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.
Viscoelastic Materials[edit]
In the previous lecture, we discussed viscoelastic materials and
wondered why the Maxwell model works even though
the effective Young's modulus $Y(\omega )$ for such materials is
analytic in the entire complex plane (except for a few isolated points).
Recall that the Maxwell model (see Figure 1)
predicts that the frequency dependent Young's modulus of the system
is given by
 $Y(\omega )=Y_{e}+\sum _{j}{\cfrac {H_{j}}{1+{\cfrac {i}{\omega \tau _{j}}}}}~.$
Figure 1. A generalized Maxwell model for viscoelasticity.

This implies that the function $Y(\omega )$ is analytic in the entire
imaginary $\omega$ plane except for poles at $\omega =i/\tau _{j}$. On the
other hand, for the frequency dependent metamaterials that we have discussed
earlier, the effective modulus is generally analytic only in the upper half
$\omega$ plane (see Figure~2). Also, for such materials,
$Y^{*}(\omega )=Y(\omega ^{*})$, where $(.)^{*}$ indicates the complex
conjugate. Note that we do not consider the mass when we derive the
modulus of the Maxwell model. The relation between viscoelastic models of
the Maxwell type and general frequency dependent materials continues to
be an open question.
Figure 2. Poles for a general frequencydependent material vs. poles for a generalized Maxwell model.

A justification of the Maxwell model can be provided by considering the
behavior of viscoelastic materials (Christ03). Consider an
experiment where a bar of viscoelastic material of length $l$ is deformed
by a fixed amount. We want to see how the stress changes with time.
Recall, that if the bar is extended by an amount $u(x)$ where $x=0$ at
one end of the bar, then the onedimensional strain is defined as
 $\epsilon ={\cfrac {du}{dx}}~.$
Therefore, the displacement in the bar can be expressed in terms of the
strain as
 $u(x)=\epsilon ~x\qquad \implies \qquad \epsilon ={\cfrac {u(l)}{l}}={\cfrac {\Delta l}{l}}~.$
Also, if $F$ is the applied force on the bar and $A$ is its crosssectional
area, then the stress is given by
 $\sigma ={\cfrac {F}{A}}~.$
Let us now apply a strain to the bar at time $t=0$ and hold the strain
fixed. Due to the initial application of the strain, the stress reaches
a value $K_{0}$ and then relaxes at time increase (due to the relaxation of
polymer chains for instance). Figure 3 shows a
schematic of this situation.
Figure 3. Onedimensional stress relaxation of a viscoelastic material.

If the strain is applied by the superposition of a two step strains as
shown in the figure, we have
 ${\cfrac {d\epsilon }{dt}}=a~\delta (tt_{1})+b~\delta (tt_{2})~.$
The stress is then given by
 $\sigma =a~K(tt_{1})+b~K(tt_{2})~.$
If the strain is applied by a series of infinitesimal steps, then we
get a more general form for the stress:
 ${\text{(1)}}\qquad \sigma (t)=\int _{\infty }^{t}K(tt')~{\cfrac {d\epsilon (t')}{dt'}}~{\text{d}}t'$
where the integral should be interpreted in the distributional sense.
Integrating by parts (and assuming that $\epsilon =0$ at $t=\infty$),
we get
 $\sigma (t)=K_{0}~\epsilon (t)+\int _{\infty }^{t}K(tt')~\epsilon (t')~{\text{d}}t'~.$
Now, $\sigma (t)$ clearly depends on past values of $d\epsilon /dt$. We
expect $\sigma (t)$ should have a stronger dependence on $d\epsilon /dt$ in
the recent past than in the distant past. More precisely, the dependence
should decrease monotonically as $\tau =tt'$ increases. This
implies that $K(\tau )$ should decrease at $\tau$ increases, i.e.,
 ${\cfrac {dK(\tau )}{d\tau }}<0\quad \forall \tau >0~~{\text{and}}~~K(\tau )>0~.$
This is the assumption of fading memmory.
