November 15, 2014 / Vol. 39, No. 22 / OPTICS LETTERS 6529 Null-field radiationless sources Elisa Hurwitz* and Greg Gbur Department of Physics and Optical Science, University of North Carolina Charlotte, North Carolina 28223, USA *Corresponding author: [email protected] Received August 19, 2014; revised October 1, 2014; accepted October 15, 2014; posted October 16, 2014 (Doc. ID 221275); published November 14, 2014 It is shown that it is in principle possible to produce combined sources of polarization and magnetization that are not only radiationless but that have any (and sometimes several) of the four microscopic or macroscopic electromag- netic fields exactly zero. The conditions that such a “null-field radiationless source” must satisfy are derived, and examples are given for several cases. The implications for transformation optics and invisibility physics in general are discussed. © 2014 Optical Society of America OCIS codes: (290.5839) Scattering, invisibility; (160.3820) Magneto-optical materials. http://dx.doi.org/10.1364/OL.39.006529 Long before the introduction of cloaking devices in 2006 ∇ × H −ikD; (4) [1,2] and the flurry of investigations they sparked [3], re- searchers had invested much effort into the understand- where k is the wavenumber. We use the definitions of the ing of simpler types of invisible objects. The earliest of auxiliary fields D and H, these is the class of so-called nonradiating sources (see, for instance, [4,5], and the review of [6]), sources D E 4πP; (5) whose fields are confined within the domain of excitation and are identically zero outside this domain. The theory B H 4πM; (6) of such sources, though relatively simple in comparison with the sophisticated mathematics of transformation and substitute these expressions into Maxwell’s equa- optics [7], allows some effects that are evidently not tions to write the latter entirely in terms of the E possible using such coordinate transformation design and H fields, as well as the polarization P and magneti- techniques. zation M, A good example of this was illustrated in a 1910 paper by Ehrenfest [8], in what may be the earliest paper dedi- ∇ · E −4π∇ · P; (7) cated to radiationless sources. Ehrenfest provided a gen- eral prescription for constructing nonradiating sources using the (not obvious) assumption that the magnetic ∇ · H −4π∇ · M; (8) field could be set identically to zero. Such null-field sources were later considered by van Bladel [9], who sug- ∇ × E ikH 4πikM; (9) gested that they could be constructed with electric and magnetic currents, and most recently, Nikolova and Rickard [10] briefly discussed the possibility of making ∇ × H −ikE − 4πikP: (10) sources with vanishing E or H. In this Letter we build on previous observations and By use of the curl of Eqs. (10) and (8), we arrive at a introduce a full theory of null-field radiationless polariza- monochromatic electromagnetic wave equation with tion and magnetization sources that can have any of the both polarization and magnetization sources, four electromagnetic fields E, H, D or B identically zero within the source domain—and often multiple fields ∇ × ∇ × H − k2H 4πk2M − 4πik∇ × P: (11) zero simultaneously. The results are derived through the use of Maxwell’s equations and verified through The source term on the right of this equation suggests the use of a Green’s dyadic formalism. Illustrative exam- an intriguing possibility: if there are no additional free- ples are given, and the implications of the results for propagating fields in the system, there will be no source invisibility physics are discussed using the known rela- of magnetic waves, and therefore no magnetic fields at tionship between the electromagnetic radiation and all, if the following condition is satisfied, scattering problems. We begin with the monochromatic macroscopic form ikM ∇ × P 0: (12) of Maxwell’s equations in Gaussian units, and assume that there exist no free currents and charges, i.e., Equation (12) may be considered our condition for a null-H source. Because a propagating electromagnetic ∇ · D 0; (1) wave requires both E and H fields, this also implies that the source must produce no electromagnetic waves ∇ · B 0; (2) outside its domain. We may also take the curl of Eq. (9) and simplify using Eq. (7); we then arrive at a similar wave equation for the ∇ × E ikB; (3) electric field E, 0146-9592/14/226529-04$15.00/0 © 2014 Optical Society of America 6530 OPTICS LETTERS / Vol. 39, No. 22 / November 15, 2014 ∇ × ∇ × E − k2E 4πk2P 4πik∇ × M: (13) E and H fields, as well as D and B, are identically zero in the far-zone. Two observations result from this equation. First, we The fields within null-field sources can also be calcu- can see that we will have a null-E source if the following lated explicitly using Hertz vectors, though care must be condition is satisfied, taken in the interchange of derivatives and integrals in the calculation, as has been long known [12]. A simpler ikP − ∇ × M 0: (14) observation comes from Eqs. (17) and (18): since a null-H source, labeled by H0, automatically has D 0 and a Second, we can see that Eqs. (12) and (14) are distinct null-E source, labeled by E0, automatically has B 0, equations, implying that, in general, a null-E source will it follows from Eqs. (5) and (6) that not be a null-H source, and vice-versa. These results are similar to those reported by van EH0 −4πP; (23) Bladel and Nikolova and Rickard previously. However, we can go further and express Maxwell’s equations entirely in terms of D and B instead of E and H, which HE0 −4πM: (24) then leads to a different pair of wave equations, To design a null-field source, the polarization and mag- ∇ × ∇ × B − k2B −4π∇ × ikP − ∇ × M; (15) netization must be chosen to satisfy one of the four conditions given above. A null-H source, for instance, may be designed by choosing functions for P and M that ∇ × ∇ × D − k2D 4π∇ × ikM ∇ × P: (16) satisfy Eq. (12). A spherical example of such a source is given by the choice In analogy with the previous results, we see that we can create null-B and null-D sources, respectively, if P rzfˆ r; (25) the polarization and magnetization satisfy ∇ P − ∇ M0 2 2 × ik × ; (17) cos πr ∕2; jrj ≤ 1; f r (26) 0; otherwise; ∇ × ikM ∇ × P0: (18) 1 On comparison of these new conditions with Eqs. (12) M r− ∇ × P r: (27) and (14), it is apparent that a null-E source is automati- ik cally a null-B source and a null-H source is automatically It is to be noted that the function f r has been taken to a null-D source, but the converse is not true. be continuous; this is not a necessary requirement, but We can confirm the nonradiating nature of sources prevents a magnetization singularity on the outer surface that satisfy these expressions by directly calculating of the sphere. Such singularities can be incorporated, the fields from the sources using Hertz vectors ([11], however, and are discussed by van Bladel [13]. An illus- Section 2.2.2), defined as tration of this source is given in Fig. 1; the electric field is Z directly proportional to the polarization. 0 0 3 0 D πe P r Gjr − r jd r ; (19) The previous source, as noted, is also a null- source. V With a slight modification, it can be converted into a source with a zero D field but a nonzero H field. To Z do so, we introduce a new magnetization M0 r 0 0 3 0 M r∇ϕ ϕ πm M r Gjr − r jd r ; (20) r , with r a function that is continuous V and that possesses a continuous first derivative (again to avoid requiring singular sources). where V is the domain of the source and G R A simple example of this is given by choosing expikR∕R is the Green’s function of the scalar Helm- holtz equation. The electric field E and the magnetic field 1 cos2 πr2∕2; jrj ≤ 1; H then satisfy ϕ r (28) ik 0; otherwise; E r∇ × ∇ × πeik∇ × πm − 4πP; (21) and the modified magnetization functions are shown in Fig. 2. H r∇ × ∇ × πm − ik∇ × πe − 4πM: (22) This example suggests other possibilities for null-field sources. If the polarization density is taken to be of the The behavior of the fields in the far-zone (kr ≫ 1) can form P r∇ϕ r, then it will automatically satisfy ˆ ~ ˆ ikr H be readily determined by noting that πe rsP kse ∕r Eq. (12), the null- condition, without any magnetization ˆ ~ ˆ ikr and πm rsM kse ∕r, where r is the distance from at all. Similarly, a source with magnetization M r the origin, and sˆ is the unit vector pointing from the ori- ∇ϕ r will satisfy Eq. (14), the null-E condition, without gin, and by using the simple far-zone rule that ∇ → iksˆ. any polarization. Such sources produce only longitudinal With these substitutions, one can demonstrate that the fields, and cannot therefore produce any transverse November 15, 2014 / Vol. 39, No. 22 / OPTICS LETTERS 6531 1 1 1 1 (a) (b) (c) 0.8 0.6 0.5 0.5 0.5 0.4 0.2 y 0 y 0 y 0 0 −0.2 −0.4 −0.5 −0.5 −0.5 −0.6 −0.8 −1 −1 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x x Fig.