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Left-handed materials: Transfer matrix method studies Peter Markos and C. M. Soukoulis Outline of Talk • • • • • • What are Metamaterials? An Example: Left-handed Materials Results of the transfer matrix method Negative n and FDTD results New left-handed structures Applications/Closing Remarks What is an Electromagnetic Metamaterial? ¾A composite or structured material that exhibits properties not found in naturally occurring materials or compounds. ¾Left-handed materials have electromagnetic properties that are distinct from any known material, and hence are examples of metamaterials. Electromagnetic Metamaterials Example: Metamaterials based on repeated cells… Material Properties ¾ All EM phenomena can be explained by Maxwell’s four Equations. ¾ When materials are present, we average over the detailed properties to arrive at two parameters: m and e. ¾ Quantities e and m are typically positive. ¾ When e and m are both negative, there are dramatically new properties. Why create metamaterials? • The electromagnetic response of naturally occurring materials is limited. Metamaterials can extend the material properties, while simultaneously providing other advantageous properties (e.g., strength, thermal, conformal) • Possibility of materials with “ideal” electromagnetic response over broad frequency ranges, that can be customized, made active, or made modulable or tunable. Electromagnetic waves interact with materials… • • • • • • • Lenses Gratings Modulators Switches Antennas Multiplexers Sources Related Terms • Artificial Dielectrics • Effective Medium Theory: Connects the macroscopic electromagnetic properties of a composite to its constituent components, whether atoms, molecules, or larger scale sub-composites. Results in e and m. • Photonic Band Gap Periodic structures, periodicity is on the order of the wavelength. • Metamaterials What’s New? • Advances in Computational Ability • Advances in Fabrication Ability • Market Needs An Example: Left-handed Materials • Predict structure and properties • Simulate and optimize structures • Fabricate structures • Measure and confirm properties • New material properties! Veselago We are interested in how waves propagate through various media, so we consider solutions to the wave equation. e,m space (-,+) k = w em (-,-) 2 ∂ E 2 — E = em 2 ∂t n = em (+,+) (+,-) Sov. Phys. Usp. 10, 509 (1968) Left-Handed Media: Introduction In 1964, V. G. Veselago contemplated the consequences of a medium having simultaneously negative permeability and permittivity. Out of the three possibilities: 1. Such a condition is precluded by other physical laws. 2. Such a condition is possible, but has no effect on wave propagation, since em is positive. 3. If such a condition occurs, it makes a considerable difference in wave propagation! Out of the three possibilities, it is the third that is predicted. Fundamental constraints on e and m While we may observe a certain range of values for the material parameters, the only fundamental limit seems to be set by causality, which imposes the∂(conditions: we r ) e r (w Æ • ) = e / e 0 Æ +1 >1 ∂w ∂(wmr ) >1 ∂w mr (w Æ • ) = m / m0 Æ +1 There appears to be no restriction on negative material constants, other than that frequency dispersion is required. Left-Handed Media: Introduction In the propagation of electromagnetic waves, the direction of energy flow is given by a right-hand rule, involving E, H, and S: S= E¥H The propagation, or “phase” velocity, is usually also determined by a right hand rule: ∂B —¥E= ∂t k ~ E ¥ B = mE ¥ H Thus, when e<0 and m<0, the medium is Left-Handed! Left-Handed Media: Consequences Group and phase velocities reversed. A Left-Handed medium can be interpreted as having a negative refractive index, n. sin q I = ± n sin qT n= e◊ m Snell’s law still holds, but the n must be interpreted accordingly: Unusual and non-intuitive geometrical optics result! Negative Refractive Index Fw 0 2 meff (w ) = 1 - 2 w - w 02 - iwG w 2p e eff (w ) = 1 - 2 w 1 n(w ) = w (w 2 ( - w b2 ) w 2 - w 2p (w 2 - w 02 ) ) “Reversal” of Snell’s Law Right Handed Media kT=nkI Left Handed Media Focusing in a Left-Handed Medium RH RH RH RH LH RH n=1 n=1.3 n=1 n=1 n=-1 n=1 Metamaterials Extend Properties w 2p e (w ) = 1 - 2 w J. B. Pendry w 2p m(w ) = 1 - 2 w - w 02 First Left-Handed