Table Of Content

Retroreflector Approximation of a Generalized Eaton Lens ∗ Sang-Hoon Kim Division of Marine Engineering, Mokpo National Maritime University, Mokpo 530-729, R. O. Korea (Dated: January 6, 2012) Weextendedapreviousstudyof theEaton lensat specificrefraction angles totheEaton lensat any refraction angle. The refractive index of the Eaton lens is complicated and has not analytical form except at a few specific angles. We derived a more accessible form of the refractive index for anyrefraction angle with some accuracy byretroreflector approximation. The findingof thisstudy will be useful for a rapid estimation of the refractive index, and the the design of various Eaton 2 1 lenses. 0 Keywords: Eatonlens;gradientindexlens;metamaterials. 2 n I. INTRODUCTION lation between the light trajectory and its RI at any im- a J pact parameter. For symmetric and spherical lenses, the 5 Trajectories of light can be controlled by prisms or RI atthe three specific refractionanglesdescribedabove a combination of mirrors, as well as by controlling the are known as4 ] refractive index(RI) of a lens, such as GRIN(Gradient s c Index) lenses like the Eaton lens, Luneburg lens, and ti Maxwell’s fish-eye lens, etc.1 It was once thought that p the control of light trajectories was unrealistic or very o difficult to realize, but recent developments of transfor- . s mation optics and high RI materials from metamaterial c techniques have provided novel methods for controlling i s wave trajectories. y FIG. 1: The trajectories of Eaton lenses at three specific an- h The Eatonlens is atypicalGRINlens inwhichthe RI gles from Ref.4. (A)90o, (B)180o, (C)360o. Reprinted with p varies from one to infinity. It has a singularity in that permission from OSA. [ the RI goes to infinity at the center of the lens and it originates from a peculiar dielectric. The speed of light 2 isreducedtozeroatthispoint,andthelenscantherefore, v 4 change the wave trajectories any direction. 1 1 5 The Eaton lens was recently studied at three specific n2 = + 1 (θ =90o), (1) 7 refractionangles: 90o(right-bender),180o(retroreflector), nr rn2r2 − 2 and 360o(time-delayer).2–4 The RI of the Eaton lens is 2 . n= 1 (θ =180o), (2) 2 given as a function of the radius, but it is not analytic rr − 1 except for at a few specific angles. Generally, the RI 1 1 1 and its trajectories can only be obtained by numerical √n= + 1 (θ =360o). (3) 1 nr rn2r2 − calculations. A simple but good approximation of the : v generalformwillbehelpfulforthedesignandapplication n = 1 for r 1, where n is the relative RI. The r is the i of Eaton lenses. ≥ X radial position between 0 and 1, and the actual radial The RI of the Eaton lens is extended to arbitrary re- position is given by ar, where a is the radius of the lens. r a fractionangles. BycombiningafewEatonlenses,wecan The importance of the above three lenses has been construct an optical triangle, square, hexagon, or any discussed previously. The right-bender in Eq. (1) has geometric shaped path without mirrors. An extremely beenstudiedinbendingsurfaceplasmonpolaritons.6The simple form of the RI is suggested within a reliable er- retroreflector in Eq. (2) is a device that returns the in- ror range for easy and practical use by utilizing a linear cident wave back to its source.7 The time-delayer in Eq. approximation of the retroreflector. (3) is a time-clocking device in which the incident wave exits in the direction of the incidence as if the lens were not present. The trajectories for the above three cases II. GENERALIZED EATON LENS was plotted by Danner and Leonhardt4 and is shown in Fig. 1. Hendi et al.5 performed studies of light trajectories at The RIofarbitraryrefractionangleswasderivedfrom central potentials. Based on the fact that the trajectory optics-mechanics analogy. RI is identical to particle tra- of the Eaton lens is an analogue of Kepler’s scattering jectories of equal total energy E in a central potential problem (“scattering tomography”), they derived the re- U(r). ThepotentialoftheEatonlenscorrespondstothe 2 3 of the relation between RI and the radius. θ=90ο,180ο, 270ο, 360ο θ(n,r) logn = . (5) 2.5 π log 1 + 1 1 (cid:16)nr qn2r2 − (cid:17) These relationsare plotted in Fig.3. A largeRI changes n 2 trajectories at a small radius. The relation between RI andtheradiusisrelativelymoreconvenienttofindusing Eq. (5)thanEq. (4),butisstillnotpracticalformaking 1.5 a rapid estimation. 1 III. RETROREFLECTOR APPROXIMATION 0 0.2 0.4 0.6 0.8 1 r FIG. 2: The refractive indexes at θ = 90o,180o,270o, and 360o from left to right. 