# Anti-reflective coating

Antireflective coatings ( AR coatings for short ) are used to suppress the reflection from optical surfaces of lenses , objectives , prisms or plates and to increase transmission. In the case of lenses and eyepieces with such a coating, one speaks of a coating , in the case of glasses, viewing windows or picture tubes of an anti-reflection coating . The Ukrainian physicist Alexander Smakula is considered to be the inventor of optical coating .

## Basics

The incident beam is reflected at A and B with a phase shift. The rays r 1 and r 2 interfere destructively to zero amplitude.

The reduction in the degree of reflection on the coated surface is achieved by destructive interference of the reflected rays.

For the simplest case of a single, homogeneous coating, we consider a ray of a certain wavelength that is perpendicular (drawn diagonally in the picture for better visibility). It is partially reflected on the surface of the coating (r 1 ), and partially it passes through the layer and is then partially reflected at the next interface (r 2 ). So that the two partial beams r 1 and r 2 interfere completely destructively, their amplitudes must be the same (amplitude condition) and in phase opposition ( phase difference ) to one another (phase condition). ${\ displaystyle \ lambda _ {0}}$ ${\ displaystyle \ pi}$

From Fresnel's formulas it follows that the refractive index of the coating ${\ displaystyle n_ {1}}$

${\ displaystyle n_ {1} = {\ sqrt {n_ {0} \ cdot n_ {2}}}}$

must be so that the amplitudes of r 1 and r 2 are the same. Here, the refractive index of the substance and the refractive index of the medium in front of the surface (usually air). It is neglected here that the beam r 2 is reflected again on the surface of the coating; in fact, it is reflected back and forth endlessly. ${\ displaystyle n_ {2}}$${\ displaystyle n_ {0}}$

Reflectance for light of wavelength λ 0  = 580 nm as a function of the angle of incidence and the thickness of a magnesium fluoride layer (MgF 2 ) on a silicon dioxide substrate.

Because of the reflection at both point A and B, there is a phase jump from , or to be more precise, a change in sign of the amplitude, which has no influence on the interference. For the necessary phase difference of , the optical path length of the beam in the coating must be ${\ displaystyle n_ {0} ${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ Delta}$

${\ displaystyle \ Delta = k {\ frac {\ lambda _ {0}} {2}} \ quad {\ text {with}} \ quad k = 1,3,5, \ ldots}$

be. If you use the thinnest possible layer ( ), you get the following for the optimal layer thickness : ${\ displaystyle k = 1}$${\ displaystyle \ Delta = 2 \; d \; n_ {1}}$${\ displaystyle d}$

${\ displaystyle d = {\ frac {\ lambda _ {0}} {4n_ {1}}}}$.

If the beam hits the surface at an angle rather than perpendicular, the optical path length in the coating changes according to Snellius' law of refraction and outside due to the laterally offset exit, so that a higher optimal layer thickness results or changes. for a given d, a shortening of the appropriate wavelength. In the case of destructive interference and applies (see also): ${\ displaystyle \ alpha}$${\ displaystyle k = 1}$

${\ displaystyle {\ frac {\ lambda _ {0}} {d}} = 4 {\ sqrt {n_ {1} ^ {2} -n_ {0} ^ {2} \ sin ^ {2} \ alpha} }}$

Different wavelengths are increasingly reflected or do not interfere completely destructively. This is the reason why the (weak) reflection from coated surfaces is colored and the color depends on the angle.

## Example of a single payment

As an example we consider the perpendicular incidence of yellow-green light (550 nm wavelength) on crown glass ( ). The external medium is air ( ). Without compensation, the reflectance is according to the Fresnel equations${\ displaystyle n_ {2} = 1 {,} 5}$${\ displaystyle n_ {0} \ approx 1 {,} 0}$

${\ displaystyle R = \ left ({\ frac {1 {,} 5-1} {1 {,} 5 + 1}} \ right) ^ {2} = 0 {,} 2 ^ {2} = 4 \ , \%}$.

According to the formula in the previous section, the ideal value would be for the AR layer , but no durable material is available for this. Magnesium fluoride (MgF 2 , ) is often used . The λ / 4-layer for the yellow-green light is thus about 100 nm thick and gives a degree of reflection at normal incidence (derivation see, inter alia, Hecht or Pedrotti) of ${\ displaystyle n_ {1} = 1 {,} 22}$${\ displaystyle n_ {1} = 1 {,} 38}$

${\ displaystyle R = \ left ({\ frac {1 {,} 5 \ cdot 1-1 {,} 38 ^ {2}} {1 {,} 5 \ cdot 1 + 1 {,} 38 ^ {2} }} \ right) ^ {2} = 1 {,} 4 \, \%}$.

## Multiple remuneration

The reflection can be reduced further and over a wider range of wavelengths and angles by using several layers with different refractive indices. There is no simple formula for the optimal layer thickness for a given choice of materials. These parameters are therefore determined with the help of simulation programs.

