I want to design a mathematical model in which the refractive index of a material is 0 or even negative. For this I need help in terms of any known naturally occurring material or
synthetic material with these characteristics? If it is possible to design such a matrix then how can I proceed and what would the starting point in the equation be?
It is not in my field, so I just suggest you to look at the two following papers and see if they help or not:
1. Robust topology optimization of three-dimensional photonic-crystal band-gap structures from Men, Lee, Freund,. Peraire and Johnson (http://arxiv.org/pdf/1405.4350.pdf) and
2.All-angle negative refraction in a three-dimensionally periodic photonic crystal Chiyan Luo,Steven G. Johnson, and J. D. Joannopoulos (http://xlab-new.berkeley.edu/MURI/publications/publications_14.pdf).
Concerning the "how to design", I suggest also to look at topology optimization works from Sigmund and Jensen on photonic-crystal band-gap structures.
Again, it is not my field, so hope some specialists in the field can give here more precise details.
Please check these two papers
Experimental realization of an epsilon-near-zero metamaterial at visible wavelengths
http://www.nature.com/nphoton/journal/v7/n11/full/nphoton.2013.256.html
Epsilon near zero based phenomena in metamaterials
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.155130
The above mentioned articles do have a lot of information regarding meta materials and their properties and for that i am obliged for the contributors who refereed me this valuable stuff. But one thing remains hidden as math model that describable meta materiel''?
Math model in this case is an application of Bloch (Bloch-Floquet) Theorem to a periodic medium which is usually solved numerically by a finite element over a Brillouin domain.
It is an advanced topic and as far as I know there is no free or commercial available software/code but if you find one please indicate it here. If you are looking for these topics, use keywords like Band gap or stop band.
Again, it is not my field, so hope some specialist in the field can give you more precise details. You may also try to contact one if in hard difficulties to find what you need.
In general the material is a periodic structure and Bloch waves are used. However when the elements and their separations are small compared to the wavelength, effective media theory can be used. Please refer to the early work of D. Smith and Pendry
Elgnaggar comment made me remember the following.
The relation between the wavelength and cell size may be a problem to the use of finite elements (theoretically we know that high frequencies present accuracy problems). But I never saw a serious study in that involving the Brillouin assumption/subject and besides it indeed I saw several studies resulting in successful prototypes that worked.
If there is some information on wavelength/BlochWave/Finite Element accuracy, please indicate some source for it here.
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