On Fractional Paradigm and Intermediate Zones in Electromagnetism: I. Planar Observation

Transcription

1 University f Pennsylvania SchlarlyCmmns Departmental Papers (ESE) Department f Electrical & Systems Engineering August 999 On Fractinal Paradigm and Intermediate Znes in Electrmagnetism: I. Planar Observatin Nader Engheta University f Pennsylvania, engheta@ee.upenn.edu Fllw this and additinal wrks at: Recmmended Citatin Nader Engheta, "On Fractinal Paradigm and Intermediate Znes in Electrmagnetism: I. Planar Observatin",. August 999. Pstprint versin. Published in Micrwave and Optical Technlgy Letters, Vlume 22, Issue 4, August 20, 999, pages Publisher URL: A prtin f the preliminary findings f this wrk was presented by the authr at the 998 IEEE Antennas and Prpagatin Sciety (AP-S) Internatinal Sympsium/USNC-URSI Radi Science Meeting in Atlanta, Gergia, June 2-26, 998. This paper is psted at SchlarlyCmmns. Fr mre infrmatin, please cntact repsitry@pbx.upenn.edu.

2 On Fractinal Paradigm and Intermediate Znes in Electrmagnetism: I. Planar Observatin Abstract In this Letter the kernel f the integral transfrm that relates the field quantities ver an bservatin flat plane t the crrespnding quantities n anther bservatin plane parallel with the first ne is fractinalized fr the tw-dimensinal (2-D) mnchrmatic wave prpagatin. It is shwn that such fractinalized kernels, with fractinalizatin parameter ν between zer and unity, are the kernels f the integral transfrms that prvide the field quantities ver the parallel planes between the tw riginal planes. With prper chice f the first tw planes, these fractinal kernels can prvide us with a natural way f interpreting the fields in the intermediate znes (i.e., the regin between the near and the far znes) in certain electrmagnetic prblems. The evlutin f these fractinal kernels int the Fresnel and Fraunhfer diffractin kernels is addressed. The limit f these fractinal kernels fr the static case is als mentined. Keywrds fractinal kernels, fractinal Calculus, fractinal paradigm, intermediate zne, electrmagnetic waves Cmments Pstprint versin. Published in Micrwave and Optical Technlgy Letters, Vlume 22, Issue 4, August 20, 999, pages Publisher URL: A prtin f the preliminary findings f this wrk was presented by the authr at the 998 IEEE Antennas and Prpagatin Sciety (AP-S) Internatinal Sympsium/USNC-URSI Radi Science Meeting in Atlanta, Gergia, June 2-26, 998. This jurnal article is available at SchlarlyCmmns:

3 On Fractinal Paradigm and Intermediate Znes in Electrmagnetism: I. Planar Observatin Nader Engheta University f Pennsylvania Mre Schl f Electrical Engineering Philadelphia, Pennsylvania 904, U.S.A. engheta@ee.upenn.edu URL: Key Wrds: Fractinal kernels, Fractinal Calculus, Fractinal Paradigm, Intermediate zne, Electrmagnetic Waves. Abstract In this Letter the kernel f the integral transfrm that relates the field quantities ver an bservatin flat plane t the crrespnding quantities n anther bservatin plane parallel with the first ne is fractinalized fr the tw-dimensinal (2-D) mnchrmatic wave prpagatin. It is shwn that such fractinalized kernels, with fractinalizatin parameter ν between zer and unity, are the kernels f the integral transfrms that prvide the field quantities ver the parallel planes between the tw riginal planes. With prper chice f the first tw planes, these fractinal kernels can prvide us with a natural way f interpreting the fields in the intermediate znes (i.e., the regin between the near and the far znes) in certain electrmagnetic prblems. The evlutin f these fractinal kernels int the Fresnel and Fraunhfer diffractin kernels is addressed. The limit f these fractinal kernels fr the static case is als mentined. Prtin f the preliminary findings f this wrk was presented by the authr at the 998 IEEE Antennas and Prpagatin Sciety (AP-S) Internatinal Sympsium/USNC-URSI Radi Science Meeting in Atlanta, Gergia, June 2-26,

