PX284 - U2 - thin film interference

considering 'weak reflections', ie. light is mostly transmitted
the reflected rays need to add coherently
therefore, considering only two reflections
at normal incidence:

\begin{matrix} r_{12} = \frac{n_{1} - n_{2}}{n_{1} + n_{2}} \\ r_{23} = \frac{n_{2} - n_{3}}{n_{2} + n_{3}} \end{matrix}

total reflection coefficient (coherent sum):

r = r_{12} + r_{23} \exp (i 2 d k_{2})

where, $k_{2}$ is the wavenumber in material 2

using a 'quarter-wave' coating: $d = λ_{2} / 4$ , so that $\exp (2 d k_{2}) = - 1$

∴ r = r_{12} - r_{23}

typical anti-reflective coating:

n_{2} < n_{2} < n_{3}

considering air: $n_{1} \sim 1$ , glass: $n_{3} \sim 1.5$ ; $r = 0$ if $r_{12} = r_{23} ⟹ n_{2}^{2} = n_{1} n_{3}$
often used material is MgF $_{2}$ with $n ≃ 1.38$
this reduces reflection losses $(\propto | r |^{2})$ , typically from $\sim 4 %$ to $\sim 1 %$ , depending on $λ$

dielectric mirror

for dielectric mirrors $(n_{1} < n_{3} < n_{2})$ $⟹ r_{12} < 0 (n_{2} < n_{2})$ and $r_{23} > 0 (n_{2} > n_{3})$

r = r_{12} + r_{23} \exp (i 2 k_{2} d)

for $d = λ_{2} / 4 :$ $r_{12} - r_{23}$
ie. add constructively due to opposite signs
eg: TiO $_{2}$ $n_{2} \sim 2.6$ used to increase reflectance of glass from $4 %$ to $\sim 40 %$ for one layer of coating or even lighter for multiple layers

soap films

soap (surfactant) stabilises surface, but does not affect optics
a layer of water $n_{2} = 1.33$ between air $n_{1} = n_{3} = 1$

\begin{matrix} r_{12} = - 0.14 \\ r_{23} = + 0.14 \\ r = r_{12} + r_{23} \exp (i 2 k_{2} d) = - 0.14 (1 + \exp (i 2 k_{2} d)) \end{matrix}

soap film under gravity:
- top: $d$ small, $\exp (i 2 k d ≃ 1)$ $⟹ r ≃ 0$ ie. transparent
- middle: $\exp (i 2 k d) \neq 1 \forall λ$ $⟹$ all visible light reflected
- bottom: $\exp (i 2 k d) = 1$ for some wavelengths, other wavelengths reflected $⟹$ colors in bands
basically, gravity causes $d$ to increase

angle dependence

phase difference between $A B C$ and $A D =$ difference in distance $\times$ wavelength:

\begin{matrix} A B C = \frac{2 d}{\cos t} \\ A D = 2 x \sin i = 3 d \frac{\sin t}{\cos t} \sin i \\ Δ ϕ = \frac{2 d}{\cos t} (k_{2} - k_{1} \sin t \sin i) \\ = \frac{2 d k_{2}}{\cos t} (1 - \sin^{2} t) = 2 d k_{2} \cos t \end{matrix}

where, $k_{1} \sin i = k_{2} \sin t$ (snell's law)

destructive interference when $2 d k_{2} \cos t = π$

λ_{2} = \frac{λ_{0}}{n_{2}} = 4 d \cos t