Electromagnetic surface and line sources under coordinate transformations

Transcription

1 PHYSICAL REVIEW A 8, Electromagnetic surface an line sources uner coorinate transformations Steven A. Cummer,* Nathan Kuntz, an Bogan-Ioan Popa Department of Electrical an Computer Engineering an Center for Metamaterials an Integrate Plasmonics, Duke University, Durham, North Carolina 2778, USA Receive 9 March 29; revise manuscript receive 5 une 29; publishe 4 September 29 Although the analysis of electromagnetic sources in the context of coorinate transformations an the implications for transformation optics have been iscusse in the literature, a correct formulation that inclues surface an line currents has not been reporte. Here we erive how surface an line currents behave uner coorinate transformations an valiate the analysis through numerical valiation of a specific example. This analysis enables transformation optics to be applie to problems that inclue singular source istributions, which is often the case for practical raiating systems an antennas, to make a given current istribution prouce the same fiels as a ifferent current istribution by surrouning it with a material with specific electromagnetic properties. DOI:.3/PhysRevA PACS numbers: Fx, 4.2.b I. INTRODUCTION The work of Greenleaf et al. an Penry et al. 2 introuce the way in which coorinate transformations of steay electric current an electromagnetic fiels, respectively, can be physically implemente with a complex meium, offering a new paraigm for the control of electromagnetic fiels aroun arbitrary objects. Less complete is the theory escribing how electromagnetic sources, namely, currents an charges, are altere by these coorinate transformation meia. Past work has examine the interaction of sources with transformation optics meia in special cases. For example, Zolla et al. 3 showe how a line source within an electromagnetic cloaking shell raiates fiels as if the source were in a ifferent location. Greenleaf et al. 4, Zhang et al. 5, an Weer 6 analyze the etaile behavior of sources in the interior of cloake regions. Luo et al. 7 examine the general problem of source behavior in transformation optics to show how currents an charges change uner coorinate transformations. Importantly, they conceptually emonstrate how a coorinate transformation meium coul be use to make one current istribution raiate like an entirely ifferent one. Conformal antennas are one potential application of such an approach, in which currents may be constraine to a given surface but one wishes to have these currents raiate as if they were in a ifferent location or ha a ifferent shape 7. Kuntz et al. 8 emonstrate this approach through numerical simulations, confirming that complex current istributions can be mae to raiate like simple ones when surroune by a properly esigne transformation optics meium. However, the analysis of 7 is incomplete in several areas an, importantly, oes not give correct expressions for the of how currents constraine to surfaces or lines transform. These singular current istributions are practically important as many physical antennas an raiating systems can be usefully escribe in terms of surface an line currents. In this work we a to the escription of the behavior of electromagnetic sources uner coorinate transformations by *cummer@ee.uke.eu eriving expressions for how surface an line current istributions transform, an thus how their raiation can be intentionally manipulate through transformation optics. The results are illustrate through two examples, namely, the mapping of a planar surface current to a nonplanar configuration an the mapping of a line current segment to a surface current on a sphere. In these examples the mechanics of transforming surface an line currents are emonstrate in two ways: treating the surface current as a vector implicitly confine to a surface, an treating the surface current as a volume current with elta functions. We show that these forms are equivalent, an which one is more useful epens on the specifics of the problem. In the former case, we present numerical simulations that emonstrate the applicability of our results to practical problems of source manipulation. II. VOLUME CURRENTS AND CURRENT CONSERVATION The manner in which electromagnetic fiel vectors transform has been analyze both mathematically an physically in recent work 2,4,7,9. In this section, we briefly summarize for completeness these results as they apply to electromagnetic sources, an we explicitly emonstrate the conservation of charge an current in transformation optics as this property plays a key role in the analysis that follows. Current ensity is a contravariant vector ensity 2, an as such it behaves in a specific way uner coorinate transformations to preserve the continuity of its normal component at an interface. Mathematically, with a coorinate transformation efine by r =Fr, the transforme current ensity can be written in terms of the original current ensity as 7 r = T v con F r or = et A A. In the former expression, we use the symbol T v con to generically enote the transformation operator for a contravariant hence the subscript con vector ensity hence the superscript v, an in the latter expression this operator is ex /29/83/ The American Physical Society