4 Intrductin In the past few years, we have been interested in develping and studying the fractinal paradigm in electrmagnetic thery [-7]. In such a paradigm, the gal is t bring the tls f fractinal calculus (see e.g., [8])--the tpic that deals with fractinal differentiatin and integratin-- and in general fractinalizatin f varius peratrs and the electrmagnetic thery tgether and t find applicatins f such fractinalized peratrs in electrmagnetic prblems. The basic idea behind this paradigm is t find the fractinal r interplated respnses that are effectively intermediate situatins between the cannical slutins t a given prblem []. If the cannical cases are labeled Case and Case 2 and if there exists an peratr L that maps Case nt Case 2, then ne wuld inquire whether the fractinalizatin f such an peratr wuld prvide us with the fractinal r interplated slutins that are cnsidered intermediate between the Cases and 2. Sme f the detail behind this idea and a general recipe fr the fractinalizatin f peratr L are described in [,2]. Here, we present anther case study that addresses the issue f the fractinalizatin f peratrs and its rles in intermediate znes in electrmagnetism. The idea is this: since fractinal derivatives/integrals effectively address the intermediate behavir fr the differentiatin/integratin peratrs, can fractinalizatin f sme apprpriate peratrs prvide us with prper tls t tackle certain electrmagnetic prblems invlving intermediate znes? It is well knwn that fr a given charge and/r current distributins, whether it is a static r dynamic case, the quantities f interest such as electric fields r ptentials resulted frm such a surce may be expressed in terms f integrals invlving the apprpriate Green functins and the given surce. Furthermre, it is als well knwn that in rder t find sme simpler expressins fr the -2-

5 quantity f interest, usually mre attentin has been paid t the analysis f the far fields and the near fields while less attentin has been aimed at the intermediate znes. In the present wrk we shw that if the apprpriate kernels f integral transfrms, which relate quantities f interest in the near zne t the quantities in the far-zne regin, are cnsidered and then are prperly fractinalized, such fractinalized kernels can act as kernels f integral transfrms that link the near-zne quantities t the quantities at the intermediate znes. This wuld prvide anther lgical interpretatin fr the intermediate znes as the regins where the integral transfrm with fractinalized kernels can give the fields and ptentials. The detail f this prblem is given in the fllwing sectin. Gemetry and Frmulatin f the Prblem Cnsider a Cartesian crdinate system ( x, y, z) and a mnchrmatic surce represented as the vlume current density J in free space. The time dependence f e i is assumed thrughut this wrk. Withut lss f generality fr presenting the idea and fr the sake f simplicity in the mathematical frmulatin here, we assume the prblem t be a tw-dimensinal (2-D) ne in which all quantities f interest are independent f y- ω t crdinate, the surce J = J( x, z) is a functin f the x- and z-crdinates, and its transverse crss sectin in the x-z plane is cnfined t a limited regin. Let us dente the ptential (r a Cartesian cmpnent f the fields) f interest with the symbl ψ ( x, z ). In the regin utside the surce, this functin satisfies the Helmhltz equatin 2 ψ + k 2 ψ = 0 where k ω µ ε with µ and ε being the permeability and permittivity f free space, respectively. We chse tw bservatin planes parallel with -3-

6 the x-y plane, ne lcated at z = z > 0 and the ther at z = z > z where z and z are bth utside the surce regin, and the regin between these tw planes is surce-free. (See Fig..) These tw values f z can be at any lcatins alng the z-axis (as lng as the space z z z is surce free), and if desired ne can assume that z is in the near zne f the surce while z is in the far-zne regin. The ptential distributins n these tw planes are dented by ψ ( x, z ) and ψ ( x, z ), respectively. Applying standard Green s therem (see e.g. [9, page 49]) t the half space z f ψ ( x, z ) using the fllwing integral transfrm z, ne can express ψ ( x, z ) in terms z + ψ ( x, z) = K( x, z; x', z) ψ ( x', z) dx' () where, after prper mathematical steps, the kernel K( x, z; x', z ) can be explicitly written as Kxz (, ; x', z) = Kx ( x', z z) = 2 i k ( z z ) 2 ( x x') + ( z z ) () 2 H k ( x x') + ( z z ) 2 e 2 j () with H af being the first-rder Hankel functin f the first kind. The spatial Furier transfrm (with respect t variable x) f the tw functins ψ ( x, z ) and ψ ( x, z ) written as ~ ψ ( kz, ) ψ ( xz, ) e ikx dx z + ~ ψ ( kz, ) z + ψ ( xz, ) e ikx dx (3) (2) are Here it is assumed that in this physical situatin, in the half space z z, ψ ( x, z) appraches zer when x. -4-

