3.3. Plane Electromagnetic Waves

Harmonic decomposition of the electromagnetic field. The EM field can be written as the superposition of plane waves (Fourier decomposition)

\[\begin{split}\begin{aligned} \mathbf{e}(\mathbf{r},t) & = \mathbf{E} e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} \\ \mathbf{b}(\mathbf{r},t) & = \mathbf{B} e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} \\ \end{aligned}\end{split}\]

Introducing this decomposition into Maxwell’s equations with no free charge and current

\[\begin{split} \begin{cases} \nabla \cdot \mathbf{d} = 0 \\ \nabla \times \mathbf{e} + \partial_t \mathbf{b} = \mathbf{0} \\ \nabla \cdot \mathbf{b} = 0 \\ \nabla \times \mathbf{h} - \partial_t \mathbf{d} = \mathbf{0} \end{cases} \end{split}\]

we obtain

\[\begin{split} \begin{cases} i \mathbf{k} \cdot \mathbf{D} = 0 \\ i \mathbf{k} \times \mathbf{E} - i \omega \mathbf{B} = \mathbf{0} \\ i \mathbf{k} \cdot \mathbf{B} = 0 \\ i \mathbf{k} \times \mathbf{H} + i \omega \mathbf{D} = \mathbf{0} \end{cases} \quad \rightarrow \quad \begin{cases} i \varepsilon \mathbf{k} \cdot \mathbf{E} = 0 \\ i \mathbf{k} \times \mathbf{E} - i \omega \mathbf{B} = \mathbf{0} \\ i \mathbf{k} \cdot \mathbf{B} = 0 \\ i \dfrac{1}{\mu} \mathbf{k} \times \mathbf{B} + i \omega \varepsilon \mathbf{E} = \mathbf{0} \end{cases} \end{split}\]

• From Gauss’ equations for the electric and the magnetic field

  \[\mathbf{k} \perp \mathbf{E} \quad , \quad \mathbf{k} \perp \mathbf{B}\]

• From Faraday and Ampère-Maxwell equations

  \[\mathbf{B} = \dfrac{\mathbf{k}}{\omega} \times \mathbf{E}\]
  \[\mathbf{E} = - \dfrac{1}{\mu \varepsilon}\dfrac{\mathbf{k}}{\omega} \times \mathbf{B}\]

It follows that:

• \(\mathbf{k}\), \(\mathbf{E}\), \(\mathbf{B}\) are orthogonal “RHS” set of vectors

• Relations between \(\mathbf{E}\), \(\mathbf{B}\), and \(\mathbf{k}\) and the speed of light

  \[\begin{split}\begin{aligned} \mathbf{B} & = \dfrac{1}{c} \, \hat{\mathbf{k}} \times \mathbf{E} \\ \mathbf{E} & = - c \, \hat{\mathbf{k}} \times \mathbf{B} \\ \end{aligned}\end{split}\]

  hold, with speed of light \(c = \dfrac{1}{\sqrt{\mu \varepsilon}} = \dfrac{\omega}{|\mathbf{k}|}\), and unit vector \(\hat{\mathbf{k}} = \dfrac{\mathbf{k}}{|\mathbf{k}|}\).

Proof using vector algebra identity

Recalling \(c^2 = \frac{1}{\mu \varepsilon}\) and

\[\mathbf{B} = \dfrac{\mathbf{k}}{\omega} \times \mathbf{E} = \dfrac{\mathbf{k}}{\omega} \times \left[ - c^2 \dfrac{\mathbf{k}}{\omega} \times \mathbf{B} \right] = - \dfrac{c^2 |\mathbf{k}|^2}{\omega^2} \hat{\mathbf{k}} \times \left( \hat{\mathbf{k}} \times \mathbf{B} \right)\]

