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

“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

“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

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

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

Uniaxial waveguiding The modes are normalized so that:

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

Coupling of the waveguide modes J S1 S2 C Using the identity: For a closed loop of total current I:

Experimental setup

Transmitted field Hz along a diameter of the hexagon Incident field Hz along a diameter of the hexagon Frequency range: 20-32 MHz Transmitted field

Transmitted field Input field of a magnetic dipole

Special frequencies Incident field Transmitted field

Frequency spectrum

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