Waves in composites and metamaterials/Airy solution and WKB solution

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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 | edit source]

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 | edit source]

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

Permittivity varies linearly with x[edit | edit source]

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 | edit source]

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 | edit source]

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

References[edit | edit source]

  • 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.