Negative Refraction and Imaging in Photonic Crystals

Transcription

1 Neative Reraction and Imain in Photonic Crystals S. Sridhar 1, P. V. Parimi, W. T. Lu, and P. Vodo, Electronic Materials Research Institute and Physics Deartment, Northeastern University 36 Huntinton Avenue, Boston, MA 115. John S. Derov, AFRL Hanscom, Bedord, MA. ABSTRACT Neative reraction and let-handed electromanetism in a hotonic crystal are demonstrated in waveuide and ree sace exeriments at microwave requencies. Precision control to achieve tailor-made reractive indices has been achieved. The neative reraction in these hotonic crystals is shown to lead to imain by a lat lens. We have also develoed a eneralized theory o lat lens imain. These results romise otential alications in a variety o otical and microwave systems or communications and imain. 1. Introduction Neative reraction (NR) is a new henomenon allowed by Maxwell s equations that has led to novel eects reardin the roaation and control o electromanetic waves. In a neatively reractin material, electromanetic waves obey a let-handed relationshi between the vectors E, H and k, and hence the media are also reerred to as lethanded metamaterials (LHM). NR has been recently demonstrated in two tyes o LHM - comosite metamaterials i, ii, iii made o slit rin resonators and wire stris, and in hotonic crystals (PC) iv,v. Neative reraction in LHM leads to new otical and electromanetic comonents that oen the door or new alications rom microwave to otical requencies. An interestin alication o neative reraction is suerlensin vi eect by a lat lens with no curved suraces that can otentially overcome the diraction limit imosed by conventional lenses. We have recently abricated a lat lens recently usin a hotonic crystal structure and demonstrated ocussin without otical axis vii. The lat lens is best suited when the source is close to the surace o the lens. For those alications where imain o ar objects is imortant, we have recently demonstrated ocusin by a lano-concave lens usin neative reraction, thus enablin several alications o ar ield imain viii. In this aer we describe the exerimental setus and results that demonstrate neative reraction in hotonic crystals. The exerimental results are in excellent areement with band structure calculations. A eneralized theory o imain by a lat lens is briely described, accomanied by numerical conirmation o imain by a lens comosed o eriodic scatterers. These results conirm the exerimental results reorted earlier in real hotonic crystal lat lens exeriments.. Exerimental setus We have carried out two tyes o exeriments that demonstrate neative reraction in PC, arallel late waveuide ix,x and ree sace exeriments inside an anechoic chamber xi. In both cases microwave reraction measurements were carried out on wede risms abricated rom PC structures, and neative reraction was clearly observed..1. Parallel Plate Waveuide exeriments The microwave hotonic crystal consists o an array o cylindrical coer rods with a radius.63 cm arraned in a trianular lattice. The ratio o the radius r to lattice constant a is ket at r / a =.. Reraction measurements are 1 s.sridhar@neu.edu

2 HP851C Neative Reraction Positive Reraction Waveuide horn Incident Beam Aerture Goniometer PC θ r Reracted Beam k i v ΓM k k t θ< k i v Γ M k θ> kt Fiure 1 (Let) Free sace exerimental setu or microwave reraction measurements. (Riht) Directions o the wavevectors and the rou velocity in the reraction exeriments or neative and ositive reraction. carried out in a metallic arallel late waveuide. The excitation inside this waveuide is transverse manetic (TM), meanin that the electric ield, is arallel to the axis o the dielectric rods. The collimated microwave beam E y roaates a distance o ~1m in the waveuide, and then is incident normally to the PC wede (see Fi. 1). The other ace o the rism, which is the surace o reraction, is at 6 to the incident beam. A diole antenna is mounted on a oniometer that runs alon the semicircular outer ede o the arallel late waveuide to detect the reracted beam. The reraction is considered ositive (neative) i the emerin beam is detected at ositive (neative) anles as shown in Fi 1. The calibration o the reraction exeriment set u is validated by the reraction exeriment erormed on a olystyrene rism o the similar dimensions to the PC rism that corresonds to a ositive reractive index n = as exected by theory... Free Sace Exeriments In addition to the arallel late exeriments described above, we also carried out ree sace (FS) exeriments usin metallic PC rism susended in air. The rincial advantae o these ree sace exeriments is the ability to searate out beams over lare transmission distances o 3m. Furthermore, the olarization can be varied and reraction exeriments were carried out or both or both TM ( E r to the rod axis) and TE ( E r to the rod axis) mode roaation. Excellent areement with band structure calculations was observed or both olarizations, showin that a PC can exhibit neative reraction with tailor made reractive indices in a lare requency rane. The microwave hotonic crystal consists o an array o cylindrical coer tubes arraned on the same eriodical trianular lattice as described in the above arallel late waveuide exeriment. Reraction exeriments were erormed in an anechoic chamber o dimensions 5 x 8 x 4 m 3 to revent relections rom the walls. A square X-band horn laced at 3 m rom the PC acts as a lane wave source (Fi.1). A iece o microwave absorber with a 6x6inch aerture in ront o the PC was used to narrow the incident beam. On the ar side another square horn was attached to a oniometer which swins around in two-deree stes to receive the emerin beam. Reraction is considered ositive (neative) i the emerin sinal is received to the riht (let) o the normal to the surace o reraction o the PC. Measurements were carried out with the incident wave vector k r i alon Γ M (,1) direction o the irst Brillion zone o the PC and in both TM and TE modes. The anle o incidence θ = 6 or Γ M is chosen in order to minimize surace eriodicity alon the surace o reraction, thus eliminatin hiher order Bra waves. 3. Results A 3D lot o the transmitted intensity measured at dierent anles in the requency window 6-1GHz or the TM mode roaation is shown i Fi. b. Also shown are olar lots o the transmitted intensity at selected requencies 6.68, and 1.77 GHz showin the emerin beams.

