A Scaling Technique for the Design of Idealized Electromagnetic Lenses Citation Baum, Carl Edward (1969) A Scaling Technique for the Design of Idealized Electromagnetic Lenses. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/N9WZ-SP03. https://resolver.caltech.edu/CaltechTHESIS:01252016-081748086 Abstract A technique is developed for the design of lenses for transitioning TEM waves between conical and/or cylindrical transmission lines, ideally with no reflection or distortion of the waves. These lenses utilize isotropic but inhomogeneous media and are based on a solution of Maxwell's equations instead of just geometrical optics. The technique employs the expression of the constitutive parameters, ɛ and μ, plus Maxwell's equations, in a general orthogonal curvilinear coordinate system in tensor form, giving what we term as formal quantities. Solving the problem for certain types of formal constitutive parameters, these are transformed to give ɛ and μ as functions of position. Several examples of such lenses are considered in detail. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Papas, Charles Herach Thesis Committee: Unknown, Unknown Defense Date: 22 August 1968 Record Number: CaltechTHESIS:01252016-081748086 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:01252016-081748086 DOI: 10.7907/N9WZ-SP03 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9540 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 25 Jan 2016 17:16 Last Modified: 25 Apr 2024 20:41 Thesis Files Preview PDF - Final Version See Usage Policy. 12MB Repository Staff Only: item control page

A SCALING TECHNIQUE FOR THE DESIGN OF IDEALIZED ELECTROMAGNETIC LENSES Tj:lesis by Carl E . Baum Capt . , USAF In Partial Fulfillment of the Requi rements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1969 (Submitte d August 22 , 1968) -ii- ACKNOWLEDGEMENT We would like to thank Professor C. H. Papas for his guidance and encouragement and Mrs. R. Stratton for typing the text. -iiiABSTRACT A technique is developed for the design of lenses for transitioning TEM waves between conical and/or cylindrical transmission lines, ideally with no reflection or di stortion of the waves. These lenses utilize isotropic but inhomogeneous media and are based on a solut i on of Maxwell's e quat ions instead of just geometrical optics. The technique employs the expres sion of the constitutive parameters, £ and µ , plus Maxwell's equations, in a general orthogonal curvilinear coordinate system in tensor form, giving what we term as formal quantities. Solving the problem for certain types of formal constitutive parameters, these are transformed to gi ve functions of position. in detail. £ and µ as Several examples of such l enses are considered -ivTABLE OF CONTENTS I. Introduction l II . Formal Vectors and Operators 5 III. Formal Electromagnetic Quantities lO IV. Restriction of Constitutive Parameters to Scalars l4 V. General Case with Field Components in All Three Coordinate Directions 16 VI . Three- Dimensional TEM Waves 19 VII. Three-Dimensional TEM Lenses 29 A. Modified Spherical Coordinates 29 B. Modi f i ed Bispherical Coordinates 33 c. Modified Toroidal Coordinates 47 D. Modified Cylindrical Coordinates 57 VIII. Two- Di mensional TEM Waves 6l IX . Two-Dimensional TEM Lens es 66 X. Conclusion 74 Appendix A: Appendix B: References Characteristics of Coordinate Systems for Field Components in Al l Th ree Coordinate Directions Characteristics of Coordinate Systems for Field Components in Two Coordinate Directions 76 82 85 -1- I. INTRODUCTION One of the techniques used in the solution of electromagnetic boundary v a lue problems consists in writing Maxwell's equations in orthogonal curvilinear coordinates and then solving, not for the physical components of the fields, but for quantities which comb ine the physical components with scale factors of the coordinate transformation. These new quantiti es are components of tensors and tensor densities r e fer re d to the orthogonal curvilinear coordinate syste m. Similarly the constitutive parame ters of the medium are combined with the sca l e factors in the resulting equations . In making such a transformation one h op e s to simplify the equations and/or boundary co nd itions in some way. One type of problem on which this technique has b een used relates to waveguides (1,2). I n this case one takes a waveguide fille d with a homogeneous isotropi c medium, and transforms t o a n orthogonal c urvi linear coordinate system in which the b ounda ry walls a r e more conveniently expressed. The res ulting transforme d constitutive parameters, however, are in general inhomogen eou s a n d a nis o t ropic . Thus while the boundaries have been simpli fie d, the medi um h a s become mo r e complicated. In t hi s report we consider an exte ns ion of this tech nique. We assume that the formal constitutive parameters , as expressed in some o rth ogonal curvilinear coordin ate system , are of a p articul a rly simple form, i. e . , homogeneous, at l eas t as they r elat e to th e a llowed field component s. Furth ermore , we assume that the con s tituti v e p a rameters, -2- before being transformed to the curvilinear system, are those of an inhomogeneous but isotropic medium. From this we find many cases of isotropic inhomogeneous media for which certain types of electromagnetic wave propagation can be simply expressed. In this approach the medium is made inhomogeneous and perfectly conducting boundaries are geometrically arranged such that when they are transformed into the appropriate orthogonal curvilinear coordinates a simpler problem results which can be solved by more standard techniques. The present approach can then be used to define geometrie s for perfectly conducting boundaries and distribution fUnctions for inhomogeneous media such that devices built to such designs will transport electromagnetic waves in certain desirable ways. In particular, we consider cases which in the c urvilinear coordinat e system corresponds to a problem of a TEM plane wave on a cylindrical transmission line. In the reference cartesian (x,y,z) coordinates the waves are still TEM, but not necessarily plane. For the examples considered the particular conductor geometries and media inhomogeneities can be used to transition waves between two transmission lines, each of which is a conical or cylindrical transmission line . Furthermore , the transition is accomplis hed with neither reflection nor distortion of the wave. Another application of such examples is for a highly directional highfrequency antenna in which the special geometry and medium inhomogeneity is us ed to launch an approximate TEM wave over a c ross section with dimensions much larger than a wavelength. -3The approach followed in this report then represents a design procedure for a certain kind of electromagnetic lens. The properties of such a l ens , combined with appropriate perfect conductors, are independent of frequency assuming that the permittivity and permeability of the medium used are real and frequency independent and that its conductivity is zero. This result is in contrast to lenses based on a geometrical opt i cs app roximation, such as the well known Luneburg lens (3), which relies on the frequency being sufficiently high. The lenses considered here, used with appropriate transmission lines, can then transmit arbitrary pulse waveforms without distortion. While the cases considered represent exact solutions to the vector wave equati on, there are , of course , approximations involved in the practical realization of such devices. For example, for pulse applications the permittivity and permeability should be frequency independent and have certain prescribed values as functions of position. Such characteristi cs can only be approximately realize d. As another example, it wi l l turn out that the lenses should, in some cases, have infinite extent and so will have to be cut off. If, however, the lens is large e n ough the relative magnitude of the fields (as compared to the magnitude of the fields near the tra nsmission line passing through the center of the lens) can be small enough that the perturbation is insi gnificant . The permittivity and permeability will be required to be infinite in some places and less than their free-space value s in others, but such positions can be made to be far from any si g nificant fie lds so that these requirements can be neglected. For certain trans- mission lines the condu ctors restrict the fields to a closed region of -4space so that no lens material at all is needed outside this region. For a particular application of these lens designs, one should consider such things as the range of permittivity and permeability required and the spatial extent of the lens required . In this report we treat the lenses from a n idealized viewpoint. In outline, this report first considers the definition of what we call formal electromagnetic fields, vector operators, and constitutive parameters used with orthogonal curvilinear coordinates. Restricting the forms of the permittivity and permeability the general, but very restrictive, case with field components in all three coordinate directions is briefly considered. This is followed by a consideration of the TEM wave case with electric field components i n two coordinate directions . Some general results are obtained f or this case a nd a few lens type s are considered . Finally, the simpler case of two-dimensional lenses i s considere d, together with a few examples. -5II. FORMAL VECTORS AND OPERATORS Let us first consider a cartesian coordinate system (x,y,z) with ~ ~ , : , and an orthogonal curvilinear coordinate x' y z unit vectors We restrict both coordinate systems to be right handed, i.e. + e + x e x + = y e (2.1) z and + + x e el + = 2 (2.2) e3 The line element is + + dr = ~ dx + e dy + ~ dz x y z where the scale factors h~1 = (~) au . 