Dielectric mirror

 

The reflectivity of a dielectric mirror is based on the interference of light reflected from the different layers of a dielectric stack. This is the same principle used in multi-layer anti-reflection coatings, which are dielectric stacks which have been designed to minimize rather than maximize reflectivity. Simple dielectric mirrors function like one-dimensional photonic crystals, consisting of a stack of layers with a high refractive index interleaved with layers of a low refractive index (see diagram). The thicknesses of the layers are chosen such that the path-length differences for reflections from different high-index layers are integer multiples of the wavelength for which the mirror is designed. The reflections from the low-index layers have exactly half a wavelength in path length difference, but there is a 180-degree difference in phase shift at a low-to-high index boundary, compared to a high-to-low index boundary, which means that these reflections are also in phase. In the case of a mirror at normal incidence, the layers have a thickness of a quarter wavelength.

 

Other designs have a more complicated structure generally produced by numerical optimization. In the latter case, the phase dispersion of the reflected light can also be controlled (a chirped mirror). In the design of dielectric mirrors, an optical transfer-matrix method can be used. A well-designed multilayer dielectric coating can provide a reflectivity of over 99% across the visible light spectrum.^[1]

Dielectric mirrors exhibit retardance as a function of angle of incidence and mirror design.^[2]

As shown in the GIF, the transmitted color shifts towards the blue with increasing angle of incidence. Regarding interference in the high reflective index ${\displaystyle n_{1}}$  medium this blueshift is given by the formula

${\displaystyle 2n_{2}d_{2}\cos {\big (}\theta _{2})=m\lambda }$ ,

where ${\displaystyle m\lambda }$  is any multiple of the transmitted wavelength and ${\displaystyle \theta _{2}}$  is the angle of incidence in the second medium. See thin-film interference for a derivation. However, there is also interference in the low refractive index medium. The best reflectivity will be at ^[3]

${\displaystyle \lambda _{\perp }/\lambda _{\theta }={\frac {cos(\theta _{2})+cos(\theta _{1})}{2}}\approx {\sqrt {1-{\frac {sin^{2}(\theta )}{n^{2}}}}}}$ ,

where ${\displaystyle \lambda _{\perp }}$  is the transmitted wavelength under perpendicular angle of incidence and

${\displaystyle n={\sqrt {\frac {2}{n_{1}^{-2}+n_{2}^{-2}}}}}$ .

 

The manufacturing techniques for dielectric mirrors are based on thin-film deposition methods. Common techniques are physical vapor deposition (which includes evaporative deposition and ion beam assisted deposition), chemical vapor deposition, ion beam deposition, molecular beam epitaxy, sputter deposition, and sol-gel deposition.^[4] Common materials are magnesium fluoride (n = 1.37), silicon dioxide (n = 1.45), tantalum pentoxide (n = 2.28) , zinc sulfide (n = 2.32), and titanium dioxide (n = 2.4).

Polymeric dielectric mirrors are fabricated industrially via co-extrusion of melt polymers,^[5] and by spin-coating^[6] or dip-coating^[7] on smaller scale.

• Distributed Bragg reflector
• Dichroic filter
• Perfect mirror
• Rugate filter

 

Wikimedia Commons has media related to Dielectric mirrors.

• Fast code for computation of dielectric mirror reflectivity and dispersion