Senso and Simulation Notes Note 55 Oct 7 Lens Design fo a Polate-Spheoidal Impulse adiating Antenna (IRA) Sehat Altunc, Cal E. Baum, Chistos G. Chistodoulou and Edl Schamiloglu Univesity of New Mexico Depatment of Electical and Compute Engineeing Albuqueque New Mexico 873 Abstact In this pape, we discuss the design pocedue fo diffeent types of dielectic lenses fo bette concentating the fields at the second focus of a polate-spheoidal IRA to incease the fields and decease the spot size. This wok was sponsoed in pat by the Ai Foce Office of Scientific Reseach.
Intoduction In this pape, we discuss the design pocedue fo diffeent types of dielectic lenses fo bette concentating the fields at the second focus of a polate-spheoidal IRA to incease the fields and decease the spot size. We have a vey fast and intense electomagnetic pulse to illuminate the taget [] which is located at the second focal point. One of the most impotant poblems with concentating the fields on the taget is eflection. We have to deal with this eflection because the dielectic popety of the taget medium and the medium though which the incident wave popagates ae diffeent. The eflection of the pulse leads to a smalle field at the second focus whee ou taget is buied. We discuss the addition of a lens to bette match the wave to the taget. We can obtain lage fields and smalle spot size []. To obtain bette concentation at the taget we can use diffeent types of lenses. The tansmission coefficient fom one medium to the anothe one is T + t, () whee t is the elative pemittivity of the taget medium. Suppose now that we have a lens in font of the taget with elative pemittivity l t. () The fields fom the eflecto ae tansmitted with a tansmission coefficient given by T + t. (3) We will have a slowe wave speed and an enhancement facto which is an incease in the impulse potion of the focal wavefom fom [] as [ μ ] v l F l t. c l, (4) Thus, fo the impulse pat of the field we will have a net incease of
F T + t (5) < l < t Suppose now that we have a lens in font of the taget with elative pemittivity. (6) We will have then two tansmission coefficients and the total tansmission coefficient is T T T 4 + + t t. (7) l + + t t l Finally, suppose we have a lens with a gaded elative pemittivity given by l, ( ), l ( l ) t. (8) The wave popagating though this takes a simila fom as that of a wave in a tansmission-line tansfome. The high fequency ealy-time tansfe function can be computed fo a continuous vaiation of as [] 4 T t. (9) t We still have the enhancement facto the tansmission enhancement F T + 4 + 4 t t t. () The tansmission enhancement of the lens, as discussed in [3] fo an exponential vaiation of the chaacteistic impedance of the tansmission line (fo constant wave speed) along the line, is somewhat optimal. In this pape we pesent diffeent types of gaded lenses fo stonge focusing at the taget. The focal point is z 37. 5 cm and the othe paametes of the polatespheoidal IRA ae defined in [4]. 3
Calculating the Optimum Numbe of Layes fo a Lens In this section we calculate the optimum numbe of layes to obtain the equied field at the focal point of a polate-spheoidal IRA based on a plane-wave appoximation. N layes of inceasing dielectic constant lenses which have the same atio of dielectic constant ae consideed fo a polate-spheoidal IRA that is based on []. The geometical illustation of this design is pesented in Figue. Figue : N layes of lens, dielectic constants and tansmission coefficients. The total tansmission coefficient can be defined as N T total T n, (.) whee T n is the tansmission coefficient between n th and n th + st laye and it can be defined as Tn n+ n+ n + n+ n + n+. (.) The atio of dielectic constant between subsequent layes ae constained to be the same, atio n + n. (.3) Fo N layes 4
5 ( ) N atio N n n +. (.4) Substituting (4) in (), we have N N T n +. (.5) Fo N layes fom () N N N T total +. (.6) If we have a continuously inceasing dielectic lens we have a total tansmission coefficient defined in () as. 4 T total. (.7) If we have an infinite numbe of layes, (.6) appoaches (.7). We should decide how many layes will be acceptable to obtain a sufficiently close tansmission coefficient to the continuously inceasing dielectic lens case.
