14C.5.1Variation with Ω

Numerical results were obtained for C = 0.5, d_p′ = 1, and p = 1.1. Figures 14C.2 and 14C.3 show the plot of q′ and ρ versus Ω for the two cases of slab motion. For obtaining the maxima and minima in the oscillatory ranges [20], it should be noted that ρ_min is zero and this occurs for Ω satisfying the equation

This is equivalent to the condition 2q′d′ = (n/2)λ′ (n even), which means that the optical path difference between two successive reflected waves (there being an infinite series of internal reflections inside the plasma layer) is an integral multiple of one wavelength, so that these waves have the same phase and reinforce each other and in this case the first reflected wave from the lower boundary exactly annuls the resultant of the remaining reflected waves, thus giving zero reflection [21].

Thus, one obtains, after some algebra,

and

The minus sign in the preceding equations gives the minima in the first oscillatory range, while the plus sign in the other range. There are thus an infinite number of oscillations in either range.

The maxima in ρ occur for Ω satisfying the transcendental equation

where

and

The maximum point between the two minima of a loop was first determined using a subroutine to solve the preceding transcendental equation. Having located this maximum, each interval between the minimum and ρ maximum points was divided into a number of parts and calculated at each point. This procedure was repeated to get the second loop and so on.

It is seen from the curves in Figures 14C.2 and 14C.3 that as q′ tends to infinity in its real range, the oscillations in ρ become more and more rapid and of increasing amplitude. In the other range (where q′ reaches its asymptotic limit C′), the oscillations slow down progressively as Ω is increased and ultimately die down. The rapidity of oscillations is due to steep rise in q′ with Ω in this range. The amplitude of oscillations is governed by the magnitude of q′. This is easily seen by noting that if we ignore the oscillations and draw a curve representing average value, this will give the ρ versus Ω curve for the semiinfinite case. For this case, reflection will be strong when the free-space fields are badly matched [22] to the plasma fields (q′ ≠ C′) and zero when the free-space fields are perfectly matched to the plasma fields (q′ = C′).

As is known, ρ + τ > 1 and the reflected wave emerges out amplified when the slab moves toward the incident wave. This is because the stored energy decreases and mechanical power has to be externally supplied to the medium against the radiation pressure exerted by the field on the medium. When the medium has a tangential velocity, the force on its surface can do no work and the time-averaged stored energy remains constant, thus giving ρ + τ = 1 for parallel motion [11].

14C.5.2Variation with β

It will be seen from the equations that the reflection and transmission coefficients are complicated functions of the slab velocity. To study these coefficients directly against β would be too involved. So, the following technique is evolved.

1. q′ is first studied against p and ρ/α plotted against p, since the maxima and minima of this curve are easily obtained as before.

2. p is studied against β. The parameter p (= ω′/ω) is significant in that it transforms the incident wave frequency to the rest system of the plasma. It is a quadratic in β and the two values of β for any p are given by

   (14C.22a)$${\left({\mathrm{\beta }}_{1,2}\right)}_x=\frac{S}{p_x^2+S^2}\pm {\left[{\left(\frac{S}{p_x^2+S^2}\right)}^2+\frac{p_x^2-1}{p_x^2+S^2}\right]}^{1/2}$$

   and

   (14C.22b)$${\left({\mathrm{\beta }}_{1,2}\right)}_z=\frac{C}{p_z^2+C^2}\pm {\left[{\left(\frac{C}{p_z^2+C^2}\right)}^2+\frac{p_z^2-1}{p_z^2+C^2}\right]}^{1/2}.$$

   The plot of p versus β for C = 0.5 is given in Figure 14C.4. It shows the variation of ω′ with β.

3. ρ/α is now plotted against β by combining the preceding two steps. With reference to a different horizontal axis (ρ/α = 1), this represents τ.

4. α versus β is studied according to Equation 14C.19a and plotted for C = 0.5 along with the ρ/α versus β curve.