2 CUMMER, KUNDTZ, AND POPA presse in terms of the acobian matrix, enote here by A, of the transformation r =Fr. Note that there are ifferent ways to express this contravariant vector ensity transformation operator T v con, incluing in terms of unit vectors an length scaling factors 2,. The et A term that appears in the enominator of Eq., which also appears in the enominator of several important expressions that follow, means that transformations that are singular in critical locations i.e., et A= will require special care in analyzing how sources behave e.g., 3. One of these examples is treate in Sec. V below, although we emphasize that the approach use may not be completely general for locally singular transformations. In this work we use vector an matrix notation in which general vectors are unerstoo to be three element column vectors. Such expressions can also be written in component form. For example, uner a general coorinate transformation a contravariant vector m transforms as m i = A i i m i, i where A i i =u i /u i an the equivalent in matrix form is m =Am =T v con m. Similarly, a covariant vector n transforms as n i = A i i n i, i an because the matrix forme from A i i is the inverse of that i forme from A i an the sum is over the first inex, the equivalent in matrix form is n =A T n =T v co n. Explicit istinctions are mae between covariant an contravariant vectors an each is transforme separately so that the metric of the transformation is not require when forming inner proucts. Note that there are two physical meanings of the expressions in Eq.. In one, the components of are the contravariant components in terms of the new basis vectors efine by the transformation. In this case, an are the same vector escribe in ifferent coorinate systems. In the other, an the one relevant for transformation optics, the components of are the contravariant components of the vector in the original basis vectors, an thus is a new vector that shows how woul change in the presence of a meium erive through transformation optics. Physically, the coorinate change escribes the translation of the current ensity from the original location to the transforme location, an the transformation operator T v con scales an rotates the vector so that it transforms as a contravariant vector ensity. For clarity in the erivations to follow we write these vector transformations in general T terms an simplify to acobian matrix terms in the en. A result useful in the analysis that follows is that charge an current-ensity flux are conserve in transformation optics. This is state without emonstration by Luo et al. 7, an because it is neee in our analysis below, it is explicitly emonstrate in the Appenix. 2 3 w III. SURFACE AND LINE CURRENTS As many antennas involve currents that are restricte to surfaces or lines, how surface current an line current vectors transform is both theoretically an practically relevant. Although surface currents were aresse by Luo et al. 7, the expressions given there are not correct in general. Close form expressions for these operations in terms of transformation operators are erive below. Since surface an line currents are limiting cases of volume currents, the irection of transforme surface an line currents must be also efine by Eq.. However, transformations of volume currents also compress or expan them into smaller or larger volumes, respectively, an the volume current magnitue must change corresponingly to conserve the total current. This iea is illustrate in Fig., in which a channel of with w containing volume current is transforme to a narrower channel of with w containing volume current. Thus to conserve the total current through this channel,, an this amplitue scaling is naturally prouce by the transformation in Eq.. Surface an line currents have zero extent in one or two irections, however, an thus these currents cannot be further compresse or expane in these irections of zero extent by a coorinate transformation. Thus, to