Research

My research areas are/have been (click for details of previous research. A brief summary of current research is attached below):

Metamaterials (Introduction)

Metamaterials are artificial materials which have physical properties which are rare, if not impossible, to find in nature. The propagation of electromagnetic waves through a medium is characterized by the electric and magnetic properties of the medium. These properties are known as the electrical permittivity (\varepsilon) and magnetic permeability (\mu). Similarly, acoustic/stress/sound waves in a medium are also characterized by the twin parameters of density (\rho) and stiffness (C).

 

In his paper on hypothetical materials, Veselago (1968) theoretically investigated the electromagnetic consequences of having a material with simultaneously negative electrical permittivity (ε) and magnetic permeability (µ). He surmised that such a material would exhibit interesting propagation characteristics such as reversed Doppler shift and Cherenkov radiation, and anomalous refraction. Since refractive index is given by n=\sqrt{\varepsilon\mu} where both ε and µ are generally positive in naturally occurring materials, materials with simultaneously negative ε and µ also give rise to a real refractive index (taken as negative). Such materials with negative index of refraction exhibit phase velocity which is anti-parallel to the group velocity. Naturally occurring materials do display negative electrical permittivity (silver, gold at optical frequencies) and negative magnetic permeability (resonant ferromagnetic and antiferromagnetic systems) but the frequency ranges where such properties are exhibited do not overlap, thereby precluding the possibility of wave transmission. Even in naturally occurring materials, negative ε and µ are a result of subwavelength resonances in the electric and magnetic fields. Since there is no natural law which prohibits the overlap of the frequencies where negative ε and µ are exhibited by a material, it is conceivable to construct artificial materials with subwavelength resonances tuned such that they give rise to effective negative ε and µ in a common frequency range. Such materials where the subwavelength microstructure results in the material exhibiting unusual effective properties are called metamaterials. Veselago's hypothetical material with simultaneously negative ε and µ was finally realized by Smith et. al. (2000) (See also Nicorovici and McPhedran 1994, Pendry et. al. 1999, Shelby et. al. 2001, Smith et. al. 2004.) The possibility of designing materials with unnatural dynamic effective properties also opened the door for research into problems which demand extreme material property profiles such a electromagnetic cloaking (See Greenleaf et. al. 2003, Alu & Engheta 2005, Leonhardt 2006, Milton & Nicorovici 2006, Pendry et. al. 2006, Schurig et. al. 2006, Milton et. al. 2006, Rahm et. al. 2008.) and superlensing (See Pendry 2000, Fang et. al. 2005.)

The theoretical and experimental successes of electromagnetic metamaterials beg the question, 'is it possible to study and design the acoustic analogues of electromagnetic metamaterials?' Electromagnetic metamaterials have been realized by designing periodic subwavelength microstructure which has a desired resonant behavior. The effective properties are defined by relating the unit cell averages of the electric and the magnetic fields. The analogous acoustic problem of wave propagation in periodic composites is complicated by the existence of both longitudinal and shear modes. Furthermore, while both effective magnetic permeability and electrical permittivity are second order tensors, their acoustical counterparts are equal or higher order tensors (effective elasticity tensor is fourth order and density is a second order tensor. See Willis 1997.) This indicates that the number of effective material parameters which must be defined for phononic transport in a periodic composite is more than photonic transport. Therefore, it is clear that in order to be able to design acoustic metamaterials with desired effective properties, the following twin problems need to be tackled,

  1. Calculation of band-structure of periodic composites in 1-, 2-, and 3-dimensions.
  2. A consistent homogenization procedure which uses the eigenvalue/modeshape information from step 1 and provides an effective constitutive law relating the unit cell averages of stress, strain, velocity, and momentum.

Once the effective constitutive tensors relating the unit cell averages of the field variables are defined, the problem of designing acoustic metamaterials for specific dynamic purposes can be tackled.

