WKB Solutions

Let us start off by examining a very simple case. Consider a plane electromagnetic wave, of angular frequency $\omega$, propagating along the $z$-axis in an unmagnetized plasma whose refractive index, $n$, is a function of $z$. Let us assume that the wave normal is initially aligned along the $z$-axis, and, furthermore, that the wave starts off polarized in the $y$-direction. It is easily demonstrated that the wave normal subsequently remains aligned along the $z$-axis, and also that the polarization state does not change. Thus, the wave is fully described by

$$E_y(z,t) \equiv E_y(z)\,\exp(-{\rm i}\,\omega\,t),$$ (6.1)

and

$$B_x(z,t) \equiv B_x(z)\,\exp(-{\rm i}\,\omega\,t).$$ (6.2)

It can readily be shown that $E_y(z)$ and $B_x(z)$ satisfy the differential equations

$$\frac{d^2 E_y}{dz^{2}} =-k_0^{2}\,n^2\,E_y,$$ (6.3)

and

$$\frac{d\,(c\,B_x)}{dz} = -{\rm i}\,k_0\,n^2\,E_y,$$ (6.4)

respectively. Here, $k_0=\omega/c$ is the wavenumber in free space. Of course, the actual wavenumber is $k=k_0\,n$.

The solution of Equation (6.3) for the case of a homogeneous plasma, for which $n$ is constant, is simply

$$E_y(z) = A\,{\rm e}^{\,{\rm i}\,\phi(z)},$$ (6.5)

where $A$ is a constant, and

$$\phi(z) = \pm k_0\,n\,z.$$ (6.6)

The solution (6.5) represents a wave of constant amplitude $A$, and phase $\phi(z)$. According to Equation (6.6), there are two independent waves that can propagate through the plasma. The upper sign corresponds to a wave that propagates in the $+z$-direction, whereas the lower sign corresponds to a wave that propagates in the $-z$-direction. Both waves propagate at the constant phase-velocity $c/n$.

In general, if $n=n(z)$ then the solution of Equation (6.3) does not remotely resemble the wave-like solution (6.5). However, in the limit in which $n(z)$ is a “slowly varying” function of $z$ (exactly how slowly varying is something that will be established later on), we expect to recover wave-like solutions. Let us suppose that $n(z)$ is indeed a “slowly varying” function, and let us try substituting the wave-like solution (6.5) into Equation (6.3). We obtain

$$\left(\frac{d\phi}{dz}\right)^2 = k_0^{2} \,n^2 +{\rm i}\,\frac{d^2\phi} {dz^{2}}.$$ (6.7)

This is a non-linear differential equation which, in general, is very difficult to solve. However, we note that if $n$ is a constant then $d^2\phi/dz^{2}=0$. It is, therefore, reasonable to suppose that if $n(z)$ is a “slowly varying” function then the last term on the right-hand side of the previous equation is relatively small. Thus, to a first approximation, Equation (6.7) yields

$$\frac{d\phi}{dz} \simeq \pm k_0\,n,$$ (6.8)

and

$$\frac{d^2 \phi}{dz^2} \simeq \pm k_0\,\frac{dn}{dz}.$$ (6.9)

It is clear, from a comparison of Equations (6.7) and (6.9), that $n(z)$ can be regarded as a “slowly varying” function of $z$ [i.e., the second term on the right-hand side of Equation (6.7) is negligible compared to the first] as long as $(dn/dz)/(k_0\,n^2)\ll 1$. In other words, the approximation holds provided that the variation lengthscale of the refractive index is far longer than the wavelength of the wave.

The second approximation to the solution is obtained by substituting Equation (6.9) into the right-hand side of Equation (6.7):

$$\frac{d\phi}{dz} \simeq \pm \left(k_0^{2} \,n^2 \pm {\rm i} \,k_0 \,\frac{dn}{dz}\right)^{1/2}.$$ (6.10)

This gives

$$\frac{d\phi}{dz} \simeq \pm k_0\,n\left(1\pm \frac{{\rm i}}{k_0 \... ...\frac{dn}{dz} \right)^{1/2}\simeq \pm k_0\,n + \frac{\rm i}{2\,n}\frac{dn}{dz},$$ (6.11)

where use has been made of the binomial expansion. The previous expression can be integrated to give

$$\phi(z)\simeq \pm k_0\! \int^z \!n\,dz' +{\rm i}\,\log\left(n^{1/2}\right).$$ (6.12)

Substitution of Equation (6.12) into Equation (6.5) yields the final result

$$E_y(z) \simeq n^{-1/2}\,\exp\left(\pm {\rm i}\, k_0\! \int^z \!n\,dz'\right).$$ (6.13)

It follows from Equation (6.4) that

$$c\,B_x(z)\simeq \mp n^{1/2}\,\exp\left(\pm {\rm i}\,k_0 \!\int^z\... ...\,n^{3/2}} \frac{dn} {dz}\,\exp\left(\pm {\rm i}\,k_0 \!\int^z \!n\,dz'\right).$$ (6.14)

The second term on the right-hand side of the previous expression is small compared to the first, and is usually neglected.

