Negative refraction and Left-handed behavior in Photonic Crystals: FDTD and Transfer matrix method studies

1. Negative refraction and Left-handed behavior in Photonic Crystals: FDTD andTransfer matrix method studies Peter Markos, S. Foteinopoulou and C. M. Soukoulis

2. Outline of Talk • What are metamaterials? • Historical review Left-handed Materials • Results of the transfer matrix method • Determination of the effective refractive index • Negative n and FDTD results in PBGs (ENE & SF) • New left-handed structures • Experiments on negative refractions (Bilkent) • Applications/Closing Remarks E. N. Economou & S. Foteinopoulou

3. What is an ElectromagneticMetamaterial? • 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. e,m space (-,+) (+,+) (-,-) (+,-) Veselago We are interested in how waves propagate through various media, so we consider solutions to the wave equation. Sov. Phys. Usp. 10, 509 (1968)

6. Left-Handed Waves • If then is a right set of vectors: • If then is a left set of vectors:

7. Energy flux in plane waves • Energy flux (Pointing vector): • Conventional (right-handed) medium • Left-handed medium

8. Frequency dispersion of LH medium • Energy density in the dispersive medium • Energy density Wmust be positive and thisrequires • LH medium is always dispersive • According to the Kramers-Kronig relations – it isalways dissipative

9. PIM RHM NIM LHM PIM RHM PIM RHM 2 2 1 1 (1) (2) (1) (2) “Reversal” of Snell’s Law

10. RH RH LH RH RH RH n=1 n=1.3 n=1 n=1 n=-1 n=1 Focusing in a Left-Handed Medium

11. PBGs as Negative Index Materials (NIM) • Veselago : Materials (if any) withe< 0 and m< 0 • ·em> 0  Propagation • k, E, H Left Handed (LHM)  S=c(E x H)/4p opposite to k • ·   Snell’s law with < 0 (NIM) • ·g opposite to k • ·Flat lenses • ·Super lenses

12. S1 S2 O΄Μ A B Ο Objections to the left-handed ideas Parallel momentum is not conserved Causality is violated Fermat’s Principle ndl minimum (?) Superlensing is not possible

13. Reply to the objections • Photonic crystals have practically zero absorption • Momentum conservation is not violated • Fermat’s principle is OK • Causality is not violated • Superlensing possible but limited to a cutoff kc or 1/L

14. Materials with e< 0 and m <0 Photonic Crystals opposite to opposite to opposite to opposite to

15. Super lenses is imaginary • Wave components with decay, i.e. are lost , then Dmax l If n < 0, phase changes sign if imaginary thus ARE NOT LOST !!!

16. Metamaterials Extend Properties J. B. Pendry

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

18. Wires alone Split rings alone e<0 m<0 m<0 m>0 m>0 m>0 m>0 e<0 e<0 e<0 Wires alone Transmission Measurements Transmitted Power (dBm) 6.0 6.5 5.5 7.0 5.0 4.5 Frequency (GHz) UCSD, PRL 84, 4184 (2000)

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

20. Measurement of Refractive Index UCSD, Science 292, 77 2001

21. Measurement of Refractive Index UCSD, Science 292, 77 2001

22. Measurement of Refractive Index UCSD, Science 292, 77 2001

23. 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

24. 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

25. Structure of the unit cell: SRR EM waves propagate in the z-direction. Periodic boundary conditions are used in the xy-plane LHM

26. Left-handed material: array of SRRs and wires Resonance frequency as a function of metallic permittivity  complex em  Real em

27. Dependence of LHM peak on metallic permittivity The length of the system is 10 unit cells

28. Dependence of LHM peak on metallic permittivity

29. PRB 65, 033401 (2002)

30. 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. UCSD and ISU, PRB, 65, 195103 (2002)

31. Effective permittivity e(w) and permeability m(w) of wires and SRRs UCSD and ISU, PRB, 65, 195103 (2002)

32. Effective permittivity e(w) and permeability m(w) of LHM UCSD and ISU, PRB, 65, 195103 (2002)

33. Effective refractive index n(w) of LHM UCSD and ISU, PRB, 65, 195103 (2002)

34. Determination of effective parameters from transmission studies 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.

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36. Another 1D left-handed structure: Both SRR and wires are located on the same side of the dielectric board. Transmission depends on the orientation of SRR. Bilkent & ISU APL 2002

37. 0.33 mm w t»w t=0.5 or 1 mm w=0.01 mm t 0.33 mm 3 mm l=9 cm 0.33 mm 3 mm ax Periodicity: ax=5 or 6.5 mm ay=3.63 mm az=5 mm Number of SRR Nx=20 Ny=25 Nz=25 Polarization: TM y E x y x B z

38. New designs for left-handed materials eb=4.4 Bilkent and ISU, APL 81, 120 (2002)

39. ax=6.5 mm t= 0.5 mm Bilkent & ISU APL 2002

40. ax=6.5 mm t= 1 mm Bilkent & ISU APL 2002

41. Cut wires: Positive and negative n

42. Phase and group refractive index • In both the LHM and PC literature there is still a lot of confusion regarding the phase refractive index np and the group refractive index ng. How these properties relate to “negative refraction” and LH behavior has not yet been fully examined. • There is controversy over the “negative refraction” phenomenon. There has been debate over the allowed signs (+ /-) for np and ng in the LH system.

43. DEFINING phase and group refractive index np and ng • In any general case: • The equifrequency surfaces (EFS) (i.e. contours of constantfrequency in 2D k-space) in air and in the PC are needed to find the refracted wavevector kf (see figure). • vphase=c/|np| and vgroup= c/|ng| • Where c is the velocity of light • So from k// momentum conservation:|np|=c kf () /. Remarks • In the PC system vgroup=venergy so |ng|>1. Indeed this holds ! • np <1 in many cases, i.e. the phase velocity is larger than c in many cases. • npcan be used in Snell’s formula to determine the angle of the propagating wavevector. In general this angle is not the propagation angle of the signal. This angle is the propagation angle of the signal only when dispersion is linear (normal), i.e. the EFS in the PC is circular (i.e. kf independent of theta). • ng can never be used in a Snell-like formulato determine the signals propagation angle.

44. Index of refraction of photonic crystals • The wavelength is comparable with the period of the photonic crystal • An effective medium approximation is not valid Equifrequency surfaces Effective index Refraction angle kx ky w Incident angle

45. Photonic Crystals with negative refraction.

46. Photonic Crystals with negative refraction. S. Foteinopoulou, E. N. Economou and C. M. Soukoulis

49. Schematics for Refraction at the PC interface EFS plot of frequency a/l = 0.58

50. Schematics for Refraction at the PC interface EFS plot of frequency a/l = 0.535