Left-handed materials: Transfer matrix method studies

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1 Left-handed materials: Transfer matrix method studies Peter Markos and C. M. Soukoulis

2 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

3 What is an Electromagnetic Metamaterial? A composite or structured material that exhibits properties not found in naturally occurring materials or compounds. Left-handed materials have electromagnetic properties that are distinct from any known material, and hence are examples of metamaterials.

4 Electromagnetic Metamaterials Example: Metamaterials based on repeated cells

5 Material Properties All EM phenomena can be explained by Maxwell s four Equations. When materials are present, we average over the detailed properties to arrive at two parameters: m and e. Quantities e and m are typically positive. When e and m are both negative, there are dramatically new properties.

6 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.

7 Electromagnetic waves interact with materials Lenses Gratings Modulators Switches Antennas Multiplexers Sources

8 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

9 What s New? Advances in Computational Ability Advances in Fabrication Ability Market Needs

10 An Example: Left-handed Materials Predict structure and properties Simulate and optimize structures Fabricate structures Measure and confirm properties New material properties!

11 Veselago We are interested in how waves propagate through various media, so we consider solutions to the wave equation. e,m space 2 E = 2 E em 2 t n = em (-,+) (+,+) k = w em (-,-) (+,-) Sov. Phys. Usp. 10, 509 (1968)

12 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.

13 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 > 1 e ( r w Æ ) = e/ e0 Æ + 1 w ( wm ) w r > 1 m w Æ m/ m0 1 r ( ) = Æ + There appears to be no restriction on negative material constants, other than that frequency dispersion is required.

14 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!

15 Left-Handed Media: Consequences Group and phase velocities reversed. A Left-Handed medium can be interpreted as having a negative refractive index, n. sinq I = ±n sinq T n = e m Snell s law still holds, but the n must be interpreted accordingly: Unusual and non-intuitive geometrical optics result!

16 Negative Refractive Index e eff w ( w) = 1 - w 2 p 2 m eff 2 Fw 0 ( w) = w - w -iwg 0 n( w ) = 1 w ( ) ( w - wb ) w - wp 2 ( w - w 2 0 )

17 Reversal of Snell s Law Right Handed Media Left Handed Media k T =nk I

18 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

19 Metamaterials Extend Properties e w ( ) = 1 - w w 2 p 2 J. B. Pendry ( ) = 1 - m w w 2 w 2 p - w 2 0

20 First Left-Handed Test Structure UCSD, PRL 84, 4184 (2000)

alone e<0 m>0 e<0 Wires alone 4.5 5.0 5.5 6.

21 Transmission Measurements Transmitted Power (dbm) m>0 e<0 m<0 e<0 Wires alone Split rings alone e<0 m>0 e<0 Wires alone Frequency (GHz) UCSD, PRL 84, 4184 (2000)

22 A 2-D Isotropic Structure UCSD, APL 78, 489 (2001)

23 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. ikd t = exp( - ) s 1 Ê 1ˆ cos( nkd ) - z + 2 Ë z z = m e ( ) = 1 - e w w 2 sin( nkd ) n = 2 w ep 2 - w + e0 ige me 0 ( ) = 1 - m w w 2 2 w mp 2 - w + m0 igm 0 UCSD and ISU, PRB, 65, (2002)

24 Measurement of Refractive Index UCSD, Science 292,

25 Measurement of Refractive Index UCSD, Science 292,

26 Measurement of Refractive Index UCSD, Science 292,

27 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 N x x N y x N z : 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

28 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 m eff <0, Re e eff <0 and Re n p <0. For the s wave, the refraction index n s = 1. Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm Typical permittivity of the metallic components: e metal = ( i) x 10 5

29 Structure of the unit cell: r w SRR EM waves propagate in the z-direction. Periodic boundary conditions are used in the xy-plane d c g LHM

30 Left-handed material: array of SRRs and wires 10 Resonance frequency as a function of metallic permittivity 0 [GHz] complex e m m Real em

31 Dependence of LHM peak on metallic permittivity Transmission i i i i i i i Frequency [GHz] The length of the system is 10 unit cells

