Spherical perfect lens: Solutions of Maxwell s equations for spherical geometry

Transcription

1 PHYSICAL REVIEW B 69, Spherica perfect ens: Soutions of Maxwe s equations for spherica geometry S. Anantha Ramakrishna Department of Physics, Indian Institute of Technoogy, Kanpur 0806, India J. B. Pendry The Backett Laboratory, Imperia Coege London, London SW AZ, United Kingdom Received 3 November 003; pubished March 004 It has been recenty proved that a sab of negative refractive index materia acts as a perfect ens in that it makes accessibe the subwaveength image information contained in the evanescent modes of a source. Here we eaborate on perfect ens soutions to spherica shes of negative refractive materia where magnification of the near-fied images becomes possibe. The negative refractive materias then need to be spatiay dispersive with (r)/r and (r)/r. We concentrate on ensike soutions for the extreme near-fied imit. Then the conditions for the TM and TE poarized modes become independent of and, respectivey. DOI: 0.03/PhysRevB PACS numbers: 78.0.Ci, 4.30.Wb, h I. INTRODUCTION The possibiity of a perfect ens whose resoution is not imited by the cassica diffraction imit has been subject to intense debate by the scientific community during the past two years. This perfect ens coud be reaized by using a sab of materia with where is the dieectric constant and the is the magnetic permeabiity. Veseago had observed that such a materia woud have a negative refractive index of n the negative sign of the square root needs to be chosen by requirements of causaity, and a sab of such a materia woud act as a ens in that it woud refocus incident rays from a point source on one side into a point on the other side of the sab see Fig.. Due to the unavaiabiity of materias with simutaneousy negative and, negative refractive index remained an academic curiosity unti recenty when it became possibe to fabricate structured metamaterias that have negative and. 3 5 Most of the negative refractive materias NRM, so far, consist of intereaving arrays of thin metaic wires that provide negative Ref. 6 and metaic spit-ring resonators that provide negative Ref. 7. Athough some initia concerns were expressed 8 that the observed effects in these experiments were dominated by absorption, the recent experiments of Refs. 9 have confirmed that negative refractive materias are today s reaity. It was demonstrated by one of us that the NRM sab acts as a ens not ony for the propagating waves for which the ray anaysis of Veseago is vaid but aso for the evanescent near-fied radiation. This phenomenon of perfect ensing becomes possibe due to the surface pasmon states 3 that reside on the surfaces of the NRM sab which restore the ampitudes of the decaying evanescent waves.,4 8 Indeed, it has been confirmed by numerica FDTD simuations that an incident puse is temporariy trapped at the interfaces for a considerabe time. 9 For a detaied description of the perfect sab ens, we refer the reader to Refs.,7,0. The perfectness of the perfect ens is imited ony by the extent to which the constituent NRM are perfect with the specified materia parameters. Absorption in the NRM and deviations of the materia parameters from the resonant surface pasmon conditions of the perfect ens causes significant degradation of the subwaveength resoution possibe. 4, 3 We have suggested some possibe measures to ameiorate this degradation of the ens resoution by stratifying the ens medium 4 and introducing optica gain into the system. 