transform a surface or line current, the amplitue scaling prouce by Eq. corresponing to current compression or expansion in the irections normal to the surface or line current must be unone. For a line current the implications are straightforwar. Because a line current has zero extent in both irections transverse to the current flow, all of the magnitue scaling from Eq. must be unone. Thus for a one-imensional line current, the transforme line current is or b a m ^ n = A PHYSICAL REVIEW A 8, = T v con T con et A w a v v m = T m con A et A = A A, ^ v ^ v ^ n = T co n/ T co n FIG.. An illustration of how a finite current channel behaves uner coorinate transformation. where again the latter expression expresses the transformation explicitly in terms of the acobian matrix A. The transformation simply rags an eforms an infinitely thin current-carrying wire without changing the line current magnitue. Since total current is the same as line current magnitue, this expression is consistent with current conservation. For a two-imensional surface current s the situation is more complicate because the current magnitue scaling must be unone in only the irection normal to the surface. Let m be the vector of length w that points across the with b

3 ELECTROMAGNETIC SURFACE AND LINE SOURCES of a thin volume current istribution from point a to point b, as in the left panel of Fig., which before transformation is parallel to the unit normal to the thin sheet nˆ. This isplacement vector m is a contravariant vector 2 an thus after transformation becomes the vector between transforme points a an b or m = T v con m. 6 Although it still extens across the thin current sheet, m is not necessarily normal to the sheet if the transformation is not orthogonal, as illustrate by the right panel of Fig.. Normal vectors are covariant vectors an thus the unit normal to this transforme thin current sheet is given by nˆ = T co v nˆ T v co nˆ. 7 Note that nˆ is not simply the transforme unit vector nˆ but has ha its length rescale so that it is still a unit vector after the transformation. We nee to etermine the factor by which the with of this thin volume current has been expane by the transformation. Let it be enote by s=w/w. Since w=m an w=m nˆ, wefin s = T con v m T v co nˆ m T v = T con v nˆ T v co nˆ co nˆ T v, 8 co nˆ where in the latter expression we have use the pretransformation relation m /m =nˆ. The transforme volume current ensity is inherently scale by s ue to the geometric expansion of the channel in the irection normal to the transforme thin current sheet. Consequently, this is the factor that must be remove in a transformation of a surface current, an therefore we multiply Eq. by s an fin that s = T con v nˆ T v co nˆ T v T v con s. 9 co nˆ where nˆ is the unit vector normal to the untransforme surface current. This can be further simplifie by noting that T v con nˆ T v co nˆ = A T nˆ T Anˆ = nˆ TA Anˆ =, which yiels s = T v co nˆ T con v s or A s = A T nˆ et A s 2 Thus, Eqs., 5, an 2 escribe how volume current, line current, an surface current, respectively, behave uner coorinate transformations. Equation 2 is consistent with the electromagnetic bounary conition associate with surface currents, namely, s =nˆ H. The vectors nˆ an H are both covariant vectors an their cross prouct is a contravariant vector ensity 2. Thus after a coorinate transformation this bounary conition transforms to T v con s = T v co nˆ T v co H. 3 While true, the above nees to have the term T v co nˆ renormalize to unit length to be useful as a bounary conition, an thus T v co nˆ T con v s = s = PHYSICAL REVIEW A 8, T v co nˆ T co v nˆ T v co H = nˆ H, 4 which gives the same expression for the transforme surface current as Eq. 2 an preserves this bounary conition uner coorinate transformations. Equation 5 for a line current is also equivalent to Eq. 2 when the transformation-inuce current compression is unone in two irections as it must be for a transforme line current. Let nˆ an nˆ 2 be orthogonal unit vectors that are each orthogonal to a line current = û 3, where û 3 =nˆ nˆ 2. The line current scaling factor in Eq. 5 can thus be rewritten as A = et A A = û 3 3 et A A et A û = A T nˆ A T nˆ = 2 A T nˆ A T nˆ. 2 5 Therefore the line current scaling factor in Eq. 5 is thus ientical to the prouct of two surface current scaling factors in orthogonal irections, as expecte. We wish to emphasize that applying the source transformation expressions in Eqs., 5, an 2 to transformations in which eta= in certain locations will likely require special care. It has been shown that such locally singular transformations