Definitions

# Anti-reflective coating

Anti-reflective or antireflection (AR) coatings are a type of optical coating applied to the surface of lenses and other optical devices to reduce reflection. This improves the efficiency of the system since less light is lost. In complex systems such as a telescope, the reduction in reflections also improves the contrast of the image by elimination of stray light. This is especially important in planetary astronomy. In other applications, the primary benefit is the elimination of the reflection itself, such as a coating on eyeglass lenses that makes the eyes of the wearer more visible, or a coating to reduce the glint from a covert viewer's binoculars or telescopic sight.

Many coatings consist of transparent thin film structures with alternating layers of contrasting refractive index. Layer thicknesses are chosen to produce destructive interference in the beams reflected from the interfaces, and constructive interference in the corresponding transmitted beams. This makes the structure's performance change with wavelength and incident angle, so that color effects often appear at oblique angles. A wavelength range must be specified when designing or ordering such coatings, but good performance can often be achieved for a relatively wide range of frequencies: usually a choice of IR, visible, or UV is offered.

Single lens coatings were invented in November 1935 by Alexander Smakula, who was working for the Carl Zeiss optics company. Anti-reflection coatings were a German military secret until the early stages of World War II. They were independently invented by Katharine Burr Blodgett in the late 1930s.

## Ophthalmic use

Opticians dispense "antireflection lenses" because the decreased reflection makes them look better, and they produce less glare, which is particularly noticeable when driving at night or working in front of a computer monitor. The decreased glare means that wearers often find their eyes are less tired, particularly at the end of the day. Allowing more light to pass through the lens also increases contrast and therefore increases visual acuity.

Antireflective ophthalmic lenses should not be confused with polarized lenses, which decrease (by absorption) the visible glare of sun reflected off of surfaces such as sand, water, and roads. The term "anti-reflective" relates to the reflection from the surface of the lens itself, not the origin of the light that reaches the lens.

Many anti-reflection lenses include an additional coating that repels water and grease, making them easier to keep clean. Anti-reflection coatings are particularly suited to high-index lenses, as these reflect more light without the coating than a lower-index lens (a consequence of the Fresnel equations). It is also generally easier and cheaper to coat high index glasses.

## Types

The simplest form of antireflection coating was discovered by Lord Rayleigh in 1886. The optical glass available at the time tended to develop a tarnish on its surface with age, due to chemical reactions with the environment. Rayleigh tested some old, slightly tarnished pieces of glass, and found to his surprise that they transmitted more light than new, clean pieces. The tarnish replaces the air-glass interface with two interfaces: an air-tarnish interface and a tarnish-glass interface. Because the tarnish has an index of refraction between that of glass and that of air, each of these interfaces exhibits less reflection than the air-glass interface did, and in fact the total of the two reflections is less than that of the "naked" air-glass interface.

### Single-layer interference coatings

The simplest interference AR coating consists of a single quarter-wave layer of transparent material whose refractive index is the square root of the substrate's refractive index. This theoretically gives zero reflectance at the center wavelength and decreased reflectance for wavelengths in a broad band around the center.

The most common type of optical glass is crown glass, which has an index of refraction of about 1.52. An optimum single layer coating would have to be made of a material with an index equal to about 1.23. Unfortunately, there is no material with such an index that has good physical properties for an optical coating. The closest 'good' materials available are magnesium fluoride, MgF2 (with an index of 1.38), and fluoropolymers (which can have indices as low as 1.30, but are more difficult to apply). On a crown glass surface, MgF2 gives a reflectance of about 1%, four times less than the 4% reflection from bare glass. MgF2 coatings perform much better on higher-index glasses, especially those with index of refraction close to 1.9. MgF2 coatings are commonly used because they are cheap, and when they are designed for a wavelength in the middle of the visible band they give reasonably good antireflection over the entire band.

### Multi-layer coatings (multicoating)

By using alternating layers of a low-index material like silica and a higher-index material it is possible to obtain reflectivities as low as 0.1% at a single wavelength. Coatings that give very low reflectivity over a broad band can also be made, although these are complex and relatively expensive. Optical coatings can also be made with special characteristics, such as near-zero reflectance at multiple wavelengths, or optimum performance at angles of incidence other than 0°.

### Absorbing AR coatings

An additional category of antireflection coatings is the so-called "absorbing AR". These coatings are useful in situations where high transmission through a surface is unimportant or undesirable, but low reflectivity is required. They can produce very low reflectance with few layers, and can often be produced more cheaply, or at greater scale, than standard non-absorbing AR coatings. (See, for example, US Patent 5,091,244. ) Absorbing ARs often make use of unusual optical properties exhibited in compound thin films produced by sputter deposition. For example, titanium nitride and niobium nitride are used in absorbing ARs. These can be useful in applications requiring contrast enhancement or as a replacement for tinted glass (for example, in a CRT display).

