Let us consider the situation depicted below,
where we define the Fresnel coefficients as the ratio of the complex amplitude of the electric fields, r = Er/Ei for the reflection coefficient, and t = Et/Ei for the transmission coefficient. The Fresnel coefficients can be obtained by considering the continuity of the tangential component of the E and H fields at the interface. The continuity of the normal components of D and B does not yield supplementary conditions and will not be considered. It is worth noting that the continuity of Hy is only valid for materials that do not sustain a surface current resulting from the addition of external charges.
For TE-polarised light, the continuity of Ey reads, Ei + Er = Et which yields, 1 + r = t. The continuity of Hx can be written as, Hicos θi − Hrcos θi = Htcos θt It is convenient to express the angles in terms of the normal component of the k-vectors, kz = nk0cos θ. for the incident, reflected, and refracted fields, so that $\cos\theta_t / \cos\theta_i=\dfrac{k_{z2}n_1}{k_{z1}n_2}$. The complex magnitude of the magnetic and electric fields are linked in each medium by the optical impedance. Its expression can be obtained from the induction law in a homogeneous medium, ∇ × E = −∂tB expressed in Fourier space as, k × E = iωB The ratio E/H is the impedance, written as, $$ \frac{E}{H}= c \mu \mu_0 / n= \sqrt{\frac{\mu\mu_0}{\varepsilon\varepsilon_0}}=Z\cdot Z_0. $$ We can define the admittance Y as the inverse impedance, $Y=\dfrac{1}{Z\cdot Z_0}$ The continuity of Hy can be expressed as, $$ Y_1 (1 - r) = Y_2 \frac{n_1k_{z2}}{n_2k_{z1}}t. $$ After rearranging, we obtain, $$ 1 - r = \frac{\mu_1k_{z2}}{\mu_2k_{z1}}t. $$ We can summarize the two continuity relations in the following system,
$$\left\{ \begin{aligned} 1 + r &= t\\ 1 - r &= \frac{\mu_1 k_{z2}}{\mu_2 k_{z1}}t \end{aligned}\right. $$ Solving for r and t yields the result, $$ t_{12}^s=\frac{2\mu_2 k_{z1}}{\mu_2 k_{z1}+\mu_1k_{z2}},\qquad r_{12}^s=\frac{\mu_2 k_{z1}-\mu_1k_{z2}}{\mu_2 k_{z1}+\mu_1k_{z2}}. $$
For TM-polarised light, it is generally easier to consider the Fresnel coefficients for the magnetic field, denoted ρ and τ. The continuity of Hy reads, 1 − ρ = τ The continuity of Ey can be written, $$ \frac{k_{z1}}{\varepsilon_1} + \rho \frac{k_{z1}}{\varepsilon_1} = \tau \frac{k_{z2}}{\varepsilon_2} $$
We can summarize the two continuity relations in the following system,
$$\left\{ \begin{aligned} 1 - \rho &=\tau\\ (1 + \rho) \frac{k_{z1}}{\varepsilon_1}&=\frac{k_{z2}}{\varepsilon_2}\tau \end{aligned}\right. $$ Solving for ρ and τ yields the result, $$ \tau_{12}^p=\frac{2\varepsilon_2 k_{z1}}{\varepsilon_2 k_{z1}+\varepsilon_1k_{z2}},\qquad \rho_{12}^p=\frac{\varepsilon_2 k_{z1}-\varepsilon_1k_{z2}}{\varepsilon_2 k_{z1}+\varepsilon_1k_{z2}}. $$
To summarize, for a single interface from 1 to 2 with normal along the z direction, the Fresnel coefficients read, $$ \begin{aligned} \rho_{12}^p&=\frac{\varepsilon_2 k_{z1}-\varepsilon_1k_{z2}}{\varepsilon_2 k_{z1}+\varepsilon_1k_{z2}},&{}& r_{12}^s=\frac{\mu_2 k_{z1}-\mu_1k_{z2}}{\mu_2 k_{z1}+\mu_1k_{z2}}\\ \tau_{12}^p&=\frac{2\varepsilon_2 k_{z1}}{\varepsilon_2 k_{z1}+\varepsilon_1k_{z2}},&{}& t_{12}^s=\frac{2\mu_2 k_{z1}}{\mu_2 k_{z1}+\mu_1k_{z2}}. \end{aligned} $$
Note that, rij = −rji. To verify the conservation of energy, one must consider the irradiance defined in terms of the electric field as $I=\frac{1}{2}Y|E|^2$, yielding, $$ R=\frac{I_r}{I_i}=|r|^2, \qquad T=\frac{I_t}{I_i}=\frac{Y_1}{Y_2}|t|^2. $$ and we can verify that, indeed, R + T = 1 for a single interface.