Table : Tansmission coefficients fo diffeent N and. The numbe of layes depends on the sensitivity of the application accuacy. In geneal building moe than layes is not pactical fo manufactuing and we ty to obtain the closest tansmission coefficient to the continuously inceasing case. Fom Table one can see that, fo layes, N, T total appoaches close to the continuously inceasing dielectic lens case. Even though layes does not give us that much impovement if we compae it with N layes, we took N layes fo ou late calculations. One can easily decease o incease the numbe of layes fo specific applications. We took the imum numbe of layes, which is N, that can be manufactued fo late calculations. 6
3 Thee Diffeent Types of Gaded Lens Design fo a Polate-Spheoidal IRA The basic design consideations fo the physical concept of thee diffeent types of inceasing pemittivity dielectic lens ae consideed. The focal point is z 37. 5 cm and the othe paametes of the polate-spheoidal IRA ae defined in (4.). The lens is a half sphee (o half ball in mathematicians tems) and its adius is, as shown in Figue. Figue : Addition of lens with polate-spheoidal IRA geomety. As discussed in [5] befoe, the exponential vaiation of the chaacteistic impedance of a tansmission line along the line is optimal, povided that the speed of popagation is constant along the line. Some modification may be useful hee since the speed vaies invesely with the squae oot of the dielectic constant. The lens elative pemittivity is at (). (3.) at 7
3. Exponential Vaiation of One suitable fom fo is an exponential function as q( ) ( ) e. (3.) As we know at the elative pemittivity is so C ( ) e, C ln( ). (3.3) If we substitute (3.3) in (3.), can be found as ln( )( ) ( ) e. (3.4) The ise time is estimated as tδ ps the distance coesponding to this ise time is l δ ctδ 3cm, (3.5) in ai. The popagation distance of the wave fom to is ct lens c d ( )d ν ( ) ln(. ) ln( )( ) e d (3.6) The nomalized ctlens is ct lens ( ). (3.7) ln( ) The distance between the souce and lens is (.375m + ). Afte this design pocedue we designed a lens that is matched to the taget dielectic. The thickness of the taget dielectic mateial should be 8
c tδ Δ n, n, (3.8) in ode to minimize the effect of the eflected wave on the impulse tem. 3. Compensated Incemental Speed (CIS) fom of As we mentioned befoe the exponential fom assumes that the popagation speed is constant. Howeve, it is not constant we need to compensate fo this assumption. Let us assume we have a plane wave poblem in an inhomogeneous (isotopic) slab with ( z ) and set the elative change in wave impedance ove a tansit time Δ τ Δ ln( ) C Δτ (3.9) with the wave impedance time can be witten as c popotional to. The distance based on the tansit c dτ dz. (3.) Fo a given ln( Δ τ the Δ z deceases as. Fo a given Δ τ the change in ) is independent of z and if we substitute (3.) in (3.9) we obtain d ln( ) C. (3.) dz Integating (3.) C d ln( z + C 3. ) e ln( ) d ln( ) C dz (3.) We can define fom (3.) as 9
at z z ( Cz + C3 ), (3.3) at z such that fom (3.3) C, C3 z. (3.4) Then, if we substitute (3.4) in (3.3) we have z ( ) +. (3.5) z How much time does the popagation of the wave take fom to the focal point in the lens? Substituting (3.5) in (3.7) ct lens d ( ) + d. (3.6) Changing the vaiable of the integal as ξ, we obtain ct lens ( ) ξ + dξ. (3.7) Let us change the vaiable of the integal as ς ( ) ξ + d ς ( ) dξ and we will have nomalized ctlens ( ) d ln( ). (3.8) Then, we can find the nomalized ct lens is ctlens ln( ). (3.9)
3.3 Linea fom of The exponential vaiation and CIS fom of ae two diffeent appoaches having some advantages and disadvantages in tems of focusing. Afte these appoaches we tied to use anothe appoach, a linealy inceasing fom of. Let us assume we have a linea vaiation as ( ) + ( ), (3.) which satisfies (.), we can find the nomalized popagation time of the wave fom to as ctlens ( )d ( + ( )) d. (3.) Let us change the vaiable of this integal as we will have [ + ( )] d ct lens. (3.) We can also change this vaiable as ξ + dξ d Using (3.), [ ] [ ]. (3.3) ct lens 3 d ( ) ( ξ ξ 3 ). (3)
