Waves in composites and metamaterials/Airy solution and WKB solution

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.

Introduction edit

Recall from the previous lecture that we assumed that the permittivity and permeability are scalars and are locally isotropic though not globally so. [1] Then we may write


The TE (transverse electric field) equations are given by


where   represents the two-dimensional gradient operator. Equation (1) can also be written as


which admits solutions of the form


and equation (2) then becomes an ODE:


The quantity


can be less than zero, implying that   may be complex. Also, at the boundary, both   and   must be continuous.

TE waves in a non-magnetic medium edit

For a non-magnetic medium,   is constant and we can write (3) as


Permittivity varies linearly with x edit

If the permittivity varies linearly with  , then we may write


where   and   are constants. Plugging this into (4) we get


Let us assume that   (this is not strictly necessary, but simplifies things for our present analysis). Let us introduce a change of variables


Then (5) becomes


Equation (6) is called the Airy equation. The solution of this equation is


where   and   are Airy functions of the first and second kind (see Abram72 for details.) A plot of the behavior of the two Airy functions as a function of real   is shown in Figure~1.

Figure 1. The Airy functions   and   as functions of  .

As   (i.e., as  ), the Airy functions asymptotically approach the values


So   corresponds to an exponentially decaying wave as   and   corresponds to an exponentially increasing waves at  . A schematic of the situation is shown in Figure 2.

Figure 2. The region where the TE wave is exponentially damped.

If there are no sources in the region   then the solution   is unphysical which implies that  . Therefore,


Now, as   (i.e., as  ), the Airy function   takes the asymptotic form


This is a superposition of right and left travelling waves (because the sine can be decomposed into two exponentials one of which corresponds to a wave travelling in one direction and the seconds to a wave travelling in the opposite direction.)

The Wentzel-Kramers-Brillouin (WKB) method edit

If we don't assume any particular linear variation of the permittivity  , we can use the WKB method to arrive at a solution for high frequency waves.

The WKB method is a high frequency method for obtaining solutions to one-dimensional (time-independent) wave equations of the form


Recall from (1) that the TE equation in a nonmagnetic medium is


Clearly this equation can be written in form (9) by setting


Recall also that the TM equation is


Equation (11) can also be reduced to the form (9). The procedure is as follows. Let us first set   to get


After expanding (11) we get




Differentiating (13) twice, we get


Substituting (12), (13) into (14) we have




Equation (16) has the same form as (9).

At this stage recall that


Let us assume that   is proportional to   which implies that   is also proportional to  , i.e.,


where   is independent of  .

In equation (16), if   is large, then   will dominate and we will end up with exactly the same equation as (9), provided variations in   are smooth (and we don't get large jumps in its derivatives).

Let us now try to solve (9). When   is constant, the solution of the equation is a traveling wave. If we assume that   varies slowly with  , we can try to get solutions of the form


and examine the phase   rather than the solution  . Differentiating (18), we get


Plugging (19) into (9), we get


If we assume that   (i.e.,   is real) we can simplify the analysis slightly at this stage (even though this is not strictly necessary).

For large  , i.e.,  , we can seek a perturbation solution of the form


Plugging (21) into (20) and using (17) we get




For large  , equation (23) reduces to




Integrating (25) from an arbitrary point   to  , we get


where   depends on the sign of the integral.

Next, collecting terms of order   in equation (22), we get


Substituting (25) into (27) we get




Integrating (28) we get


Plugging (26) and (29) into (21) (and ignoring terms containing powers of   and higher) we get


This implies that the solution (18) has the form


Equation (31) is the WKB solution assuming  . Note that when  , a solution does not exist.

Also note that since   is proportional to  ,






Therefore, the restriction is that   is large and that   is smooth with respect to  .

Now, consider for example the profile shown in Figure 3. In region I, the WKB solution is valid since  . At the point where the profile meets the   axis, a solution does not exist since  . However, if the profile is smooth enough, we can assume that   is linear and we can use the Airy solution for the region II around this point. When the profile goes below the   axis,  . However, the WKB solution is valid in this region (III) too as equation 32 can still be satisfied with  .

Figure 3. Regions of validity of the linear solution and the WKB solution for large  .

There is an area of overlap between the regions where the WKB solution is valid and the region where the Airy solution is valid. In fact, the unknown parameters in the two solutions can be determined by matching the solutions at points in this region of overlap.

To do this, let   be the point on the  -axis where  . Then, in region I, the solution is


If there are no sources in region III the solution decays exponentially in the   direction. Then the WKB solution with   is


where the coefficient  .

In region II, since   or   vary linearly with  , we may write


Then, from (7)


When   is high, the region I, II, and III overlap. Also, from (35) we observe that  . Hence, the large   expansion (equation (8)) for the Airy function can be used in the overlap region, i.e.,


Substituting for   and using the identity


we get


Also, in the neighborhood of region II,




Therefore,   becomes


Comparing (37) with (36) we get


Similarly, by comparing   and   in the region of overlap, we get


Footnotes edit

  1. This content is based on the exposition in Chew95. Please consult that text for further details.

References edit

  • M. Abramowitz and I. A. Stegun. Airy functions. In Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, pages 446--452. Dover, New York, 1972.
  • W. C. Chew. Waves and fields in inhomogeneous media. IEEE Press, New York, 1995.