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Fundamental Causality and a Criterion of Negative Refraction with Low Optical Losses Mark I. Stockman Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA Phone: 404-651-2779, e-mail:
[email protected], web: http://www.phy-astr.gsu.edu/stockman
Abstract: From the fundamental requirement of causality, we derive a rigorous criterion of negative refraction (lefthandedness). This criterion imposes the lower limits on the electric and magnetic losses in the region of the negative refraction. If these losses are eliminated or significantly reduced by any means, including the compensation by active (gain) media, then the negative refraction will disappear. This theory can be particularly useful in testing the feasibility of the left-handed materials design. ©2006 Optical Society of America OCIS codes: 260.3910 Metals, optics of; 160.4670 Optical materials
1. Introduction There has been recently a significant attention devoted to the so called left-handed materials (LHM), which are also called negative-refraction media [1-13]. In such materials, the directions of energy transfer and wave-front propagation are opposite. This leads to remarkable electromagnetic (optical) properties such as refraction at surfaces that is described by a negative refraction index n .This, in turn, causes a flat slab of a left-handed material with n = −1 to act as a “perfect’’ lens creating, without reflections at the surfaces, a non-distorted image. This is a socalled Veselago lens [14]. Moreover, such a lens can also build an image in the near field [15]. Optical losses in LHMs are detrimental to their performance. These losses for LHMs in the near-infrared and visible region are significant [8-11], which drastically limits their usefulness. There have been proposals to compensate these losses with optical gain [16, 17], which appears to be a way to resolve this problem. In this paper [18], we show that compensating the optical losses, which implies significantly reducing the imaginary parts of dielectric permittivity ε and magnetic permeability µ will necessarily change also the real parts of these quantities in such a way that the negative refraction disappears. This follows from the dispersion relations, i.e., ultimately, from the fundamental principle of causality. 2. Causality and Negative Refraction We consider a material to be an effective medium characterized by macroscopic permittivity ε (ω ) and permeability
µ (ω ) . The squared complex refraction index n 2 (ω ) = ε (ω ) µ (ω ) has exactly the same analytical properties as ε (ω ) and µ (ω ) separately: n 2 (ω ) does not have singularities in the upper half plane of complex ω and n 2 (ω ) → 1 for ω → ∞ . Therefore, absolutely similar to the derivation of the Kramers-Kronig relations for the permittivity or permeability [19], we obtain a dispersion relation for n 2 (ω ) , ∞ Im n 2 (ω1 ) 3 2 (1) ω1 dω1 , Re n 2 (ω ) = 1 + P ∫ π 0 (ω12 − ω 2 )2 where P denotes the principal value of an integral. Now, we assume that the losses at the optical frequency are negligible. Then by multiplying (1) by ω 2 and differentiating over ω , we obtain an exact consequence of this dispersion relation (i.e., of the causality), ∞ 1 1 2 ε ′′(ω1 ) µ ′(ω1 ) + µ ′′(ω1 )ε ′(ω1 ) 3 (2) − 2 = 2∫ ω1 dω1 , 2 πc 0 v pvg c ω12 − ω 2
(
)
where prime and double prime denote the real and imaginary part of the corresponding quantities; . v p is the phase velocity and v g is the group velocity. In the case of the negative refraction, the directions of the phase and energy propagation are opposite, therefore. Consequently, we obtain from this equation a rigorous criterion of the negative refraction with no (or low) loss at the observation frequency ω as
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∞ 2 ε ′′(ω1 ) µ ′(ω1 ) + µ ′′(ω1 )ε ′(ω1 ) 3 ω1 dω1 ≤ −1 . 2 πc 2 ∫0 ω12 − ω 2
(
)
(3)
This criterion directly imposes the lower bounds on the dielectric losses ε ′′(ω1 ) > 0 overlapping with the magnetic plasmonic behavior µ ′(ω1 ) < 0 and the magnetic losses µ ′′(ω1 ) > 0 overlapping with the electric
(
plasmonic behavior ε ′(ω1 ) < 0 . The denominator ω12 − ω 2
)
2
makes the integral to converge for ω − ω1 large; it
would have diverged at ω − ω1 → 0 if the integrand did not vanish at that point. Thus, the major contribution to (3) comes from the lossy, overlapping electric and magnetic resonances close to observation frequency ω . Note that the compensation of losses imply that ε ′′(ω1 ) → 0 and µ ′′(ω1 ) → 0 ., which is incompatible with the negative refraction in view of criterion (3). 3. Examples of Exactly Solvable Negative Refraction Systems in Surface Plasmon Polaritonics We discuss three examples of exactly solvable systems, which possess spectral regions and excitations branches that are negative refracting. These are: metal nanolayer in dielectric, dielectric nanolayer in metal, and thin dielectric layer on metal surrounded by vacuum [20]. In all these cases, considering silver as a metal, the propagation losses of the surface plasmon polaritons (SPPs) are relatively low, except in the domain of the negative refraction. In the latter case, the losses in the optical region become so large that the SPPs cease to be well-defined, propagating modes. 3. Conclusions From the fundamental principle of causality, deriving the corresponding dispersion relations (1) and (2), we have shown that the negative refraction in metamaterials is necessarily related to the propagation losses at and near the observation frequency. The corresponding criterion (3) is an exact consequence of the fundamental principle of the causality. Eliminating or compensating those losses would necessarily eliminate the negative refraction itself. Acknowledgments This work was supported by grants from the Chemical Sciences, Biosciences and Geosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, a grant from National Science Foundation, and a grant from US-Israel Binational Science Foundation. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
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