can result in unusual material properties or wave behavior in these regions 3,4, an placing singular or nonsingular sources in these regions seems likely to lea to further anomalous behavior. The precise form of this behavior appears to epen on the etails of the transformation 3,4, an thus we o not attempt to treat this issue here in a general way. IV. SURFACE CURRENT UNDER A NONORTHOGONAL TRANSFORMATION We emonstrate an valiate through simulation the above analysis on a surface current confine to y=,onthe surface of a perfect magnetic conuctor PMC half-space, which is transforme to the triangular bump illustrate in Fig. 2. The overall transformation is limite to a twoimensional box with sies x an y an is escribe by x = x, y = a xy + a x, z = z. 6 For this transformation the acobian matrix an its inverse transpose are

4 CUMMER, KUNDTZ, AND POPA PHYSICAL REVIEW A 8, y (-,) (,) transformation region z x (,a) (-,) (,) FIG. 2. An illustration of the coorinate transformation escribe by Eq. 6. = = A c b A T c/b /b, 7 with b= a x an c=a sgnxy / b for convenience. Note that eta=b. To illustrate the analysis, we consier two ifferent vector irections for the original y= surface current, s = ẑ an s2 = xˆ. These currents will raiate orthogonally polarize uniform plane waves when on a PMC surface. In both cases the unit normal to the original surface current is nˆ =ŷ. Applying Eq. 2, the resulting general transforme surface current vector is s = A b s A T ŷ = A c 2 + s, 8 since A T ŷ=b c 2 +. We now revert to the original basis an coorinates by ropping the primes an thus let the above expression represent a new surface current in the original coorinates. The resulting transforme current ensities s an s2 are confine in both cases for x to the y=a x surface the transforme y= plane an are given by s = c 2 + ẑ, s2 = xˆ + cŷ. c On the y=a x plane, c= a sgnx an we thus have on the y=a x surface for x s = a 2 + ẑ, ^ s = z.86 ^ ^ s = z s = z FIG. 3. Color online The numerically simulate electric fiel prouce by a nonuniform surface current on the lower bounary of the omain. In the presence of the transformation electromagnetic meium containe in the region boune by the ashe lines, the raiate fiels are ientical to those prouce by a uniform an flat surface current in free space. The compute power flow irection inicate by the gray lines remains purely in the y irection, even in the anisotropic transformation meium. s2 = xˆ a sgnxŷ. a The surface currents are thus ragge to a new location an irection by the transformation. For case, the total transforme current flowing in the z irection between x = an x= is 2 an equals the untransforme current. For case 2, the total transforme current per unit z with is an also equals the untransforme current per unit with. Thus, in both cases, the total current is conserve, an the A T ŷ scaling factor from Eq. 2 plays a critical role in correctly scaling the transforme current. Case is straightforwar to simulate numerically using the COMSOL Multiphysics solver an we o so now to emonstrate an valiate the analysis. The relative permittivity an permeability of the transformation electromagnetic meium insie the region efine by x an y is given by 5 r = r = AAT et A = b c c c 2 + b 2. 2 We choose the specific value a=.6, which results in the transforme current ensity s =.86 ẑ along the triangular bump from x. The source frequency is.5 GHz. Figure 3 shows a time snapshot of the resulting electric fiel E z istribution prouce by this nonuniform surface current istribution raiating on a PMC region. In the free space region outsie the transforme area, the electric fiel is exactly the uniform plane wave that woul be prouce by a uniform surface current along the y= plane raiating in free space. Small nonuniformities are present that we attribute to the iscrete finite element approach use in the simulation. This confirms that the combination of the transformation meium from Eq. 2 with the eforme an appropriately scale surface current from Eq. 2 yiels the expecte result. If the correct scaling factor of.86 from Eq. 2 is not applie to the transforme surface current, the resulting simulate fiels not shown are clearly not uniform in amplitue an are not equal to the fiels prouce by the untransforme uniform surface current in free space. We note that the resulting electric an magnetic fiel irections are also those expecte from theory e.g.,. Both E an H are covariant vectors an thus transform as E =A T E. For the untransforme problem, the raiate plane-wave fiels are Ey= ẑe exp jky an Hy= xˆh exp jky. As seen in Fig. 3, the electric fiel insie the transformation meium is spatially compresse in the y irection as expecte from the original coorinate transformation. The irection of E insie this region is given by A T E ẑ=e ẑ an is thus unaltere except for the spatial compression. Similarly, the irection of H insie the transformation meium is given by A T H xˆ =H xˆ an is thus

