Since metamaterials are sub-wavelength particles whose size is much smaller than 1/4th of the wavelength, is there any technique or formula to determine the shape and dimensions of the metamaterial to operate at single or multiple frequencies.
The question is 'what wavelength'? If your wavelength of interest is say radio waves, such structures were fabricated some time ago. For the rest the question is what are your 'multiple' frequencies of interest. Conceivably you could have 2 layers, with the outer layer effective against frequency A. If frequency B penetrates that layer, it could be handled by the inner layer, which woud be desiged to operate for frequency B
There is no simple, direct answer to this. There are many approaches. It really depends a lot on the wavelength of application, like S. Alexiou said, on the bandwidth you are interested in, on your limitations in terms of materials and fabrication capabilities...Without you providing more information, it isn't easy to give more detailed advice. A reasonable approach is to find someone's work who did something similar to what you want to do, and use that as a starting point for your design. Then, you can run finite-element or finite-difference simulations to examine how the metamaterial element response varies as you change one or more geometrical parameters. This can direct your search for a good design for your application.
Optical meta-materials (since you mentioned wavelength which rules out mechanical meta materials and others..) are essentially a carefully designed and arranged ensemble of particles with different resonance properties. Using the spring-mass model of a resonator, we know that the resonance properties of a particle are determined in a big part by the restoring force, which in this case would be the built up local charge differentials every time electrons in a material are pushed away from their neutral position by an external field (Separating electrons from their neutral positions creates a net positive charge that tries to attract them back). Changing the dimensions, shape or material directly affects how many electrons are displaced every time and how far they can be displaced, which consequently changes the "restoring force" thus shifting the particle resonance frequency. When light falls on a resonant particle, the electrons inside oscillate together at the incident frequency which effectively makes the particle re-radiate light, essentially acting like a dipole source. The relative phase, polarization and amplitude of the re-radiated light compared to the incident light is determined by the shape of the nano-particle and how far or close to the resonance frequency the incident light is (Read up on the lorentz oscillator). So for a specific arrangement of many resonators ( essentially dipole sources when illuminated), the overall resultant response of the meta material to the incident light is a superposition of the re-radiated light from all the resonators. Thus if you have a desired resultant meta material response to an incident field ( for example desired reflected or transmitted phase profile), you can reconstruct that by spatially placing nano-resonators (with resonance frequency determined by their shapes, dimensions and material properties) in the desired arrangement. For multiple frequencies you would have to do this for a range of incident frequencies. You can determine the response of each nano resonator at a specific frequency by full-wave simulations (For petty much any shape) or analytic approximations based on the lorentz oscillator model (For well studied shapes).
There are various ways to design metamaterials. I suggest you start by getting familiar with the type of waves for which you are designing the metamaterial, for example elastic, acoustic, electromagnetic, etc. You then need to understand the mechanism by which you can create an interaction between the travelling wave and the geometry of the metamaterial. The design and size of the metamaterial will then naturally follow. Check my article for designing metamaterials that target attenuation of elastic waves here: https://www.nature.com/articles/s41598-019-47644-0, this article for acoustic waves: Article Acoustic band gaps and elastic stiffness of PMMA cellular so...
or this article for optical waves:Article The Optical Properties of Metamaterial-Superconductor Photon...
. Cheers!
Priyanka Garg
In the case of dielectrics/semiconductors, you have to compare not he physical but the optical size to the wavelength (physical times refractive index).
The size of the cell in metamaterial should be less than a 1/10 of the wavelength, not 1/4th of the wavelength
Conference Paper Metamaterials on antenna solutions
@Waiel Elmadih: can you provide any reference articles to design MM structure for electromagnetic wave please?
Dea Vaibhav Bhavsar I'd like to suggest for your attention these reference articles regarding the design of MM structure for electromagnetic waves:
Conference Paper Metamaterials on antenna solutions
Conference Paper Parametric Synthesis of 3D Structure of SRR Element of the M...
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