Sameh and Maziar, there is a distinct difference between a negative index material (left-handed material) and a metamaterial. A neg. index material is a specifically defined material, featuring a negative index of refraction and is considered a metamaterial. However, metamaterial does not necessarily infer negative permittivity, much less negative index. A metamaterial is loosely defined as a material, whose resultant optical properties are not governed by the law of mixtures, e.g., you can't just take the weighted average of the constituent materials and determine the optical properties of the final material. In most cases, this incorporates nanostructures of high index dielectrics, metals, polar crystals, etc... where the metamaterial optical properties are governed by both the optical properties (material dependent) and the geometrical structure (size, shape) of the constituent nanoparticles. Thus, metamaterials is much more broadly defined definition.
I believe both permittivity and permeability should be negative. The reason can be found in Veselago's original paper.
D Smith and others realized the first negative refractive material using SRRs which give the negative effective permeability together with wires which realize the effective permittivity.
Thank you sammeh, but in Hyperbolic Metamaterial the permeability isn't negative (is non-magnetic) and one of the diagonal effective permittivity tensor is negative, Why we name this medium "Metamaterial" where the refractive isn't negative?
In anisotropic materials you can have a negative refraction without having negative permeability. See Phys. Status Solidi B 249, No. 6, 1110–1118 (2012) / DOI 10.1002/pssb.201100747
Sameh and Maziar, there is a distinct difference between a negative index material (left-handed material) and a metamaterial. A neg. index material is a specifically defined material, featuring a negative index of refraction and is considered a metamaterial. However, metamaterial does not necessarily infer negative permittivity, much less negative index. A metamaterial is loosely defined as a material, whose resultant optical properties are not governed by the law of mixtures, e.g., you can't just take the weighted average of the constituent materials and determine the optical properties of the final material. In most cases, this incorporates nanostructures of high index dielectrics, metals, polar crystals, etc... where the metamaterial optical properties are governed by both the optical properties (material dependent) and the geometrical structure (size, shape) of the constituent nanoparticles. Thus, metamaterials is much more broadly defined definition.
There is an interesting paper from R. W. Ziolkowski which shows all the cases and how to compute the square root which gives the index of refraction.
There are several papers where you can found further information of this topic. But getting negative refraction index by negative permitivity and permeability is not the only way. I hope the following papers give you a new insight in it.
Microwave Opt. Tech. Lett. 50, 1804 (2008)
On mediums with negative phase velocity: a brief overview
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