3 As can be seen rom the iure between 6 and 7.1 GHz the sinal emeres on the ositive side o the normal to the surace corresondin to ositive reraction. No transmission is observed between 7.1 and 8.3 GHz. Above 8.3 GHz and 11 GHz two sinals are observed on the ositive and neative sides o the normal corresondin to 1 st and th order Bra wave reraction. The neatively reracted sinal is stronest rom GHz and ositively reracted sinal around GHz and in the requency window GHz both sinals have the same strenth. So with the increase in requency the ositive sinal ets weaker while neative sinal ets stroner. The exerimental reractive index n is obtained rom alyin Snell s law n = sin( θ r ) / sin( θi ) to each emerin beam. The validity o Snell s law has been established earlier in metallic PCs (ix). 3 = 6.68 GHz -3 a1) b1) 6-6 Frequency (GHz) 9-9 TM Mode, r/a=. 3 =9.315 GHz -3 a) =1.77 GHz 3-3 a3) Transmission ( arb. units) b) Frequency (GHz) Reractive anle Fi. Fiures a1, a, a3 show olar lots o wave directivity or reresentative requencies o 6.68 GHz, GHz and 1.77GHz. Fiure 3b1 shows the band structure o the trianular lattice o metallic PC with ackin density r/a=. or TM Mode and Fiure b shows the 3D exerimental ma o the reracted ield versus requency and anle o reraction. 3

4 Transmission (arb. units) a) b) Frequency (GHz) Frequency (GHz) Reractive anle Fi. 3 3 a shows a 3D exerimental lot o the manitude o the ield versus requency and anle o reraction and Fi. 3 b shows the band structure calculation or trianular lattice PC with r/a=. in TE mode We have also carried out measurements o reraction or TE mode roaation. The band structure or TE mode is shown in the iure 3. Here neative reraction is observed between GHz and ositive reraction between GHz. It is imortant to note that neative reraction is ossible or both TM and TE modes; such a reedom in the choice o modes rovides a crucial advantae o usin the metallic PC over the slit rin wire array metamaterial. Γ X M TE Mode, r/a=. Γ 3.1. Band Structure Calculations The hysical rincile behind the resent results can be understood rom the band structure o the metallic PC. We have calculated the band structure o the trianular lattice PC emloyin a inite dierence method. For the TM modes, the electric ield can be identiied with the wave unction o a D eriodic δ-shell otential barrier, thus a lane wave exansion method can be used to calculate the band structure. The D band structures o both TM and TE are shown resectively in Fis. (b1) and 3(b). For a lane wave with incident wave vector k r and requencyω incident normally on an air-pc interace, the wave vector k r inside the PC is arallel or anti-arallel to k r i as determined by the r r r r band structure. For a eneral case the hase velocity v = ( c / n ) kˆ with kˆ = k / k and the rou velocity v = rω can have any anle between them. Thus distinction must be made between neative wave vector reraction and neative rou reraction. r r r An eective reractive index can be deined n = sn( v k ) c k / ω and calculated rom the band structure. The sin o n is determined rom the behavior o the EFS. Neative wave vector reraction will haen i v k < while neative rou reraction will be resent only i v ( k κ) < with κ is the center o the equirequency surace (EFS) ellisoid. Note that the direction o the rou velocity v r in an ininite PC coincides with that o the enery low. I only the zero-th order Bra wave reraction is resent, conservation o the the surace o reraction would result in ositive or neative wave vector reraction, deendin on whether or anti-arallel to rou velocity. I hiher order Bra wave reraction is resent, the emerin beam can be written as i t k r comonent alon ik r k k r is arallel ik r 1 t t1 Ψ = a e + a e where k t and k t1 reresent the reracted wave vectors corresondin to the th and 1 st order Bra wave-vectors o the 4