Vector identity

\[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \varepsilon_{ijk} a_j \varepsilon_{klm} b_l c_m = \left( \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} \right) a_j \, b_l \, c_m = b_i a_m c_m - c_i a_m b_m = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}\]

applied to \(\hat{\mathbf{k}} \times \left( \hat{\mathbf{k}} \times \mathbf{B} \right)\) gives

\[ \hat{\mathbf{k}} \times \left( \hat{\mathbf{k}} \times \mathbf{B} \right) = \underbrace{\left( \hat{\mathbf{k}} \dot \mathbf{B} \right)}_{=0 \text{ since $\mathbf{k} \perp \mathbf{B}$}} \hat{\mathbf{k}} - \underbrace{\left( \hat{\mathbf{k}} \cdot \hat{\mathbf{k}} \right)}_{= 1} \mathbf{B} = - \mathbf{B} \ , \]

and the original relation gives

\[\mathbf{B} = \mathbf{B} \dfrac{c^2 |\mathbf{k}|^2}{\omega^2} \ ,\]

and the relation between pulsation \(\omega\), wave vector \(\mathbf{k}\) and speed of light (EM radiation) \(c\),

\[c = \dfrac{\omega}{|\mathbf{k}|} \ .\]

3.3.1. Snell’s Law at an Interface

Snell’s law is derived here assuming isotropic linear media, so that

\[\begin{split}\begin{cases} \mathbf{d}(\mathbf{r},t) = \varepsilon \mathbf{e}(\mathbf{r},t) \\ \mathbf{b}(\mathbf{r},t) = \mu \mathbf{h}(\mathbf{r},t) \end{cases}\end{split}\]

and for harmonic plane EM waves

\[\begin{split}\begin{cases} \mathbf{e}(\mathbf{r}, t) = \mathbf{E}_{a} \, e^{i \left( \mathbf{k}_a \cdot \mathbf{r} - \omega t \right)} \\ \mathbf{b}(\mathbf{r}, t) = \mathbf{B}_{a} \, e^{i \left( \mathbf{k}_a \cdot \mathbf{r} - \omega t \right)} \\ \end{cases}\end{split}\]

\[\begin{split}\begin{aligned} \mathbf{B}_a & = \dfrac{1}{c} \, \hat{\mathbf{k}}_a \times \mathbf{E}_a \\ \mathbf{E}_a & = - c \, \hat{\mathbf{k}}_a \times \mathbf{B}_a \\ \end{aligned}\end{split}\]

being index \(a\) representing the media involved: \(a = 1\) for the medium with incident and reflected waves, \(a = 2\) for the medium with the refracted wave.

Jump conditions of electromagnetic field at an interface with no charge or current surface density are given by conditions (2.2),

\[\begin{split}\begin{cases} \varepsilon_1 e_{n,1} = \varepsilon_2 e_{n,2} \\ e_{t_{\alpha},1} = e_{t_{\alpha},2} & \quad , \quad \alpha=1:2 \\ b_{n,1} = b_{n,2} \\ \dfrac{1}{\mu_1} b_{t_{\alpha},1} = \dfrac{1}{\mu_2} b_{t_{\alpha},2} & \quad , \quad \alpha=1:2 \\ \end{cases}\end{split}\]

Definition of some vectors: \(\hat{\mathbf{n}}\) unit normal vector, \(\mathbf{k}\) wave vector, \(\hat{\mathbf{b}} = \dfrac{\hat{\mathbf{n}} \times \mathbf{k}}{|\hat{\mathbf{n}} \times \mathbf{k}|}\) (singular only for normal incident ray), \(\hat{\mathbf{c}} = \dfrac{\hat{\mathbf{b}} \times \mathbf{k}}{|\hat{\mathbf{b}} \times \mathbf{k}|}\), \(\hat{\mathbf{t}} = \dfrac{\hat{\mathbf{b}} \times \hat{\mathbf{n}}}{|\hat{\mathbf{b}} \times \hat{\mathbf{n}}|}\)

Incident angle \(\theta_{1,i}\) is the angle between \(\hat{\mathbf{n}}\) and \(\mathbf{k}\), s.t. \(\hat{\mathbf{n}} \times \mathbf{k} = \hat{\mathbf{b}} \, k \, \sin \theta_{1,i}\).

\[\begin{split}\begin{cases} \hat{\mathbf{k}} = \quad \cos \theta_{1,i} \hat{\mathbf{n}} + \sin \theta_{1,i} \hat{\mathbf{t}} \\ \hat{\mathbf{c}} = -\sin \theta_{1,i} \hat{\mathbf{n}} + \cos \theta_{1,i} \hat{\mathbf{t}} \end{cases} \quad , \quad \begin{cases} \hat{\mathbf{n}} = \cos \theta_{1,i} \hat{\mathbf{k}} - \sin \theta_{1,i} \hat{\mathbf{c}} \\ \hat{\mathbf{t}} = \sin \theta_{1,i} \hat{\mathbf{k}} + \cos \theta_{1,i} \hat{\mathbf{c}} \end{cases}\end{split}\]