In a series of papers (Nemat-Nasser et. al. 2011, Nemat-Nasser and Srivastava 2011, Srivastava and Nemat-Nasser 2011, Srivastava and Nemat-Nasser 2011) our group has started making progress on the twin fronts described above. We now have a consistent definition of effective properties and an increasingly better understanding of the geometry and architecture of the micro-structure required for these effective properties to assume extreme values. Research in the immediate future will aim at experimentally demonstrating some of the artifacts of negative effective properties like negative refraction and acoustic focusing. Some relevant references are listed below,

 

  • Alu, A., Engheta, N. 2005, Phys. Rev. E 72, 016623.
  • Amirkhizi, A.V., Nemat-Nasser, S. 2007, Comptes Rendus Mecanique 336, 24.
  • Amirkhizi, A.V., Nemat-Nasser, S. 2008, Smart Mater. Struct. 17, 015042.
  • Avila, A., Griso, G. Miara, B. 2005, C. R. Acad. Sci. Paris Ser. I 340, 933.
  • Babuska, I., Osborn, J.E. 1978, Mathematics of Computation 32.
  • Bakhvalov, N.S. Panasenko, G.P. 1984 Homogenization: averaging processes in periodic media. Dordrecht, The Netherlands: Kluwer.
  • Beran, M. J. 1968, Statistical continuum theories. New York, NY: Interscience Publishers.
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  • Fang, N., Lee, H., Sun, C., Zhang, X. 2005, Science 308, 534.
  • Greenleaf, A. Lassas, M. Uhlmann, G. 2003, Physiological Meas. 24, 413.
  • Leonhardt, U. 2006, Science 312, 1777.
  • Liu, Z., Chan, C.T., Sheng, P. 2005, Phys. Rev. B 71, 014.
  • Milton, G.W. Nicorovici, N.A.P. 2006, Proc. Royal. Soc. A 462, 3027.
  • Milton, G.W., Briane, M. Willis, J.R. 2006, New Journ. Phys. 8, 248.
  • Milton, G.W., Willis, J.R. 2007, Proc. Royal Soc. A 463, 855.
  • Minagawa, S., Nemat-Nasser, S. 1976,  Int. J. Sol. Struct. 12, 769.
  • Nemat-Nasser, S. 1972, J. Elasticity 2, 73.
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  • Nemat-Nasser, S. and Hori, M. 1999, Micromechanics: overall properties of heterogeneous materials, Elsevier: The Netherlands.
  • Nemat-Nasser, S., Srivastava, A. 2011, J. Mech. Phys. Solids, submitted.
  • Nemat-Nasser, S., Willis, J.R.W, Srivastava, A., Amirkhizi, A.V. 2011, Phys. Rev. B 83, 104103.
  • Nicorovici, N.A., McPhedran, R.C., Milton G.W. 1994, Phys. Rev. B 49, 8479.
  • Pendry, J.B., Holden, A.J., Robbins, D.J., Stewart, W.J. 1999, IEEE tran. on Microwave Th. and Tech., 47, 2075-2084.
  • Pendry, J.B. 2000, Phys. Rev. Lett. 85, 3966.
  • Pendry, J.B., Schurig, D., Smith, D.R. 2006, Science 312, 1780.
  • Rahm, M., Cummer, S.A., Schurig, D., Pendry, J.B., Smith, D.R. 2008, Phys. Rev. Lett. 100, 63903.
  • Rytov, S.M. 1956, Sov. Phys. Acoustics 2, 6880.
  • Schurig, D., Mock, J.J., Justice, B.J., Cummer, S.A., Pendry, J.B., Starr, A.F., Smith, D.R. 2006, Science, 314, 977.
  • Shelby, R.A., Smith, D.R., Schultz, S. 2001, Science 292, 77.
  • Sheng, P., Zhang, X.X., Liu, Z., Chan, C.T. 2003, Phys. B. Condens. Matter 338, 201.
  • Shuvalov, A.L., Kutsenko, A.A., Norris, A.N., Poncelet, O. 2011, Proc. Royal Soc. A.
  • Smith, D.R., Padilla, W.J., Vier, D.C., Nemat-Nasser, S.C., Schultz, S. 2000, Phys. Rev. Lett. 84,4184.
  • Smith, D.R., Pendry, J.B., Wiltshire, M.C.K. 2004, Science 305,788.
  • Smith, D.R. Pendry, J.B. 2006, J. Opt. Soc. America 23, 391.
  • Srivastava, A., Nemat-Nasser, S. 2011, Proc. Royal Soc. A., submitted.
  • Veselago, V.G. 1968, Physics Upsekhi 10, 509.
  • Willis,J.R. 1981a, Adv. Appl. Mech. 21, 1.
  • Willis, J.R. 1981b, Wave Motion 3, 1.
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  • Willis, J. R. 1997, Dynamics of composites. In Continuum micromechanics. CISM Lecture Notes, 265–290.
  • Willis, J.R.W. 2008, Mech. Mat. 41, 385.
  • Willis, J.R. 2011, Proc. Royal Soc. A.