3.3 The coefficients of reflection (R) and transmission (T) and conservation of energy (R þ T ¼ 1)

$$\mathbf{R} = \left[\frac{\mathbf{S}\_{\text{r \ av}} \cdot \mathbf{n}}{\mathbf{S}\_{\text{i \ av}} \cdot \mathbf{n}}\right] = \frac{\mathbf{E}\_{\text{0 \ r}}^2}{\mathbf{E}\_{\text{0 \ t}}^2} \tag{29}$$

$$\mathbf{T} = \left[\frac{\mathbf{S\_{t}} \cdot \mathbf{n}}{\mathbf{S\_{i}} \,\mathrm{av} \cdot \mathbf{n}}\right] = \left(\frac{\mathbf{e\_{r}} \,\mathrm{a}}{\mathbf{e\_{r}} \,\mathrm{a}}\right)^{1/2} \frac{\mathbf{E\_{0}^{2}} \,\mathrm{c} \cos\theta\_{\mathrm{t}}}{\mathbf{E\_{0}^{2}} \,\mathrm{c} \cos\theta\_{\mathrm{i}}} = \frac{\mathbf{n}\_{2} \mathbf{E\_{0}^{2}} \,\mathrm{c} \cos\theta\_{\mathrm{t}}}{\mathbf{n}\_{1} \mathbf{E\_{0}^{2}} \,\mathrm{c} \cos\theta\_{\mathrm{i}}} \tag{30}$$

$$\mathbf{R}\_{\rm N} = \left\{ \frac{\left(\frac{\mathbf{n}\_{\rm l}}{\mathbf{n}\_{\rm l}}\right) \cos \theta\_{\rm i} - \cos \theta\_{\rm t}}{\left(\frac{\mathbf{n}\_{\rm l}}{\mathbf{n}\_{\rm l}}\right) \cos \theta\_{\rm i} + \cos \theta\_{\rm t}} \right\}^2 \tag{31}$$

$$\mathbf{T\_N} = \frac{4\left(\frac{\mathbf{n\_i}}{\mathbf{n\_2}}\right)\cos\theta\_\mathbf{i}\cos\theta\_\mathbf{t}}{\left\{\left(\frac{\mathbf{n\_i}}{\mathbf{n\_2}}\right)\cos\theta\_\mathbf{i} + \cos\theta\_\mathbf{t}\right\}^2} \tag{32}$$

#### Figure 8.

Both media are dielectric and magnetic, and the magnetic field component H<sup>i</sup> is parallel to the interface, and so the reflected and the transmitted magnetic components, Hr, Ht, are also parallel to the interface.

$$\mathbf{R}\_{\rm P} = \left\{ \frac{-\cos\Theta\_{\rm i} + \left(\frac{\mathbf{n}\_{\rm i}}{\mathbf{n}\_{\rm i}}\right)\cos\Theta\_{\rm t}}{\cos\Theta\_{\rm i} + \left(\frac{\mathbf{n}\_{\rm i}}{\mathbf{n}\_{\rm i}}\right)\cos\Theta\_{\rm t}} \right\}^2 \tag{33}$$

$$\mathrm{T\_P} = \frac{4\left(\frac{\mathrm{n\_1}}{\mathrm{n\_2}}\right)\cos\Theta\_\mathrm{i}\cos\Theta\_\mathrm{t}}{\left\{\cos\Theta\_\mathrm{i} + \left(\frac{\mathrm{n\_1}}{\mathrm{n\_2}}\right)\cos\Theta\_\mathrm{t}\right\}^2} \tag{34}$$

And obviously RN þ TN ¼ 1 , and Rp þ Tp ¼ 1. With these equations we calculate the energy content in the reflected wave and in the transmitted wave in terms of the incident wave for both cases: the so-called parallel incidence and the normal incidence. For normal incidence as in geophysics altimeter radars, θ<sup>i</sup> = θ<sup>r</sup> = θ<sup>t</sup> = 00 , and the R and T coefficients become R = {(n1 – n2)/(n1 + n2)}<sup>2</sup> and T = (4n1)/{(n1 + n2)}<sup>2</sup> .

The whole subject of reflection, refraction, and absorption of microwaves at the interface of air and a conducting (σ), magnetic (μ), and dielectric (ϵ) medium is summarized qualitatively in Figure 4. The most relevant mathematical expressions are given above. Depending on the application at hand, reflection and transmission count with sensitive methods for their measurement, and in consequence quantitative determinations of absorption of microwave power is readily available.