From the viewpoint of ray optics, a thin layer will support an infinite number of internal reflections (absorption and irregularities will however reduce the intensity in a physical situation). The infinite series of reflected orders can be expressed in the form a geometric sum, leading to a closed form formula as shown below.
An incident plane wave with amplitude A impinges on the first interface.
It can be reflected, B = r01A,
or transmitted. The total response of the slab can be obtained by
following each order of reflection inside the slab (C
,
D
, …).
Upon transmission at the first interface, the wave amplitude is t01A.
Application of Fermat’s principle yields a phase change Δϕ = kz1d
when the wave hits the second interface. The reflection coefficient at
this interface is r12. The partial wave
reflected from this path, noted C
, is therefore C = t10t01r12exp (2ikz1d)A.
Similarly, in D
, D = t10t01r10r122exp (4ikz1d)A
And, for the jth
partial wave, t10t01r12jr10j − 1exp (2ijkz1d)A
The wave reflected by the slab is the sum of these contributions, $$
r_{\text{slab}}A=B+C+D+\dots=\left[r_{01}+t_{10}t_{01}r_{12}\sum_{j=0}^{\infty}r_{12}^jr_{10}^{j}\exp(2ji
k_{z1} d)\right]A
$$ For clarity, I write β = r12r10exp (2ikz1d).
The summation of all partial waves is thereby expressed as a geometrical
sum, $$
r_{\text{slab}}=r_{01} + \left(t_{10}t_{01}r_{12}\exp(2i
k_{z1}d)\right)\sum_{j=0}^{\infty}\beta^j
$$ Recalling that the sum of a geometric series of common
ratio q is $\frac{1}{1-q}$, we can write, $$
r_{\text{slab}}=r_{01} + \frac{t_{10}t_{01}r_{12}\exp(2i
k_{z1}d)}{1-r_{12}r_{10}\exp(2i k_{z1}d)}
$$
We note that for either polarisation we have the following identity, $$ t^s_{ij}t^s_{ji} = (1-r^s_{ij})(1+r^s_{ij})= 1-(r^s_{ij})^2 ,\qquad t^p_{ij}t^p_{ji} =\frac{Y_i}{Y_j}(1-r^p_{ij})\frac{Y_j}{Y_i}(1+r^p_{ij}) = 1-(r^p_{ij})^2 $$ Using this equation and the substitution r10 = −r01 we finally obtain, $$ r_{\text{slab}}=\frac{r_{01}+r_{12}\exp(2i k_{z1}d)}{1+r_{01}r_{12}\exp(2i k_{z1}d)}. $$
For the transmission, one obtains,
$$ t_{\text{slab}}=\frac{t_{01}t_{12}\exp(i k_{z1}d)}{1+r_{01}r_{12}\exp(2i k_{z1}d)}. $$
When N layers are stacked together, the reflection coefficient of the structure can be found by applying recursively the preceding formula for a single layer. This amounts to considering one of the reflection coefficients to be the effective reflection accounting for all the layers behind. Let us consider explicitly the case of a 3 layer system. The procedure is as follows,