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# Antireflective textured coatings

Elimination of undesired reflection from optical surfaces is important for many technologies. In photovoltaics reduction of reflectance from solar cells leads to enhancement of their efficiency. In telescopes and similar optical devices elimination of reflection is required to achieve better quality of image. Antireflective coatings allow to reduce the glint from a covert viewer's binoculars or telescopic sight.

To reduce reflection one can use single-layer quarter-wave coatings. The reduction of reflection is caused by destructive interference in the beams reflected from the interfaces, and constructive interference in the transmitted beams. However, as a result, single-layer coatings possess antireflective properties only for limited range of wavelengths and incidence angles.

To extend this range multi-layer coatings can be used. They are based on the same principle as single-layer coatings: destructive interference between beams reflected from different layers. Layers thickness and refractive indices should be chosen to achieve minimal reflectance in a wide wavelengths or incident angles range. The disadvantage of multi-layer coatings is difficulty to find materials with required refractive indices.

Alternative to multi-layer coatings are layers with continuously changing refractive index. Their application allows to achieve small reflectance in a wide spectral range. However, their manufacturing encounters problems with thermal mismatch, adhesion and stability of thin-film stack.

An alternative method for the reflection reduction consists in texturing the surface with 3D pyramids or 2D grooves (gratings). This method was discovered in nature by studying structure of moths' eyes. Surfaces of moths' eyes are covered with a natural nanostructured film. The film structure consists of a hexagonal pattern of bumps, each roughly 200 nm high and spaced on 300 nm centers. Since bumps are smaller than the wavelength of visible light, the light sees the surface as having a continuous refractive index gradient between the air and the medium. It leads to reflection reduction by effectively removing the air-lens interface. This allows the moth to see well in the dark, without reflections to give its location away to predators.

*Moth eye.*

Textured coatings have antireflective properties for wavelengths much smaller then texture size as well. In this case reflection reduction can be illustrated geometrically: rays should be reflected many times until being reverted back. At the same time transmitted rays deviate from the incident direction that leads to light trapping effect used in solar cells.

Since the seventies of the last century there is a lot of experimental studies of the properties of textured surfaces. Many reports on successful fabrication of the antireflective nanostructured surfaces appeared recently. Their use in the solar cell technology may lead to one or two orders of magnitude reduction of the surface reflection.

In the following we discuss antireflective properties of textured coatings at the whole range of their sizes including long and short wavelength limits.

## Long wavelength limit

If wavelength is much larger then texture size, optical properties of the textured coating can be described by effective medium approximation. According to this approximation, electromagnetic wave propagates in textured coating as in anisotropic medium with some effective dielectric permittivity. Below we describe how to calculate effective permittivity for following structures:

- structures which are infinite in -direction (dielectric permittivity depends only on and ).

- structures which are finite -direction.

- multi-layered structures and structures with gradually changing profile (textured coatings).

In the first case, dielectric permittivity of the structure is periodic in -plane and does not depend on . One can choose and axis in such way that effective permittivity will be described by tensor with 3 nonzero diagonal elements , and . If structure possesses central symmetry in -plane, 2 components of this tensor should be equal .

As an example, let us consider sequence of parallel plates ( depends only on one coordinate). is width of each plate, is distance between them, is plates dielectric permittivity, is dielectric permittivity of the enviroment.

*Sequence of parallel plates.*

If distance between plates is small compare to wavelength , electric field can be approximated as a constant within a plate and between two closest plates. At the plate interface, normal component of vector and tangential component of vector should be continuous. It leads to the following expressions for components of the tensor , corresponding to directions perpendicular and parallel to plates:

where and are volume fractions for plates and for environment.

Now lets consider square parallelepipeds which are infinite in -direction. Parallelepipeds are packed in square lattice.

*Square parallelepipeds which inifinite in <tex>z</tex>-direction. Parallelepipeds are packed in square lattice.*

Brauer and Bryngdahl proposed the following empirical expression for effective permittivity in and directions:

where is refractive index averaged value, and and are square roots from the following expressions

where , and should be calculated using expressions for plates which are given above.

For infinite cylinders effective permittivity can be calculated using Maxwell-Garnett expression:

where is cylinders filling fraction.

# Geometry optimization

In the following we consider surfaces coated by a periodic pyramid-type texture with height .

Pyramids bases have the shape of triangles, squares, hexagons and circles (in the last case pyramid is in fact a cone) with the distance between the base side and its center . The pyramids are closely packed on the substrate in the triangular or square lattice with the period .

In the following we specially distinguish two cases: complete tiling case when pyramids bases touch each other along their whole perimeter (this corresponds to the polygon base pyramids in our study) and incomplete tiling case when there are gaps between bases (this corresponds to cones). We consider normal light incidence case.

Textured surface is made from glass (the refractive index ).