3.4 Conclusion A dielectic exponentially inceasing dielectic constant, CIS, and linea inceasing lens designs wee discussed. One can see fom Figue 3 that the wave popagates faste fo the CIS fom of. We can see fom Figue 4 a-d) that if inceases, the wave popagates slowe as expected. vaies fom to 8 (with 8 coesponding to wate, which is the highest that is used in biological applications). If we incease fom 36 to 8, the CIS design of has the deepest cuvatue. The focusing popety of the lens inceases fom the CIS to the linea design because fo the same we have an incease in, we expect the lens to become moe effective. Also fom [] if we incease the spot size deceases while the wave impedance 4 deceases and the amplitude of the wavefom inceases by a facto of. This ough calculation has to extend out some distance fom the taget fo effective focusing to occu and thus equies moe detailed calculations. Figue 3: ct fo linea, exponential and CIS foms of. lens
Figue 4: values fo linea, exponential and CIS foms of fo diffeent espect to. with 3
4. Spatially Limited Exponential Lens Design fo Bette Focusing an Impulse A spatial limited exponential lens design is discussed and an analytical fomulation has been used to examine the pulse doop in ode to minimize it. A fomulation in [3] has been used to examine the pulse doop fo a tansmission line with an exponentially tapeed impedance pofile. We wish to minimize this doop, o ask how long the tansmission line should be fo a given doop. The exponentially tapeed tansmission line has an optimal tansfe function in tems of ealy-time voltage gain and impoved doop chaacteistics. We apply this esult to an exponentially tapeed dielectic constant of a focusing lens. We find the equied lens dimensions fo a given doop. The lens geomety and incoming spheical wave ae pesented in Figue 5. Ou calculations ae based on a one-dimensional plane-wave appoximation (Figue 6). This will not diectly give an estimate of spot size, only the tansmissioneflection by the lens. Othe consideations also apply []. Figue 5: Lens geomety and incoming spheical wave. 4
Figue 6: Equivalent plane wave geomety. 4. Equivalent Tansmission-Line Model (One Dimensional) of Lens As discussed in [3], the exponentially tapeed lens has a minimized doop and the optimal tansfe function fo the case of unifom popagation speed. Hee we adapt this solution to a dielectic lens, noting that the popagation speed slows as the wave popagates in highe-pemittivity media. This model does not include any infomation about spot size. We can define the lens wave impedance as follows: z spatial coodinate, modified space coodinate. We have a new coodinate whee the wave popagates with a constant v speed and has an exponential wave impedance vaiation though the lens. We use a plane wave appoximation and this appoximation is valid up to the case when the wavelength is still small compaed to the coss section of the beam tansit time to z and hence c. (4.) Let ( ) e, (4.) whee is the wave impedance at the beginning of the lens; which is ou case. 377Ω in e, (4.3) whee is the wave impedance at the end of the lens. 5
μ ( ) ( ) ( ). (4.4) The popagation speed can be defined as v v ( ), (4.5) [ μ ( )] whee v is the popagation speed befoe the lens, which is typically c. The tansit time though the lens can be defined as z - ' ' - t t v ( z ) dz v d v z. (4.6) Taking the deivative of both sides of (4.6), we have dt dz v v (z), d d dt v (z), (4.7) dz d v ( ) e. dz v(z) Using (4.7) to solve fo the spatial coodinate z in tems of modified space coodinate can be find as z ( ' ) d '. (4.8) Fom (4.) and (4.4) we can wite (4.8) as ' ' z e d e. (4.9) [ ] We can see fom (4.9) as, z and this does not continue to gow. This gives us a spatially limited lens. This is convenient fo puposes of implementation. The wave popagation can be descibed by the souce-fee telegaphe equations ((.3) in [3]). We can tansfom the D wave equation to an equivalent space coodinate as 6