5 It appears that obtaining negative refractive materias with sufficienty ow eves of dissipation wi be the greatest chaenge. We note that athough the phenomenon of subwaveength focusing using NRM is yet to be experimentay demonstrated, there is some experimenta evidence for the ampification of evanescent waves 6 and we expect that there are good chances for reaizing this using NRM at microwave frequencies. The image formed by the NRM sab ens is identica to the object and hence there is no magnification in the image. Lenses are mosty used to produce magnified or demagnified images and the ack of any magnification is a great restriction on the sab ens on which most of the attention in the iterature has been focused. The sab ens is invariant in the transverse directions and conserves the parae component of the wave vector. To cause magnification this transverse FIG.. Radiation from a point source on one side of a sab of materia with i and is refocused into a point on the other side. The rays representing propagating waves are bent on to the other side of the norma at the interfaces due to the negative refractive index of the sab /004/69/557/$ The American Physica Society

2 S. ANANTHA RAMAKRISHNA AND J. B. PENDRY PHYSICAL REVIEW B 69, invariance wi have to be broken and curved surfaces necessariy have to be invoved. The perfect ens effect is dependent on the near degeneracy of the surface pasmon resonances to ampify the near fied, and curved surfaces in genera have competey different surface pasmon spectrum. 7 It was recenty pointed out by us that a famiy of near-fied enses in the quasistatic approximation in two dimensions can be generated by a conforma mapping of the sab ens. 0 Thus, a cyindrica annuus with dieectric constant was shown to have a ens-ike property of projecting in and out images of charge distributions. Simiary in Refs. 8 and 9, it was shown how a genera method of coordinate transformations coud be used to map the perfect sab ens soution for the Maxwe s equations into a variety of situations incuding the cyindrica and spherica geometries, respectivey. In this paper, we eaborate on the perfect ens soutions in the spherica geometry and show that media with spatiay dispersive dieectric constant (r)/r and magnetic permeabiity (r)/r can be used to fabricate a spherica perfect ens that can magnify the near-fied images as we. In Sec. II of this paper, we wi present these perfect ens soutions of the Maxwe s equations for the spherica geometry. In Sec. III, we wi examine the soutions in the extreme near-fied imit or the quasistatic imit which is usefu when the ength scaes in the probem are a much smaer than a waveength. Then the requirements for TM and TE poarizations depend ony on /r or /r, respectivey. This is usefu at frequencies where we are abe to generate structures with ony one of or negative. We wi investigate the effects of dissipation in the NRM and point out the connections to the one-dimensiona D sab ens soutions. We wi present our concuding remarks in Sec. IV. FIG.. A spherica she with negative (r)/r and (r)/r images a source ocated inside the she into the externa region. The media outside have positive refractive index, but (r)/r and (r)/r. The ampification inside the spherica she of the otherwise decaying fied is schematicay shown. II. A PERFECT SPHERICAL LENS Consider a sphericay symmetric system shown in Fig. consisting of a spherica she of NRM with the dieectric constant (r) and (r) imbedded in a positive refractive materia with (r) and (r). First of a we wi find the genera soutions to the fied equations with spatiay inhomogeneous materia parameters: Ei 0 rh, Hi 0 re D0, H0, DrE, BrH. 3 Under these circumstances of spherica symmetry, it is sufficient to specify the quantities (r E) and (r H) which wi constitute a fu soution to the probem. Let us now ook at the TM poarized modes r H0, impying that ony the eectric fieds have a radia component E r. Operating on Eq. by, we have Ei 0 rh, r rrei c r E. Using Eqs. and 3 we have D re r Er E0, 5 and if we assume (r)(r) and (r)(r), we have E r r r E rr rr re r. 6 We note the foowing identities for ater use: E E E, r Er E E, and using Eq. 6 we aso note that r Er r r r E, r r r r E r, r r r re r r r E r. We now take a dot product of r with Eq. 4, and use the Eqs. 6 9 to get an equation for (re r ) as: re r r r r re r r rr re r rr c re r

