Waves in composites and metamaterials/Perfect lenses and negative density materials

The content of these notes is based on the lectures by Prof. Graeme W. Milton (University of Utah) given in a course on metamaterials in Spring 2007.

Perfect Lenses edit

Recall the interface separating media with permittivities   and permeabilities   that "behaves like a mirror" (see Figure 1).

Figure 1. "Reflection" at an interface.

For the situation shown in the figure, on the left hand side (LHS) of the interface, let   and   solve Maxwell's equations


Let the solutions on the LHS be of the form


Also, on the right hand side, let


On the right hand side of the interface, let   and   solve the Maxwell equations


If the interface acts as a mirror such that the right hand side (RHS) of the interface has reflected fields, then


Then, to the right of the interface, we have




Similarly, on the RHS, we have


For continuity of the fields at the interface, we must have


This implies that negative permeability and permittivity have the bizzare property of reflecting the fields.

A series of interfaces edit

Let us consider a slab of one material immersed in another medium. Let the permittivity and permeability of the surrounding medium be   (normalized with respect to the values for free space). Let the normalized permittivity and permeability of the slab be  . Hence, the first interface between the medium and the slab acts as a mirror in that it reflects the electric field  . The second interface also acts as a mirror and reflects the field   to the original orientation (see Figure 2).

Figure 2.The effect of a slab of material with   on the field  .

If the source is located at a distance   from the first interface, and the slab has a thickness  , we have


Therefore, the effect of the slab is just a translation. The same is true for the  ,  , and   fields.

Let   be the field which solves the electromagnetic problem with the slab removed for a given source. Let us now insert a negative index slab in the field. The effect of the slab is that the fields appear to move to the right of the slab, i.e., to the right of the slab it appears as if all the fields have been moved a distance  . In other words, it appears that the source has been moved a distance   to the right (see Figure 3).

Figure 3. The effect of a negative index slab is to make the source appear to be at the right of the slab.

This implies that the slab works as a "perfect lens" in the sense that the image to the right of the slab is not diffraction limited. This observation was first made by Pendry [Pendry2000] and was a surprising result because most lenses were though to be diffraction limited.

Consider for instance the ordinary lens shown in Figure 4(a). From geometric optics we expect the rays from the source to be focussed at a point. However, if we consider the wave nature of electromagnetic radiation, several Fourier components of the wave are superimposed at the focal point and the maximum resolution of the image can never be greater than   where   is the wavelength.

Figure 4. Focussing due to an ordinary lens versus a "perfect" lens.

On the other hand, the lensing effect with a slab of negative index material is expected to lead to a point source being exactly represented at the focal point. This idea dates back to Veselago [Veselago1968]. From Figure 4(b) we observe that there will appear to be sources inside the lens and at the focal point when a negative index slab is used as a lens. However, a point source leads to a singularity in the Maxwell equations and there should be no singularities where there no physical point sources. This is a paradox.

The paradox can be resolved by observing that, in fact, a solution does not exist to the time harmonic equations if   in the slab. However, if we let


and let  , then we do have a solution. This is equivalent to assuming that there some loss in the material due to the electrical conductivity of the material.

For   (see Figure 5), the fields blow up to infinity within a strip of width   starting from the focal point within the slab to the focal point outside the slab. They develop more and more oscillations (in space), i.e., at finer and finer length scales. In the remaining regions, the field converge to Pendry's solution. Therefore, the image looks like a point source only on one side of the lens if  . However, in the limit that  , the image also looks like a point source.

Figure 5. Behavior of fields around and inside a "perfect" lens.

If we look at the wave vectors of the electromagnetic waves, then the from the reflected direction of the wave vector inside the lens it appears that light travels backwards inside a negative refractive index lens. But one has to remember that it is the wave crests that are travelling backwards and transport of energy is in the direction propagation of the EM waves (and therefore it is not useful to think of the direction of the wave vector as the direction of the Poynting vector). Note that the phase velocity is negative if the refractive index of the material is negative. Figure 6 illustrates this point.

Figure 6. Direction of the wave vector and the wave packets inside a negative refractive index lens.