The electromagnetic field can be written as

\[\begin{split}\begin{aligned} \mathbf{E} & = E_b \hat{\mathbf{b}} + E_c \hat{\mathbf{c}} = \\ & = E_b \hat{\mathbf{b}} - E_c \sin \theta_{1,i} \hat{\mathbf{n}} + E_c \cos \theta_{1,i} \hat{\mathbf{t}} \\ \mathbf{B} & = B_b \hat{\mathbf{b}} + B_c \hat{\mathbf{c}} = \\ & = \frac{E_c}{c} \hat{\mathbf{b}} - \frac{E_b}{c} \hat{\mathbf{c}} = \\ & = \frac{E_c}{c} \hat{\mathbf{b}} + \frac{E_b}{c} \sin \theta_{1,i} \hat{\mathbf{n}} - \frac{E_b}{c} \cos \theta_{1,i} \hat{\mathbf{t}} \ . \end{aligned}\end{split}\]

so that jump relations become

\[\begin{split}\begin{cases} b: & \quad E_{b,1} = E_{b,2} \\ n: & \quad \dots \\ t: & \quad \dots \\ \end{cases} \quad , \quad \begin{cases} b: & \quad \dots \\ n: & \quad \frac{E_{b,1}}{c_1} \sin \theta_{1,i} = \frac{E_{b,2}}{c_2} \sin \theta_{2,i} \\ t: & \quad \dots \\ \end{cases}\end{split}\]

thus Snell’s law follows

\[\frac{\sin \theta_{1,i}}{\sin \theta_{2,t}} = \frac{c_2}{c_1} = \frac{n_1}{n_2} \ .\]

Incident, Reflected, and Refracted Wave. The wave at the interface in medium 1 has the contribution of the incoming incident wave and the reflected one.

\[\begin{split}\begin{aligned} \mathbf{e}_1(\mathbf{r},t) & = \mathbf{e}_i(\mathbf{r},t) + \mathbf{e}_r(\mathbf{r},t) = \\ & = \mathbf{E}_{i} e^{i \left( \mathbf{k}_i \cdot \mathbf{r} - \omega t \right)} + \mathbf{E}_{r} e^{i \left( \mathbf{k}_r \cdot \mathbf{r} - \omega t \right)} = \\ & = \left( \mathbf{E}_{i} e^{i \mathbf{k}_i \cdot \mathbf{r}} + \mathbf{E}_{r} e^{i \mathbf{k}_r \cdot \mathbf{r} } \right) e^{-i \omega t} \end{aligned}\end{split}\]

with

\[\begin{split}\begin{aligned} \mathbf{k}_i & = k_{i,n} \hat{\mathbf{n}} + k_{i,t} \hat{\mathbf{t}} \\ \mathbf{k}_r & = k_{r,n} \hat{\mathbf{n}} + k_{r,t} \hat{\mathbf{t}} \\ \end{aligned}\end{split}\]

At the interface, \(\mathbf{r}_s \cdot \hat{\mathbf{n}} = 0\), and thus

\[\begin{split}\begin{aligned} \mathbf{e}_1(\mathbf{r}_s, t) & = \left( \mathbf{E}_i e^{i k_{i,t} x_t} + \mathbf{E}_r e^{i k_{r,t} x_t} \right) e^{-i\omega t} \\ \mathbf{e}_2(\mathbf{r}_s, t) & = \mathbf{E}_t e^{i k_{t,t} x_t} e^{-i \omega t} \\ \end{aligned}\end{split}\]

In order for the boundary conditions to be satisfied at all the points of the interface at each time,

\[k_{i,t} = k_{r,t} = k_{t,t} \ .\]

