How can metamaterials be viable for modulus of refractive index |n| < 1 when the curves indicate V_group =V_phase
= –3c (speed of light)?

This can be seen where:
dn/dλ = 0
for V_g = c[n–λ (dn/dλ)]^–X
where λ = wavelength in vacuum

This also invalidates signal and energy velocity. We now have much literature with curves that correspond to the profoundly researched anomalous dispersion of Sommerfeld – Brillouin- Stratton, (modulus of n < unity for both positive and negative refractive index n (GREATER THAN THE SPEED OF LIGHT!!!). How can you have negative refractive index without negative wave impedance, which implies a source of energy?

All the demonstrations that are phased arrays of Hertzian dipoles can bend a beam in any direction you wish. (Heinrich Hertz invented the split ring resonator also). There are also problems with energy density in both classical and quantum electrodynamics. There are increasing numbers of papers in which scientists claim to have proven extraordinary phenomena by applying the concept of group velocity to the anomalous dispersion of waves.

Two of the greatest wave theorists of all time, Arnold Sommerfeld and Léon Brillouin, have dealt with the subject. In separate papers, Sommerfeld and Brillouin wrote that, in anomalous dispersion, the group velocity cannot be the signal velocity.¹ Indeed, in anomalous dispersion, the group velocity goes through both negative and positive infinite values. It also goes through values greater than the speed of light (as does the phase velocity).²

Metamaterials and the possibility of negative refraction are interesting, but before the industries based on classical and quantum electrodynamics can take them seriously, these questions must be addressed. In anomalous dispersion of Sommerfeld – Brillouin-Stratton (modulus of n < unity), the incident wave and the resonances interfere to give apparent speed > c, and/or traveling backward, in a way analogous to the old chestnut about the speed of intersection of two searchlight beams. The published values of +0.9 > n > –0.6 cannot be physical—especially where V_group = V_phase equal speeds greater than light, thus invalidating signal and energy velocity.

Dr. M.J. Lazarus
University of Lancaster
e-mail: M.Lazarus@lancaster.ac.uk

REFERENCES
1. A. Sommerfeld, Annalen der Physik 44, 177 (1914); L. Brillouin, Annalen der Physik 44, 203 (1914). For a lucid English-language digest of the two papers, see ref. 2, p. 334.
2. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York (1941), p. 339, Fig. 63.