3 SPHERICAL PERFECT LENS: SOLUTIONS OF... This equation is separabe and the spherica harmonics are a soution to the anguar part. Hence the soution is (re r ) U(r)Y m (,), where the radia part U(r) satisfies r r r U r r U r r r U r rr U rr c U0. If we choose (r)r p and (r)r q, we can have a soution U(r)r n and we get nnpnpr n impying pq and /c r pqn 0, n pp 44 /c. 3 Hence the genera soution can be written as E r r,m n A m r n n B m r n Y m,. 4 A simiar soution can be obtained for the TE modes with r E0. Now assuming an arbitrary source at ra 0, we can now write down the eectric fieds of the TM modes in the different regions for the negative spherica she of Fig. as E () r n A () m r n n B () m r n Y m,,,m a 0 ra, E () r n A () m r n n B () m r n Y m,,,m a ra, E (3) r n A (3) m r n n B (3) m r n Y m,,,m a r, and simiary for the magnetic fieds. Note that B () m correspond to the fied components of the source ocated at r a 0. For causa soutions A (3) m 0. Now the tangentia components of the magnetic fieds and the norma components of the dispacement fieds have to be continuous across the interfaces. Under the conditions p, q, (a ) (a ), and (a ) (a ), we have A () m 0, 8 A () m a () /c B () m, 9 B () m 0, 0 B (3) m PHYSICAL REVIEW B 69, a a () /c B () m. The ensike property of the system becomes cear by writing the fied outside the spherica she as E (3) r r a a r () /c B () m Y m,. Hence apart from a scaing factor of /r, the fieds on the sphere ra 3 (a /a )a 0 are identica to the fieds on the sphere ra 0. There is aso a spatia magnification in the image by a factor of a /a. Let us note a coupe of points about the above perfect ens soutions in the spherica geometry. First, for ra 3, i.e., points outside the image surface the fieds appear as if the source were ocated on the spherica image surface (r a 3 ). However, this is not true for points a ra 3 within the image surface. Second, given that (a ) (a ), we have the perfect ens soutions if and ony if n n which impies that p in Eq. 3. Athough the soutions given by Eq. 4 occur in any medium with /r, the perfect ens soutions ony occur for /r. Here we have written down the soutions for the TM modes. The soutions for the TE modes can be simiary obtained. III. THE SPHERICAL NEAR-FIELD LENS As it has been pointed out in the preceding section, the power soutions are good for any (r)r p and (r) r q such that pq. However, the perfect ens soutions for the Maxwe s equations resut ony for the singe case of pq. In the quasistatic imit of 0 and p, q, we can reax this condition. In particuar, by setting (r)/r and (r) constant, we can have a perfect ens for the TM modes aone. Simiary, we can have a perfect ens for the TE poarization by having /r and constant. This extreme near-fied imit is both important and vaid for situations when a ength scaes in the probem are much smaer than a waveength of the radiation. This becomes usefu at frequencies where we can ony generate media with either negative and positive, or, negative and positive. Exampes are the siver sab ens at optica frequencies, the metamaterias Swiss ros used for magnetic resonance imaging at radio frequencies. 30 Particuary at radio and microwave frequencies, we currenty can practicay engineer the required metamaterias with spatiay dispersive characteristics at the corresponding ength scaes. Further it aso ifts the restriction that the system has to have a spatiay dispersive materia parameters even outside the spherica 55-3

4 S. ANANTHA RAMAKRISHNA AND J. B. PENDRY PHYSICAL REVIEW B 69, she of NRM. In this section we wi work in this extreme near-fied imit. Then it is sufficient to sove the Lapace equation and we present ensike soutions to the Lapace equation beow. Consider the spherica she in Fig. to be fied with a materia with (r)c/r with the inner and outer regions fied with constant dieectrics of and 3, respectivey. Let everywhere. Now pace a charge q at the center of the concentric spheres and a charge q at a distance a 0 from the center inside region. We wi consider the z axis to be aong the dipoe axis and make use of the azimutha symmetry here, athough it is cear that our resuts do not depend on any such assumption of azimutha