2 θ=90ο 4 1.8 θ=180ο θ=270ο 3.5 1.6 θ=360ο 3 p 2.5 1.4 θπ/ 2 n=2.5 n=2.0 n=1.5 n=1.2 1.2 1.5 1 1 0 0.2 0.4 0.6 0.8 1 r 0.5 0 FIG. 4: p(r) as a function of radius. θ=90o,180o,270o, and 0 0.2 0.4 r 0.6 0.8 1 360o from top to bottom. p(r) is linear for θ=180o. FIG. 3: The refraction angles at various refractive index and Introducing a new radial function p(r), the nonlinear radius. Eqs. (4)and(5)areusedasamoreaccessibleform. From Eq. (4) we obtain n(r) at two boundaries. As r 0, gravitation-like U 1/r.2 On the other hand the po- then n (2/r)θ/(π+θ). As r 1, then n (1/r)θ/(π∼+θ). ∝ − ∼ ∼ ∼ tentialoftheLuneburglenscorrespondstotheharmonic- Therefore, we can write n(r) for the range of 0 < r 1 likeU r2.8Fromtheconservationofmechanicalenergy in the following compact form ≤ ∝ insideandoutsidethelens,thekineticenergyofaparticle θ inthemediumofRIniswrittenas(1/2)mn2v2 =E U, p(r) π+θ − n(r)= , (6) where m is the mass and v is the velocity of the particle (cid:26) r (cid:27) inside the lens. Then, nds or √E Uds should be − where 0 < p(r) 1. The new variable p(r) is plotted in stationary bases on FermRat’s prinRciple. ≤ Fig. 4. Replacing χ+π = θ θ into θ of Hannay final − initial The time-delayer with θ =2Nπ is an invisible sphere. and Haeusser’s equation2 (See reference for the detailed The general form with an N turn can be represented derivation),weobtainageneralizedformoftheRIofthe easily using p(r) in Eq. (6). When it has an N turn, the Eaton lens for arbitrary refraction angles as time delay τ is obtained as △ 1 1 2N nπ/θ = + 1, (4) 2Nπrn 2Nπr ap(r) 2N+1 nr rn2r2 − τ = = , (7) △ v v (cid:26) r (cid:27) o o where θ is any radian angle and n = 1 for r 1. The where v is the backgroundvelocity outside the lens and ≥ o refraction angle can be generalized as θ = (2N +1/2)π a is the radius of the lens. If N 1, then 2Nπa/v o fortheright-bender,θ =(2N+1)π fortheretroreflector, t 4Nπa/v . Therefore, a an≫d N are the two mai≤n o andθ =2Nπforthetime-delayer,whereN isaninteger. △facto≤rsthatdecide the time cloakingordelayproperties. Note that θ = π produces the same result with Eq. As p(r) is a function of the radius, we can introduce ± (2). We calculated RI numerically and plotted it at four an approximation. From Eq. (6) p(r) can be written as specific angles in Fig. 2 π+θ Using the logarithm provides a more convenient form p(r)=rn(r) θ . (8) 3 3 Substituting Eq. (9) into Eq. (6), we obtain the simple form of the RI at any refraction angle as θ=90ο θ=270ο 2.5 θ 2 π+θ appr. n(r) 1 . (10) ≃(cid:18)r − (cid:19) n 2 exact exact The effectiveness ofthe approximationis examinedby appr. 1.5 comparison with the exact values obtained from the numerical calculations in Fig. 5 and the optical path was plotted for θ = 60o in Fig. 6. The error of Eq. (10) is 1 0 0.2 0.4 0.6 0.8 1 less than 4% at θ = 90o, 5% at θ = 270o, and 10% at r θ =360o. Theerroraccumulatesformorethan360o and less than 0o. Therefore, this calculation is useful at the FIG. 5: Comparison between the exact value obtained by numericalcalculationandtheretroreflectorapproximationfor conventional angle ranges of 0 < θ < 2π within a 10% θ =90o,270o. The error is less than 4% at θ =90o, and less error. than 5 % at θ =270o. They match exactly at the both ends of r→0,1 at every angle. IV. CONCLUSIONS The Eaton lens is a typical GRIN lens with a compli- catedRI.WeextendedpreviousstudiesoftheEatonlens at specific refraction angles to any refraction angle. The RI refractive index of the Eaton lens is complicated and notanalyticalexceptforafewspecificangles. Wederived amoreaccessibleformoftheRIforanyrefractionangles using the retroreflector approximation. This method is applicable at conventional angle ranges of 0 < θ < 2π within a 10% error. It is extremely simple but useful for arapidestimationofopticalpropertiesobtainedfromthe RI. FIG.6: Opticalpathatθ=60o bytheretroreflectorapproxi- mationinEq. (10). TheplotwasgeneratedusingaCOMSOL simulator. Acknowledgments It is a bounded and analytic function of the radius between 1 and 2. Every line is monotonically decreasing The author would like to thank S. H. Lee and M. fromr =0 to r =1 andnearlylinear. Therefore,we can P. Das for useful discussions. This research was sup- take the retroreflectorapproximationor a linear approx- ported by Basic Science Research Program through the imation as NationalResearchFoundationofKorea(NRF)fundedby theMinistryofEducation,ScienceandTechnology(2011- p(r) 2 r. (9) 0009119) ≃ − ∗ Electronic address: [email protected] lens and invisible sphere by transformation optics with 1 V.N.Smolyaninova,I.I.Smolyaninov,A.V.Kildishev,and no bandwidth limitation, 2009 Conference on Lasers and V.M.Shalaev,Maxwellfish-eyeandEatonlensesemulated Electro-Optics(CLEO), Baltimore, MD, U. S.A. (2009). bymicrodroplets, Opt. Lett. 35 (2010) 3396. 5 A. Hendi, J. Henna, and U. Leonhardt, Ambiguities in the 2 J. H. Hannay and T. M. Haeusser, Retroreflection by re- Scattering Tomography for Central Potentials, Phy. Rev. fraction, J. Mod. Opt. 40 (1993) 1437. Lett. 97 (2006) 073902. 3 T.TycandU.Leonhardt,Transmutationofsingularitiesin 6 T. Zentgraf, Y. Liu, M. H. Mikkelson, J. Valentine, and optical instruments,N. J. Phys. 10 (2008) 115038. X. Zhang, Plasmonic Luneburg and Eaton lenses, Nature, 4 A.J.DannerandU.Leonhardt,LosslessdesignofanEaton Nano, 6 (2011) 151. 4 7 Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, An om- 8 S.P.Morgan,GeneralSolutionoftheLunebergLensProb- nidirectional retroreflector based on the transmutation of lem, J. Appl. Phys. 29 (1958) 1358. dielectric singularities, Nature, Mat. 8(2009) 639.