7 E. e E e d,s ) dh( H, e d,s ) de( μ (4.) 4.4 Solution of the Tansmission-Line Equations We solve an equivalent poblem of [3], but instead of an incease in the tansmission-line impedance we have a decease in wave impedance, but the equations in [3] still apply. One can define the tansmission coefficient fo high h T and low fequency l T as follows T, T h + l (4.) Calculate the diffeence between these two coefficients as [ ] [ ] > + + + T T h l. (4.) This is always positive except at. Thus, thee is a doop (positive, i.e. a decease) fom initial to final value fo both inceasing and deceasing impedances. The impedance is deceasing but thee is still a doop. We can use the exact solution of the tansfe function in (3.8) of [3], ( ) ( ) ( ) + + + + + G S ~ G S sinh G S S G S cosh e T, (4.3) whee S is the nomalized complex fequency t ) j ( s t S ω Ω +. (4.4) The high-fequency gain is defined in (3.4) of [3] as 4 G e g. (4.5) One can define the tansit, nomalized and doop time ((3.) of [3]) paametes as follows
t τ t t τ d ln t d t ( g ) v. tansit time though lens, nomalized time, nomalized doop time (4.6) t d is the doop time, the step-esponse fom is defined as (3.3) of [3] [ + τ τ + O( τ )] as τ R( τ ) g d. (4.7) 4.3 Example Now we can calculate the lens thickness fo a given dielectic taget pemittivity. Setting t td. 5 and., and using a t ps pulse width(imum time of inteest) fom (4.4) td ns and ns, t d. ln ( g ) (4.8) Fom (4.3) and (4.6) ctd ln ln( e ( g ) ), (metes). (4.9) Substituting (4.8) in (4.9) we have z [ ct ] d ln ( g ) e (metes) ln( ). (4.) 8
4.4 Conclusion We might design a spatially limited exponential lens based on [3]. This lens is designed fo a biological application [6]. Fom (5) we can find the z values fo diffeent biological tissues, which ae summaized in Table. Table Design paamete values fo diffeent biological tissues [7,8]. One can see fom Table. that, if we have lowe dielectic constant fo taget biological tissue, we need a smalle lens. This is not the only consideation. A lage dielectic constant in the lens exit esults in a smalle spot size and highe fields. The smalle spot size concentates the enegy in the vicinity of the skin cance. One can find how changes as a function of and z fom (4.9) and (4.9) ( ) e, ( z ). z (4.) Let us conside td ns and find ( ) and ( z ) with espect to and z fo diffeent dielectic tissues. These ae pesented in Figues 7 and 7.8. 9
Figue 7: ( ) values fo diffeent dielectic tissues. Figue 8: (z) values fo diffeent dielectic tissues. The compession of the coodinates fo t ns and 8 is pesented in Figue 9. d
Figue 9: Compession of the coodinates fo t ns and 8. d
5. Lens Design fo Incoming Spheical Wave In this section an altenate lens design pocedue is discussed to obtain bette focusing fom a polate-spheoidal in which the lens is not a sphee. The lens design consideations ae based on [9]. N layes of an inceasing dielectic lens, which have the same atio of dielectic constants between adjacent layes, ae consideed fo a polatespheoidal IRA. Instead of using a half-spheical lens, a new appoach is poposed fo incoming spheical waves to obtain bette focusing fo a polate-spheoidal IRA 5. Design Consideations layes of inceasing-dielectic-constant lens ae used based on the calculations in Section. We use the same atio of dielectic constant between subsequent layes as atio atio n + N. n, ( ) n + n N N atio, (5.) We use N layes and 8 fo the wost case scenaio fo biological applications. We stat fom fee space and ou taget dielectic is 8 and atio.55 between subsequent layes. The fist shell of the lens fo incoming spheical wave is illustated in Figue.