1 The h. 1 h = 0, i 00 2 + ( ~) au. 2 1 + h. + a re gi ven for 1 + ( ~) au . + h du e + h du e + h du e 1 1 1 2 2 2 3 3 3 = 2 i =l, 2 ,3 au . 2 = a~) [( 1 au. 2 + ( a~) ( 2.3 ) as au. 2]-1 +( a~) ( 2 . 4) are tal~en positive and we exclude singular points where for any i=l, 2,3 from our consideration. The line eleme nt is a lso often written using the metric tensor ( g . . ) as 1J + + dr • dr 3 = l i=l g .. (du.) 11 2 (2.5) l where for orthogonal curvilinear coordinates the met ric tensor h as the s imple form 0 { g .. ) =(g .. Q•• ) 1J 11 1J = 0 0 0 : 0 h~ ) (2 . 6 ) -6For later use we define some combinations of the H - h. 1. as (2 . 7) hlh2h3 ( .. r. . ) = (aij) - (6 1.J ::... H <sij) - (6 . . h) = 1.J i and hl 0 0 0 h2 0 0 0 h3 (2. 8) h 2h3 0 0 0 h3hl 0 0 0 hlh2 0 0 (2.9) h2h3 hl ( y ij) - (6ij h3hl !:!_) 0 h~1. ( 2.10) 0 h2 h lh2 0 0 h3 where 6 .. is the Kronecker delta funct i on . 1. J In the u. 1. coordinate system t h e standard vector ope ration s are the g radient 'il<l> 34> + 1 34> + 1 34> + 1 -- e + - e = --- e + h 3u 2 h au 3 hl 3ul 1 2 2 3 3 'i7 x x = H1 {hl [ 3~2 (h3X 3 ) (2.11) curl + + 3 h 2 [3u 3 11 3 [ 3u 1 + (h2X2 )] el 3 (h1X1) + 3 3u ( hl3)] e2 1 (h2X2 ) + 3 (hlXl)J e3 3u2 3 + 3 3u (2.12 ) -7and divergence (2.13) The X. l vectors and Y. l are referred to as the physical components of the X and Y which have the representations and (2.14) Other common operations such as the scalar and vector Laplacians are formed as combinations of the above operations. Now we define another set of vectors and operators which we call formal vectors and formal operators and symbolize by the addition of a prime to the standard symbols. Related to + X we define -+ (2.15) -(a . . ) ·X lJ Thus the components of X' and X' Related to + Y X! l are related as (2.16) h .X . i In tensor language the -+ X l l are the covariant components of + X we define (2.17) Thus the components of + Y' and + Y are related as -8H = h. Y!J. Y' In tensor language the i Y.J. (2.18) J. are the components of a relative contra- variant tensor of weight +l which can also be called ~ relative contravariant tensor density vectors and + Y + X' and + Y' (4). Note t hat we have defined the formal differently because of the different ways that ~ X appear in equations 2.12 and 2 .13. Now we define formal vector operators by (2 .19) 'JI <f> I (2.20) and 'JI + • y I ( 2.21) Note that the formal vector operators have precisely the same form in orthogonal curvilinear coordinates as the standard ve ctor operators have in cartesian coordinates. These formal operators are related to the standard ones by l/hl 'V <I> (etij)-1 • 'JI <f> I = ( 0 0 0 l/h,, 0 0 l/h3 c. 0 . 'J I <l> I (2.22) -9- where the potential function is related to <1> ' <1> = <1> by (2.23) <I>' and by (2.21+) and -+ v • y Again in tensor l a n g uage <1> = 1 V' H Y' (2.25 ) is an invariant scalar, the components of are the covariant components of V' <1>' • V<t> , the components of V' -~ x X' are the components of a relative contravariant tensor of weight +l, a nd -+ V • Y is a relative scalar of weight +l. Finally we define a formal matrix (vij) related to ( vij) by (v~ . ) lJ The vj_j weight +1. (2.26 ) are the components of a relative contravari ant tensor of This transformation will be used later for the cons ti tuti v e parameters in Maxwell's e~uations. For the special case that ( v . . ) is lJ diagonal we have (2 . 27) It is this latter case which will be of concern to us in this report. -10- III. FORMAL ELECTROMAGNEI'IC QUANTITIES Now consider Maxwell ' s equations + vxE=--~ (3.1) v x ii (3.2) at n= P (3.3) v • n= a ( 3 . 4) v • and together with the constitutive relations n = ( £ij ) . E+ ( 3. 5) . ii (3. 6) and + B = ( µij) and the equation of continuity IJ • Note that p j ( 3. 7 ) is the "free" charge dens ity and does not inc] u de charge displacement conventionally included in(£ . . ). lJ In writing the above equations we have assumed that (£ij) and (µij) are real constant matrices, independent of frequency; they may, however, be functions of position . If we had written the above equations in the frequency domain, then(£ .. ) and(µ . . ) could eas ily have been taken as complex functions lJ of f r e quency. lJ -11- Equations 3.1 through 3.7 are assumed to be expressed in terms of the u. coordinates and the l + e. l unit vectors as in Section II . So now we make some appro?riate definitions of formal electromagnetic quantities. + Since E and + H appear with t he curl operator, we define, as + in the case of X ->- + Since + + B, D, and in the case of J p E H' - + H (aij) (3.8) appear with the divergence operator, we defi ne, as Y, B' - ( Bij ) • B , Now + + E':: (a . . ) lJ D' - ( Bij ) • D ' (3,9) equals a dive r gence in e quation 3 . 3 so we define p' so that p Hp ( 3.10 ) is a relative scalar of weight +l . Substituting for + + E, H, B, and D from e quations 3.8 and 3.9 into e quations 3.5 and 3.6 and requiring B ' =(µ~.)·tt' lJ (3.11) shows that for the formal constit utive parameter matrices we s hould define ( £i_j) - ( B. . ) ( £ .. ) ( aij )-1 ( µi j ) - ( Bij) . ( µij) (aij)-1 lJ lJ ( 3. 1 2 ) -12- For some problems one might include a conductivity matrix that j includes a conduction current density • -+ E (oij) so Then we would define (3.13) If (o. . ) ( e: . • ) , ( µ .. ) , and lJ lJ are required to be diagonal, equations lJ 3.12 and 3.13 reduce to ( 3 .14) The formal electromagnetic quantities defined in equations 3.8 through 3 .12 can now be substituted into Maxwell's equations, the constitutive relations and the equation of continuity. The curl and di- vergence operators can be replaced by the forma l ope rators from equations 2.24 and 2.25. Equations 3.1 through 3.7 can the n be rewritten as I/' x ;)B' E' - - at (3.1 5) + \J I II' • I/' x tt' n' = p . B' = ClD' j1 + - at ( 3 .16 ) (3.17) I 0 (3.18) -+ = ( e:j_j ) . E' (3.19) -+ = ( )J lj ) . H' (3.20) D' B' -+ -13and 'i/ I -+ • (3.21) JI Note that equations 3.15 through 3.21 are of the same form as equations 3.1 through ' 3.7. All electromagnetic quantities and operators are replaced with primed symbols, except for unchanged. u. l. t whi ch has remained However, the formal curl and divergence operators, using the coordinates, have the same mathematical forms as have the standard operators, using the x,y ,z cartesian coordinates. formally think of the u. Suppose that we as a cartesian coordinate system and think l. of the primed quantities as the electromagnetic fields, constitutive parameters, etc. Then we can take a known solution of Maxwell's equa- tions related to cartesian coordinates, directly substitute primed for u. unprimed quantitie s and the l. for the cartesian coordinates, and thereby construct a solution of the above equations. Transforming the formal quantities back to the standard o nes by equations 3.8 through 3.13, we then have a solution of Maxwell's equations for which(£ .. ), l. J (µ:.), and/or (o .. ) may be a nisotropic and/or inhomogeneous. l.J l.J The ide a is then to pick(£! . ), (µ! .) , a nd (cr! .) of s ome p a rticularly convenient l.J l.J l.J form and also to choose any boundary surfaces to have convenient forms in the u. l. coordinate system so that we can obtain a solution in terms of the formal electromagnetic quantities . relationship between the u. l. Choosing some particular coordinates and x, y, and z, the par1:llll- eters (£ij), (µij), and (oij) as well as the geometry of the boundary surface s are det ermi ned and the solution i s applied to the particular cas e . -14- I V. RESTRICTION OF CONSTITUTIVE PARAMETERS TO SCALARS In this report we are only concerned with problems related to inhomogeneous isotropic media. utilize such media. The l ater examples of lenses will Thus we restrict t he constitutive parameter and conductivity matrices to be of the forms (£. . )=do . . ) , lJ where (µ .j ) l lJ a £, µ, and = µ(6 lJ .. ) a( 6 .. ) (4.1) lJ are scalar functions of the coordinates . From equations 3 .14 the formal constitutive parameters then have the forms ( 4. 2 ) a( Y •• ) lJ Also, we restrict a ~ 0 frequency independent . the coordinates . and assume that However £ and £ µ and µ are real a nd may, in general, depend on The formal constitutive parameters ( £ ! . ) a nd(µ!.) lJ lJ are now diagonal matrices with the three diagonal terms possibly funct ions of the coordinates. Thus we a r e led to consider some possible forms for diagonal ( £ ! . ) and ( µ ! . ) which are cons ist ent with equations ( 4 . 