5 ELECTROMAGNETIC SURFACE AND LINE SOURCES also unaltere except for the spatial compression. This results in the EH power flow irection in the transformation meium being solely in the y irection, while the wave vector or phase normal clearly has an x component as well. The anisotropy of the transformation meium is responsible for this effect. Note that if one efines the original surface current in terms of elta functions, for instance, r = yxˆ, 22 then this is an expression for volume current ensity, an Eq., not Eq. 2 must be use to compute the transforme current. Applying Eq. to case 2 above with the coorinate transformation in Eq. 6 yiels, for the transforme current ensity, r = y a x xˆ + cŷ. 23 b b This shifts the surface current from y= to y=a x, as expecte. Accounting for the scaling properties of the elta function 6 an noting again that c= a sgnx along y =a x gives r = y a x xˆ a sgnxŷ, 24 which becomes r = a 2 + y a x xˆ a sgnxŷ 25 a 2 + after the elta function is scale to unit amplitue. This expression for the transforme surface current is ientical to that for case 2 in Eq. 2. The scaling properties of the elta function thus naturally yiel the same scaling factor to the surface current magnitue that is given explicitly in Eq. 2. V. LINE TO SURFACE CURRENT TRANSFORMATION Some source transformations, for example one that transforms a line to a surface current, are perhaps more easily hanle through the elta function approach an applying Eq.. Luo et al. 7 consiere a three-imensional transformation of a finite line current to the surface of a sphere via x = arcos sin, y = arsin sin, z = brcos, 26 ar = R 2 R 2 R r R = k r R, br = R 2 /2 r R + R 2 R 2 = k 2r R + 2, 27 in which the constants k an k 2 are efine implicitly. Note that Luo et al. 7 i not specify the mapping of the polar or azimuthal angles in their presentation of this transformation. For this transformation the inverse acobian matrix is given by r A =x y r z r x y z PHYSICAL REVIEW A 8, x y z = k cos sin a sin sin a cos cos k sin sin a cos sin a sin cos k 2 cos b sin. 28 To apply Eq. we will nee A/etA. For our particular problem, the source an transformation possess complete azimuthal symmetry an thus we can erive A/etA by making the convenient assumption that =, which results in ab sin 2 a 2 sin cos A et A = k b sin 2 k 2 a cos k 2 a sin cos k a sin 2 As in Fig. 4, we begin with an infinitely thin linear volume current ensity in the original omain of r = ẑ 2 I z, 3 where = x 2 +y 2 an the unit pulse or rect function x = for 2 x 2 an = otherwise 6. This creates a line current of total current I that extens to /2 in the z irection. Noting that the transformation efines =arsin an z=brcos, an following Eq., the original volume current can be written to r = A a sin et A F r = I 2a sin b cos a2 sin cos, 3 k a sin 2 an when we revert to the original coorinates an basis by

6 CUMMER, KUNDTZ, AND POPA ropping the primes, the original current ensity transforms to the new current ensity r = I a sin 2a sin r=r 2 r=r s s FIG. 4. An illustration of the transformation in Eq. 26, which maps a finite length line current to a spherical surface. b cos a2 sin cos k a sin This can be simplifie through several steps. First, we use the scaling properties of the elta function to fin PHYSICAL REVIEW A 8, a sin = k r R sin = r R k sin. 33 When all of the terms are combine, the first component of the transforme current has a raial epenence proportional to r R r R =, an thus this term vanishes except at = because of a remaining sin term in the enominator, but this is not significant. After fully expaning the remaining thir component an simplifying, we fin that the transforme current istribution is r = I 2 r R brcos. 