5 ield inside the PC. nd and hiher order Bra wave reraction is inored. Let the comonent o k t alon the normal to the surace o reraction o the rism be kt. One has kt = ω / c k sin θ and kt1 = ω / c (π / a k sinθ ) where θ=π/3 is the incident anle at the second surace o the rism and a is the lattice eriodicity. The arallel comonent is k t = k sinθ. The reracted anle can be obtained rom θr = tan ( kt / kt ) or each beam. From the band structure and the EFS or TM mode o roaation neative reraction is redicted or the second and third band reions, with ositive reraction in the 1 st band. In the 1 st band between 6-7.1GHz the EFS move outward with increasin requency, so that r r v k >. In the second band between GHz, the EFS move inward r r with increasin requency, consistent with n < corresondin to v k < ( v r anti-arallel to k r ). The band a is in the requency rane GHz between the 1 st and nd ass-bands and rom GHz between the second and third bands Discussion TM MODE The areement between exerimental results o reraction and those deduced rom band structure calculation is remarkable. In Fi 3(c) the reresentative exerimental results show that neative and ositive reraction are resent in dierent requency ranes. I. In the irst band between GHz the reraction is ositive, consistent with > rom the band structure n and corresondin to RHE v k is larely in the ositive direction. >. The reresentative lot at = 6.68 GHz, Fi. (a1) shows that the sinal II. In the second band between GHz, the reraction is neative or the zero th order Bra wave reraction, consistent with n < rom the band structure and corresondin to LHE v k <.There is observed also a irst order Bra wave that is ositively reracted in the same requency window. This branchin o the beams observed in metallic rods in air desin can be eliminated in a dielectric PC desin, where only a sinle beam can be observed. Two illustrative lots corresondin to and 1.77 GHz show that the sinal is stronly eaked in the neative direction and becomes stroner with the increase in requency. III. A band a is clearly observed in the exeriment, which matches with the band structure, in the reion between GHz. As can be seen rom the 3D lot iure the transmission throuh the PC is near zero. TE MODE It is imortant to note that neative reraction is ossible or TE mode as well. Such a reedom in the choice o modes is a crucial advantae o usin the metallic PC over the slit rin wire array metamaterial. I. In the irst band between GHz the zero th order Bra wave reraction is neative, consistent with n < and v k < rom the band structure and corresondin to LHE. The bandwidth o this neative band is 39% which is quite a remarkable imrovement. 5

6 .8 QM square lattice, r/a=., γ=-4.5 (δ-otential) ka/π ΓXMΓ The intensity o the transmitted wave unction. Fi. 4 a) Band structure o a eriodic deltashell otential o a square lattice. The straiht lines are liht lines b) Flat lens imae o a eriodic delta-shell otential c) The intensity o the transmitted wave unction. Flat lens imae o a eriodic delta-shell otential II. In the second band between GHz, the reraction is larely ositive, consistent with n > rom the band structure and corresondin to RHE v k >. III. The absence o a band a rom 6-1GHz alon GM Brillion zone roaation direction as redicted by the band structure calculation is in comlete areement with exerimental results shown in Fi 3a Imain by a Flat Lens Neative reraction usin LHM leads to entirely new and ascinatin concets or imae ormation. One o the most strikin consequences o NR is the ability o lat suraces to ocus a diverent beam and hence leads to imain o an object laced near the lens. A neative index lat lens reconstructs not only the ar ield roaatin waves, but can also reconstruct evanescent waves. 6