2 ) . lJ lJ We would like ( £ !.)and(µ!.) to have rather simple forms so that electromagnetic lJ lJ waves, as expressed using the formal electromagnetic quantities and coordinates, have desired forms . A first case to consider is defined by requiring with terms (£ij) and £1 and (µij) µ' to be expressible as £ ' ( 6 .. ) lJ independent of the coordinates. of the formal quantities, this corresponds to a homoge neous me dium problem for which many types of sol ut ions of and In u. l -15Maxwell ' s equations are available. This first case is considered in Section V and Appendix A. It is not necessary, however , for (Eij) and (µij) to each have their three diagonal components equal and independent of the the problem to correspond to one of a homogen eous medium . u. J. for I n par ticu- lar, suppose that for each matrix just the first two of the di agonal components are constrained to be equal and independent of the coordinates. An inhomogeneous TEM wave with formal field components with only subscripts 1 and 2 h as no interaction with £3 3 or µ3 3 , and so £3 3 and µ3 3 are unimportant in the case of such a wave . Such TEM solutions are used to define lenses to match waves onto c ylindrical and/or coni cal transmission lines . This second case is considered in Sections VI and VI I and Appendix B. As a fUrther simplification we consider the two- dimensional problem in which u 3 = z , one of the formal e l ectromagnetic fields has only a u3 component, and the other formal electromagnetic field has only a u2 component . With appropriate restrictions on the comp on- ents of ( Ei_j ) and ( µij) this defines a third case considered in Sections VIII and IX. Solutions for this case are used to define lenses for launching TEM waves o n two p arallel pe r fect ly conduc ting plates . -16V. GENERAL CASE WITH FIELD COMPONENTS IN ALL THREE COORDINATE DIRECTIONS Now consider the case in which have all three formal components. -+ E' and -+ H' are both allowed to For this case we constrain the constitutive paramete~s to have the forms ( 5 .1) 0 and µ' 0 are both independent of the where E' nates. In terms of ~he formal electromagnetic quantities we have a > > homogeneous medium problem. u.}_ coordi- One might then apply many known solutions for homogeneous media to this case. With (E! . ) and(µ!.) each constrained by both e qua tions 5 .1 and 1-J 1-J 4.2, we have h2h3 hl ( y ij) 0 - 0 0 h3hl E' =E 0 h2 .!!...'._ ( 6 ij ) = µ ( cS ij ) (5 . 2 ) hlh2 0 0 h3 where E and µ are both assumed nonzero at positions of interest. This implies = From we obto.in E I E = .!!...'._ µ ( 5. 3) -17( 5. 5) Since the h. l are all taken positive, then we have them all equal which we express as ( 5. 6) Then from equation 5.3 E and are given by µ µh = µ' Eh so that Eh and µh are both independent of the coordinates. However, we cannot just choose coordinates. for h ( 5. 7) h to be any function of the In Appendix A we show that there are two general forms which satisfy the restriction imposed by equat i on first is given by h equals a constant for which the tesi an coordinate system. For this case E and µ u. l 5.6. The form a car- are constant so that the medium is homogeneous. The second form of h , from equations A. 27 and A. 31, g ives an inhomogeneous medium described by a x where a i 0 + y is a real constant. of coordinate system . r we have 2 2 2 + z 2 (5. 8 ) This corresponds t o a 6 -sphe r e type Defining the radius 2 x 2 + y 2 + z 2 (5. 9 ) -18- £ a 1 µ "2"" = µT = -h = r 2 (5.10) 2 If one were to attempt to construct such a medium for frequency independent £ and µ , then £ and µ would be constrained to be at least as l arge as their free space values. µ' , £' , a 2 r for which neighborhood of r =0 is excluded becaus e of th e singularity in and Thus there are restrictions on reali zing such a medium. µ there . With the h. l µ restricted as in equati o n can be realized . and there is a maximum £ and For fixed. Also, a E 5.6, the as s ociate d class of inhomogeneous media is then very restricted, being limited to spherically stratified media of the form given by equation 5.8. next section we l oose n somewhat this restriction on the h. l In the -19 - VI. THREE-DIMENSIONAL TEM WAVES Now we restrict our attention to waves of a ce rt ain form. Consider inhomogeneous TEM p lane waves such as propagate on ideal cylindrical transmission lines , including coaxial cables, s t r i p l i nes, etc . Such a structure supports TEM plane waves which propagate parallel to some fixed di r ection , say the z axis. It has two or more separate perfect conductors which fo rm a cross section ( i n a plane perpendicular to the z axis)which i s independent of z . Als o, let the medium in which the perfect conductors are placed be homogeneous. Next appl y th i s type of inhomogeneous TEM wave solution to the formal fields discussed in Section III. +u 3 Let the wav e propagate in the direction and let the formal cons titut ive parameters have the forms ~. ) 0 £ I 0 > 0 and µ' ( µij) - ( µ~' 0 .) (6.1) µ' 0 3 > : 0 are cons tants but 3 µ' and where £' fied . Since we shall only consider waves with no field components parallel to the direction, then £ I 3 £ I 3 µ' and 3 3 nowhere enter the formal constitutive relations, equati ons 3.19 and 3.20 . dependence of be ignored. £ ' 3 and µ' 3 Then the on the coordinates is irrelevant a nd can For this TEM wave th e medium can then be formally consi- dered isotrop ic and homogeneous since only cant. a r e unspeci- £ 1 a nd µ' are signifi- - 20- Specifically, cons ider formal f i e lds of the fo rm u3 u3 E' E' E' f( t - ;t) E' 1 = 1 (~,u2) ' 2 = 2 (ul,u2) f(t - ; t). 3 - 0 E' 0 (6.2) 0 and H' 1 u~ u3 H' (ul,u2) f(t - - ) • H' = H' (ul,u2) f(t - c ~). H' - 0 c' 1 2 2 3 0 ( 6 . 3) 0 where we define c' - 1 c - 1 0 and where we can choose the form of (6.4) /µ£ 0 u3 f (t - - ) c' to specify the waveform. This is the well-known form of TEM waves on cylindrical transmis sion lines (5). The formal field component s are r elated by E' = Z' H' 0 2 ( 6. 5) E' = - Z' H' 1 0 (6 . 6) 1 and 2 wh ere Z' is the formal wave impedance defined by 0 Z' 0 F; (6.7) I Equations 6 . 5 and 6. 6 express the orthogonality of E' · tt' Also -+ E' a nd -+ H' , i . e . (6.8) 0 E and H can be derive d from scalar potential functions as -·>- E' u3 = f (t - -c , ) V' <Ii e (u ,u,) ) , .1 L. - 21- (6.9) and where v12 = V' • V' ~h both satisf'y the Laplace equation (using the operator). These potential functions can be combined ~e to form a complex potential + i~h which allows one to use confo rmal transform techniques with the complex variable equations, u 1 + iu 2 . All these 6.2 through 6.9, are merely the direct application of known results for cylindrical transmission lines to their formal e quiva l ents using formal field compon.e nts and the u. coordinates in place of l physical field components a n d cartesian coordi nates. Note, of course, that while the results for cylindrical transmissi on lines ass ume constant £ and formal quantities ass ume con stant µ , the present results using the £ 1 and µ' . Likewise the present results require that th e two or more perfect conductors forming the transmis sion line intersect surfaces of con s tant that the r epresentation in terms o f u 3 . and in such a mann er is indepen dent of Put simply, t h ese perfectly conducting boundari es can be repre - sented in t e rms of only their and coordinates. The important feature of these TEM waves is that we only need restrict the first two diagonal components of the formal con stitutive parameter matri ces as in equations 6 . 1. We st i ll assume that ( £ . • ) and lJ (µ ij) correspond to isotropic but inhomogeneous media h aving the forms as in equ at ions 4.1 ( £ .. ) = lJ where £ > 0 do . . ) (6 . 10) lJ and µ > 0 may be functions of the coordinate s . 'l'hen a s -22- in equations 4.2 the forma l constitutive parameters have the forms (6.11) Combining equations 6.1 and 6 .11 t hen gives 0 0 1 0 £ = 1:..£ 0 0 0 £ 0 0 0 1 ) (µ' 1 0 0 µ' : 0 0 µ' =µ 0 ) ( 6.12) ( £ 0 0 1 3 3 This implies h2h3 h3hl = hl h2 £ I -= = £ .!:..'... (6 . 13) µ and µ' £ ' hlh2 = -1 -1 (6.14) µ £ h3 From equation 6. 1 3 we find that the first two scal e factors are equal which we express as h Note that h 3 h 1 = is not included in this equation. greater degree of freedom in choosing o ur Now £ (6.15) and µ £ I u. l Th i s will a llow u s a coordinate s ystems. are given by (6 . 1 6) -23so that Eh and 3 µh are both indepen dent of the coordinates. 3 Then the formal wave impedance from equation 6 . 7 is the same as the physical wave impedance because Z' 0 Since f}Z_ = -yE' z0 = and µ' (6 . 17) are arbitrary , then any orth ogonal curvi l inear 3 coor dinate system which satisfies equation 6 . 15 is acceptable. that results defines E and µ by equat i ons 6 . 16. Not just any ortho- gonal system, h owever, satisfies equation 6 . 15 . that surfaces of constant u 3 The In Appendix B we s h ow can only be planes or spheres (with res - pect to an x,y,z cartesian coordinate system). Two examples of such coordinate systems h ave already appeared in Section V (and Appendix A), namely cartesian coordinates and 6-sphere coordinates. examples all three h. l u. l In those surfaces are planes or spheres, since all three were made equal. In the next section several examples of orthogonal curvilinear coordinate systems satisfying equation 6.15 are considered. These are used to define types of inhomogeneous lenses which are then combined with conical and/or cylindrical transmission lines . Some of these lenses have rotational symmetry, while the associated system is not a rotational system. u. l coordinate For conve nience in such cases we then introduce an additional orthogonal curvilinear coordinate system v ,v ,v 1 2 3 which i s both right handed and rotational. cylindrical coordinates p,$ , z with We define the -24- - ( x 2 + y 2 )1/2 (6.18) tan(<j>) y/x (6.19) p and where <j> = O is taken from the xz plane for x positive . To make the vi coordinate system a rotational system we define (6.20) v. In order to distinguish the scale factors for the 1 coordinate where and , h~, a nd h may be vl o/ v3 replaced by other symbols for a parti cular rotational coo rdinate sys- system we write them as tem. h There are many well- known rotational coordinat e systems for which the h V. are tabulated ( 6) . 