5. Finally, ρ is obtained as a function of β using steps 3 and 4.

It should be noted that since α = 1 for parallel slab motion, the development of the ρ versus β curve in this case involves only three steps, q′ versus p gives interesting results. q′ is seen to vary in a different manner with p for various ranges of Ω given by Ω < 1/C, Ω = 1/C, and Ω > 1/C for slab motion parallel to the interface and Ω ≤ 1/S and Ω > 1/S for normal motion.

14C.5.3Slab Motion Parallel to Interface ($\overline{\mathrm{\upsilon }}={\mathrm{\upsilon }}_0\widehat{x}$)

Numerical results were obtained for C = 0.5, $d_{\text{p}}^{\prime }=1$ in the three ranges of values of aforementioned Ω. For Ω = 1/C, q′ = 0 and it can be shown that ρ_x for this case is given by

For the other two cases, namely Ω < 1/C and Ω > 1/C, curves were obtained with Ω = 1.2 and 2.5, respectively (1/C = 2).

The plot of $q_x^{\prime }$ and ρ_x against p_x has been shown in Figure 14C.5 (for Ω < 1/C). In the oscillatory range, (ρ_min)_x is zero and occurs for p_x given by

Thus, one obtains

It should also be noted that there is an additional minimum point at p_x = C. This is readily seen by setting B_x = α in Equation 14C.18c. For this condition, q′ = C′ = 1 representing a perfect match for the free-space and plasma fields and the incident wave is wholly transmitted.

The maxima in ρ_x occur for p_x satisfying the transcendental equation tan A_x = (f_p)_xA_x, where

Case 1: Ω < 1/C.

It should be noted in Equation 14C.23 that n = 0 is not a permissible even integer since this will correspond to a zero value of p_x. The lowest permissible even integer may be determined from the minimum value of p_x that equals C. This value of p_x corresponds to n = n′ = 4d_pΩC. Since this value may not be an even integer, the lowest permissible even integer is the one that is greater than n′. The highest permissible value for n is infinity when p_x = 1/Ω. Beyond the oscillatory region, ρ_x is monotonic.

The ρ_x versus p_x curve is finally developed into the ρ_x versus β curve (Figure 14C.6) with the help of p_x versus β curve that has been earlier explained. As there are two values of β for any p_x, one loop in the ρ_x versus p_x curve leads to two loops in the ρ_x versus β curve, the two loops being one on each side of β = S (corresponding to the starting point p_x = C in the ρ_x versus p_x curve).

It is seen from the curve that ρ_x is oscillatory in the range of β from 0.23 to 0.98 and for other values of β, ρ_x is monotonic. The rapidity of the oscillations (of increasing amplitude) increases as one approaches the end of the oscillatory region on both sides of β = S. The oscillations are even more rapid to the right of β = S than to the left. This is because the wavelength λ′ decreases much faster in the right range than the left range as can be seen from an inspection of the p_x versus β curve.

Case 2: Ω > 1/C

Referring to Equation 14C.23 again, it should be noted that n cannot assume all even integer values. The limits on n are set from the requirement that p_min has to lie between C and ∞. p_x = C corresponds to n = n′ = 4d_p′ΩC and p_x = ∞ corresponds to

Noting that n″ < n′, the lowest permissible value for n is an even integer which is greater than or equal to n″. The highest permissible value of n is again an even integer that is less than or equal to n′.

For the chosen data of $d_{\text{p}}^{\prime }$ = 1, C = 0.5, and Ω = 2.5, range of n as given by Equation 14C.25 is 3–5 and as only even n is allowed, the value to be used is 4. This corresponds to p_x = 0.6. Thus, we have only two minima in ρ_x, one at p_x = 0.5 and the other at p_x = 0.6, giving one loop in the region.