32 Dependence of LHM peak on metallic permittivity

33 Dependence of LHM peak on L and Im e m Transmission peak i i i i i i i Length of the system

34 Transmission depends on the orientation of SRR Transmission # of unit cells N=5 N=10 N=15 N=20 Re(n)<0 Df=0.8GHz Frequency [GHz] Transmission 10 0 # of unit cells: 10 2 N=5 N=10 N=15 N=20 SRR: x3x3 mm 10 6 Re(n)<0 Df=1.2GHz Frequency [GHz] Transmission properties depend on the orientation of the SRR: Lower transmission Narrower resonance interval Lower resonance frequency Higher transmission Broader resonance interval Higher resonance frequency

35 Determination of effective refraction index from transmission studies Refraction index Im (n) = 10 2 e metal =( i)x Re n Im n 4 Re(n) < Frequency [GHz] Refraction index Im (n) = 10 2 Re(n) <0 e metal =( i)x10 5 Size of the unit cell: 3.3 x 3.67 x 3.67 mm Re n Im n 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.

36 How do the Left-handed EM waves look like? Real part of the transmission 0.2 f=9.8 GHz n= i System length [#unit cells] Real part of the transmission f=11 GHz n= i System length [#unit cells] Real part of the transmission 1.0 f=10.5 GHz n= i System length [#unit cells] Propagation through 300 unit cells was simulated. The system length is 1.1 m. Transmission in the LH material is very good.

37 Length dependence of the transmission Transmission System length [# of unit cells] Outside the resonance interval, transmission t pp 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: t pp =t pp (0) +t ps t ss t sp + The first term, t pp (0) decreases exponentially, the second term does not depend on the system length.

38 Anisotropy is important outside the LH interval Real (t) f=9.7 GHz p wave p wave x 100 p wave has the frequency of s wave! s wave System length [# unit cells] Transmission t pp decreases exponentially only for short system length. For longer systems, t pp =const. There is nonzero transmission t ps of the p wave into the s wave and back. The s wave propagates through the structure without decay. Re n s =1, Im n s =0 t pp =t pp (0) +t ps t ss t sp Therefore, only data for short systems are relevant for the estimation of refraction index of the p wave.

39 Dependence on the incident angle Incident angle [p] Incident angle [p] 10 2 LHM, 0 LHM, LHM, 0 LHM, 0.05 Transmission LHM, 0.10 SRR, 0 SRR, 0.05 Transmission LHM, Frequency [GHz] Frequency [GHz] Transmission peak does not depend on the angle of incidence! Transition peak strongly depends on the angle of incidence. This structure has an additional xz - plane of symmetry

40 Dependence of the LHM T peak on the Im e Board Transmission peak Transmission Frequency [GHz] Im e Board Re e Board =3.4 In our simulations, we have: Periodic boundary condition, therefore no losses due to scattering into another direction. Very high Im e metal therefore very small losses in the metallic components. Losses in the dielectric board are crucial for the transmission properties of the LH structures.

41 Unit cell: 5x3.3x5 mm SRR LHM e metal =( )x10 5 SRR LHM Another 1D left-handed structure: Transmission Transmission Frequency [GHz] Frequency [GHz] Both SRR and wires are located on the same side of the dielectric board. Transmission depends on the orientation of SRR.

42 Photonic Crystals with negative refraction.

43 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= The radius of rods is r=0.35a. a is the lattice constant.

44 Photonic Crystals with negative refraction. t 0 =1.5T T=l/c

45 Photonic Crystals with negative refraction.

46 Photonic Crystals with negative refraction.

47 Photonic Crystals with negative refraction.

$Photonic Crystals: negative refraction The EM wave is trapped temporarily at the interface and after a long time, the wave front moves$

48 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= a (a is the lattice constant of PC)

49 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 m eff and e eff 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, (2002) P. Markos and C. M. Soukoulis, Phys. Rev. E 65, (2002) D. R. Smith, S. Schultz, P. Markos and C.M.Soukoulis, Phys. Rev. B 65, (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)

50 Metamaterials: New Material Responses for Applications Compact cell phone antennas Improvements to base station antenna Multiple degree-of-freedom antennas for MIMO Active materials for beam steering/smart antennas (SDMA) Phased array antennas Military applications

Electromagnetic Metamaterials

Electromagnetic Metamaterials Dr. Alkim Akyurtlu Center for Electromagnetic Materials and Optical Systems University of Massachusetts Lowell September 19, 2006 Objective Outline Background on Metamaterials