1 To construct the ti on from the u. coordi nate systems we consider a transforma - 1 v . system of the form 1 ul = A.(vl ) cos ( <I>) (6.21) u2 = A. (vl) sin ( <I>) ( 6. 22 ) u3 = ~ (v3) (6. 23 ) and with A.(v ) assumed non- negative. 1 There are several reasons for con- sidering this type of transformat ion. also surfaces of constant v 3 spheres . The functional form choosing h and will make 3 Surfaces of constant are which must then also be plane s or ~ (v ) 3 which in turn defines gives us some flexibility in £ and \J 'l'he ch o ice for The fUnctional f orm >. (v ) 1 is used to -25gain flexibility in trying to make surfaces of constant orthogonal. faces of constant v ,<J>,v 2 t,;(z) =z Then and sur- As an illustrative example, let be cylindrical coordinates 1 ul p,$,z is j ust and l et x,y ,z A.(p) =p ' This corresponds to the well-known case of a TEM wave propagating in the +z direction on a cylindrical transmission line in a homogeneous medium. Surfaces of constant hypothes is. while v , v 1 Then since neither is a function of u3 2 and v nor ul u2 are f uncti ons of only, surfaces of consta nt v3 orthogonal both to surfaces of constant st ant are mutually orthogonal by 3 ul u3 v3 ' are and to surfaces of con- This leaves the question of the mutual orthogonality of surfaces of constant and surfaces of constant gonality of constant and constant -+ or J avl -+ 21 - - + ar au <l<J> aul 1 ar -+ av Since the 1 v. l. surfaces we need -+ ar ar aul • au l For ortho- = 2 (6.24) 0 e; J avl Clr -+ ~ Clu + Cl$ au 1 2 2 Clv 0 (6.25) surfaces are orthogonal we have -+ Clr 3$ ( 6.26 ) 0 so that equation 6 . 25 becomes 3v 3v -+ ; 3vl 3vl Clul Clu2 -+ 3r . 3 -- - - -1- -1+ -3r . 3$ or 0 (6.27) - 26- (6.28) 0 Using the relations u2 arc tan( - ) ul = (ul2 + u22)1/2 (6 . 29) we find ~ sin(<j>) au - >1 ~ _ au 2 - cos(cj>) >- (6.30) and (6. 31) si n For hcj> we have p (6. 32 ) Substituting these results in equation 6.28 gives (6 . 33 ) or ( 6. 34) Now we have -27- (~)2]1/2 av 1 (6.35) /h ~ is independent of ¢ . In Appendix 3 (equation B. 12) vl "' we find, from the requirement that surfaces of constant v be spheres so that h 3 /h~ is vl "' Then e quation 6 . 34 can be integrated to obtain is independent of or planes , that only a function of A v . 1 as only a function of v . 1 v with surfaces of constant u3 h Hence 3 Thus, from an orthogonal system v ,¢,v 1 3 spheres or p l anes only, equations 6.21 3 u. through 6 . 23 defi n e an orthogonal st ant v 1 system in which surfaces of con- are sph e r es or planes only. Next consider the and relate t h e m to the h. 1 h v. For 1 have -+ ar au h2 = 3 -+ 3 -+ -+ dv3 2 ar .£.!:.._ • ar () . -= av du au av 3 3 3 ( 6 . 36) 3 or h3 = For hl h v3 ldv31 du 3 ( 6 . 37) we have (using equation 6.26) -+ -+ h 2 = .£.!:.._ • ar 1 aul aul -+ ar = avl -+ h -+ 1 2 dvl 2 2 2 2 + .Q_ sin ( ¢) cos (<I>) hvl (fil) A2 Substituting for -+ ar (-avl )2 + ar • .£.!: (l!.L)2 . -a cp acp au avl aul (6.38) from equation 6.33 s i mplifies this last result to vl p ); = ( 6. 39) -28For h 2 we have similarly -+ h2 = ar 2 au2 = -+ -+ -+ -+ av ~. ar (2 )2 + -ar • !!:. (--1..<E.)2 aqi a u au av av a 4> 2 1 2 1 -+ ar au2 = dv p2 2 h2 (2)2 sin 2 (qi) + 2 cos (ij>) v d).. ,>. 1 (6.40) which, using equation 6.33, simplifies to (6.41) Thus the form of the u. l given by equations 6.21 through 6.23, with ,>..(v ) satisfying equation 6.33, also satisfies the requirement of 1 We then have an acceptable equation 6.15 that Note that the u. l u. l system. system defined by equations 6.21 through 6.23 and equation 6 . 33 is based on a rotational system, v ,ip,v , 1 3 propagation in the ±.v spheres or planes . This is not the only way to define an acceptable 3 u. system . u. system differently . l l direction where surfaces of constant with are The last example in the next section will construct the -29VII. THREE-DIMENSIONAL TEM LENSES In this section we consider some examples of lenses for transporting TEM waves of the form considered in Section VI . These inhomogeneous TEM waves propagate on transmission lines with two or more independent perfe ctly conducting boundaries de sc r ibed in the fo rm (7 .1) so that the boundaries are in dependent of of this case i s g iven by u ,u ,u 1 2 3 u equal to 3 The simp l es t example x,y,z respective ly wh ich corresponds to a cylindrical transmission l ine with a homogeneous medium . We first consider t h e example of conical transmission lines as a simple illustration of t he method developed in the last section. This is followed by two inhomogeneous lenses based on bispherical and toroida l coordinate s ystems. We also show how these can be us ed t o transition TEM waves between conical and/or cylindrical transmission lines. The b i spherical lens can be thought of as a converging l e ns and t he toroidal l ens as a diverging len s . Finally we con s i der a l ens, based on cylindrical coordinates, which can be u sed to trans iti on TEM waves between two d iffere nt cylindri cal transmission l ines whi ch have their propagat ion axes poi nt ing in two different directions. A. Modi fied Spherical Coordinates As a fi rst example start with a rota ti onal orthogonal curvilinear coordinate system v ,$,v 1 3 gi ven by the spherical coordinates G,$,r illus trate d in Fi gure 1 and defined by x - r s in( G) cos ($) (7.2) - 30- r I I I I I I I y I I I _ _ _J x Fig . 1 . Sph e ri cal Coo rdi nat es -31(7.3) z - z is a constant we can choose later. where constant the (7.4) - r cos(G) 1 v 3 =r are spheres and we are considering ~ropagation in r direction. hg =r '!'he scale factors are h~ = r sin(G) = p Next we construct the ~(r) Note that surfaces of for equations u. 1. h = ~(r) =1 (7. 5) system for which we n eed 6.21 through 6.23. For convenience we choose r and From equation 6.37 we have 1~:31 = r + r0 A (G) (7.6) , where r 0 is a constant we can choose later, giving u Now we find a 3 = r + r A from integrating equation 6. 34 as g ~ I 2z where z 0 > 0 (7.7) 0 0 d\' AI dG' = (7 . 8) J sin(G') TT /2 is a constant for late r use. This gives (7.9) or (7.10) -32- so that = 2z 0 tan(~) sin( <I>) = 2z 0 From equation 6 . 41 the associated scale factor is ~A - ~r~s_i_n_(~G_)~ = 2~ h 2z 0 < h Note that G 0 < tan ( 2) 00 [ l + cos(G) ] ( 7 . 12) o on the +z axis (G = O, r > 0) so that the coordinate system is well behaved there, even though the s i ngular there. We call this u. v. l u. l system is system modified spherical coordinates . l The requi red const i tutive paramete r s are given by equation:; 6 . 16 as e: £ I L jJ I = Thus for the present choice of 1 u. l coordinates the medium is homogene- ous. For convenience we might choose also e: ,µ 0 0 (7.13) e:' ,µ ' so that the medium is free space. as e: , µ 0 0 e:,µ making The structure defined by the perfect conductors satisfying equation ( . 1 i s called a conical transmission l ine. giving the The transformations of equations 7 . 7 and 7 . 11, ui, are the well-known transformation for finding the TEM waves on such a conical structure(() . The present example for the u . is l then a comparatively simple one and the resulting medium is homogeneous. However, this example i l lustrates how to construct the systems. u. l In addition the coni cal transmission line is u sed l a t e r in conjunction with inhomogeneous lenses . -33B. Modified Bispherical Coordinates For constructing an example of an inhomogeneous lens, start with the rotational system ljJ,$,n with v 1 ,qi,v 3 given as bispherical coordinates as illustrated in F'igure 2 and defined by (6) x - a sin(~) cos($) cosh(n + cos(ijJ) (7 . 14) y - a sin(ljJ) sin($) cosh(n) + cos ( 1jJ) (7.15 ) z - a sinh(n) cosh(n) + cos ( 1jJ) (7 .16 ) 0~1jJ0:1T and < n < 00 -00 spheres; surfaces of constant in circles. h n v1 = 1jJ Surfaces of constant v 3 = n intersect plane s of constant are <P The scale factors are h <P = a sin(lj!) cosh(n) + cos ( 1jJ) = = hljJ = p a = cosh(n) + cos ( ljJ) sin(ijJ) Next construct the u. l system. p Firs t we c a lculate (7.17) (7.18) I- from equation 6.34 as ljJ ;\ f a d,\' - - = ,\ ' I dljJ' sin ( lj! 1 ) ( 7 . 19) a t an(~) ('(. 2 0) 1T I 2 0 which gives I- 0 - 34z ~ Rotate from 0 to 2n to g ive surfaces of con stan t T) .1J! • z =a y I I I I I I l"ig . ~' . B.isph e ri.c nl Coo r dinat es x - 35so that (7.21) where a 0 0 > is a constant we can choose. From equation 6 . 41 the associated scale factor is h hcjl a sin(w) 1 a = >:" = ~ tan ( ljJ / 2 ) _c_o _s -h-,-(-n.,....)_+_c_o_s__,(_tji-.-) = a 0 0 1 + cos(tji) cosh ( n) + cos ( ljJ) (·r.22) which has u. l 0 < h < 00 for -a < z < a on the z axis so that the system is well behaved the r e . From equation 6 . 37 we h ave (7 . 23) Now h i s rel ated to the constitutive parameters by 3 (7 . 24) For convenience let µ ~ µ 0 • = E: o' µ' E:' and also restrict E: ~ E: 0 This impli es the restriction 1 Next observe for 0 ~ ljJ < TI (7 . 25 ) and :for fixed n that h., is a monotoni - .J cally increasing function of interest and call it q1 0 with qi Then consider some maximtu11 0 < ljJ 0 occupied by the inhomogeneous medium to < TI q1 of Th en restrict th e space Then to minimize - 36- the magnitudes of £ and µ h3 = l required , set on 1jJ = ljlo This gives h3 = cosh ( n) + cos(ij1 0 ) (7. 26) cosh(n) + cos(v1) and we choose ~ = !£ cosh ( n) + cos ( ljJ ) ] a o du (7.27) 3 ~ du 3 The present choice is for the sake of convenience a nd definiteness . Note that there are many other forms that one could choose for From equation 7 . 27 we then calculate u 3 as (8) dn ' cosh(n') + cos(ljl ) 0 2a n ljlo sin(ljl ) arctan [tanh(2) tan( 2 )J (7 . 