurtely descries the development of the min em through the wveguide, ut diffrctions re not included. It is epected tht the method hs similr dvntges nd limittions for em wveguide with lenses, ut the computtions hve not yet een crried out. 3 Emples A plno-conve lens is used s computtionl emple. The geometry is defined elow nd the nlysis is crried out y MoM, nd. 3. Lens geometry The plno-conve lens used in the following clcultions is shown in Figure ( ) n f (4) In ll the following test cses the incident field is n idel Gussin em (comple Huygens source) with rdius of curvture of R=50 mm t the plnr surfce, such tht the output em will hve its wist just t the right hnd side of the curved surfce of the lens. The width of the incident em is w=0 mm corresponding to n edge tper of pproimtely -20 db. 3.2 Lens Dimeter = 0 The wvelength is 3 mm corresponding to frequency close to 00 GH. The lens is locted in free spce nd it is seen tht the generl greement etween MoM, nd is good, ut tht the st sideloe is predicted too low y MoM (,0) ( 2, 2 ) dbi 0 D=30 mm n r.5 f=50 mm d=6 mm Figure 6: Plno-conve lens tht trnsforms sphericl wve into plne wve. The lens is defined y the focl length f, thickness d, dimeter D nd the inde of refrction n. It is designed y such tht it trnsforms sphericl wve into plne wve. A lens of this type is e.g. useful for correcting the phse of corrugted horn. The ctul dimensions re shown on the drwing nd the curved surfce is given y the following equtions from [7] tht reltes point on the plnr surfce to the corresponding point, long refrcted ry, on the curved surfce Figure 7: Comprison of MoM, nd for D= Lens Dimeter = 5 The wvelength is now 2 mm corresponding to frequency of pproimtely 50 GH. It is seen tht MoM nd grees very well wheres the level of the st sideloe with is 2 db lower. ( n ) d f f 2 2 n f n n ( f ) ( ) (3)

5 dbi MoM Figure 8: Comprison of MoM, nd for D= Lens Dimeter = 40 The MoM clcultions re only possile up do D=5 on norml PC worksttions, ut nd cn e computed t much higher frequencies. Due to the perfect focusing of the lens, the ptterns do not chnge very much (ecept for scling) when the frequency is incresed. As n emple, the ptterns for D=40 re shown elow for the lens mounted in n infinite conducting screen. It is seen tht the sideloes re still out db lower thn the more ccurte, ut otherwise the ptterns gree well Conclusion It is shown tht the ccurte MoM nlysis is useful nd fesile for nlysis of dielectric lenses with dimeters up to 5 with stndrd computers. For this sie of the lens the MoM is in good greement with the much fster nd. If the sie of the lens is lrger thn 00 the clcultions strts to ecome time consuming nd the simpler nd fster should e used. All of the nlysis methods cn e formulted in terms of equivlent currents which llow the field from lens in free spce to e clculted s well s the rdition from lens in conducting screen. References [] E. Jørgensen, J. L. Volkis, P. Meincke, O. Breinjerg. Higher Order Hiertchicl Legendre Bsis Function for Electromgnetic Modeling, IEEE Antenns nd Propgtion, Vol. 52, No., pp , (2004). [2] K. Pontoppidn, GRASP9 technicl description, TICRA, Copenhgen, Denmrk, (2005), downlodle from [3] S. B. Sørensen, H.-H. Viskum, K. Pontoppidn, Accurte nlysis of qusi-opticl network with polrition grid, IEEE Antenns & Propgtion Symposium, Honolulu, Hwii, 2007, pp [4] T. Bondo, S. B. Sørensen, Physicl Optics Anlysis of Bem Wveguides Using Auiliry Plnes, IEEE Antenns nd Propgtion, Vol. 53, No. 3, pp , (2005). [5] K. K. Chn, S. K. Ro, G. A. Morin, M. Q. Tng, Tringulr Ry-tue Anlysis of Dielectric Lens Antenns, Vol 45, No. 8, pp , (997). dbi 5 5 [6] W. A. Imrile, D. J. Hoppe, Recent trends in the Anlysis of Qusiopticl Systems, Proc. AP2000, Dvos, Switerlnd, April [7] R. Johnson, H. Jsik, Antenn Engineering Hndook,McGrw-Hill, Figure 9: Comprison of nd for D= Computtion time The MoM nlysis tkes 8 minutes with D=0 nd 28 minutes with D=5 on modern PC worksttion. In comprison the nd nlysis shown in the pper ll use less tht 5 seconds. For fied numer of sideloes the computtion time is nerly independent of the frequency, wheres the time for strts to grow noticely for D>40. With D=40 the clcultion is 3 times fster thn, wheres it is 20 times fster for D=80.