34 The basis vectors for the above expression are efine by the transformation from Cartesian to spherical coorinates using A cs /eta cs accoring to Eq. an where A cs is the acobian matrix for the Cartesian to spherical transformation. One coul thus either multiply Eq. 34 by A cs /eta cs to convert back to a Cartesian basis or use the thir column of the matrix A cs /eta cs as the neee basis vector in Eq. 34. Both result in r = I 2 r R brcos cos cos xˆ sin cos ŷ + sin ẑ I = ˆ r sin 2r sin r R brcos, 35 an when we substitute in r=r to reflect the elta function cos istribution, the pulse istribution in theta becomes 2 an thus simply extens across the entire omain of = to. The transforme current istribution then can be written simply as I r = ˆ 2R sin r R. 36 Thus, the original line current of magnitue I is now a surface current sprea uniformly over the surface of the r=r sphere with the same total current through any transforme contour. VI. CONCLUSIONS Builing on the ieas first escribe by Luo et al. 7 an numerically simulate by Kuntz et al. 8, we have erive how surface an line currents behave uner coorinate transformations as applie in transformation electromagnetics. These cases require special hanling because they cannot be stretche by transformations in their irections of zero extent. This effect appears as aitional scaling factors that must be ae to the expression for the transformation behavior of volume current. The relevance an correctness of these scaling factors is illustrate through a specific example supporte by numerical simulations. We have also shown that surface an line current transformations can be obtaine by treating these singular istributions as volume current istributions containing explicit elta functions. This approach, emonstrate in two specific cases, can be easier to apply in situations where the transformation changes the character of the current istribution, such as one that maps a line current to a surface current. Collectively, the analysis presente here contributes to the theoretical picture of how a current istribution can be mae to prouce the same fiels as a ifferent current istribution by surrouning it with a material with electromagnetic properties etermine by transformation optics. APPENDIX: CHARGE AND CURRENT CONSERVATION IN TRANSFORMATION ELECTROMAGNETICS Here we emonstrate charge an current conservation uner transformation electromagnetics. Given a charge-ensity istribution r an a volume efine in terms of a function vr such that vr = insie the volume an vr = outsie, then the charge in this volume is given by

7 ELECTROMAGNETIC SURFACE AND LINE SOURCES Q = r vr V, A where the integral is over all space. After applying the coorinate transformation r =Fr through the transformation optics approach, the new charge-ensity istribution is given by Luo et al. 7, r = et A F r. A2 This form results from being a scalar ensity 2, an represents the physical isplacement of the charge from one location to another an the magnitue scaling that result from the transformation. The new charge in the transforme volume is thus given by Q = et A F r v F r V, A3 where again the integral is over all space an the original volume has been transforme to a new volume efine by v(f r )=. Now change coorinates to r =Fr an, in oing so, V becomes Vet A an thus Q = F r v F r V = r vr V = Q, A4 an thus total charge in a volume is conserve through the translation an scaling of the volume an the charge ensity. PHYSICAL REVIEW A 8, If charge is conserve an the Maxwell equations are still satisfie after the transformation, then total current must also be conserve. But this can also be shown irectly by a similar approach in which we efine I = sr r s, A5 where sr = efines the integration surface. After transformation, the total current through the new s(f r )= surface is I = s F r et A A F r s, A6 where Eq. has been use to give the new current ensity after the transformation. Again change coorinates to r =Fr, an because the infinitesimal surface element is a covariant vector capacity 2, s becomes et AA T s. Thus I = sr Ar A T s = sr r s = I, A7 where we have use A A T s = T A T A T s= s. Thus, total current through a surface is conserve through the translation an scaling of the surface an the vector current ensity. A. Greenleaf, M. Lassas, an G. Uhlmann, Math. Res. Lett., B. Penry, D. Schurig, an D. R. Smith, Science 32, F. Zolla, S. Guenneau, A. Nicolet, an. B. Penry, Opt. Lett. 32, A. Greenleaf, Y. Kurylev, M. Lassas, an G. Uhlmann, Commun. Math. Phys. 275, B. Zhang, H. Chen, B.-I. Wu, an. A. Kong, Phys. Rev. Lett., R. Weer,. Phys. A 4, Y. Luo,. Zhang, L. Ran, H. Chen, an. A. Kong, IEEE Antennas Wireless Propag. Lett. 7, N. Kuntz, D. A. Roberts,. Allen, S. A. Cummer, an D. R. Smith, Opt. Express 6, U. Leonhart an T. G. Philbin, New. Phys. 8, S. A. Cummer, M. Rahm, an D. Schurig, New. Phys., A. V. Kilishev, W. Cai, U. K. Chettiar, an V. M. Shalaev, New. Phys., G. Weinreich, Geometrical Vectors University of Chicago Press, Chicago, IL, A. Greenleaf, Y. Kurylev, M. Lassas, an G. Uhlmann, Opt. Express 5, Z. Ruan, M. Yan, C. W. Neff, an M. Qiu, Phys. Rev. Lett. 99, D. Schurig,. B. Penry, an D. R. Smith, Opt. Express 4, R. N. Bracewell, The Fourier Transform an Its Applications McGraw-Hill, New York,