Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

With $θ_{r} = θ_{i}$ and $β_{1} sin θ_{i} = β_{2} sin θ_{t}$ the three exponentials in Eq. (6.4.7) and Eq. (6.4.8) are the same function of $y$ and cancel. What is left is

E_{0}^{i} + E_{0}^{r} = E_{0}^{t}, - \frac{E _{0}^{i}}{η _{1}} cos θ_{i} + \frac{E _{0}^{r}}{η _{1}} cos θ_{i} = - \frac{E _{0}^{t}}{η _{2}} cos θ_{t} .

Define, as for normal incidence, the reflection and transmission coefficients $Γ_{⊥} = E_{0}^{r} / E_{0}^{i}$ and $τ_{⊥} = E_{0}^{t} / E_{0}^{i}$ . Dividing by $E_{0}^{i}$ , and the second equation also by $- cos θ_{i} / η_{1}$ ,

1 + Γ_{⊥} 1 - Γ_{⊥} = τ_{⊥}, = \frac{η _{1} cos θ _{t}}{η _{2} cos θ _{i}} τ_{⊥} . (6.4.12)

Adding the two equations eliminates $Γ_{⊥}$ : $2 = τ_{⊥} (η_{2} cos θ_{i} + η_{1} cos θ_{t}) / (η_{2} cos θ_{i})$ , which gives $τ_{⊥}$ , and then $Γ_{⊥} = τ_{⊥} - 1$ .

Fresnel coefficients, perpendicular (S) polarisation

Γ_{⊥} τ_{⊥} = \frac{E _{0}^{r}}{E _{0}^{i}} = \frac{η _{2} cos θ _{i} - η _{1} cos θ _{t}}{η _{2} cos θ _{i} + η _{1} cos θ _{t}}, = \frac{E _{0}^{t}}{E _{0}^{i}} = \frac{2 η _{2} cos θ _{i}}{η _{2} cos θ _{i} + η _{1} cos θ _{t}}, (6.4.13)

with $θ_{t}$ from Snell's law Eq. (6.4.10), and $τ_{⊥} = 1 + Γ_{⊥}$ .

In plain words

The formulas are those of normal incidence, $Γ = (η_{2} - η_{1}) / (η_{2} + η_{1})$ , with each impedance divided by the cosine of the angle in its own medium: the S wave sees the impedances $η_{1} / cos θ_{i}$ and $η_{2} / cos θ_{t}$ . Dividing numerator and denominator of Eq. (6.4.13) by $cos θ_{i} cos θ_{t}$ shows this at once.

Three limits check the result. At normal incidence, $θ_{i} = θ_{t} = 0$ , both coefficients reduce to those of Section 6.3. If medium 2 is a perfect conductor, $η_{2} = 0$ and $Γ_{⊥} = - 1$ , $τ_{⊥} = 0$ at every angle: the tangential $E$ , which is the whole of $E$ for an S wave, vanishes on the conductor. At grazing incidence, $θ_{i} \to 9 0^{\circ}$ , the term $η_{2} cos θ_{i}$ vanishes and $Γ_{⊥} \to - 1$ for any pair of media. Figure 6.4.4 shows the coefficients for light going from air into glass: $Γ_{⊥}$ starts at $- 0.2$ and grows steadily in magnitude, $- 0.303$ at $4 5^{\circ}$ , $- 0.734$ at $8 0^{\circ}$ , to $- 1$ at grazing incidence. Every surface becomes a mirror for S waves at grazing angles, as anyone who has looked along a window pane or a lake knows.

The S coefficients for a wave going from air into glass (n_1=1, n_2=1.5), computed from Eq. (6.4.13). \Gamma_\perp is negative at every angle and tends to -1 at grazing incidence, while \tau_\perp=1+\Gamma_\perp falls from 0.8 to zero. — **Figure 6.4.4.** The S coefficients for a wave going from air into glass ( $n_{1} = 1$ , $n_{2} = 1.5$ ), computed from Eq. (6.4.13). $Γ_{⊥}$ is negative at every angle and tends to $- 1$ at grazing incidence, while $τ_{⊥} = 1 + Γ_{⊥}$ falls from $0.8$ to zero.

6.4.4 The Fresnel coefficients for S polarisation