1. A THEORETICAL MODEL for MAGNETIC FIELD IMAGING with “SWISS ROLLS” STRUCTURES V. Yannopapas J. B. Pendry M. C. K. Wiltshire J. Hajnal

2. “Swiss Roll” structure
Highly anisotropic (uniaxial) metamaterial:
Finite slab of a “Swiss Roll” structure with finite (hexagonal) cross-section
The period of the structure and the dimensions of the individual rolls
are much smaller than the wavelength of the EM field:
Effective-medium
approximation

3. “Swiss Roll” structure
Important issues:
Does the effective-medium approximation describe adequately the propagation properties of the system?
How do the material-air boundaries affect the imaging of the magnetic field?
How does the image of the magnetic field vary with frequency?
Theoretical model:
The periodic array of rolls is substituted by a uniaxial effective medium
The air-material interfaces are treated as totally reflecting walls
Near-field limit: the magnetic and electric-field components of the EM field are decoupled from each other

4. Propagation in a uniaxial medium
Maxwell’s equations in a uniaxial medium
with
Combining the above we obtain
For a waveguide geometry:

5. EM modes in a uniaxial medium
Ordinary waves:
Extraordinary waves:
μz leads to the exotic magnetic
effects
Equivalent of the TE modes in a
isotropic medium where
in the near-field limit:
H>>E

6. Uniaxial waveguiding
The modes are normalized so that:

7. Transmission and reflection
In the extreme near-field limit, in free-space:
By requiring the continuity of Ht and Bz at the interfaces:

8. Coupling of the waveguide modesJ S1 S2 C
Using the identity:
For a closed loop of total current I:

9. Experimental setup

10. Transmitted field Hz along a diameter of the hexagon
Incident field
Hz along a diameter of the hexagon
Frequency range: MHz
Transmitted field

11. Transmitted field
Input field of a magnetic dipole

12. Special frequencies
Incident field
Transmitted field

13. Frequency spectrum

14. Conclusions – Further work
Understand the nature of the resonances in the frequency spectrum
Role of the finite size of the lens
Comparison with experiment
Corrections to the model