Test Structure UCSD, PRL 84, 4184 (2000) Transmitted Power (dBm) Transmission Measurements Wires alone Split rings alone m>0 e<0 m<0 e<0 m>0 e<0 e<0 Wires alone 4.5 5.0 5.5 6.0 Frequency (GHz) 6.5 7.0 UCSD, PRL 84, 4184 (2000) A 2-D Isotropic Structure UCSD, APL 78, 489 (2001) Example of Utility of Metamaterial The transmission coefficient is an example of a quantity that can be determined simply and analytically, if the bulk material parameters are known. exp(- ikd ) ts = 1ˆ 1Ê z + sin( nkd ) cos( nkd ) z¯ 2Ë z= m e n = me 2 w ep e (w ) = 1 - 2 w - w e20 + iGe0 2 w mp m(w ) = 1 - 2 w - w m2 0 + iGm 0 UCSD and ISU, PRB, 65, 195103 (2002) Measurement of Refractive Index UCSD, Science 292, 77 2001 Measurement of Refractive Index UCSD, Science 292, 77 2001 Measurement of Refractive Index UCSD, Science 292, 77 2001 Transfer matrix is able to find: • Transmission (p--->p, p--->s,…) p polarization • Reflection (p--->p, p--->s,…) s polarization • Both amplitude and phase • Absorption Some technical details: • Discretization: unit cell Nx x Ny x Nz : up to 24 x 24 x 24 • Length of the sample: up to 300 unit cells • Periodic boundaries in the transverse direction • Can treat 2d and 3d systems • Can treat oblique angles • Weak point: Technique requires uniform discretization Structure of the unit cell EM wave propagates in the z -direction Periodic boundary conditions are used in transverse directions Polarization: p wave: E parallel to y s wave: E parallel to x For the p wave, the resonance frequency interval exists, where with Re meff <0, Re eeff<0 and Re np <0. For the s wave, the refraction index ns = 1. Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm Typical permittivity of the metallic components: emetal = (-3+5.88 i) x 105 Structure of the unit cell: r w EM waves propagate in the z-direction. Periodic boundary conditions are used in the xy-plane dc SRR g LHM Left-handed material: array of SRRs and wires 10 [GHz] 9 8 0 Resonance frequency as a function of metallic permittivity 7 6 V complex em 5 1000 10000 | m| { Real em Dependence of LHM peak on metallic permittivity Transmission 10 0 10 −2 10 −4 10 −6 10 −8 10 −10 10 −12 −588000 −300000+588000 i 1+588000 i 1 +168000 i 1 + 68000 i −38000 + 38000 i 1+38000 i 1+18000 i 6 7 8 9 Frequency [GHz] The length of the system is 10 unit cells 10 Dependence of LHM peak on metallic permittivity Transmission peak Dependence of LHM peak on L and Im em 10 1 10 0 10 −1 10 −2 10 −3 10 −4 10 −5 10 −6 10 −7 −588000 −300000+588000 i 1+588000 i 1+168000 i 1+68000 i 1+38000 i 1+18000 i 1+8000 i −8 10 0 1 2 3 4 5 6 7 8 Length of the system 9 10 11 Transmission depends on the orientation of SRR 10 0 10 0 # of unit cells # of unit cells: −2 10 −4 10 −6 10 −8 10 −10 10 −12 N=5 N=10 N=15 N=20 10 Transmission Transmission 10 Re(n)<0 N=5 N=10 N=15 N=20 −2 10 −4 10 −6 10 −8 SRR: 3x3x3 mm Re(n)<0 Df=0.8GHz 7.5 8 8.5 9 9.5 Frequency [GHz] Df=1.2GHz 10 10.5 8 9 10 11 Frequency [GHz] 12 13 Transmission properties depend on the orientation of the SRR: ¾Lower transmission ¾Narrower resonance interval ¾Lower resonance frequency ¾Higher transmission ¾Broader resonance interval ¾Higher resonance frequency Determination of effective refraction index from transmission studies 3 emetal=(−3+5.88 i)x10 5 4 emetal=(−3+5.88 i)x10 3 1 Im (n) = 10 0 −1 −2 −3 Re n Im n −4 −5 Re(n) < 0 8 8.5 9 Frequency [GHz] 9.5 10 5 Size of the unit cell: 3.3 x 3.67 x 3.67 mm −2 Refraction index Refraction index 2 2 1 −2 Im (n) = 10 0 −1 −2 −3 −4 Re n Im n Re(n) <0 9 9.5 10 10.5 11 11.5 12 Frequency [GHz] From transmission and reflection data, the index of refraction n was calculated. Frequency interval with Re n<0 and very small Im n was found. How do the Left-handed EM waves look like? 0.2 1.0 f=11 GHz n=−0.378 + 0.008 i 0.0 −0.1 −0.2 200 225 250 275 300 System length [#unit cells] Real part of the transmission Real part of the transmission f=9.8 GHz n=−3.26 + 0.0155 i 0.1 0.5 0.0 −0.5 −1.0 0 100 200 300 System length [#unit cells] 1.0 Real part of the transmission f=10.5 GHz n=−1.31 + 0.005 i 0.5 0.0 −0.5 −1.0 200 225 250 275 System length [#unit cells] 300 Propagation through 300 unit cells was simulated. The system length is 1.1 m. Transmission in the LH material is very good. Length dependence of the transmission Transmission 10 0 10 −2 10 −4 10 −6 10 −8 10 −10 10 −12 8.00 8.50 8.70 9.00 