7 Imae ormation is wave reconstruction o both ar and near ields. We have develoed a new aroach to imain by a lat lens made o a roosed artiicial metamaterial [xii]. The key asect o a lat lens usin LHM is the absence o an otical axis. In contrast, current lat lenses like the Fresnel lens, which use radient-index reractive material (GRIN), and all have an otical axis. The conditions or imae ormation are determined by the requirement that the hase shit Φ = k dl rom object to imae is indeendent o wave vector or all directions. Flat lens imain can be achieved by abricatin a material which has aroriate disersion relations ω (k). The intrinsic arameters that deine the new material are σ, which is determined by the disersion characteristics o the material, and κ, which is a measure o the hase shit ex( iσκ d) by the lat slab. This is a eneralization o Pendry s lat lens with ellitic disersion k x + σ ( k κ) = ω / c and neative rou reraction v ( k κ) <. The lens equation or a slab o thickness d is z determined to be u+v=σd. Here u and v are the distances o the object and the imae rom the nearest suraces. Unlike GRIN, the eective reractive index is anle deendent n (θ ), not satial deendent n(x). The lens constant κ determines whether the lens has ositive or neative reractive index n and ives rise to a hase shit o the imae ex( iσκ d). Simulations o imae ormation by such a material are shown in Fi.7. The sace inside the slab aears to be stretched by a actor o σ. Perect imaes are achieved when the material is suitably matched to its environment as shown in the iure. We have shown that this material can be realized aroximately usin hotonic crystals to achieve the desired ω (k).this reliminary theoretical investiation o a secial material or lat lens imain is entirely novel and can lead to new desins or lat lenses. The theoretical rinciles that we have obtained will be used to desin novel lat lenses made o metamaterials and hotonic crystals. The main oal is to desin a material with suitable characteristics leadin to minimal relection and erect imain. In our lat lens exeriment the imae is imerect and reveals the resence o satellites this is due to the act that the roaation is alon the so-called Γ X or a square lattice in the nd band where the inside the crystal the 1 st Bra wave-vector is involved. A better imae should be achieved with roaation alon the Γ M in the irst band where also the neative rou reraction condition can be satisied. Under this condition a sinle imae should be ossible. More enerally the best conditions o imain are achieved when the equirequency suraces are circular or nearly so, and consistent with neative reraction. NUMERICAL SIMULATIONS OF FLAT LENS IMAGING To understand lat lens imain, we consider a model system which is a eriodic δ-shell otential V r) = γ δ ( r R j r ). For γ, the wave unction must vanish on the disk boundary. The wave unction is ( j equivalent to the electric ield o the TM modes o a metallic PC. We have used this model to simulate neative reraction in metallic PC. For attractive otential γ <, and within aroriate enery window, neative reraction and lat lens imain o waves can be achieved. Below is a band structure o a square lattice or γ=-4.5 and r /a=. with lattice sacin a. It is redicted that above ka/π=.14, all-anle-neative-reraction can be achieved alon the ΓM direction. i 4 The lat lens imain o a oint source H kr) is shown below at k a/ π =.14 with a distance u =.4a away rom ( the irst surace. A clear imae is seen on the other side o the slab at v=4.7a. This ives an eective σ e =1.3 which is consistent with the EFS. When the source is moved away rom the irst surace o the slab, the imae will move toward the second surace. When u > σd, virtual imae will be seen throuh ray-tracin. The mechanism o sub-wavelenth imain is due to the bounded states o the δ-shell otential. The surace waves are not resent i no δ-shell is broken. For real hotonic crystals or the above δ-shell otential model, the eective reractive index σ is anle deendent. At certain requency and with aroriate desin, the EFS can be ellitic with neative rou reraction, the imae can 7

8 have hih quality with small aberration. Otherwise, the imae will be blurred due to aberration. Our lat lens theory can be used to desin hih quality lat lens. 4. Conclusion The henomenon o neative reraction in eriodic media embodied by hotonic crystals is well established by the exeriments described above. The imain and ocusin exeriments also show that entirely novel concets are ossible due to NR. Otical comonents utilizin NR are exected to have several advantaes such as enablin sub-wavelenth resolution, lat lenses without otical axis thus avoidin otical alinment, and imroved erormance such as due to reduced aberration or the same structural dimensions. In addition liht weiht and comact structures oer additional advantaes in a variety o alications. We thank Beverly Turchenetz and Richard Win or invaluable contributions. This work was suorted by the National Science Foundation and the Air Force Research Laboratories, Hanscom AFB (contract #F ). i R. A. Shelby, D. R. Smith, and S. Schultz, Science 9, 77 (1). ii C. G. Parazzoli, R. B. Greeor, K. Li, B. E. C. Koltenbach, and M. C. Tanielian, Phys. Rev. Lett. 9, 1741 (3). iii A. A. Houck, J. B. Brock, and I. L. Chuan Phys. Rev. Lett. 9, (3). iv M. Notomi, Phys. Rev. B 6, 1696 (); Ot. Quant. Elect. 34, 133 (). v C. Luo, S. Johnson, J. Joannooulos, and J. Pendry, Phys. Rev. B 65, 114 (). vi J. B. Pendry, Phys. Rev. Lett. 85, 3966 (). vii P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, Nature(London) 46, 44 (3). viii P.Vodo, P.Parimi, W.T.Lu, S.Sridhar, A. Phys. Lett. 86, 118 (5). ix P. V. Parimi, W. T. Lu, P. Vodo, J. B. Sokolo, and S. Sridhar, Phys. Rev. Lett. 9, 1741 (4).. x E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinooulou, and C. M. Soukoulis, Nature (London) 43, 64 (3) xi P.Vodo, P.Parimi, W.T.Lu, S.Sridhar, A. Phys. Lett. 85, 1858 (4). xii W.T. Lu and S. Sridhar, Flat lens without otical axis: Theory o imain, arxiv: cond-mat/51715 (5). 8