28 ) 0 This last result can be verified by first observing that n = 0 u 3 =0 for and by second differentiating the result and using the half an g le formulas for the trigonometric a nd hyperbolic functions . have a ll the u. l We now coordinates which for the present geometry we call modifi ed bispherical coo rdinat es . Now that the ui coordinates a nd hi scale factors are calcu- lated, cons ider the comb ination of this bispherical lens with a cylindrical transmission line. and u 3 On the plane z = 0 , on which n = 0 = 0 , we h ave from equations 7.14, 7 . 15 and 7 . 21 , and defining (7. 29) -37- Let the lens material modifying (corresponding to and µ n < 0, z < 0). be present only for £ z > 0 Then for free space with constitutive parameters £ 0 ,µ u 3 < 0 let the medium be Next let there be two 0 or more perfect conductors forming a transmission line described in the form (7.30) Since on the dividing plane we have ~ = x, u tors are continuous through this interface. 2 = y, then the conduc- For z ~ 0 these conduc- tors form a cylindrical transmission line on which a TEM mode has the form (from equations 6.9) E = f(t - ~) v~ (x,y) c e (7 . 31) The potential functions solve 2 V ~(x,y) =0 subject to appropriate boundary conditions from equations 7.30. Similarly for u 3 ~ ~ z 0 , making O, there is a corresponding TEM mode of the form (7.32) We purposely use ~ e and ~h for both z ~ 0 and u 3 ~ 0 because they solve the same Lapl ace equation and boundary conditions on bot h sides of other. z =0 with on one side exchanged for Note that from equations x,y on the + 6.5 and 6 .6 the components of E H are related by the wave impedance. Since we want both the same on both sides of the boundary then we must have ~e and and ~h - 38- z0 Z' (7,33) 0 which we h ave a lready required . and Now righ t z = 0 at we have h = 1 so that (7~34) Thus tangential E and H are continuous across TEM waves are exactly matched there. z =0 and the two Then a 'r EM wave as in equations 7.32 in the inhomogeneous lens wi ll propagate into free space in the form of equation s 7. 3:L with no reflection . An alternative approach to matching the TEM mode through the z = 0 interface is to define one positive and negative for z ~ 0 a0 - a let Then z . u ,u ,u 1 2 3 h For u. coordinate s ystem for both l z ~ 0 let (u , u ,u ) 1 2 3 = (x,y,z) while be defined by equations 7 . 21 and 7.28 with is continuous at continuity there, since for z > 0 z =0 we have has a step dis h = 1 . Note that describing the combina tion of the lens with free space by a single u. l coordinate system automatically poses the restriction of equation 7,33 in that the ratio µ/ £ must be the same at all positions of interest in order to satisfY equations 6.16. In terms of this composite u. l coordinate system the TEM wave is then described by equations 7.32 . This type of lens - t r ansmiss i on-line combination is illustrated in Figure 3 in which the cylindrical transmission l ine for strip line . z ~ 0 is taken as a The lens is stopped a little before the singularity at (x,y,z) = (0,0, - a) is reached. - 39- x cylindrical transmission-line conductors Fig. 3 . Bi sph erical Lens with Cylindrical Transmission Line -40n = n Next intro duce a s econd inte rface at face is a sphere describ ed by x 2 + y 2 0 < 0 Such a sur- (6 ) + (z - a coth ( n ) ) 2 a 2 = 0 2 (7 . 35) sinh (n ) 0 This sphere is centered on the z axis at radius a ls inh(n 0 )! -1 . illust rated in Figure n = n 0 = a coth( n0 ) and has a A cross section of the lens in the zx plane is 4 and a perspective view with the transmission line conductors is illust r a ted in Figure 5 , sphere z The region inside the is assumed to be free space and in this region we p l ace a conical transmis s i on line with conductors matching to those in the lens . Recall the conical t r ansmiss i on line discuss ed in Section VIIA. In order to center the apex of the conical li ne at the cent er of the n = n 0 sphere we choose z i n equation 7.4 as 1 ( 7 . 36) From equations 7 .11 we h ave fo r the conical line = 2z 0 ( 7 . 37) while from equations 7 . 21 and a 0 =a we have for the l ens (7 , 38) We would like and Thus we need o n to be cont inuous across the surface n = ( 7. 39) - 41- x r u = canst. 1 free space lens f ree space z z = a coth ( n ) 0 Fig. 4. Bispherical Len s -42- conical transmission-line conductors x lens y Fig. 5. cylindri cal transmi ssionline conductors Bispherical Lens with Cylindrical and Conical Transmission Li nes -43To do this consider P p For the conical line we have on = sin~(n ) (7.40) sin(Q) 0 while for the lens we have p a sin(lji) = cosh ( n ) + cos ( ljJ) 0 The G and ljJ coordinates are then related on this surface by sin(G) = - sin(lji) sinh(n0 ) cosh ( n ) +cos ( ljJ) A (7.42) 0 Then we have 1 1 + where, after some manipulation, we obtain 1 + cosh(n ) cos(lji) 0 (7.44) cosh(n ) + cos(lji) 0 Substituting from equation 7. 44 into equation 7. 43 and using the half angle f o rmulas for the trigonometric and hyperbolic functions gives 2 g tan (2) 2 n tan = tanh (.....£.) 2 2 (f) (7.45) f) (7.46) or g tan(2) no tanh(2) tan( whe r e the minus s i gn is used b e caus e n 0 < 0 . 'l'heref'o r c we de fin e -44- a 2z (7.47) 0 so that equations 7.46 and 7.39 are made equivalent. and continuous across the spherical surface u2 In order to make ti on 7.7 that on continuous at u3 This makes n = no n = no we note from equa- we have n = no - - ~~a_,..~~ + r sinh ( n ) o (7.48) 0 while from equation 7.28 we have u 3 2a = no arctan[ tanh(2) (7.49) sin ( tjJ ) 0 Equating these results gives r 0 a 1 sinh( n ) (7.50) 0 as our definition of With r 0 and constant continuous across 0 , a surface of is automatically continuous there. h a step discontinuity at this surface. at n = n as be for e at However, h 3 has Then we have the same condi tions n = 0, namely the TEM wave passes through this surface without reflection and is described by equati ons 7.32. In summary, inside the n = n 0 sphere the u. l are given by equat i ons 7.7 and 7.11 and the constitutive paramet ers are just the lens bounde d by n = n 0 , n = O and tjJ = tjJ 0 E 0 the and In u. a re given l by equations 7.28 and 7 . 38, a nd the constitutive parameters are given -45£ 1 =£ by equations 7.24 and 7.26 with are x,y ,z o' 11 1 =11 For o z ~ 0 and the constitutive parameters are The transmission line conductors in all three regions have exactly the same description as functions of u 1 and only as in equation From equations 7.24 and 7.26 the constitutive parameters for the lens are given by = cosh(n) + cos(~) cosh(n) + cos(~ ) ('(. 51 ) 0 For convenience one mi ght prefer to have this relation expressed in terms of p and z To do this we form comple x variables from equa- tions 7.16 and 7.17 as p sin(~) + i sinh(n) cos(~) + cosh(n) + iz a = + in 2 = arctan ( P +a iz) = tan(~ + in) ( 7. 52) 2 so that ~ kn =2 where k 2 ( 1 2aP · )2 + 2 a r ctan [ 2 2 2 ] + ~ Q,n[p2+ z+a 2 ] a - p - z p + (z-a) (9 ) . is an integer or zero imaginary parts gives ~ and n (7. 53) Separately equating real and as function s of p and z . Then we have cosh(n) = ~{[p:+ (z+a) : Jl/ 2 + lp:+ (z+a):_i-1/ 2 } p + (z-a) and p + ( z- a ) J (7.54) -46- (7.55) cos ( tjJ) These can be substitut e d in equation 7.51 to find tions of p and that since occur at 0 tjJ ~ =0 for a given value of z tjJ £. tjlo < 1T Si nce see that the maximum Assuming n = 0 £ cos ( tjJ ) < 1 0 and and £ µ as fun c - 7.51 for any fixe d the n varying T1 occur at the minimum of µ µ Note f rom equation tjlo the maximum and £ for tjJ T1 =0 we co sh ( n) is in the regi on of i nterest , the minimum occurs there an d we have 2 _ l _+_c_ o_ s_,.(-tj!~) = 1 0 The minimum occur on tjJ and µ are, by previous choice, = tjJ 0 , the maximum Referri ng to F i g ures £ 0 an d which for the region of interest . tjJ 3 through 5 one can better appreciate the approximation involve d in placing a boun dary on the lens at ijJ = t/! 0 In these fi gures we have used a strip line to illustrat e a typical cylindrical transmission line. For s u ch a transmission line t h e fields for the TEM mode extend over the entire cross - section surface , a plan e of constant z, or more generally a surface of constant u 3 . However, these fi e lds fall off in amplitude with distance fro m the conductors, for l arge distances. Thus we require tha t that the f ields in the TEM mode for to the fields near the conductors . tjJ ~ tjJ tjl 0 0 be chosen l a rge enough are insignifi cant compared For certain types of cylindrical transmission lines , such as coaxi al lines , the fields are ze ro outside -47some c l osed outer perfectly conducting boundary. For s uch cases the lens material is not needed outside the outer conducting bou ndary and stopping the l ens at some external creates no d i sturbance in the fields. This lens, based on a bispherical coordinate system, can be classified as a converging l ens . Referring to Figure 5, a spherical TEM wave launched near the apex of the conical transmission line i s converted into a plane TEM wave on the cylindrical transmission l ine . C. Modified Toroidal Coordinates For an example of an inhomogeneou s diverging l ens define the rotational system v ,$,v 1 3 illustrated in Figure x - as toroidal coordinates v,~, s as 6 and defined by (6) a sinh(v) cos($) (7. 57) cosh( v) + cos(d y z - a sinh(v) sin(tl (7.58) cosh( v) + cos( d a sin(d - (7 . 59) cosh(v) + cos(~) wi th - lT < s ~ lT and 0 £° V<co Surfaces of constant are toroids . are spheres ; surfaces of constant v 3 =s The scale factors are h$ = a sinh(v) ~~~~~-------~~ cosh(v) + cos (s) = p (7.60) -48- Rotate z ~ f r om 0 to 2n to give s urfaces of constant [, ' v . I;; = / / -- -- - ~ / / / Fig . 6. 