For any other data, the number of the oscillatory loops will change. For example, if d_p′ = 10, C = 0.5, and Ω = 2.1, we have n′ = 42 and n″ = 12.8. So n″ has to be taken as 14 (the next higher even number), giving the range of n as 14–42. This leads to 14 oscillatory loops and as n = 14 corresponds to p_x = 1.18, it will be seen that all these loops will be in the range of p_x from 0.5 to 1.18, which means intense crowding of the oscillations in the region.

As p_x tends to infinity, A_x tends to $2\mathrm{\pi }{d}_p^{\prime }{\left({\mathrm{\Omega }}^2{C}^2-1\right)}^{1/2}$ and B_x tends to 4Ω²C²(Ω²C² – 1) (refer Equation 14C.18). Thus, as p_x increases beyond the oscillatory region, ρ_x rises asymptotically to a value given by

and

Only the final plot of ρ_x versus β is shown (Figure 14C.7), from which it is seen that after some oscillation, the reflection coefficient increases to its final value on either side of β = S, the rise to the right of β = S being extremely steep when compared to the slow rise in the left region. A physical explanation has already been given.

14C.5.4Slab Motion Normal to Interface ($\overline{\mathrm{\upsilon }}={\mathrm{\upsilon }}_0\widehat{z}$)

Curves were obtained for Ω = 0.5 (<1/S) and Ω = 2.0 (>1/S) taking $d_{\text{p}}^{\prime }$  = 1 and C = 0.5. Figures 14C.8 and 14C.11 show the plot of q_z and ρ_z/α_z versus p_z for the two ranges of Ω. The maxima and minima in ρ_z/α_z are obtained as before and occur for p_z given by

(the minus sign before the square-root sign is not permissible for Ω > 1/S as in this case $q_z^{\prime }$ is real in one range only, that is, [S² + (1/Ω²)]^1/2 < p_z < ∞) and tan A_z = (f_p)_zA_z, where

Case 1: Ω < 1/S.

The minimum value of p_z is S and this corresponds to n = n′ = 4d_p′ΩS/(1 – Ω²S²)^1/2. Since this value may not be an even integer, the lowest permissible even integer is the one that is greater than n′. The highest permissible value of n is, of course, infinity corresponding to p_z = 1/Ω.

The development of the ρ_z versus β curve is illustrated in Figure 14C.8, Figure 14C.9 and Figure 14C.10 following the procedure described before. The significant point to be noted here is the appearance of an additional factor α_z in the expression for reflection coefficient. This is actually the reflection coefficient for a perfect moving minor [11] and the multiplier 1/(1 + B_z cosec² A_z) is because of the uniaxial anisotropy of the slab.

It is seen (Figure 14C.9) that starting from infinity α_z reduces to unity as β varies from −1 to 0. Thus the reflected wave emerges out amplified in this range (physical explanation given before). In the range 0 < β < (1 − S)/C, α_z reduces from 1 to 0. So, the reflection coefficient is less than unity in the above range, finally becoming zero at β = (1 – S)/C. In this range, the energy balance may be shown as

Thus, when the medium recedes from the incident wave, field supplies work to the medium. At β = (1 – S)/C, the angle of reflection θ_R = 90°, thus explaining the zero-reflected power (Figure 14C.11).

Oscillations in the reflection coefficient have been explained before.

Case 2: Ω ≥ 1/S.

ρ_z versus β is shown in Figure 14C.12 for Ω = 2.0 (>1/S) taking $d_{\text{p}}^{\prime }$  = 1 and C = 0.5. This may be compared with the previous curve for Ω < 1/S (Figure 14C.10) noting that the difference is because of just one oscillatory range for ρ_z in the former in contrast to the two oscillatory ranges in the latter.

The curve for Ω = 1/S is similar to the preceding and hence not given.

At β greater than or equal to C, the incident wave from below (Figure 14C.1) has no interaction with the slab since the normal component of the phase velocity of the wave is less than or equal to that of the slab. However, the results given here are applicable even when β > C if the problem is understood to be that the slab impinges on the portion of an existing incident wave above the slab. This aspect of the problem will be reported later.