### Moth eye

Moths' eyes have an unusual property: their surfaces are covered with a natural nanostructured film which eliminates reflections, allowing the moth to see well in the dark, without reflections to give its location away to predators. This kind of antireflective coating works by creating a refractive index gradient between the air and the medium, which decreases reflection by effectively removing the air-lens interface. Engineers have succeeded in making practical antireflection films using this effect. This is a form of biomimicry.

## Theory

There are two separate causes of optical effects due to coatings, often called thick film and thin film effects. Thick film effects arise because of the difference in the index of refraction between the layers above and below the coating (or film); in the simplest case, these three layers are the air, the coating, and the glass. Thick film coatings do not depend on how thick the coating is, so long as the coating is much thicker than a wavelength of light. Thin film effects arise when the thickness of the coating is approximately the same as a quarter or a half a wavelength of light. In this case, the reflections of a steady source of light can be made to add destructively, and hence reduce reflections by a separate mechanism. In addition to depending very much on the thickness of the film, and the wavelength of light, thin film coatings depend on the angle at which the light strikes the coated surface.

### Reflection

Whenever a ray of light moves from one medium to another (for example, when light enters a sheet of glass after travelling through air), some portion of the light is reflected from the surface (known as the interface) between the two media. This can be observed when looking through a window, for instance, where a (weak) reflection from the front and back surfaces of the window glass can be seen. The strength of the reflection depends on the refractive indices of the two media as well as the angle of the surface to the beam of light. The exact value can be calculated using the Fresnel equations.

When the light meets the interface at normal incidence (perpendicularly to the surface), the intensity of light reflected is given by the reflection coefficient or reflectance, R:

$R = left\left(frac\left\{n_0 - n_S\right\}\left\{n_0 + n_S\right\} right\right) ^2$,
where n0 and nS are the refractive indices of the first and second media, respectively. The value of R varies from 0.0 (no reflection) to 1.0 (all light reflected) and is usually quoted as a percentage. Complementary to R is the transmission coefficient or transmittance, T. If absorption and scattering are neglected, then the value T is always 1–R. Thus if a beam of light with intensity I is incident on the surface, a beam of intensity RI is reflected, and a beam with intensity TI is transmitted into the medium.

For the simplified scenario of visible light travelling from air (n0≈1.0) into common glass (nS≈1.5), value of R is 0.04, or 4% on a single reflection. So at most 96% of the light (T=1–R=0.96) actually enters the glass, and the rest is reflected from the surface. The amount of light reflected is known as the reflection loss.

In the more complicated scenario of multiple reflections, say with light travelling through a window, light is reflected both when going from air to glass and at the other side of the window when going from glass back to air. The size of the loss is the same in both cases. Light also may bounce from one surface to another multiple times, being partially reflected and partially transmitted each time it does so. In all, the combined reflection coefficient is given by 2R/(1+R). For glass in air, this is about 7.7%.)

### Rayleigh's film

As observed by Lord Rayleigh, a thin film (such as tarnish) on the surface of glass can reduce the reflectivity. This effect can be explained by envisioning a thin layer of material with refractive index n1 between the air (index n0) and the glass (index nS). The light ray now reflects twice: once from the surface between air and the thin layer, and once from the layer-to-glass interface.

From the equation above, and the known refractive indices, reflectivities for both interfaces can be calculated, and denoted R01 and R1S, respectively. The transmission at each interface is therefore T01 = 1-R01 and T1S = 1-R1S. The total transmitance into the glass is thus T1ST01. Calculating this value for various values of n1, it can be found that at one particular value of optimum refractive index of the layer, the transmittance of both interfaces is equal, and this corresponds to the maximum total transmittance into the glass.

This optimum value is given by the geometric mean of the two surrounding indices:

$n_1 = sqrt\left\{n_0 n_S\right\}$.

For the example of glass (nS≈1.5) in air (n0≈1.0), this optimum refractive index is n1≈1.225, the optimum refractive indices for a multi-layer coating can be computed by the procedure given in . The optimum refractive indices for a multi-layer coating at angles of incidence other than 0° is given by Moreno et al. (2005).

The reflection loss of each interface is approximately 1.0% (with a combined loss of 2.0%), and an overall transmission T1ST01 of approximately 98%. Therefore an intermediate coating between the air and glass can halve the reflection loss.

### Interference coatings

The use of an intermediate layer to form an antireflection 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 antireflection 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 thickness is controlled precisely, such that it is exactly one quarter of the light's wavelength thick (λ/4). The layer is then called a quarter-wave coating. For this type of coating the 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. 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%. Practical details include correct calculation of the layer thickness; since the wavelength of the light is reduced inside a medium, this thickness will be λ0 / 4n1, where λ0 is the vacuum wavelength. 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 antireflection. 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 antireflection 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 antireflection 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.