7 where k is the spatial angular frequency (Furier variable). Cnsidering the spaceinvariance prperty f the kernel K( x x', z z ) given in Eq. (2), Eq. () can be written in the spatial Furier dmain as ~ ~ ψ ( kz, ) = Kkz (, z) ~ ψ (, kz) (4) where ~ K( kz, z) is the spatial Furier transfrm f the kernel K( x, z z ). Cnsidering a suitable integral representatin f the Hankel functins (see e.g., [0, page 89]) and the expressin fr the kernel K( x x', z z ) given in Eq. (2), ne can find ~ K( kz, z) t be ~ 2 2 i k k ( z z ) Kkz (, z ) = e (5) which is cnsistent with the prpagatr in the psitive z directin fr the Helmhltz 2 2 peratr + k. (See e.g., [, p. 50]) Returning t the ntin f cannical cases dented as Case and Case 2 as specific slutins t a given prblem [], here fr the prblem at hand, which is a 2-D wave prpagatin, we can label the functin ψ ( x, z ) n the plane z = z as the Case and the functin ψ ( x, z ) n the plane z = z as the Case 2, bth f which are bviusly slutins t the Helmhltz equatin. Then searching fr a mapping that links the Case t Case 2, ne finds Eq. () as such mapping. Therefre, the linear peratr L that maps ψ ( x, z ) int ψ ( x, z ) is indeed the integral transfrm with the kernel K( x x', z z ) S we can write z + L Kx ( x', z z) L dx'. (6). Nw we pse the fllwing questin: If we prperly fractinalize this peratr L and symblically shw such a fractinalized peratr as L ν where ν is a fractinalizatin -5-

8 parameter with values between zer and unity, wuld applying f L ν n the functin ψ ( x, z ) prvide us with a functin that is the slutin t ur Helmhltz equatin fr bservatin pints n a plane lcated smewhere between z and z? In ther wrds, wuld this fractinal peratr act as an integral transfrm t find the slutins in the intermediate zne? T find answers t these questins, we first need t fractinalize the peratr L in Eq. (6). We have presented elsewhere a recipe fr fractinalizatin f sme class f linear peratrs [2,]. The detail is nt repeated here, but a brief mentin f the relevant parts f that recipe is given here. The first step in fractinalizatin f an peratr is t find the eigenfunctins and eigenvalues f the peratr. T that end, we write L f ( x) = a f ( x) m m m (7) where fm( x) and a m are the eigenfunctins and eigenvalues f the peratr L. Here m =,2, 3, L, n with n being the dimensin f the space f the dmain (and the range) f the peratr L. Fr the peratr L given in Eq. (6), we then have z + Kx ( x', z z) fm( x') dx' = amfm( x). (8) T describe the eigenfunctins and eigenvalues f this peratr, we take the spatial Furier transfrm f Eq. (8) (with respect t variable x) and btain ~ ~ ~ K( k, z z ) f m ( k) = a m f m ( k), (9) where ~ fm( k) represents the spatial Furier transfrm f fm( x). Substituting Eq. (5) int Eq. (9), ne gets the fllwing equatin fr ~ f ( k) L NM 2 2 O m m = QP m i k k ( z z) ~ e a f ( k ) 0. (0) -6-