symmetry. Thus, a our charge and their images wi now ie aong the z axis. Now we wi cacuate the potentias in the three regions, which satisfies the Lapace equation and the continuity conditions at the interfaces. The potentia in region (ra ) can be cacuated to be using the azimutha symmetry V r q 4 A r P cos a 0 r P cos. 3 Note that the second term in the above expansion arises due to the dipoe within the sphere. It can be shown see the Appendix, that the genera form of the potentia in region (a ra ) where the dieectric constant varies as /r is V r q 4 0 A r () P cos B P r cos. 4 In region 3 (ra ), the potentia is given by V 3 r q 4 3 B 3 r P cos. 5 Now we must match the potentias at the interfaces at r a and ra put 0 ) to determine the A and B coefficients. The conditions of continuity of the potentia and the norma component of D at the interfaces are V a r V a V a, V a V 3 a, 6 V a r, V a r V 3 a 3. r 7 We determine the coefficients from these conditions to be a 0 a 3 a a 3 a a a A a a 3 a a a 3 a, a 3 a 0 a A a a 3 a a a 3 a, a 3 a a () 0 a B a a 3 a a a 3 a, 3 a a 0 a () a B 3 a a 3 a a a 3 a Under the perfect ens conditions a, a 0 A a, () 34 and B 0, 35 we have a 3, 3 B 3 3 a () a a A 0, 33 Hence the potentia outside the spherica she for ra is 55-4

5 SPHERICAL PERFECT LENS: SOLUTIONS OF... q 3 V 3 r 4 3 a () a a 0 r P cos, 37 which is the potentia of a dipoe with the positive charge at the origin and the negative charge at a 3, where and of strength a 3 a a a 0 q 3 a a qq, as 3 / (a /a ). Thus, on one side of the image the region ra 3 ) the fieds of a point charge ocated at a 3 are reproduced. However, it shoud be pointed out that there is no physica charge in the image ocation, and the fieds on the other side of the image i.e., in the region a ra 3 )do not converge to the fieds of the object and cannot do so in the absence of a charge in the image. Further there is no change in the strength of the charge either. There is a magnification in the image formed by a factor of (a /a ). Now et us consider the case of a point source paced at a 3 in the outer region. Again assuming the z axis to pass through a 3, we can write the potentias in the three regions as V r q 4 0 A r P cos ra, 40 V r q P 4 cos 0 A r B r a ra, 4 V 3 r q r a B 3 P cos, 3 r a ra 3, 4 where the first term in V 3 (r) comes from the point source at a 3. Now appying the conditions of continuity at the interfaces, we can simiiary obtain for the coefficients as before. In the imiting case of (a ) and (a ) 3,we have A 3 a a a, 3 A 0, a B 3 a, V r q PHYSICAL REVIEW B 69, a a r a 3 P cos, 47 i.e., that of a point charge of strength q ( / 3 )(a /a ) q at a 0 a 3 (a /a ). As before, for the inner region of ra 0, the system behaves as if there were a singe charge of strength q ocated at ra 0. Thus, the she has a ensike action. We note that there is a demagnification of (a /a ) in this case. A. Simiarities to the D sab ens Let us point out the simiarities to the panar sab ens. In both cases, the eectromagnetic fied grows in ampitude across the negative medium when the perfect ens conditions are satisfied at the interfaces: as an exponentia (expk x z) in the panar ens and as a power of the radia distance r in the spherica ens. The decaying soution away from the source is absent in the negative medium in both cases. Further, when the perfect ens conditions are matched at both the interfaces, there is no refected wave in both the panar sab as we as the spherica ens, i.e., the impedance matching is perfect as we. In addition this mapping preserves the strength of the charge. The key differences, however, are the different dieectric constants on either side of the spherica she of the negative medium. This is a direct consequence of the spatia /r dependence of the negative dieectric constant which reates the two positive dieectric constants to be (a /a ) 3. But this need not be a particuar restriction as we can use the ideas of the asymmetric ens to terminate the different positive media at some radii beyond. 