'I'oroidal Coordin ates - 49p a ~~..,.-.,.._~~-.,..---:- cosh(v) + cos(~) To construct the u. l. = sinh(v) ,, system f'irst calculate f'rom equation 6.34 as v A. dA I A.' f a = dv ' sinh(v ' ) f (7 . 62) 00 0 which gives A. R.n( - ) = a R.n [ t anh ( ~) ] 0 or A. = a v tanh(2) 0 (7. 64) so that (7.65) where a 0 > 0 is a constant which i s chosen later. We obtain h f'rom equation 6.41 as sinh(v) l v cosh(v) +cos(~) tanh(2) a cosh(v) + l ; - cosh(v) + cos(~) (7. 66) 0 From equation 6 . 37 we have As b ef'ore we set € 1 = £ , µ' = µ whi ch together require 0 h 3 ~ l . 0 and re st ri ct Observe that f'or € ~ € 0 v ~ 0 and f'or -50fixed of i; with -TI < i; < TI , h is a monotonically decreasing function 3 is a maximum for Thus for fixed \) Then to minimize the required E and µ set h 3 = 0 , the z axis. \! =1 on \! =0 This gives 1 + cos(i;) (7.68) = cosh(\J) + cos(i;) and we choose .9£._ = 1 [l + cos(i;)] du a 3 could have many other forms. Again di;' 1 + COS (/;I ) We now have all the u. 1 Then is calculated as = (7.70) coordinates and call them modified toroidal coordinates. u. Having the 1 and h. 1 for this toroidal lens we now join cylindrical and conical transmission lines to the lens. z = 0 surface for the lens is taken as the plane =0 a 0 _ a Combining equations giv es for z One boundary on which i; = 0 , 7.57, 7.58, and 7.65 and defining =0 , (7.71) Let th e lens material b e present only for u i; > 0, z be free space with consti- > 0). Let the medium for tutive parameters x,y,z. and let z < 0 3 > for 0 (corresponding to z ~ 0 be simply The transmission line conductors are constrained by equation -51 - 7. 1 for all considered. z ~ 0 Thus, for we have a cylindrical transmission line, while in the lens the conductors are curved to satisf'y equations 7.1 and 7.65 with u. singularity in the at l p a =a . 0 = a, z = 0 Note that there is a corresponding to v=-+=. Thus for t h e toroidal coordinates we confine our interest t o f'ying = 0 \I -£ \I 0 v = v We let < +oo • v satis- be a boundary for the lens 0 material. The u., h, and the transmission line conductors are continuous l through the plane = O. z We have a TEM wave, as before, of the form u3 E' = f(t - -c' ) 'V ' ~e ( ul 'u2 ) u3 H' = f(t - -c' ) 'V' ~h ( ul 'u2 ) Since is continuous through h continuous through for z < 0 z and that z (7.72) = 0 , tangential =0 as required . h has a step discontinuity at 3 -+ E and Note, however, that Introduce another lens s urface at -+ H h are 3 =h =1 z =0 with This surface is a sphere described by x 2 + y 2 + ( z + a cot ( s 0 )) This sphere is centered on the z axis at radius ajsin(s ) j- l . 0 2 = z a 2 (7.7 3 ) = -a cot(s 0 ) and h as a Figure 7 illustrates a l ens cross section in the zx plane and Figure 8 gives a p erspective view with the transmis sion line conductors. The r egion outside the sphere described above is assumed to be free space and contains a conical transmission line with - 52- x 1 u 1 = canst. - z =- a cot(s ) 0 a sin(s ) 0 z free space s=O Fig. 7. Toroidal Le ns l e ns free s p ace -53- conical transmis sion- line conductors x cylindrical transmission-line conducto r s Fig . 8. Toroidal Lens with Cylindrical and Con i ca l Tr ansmission Lines -54conductors matched to those in the lens. Now we match the u. coordinates at l s = s0 using the modified spherical coordinates of Section VIIA to describe the continuation of the coordinates past u. l s = so . Center the apex of the conical line at the center of the sphere corresponding to zl s = so by choosing in equation 7.4 as (7.74) - a cot ( s ) 0 For the conical line we have, from equations 7 . 11 (7.7 5) and for the lens u 1 = a tanh(~) cos(cj>) Thus on a tanh ( ~) sin ( cj>) (7 . 76) we need (7.77) Then considering p on P s s we have for the conical line 0 a = sin(s ) sin(G) (7 . 78) 0 and for the l ens p a sinh(v) = cosh (v) + cos(s ) (7.79) 0 The Q and v coordinates are then related on this surface by sinh(v) sin( s ) sin(Q) = 0 cosh(v) + cos(s ) 0 (7.80) -55n This has the same form as equation 7.42 if we replace 1jJ by - I:; 0 • by 0 v and Then from equation 7.46 we have the result (7.81) Therefore we define for this case I; a 2z (7.82) tan(;) 0 making equations 7.81 and 7. 77 equivalent. Then are con- and tinuous across the spherical surface From equation 7 . 7 we have, on a sin(z; 0 ) + r I; = I; 0 , (7.83) 0 while from equation 7. 70 we have (7.84) = These results g ive, as a definiti on of r 0 0 for this case, - cot(z; a 0 ) are continuous across and Now r also continuous there . However, h 3 so that h is has a step discontinuity there. Then the TEM wave described b y equations 7. 72 passes through this surface without reflection. to x,y,z In sununary, for z ~ 0 and the constitutive parameters are lens, bounded by i:; =0 , i:; = i:; 0 , and u ,u ,u 1 2 3 we have v = v £ 0 0 and , the j.J u. 1 0 equal • In the are given by - 56equations l E 0 ,µ equa- 7 . 24 and 7.68 with For are given by equations 7.7 and 7.11 with constitutive parameters tions u. 7.70 and 7.76; the constitutive parameters are u 3 ~ a the The transmission line conductors are described by equation • 0 7. 1 of interest. for a l l values of The constitutive parameters for the l ens are given by E - = .!:!._= L- cosh ( v ) + cos(r,) µ h 1 + cos(r,) E (7. 86 ) 3 0 0 To express thi s result in terms of p and z , as in Section VIIB , we form a compl ex variable z + iP + i sinh(v) -a - = sin(r,) cos ( r, ) + cosh (v) so that, just as in equations (7. 87) 7. 52 and 7. 53 r, + iv = arctan ( z + ip) 2 a 2kn + 21 arctan[ 22 az 2 (7.88) a - z - with k an integer or zero. cosh(v) = ~ 2 From thi s we obtain 2 1/2 [z/ (p+a)) { z + ( p- a) j 2 + 2 -1/2 [z 2 +( p+a)~J } (7 . 89) z +(p- a) and c os( r, ) ± ( 1 + tan ~)- (l_. ) ] - 1/2 ('"(. 90 ) -57These can be substituted in equat i on 7 . 86 to find of l;, E and \) ~ 0 the maximum ' 0 fo r l;, µ From equation 7.86 the maximum \) = o ccur at varying . z and p occur at .'.; ~ l;, l;, 0 = so < \) = E and in te r ms µ µ for f ixe d con side red for the lens. \) \) = for 1T and E \) ' 0 \) ' 0 Then we finC. that the maximum for which we h a ve cosh( v ) + c o s ( l;, ) 0 0 (7. 91) 1 + cos( l;, ) 0 The minimum e quivalently E and µ were made E 0 and µ 0 on the z axis, or v = 0 Referri n g to Figures 7 and 8 , not e that for this toroidal lens, as in the previ ous case , we require tha t the f iel ds in th e 'l'EM mode for v ~ v 0 be negl igib l e compared to the fi e lds near the conduct ors so that the TEM mode is not s i gnifi cantly distur bed . based on toroida l Th e pre sent lens, coo rdin ates , can be c l assifi ed a s a diverging l e ns. Referring to F i gure 8, a plane wave o n the cylindri cal transmis sion line is conve rt ed on passing through the l ens ,into an expandin g spheri cal wave on the conical transmission line. D. Modi fied Cylindrical Coordinates As a last example we consi der a lens which can be u s ed t o tra.n s i - ti on TEM waves between two cylindri cal transmission l ines with different propagation directions. Specifi cally, c h oose the u. l based on cy l in- drical coordinat e s as z c·r. 92 ) -58- p u2 - u3 corresponds to v. J. y (7.93) tan <P - y/x po<f> 0 ~ <f> < 2n O ~ p < oo with - 2 (x + y2)1/2 - where = O, x > 0 p 0 > ( 7 . 94) 0 is a constant a nd u 3 h 1 a nd h Surfaces of con- The scale factors are are planes. h - h Since =0 Note that the intermedi ate rotational coordinate system is not used for this example . stant <P = l (7 . 95 ) p = h3 = h<I> ~ du Po 3 (7. 96 ) = h l 2 are equal we have an acceptable coordinate system 2 satisfYing equation 6 .15 . The res ulting l ens and transmission lines are illustrated in Figure 9. We call the u.). modified cylindrical coordinates. The constitutive parameters are given from equations 6 .16 with e:' = E0 ' µ ' = µ 0 as E -= E Consider p = Po the l e ns to make p = pl with with z2 > 0 (7. 97) 0 one surface of the lens and constrain E ~ E ' 0 µ ~ <P = 0, 0 < pl < Po' 'l'he maximwn <P t: a nd ~Jmax LI = µo max for Fix other lens surfaces as µo = p ~ Po = <Po µ Po pl with 0 < the n oceur ·~ 0 for < 2 11 ' p = pl z = ±z 2 g .i vi.ng (7 . 98) - 59- z x c y li ndri ca l transmi s s i on- line condu c tors Fig . 9. Cylindrical Le n s with Cylindrical Tran smissio n Li nes -60Including the cylindrical transmi ssion l i nes in the nate system, we define , for 0 ~ u ~ p cj> 3 0 0 u 3 £ 0, as u. coordi- l z , x,y For 7.92 through 7 . 94 . the u. u.l are defined (from a rotation in the xy plane) by l are defined by equations For - x cos(¢ ) + y sin (cj> ), 0 0 - -x sin(¢ ) +y cos(<j> ) +p <j> 0 With these definitions the =0 <I> that and h cl> = cl> 0 • u. l For h = h 3 = 1 is continuous across these latter two lens surfaces while three regions of > n (7.99) 0 outside the lens we have u 3 so h 3 The transmission line conductors are desc ribed by equation 7.1 for all 0 0 are all continuous across th e surfaces has step discontinuities there . cj> 0 u 3 of interest . is described by equations The TEM wave in all 6.9. Note that if then one or both cylindrical transmission lines may need to be cut short to prevent their intersecting each other. Referring to Figure 9 we require for this cyl indrical l e n s that the fields in the TEM mode for p ~ pl , for P ~ P 0 and for lzl ~ z2 (separately) be negl i gible compared to the fields near the tra n s mis sion line conductors . This lens is neither a converging nor a diverging lens but might be better termed a prism or a redirectin g l e ns. -61VIII. TWO - DIMENSIONAL TEM WAVES Now consider a restricted f o rm of the u. c oordinates by defining l u3 - z (8.1) h3 = 1 (8. 