Figue : Lens fo incoming spheical wave [9]. In Figue θ actan( b z ), θ θ. (5.) Equation (54) epesents the ange of inteest of angles fo the incoming wave fom the polate-spheoidal IRA which has the dimensions as given in []. Fom (5.) and (5.3) o fo the fist shell θ 53.3. Inside the lens the ays change thei diection to the angle of θ with espect to the z ' -axis and θ π fo geometical design puposes. l and l ae the distances on the z ' -axis, h is the height of the lens. The nomalized l and l paametes ae defined fom (4.7) in [9] as l h l h sin( θ θ ) + ( )sin( θ [ sin( θ θ ) + sin( θ )] ( )sin( θ sin( θ )sin( θ )sin( θ ) sin( θ ) sin( θ ) ), ). (5.3) 3
To find θ as a function of θa quadatic equation in eithe cos( θ ) o sin( θ ) can be solved fom (4.8-5.) in [9] as cos( θ ) sin( θ ) A ( l l ABsin ( θ ) + Bcos( θ ) A A( A Bcos( θ )) + Bsin( θ ) ), B ( l l ). B AB cos( θ ) + A B AB cos( θ ) + A B AB cos( θ ) + A B AB cos( θ ) + A A sin ( θ ) A sin ( θ ),, (5.4) A lens bounday cuve can be defined by the coodinates of z ' and Ψ as a function of θ and θ fom (4.) and (4.) in [9] as z ' ( l l ) htan( θ ) h tan( θ ) tan( θ ) Ψ z ( l ) htan( )tan( ) tan( ) l θ θ θ. h h tan( θ ) tan( θ ) (5.5) This appoach is just fo the fist shell, but we can expand it to the othe shells. atio.55 and we will have diffeent l, l, θ and θ fo each laye. We can define a new coodinate system which is centeed at z z. We will call this system z' and it can be defined as z ' h ( z z ) h. (5.6) The IRA and lens geomety ae pesented in Figue. The angles of θ and θ ae in given Figue 4
Figue : IRA and lens geomety. 5
Figue : θ and θ values. We use N layes and Δθ is the change in the angle as one goes fom one laye to the next. This is constant and is given by Δθ ( θ θ ) N N. (5.7) We design the lens fo two diffeent θ angles as: o 9 ( π ) θ. (5.8) o 85 Fo theπ case o Δθ 3.7 and fo the o 85 case distance between each laye-beginning point on the z ' -axis. ' distances on the z -axis which is shown in Figue. o Δ θ 3.. Δ z ' h is the nomalized z ' n n h is the sum of the n 6
5. Concluding Remaks fo the Lens Design fo Incoming Spheical Wave We have designed a lens fo incoming spheical waves to obtain bette focusing fom a polate-spheoidal IRA. This design is based on the same pocedue as in [9].In this design, howeve, just a single laye was used. We extended this design to N layes. In this case we have diffeent l h, h,,, h h, z ' l θ θ n and n h. We calculated these values fo the fist laye. Then we coect the values fo the othe layes. Fist we calculate the Ψ h and z' h values fo the fist laye, then fo the second laye we calculate Ψ h and z' h. We coect them by multiplying with hcoected h n hvalue, then we add Δ z ' n h fo each laye to find the coected z ' n h values at that laye. o As one can see fom Figues 3 and 7.4, fo θ 85 case we obtain bette focusing. We call h the adius of the shell, it is a univesal nomalization paamete. But this calculation is not detemining h because it is an optical calculation (infinite fequency). To detemine how lage h should be is a difficult poblem. Clealy hc must be much geate than the focus pulse width at the focus, and the ise-time of the incoming wave, othewise it does not focus, λ << othe dimensions of the lens. h should be smalle than the adius of the eflecto as well ctδ 3 cm<< h < b 5 cm. (5.9) 7