Exploiting the relation between the pulsation, the wave-length, and the speed of light in media, \(c_a = \frac{\omega}{|\mathbf{k}_a|} = \frac{c}{n_a}\),

\[|\mathbf{k}_i| = |\mathbf{k}_r| \qquad \rightarrow \qquad k_{r,n} = - k_{i,n}\]

\[\frac{|\mathbf{k}_2|}{|\mathbf{k}_1|} = \frac{c_1}{c_2}\]

\[\frac{k_{t,t}^2 + k_{t,n}^2}{k_{i,t}^2 + k_{i,n}^2} = \frac{c_1^2}{c_2^2}\]

\[\begin{split} \begin{aligned} k_{i,n} & = \ \ \ |\mathbf{k}_i| \, \cos \theta_i \\ k_{r,n} & = - |\mathbf{k}_r| \, \cos \theta_r \\ k_{t,n} & = \ \ \ |\mathbf{k}_t| \, \cos \theta_t \\ \end{aligned} \quad , \quad \begin{aligned} k_{i,t} & = \ \ \ |\mathbf{k}_i| \, \sin \theta_i \\ k_{r,t} & = \ \ \ |\mathbf{k}_r| \, \sin \theta_r \\ k_{t,t} & = \ \ \ |\mathbf{k}_t| \, \sin \theta_t \\ \end{aligned} \end{split}\]

\[\begin{split}\begin{cases} E_n: & \quad \varepsilon_1 \left( E_{i,c} \sin \theta_i + E_{r,c} \sin \theta_r \right) = \varepsilon_2 E_{t,c} \sin \theta_{t} \\ E_t: & \quad E_{i,c} \cos \theta_i - E_{r,c} \cos \theta_r = E_{t,c} \cos \theta_{t} \\ E_b: & \quad E_{i,b} + E_{r,b} = E_{t,b} \\ B_n: & \quad B_{i,c} \sin \theta_i + B_{r,c} \sin \theta_r = B_{t,c} \sin \theta_{t} \\ B_t: & \quad \frac{1}{\mu_1} \left( B_{i,c} \cos \theta_i - B_{r,c} \cos \theta_r \right) = \frac{1}{\mu_2} B_{t,c} \cos \theta_{t} \\ B_b: & \quad \frac{1}{\mu_1} \left( B_{i,b} + B_{r,b} \right) = \frac{1}{\mu_2} B_{t,b} \\ \end{cases}\end{split}\]

Writing the magnetic field as a function of the wave-vector and the magnetic field, it’s possible to write 2 decoupled systems of equations

\[\begin{split}\begin{cases} E_n: & \quad \varepsilon_1 \left( E_{i,c} \sin \theta_i + E_{r,c} \sin \theta_r \right) = \varepsilon_2 E_{t,c} \sin \theta_{t} \\ E_t: & \quad E_{i,c} \cos \theta_i - E_{r,c} \cos \theta_r = E_{t,c} \cos \theta_{t} \\ B_b: & \quad \frac{1}{\mu_1} \left( \frac{E_{i,c}}{c_1} + \frac{E_{r,c}}{c_1} \right) = \frac{1}{\mu_2} \frac{E_{t,c}}{c_2} \\ \end{cases}\end{split}\]

\[\begin{split}\begin{cases} E_b: & \quad E_{i,b} + E_{r,b} = E_{t,b} \\ B_n: & \quad \frac{E_{i,b}}{c_1} \sin \theta_i + \frac{E_{r,b}}{c_1} \sin \theta_r = \frac{E_{t,b}}{c_2} \sin \theta_{t} \\ B_t: & \quad \frac{1}{\mu_1} \left( \frac{E_{i,b}}{c_1} \cos \theta_i - \frac{E_{r,b}}{c_1} \cos \theta_r \right) = \frac{1}{\mu_2} \frac{E_{t,b}}{c_2} \cos \theta_{t} \\ \end{cases}\end{split}\]

The equations \(E_n\) and \(B_b\) are equivalent; \(E_b\) and \(B_n\) are equivalent as well, because of Snell’s law. Thus, defining

\[\begin{split} \begin{aligned} r_c & := \dfrac{E_{r,c}}{E_{i,c}} \\ t_c & := \dfrac{E_{t,c}}{E_{i,c}} \\ \end{aligned} \quad , \quad \begin{aligned} r_b & := \dfrac{E_{r,b}}{E_{i,b}} \\ t_b & := \dfrac{E_{t,b}}{E_{i,b}} \\ \end{aligned} \end{split}\]

and \(\alpha_i := \frac{1}{\mu_i c_i}\). These systems of equations can be written as two uncoupled linear systems of equations,