From equation (1) the rate of change of $\sigma$ is given by
 ${\cfrac {d\sigma (t)}{dt}}=\int _{\infty }^{t}\left.{\cfrac {dK(\tau )}{d\tau }}\right_{\tau =tt'}~{\cfrac {d\epsilon (t')}{dt'}}~{\text{d}}t'~.$
Again, we expect $d\sigma /dt$ to have a stronger dependence on
$d\epsilon (t')/dt'$ in the recent past than in the far past, i.e,.
 $\left{\cfrac {dK(\tau )}{d\tau }}\right~~{\text{should decrease as}}~\tau ~{\text{increases.}}$
Now,
 ${\cfrac {dK(\tau )}{d\tau }}<0\quad \implies \quad {\cfrac {d^{2}K(\tau )}{d\tau ^{2}}}>0\quad \implies \quad {\cfrac {d^{3}K(\tau )}{d\tau ^{3}}}<0\quad \implies \quad {\cfrac {d^{4}K(\tau )}{d\tau ^{4}}}>0\quad \dots \qquad \forall ~~\tau >0~.$
Such functions are said to be completely monotonic. An example is
 $K(\tau )=e^{\tau /\tau '}\qquad \implies \qquad {\cfrac {dK(\tau )}{d\tau }}<0~,~~{\cfrac {d^{2}K(\tau )}{d\tau ^{2}}}>0~{\text{etc.}}$
More generally,
 $K(\tau )=K_{\infty }+\int _{0}^{\infty }H(\tau ')~e^{\tau /\tau '}~{\text{d}}\tau '$
is completely monotonic if $H(\tau ')\geq 0$ for all $\tau$ and
$K_{\infty }\geq 0$. The function $H(\tau ')$ is called the {\bf relaxation
spectrum.}
Conversely, any completely monotonic function can be written in this form
(Bernstein28).
Specifically, if
 $\epsilon (t)={\text{Re}}({\widehat {\epsilon }}~e^{i\omega t})$
then
 ${\cfrac {d\epsilon (t)}{dt}}={\text{Re}}(i\omega ~{\widehat {\epsilon }}~e^{i\omega t})~.$
Therefore,
 $\sigma (t)={\text{Re}}\left(i\omega ~{\widehat {\epsilon }}~\int _{\infty }^{t}K(tt')~e^{i\omega t'}~{\text{d}}t'\right)~.$
Let $\tau tt'$. Then
 $\sigma (t)={\text{Re}}\left(i\omega ~{\widehat {\epsilon }}~e^{i\omega t}~\int _{0}^{\infty }K(\tau )~e^{i\omega \tau }~{\text{d}}\tau \right)~.$
Define
 ${\text{(2)}}\qquad Y(\omega ):=i\omega ~\int _{0}^{\infty }K(\tau )~e^{i\omega \tau }~{\text{d}}\tau ~.$
Then we have
 $\sigma (t)={\text{Re}}\left(Y(\omega )~{\widehat {\epsilon }}~e^{i\omega t}\right)~.$
If we define
 ${\widehat {\sigma }}:=Y(\omega )~{\widehat {\epsilon }}$
we get
 $\sigma (t)={\text{Re}}\left({\widehat {\sigma }}~e^{i\omega t}\right)~.$
Now, let $K(\tau )$ be a completely monotonic function of the form
 $K(\tau )=K_{\infty }+\int _{0}^{\infty }H(\tau ')~e^{\tau /\tau '}~{\text{d}}\tau '~.$
Then from equation (2) we get
 $Y(\omega )=i\omega ~\int _{0}^{\infty }K_{\infty }~e^{i\omega \tau }~{\text{d}}\tau i\omega ~\int _{0}^{\infty }{\text{d}}\tau '~H(\tau ')~\int _{0}^{\infty }{\text{d}}\tau ~e^{\tau (i\omega 1/\tau ')}~.$
Assume that $\omega$ has a very small poistive imaginary part (which implies
that $\epsilon (t)$ increases very slowly as $t$ goes to $\infty$). Then
 $Y(\omega )=i\omega ~\left({\cfrac {K_{\infty }}{i\omega }}\right)i\omega ~\int _{0}^{\infty }{\text{d}}\tau '~H(\tau ')~\left({\cfrac {1}{i\omega 1/\tau '}}\right)$
or,
 $Y(\omega )=K_{\infty }+\int _{0}^{\infty }{\cfrac {H(\tau ')}{1+{\cfrac {i}{\omega \tau '}}}}~{\text{d}}\tau '~.$
This is the generalized Maxwell model.
