1. Negative refraction and Left-handed behavior in Photonic Crystals:
FDTD and Transfer 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 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. Veselago
We are interested in how waves propagate through various media, so we consider solutions to the wave equation.
(+,+)
(-,+)
(-,-)
(+,-)
e,m space
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 W must be positive and this requires
LH medium is always dispersive
According to the Kramers-Kronig relations –
it is always dissipative

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

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

11. k, E, H Left Handed (LHM)  S=c(E x H)/4p opposite to k
PBGs as Negative Index Materials (NIM)
Veselago : Materials (if any) with e < 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. Objections to the left-handed ideas
Parallel momentum is not conserved A B
O΄Μ Ο
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 Wave components with decay, i.e. are lost , then Dmax  l
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. Transmission Measurements
Wires alone
e<0
Wires alone
Split rings alone
Transmitted Power (dBm)
4.5
5.0
5.5
6.0
6.5
7.0
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,

21. Measurement of Refractive Index
UCSD, Science 292,

22. Measurement of Refractive Index
UCSD, Science 292,

23. Transfer matrix is able to find: Some technical details:
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 = ( 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, (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, (2002)

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

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

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

35. ???

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. w t»w t=0.5 or 1 mm t w=0.01 mm l=9 cm 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. Remarks DEFINING phase and group refractive index np and ng
In any general case:
The equifrequency surfaces (EFS) (i.e. contours of constant frequency 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.
np can 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 formula to 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
Incident angle w

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

51. Negative refraction and left-handed behavior for a/l = 0.58

52. Negative refraction but NO left-handed behavior for a/l = 0.535

53. Superlensing in 2D Photonic Crystals Lattice constant=4.794 mm
Dielectric constant=9.73
r/a=0.34, square lattice
Experiment by Ozbay’s group

54. Negative Refraction in a 2d Photonic Crystal

55. Band structure, negative refraction and experimental set up
Frequency=13.7 GHz
Negative refraction is achievable
in this frequency range for
certain angles of incidence.
Bilkent & ISU

58. Superlensing in photonic crystals

59. Subwavelength Resolution in PC based Superlens
The separation between the two point sources is l/3

60. Photonic Crystals with negative refraction.
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.

61. Photonic Crystals with negative refraction.
t0=1.5T
T=l/c

62. Photonic Crystals with negative refraction.

63. Photonic Crystals with negative refraction.

64. Photonic Crystals with negative refraction.

65. 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)

66. Conclusions Simulated various structures of SRRs & LHMs
Calculated transmission, reflection and absorption
Calculated meff and eeff and refraction index (with UCSD)
Suggested new designs for left-handed materials
Found negative refraction in photonic crystals
A transient time is needed for the wave to move along the - direction
Causality and speed of light is not violated.
Existence of negative refraction does not guarantee the existence of
negative n and so LH behavior
Experimental demonstration of negative refraction and superlensing
Image of two points sources can be resolved by a distance of l/3!!!
$$$ DOE, DARPA, NSF, NATO, EU

67. 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, Phys. Rev. E 66, (R) (2002)
S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL, accepted (2003)
S. Foteinopoulou and C. M. Soukoulis, submitted Phys. Rev. B (2002)
P. Markos and C. M. Soukoulis, submitted to Opt. Lett.
E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, submitted to Nature
P. Markos and C. M. Soukoulis, submitted to Optics Express

70. The keen interest to the topic
Terminology
Left-Handed Medium (LH)
Metamaterial
Backward Medium (BW)
Double Negative Medium (DNG)
Negative Phase Velocity (NPV)
Materials with Negative Refraction (MNR)

71. Collaboration between Crete, Greece and Bilkent University, Turkey

72. GM

73. Image Plane

74. Menu
Display
LHM
Cal
Store
Network Analyzer
Experimental Setup

75. Scanned Power Distribution at the Image Plane

76. Dependence of LHM peak on L and Im em

78. Dependence on the incident angle
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

79. Transmission depends on the orientation of SRR
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

80. Dependence of the LHM T peak on the Im eBoard
In our simulations, we have:
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.

81. New / Alternate Designs

82. Superprism Phenomena in Photonic Crystals Experiment
H.Kosaka, T.Kawashima et. al. Superprism phenomena in photonic crystals, Phys. Rev. B 58, (1998)

83. Scattering of the photonic crystal Hexagonal 2D photonic crystal
M.Natomi, Phys. Rev. B 62, (2000)
Using an equifrequency surface (EFS) plots
Vanishingly small index modulation
Small index modulation

84. Photonic crystal as a perfect lens
C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry
Phys. Rev. B, 65, (2002)
Resolution limit 0.67 l