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Bounded electromagnetic problems present additional constraints on the assignment of the propagation vector in each region. The phase matching boundary condition on electromagnetic waves requires that the tangential components of the propagation vector be continuous across layer boundaries. In the geometry we are interested in this means that kx and

ky are continuous. Whether we are considering reflection and transmission problems or source

specific problems involving the Green’s functions, the transverse components are fixed and common to all of the propagation vectors. Therefore, for layered problems we want to compute

kz in the biaxial medium given kx and ky to evaluate the electric field vectors. We will use the

Booker quartic equation derived by Pettis [1] for kz, given by

0 2 3 4_{   } z z z z zzk k k k  (2.1.10)

where the coefficients εzz, Δ, Σ, Χ, and Γ are defined by Pettis [1, Appendix I]. The solution of

this Booker quartic yields four unique roots for kz: two roots correspond to the upward

22 for the b-wave ( bu

z

k ) and one for the a-wave ( au z

k ). Similarly, there will be one downward propagating b-wave root ( bd

z

k ) and one downward propagating a-wave root ( ad z

k ). The way we assign these roots is important in understanding the way the a- and b-waves propagate.

We can see from Figure 2-1 that when all four roots are real, the magnitude k is larger for the b-wave than the a-wave. With the transverse components common to both waves the magnitude of b

z

k is greater than the magnitude of a

z

k for four real roots. However, when kx and ky

get large, as they will when computing the Green’s function, the roots become complex and their assignment is less intuitive.

If we track the four roots, we start with the a z

k being smaller than the b z

k roots. This

means that the real a-wave roots approach zero before the real b-wave roots do as kρ

(or 2 2

y

x k

k  ) increases. As kρ increases beyond some point, the a-wave roots will become

complex. This will happen before the b-wave roots become complex. Increase kρ further and all

four roots will be complex. In this case, the a-wave propagation constant will be larger (although complex) than the b-wave propagation constant because the imaginary part is greater for the a- wave.

In defining the orientation of a biaxial medium (with rotated permittivity tensor) Pettis used three rotation angles. Three angles are necessary for the unbounded biaxial medium to be arbitrarily oriented, however in the bounded case, two angles are sufficient as the normal to the boundary is fixed by the geometry. Therefore, we will use the two angle orientation of the biaxial medium discussed in Chapter 1.

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The unrotated permittivity tensor is similar to the one given by equation (2.1.9) only we switch εy and εz resulting in

           8 4 2 r  (2.1.11)

This permittivity tensor is put under a rotation of (ψ1 ψ2) = (30˚, 75˚). We first look at the behavior of the Booker quartic roots for a fixed kx (kx = 0.5ko) as ky is varied. The resulting plot

of the propagation constant in the ky-kz plane is shown in Figure 2-3. In Figure 2-3, we see that

for small values of ky, all the roots are purely real. As ky reaches approximately 1.75ko, we see that ka is no longer purely real. The real part of k_zau and k_zadconverge to zero as the imaginary

components grow from zero. As ky approaches 3ko, we see that all four roots are complex. The real parts of bu

z

k and bd z

k converge to zero and the imaginary parts grow from zero. We also note that when the roots are purely real, b

z

k is greater than a

z

k . However, when the roots become complex the imaginary part of a

z

k is greater than the imaginary part of b z

k . The logic for

assigning the roots is summarized in Table 2-1.

Table 2-1: Booker Quartic Root Assignment Summary

Root Type Action Assignment

4 purely real roots Sort (descending) on real roots bu z k _, au z k _, ad z k _, bd z k

2 purely real roots

2 complex roots Two real roots:

b z

k roots

Two complex roots: a z

k roots Larger real root is

bu z

k _{, smaller is} bd z

k

Larger complex root iskzau, smaller is ad z

k

4 complex roots Sort (descending) on imaginary parts of roots

au z k _, bu z k _, bd z k _, ad z k

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Figure 2-3: Booker quartic root assignment for biaxial medium, kz vs. ky (permittivity

tensor (εx, εy, εz) = (2, 4, 8), rotated by (ψ1 ψ2) = (30˚, 75˚))

Using the root assignment rules shown in Table 2-1, we also show the kz roots plotted

as kx is varied in Figure 2-4. Here, we choose to fix ky at ko and therefore the roots become complex at a lower value of kx than was observed for ky in Figure 2-3.

Figure 2-4: Booker quartic root assignment for biaxial medium, kz vs. kx (permittivity

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We can also plot the wave vector surface for this medium. The a- and b-wave vector surfaces are plotted as kx and ky are varied. These wave vector surfaces are shown in Figures 2-5

through 2-8.

Figure 2-5: Wave vector surface: wave vectors computed using Booker quartic

(permittivity tensor (εx, εy, εz) = (2, 4, 8), rotated by (ψ1 ψ2) = (30˚, 75˚))

Figure 2-6: Wave vector surface showing umbilical point and optic axis 2: wave vectors

computed using Booker quartic (permittivity tensor (εx, εy, εz) = (2, 4, 8), rotated by (ψ1 ψ2)

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In Figure 2-5 and Figure 2-6 the entire wave vector surface is shown. It is difficult to clearly see the umbilical point and optic axes in these plots. Figure 2-7 and Figure 2-8 more clearly illustrate the behavior around the umbilical point by limiting the angular sweep of the wave vector surfaces.

Figure 2-7: Umbilical point at optic axis 1, (εx, εy, εz) = (2, 4, 8), (ψ1 ψ2) = (30˚, 75˚)

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The complex kz values represent evanescent waves in the medium. It is as if a wave is incident from some angle beyond 90˚ and the inverse sine of kz is greater than one. When we evaluate the Green’s function (discussed in Chapter 3), we perform a doubly infinite integral over kx and ky so the assignment of kz from the Booker quartic becomes important in this

complex region.