This brings up the question: Is the assumption of fading memory always
correct?
Recall the model of the Helmholtz resonator shown in
Figure~4.
Figure 4. A model of the Helmholtz resonator. [hb]

If we apply a strain in the form of a step function to this model, the
resulting stress response is not a monotonically decreasing function
of time. Rather if oscillates around a certain value and may damp out
over time. A similar oscillatory behavior is expected in other springmass
systems and $K(\tau )$ will, in general, not be monotonic.
A short interlude: Maxwell's equations in Elasticity Form[edit]
In this section, we discuss how Maxwell's equation can be reduced to the
form of the elasticity equations. Recall that, at a fixed $\omega$,
Maxwell's equation take the form
 ${\boldsymbol {\nabla }}\times \mathbf {E} =i\omega {\boldsymbol {\mu }}(\mathbf {x} )\cdot \mathbf {H} (\mathbf {x} )~;~~{\boldsymbol {\nabla }}\times \mathbf {H} =i\omega {\boldsymbol {\epsilon }}(\mathbf {x} )\cdot \mathbf {E} (\mathbf {x} )~.$
Therefore,
 $\omega ^{2}~{\boldsymbol {\epsilon }}\cdot \mathbf {E} =i\omega ~{\boldsymbol {\nabla }}\times \mathbf {H} ={\boldsymbol {\nabla }}\times [{\boldsymbol {\mu }}^{1}\cdot ({\boldsymbol {\nabla }}\times \mathbf {E} )]~.$
Recall that, in index notation and using the summation convention, we
have
 $[{\boldsymbol {\nabla }}\times \mathbf {a} ]_{i}={\mathcal {E}}_{ijk}~{\frac {\partial a_{k}}{\partial x_{j}}}$
where ${\mathcal {E}}_{ijk}$ is the permutation tensor defined as
 ${\mathcal {E}}_{ijk}={\begin{cases}1&{\text{for even permutations, i.e., 123, 231, 312 }}\\1&{\text{for odd permutations, i.e., 132, 321, 213 }}\\0&{\text{otherwise}}.\end{cases}}$
Therefore,
 ${\begin{aligned}\left[\omega ^{2}~{\boldsymbol {\epsilon }}\cdot \mathbf {E} \right]_{j}&={\mathcal {E}}_{jim}~{\frac {\partial }{\partial x_{i}}}[{\boldsymbol {\mu }}^{1}\cdot ({\boldsymbol {\nabla }}\times \mathbf {E} )]_{m}\\&={\mathcal {E}}_{jim}~{\frac {\partial }{\partial x_{i}}}[({\boldsymbol {\mu }}^{1})_{mn}~({\boldsymbol {\nabla }}\times \mathbf {E} )_{n}]\\&={\mathcal {E}}_{jim}~{\frac {\partial }{\partial x_{i}}}[({\boldsymbol {\mu }}^{1})_{mn}~{\mathcal {E}}_{nkl}~{\frac {\partial E_{l}}{\partial x_{k}}}]\\&={\frac {\partial }{\partial x_{i}}}[C_{ijkl}{\frac {\partial E_{l}}{\partial x_{k}}}]\quad {\text{where}}\quad C_{ijkl}:={\mathcal {E}}_{jim}~{\mathcal {E}}_{nkl}~[{\boldsymbol {\mu }}^{1}]_{mn}\end{aligned}}$
or,
 ${\omega ^{2}~{\boldsymbol {\epsilon }}\cdot \mathbf {E} ={\boldsymbol {\nabla }}\cdot ({\boldsymbol {\mathsf {C}}}:{\boldsymbol {\nabla }}\mathbf {E} )~.}$
This is very similar to the elasticity equation
 $\omega ^{2}~\rho ~\mathbf {u} ={\boldsymbol {\nabla }}\cdot ({\boldsymbol {\mathsf {C}}}:{\boldsymbol {\nabla }}\mathbf {u} )~.$
The permittivity is similar to a negative density and the electric field
is similar to the displacement. The equations also hint at a tensorial
density. However, continuity conditions are
different for the two equations, i.e., at an interface, $\mathbf {u}$ is
continuous while only the tangential component of $\mathbf {E}$ is continuous.