(for P-polarization todo *change index from \(c\) to \(p\); for S-polarization todo change index from \(b\) to \(s\))

\[\begin{split} & \begin{cases} E_t: & \quad \cos \theta_i - \cos \theta_r \, r_c = \cos \theta_{t} \, t_c \\ B_b: & \quad \alpha_1 + \alpha_1 \, r_c = \alpha_2 \, t_c \\ \end{cases} \\ & \begin{cases} E_b: & \quad 1 + r_b = t_b \\ B_t: & \quad \alpha_1 \, \cos \theta_i - \alpha_1 \, \cos \theta_r \, r_b = \alpha_2 \, \cos \theta_{t} \, t_b \\ \end{cases} \end{split}\]

Calling \(\theta_i = \theta_r = \theta_1\), \(\theta_2 = \theta_t\), these linear systems can be written using matrix formalism,

\[\begin{split} & \begin{bmatrix} -1 & 1 \\ 1 & \frac{\alpha_2}{\alpha_1} \frac{\cos \theta_2}{\cos \theta_1} \end{bmatrix} \begin{bmatrix} r_b \\ t_b \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \\ & \begin{bmatrix} 1 & \frac{\cos \theta_2}{\cos \theta_1} \\ -1 & \frac{\alpha_2}{\alpha_1} \end{bmatrix} \begin{bmatrix} r_c \\ t_c \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \end{split}\]

todo Analysis of the total reflection, forcing \(t_x = 0\). Check signs before

\[\begin{split} \begin{bmatrix} 1 & \frac{\cos \theta_2}{\cos \theta_1} \\ -1 & \frac{\alpha_2}{\alpha_1} \end{bmatrix} \begin{bmatrix} r_c \\ t_c \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \qquad \rightarrow \qquad \begin{bmatrix} r_c \\ t_c \end{bmatrix} = \dfrac{1}{\frac{\alpha_2}{\alpha_1} + \frac{\cos \theta_2}{\cos \theta_1}} \begin{bmatrix} \frac{\alpha_2}{\alpha_1} & - \frac{\cos \theta_2}{\cos \theta_1} \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{\alpha_2 \cos \theta_1 - \alpha_1 \cos \theta_2}{\alpha_2 \cos \theta_1 + \alpha_1 \cos \theta_2} \\ \frac{2 \alpha_1 \cos \theta_1}{\alpha_2 \cos \theta_1 + \alpha_1 \cos \theta_2} \end{bmatrix} \end{split}\]

\[\begin{split} \begin{bmatrix} -1 & 1 \\ 1 & \frac{\alpha_2}{\alpha_1} \frac{\cos \theta_2}{\cos \theta_1} \end{bmatrix} \begin{bmatrix} r_b \\ t_b \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \qquad \rightarrow \qquad \begin{bmatrix} r_b \\ t_b \end{bmatrix} = \dfrac{1}{-\frac{\alpha_2}{\alpha_1} \frac{\cos \theta_2}{\cos \theta_1} - 1} \begin{bmatrix} \frac{\alpha_2}{\alpha_1} \frac{\cos \theta_2}{\cos \theta_1} & -1 \\ -1 & -1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{\alpha_1 \cos \theta_1 - \alpha_2 \cos \theta_2}{\alpha_1 \cos \theta_1 + \alpha_2 \cos \theta_2} \\ \frac{2 \alpha_1 \cos \theta_1}{\alpha_1 \cos \theta_1 + \alpha_2 \cos \theta_2} \end{bmatrix} \end{split}\]

that can be recast with the wave impedance \(Z\),

\[\alpha_1 = \frac{1}{\mu_1 c_1} = \frac{\sqrt{\mu_1 \varepsilon_1}}{\mu_1} = \sqrt{\dfrac{\varepsilon_1}{\mu_1}} =: \frac{1}{Z_1} \ ,\]

\[\begin{split} \begin{bmatrix} r_c \\ t_c \end{bmatrix} = \begin{bmatrix} \frac{Z_1 \cos \theta_1 - Z_2 \cos \theta_2}{Z_1 \cos \theta_1 + Z_2 \cos \theta_2} \\ \frac{2 Z_2 \cos \theta_1}{Z_1 \cos \theta_1 + Z_2 \cos \theta_2} \end{bmatrix} \end{split}\]

\[\begin{split} \begin{bmatrix} r_b \\ t_b \end{bmatrix} = \begin{bmatrix} \frac{Z_2 \cos \theta_1 - Z_1 \cos \theta_2}{Z_2 \cos \theta_1 + Z_1 \cos \theta_2} \\ \frac{2 Z_2 \cos \theta_1}{Z_2 \cos \theta_1 + Z_1 \cos \theta_2} \end{bmatrix} \end{split}\]