9 Frm this equatin, a set f slutins fr ~ f ( k) m and their crrespnding eigenvalues a m is fund t be ~ f ( k) = δ ( k k ) m m () with a m = e 2 2 m i k k ( z z ) where δaf is the Dirac delta functin, and k m is any given value in the spatial Furier dmain. Taking the inverse spatial Furier transfrm f Eq. () results in f ( x)= e m 2π ik x m (2) which is in ttal agreement with the well-knwn fact in the system thery that fr a spaceinvariant linear system the eigenfunctins are cmplex expnential functins as e ik x m. Since k m is any given pint alng the spatial Furier variable k, we chse the ntatin h instead f k m t shw the cntinuus nature f k m, and therefre use the symbls f x h ( ) and a h instead f f x m( ) and a m, respectively. S we have the eigenfunctins and eigenvalues f ur peratr L as ih x f ( x)= e, (3a) h a h = e 2 2 i k h z z ( ). (3b) The eigenvalues here indeed equal t the Furier transfrm f the kernel K, i.e., ~ a = K ( h, z z ). Having expressed the eigenfunctins and eigenvalues f the peratr h L, the next step is t define the fractinalizatin f the peratr L as the new peratr L ν whse eigenfunctins are the same as thse f L, but whse eigenvalues are a ν h where ν the fractinalizatin parameter here is between zer and unity. That is ν ν L f = a f. (4) h h h If we write the new fractinal peratr L ν as the fllwing integral peratr -7-

10 L ν z + Kν ( x x', z z) L dx' (5) we need t find the kernel K ν ( x x', z z ), which we name fractinal kernel here. Having the expressin fr the eigenvalues f the fractinal peratr L ν, we can find the Furier transfrm f this fractinal kernel as ~ 2 2 i k h ( z z ) K ( h, z z ) ν ν = a = e. (6) ν h The expressin fr K ν ( x x', z z ) can then be btained by inverse spatial Furier transfrming Eq. (6). That results in iν k( z z) () Kν ( x x', z z ) = H k ( x x') + ν ( z z) e j, (7) 2 ( x x') + ν ( z z ) with 0 ν. 2 In the next sectin, we will discuss sme f the ntable features f this fractinal kernel. Physical Remarks When ne cmpares Eqs. (2) and (7), ne ntices that νb g b g. (8) K x x ', z z = K x x ', ν ( z z ) It can be easily seen that the integral transfrm with such fractinal kernel satisfies the general features f fractinalized peratrs discussed in ur previus wrk [,2]. Specifically it can be shwn that when the fractinalizatin parameter ν appraches b g becmes ur riginal kernel unity, the fractinal kernel K ν x x', z z Kx b x', z z g; When the parameter ν ges t zer, the fractinal kernel in Eq. (7) can be shwn t apprach the Dirac delta functin δ x x' f, as expected, since b g a f becmes a kernel that maps the functin ψ ( x, z ) K x x', z z = K x x', 0 0 a at the -8-

11 plane z = z nt itself; and finally it can be shwn that the integral transfrm with this fractinal kernel (i.e., Eq. (5)) satisfies the additivity prperties in fractinal parameter ν, i.e., L L = L ν ν ν + ν 2 2 where L ν is described in Eq. (5). This is because we can shw that K K = K + where the symbl here dentes the cnvlutin. Frm Eq. (8), ν ν ν ν 2 2 we can bserve that when 0 < ν <, the argument ν ( z z ) can be effectively interpreted as the distance between the riginal plane at z and anther plane between z and z. If the z-crdinate f this intermediate plane is dented by z ν, we can then write zν z = ν ( z z) frm which z ν can be explicitly given as zν = z + ν ( z z). (9) and thus the fractinal kernel K ν can be explicitly written as i k( zν z) () 2 2 Kνbx x', z zg = Kbx x', zν zg = H k ( x x') + ( zν z) 2 2 e j 2 ( x x') + ( zν z ) (20) We ntice again that when ν 0, the intermediate plane wuld apprach z, and when ν, it will g t the riginal plane at z. Therefre, the fractinal kernel K ν given in Eq. (20), which is the result f fractinalizatin f the riginal kernel K in Eq. (7) cnnecting the functin at the plane z t the crrespnding functin at the plane z, is a kernel that links the functin at the plane z t the functin at an intermediate plane at z ν. As the fractinalizatin parameters varies between zer and unity, the lcatin f intermediate plane changes frm z t z. If the riginal kernel is the mapping between the near-zne and the far-zne fields, the fractinal kernel with parameter ν between zer and unity wuld prvide anther way t interpret the fields in the intermediate znes. 2 Fr nw, we are still taking ν t be between zer and unity. Later in ur discussin, this restrictin will be remved. -9-

