Anti-reflective Coating - Theory - Interference Coatings

Interference Coatings

The use of an intermediate layer to form an anti-reflection coating can be thought of as analoguous to the technique of impedance matching of electrical signals. (A similar method is used in fibre optic research where an index matching oil is sometimes used to temporarily defeat total internal reflection so that light may be coupled into or out of a fiber.) Further reduced reflection could in theory be made by extending the process to several layers of material, gradually blending the refractive index of each layer between the index of the air and the index of the substrate.

Practical anti-reflection coatings, however, rely on an intermediate layer not only for its direct reduction of reflection coefficient, but also use the interference effect of a thin layer. Assume the layer's thickness is controlled precisely, such that it is exactly one quarter of the wavelength of light in the layer (λ/4 = λ₀/(4n₁), where λ₀ is the vacuum wavelength). The layer is then called a quarter-wave coating. For this type of coating a normally incident beam I, when reflected from the second interface, will travel exactly half its own wavelength further than the beam reflected from the first surface, leading to destructive interference. This is also true for thicker coating layers (3λ/4, 5λ/4, etc.), however the anti-reflective performance is worse in this case due to the stronger dependence of the reflectance on wavelength and the angle of incidence.

If the intensities of the two beams R₁ and R₂ are exactly equal, they will destructively interfere and cancel each other since they are exactly out of phase. Therefore, there is no reflection from the surface, and all the energy of the beam must be in the transmitted ray, T. In the calculation of the reflection from a stack of layers, the transfer-matrix method can be used.

Real coatings do not reach perfect performance, though they are capable of reducing a surface's reflection coefficient to less than 0.1%. Also, the layer will be the ideal thickness for only one distinct wavelength of light. Other difficulties include finding suitable materials for use on ordinary glass, since few useful substances have the required refractive index (n ≈ 1.23) which will make both reflected rays exactly equal in intensity. Magnesium fluoride (MgF₂) is often used, since this is hard-wearing and can be easily applied to substrates using physical vapour deposition, even though its index is higher than desirable (n = 1.38).

Further reduction is possible by using multiple coating layers, designed such that reflections from the surfaces undergo maximum destructive interference. One way to do this is to add a second quarter-wave thick higher-index layer between the low-index layer and the substrate. The reflection from all three interfaces produces destructive interference and anti-reflection. Other techniques use varying thicknesses of the coatings. By using two or more layers, each of a material chosen to give the best possible match of the desired refractive index and dispersion, broadband anti-reflection coatings which cover the visible range (400–700 nm) with maximum reflectivities of less than 0.5% are commonly achievable.

The exact nature of the coating determines the appearance of the coated optic; common AR coatings on eyeglasses and photographic lenses often look somewhat bluish (since they reflect slightly more blue light than other visible wavelengths), though green and pink-tinged coatings are also used.

If the coated optic is used at non-normal incidence (that is, with light rays not perpendicular to the surface), the anti-reflection capabilities are degraded somewhat. This occurs because the phase accumulated in the layer relative to the phase of the light immediately reflected decreases as the angle increases from normal. This is counterintuitive, since the ray experiences a greater total phase shift in the layer than for normal incidence. This paradox is resolved by noting that the ray will exit the layer spatially offset from where it entered, and will interfere with reflections from incoming rays that had to travel further (thus accumulating more phase of their own) to arrive at the inteface. The net effect is that the relative phase is actually reduced, shifting the coating, such that the anti-reflection band of the coating tends to move to shorter wavelengths as the optic is tilted. Non-normal incidence angles also usually cause the reflection to be polarization dependent.