2) which implies while a re t a k en independen t of and z . Also l et e ither the u .., formal e l ectri c field or formal magneti c field have only a com- .) ponent and let the remaining formal fie ld h ave o n ly a Let the forma l field components b e o nly functions of wave propagate in the +u 1 direction . u u In terms of the component. 2 and let the 1 u. l and the formal fiel d components this represents a uniform TEM wave . Aga in we assume, f or the c onstit uti ve p a rameters , that id olJ ..) (8. 3) with t h e condu ctivity zero. general, inhomogeneous. Thus the medium is i sotropic but , in The formal constitutive paramet ers are assumed to have the forms ( £ ! . ) lJ ~( £' 0 0 £I 0 0 0 s' 1 0 2 ) ( µj_j ) (:i 3 0 0 µ' 2 0 0 µI ) (8 .4 ) 3 We also h ave ( sj_j ) dY .. ) lJ ( µj_j ) = µ ( y ij ) ( 8. 5 ) - 62where, because of equation 8.2, (8.6) Note in equations 8 . 4 that the diagonal components may be all unequal. o:: ( £ ! . ) and ( µ ~ . ) lJ lJ Howe ver, since the formal electric and magnetic fields are each assumed to have only one component, then only one of £ ' £! and l µ! and one of the the i µ! l will be significant. These s i gnificant will be assumed independent of the coordinates so that l in terms of the u. coordinates and formal fields the mecli um is effec- 1 tively homogeneous. We have two cases to consider. Call the case with the electric field parallel to the z axis Case l; call the case with the magnetic field parallel to the z axis Case 2. For Case 1 we assume a wave of the form + + = e3 E'3 E' with ul 0 J' where 30 - E' and 30 H' 2 £; H'2 c' 0 = f(t 0 ) - c' 1 (8. 8) j 2 3 µ I (8.7) I £ a re independent of the coordinates . The n for 0 Case 1 we ass ume that Then from equations = e2 H'2 H' µ2 E' ul + ->- ) - c' f(t µI 2 > 0 and £I 3 > 0 8.4 through 8. 6 we h a ve are independent of the u. l -63- (8.9) £ -+ Note for Case 1 that since E is parallel to the z ~<is, perfectly conducting planar sheets can be placed perpendicular to the z axis and used as boundaries for t h is TEM wav e . For Case 2 we ass ume a wave of the form ul f(t - - ) -+ -+ H' c' e 3 H' 30 ul f(t - - ) (8.10) c' with E' 2 E' 2 where = c' 1 = (8 . 11) 0 and H' 0 then assume that are independent of the 30 µ' 3 > 0 and u. . l For this case we E2 > 0 are independent of the We u. l also have £ = For Case 2 since ( 8 . 12) -+ E is perpendicular to surfaces of con stant u 2 , perfectly conducting sheets can be placed along these generally curved surfaces and used as boundaries for this TEM wave . There are many possible ways to choose u (x,y) 1 form an orthogonal curvilinear coordinate system . and 11 2 one can find e: and µ and u (x,y) and 2 Then calculating h 1 from equations 8.9 or 8. 1 2. For the examples in the next section we consider coordinate systems with ( 8 . 13) -64Define the compl ex variables p E X + iy (8.14) Then we h a ve for the line element ( 8 .1 5) Thus i f we are give n a conformal transformati o n o f the f o rm its inverse, we can calculate an h The n from = 1 (Cl . 16 ) we can also obtain q(p) or as h l~I = 1£;1- q(p) and Wi t h the restri ction of equa ti o n 8.13 , l ook again a t Ca se 1 . 8.9 b e come Equa ti o ns = € so that µ µ' µ ( 8 .17 ) 2 is h omogen eous fo r Case 1 . Similarly for Case 2 , e qua ti o ns µ ( 8 . 18 ) 8 . 1 2 b ecome € so that € ' 2 is h omogeneous f o r Cas e 2 . E: Fo r c onve ni e n ce we choo se E: ' 2 = E: 0 ' µ ' 3 = µ0 fo r Ca s e 2 . E: ~ = s0 , µ~ =µ0 fo r Case 1 , a n d The n f or each case o n e o f t h e cons t i - tut i ve paramet e rs is the same as f o r free spac e . Requi r ing € ~ € 0 ' -65µ ~ µ , 0 then for both cases we require h ~ 1 . In the next section we choose examples of two-dimensional lenses which might be appropriate for l a unching TEM waves between wide perfectly conducting parallel sheets. After defining the conformal transformation, giving u , regions with 2 h > 1 are exclude d from consideration. u 1 and - 66IX. TWO-DIMENSIONAL TEM LENSES As a first example of a coordinate system for a two- dimens i onal lens consider the conformal transformation defined by 9.. a q n- p l n- E. - l ~n[e a+ l] ~n[ e a - l] 1T a This is illustrated in Figure 1 0 . n (9.l) This transformation also describes the potential distribution around a uniformly charged 1-rire grid (in a homogeneous medium) termina ting a uniform electric field for From equations 9.1 we have , for 2nx x >> 0 . and 1TX a [ a = 1T tn e 2e a cos( ny) + l] a 2 (9.2) 1TX a n where k e arctan [ = 0, ±1 . a . ( ny) sin --; I e nx a cos ( Note that '!!}[_ a a ) - 1 ] + ak ( 9 . 3) is just a parame t er which can be used to scale the dimensions . The scale factor h g iven by equation 8.16 is (9.4) In terms of x and y this is ( 9. 5) Since only regions with h = 1 given by h ~ 1 are of interest we find the contour for l 1 T u/a = 0 \. 5 11 \ I 0 -;--- = ±1 12 .5 u u2 y I 1. 5 I CD 2 = 0 I \ I,-+1--1 a ____. x / a a ! 0\ --.J I -. 5 -1 -1 -1 , 0 2 -'- x a u + i u,_, -1 c:. a Fig . 10. 1 = - :r ?,n r :-(x+iy ) Le a. 11 .l 2 oordinates for ?'irs t Exa..rr:ple -68- e = which is indicated in Figure 10. -rrx/a (9 . 6) In terms of and the scale factor has the form - 2rru/a l (9.7) + e -= Concentrating our att e nti o n o n the region ne ar the positive x axis, we Iu 2 / a I s ee from this last equation that if we restrict this will assure having h f l . £ 1/2 , t hen For Case 2 (magnetic fi e ld paralle l t o the z axis ) one might place perfectly conduct ing boundaries on surfaces within t his restriction. of c o nstant the maximum h -2 or E J.J , as appropriate, is related to the max imum of in the reg ion of interest. ul of interest. on u2 = 0 maximum of h -2 Consider some Note that the maximum of for which From equ at ions 8.17 or 8.18 y = 0 occurs at h Then varying a s the mi nimum ul = u l -2 ul 0 for fixed ul occurs we find that the so that ul 0 /a] 2 ~2 1max e Also note that as h -+ 1 -TfUl x ->- oo o (9.8) we have , (9 .9 ) On e of th e constitut ive par:unet e r s of the l e n s is the some as free space; t h e oth er t e nds to the free - space value as ciently l a r ge x the l e n s ma t erial x - > "' 'l'h e n for suffi- can be s t opped without s ignificantly -69distorting the TEM wave. As a second example, consider the conformal transformation defined by .9:.. - ~ R.n[sinh(~ £.)] a TI 2 a £. = .?arcsinh[e Tiq/ 2 a] a TI This is illustrate d in Figure 11 . (9.10) This transformati on also describes the potenti al distribution around a uniformly charged wire grid (in a homoge neous medium) terminating uniform, equal but opposite, electric fields for x >> 0 x << 0 and Obtaining and from equa- tions 9.10, we have ( 9.11) 2a ( (TI X) TI v = 1T arctan coth 2 a tan( 2 ~) J + 2ak (9.12) k = 0,±1 where The scale factor h is g iven by = 11 + e-nq/al-1/2 = ltanh(~ £.)I 2 a In terms of h2 h x 1 and y this is TIY + e - 2 e-Tix/a cos(-) a 1 + 2e The contour for (9.13) - 2Tix/a -TIX /a ny - 2Tix/a cos(-) + e a h = 1 is '!... = ± 1 a 2 (9.14) (9.15) -70- .5 .5 l . c-) l u2 -= O y_ 0,---1ir.--1-~~~~-t~~~~-t~~~~~-+~a~~~~-t-~~~.. ~ ~ a a -. ) -1 x a 0 l . (TI -2 .'ln ( s1nh - x + iy) ) TI 2 a Fig. 11. Coordinat es for Secon d Example 2 -71which is indicated in F'igure 11. 1 1 + 2e = h4 In terms of and u we h ave 2 -nu /a Tiu 2 1 cos( ~-) (9.16) a Consj_dering the region near the positive x axis, not e that by restri ct ing ju /aj ~ 1/2 2 example consider as the minimum minimum u h h ~ 1 . this will assure having occurs on of interest and note that the = 0 (for which 2 For this second y = 0) and at u 1 u 1 , so 0 that =1 + e -nu1 /a 0 The maximum E or Also as x + 00 we have h + 1 ' u1 can then be found from equations µ + (9 . 17) x - 2 8.17 or 8.18. (9.18) an in(2) , Then, as in the first example, the lens material can be stopped at sufficiently large x without significantly distorting the TEM wave. Figure 12 illustrates the present types of two-dimensional lenses together with appropriate parallel-plate transmission lines for both cases of field polarization discussed in Section VIII. Note that the conductors and the inhomogeneous medium are stopped before reaching the singularity on the z axis ; sources to launch the TEM wave might be plac ed here. The perfectly conducting sheets and the inhomoge n eo us medium are also stopped on surfaces of constant u 2 . particularly near the edges of the sheets. 'rhi s distorts the TEM wuve, However, the sheets are -12- ~--/ -- transmission-l ine conductors A. Case 1: -+ E par allel to z axis len B. Fig. 1 2. Case 2: -+ H .....