12 Therefre, fr bservatin pints n parallel flat planes (i.e., planar bservatin) in the Cartesian crdinate system as described here, the fractinalizatin f the near-zne t far-zne mapping prvides us with the kernel fr the intermediate znes. In the general expressin fr the fractinal kernel, when the parameter ν is specified at a nn-integer value between zer and unity, the value f z ν is then determined frm Eq. (9). Cnversely, if we are interested in determining the fractinal kernel fr a specified plane f bservatin at z ν, then we can find the value f ν frm the expressin ν = z z ν z z. (2) Knwing ν, we can then determine the apprpriate fractinal kernel frm Eq. (7) (r needless t say, alternatively, this kernel can be btained directly frm Eq. (20) when z ν is specified.) The fractinalizatin f such kernels and the relatinship between the fractinalizatin parameter ν and the lcatin f the bservatin plane z ν becme even mre evident, if the z-crdinate is chsen such that the riginal plane is at z = 0. 3 In this case, z ν = ν z, and it can be easily seen that as ν varies frm zer t unity, z ν evlves frm zer t z. It must be nted that althugh the fractinalizatin parameter ν is taken t be between zer and unity in the afrementined discussin, it des nt have t be limited t this range and it can take values larger than unity in the expressin fr the fractinal kernel in Eq. (20) (r Eq. (7) r (8)). Fr ν >, the value f z ν in Eq. (9) becmes 3 In this case, the surce J shwn in Fig. shuld be mved t the left (r the Cartesian crdinate shuld be mved t the right) s that the regin z 0 shuld still be surce-free. -0-

13 greater than z, which implies that Eq. (20) can als bviusly prvide us with the kernels apprpriate fr the planes beynd the regin z z z. It is wrth nting that when ν k ( z z ) >> the fractinal kernel in Eq. (20) can be apprximately written as 3π i k i( z z e ν ) ik ( x x') + ( zν z) Kνbx x', z zg = Kbx x', zν zg 2π 3 e. (22) ( x x') + ( z z ) where ν ( z z ) is substituted by z ν z. Additinally, if we nw use the standard Fresnel apprximatin (see, e.g., [, p. 59]), fr which here as a sufficient cnditin ne wuld assume that ν 3 3 k 4 ( z z) >> 8 ( x x') max, we btain ik( zν z ) k e i 2 2( z x x ν z ) ( ') νb g b ν g e (23) iλ ( zν z) K x x', z z = K x x', z z which is the usual kernel in the regin f Fresnel diffractin fr the 2-D prpagatin π when ν ( z z ) is substituted by z ν z, and λ = 2. In the Fraunhfer diffractin 2 regin (see e.g., [, p. 6]), where we can assume 2ν ( z z) >> k( x x') max, the fractinal kernel can be further simplified as k ik z z i x 2 ( ) ( z z ) k i xx' zν z b g b g, (24) ν ν 2 ν e e Kν x x', z z = K x x', zν z e iλ ( z z ) which is the cnventinal Fraunhfer diffractin kernel fr the 2-D prpagatin prblem. S as the parameter ν increases frm zer and gets increasing larger, the fractinal kernel k ν given in Eq. (20) evlves frm the Dirac delta functin δ( x x' ) fr ν = 0, and becmes the kernels fr the intermediate zne, and as ν keeps increasing, this fractinal kernel evlves int the kernel which, fr a certain cnstraint n x, x', z ν, and z, becmes the kernel fr the Fresnel diffractin regin, and eventually when ν increases even further it will get t the kernel that again fr sme specific cnstraint n x, x', z ν, and z becmes --

