FORTH - Modelling Issues addressed Heraklion 18/9/2003 Issues addressed Development/adaptation of the Finite Difference Time Domain Method for lossy and for dispersive media Microwave Studio SRR parametric study CMM parametric study (based on the structures constructed at FORTH)

Modelling tools: Finite Difference Time Domain (FDTD) method Implementation for lossy and for frequency dispersive materials, in 2D and 3D Lossy dielectrics: (through an equivalent current)

Dispersive materials (more equations)

Problems  For stability time step must be comparable to 1/pe Large structures  low frequencies  periods much larger than 1/pe  Difficult to apply to metallic structures with u.c. size larger than 10 m Application to “equivalent” smaller structures?? Are equivalent (scalability of the problem)?? ( different) For relatively thick wires YES…. (structures 3 orders smaller  characteristic frequencies “almost” 3 orders larger and relative relation the same)

Microwave Studio (commercial software) Time Domain Solver & Frequency domain solver & Eigenmode solver Treats the metals as perfect conductors or as lossy media with im=i/ or as dispersive media Advantage: Possibility for very thin wires Disadvantages: No periodic boundary conditions are possible  Difficult to treat large systems For dispersive media works only for small structures (like FDTD) Frequency domain solver not complete yet

Metamaterial characteristics vs system parameters SRR parameters Wires parameters Based on the experimental systems - with aim to optimise them

FORTH GaAs structure (in plane) Board: GaAs =12.3, thickness=0.3 mm Metals: Cu and Ag At 10 GHz: Skin depth=0.6 m At 100 GHz: Skin depth=0.2 m Layer of 20x20 unit cells Unit cell of 0.5 x 0.5 x 0.3 mm f  30 GHz

FORTH PCB structure (in-plane) Board: PCB =4.4, thickness=1.5 mm Metals: Cu and Ag Thickness=0.03 mm At 10 GHz: Skin depth=0.6 m At 100 GHz: Skin depth=0.2 m Layer of 20x20 unit cells Unit cell of 5 x 3.63 x 5.5 mm f  10 GHz

Dependence of the SRR dip on: Wires width (w) Rings distance (s) 1 SRR unit cell g w s d Dependence of the SRR dip on: Wires width (w) Rings distance (s) Wires depth (d) Rings gaps (g) Orientation relative to E

1 SRR: FIELDS on resonance k E

1 SRR: Influence of wires width (w) and distance (s) g w s d Reduction of rings width (50%)  reduction of dip freq. (13%) Reduction of rings distance (70%)  reduction of freq. (25%) Depth (thickness) of rings does not affect a lot the SRR dip

1 SRR: Influence of gaps width (g) Smaller gaps  smaller dip-frequency

1 SRR u.c.: Influence of rotation g w s d E k g w s d E k No considerable change in the magnetic dip Change in the electric response

To reduce the magnetic dip frequency (m)? Thin rings Rings close together Small gaps Large area of the outer ring g w s d E Qualitative agreement with

Electric response ? Wires ? Wires + cut-SRRs ? Wires + closed-SRRs ? Aim: Isolate and control the electric response

SRR: Fields SRR off-resonance Closed SRR k E Cut-SRR

GaAs system x 10-2 Cut-SRR Closed-SRR Wires + Closed-SRRs: smaller cut-off ('p) than wires + cut-SRRs or wires

GaAs system (x 10-2): Influence of the wire width Increase of width  increase of p Influence larger than the expected from the logarithm relation

Wires + closed-SRRs: Influence of the rings width on 'p Double rings width The SRR rings width does not affect the electric response (p’)

Changing the metal depth (GaAs system) Thin (depth =1m) Thick (depth =20 m) Increasing the depth  Cut off freq. is increased The depth of the closed-SRRs does not contribute to the change of p’

To increase the cut-off frequency ('p)? Wide continuous wires Additional wires Thick metal

Possibility of separate control of 'p and m Conclusion Possibility of separate control of 'p and m To lower the SRR dip frequency without affecting the 'p: Rings thinn and close together Gaps small To lower the 'p without affecting the SRR dip Wide and thick wires (only the continuous wires)