__ _ _ transmi s sion- line conductors parallel to z axis 'I'wo- Dimensi onal Lenses with •rr ansmission Lines -73assumed to be much wider than the sheet separation to minimiz e the influence of this distortion. -74X. CONCLUSION In sununary there appear to be many ways of specifying inhomogeneous media such that simple electromagnetic waves, such as the TEM waves used here, can propagat e in the medium. These types of inhomogeneous medi a can be used to define lenses for transitioning TEM waves, without reflection or distortion, between conical and/or cylindrical transmission lines. Of course, there are practical limitations in the realization of such lenses . For example, in some cas es the lens should ideally be infinite in extent; limiting the extent of the l e ns can introduce perturbations into the desired pure TIM wave, and care will have to be taken to insure that these perturbations are small. Another limitation lies in the chara cterist i cs of prac tical materials used to realize the desired permittivity and permeability of the inhomogeneous medium . The available range of these parameters will be limited and their frequency dependence imperfect. are not really necessary. Of course, perfect characteristics Elsewhere we have proposed a lens based o n geometrical optics for transitioning TEM waves b etween conical and cylindrical transmission line s (10). The lenses discussed in the present r eport, however, have the advantage that, within the limitatio n s menti oned above, the TEM wave passes through the l e ns undistorted, based on a solution of Maxwell 's equations . In this report we only consider isotropic inhomogeneous media for the l enses. Within this area we consider a few examples each of th ree- dimensional and two-dimensional lenses . •ro extend t h e present work one might consider several other such examples in orde r to hav e -75avai lable other types of geomet ries a nd inhomogeneiti es . extens i on one might inhomogeneou s . For a wider allow the medium to be anisot rop ic as well as This would r emove some of the r es trictions on the coor- dinate systems which could be used. However, such a~ a nis otropic inhomogeneous medium might be more d ifficult to realize. -76APPENDIX A: CHARACTERISTICS OF COORDINATE SYSTEMS FOR FIELD COMPONENTS IN ALL THREE COORDINATE DIRECTIONS In Section V, when considering the case of forma l field components in all three coordinate directions, we require that ( £i j) and (µ!.)reduce to constant scalars times the identity matrix . lJ Combining this with the earlier requirement that (£ . . )and(µ .. ) reduce to lJ lJ scalars time s the identity matrix le ads to the result of equation 5.6, namely (A.l) This very restrictive form of the what forms of h(x,y,z) Now the hypothesis. u. l or h. l leads to the natural question of h(u ,u ,u ) 1 2 3 are possible. form an orthogonal curvilinear coordinate system by It is then necess a ry and sufficient that the h. l satisfy the Lame equations, namely 2 a h. l au. a~ J L Clh. Clh. _J_ _1_ h . a~ au. J J Clhk Clh i 1 -au. --- = 0 hk (J\.2) J a~t and 1 Clh . Clh. 1 Clhi Clhj a (1 - _J_) a l) + - - - - - = + - - ( - __ au. h. au. au. hj Cluj h2 a~ a~ l l l J k where i,j,k tions (11). 0 (A.3) is a permutation of 1, 2,3 yielding six independent equaSubstituting from equation A.l gives - 77and (A. 5) 0 Introduce a change of variable defined by 1 v v = 0, 00 where any points with Equations (A.6) h are excluded from our consideration . A. 4 and A. 5 then become, respectively 0 and 2 1 av ---+ 1 v au.2 l v 2 1 av - --+ .L (~)2 + 1 (~v ) 2 = 2 au. 2 oQ v au.2 V J V K (~)2 2 au. l 0 (A . 8) J Rewrite equat i on A. 8 as 2 a v 2 au. l Since this holds for Cl2v 2 au. l 2 a v 2 i=l auR, 3 = l 1 v 3 (~)2 l auR, i=l (A. 9) i=l,2, 3 we deduce 2 a v 2 au. (A.10) J Also, from equation A.7 we have, for 0 i :f j (A.11) -78Now from e quation A.11 a 2 v/au~l But then av/aui is at most a function of i s at most a function of we have a function of l equal to a function of u. l are a real constant, say u. . ui . From equation A.10 u. J and thus both c , i.e. 1 = (A.12) Integrating we obtain (A.13) where d. is a real constant, since u. Integrating again gives l av/au. i s at most a function of = - 2- + d l. u l. + e l. (uJ. ' 1\.) (A.14) l l 2 c u. v 1 1 where, as indicated, e . l i,j,k distinct. is at most a function of Sununing over i u. J and for (from 1 to 3) on both s ides of equ a - tion A.9 gives 2 3 v 3 l ( A. 15) R.= l Substituting in this last equation from equations A.12 and A. 13 gives (A.16) We now consider two cases . - 79For the first case assume that c 1 = O. Then from equation A.16 we have 3 l 0 = (A . 17) t =l which implies for i =l,2,3, = d. l (A.18) 0 Thus equation A.14 becomes = e.(u . ,u.) l J K v v so that u. is independent of i s a constant, and thus h for l 1 c 2 # 0 This implies that v as well, i.e. h = -.;; = c where i=l,2,3. (A.19 ) is a real constant . 2 (A.20) 2 Then for this first case the u. l are just a cartesian coordinate system. This is the trivi al case of a homogeneous medium in which are independent of the coor- and £ µ dinates. For t he second case assume that c 1 # 0 . Then from equation A.16 we h ave 3 l v = (A. 21) t=l Note that c 1 > 0 since bt , the general f o rm for v > h 0 . is Defining new real constants a and -80- 1 h=-= v where a 'f. 0 since coordinates we make 1-l u 3l (....&.+b )2 [ R,=l a R. (A . 22) cl 'f. 0 Now by a simple l inear shi:rt of the =0 giving the simple symmetrical result b h = R, a u R, 2 (A.23) 2 2 2 ul + u2 + u3 From equation 2.5 we can write the line element as (A . 24) Rewrite this as So momentarily regarding the u.1 as cartesian coordinates, as the orthogonal curvilinear coordinates, and l/h x,y,z as the scale fac- tor, we can repeat the foregoing derivation from the Lame equations and, by interchanging the quantities in equation A.22, obtain the result 1 - = (A . 26) h where a' and b' R, are real constants and shi:rt in the x,y , z coordinates so that b' R, a' 'f. 0 . =0 Make a linear giving (A.27) - 81This type of h in equations A.23 and A. 27 corresponds to 6- sphere coordinates or the inversion of cartesian coordinates, given in one of its forms by (6) x y z - =- - 2 a u 1 2 2 2 ul + u2 + u3 (A.28) 2 a u 2 2 2 2 ul + u2 + u3 (A . 29) 2 a u 3 2 2 2 ul + u2 + u3 (A.30) We have included minus signs in these equations to make the The scaling constant dinate system right handed . a 2 u. l coor- is required for these equations to be cons i stent with equations A.23 and 2.4. We also have a 2 = a' 2 (A . 31) which comes from equating the right sides of equations A.23 and A.27 and using equations A . 28 through A.30 to relate Thus for this special types of solutions . The of coordinate systems. h u. l x ,y ,z and ul,u2,u3 . given by equation A. l there are two form either cartesian or 6-sphere type -82APPENDIX B: CHARACTERISTICS OF COORDINATE SYSTEMS FOR FIELD COMPONENTS IN TWO COORDINATE DIRECTIONS In Section VI we consider the case of formal field components in only the and coordinate directions. Imposing appropriate requirements on the constitutive parameters and on the formal consti tutive parameters (equations 6 .10 and 6. 1, respectively) l eads to the result of equation 6 .15, namely (B. l) In this appendix we consider a restriction which this imposes on the orthogonal curvilinear coordinate systems . Eisenhart (12) defines the second fundamental form of a surface (which we take as defined by any particul ar u ) as the quadratic d i f 3 ferent ial form (B . 2) wh i c h for an o rthogonal curvilinear system reduces to (B . 3) with D" 3 The first fundamental form of a (B.4) surface is just the line elemen t given by (B. 5) -83However, substituting h _ h 1 = h2 into equations B.4 gives (B. 6) D" 3 The first and second fundamental coefficients are then in proportion, i.e. (B.7) We then apply a result of Eisenhart (12) that these coefficients are in proportion if and only if the surface (given by constant this case) is a plane or a sphere. in Thus for this restriction on the coordinate system, given by equation B.l, surfaces of constant only be planes or spheres (with respect to an x,y,z u 3 can cartesian coordi- nate system) . Also in Section VI another orthogonal curvilinear coordinate system with coordinates is introduced. This s y s tem is rota- tional with = tan(<j>) y x h <P = p - ( x 2 + y 2 )1/2 from equations 6 .18 thro u gh 6 . 20 and equation 6.32 . scale factors for the v. system are designated by i equation 6 . 23 relating u 3 and Note that the h vi v , surfaces of constant v 3 planes or spheres . Since surfaces of constant v 3 (B.8) Because of 3 are also are planes or spheres , we again invoke the result of Eisenhart that the first and second fundamental coeffi cie nt s for a s urfac e must be i n proportion, - 84which we express as D D" v3 h2 vl where h D v3 - -h v3 = (B . 9) h2 cp ah vl vl av 3 v3 D" v3 ~ l - - h v3 av (B. 10) 3 Combi ning equat ions B. 9 and B.10 gives a R.n (h ) vl av 3 a R,n ( h cp ) = av (B.11) 3 But t hi s impli es hvl a R,n ( h <I> ) av a R.n(h = ) a R.n (h cp ) = av 3 av3 3 or, in other words, h vl is independent of /h~ vl "' used in const ructing the u. l from the v. l 0 (B . 12) This result i s -85REFERENCES l. Borgnis, F. E. and Papas, C. H., "Electromagnetic Waveguides and Resonators", in Handbuch der Physik, Vol. XVI, Springer-Verlag, Berlin, 1958, pp. 358-364. 2. Tischer, F. J., "Conformal Mapping Applied to Three-Dimensional Wave Problems", Proc. IEEE 21, Feb. 1965, p. 168. 3. Luneburg, R. K., Mathematical Theory of Optics, U. Calif. Press, Berkeley, 1966, pp. 182-188. 4. Sokolnikoff, I. S., Tensor Analysis Theory and Applications to Geometry and Mechanics of Continua, Second Edition, John Wiley & Sons, New York, 1964, pp. 69-71. 5. Collin, R. E., Field Theory of Guided Waves, McGraw-Hill, New York, 1960, Chapter 4. 6. Moon, P. and Spencer, D. E., Field Theory Handbook, Springer-Verlag, Berlin, 1961, Sections I and IV. 7. Smythe, W.R., Static and Dynamic Electricity, Second Edition, McGraw-Hill, New York, 1950, p. 479. 8. Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals , Series, and Products, Fourth Edition, Academic Press, New York, 1965, 2.444-2. 9. Abramowitz, M. and Stegun, I. A., ed., Handbook of Mathematical Functions, U.S. Govt. Printing Office, Washington, D.C., 1964, 4.3.57 and 4.4.39. 10. Baum, Capt. C.E., A Lens Technique for Transitioning Waves between Conical and Cylindrical Transmission Lines, Electromagnetic Pulse Sensor and Simulation Notes, Note 32, DASA 1880-2 , Jan. 1967. 11. Eisenhart, L.P., A Treatise on the Different ial Geometry of Curves and Surfaces, Dover, New York, 1960, p. 449 . 12. Eisenhart, op. cit., pp. 114-117, 446-449.