14 the kernel fr the Fraunhfer regin. As is well knwn, aside frm the term e e ik ( z z i ν ) 2( zν ) iλ ( z z ) ν k x 2 z in the kernel fr the Fraunhfer diffractin regin, this kernel represents the Furier transfrm kernel. In the ptics literature, it has been shwn that while Fraunhfer diffractin prduces the Furier transfrm f the bject plane, the Fresnel diffractin may result in Fractinal Furier transfrm f the bject [2-4]. The reader interested in the cncept f fractinal Furier transfrmed is referred t the excellent wrk f Ozaktas, Mendlvic, Lhmann and their c-wrkers in the literature (see, e.g., [3-8].) Our fractinal kernel given in Eq. (20) (r Eq. (7) r (8)) expresses the exact kernels fr all pssible intermediate planes starting frm the surce/aperture (at z ) evlving all the way twards the bservatin planes in the far zne when the fractinal parameter ν varies frm zer t. S when the fractinalizatin parameter ν in Eq. (20) (r Eq. (7)) becmes large such that the Fresnel apprximatin may be applicable, then the fractinal kernel wuld apprach the kernel f the Fractinal Furier transfrm. Hwever, it must be nted that when ν is taken a small value and the term ν ( z z) zν z is n lnger satisfies the cnditins ν k( z z) >> and the Fresnel apprximatin, the kernel fr the Fresnel diffractin regin cannt be used, and instead the exact frm f the fractinal kernel given in Eq. (20) (r Eq. (7)) shuld be utilized. S fr the fractinal kernel given in this reprt, n cnstraint n x, x', z ν, and z are necessary. In fact, the cncept f fractinal kernel can even be extended t the static prblems. If ne is interested in the fractinal kernels fr the 2-D electrstatic prblem, ne can find such kernels by evaluating the limit f Eq. (7) (r Eq. (20)) when k 0. These fractinal kernels fr the static case can then be explicitly written as -2-

15 K ( x x', z z ) = ν k 0 ν ( z z) zν z = (25) π ( x x') + ν ( z z ) π ( x x') + ( z z ) The fractinal kernels presented in this Letter have been fr the case f mnchrmatic 2-D wave prpagatin, and fr bservatin pints n parallel flat planes (i.e., planar bservatin) in the Cartesian crdinate system. The fractinal kernels fr the cases f bservatins ver the cylindrical and spherical bundaries, which encunter smewhat different mathematical features, have als been studied [9], and will be ν reprted in a future publicatin. The analysis reprted here can be easily and straightfrwardly extended t the three-dimensinal (3-D) case fr planar bservatins. Amng the prblems f interest fllwing the analysis reprted in this Letter is the effect f cmplexificatin f the fractinalizatin parameter ν. Explring ptential applicatins f the fractinalizatin f kernels in treating and prcessing the fields in the intermediate znes f electrmagnetic radiatin and scattering prblems is anther prblem f interest t pursue. These are currently under study by the authr. Acknwledgements This wrk is supprted in part by the U.S. Natinal Science Fundatin Grant N. ECS References [] N. Engheta, Fractinal Paradigm in Electrmagnetic Thery, a chapter in the bk entitled Frntiers f Electrmagnetics, D. H. Werner and R. Mittra (eds.), IEEE Press, t appear in