9.30 9.50 10.00 0 5 10 15 System length [# of unit cells] 20 Outside the resonance interval, transmission tpp decreases exponentially as the length of the system increases. However, it never decreases below a certain limit. This is due to the non-zero transmission of the p wave into the s wave: tpp=tpp(0)+tps tss tsp + … The first term, tpp(0) decreases exponentially, the second term does not depend on the system length. Anisotropy is important outside the LH interval 1 f=9.7 GHz 0.5 p wave 0 −0.5 Real (t) −1 1 p wave x 100 0.5 0 −0.5 p wave has the frequency of s wave! −1 1 0.5 s wave 0 −0.5 −1 0 10 20 30 40 50 System length [# unit cells] Transmission tpp decreases exponentially only for short system length. For longer systems, tpp=const. There is nonzero transmission tps of the p wave into the s wave and back. The s wave propagates through the structure without decay. Re ns =1, Im ns=0 tpp=tpp(0)+tps tss tsp Therefore, only data for short systems are relevant for the estimation of refraction index of the p wave. Dependence on the incident angle 10 0 10 0 −2 10 −4 10 −6 10 −8 10 −10 10 −12 Incident angle [p] LHM, 0 LHM, 0.05 LHM, 0.10 SRR, 0 SRR, 0.05 8 8.5 9 9.5 Frequency [GHz] 10 Transmission Transmission Incident angle [p] 10 10.5 Transmission peak does not depend on the angle of incidence ! This structure has an additional xz - plane of symmetry 10 −2 10 −4 10 −6 10 −8 10 LHM, 0 LHM, 0.05 LHM, 0.15 −10 9 10 11 12 Frequency [GHz] 13 Transition peak strongly depends on the angle of incidence. Dependence of the LHM T peak on the Im eBoard 0 10 −2 10 −4 10 −6 10 −4 10 −8 10 10 In our simulations, we have: 0 10 Transmission Transmission peak 10 Re eBoard=3.4 −12 6 7 8 Frequency [GHz] 9 −8 10 −3 10 −2 10 Im eBoard −1 10 0 Periodic boundary condition, therefore no losses due to scattering into another direction. Very high Im emetal therefore very small losses in the metallic components. Losses in the dielectric board are crucial for the transmission properties of the LH structures. Another 1D left-handed structure: 0 −2 10 −4 10 −6 10 −8 10 0 10 −2 10 −4 10 −6 10 −8 Unit cell: 10 −10 10 −12 10 −14 Transmission Transmission 10 10 5x3.3x5 mm SRR LHM 6 7 8 Frequency [GHz] 9 emetal=(−3+5.88)x10 5 SRR LHM 10 10 −10 8 8.5 9 9.5 Frequency [GHz] 10 Both SRR and wires are located on the same side of the dielectric board. Transmission depends on the orientation of SRR. Photonic Crystals with negative refraction. Photonic Crystals with negative refraction. Photonic Crystal vacuum FDTD simulations were used to study the time evolution of an EM wave as it hits the interface vacuum/photonic crystal. Photonic crystal consists of an hexagonal lattice of dielectric rods with e=12.96. The radius of rods is r=0.35a. a is the lattice constant. Photonic Crystals with negative refraction. t0=1.5T T=l/c Photonic Crystals with negative refraction. Photonic Crystals with negative refraction. Photonic Crystals with negative refraction. Photonic Crystals: negative refraction The EM wave is trapped temporarily at the interface and after a long time, the wave front moves eventually in the negative direction. Negative refraction was observed for wavelength of the EM wave l= 1.64 – 1.75 a (a is the lattice constant of PC) Conclusions • Simulated various structures of SRRs & LHMs • Predicted how resonance transmission frequency depends on the parameters of the system • Calculated transmission, reflection and absorption • Calculated meff and eeff and refraction index (with UCSD) • Analyzed transmission properties of 2d and 3d structures • Found negative refraction in photonic crystals DOE, DARPA, NSF Publications: P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 033401 (2002) P. Markos and C. M. Soukoulis, Phys. Rev. E 65, 036622 (2002) D. R. Smith, S. Schultz, P. Markos and C.M.Soukoulis, Phys. Rev. B 65, 195104 (2002) M. Bayindir, K. Aydin, E. Ozbay, P. Markos and C. M. Soukoulis, Appl. Phys. Lett. (2002) P. Markos, I. Rousochatzakis and C. M. Soukoulis, submitted (2002) S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, submitted (2002) Metamaterials: New Material Responses for Applications ¾Compact cell phone antennas ¾Improvements to base station antenna ¾Multiple degree-of-freedom antennas for MIMO ¾Active materials for beam steering/smart antennas (SDMA) ¾Phased array antennas ¾Military applications