Also, the tensor ${\boldsymbol {\mathsf {C}}}$ has different symmetries for the two situations.
Interestingly, for Maxwell's equations
 $C_{ijkl}=C_{jikl}=C_{ijlk}=C_{klij}~.$
Waves in Layered Media[edit]
A detail exposition of waves in layer media can be found in Chew95.
In this section we examine a few features of electromagnetic waves in
layered media.
Assume that the permittivity and permeability are scalars and are
locally isotropic though not globally so. Then we may write
 ${\boldsymbol {\epsilon }}=\epsilon (x_{3})~{\boldsymbol {\mathit {1}}}\quad {\text{and}}\quad {\boldsymbol {\mu }}=\mu (x_{3})~{\boldsymbol {\mathit {1}}}~.$
The TE (transverse electric field) equations are given by
 ${\text{(3)}}\qquad {\overline {\boldsymbol {\nabla }}}\cdot \left({\cfrac {1}{\mu }}~{\overline {\boldsymbol {\nabla }}}E_{1}\right)+\omega ^{2}~\epsilon ~E_{1}=0$
where ${\overline {\boldsymbol {\nabla }}}$ represents the twodimensional gradient operator.
Multiplying (3) by $\mu (x_{3})$, we have
 ${\text{(4)}}\qquad \left[{\frac {\partial }{\partial x_{2}^{2}}}+\mu (x_{3})~{\frac {\partial }{\partial x_{3}}}\left({\cfrac {1}{\mu (x_{3})}}~{\frac {\partial }{\partial x_{3}}}\right)+\omega ^{2}~\epsilon (x_{3})~\mu (x_{3})\right]~E_{1}=0~.$
Equation (4) admits solutions of the form
 $E_{1}(x_{2},x_{3})={\tilde {E}}_{1}(x_{3})~e^{\pm i~k_{2}~x_{2}}$
and equation (4) then becomes an ODE:
 ${\text{(5)}}\qquad \left[\mu (x_{3})~{\cfrac {d}{dx_{3}}}\left({\cfrac {1}{\mu (x_{3})}}~{\cfrac {d}{dx_{3}}}\right)+\omega ^{2}~\epsilon (x_{3})~\mu (x_{3})k_{2}^{2}\right]~{\tilde {E}}_{1}=0~.$
The quantity
 $k_{3}^{2}:=\omega ^{2}~\epsilon (x_{3})~\mu (x_{3})k_{2}^{2}$
can be less than zero, implying that $k_{3}$ may be complex. Also,
at the boundary, both ${\tilde {E}}_{1}$ and $1/\mu \partial {\tilde {E}}_{1}/\partial x_{3}$
must be continuous.
Similarly, for TM (transverse magnetic) waves, we have
 $H_{1}(x_{2},x_{3})={\tilde {H}}_{1}(x_{3})~e^{\pm i~k_{2}~x_{2}}$
and the ODE is
 ${\text{(6)}}\qquad \left[\epsilon (x_{3})~{\cfrac {d}{dx_{3}}}\left({\cfrac {1}{\epsilon (x_{3})}}~{\cfrac {d}{dx_{3}}}\right)+\omega ^{2}~\epsilon (x_{3})~\mu (x_{3})k_{2}^{2}\right]~{\tilde {H}}_{1}=0~.$
References[edit]
 S. Bernstein. Sur les fonctions absolument monotones. Acta Mathematica, 52:166, 1928.
 W. C. Chew. Waves and field in inhomogeneous media. IEEE Press, New York, 1995.
 R. M. Christensen. Theory of viscoelasticity: 2nd Edition. Courier Dover Publications, London, 2003.