Energy Balance and Transmission Coefficients. Energy balance for a domain collapsing on the interface reduces to power flux balance, namely

\[\oint_{\partial V} \mathbf{s} \cdot \hat{\mathbf{n}} = 0 \ ,\]

with \(\mathbf{s} = \mathbf{e} \times \mathbf{h}\) the Poynting vector. For harmonic plane waves,

\[\begin{split}\begin{aligned} \mathbf{s}(\mathbf{r},t) & = \mathbf{e}(\mathbf{r},t) \times \mathbf{h}(\mathbf{r},t) = \\ & = \frac{1}{\mu} \left[ \mathbf{E} e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} + \mathbf{E}^* e^{-i(\mathbf{k} \cdot \mathbf{r} - \omega t)} \right] \times \left[ \mathbf{B} e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} + \mathbf{B}^* e^{-i(\mathbf{k} \cdot \mathbf{r} - \omega t)} \right] = \\ & = \frac{1}{\mu} \left[ \, \mathbf{E} \times \mathbf{B} \, e^{i 2(\mathbf{k} \cdot \mathbf{r} - \omega t)} + c.c. \, \right] + \frac{1}{\mu} \left[ \, \mathbf{E} \times \mathbf{B}^* + c.c. \, \right] = \\ & = \dots + \frac{1}{\mu} \mathbf{E} \times \left( \frac{1}{c} \hat{\mathbf{k}} \times \mathbf{E} \right)^* = \\ & = \dots + \frac{1}{\mu c} \left( \mathbf{E} \cdot \mathbf{E}^* \right) \hat{\mathbf{k}} = \\ & = \dots + \frac{1}{\mu c} | \mathbf{E} |^2 \hat{\mathbf{k}} \ . & = \dots + \alpha | \mathbf{E} |^2 \hat{\mathbf{k}} \ . \end{aligned}\end{split}\]

For each one of the two polarizations, the following holds (\(\cos \theta\) comes from the dot product \(\hat{k} \cdot \hat{n}\) appearing in the surface integral),

\[\alpha_1 \cos \theta_1 = \alpha_1 r_x^2 \, \cos \theta_1 + \alpha_2 t_x^2 \, \cos \theta_2 \ ,\]

i.e., the sum of reflected and transmitted power equals the incident power.

Proof of the power balance, for P-polarization

todo Here \(P\) is index \(c\)

Dividing by \(\alpha_1 \cos \theta_1\)

\[\begin{split}\begin{aligned} & \frac{1}{\alpha_1 \cos \theta_1} \left( \alpha_1 r_p^2 \, \cos \theta_1 + \alpha_2 t_p^2 \, \cos \theta_2 \right) = \\ & = \frac{\left(\alpha_1 \cos \theta_1 - \alpha_2 \cos \theta_2\right)^2}{\left(\alpha_1 \cos \theta_1 + \alpha_2 \cos \theta_2\right)^2} + \frac{\alpha_2 \cos \theta_2}{\alpha_1 \cos \theta_1} \frac{\left( 2 \alpha_1 \cos \theta_1 \right)^2}{\left( \alpha_1 \cos \theta_1 + \alpha_2 \cos \theta_2 \right)^2} = \\ & = \dfrac{1}{\left( \alpha_1 \cos \theta_1 + \alpha_2 \cos \theta_2 \right)^2} \left[ \alpha_1^2 \cos^2 \theta_1 - 2 \alpha_1 \alpha_2 \cos \theta_1 \cos \theta_2 + \alpha_2^2 \cos^2 \theta_2 + 4 \alpha_1 \alpha_2 \cos \theta_1 \cos \theta_2 \right] = \\ & = 1 \ . \end{aligned}\end{split}\]