## Manufacturing

The production of anti-reflective coatings is carried out using coating methods of thin-film technology . The most commonly used methods include physical vapor deposition , such as thermal evaporation and sputter deposition . The choice of the coating method depends mainly on the desired layer material, for example there are materials that are not suitable for thermal evaporation.

The anti-reflective layers place high demands on the uniformity of the layer thickness. Because unevenly coated glasses show color gradients or even Newtonian rings and are therefore useless for many applications. Coated surfaces are also sensitive to soiling (fingerprints, residues of cleaning agents), as these are also a thin layer and thus affect the optical properties.

## Areas of application

### Lenses and Lenses

Lenses with ten or more lenses, such as zoom lenses , would practically not be usable without anti-reflective coatings, because about 8% of the incident intensity per lens is lost due to the reflections. Above all, however, after two reflections, light can exit the lens together with the useful light and cause disruptive light spots (reflections) on the image or, as a diffuse veil, reduce the contrast of the image. On lens surfaces, there are pairs of faces where this can occur; That is, the effect grows quadratically with the number of areas. ${\ displaystyle k}$${\ displaystyle k (k-1) / 2}$

Reducing the degree of reflection on the individual surfaces by a factor reduces the intensity of the reflections by a factor because the light is always reflected twice. The positive effect of the remuneration also has a quadratic effect. ${\ displaystyle f}$${\ displaystyle f ^ {2}}$

With good camera lenses, all air-glass surfaces are multi-coated. Coated lenses have been used in camera lenses since the 1930s, but it was not until the 1970s that multi-layer coating became established for high-quality lenses and is now standard except for very simple cameras and lenses.

Some optical materials for the infrared spectral range, for example in thermography cameras, have a high refractive index, e.g. B. monocrystalline germanium or zinc selenide , and therefore have a high degree of reflection uncoated .

In the case of lenses for photography , the type of coating used in modern MC layers has only a very minor effect on color rendering, because the proportion of the light that is still reflected in the total energy of the radiation is very small and because different coatings are combined in this way within one lens that the total reflection on all surfaces is only slightly dependent on the wavelength. However, due to their specific overall transmission, lenses can draw a bit “warmer” or “colder”, which in practice is mostly only of importance in slide photography . Changing the layer thickness influences the dependence on the wavelength, which creates the color impression of the objective front lenses.

With glasses colored reflections are particularly undesirable. That is why broadband effective multiple remuneration is used there. Anti-reflective glasses are particularly important for driving in the dark.

### High performance optics

The materials used for the coating usually have higher absorption than the materials of the optical components. The damage threshold of an anti-reflective coating due to thermal stress is therefore generally lower than that of an uncoated interface. In the case of high-performance fiber optic cables, the beam is therefore first continued in a glass block with the same refractive index before entering or after leaving the fiber until it has a larger diameter. An interface can then lie there, which can also be coated with an anti-reflective coating.

The damage threshold of coated and uncoated laser optics is given for continuous radiation (CW) with a maximum power density (e.g. watt per cm 2 ) and for pulsed radiation with a maximum energy density (e.g. joules per cm 2 ).

## Continuously varying index of refraction

A smooth transition in the refractive index reduces the degree of reflection without strong dependence on wavelength and angle. For the transition to n  = 1, however, a refractive index close to 1 is necessary. A research team from the Rensselaer Polytechnic Institute has developed a coating made of silicon rods (see black silicon ) that has a refractive index of just 1.05. The anti-reflective coating through nanostructures on the surface is also known as the moth-eye effect .

## Individual evidence

1. Josef Reiner: Fundamentals of Ophthalmic Optics . BoD - Books on Demand, 2002, ISBN 3-8311-2767-0 , pp. 72 ( limited preview in Google Book search).
2. ^ A b F. Pedrotti, L. Pedrotti, W. Bausch, Hartmut Schmidt: Optics for engineers: Fundamentals . 3rd, arr. u. updated edition. Springer, Berlin / Heidelberg 2005, ISBN 3-540-22813-6 , pp. 295-296 .
3. ^ Eugene Hecht, Alfred Zajac: Optics . 4th edition. Addison-Wesley Longman, Amsterdam, 2003, ISBN 0-321-18878-0 , pp. 402 .
4. ^ Eugene Hecht, Alfred Zajac: Optics . 4th edition. Addison-Wesley Longman, Amsterdam, 2003, ISBN 0-321-18878-0 , pp. 425 .
5. F. Pedrotti, L. Pedrotti, W. Bausch, Hartmut Schmidt: Optics for engineers: Fundamentals . 3rd, arr. u. updated edition. Springer, Berlin / Heidelberg 2005, ISBN 3-540-22813-6 , pp. 559-601 .
6. J.-Q. Xi, Martin F. Schubert, Jong Kyu Kim, E. Fred Schubert , Minfeng Chen, Shawn-Yu Lin, W. Liu, JA Smart: Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection . In: Nat Photon . tape 1 , no. 3 , February 2007, p. 176-179 , doi : 10.1038 / nphoton.2007.26 .
7. Fred Schubert: New Nano Coating Is Virtual Black Hole for Reflections . Physorg.com, March 1, 2007.