We can test to what extent expression (6.13) is a good solution of Equation (6.3) by substituting this expression into the left-hand side of the equation. The result is

$$\frac{1}{n^{1/2}}\left[ \frac{3}{4}\left(\frac{1}{n} \frac{dn}{dz... ...\frac{1}{n} \frac{dn}{dz}\right)^2 -\frac{1}{2\,n}\frac{d^2 n}{dz^2}\right]E_y.$$ (6.15)

This quantity needs to be small compared to $k_0^{2}\,n^2\,E_y$. Hence, the condition for Equation (6.13) to be a good solution of Equation (6.3) becomes

$$\frac{1}{k_0^{2}}\left\vert \frac{3}{4}\left(\frac{1}{n^2} \frac{dn}{dz}\right)^2 -\frac{1}{2\,n^3}\frac{d^2 n}{dz^2}\right\vert \ll 1.$$ (6.16)

The solutions

$$E_y(z) \simeq n^{-1/2}\,\exp\left(\pm {\rm i}\, k_0 \!\int^z \!n\,dz'\right),$$ (6.17)

$$c\,B_x(z) \simeq \mp n^{1/2}\,\exp\left(\pm {\rm i}\,k_0\! \int^z\! n\,dz'\right),$$ (6.18)

to the non-uniform wave equations (6.3) and (6.4) are usually referred to as WKB solutions, in honor of G. Wentzel (Wentzel 1926), H.A. Kramers (Kramers 1926), and L. Brillouin (Brilloiun 1926), who are credited with independently discovering these solutions (in a quantum mechanical context) in 1926. Actually, H. Jeffries (Jeffries 1924) wrote a paper on WKB solutions (in a wave propagation context) in 1924. Hence, these solutions are sometimes called the WKBJ solutions (or even the JWKB solutions). To be strictly accurate, the WKB solutions were first discussed by Liouville (Liouville 1837) and Green (Green 1837) in 1837, and again by Rayleigh (Rayleigh 1912) in 1912. In the following, we refer to Equations (6.17) and (6.18) as WKB solutions, because this is what they are most commonly called. However, it should be understood that, in doing so, we are not making any definitive statement as to the credit due to various scientists in discovering them. More information about WKB solutions can be found in the classic monograph of Heading (Heading 1962).

If a propagating wave is normally incident on an interface at which the refractive index suddenly changes (for instance, if a light wave propagating through air is normally incident on a glass slab) then there is generally significant reflection of the wave (Fitzpatrick 2013). However, according to the WKB solutions, (6.17) and (6.18), when a propagating wave is normally incident on a medium in which the refractive index changes slowly along the direction of propagation of the wave then the wave is not reflected at all. This is true even if the refractive index varies very substantially along the path of propagation of the wave, as long as it varies sufficiently slowly. The WKB solutions imply that, as the wave propagates through the medium, its wavelength gradually changes. In fact, the wavelength at position $z$ is approximately $\lambda(z)= 2\pi/ [k_0\,n(z)]$. Equations (6.17) and (6.18) also imply that the amplitude of the wave gradually changes as it propagates. In fact, the amplitude of the electric field component is inversely proportional to $n^{1/2}$, whereas the amplitude of the magnetic field component is directly proportional to $n^{1/2}$. Note, however, that the energy flux in the $z$-direction, which is given by the the Poynting vector $-(E_y\, B_x^{\,\ast} +E_y^{\,\ast} \,B_x)/(4\,\mu_0)$, remains constant (assuming that $n$ is predominately real).

Of course, the WKB solutions (6.17) and (6.18) are only approximations. In reality, a wave propagating through a medium in which the refractive index is a slowly varying function of position is subject to a small amount of reflection. However, it is easily demonstrated that the ratio of the reflected amplitude to the incident amplitude is of order $(dn/dz)/(k_0\,n^2)$ (Budden 1985). Thus, as long as the refractive index varies on a much longer lengthscale than the wavelength of the radiation, the reflected wave is negligibly small. This conclusion remains valid as long as the inequality (6.16) is satisfied. This inequality obviously breaks down in the vicinity of a point where $n^2=0$. We would, therefore, expect strong reflection of the incident wave from such a point. Furthermore, the WKB solutions also break down at a point where $n^2\rightarrow\infty$, because the amplitude of $B_x$ becomes infinite.