Negative, Complex Anisotropic Density Materials edit

An important question in elasticity is how can we get materials with negative moduli. The answer is "through resonance". In this section we approach the problem by exploring the difference between "dynamic" density and "static" density. The static density is defined as the mass per unit volume whereas the dynamic density is defined as the inertial density that appears in Newton's law  . Is the dynamic density the same as the static density?

This question was first addressed by Sheng and coworkers [Sheng2003]. This work was extended by Liu et al. [Liu2005] and the mathematical analysis by Avila et al. [Avila2005]. More recently, several models have been explored by Milton and Willis [Milton2007]. The following discussion is based on one of the models presented by Milton and Willis.

Consider a rigid bar with   voids each of width   as shown in Figure 7. Each cavity contains a spherical ball of mass   and radius  . The ball is attached to the walls of the cavity by springs with sprint constant   (the spring constant may be complex valued to allow for materials with viscous damping). The spring attached to the left wall of the cavity exerts a force   on the wall while the spring attached to the right wall exerts a force of   on the wall. A force   is applied on the left side of the rigid bar. Our aim is to find the response of the bar as a function of time.

Figure 7. A rigid bar containing   voids. Each void contains a spherical ball that is attached to the bar by springs.

Let us assume that all quantities depend harmonically on time. Let us also assume that a one-dimensional approximation of the problem is adequate. Then the forces acting on the bar are given by


where the amplitudes  ,  , and   are generally complex.

From Newton's second law,


where   is the linear momentum which is of the harmonic form


Therefore, plugging equations (1) and (3) into equation (2), we get




Let the time-dependent position of the left side of each cavity be given by


where   is the initial position and   is the complex valued displacement of the bar.

Similarly, the position of each ball is assumed to be given by


where   is the complex valued displacement of each ball.

Then the velocity of the rigid bar is


Assume that the rigid bar has mass  . Therefore, the linear momentum of the rigid bar is


Similarly, the velocity of each sphere is


If there are   balls, the total linear momentum of the balls is


Therefore, the total linear momentum of the system is


From Newton's second law, the applied force equals the rate of change of linear momentum. Therefore,




But   is unobservable since it is in the hidden part of the bar and we need to relate   directly to the observable velocity  .

Let us now consider the free-body diagram of each spring inside a cavity (see Figure 8).

Figure 8. Free-body diagrams for the spring-mass system and for each spring.

Hooke's law for each spring implies that (note that   is positive in the positive   direction)


where   is the complex spring constant.

Recall (from equation 4) that the displacement of the spring is given by


Using the assumed harmonic forms of  ,   and  , we then have




Next, considering the free body diagram of the sping-mass system (see Figure 8), the balance of linear momentum for the spring-mass system implies that


Therefore, substituting in the harmonic forms of   and  , we get


From equations (6) and (7), we have




Now   and  . Hence,


Plugging equation (8) into equation (5), we get


where   is the effective mass. Clearly, the effective mass depends on the frequency   and is different from the static mass.

A normalized plot of the effective mass versus the frequency in shown in Figure 9.

Figure 9. The effective mass of the bar as a function of the frequency.   is the resonant frequency.

At  , the effective mass is equal to the rest mass  . The resonant frequency is given by


Close the resonant frequency, the effective mass can either take high positive values or negative values.

References edit

A. Avila, G. Griso, and B. Miara. Bandes photoniques interdies en elasticite linearisee. C. R. Acad. Sci. Paris Ser. I, 340:933--938, 2005.

Z. Liu, C. T. Chan, and P. Sheng. Analytic model of phononic crystals with local resonances. Physical Review B. Solid State, 71:014103, 2005.

G. W. Milton and J. R. Willis. On modifications of newton's second law and linear continuum elastodynamics. Proc. R. Soc. London A, 463:855--880, 2007.

J. B. Pendry. Negative refraction makes a perfect lens. Physical Review Letters, 85(18):3966--3969, 2000.

P. Sheng, X. X. Zhang, Z. Liu, and C. T. Chan. Locally resonant sonic materials. Physica B. Condensed Matter, 338:201--205, 2003.

V. G. Veselago. The electrodynamics of substances with simultaneous negative values of   and  . Soviet Physics Uspekhi, 10:509--514, 1968.