## 1 Introduction  
Metamaterials are patterned materials whose dispersive properties are controlled by their internal structure. The distinctive feature is that signal propagation within the metamaterial is governed by micro or nano resonators smaller than the smallest wavelength of the propagating signal. Familiar examples include stained glass and colloidal suspensions which derive their coloration from the plasmonic resonances of gold nanoparticles at optical frequencies, see [1]. Experimental validation of negative index metamaterials created using the negative effective magnetic permeability of split ring resonators [2] together with the negative effective dielectric constant from metal posts [3] proposed in [4] is given in the article of [5]. Refraction of Gaussian pulses by negative index slabs are theoretically investigated in [6]. Subsequent work has delivered several new designs using different configurations of metallic resonators and corrugated wave guides for double negative and effective magnetic behavior [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20] in the microwave regime.  
For higher frequencies in the infrared and optical range, new strategies for generating materials with double negative bulk properties rely on Mie resonances. One scheme employs coated rods made from a high dielectric core coated with a frequency dependent dielectric plasmonic or Drude type behavior at optical frequencies [21; 22; 23]. A second scheme employs small rods or particles made from dielectric materials with large permittivity, [24; 25; 26]. Alternate strategies for generating negative bulk dielectric permeability at infrared and optical frequencies use special configurations of plasmonic nanoparticles [27; 28]. The list of metamaterial systems is growing and reviews of the subject can be found in [29; 30]. Studies on random configuration of dielectric materials has been pursued recently in [31; 32; 33].  
In the present work we are interested in the refraction of signals of a given center frequency and bandwidth traveling from free space into a metamaterial. We construct our metamaterial from non-magnetic sub-wavelength arrays of scatterers that exhibit both local Mie resonances as well as local plasmon resonances associated with a Drude dielectric. These resonances interlace and generate the passbands and stop bands of the metamaterial [34; 35; 36; 37]. The resonances are shown to control the signs of the effective magnetic permeability and dielectric constant. For frequency bands where the effective properties are simultaneously positive one has a positive dispersion while one has a negative dispersion for signals over frequency bands where effective properties are simultaneously negative, see [34; 35; 36; 37]. The Mie resonances are generated by high dielectric fibers coated with a frequency dependent dielectric. This patterned sub-wavelength heterogeneous media delivers an effective index of refraction that can either be positive or negative across different frequency bands in the near infrared and optical regime. In this paper we outline the explicit connection linking these frequency bands to the sub-wavelength geometry and associated resonances. We illustrate the resulting positive to negative index metamaterial and the associated positive to negative angle of refraction for signals transmitted into the metamaterial over different frequency bands.