Figue 3: Ψ h vs z' h fo θ. π Table 3: h Δ ' ' θ and θ values fo π n h, zn h, zn h,, θ. 8
' o 85 Figue 4: Ψ h and z h fo θ. Table 4: h ' o Δ θ and θ values fo θ. ' n h, zn h, zn h,, 85 9
5.3 Lens Design fo Incoming Spheical Wave fo Diffeent Biological Dielectic Tissues Five diffeent biological dielectic tissues ae used as diffeent taget dielectics and we ty to obtain bette focusing fom a polate-spheoidal IRA fo an incoming spheical wave fom the eflecto fo these tissues. This subsection is an extension of the pevious one. We use 5 diffeent taget dielectic tissues compising wate, muscle, tumo, skin and fat. Ten layes of an inceasing dielectic constant lens that have the same atio of dielectic constants between adjacent layes ae consideed fo a polatespheoidal IRA.We use the same atio of dielectic constant between subsequent layes as atio N, (5.) whee and values fo diffeent human tissues ae pesented in Table 5. Table 5: atio atio and values fo diffeent human tissues [7,8]. A lens is designed fo incoming spheical waves to obtain bette focusing fom a polate-spheoidal IRA fo diffeent dielectic human tissues. We obtain bette focusing fo the highe dielectic lens. h vs z' o Ψ h values fo θ π and 85 fo diffeent ae pesented in Figue 3. and Figue 3.. One can see fom Figue 5 and Figue 6 that fo smalle, the fist shell moves left. We have fixed the vetical ( Ψ h ) axis values to incement by a unifom value of., leaving some vaiation (small) in the location along the hoizontal coodinate. 3
Figue 5: Ψ h vs z' o h fo θ 9 and diffeent. 3
Figue 6: Ψ h vs z' h o foθ 85 and diffeent. 3
Refeences [] C.E. Baum, Focal wavefom of a polate-spheoidal impulse-adiating antenna (IRA), Radio Science, accepted fo publication. [] C.E. Baum, Addition of a Lens Befoe the Second focus of a Polate-Spheoidal IRA Senso and Simulation Note 5, Apil 6. [3] C.E. Baum and J.M. Leh, Tapeed tansmission-line tansfomes fo fast highvoltage tansients, IEEE Tans. Plasma Science, vol. 3,, pp. 7-7. [4] S. Altunc and C.E. Baum. Extension of the analytic esults fo the focal wavefom of a two-am polate-spheoidal impulse-adiating antenna (IRA), Senso and Simulation Note 58, Nov. 6. [5] C.E. Baum, Some consideations concening analytical EMP citeia wavefoms, Theoetical Note 85, Oct 976. [6] K.H. Schoenbach, R. Nuccitelli and S.J. Beebe, AP, IEEE Spectum, Aug 6, pp. -6. [7] M. Convese, E.J. Bond, B.D.V. Veen and S.C. Hagness, A computational study of ulta-wideband vesus fo beast cance teatment, IEEE Tanscations on Micowave and Techniques, vol. 54, 6, pp 69-8. [8] C. Gabiel, S. Gabiel and E. Cothout, The dielectic popeties of biological tissues: I. Liteatue Suvey, Phys. Med. Boil.vol. 4, 996, pp 3-49.. [9] C.E. Baum, J.J. Sadle and A.P. Stone A unifom dielectic lens fo launching a spheical wave into a paaboidal eflecto, SSN 36, July 993. [] S. Altunc and C.E. Baum. Extension of the analytic esults fo the focal wavefom of a two-am polate-spheoidal impulse-adiating antenna (IRA), Senso and Simulation Note 58, Nov. 6. 33