16 [2] N. Engheta, Fractinal Curl Operatr in Electrmagnetics, Micrwave and Optical Technlgy Letters, Vl. 7, N. 2, pp. 86-9, February 5, 998. [3] N. Engheta, "On the Rle f Fractinal Calculus in Electrmagnetic Thery," in IEEE Antennas and Prpagatin Magazine, Vl. 39, N. 4, pp , August 997. [4] N. Engheta, "Electrstatic "Fractinal" Image Methds fr Perfectly Cnducting Wedges and Cnes," IEEE Trans. Antennas & Prpagatin, Vl. 44, N. 2, pp , Dec [5] N. Engheta, "On Fractinal Calculus and Fractinal Multiples in Electrmagnetism," IEEE Trans. Antennas & Prpagatin, Vl. 44, N. 4, pp , Apr Erratum: Vl. 44, N. 9, p. 307, September 996. [6] N. Engheta, "Use f Fractinal Integratin t Prpse Sme "Fractinal" Slutins fr the Scalar Helmhltz Equatin," a chapter in Prgress in Electrmagnetics Research (PIER) mngraph Series Vl. 2, Jin A. Kng (ed.), EMW Pub., Cambridge, MA, pp , ch. 5, 996. [7] N. Engheta, A Nte n Fractinal Calculus and the Image Methd fr Dielectric Spheres," J. f Electrmagnetic Waves and Applicatins, Vl. 9, N. 9, pp , September 995. [8] K. B. Oldham and J. Spanier, The Fractinal Calculus, Academic Press, New Yrk, 974. [9] A. Ishimaru, Electrmagnetic Wave Prpagatin, Radiatin, and Scattering, Prentice-Hall, Inc., New Jersey, 99. [0] A. Smmerfeld, Partial Differential Equatins in Physics, (Lectures n Theretical Physics, Vl. VI), Academic Press, New Yrk,

17 [] J. W. Gdman, Intrductin t Furier Optics, McGraw-Hill Pub., New Yrk, 968. [2] P. Pellat-Finet, Fresnel Diffractin and the Fractinal Furier Transfrm, Optics Letters, Vl. 9, N. 8, pp , September 5, 994. [3] 4 H. M. Ozaktas, M. A. Kutay, and D. Mendlvic Intrductin t the Fractinal Furier Transfrm and Its Applicatins, in Advances in Imaging and Electrn Physics, Vl. 06, Peter W. Hawkes (ed.), Academic Press, Califrnia, 999, pp [4] H. M. Ozaktas and D. Mendlvic, Fractinal Furier Transfrm, Jurnal f Optical Sciety f America A, Vl. 2, N. 4, pp , April 995. [5] H. M. Ozaktas and M. F. Erden, Relatinship amng Ray Optical, Gaussian Beam, and Fractinal Furier Transfrm Descriptins f First-Order Optical Systems, Optics Cmmunicatins, Vl. 43, pp , Nvember, 997. [6] A. W. Lhmann, "Image Rtatin, Wigner Rtatin, and the Fractinal Furier Transfrm," J. Opt. Sc. Am. A, Vl. 0, N. 0, pp , 993. [7] D. Mendlvic, H. M. Ozaktas, and A. W. Lhmann, "Graded-Index Fibers, Wigner-Distributin Functins, and the Fractinal Furier Transfrm," Applied Optics, Vl. 33, N. 26, pp , 0 Sept [8] H. M. Ozaktas, B. Barshan, D. Mendlvic, and L. Onural, Cnvlutin, Filtering, and Multiplexing in Fractinal Furier Dmains and Their Relatin t Chirp and Wavelet Transfrms, Jurnal f the Optical Sciety f America, A., Vl., N. 2, pp , February We thank Prfessr H. M. Ozaktas f Bilkent University fr sharing with us Ref. [3] and sme f his ther publicatins n the tpic f Fractinal Furier Transfrm. -5-

18 [9] N. Engheta, Fractinalizatin f Kernels fr Electrmagnetic Intermediate-Zne Fields in Cylindrical and Spherical Gemetries, a talk presented in the Prgress in Electrmagnetic Research Sympsium (PIERS 98), Nantes, France, July 3-7, 998. The ne-page abstract appeared in Vl. 33 f the Prceedings f this cnference, page

19 Figure Captin Fig.. Gemetry f the Prblem fr the tw-dimensinal (2-D) wave prpagatin. A mnchrmatic current surce J( x, y) is lcated in a Cartesian crdinate system ( x, y, z ). All quantities f interest are independent f y-crdinate. The riginal tw bservatin flat planes are at z = z and z = z. -7-

20 x J z z = z z = z FIGURE -8-