7 The net resut is that the image can now be magnified or demagnified when the image of the charge source is projected out of or into the spherica she, which is true in the D cyindrica ens as we. 0 B. Possibiity of the asymmetric ens In the case of a panar sab, it was possibe to have the perfect ens effect by satisfying the required conditions at any one interface not necessariy at both interfaces. 7 Particuary, in the imit of very arge parae wave vectors the ensing is indeed perfect, athough the image intensity differed from the source by a constant factor. Simiary et us now investigate the effects of having the perfect ens conditions in the case of the spherica ens at ony one of the interfaces. Let us consider first, the case of projecting out the image of a point source from inside the spherica she to outside and enabe the perfect ens conditions ony at the outer interface (ra ) 3 and have an arbitrary. Now the A and B coefficients come out to be B 3 0. Hence the potentia inside the inner sphere is 46 A a 0 a a a,

6 S. ANANTHA RAMAKRISHNA AND J. B. PENDRY PHYSICAL REVIEW B 69, a 0 A a a, B 0, 3 a 0 B 3 a /a 3 a a Ony the growing soution within the negative spherica she remains. The coefficient of the decaying soution (B ) remains stricty zero. Thus, ampification of the decaying fied at east is possibe in this case as we. But there is a finite refectivity in this case. However, the soution outside for r a is not the exact image fied of the point source as the coefficient B 3 has an extra dependence on through the dependence on the dieectric constants. Moreover, the process does not preserve the strength of the charge due to the different dieectric constants invoved. This shoud be compared to the soution of the panar asymmetric sab ens where, at east in the eectrostatic imit, the system behaved as a perfect ens. In this case, the system behaves as a spherica asymmetric perfect ens ony in the imit of arge. The soution outside the spherica she is the same when we meet the perfect ens condition on the inner interface just as in the case of the panar sab ens. However, the refection coefficient is again nonzero, but different to the earier case. In either case, the fieds are argest at the interface where one meets the perfect ens conditions or the interface on which the surface pasmons are excited. C. Effects of dissipation Media with negative rea part of the dieectric constant are absorptive as a metas are, and hence we can write the dieectric constant (r)c/r i i (r) note that i /r as we for us to be abe to write the soution in the foowing form. Consider the first case of projecting out the image of a dipoe ocated within the spherica she where the potentia outside the she is given by Eq. 5 and B 3 is given by Eq. 3. When we have a dissipative negative medium and have the perfect ens conditions at the interfaces on the rea parts of the dieectric constant aone, (a ) i i (a ) and (a ) 3 i i (a ). In parae with the case of the panar ens, we note that the denominator of B 3 consists of two terms, one containing a power of the smaer radius a and the other containing a power of the arger radius a. Cruciay the ampification of the evanescent fieds depends on the possibiity that the smaer power dominates by making the coefficient of the arger term as cose as possibe to zero. The presence of the imaginary part of the dieectric constant woud not aow the coefficient to be zero and the image restoration is good ony as ong as the term containing a dominates in the denominator of B 3, i.e., i a i a a i i a 3 i i a a. 5 Hence a usefu estimate of the extent of image resoution can be obtained by noting the mutipoe for which the two terms in the denominator are approximatey equa. 4 We obtain for this vaue max n3 3 / i a i a. 53 na /a Higher-order mutipoes are essentiay unresoved in the image. We can simiary obtain the same criterion by considering the second case of transferring the image of a charge ocated outside the spherica she into the inner region. Again, we can consider the effects of deviating from the perfect ens conditions on the rea part of the dieectric constant as we and obtain a simiar imit for the image resoution due to those deviations. IV. CONCLUSIONS In concusion, we have presented a spherica perfect ens which enabes magnification of the near-fied images. The perfect ens soution requires media with (r)/r and (r)/r and the conditions (a, ) (a, ) and (a, ) (a, ) at the interfaces of the spherica she of the NRM. We have shown that in the quasistatic imit of sma frequencies ( 0) and high-order mutipoes p, this condition can be reaxed and the two poarizations TE and TM modes decoupe. Thus, a she with negative dieectric constant (r)/r with constant can act as a near-fied ens for the TM poarization whie (r) /r with (r)constant acts as a near-fied ens for the TE modes. We have shown that dissipation in the ens materia, however, prevents good resoution of higher-order mutipoes. Thus, whie the near-fied enses work best for the higher-order mutipoes, dissipation cutsoff the higher-order mutipoes. Further the spherica ens works in the asymmetric mode ony in the imit of high-order mutipoes. Thus, one has to find an intermediate regime where dissipation does not wipe out the near-fied image information and yet the metamaterias work. This is the design chaenge invoving these near-fied enses. APPENDIX: SOLUTION OF THE LAPLACE EQUATION IN A SPATIALLY VARYING MEDIUM We have to sove the Maxwe equations in materia media D0, E0. A Using E V, where V(r) is the eectrostatic potentia we have: r V r V0. A If (r) has ony a radia dependence as in our case /r ), then (r)rˆ(/r) and we can separate the soution as V(r) U(r)/r Y m (,), where the Y m is the spherica harmonic and the radia part U(r) satisfies 55-6

7 SPHERICAL PERFECT LENS: SOLUTIONS OF... r d U U d dr r dr du dr U r 0. A3 To have a soution as a singe power of r, the ony choices possibe for the dieectric constant are either C, a constant the usua case, or C/r. In the atter case the soution is U(r)r or U(r)r (). The fu soution can then be written as Vr 0 PHYSICAL REVIEW B 69, A m r B m r Y m,. A4 J.B. Pendry, Phys. Rev. Lett. 85, V.G. Veseago, Usp. Fiz. Nauk 9, Sov. Phys. Usp. 0, D.R. Smith, W.J. Padia, D.C. Vier, S.C. Nemat-Nasser, and S. Schutz, Phys. Rev. Lett. 84, R.A. Sheby, D.R. Smith, and S. Schutz, Science 9, A. Grbic and G.V. Eeftheriades, J. App. Phys. 9, J.B. Pendry, A.J. Hoden, W.J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, ; J.B. Pendry, A.J. Hoden, D.J. Robbins, and W.J. Steward, J. Phys.: Condens. Matter 0, J.B. Pendry, A.J. Hoden, D.J. Robbins, and W.J. Steward, IEEE Trans. Microwave Theory Tech. 47, N. Garcia and M. Nieto-Vesperinas, Opt. Lett. 7, C.G. Parazzoi, R.B. Greegor, K. Li, B.E.C. Kotenbah, and M. Taniean, Phys. Rev. Lett. 90, A.A. Houck, J.B. Brock, and I.L. Chuang, Phys. Rev. Lett. 90, A. Grbic and G.V. Eeftheriades, App. Phys. Lett. 8, P.V. Parimi, W.T. Lu, P. Vodo, J. Sokooff, and S. Sridhar, cond-mat/ unpubished. 3 H. Raether, Surface Pasmons Springer-Verag, Berin, D.R. Smith, D. Schurig, M. Rosenbuth, S. Schutz, S.A. Ramakrishna, and J.B. Pendry, App. Phys. Lett. 8, F.D.M. Hadane, cond-mat/00640 unpubished. 6 G. Gomez-Santos, Phys. Rev. Lett. 90, S.A. Ramakrishna, J.B. Pendry, D.R. Smith, D. Schurig, and S. Schutz, J. Mod. Opt. 49, X.S. Rao and C.N. Ong, Phys. Rev. B 68, S. Foteinopoou, E.N. Economou, and C.M. Soukouis, Phys. Rev. Lett. 90, J.B. Pendry and S.A. Ramakrishna, J. Phys.: Condens. Matter 4, J.T. Shen and P. Patzmann, App. Phys. Lett. 90, Z. Ye, Phys. Rev. B 67, N. Fang and X. Zhang, App. Phys. Lett. 8, S.A. Ramakrishna, J.B. Pendry, M.C.K. Witshire, and W.J. Stewart, J. Mod. Opt. 50, S.A. Ramakrishna and J.B. Pendry, Phys. Rev. B 67, 00R Z. Liu, N. Fang, T-J. Yen, and X. Zhang, App. Phys. Lett. 83, V.V. Kimov, Opt. Commun., J.B. Pendry, Opt. Express, J.B. Pendry and S.A. Ramakrishna, J. Phys.: Condens. Matter 5, M.C.K. Witshire, J.B. Pendry, I.R. Young, D.J. Larkman, D.J. Giderdae, and J.V. Hajna, Science 9, ; Opt. Express,