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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.

Corresponding results can be also found in: http PDF

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 tex:z-direction (dielectric permittivity tex:\varepsilon depends only on tex:x and tex:y).

- structures which are finite tex:z-direction.

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


In the first case, dielectric permittivity of the structure tex:\varepsilon(x,y) is periodic in tex:xy-plane and does not depend on tex:z. One can choose tex:x and tex:y axis in such way that effective permittivity will be described by tensor tex:\hat\epsilon with 3 nonzero diagonal elements tex:\epsilon_{x}, tex:\epsilon_{y} and tex:\epsilon_z. If structure possesses central symmetry in tex:xy-plane, 2 components of this tensor should be equal tex:\epsilon_{x}=\epsilon_{y}.

As an example, let us consider sequence of parallel plates (tex:\varepsilon depends only on one coordinate). tex:t is width of each plate, tex:\Lambda is distance between them, tex:\varepsilon_s is plates dielectric permittivity,tex:\varepsilon_i is dielectric permittivity of the enviroment.


Sequence of parallel plates.

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

tex:\varepsilon_{\perp}=\frac{\vec D}{\vec E}=\left(\frac{f_s}{\varepsilon_s}+\frac{f_i}{\varepsilon_i}\right)^{-1},

tex:\varepsilon_{\parallel}=\frac{\vec D}{\vec E}=f_s\varepsilon_s+f_i\varepsilon_i,

where tex:f_s=t/\Lambda and tex:f_i=(\Lambda-t)/\Lambda are volume fractions for plates and for environment.

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


Square parallelepipeds which inifinite in z-direction. Parallelepipeds are packed in square lattice.

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

tex:n_{2D}=[\bar n + 2\hat n_{2D} + 2\check n_{2D}]/5,

where tex:\bar n is refractive index averaged value, and tex:\hat n_{2D} and tex:\check n_{2D} are square roots from the following expressions

tex:\hat \varepsilon_{2D}=(1-f)\varepsilon_{i}+f\varepsilon_{\perp},

tex:1/\check \varepsilon_{2D}=(1-f)/\varepsilon_{i}+f/\varepsilon_{\parallel},

where tex:f=t/\Lambda, tex:\varepsilon_{\perp} and tex:\varepsilon_{\parallel} should be calculated using expressions for plates which are given above.

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

tex:\varepsilon_{2D}=\epsilon_i+2f\epsilon_i\frac{\epsilon_s-\epsilon_i}{\epsilon_s+\epsilon_i-f(z)(\epsilon_s-\epsilon_i)},

where tex:f is cylinders filling fraction.

For last two cases effective permittivity is tensor tex:\hat\epsilon with equal diagonal components tex:\epsilon_z and tex:\epsilon_{x}=\epsilon_{y} due to the central symmetry in tex:xy-plane. Component tex:\epsilon_z is dielectric permittivity averaged value (it follows from continuity of electric field tangential component, as in derivation for tex:\varepsilon_{\parallel} in the case of parallel plates).

Before we were assuming that structure is infinite along tex:z-direction. However, if the thickness of the structure in tex:z-direction is more than critical value tex:\lambda/10, effective permittivity can be calculated as in “infinite” case.

If structure is multi-layered, than each layer is characterized by its own effective permittivity. One can use effective medium approximation for structures with gradually changed profile (like textures coatings) as well. Optical properties of this structure should be similar to properties of the film gradually changing dielectric permittivity. There are two methods to calculate reflectance from such type of film.

In one method, film is approximated by a multi-layered structure with great number of very thin layers. Dielectric permittivity of each layer is assumed to be constant. Each layer is characterized by transfer matrix and product of such matrices is transfer matrix for the whole structure. This resulted transfer matrix can be used to calculate reflectance. Description of transfer matrix method can be found here1).

Another method to find reflectance, which consists in solving of Maxwell equations in inhomogeneous medium, can be found here 2).

Geometric optics approximation

If wavelength is much smaller then texture size, geometric optics approximation can be used. In this case on can apply ray tracing technique to calculate reflectance. Modeled rays propagate in straight lines inside the texture and are reflected or refracted from the texture interface according to Fresnel equations.

Geometry optimization

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


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 tex:L. The pyramids are closely packed on the substrate in the triangular or square lattice with the period tex:\Lambda.


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 tex:n=1.5).

Effective medium approximation

As we showed before, in the long wavelength limit electromagnetic waves propagate in textured surface as in film with gradually changing effective dielectric permittivity tex:\varepsilon(z). Here tex:z direction is aligned along the pyramid axis, with tex:z=0 at the pyramids tops and tex:z=d at the pyramids bases.

Increase of the pyramids height tex:d and decrease of the optical contrast between the incident medium and the texture reduces the reflection.

In the case of complete tiling, the filling fraction at the top of the pyramids is tex:f(0)=0 and at the base of the pyramids is tex:f(d)=1, therefore tex:\varepsilon(0)=\varepsilon_i and tex:\varepsilon(d) = \varepsilon_s. Here tex:\varepsilon_s is the pyramids permittivity, tex:\varepsilon_i is the incident medium permittivity, and tex:f(z) is the filling fraction occupied by pyramids at tex:z, that is equal to the ratio between cross sectional area of the pyramid and area of the unit cell of the lattice.

Some special profiles tex:f(z) can be chosen to reduce the reflection with the increasing tex:d.


For example, if tex:f(z) is polynom of degree tex:(2k'-1) with zero derivatives tex:f^{(i)}(0) = f^{(i)}(d) = 0, 0 < i < k', then tex:R \sim (d/\lambda)^{-2k'}. In particular, for profiles tex:f(z)=3z^2-2z^3 and tex:f(z)=10z^3-15z^4+6z^5 (we assume that tex:d=1) have get tex:R \sim (d/\lambda)^{-4} and tex:R \sim (d/\lambda)^{-6} correspondingly.

Let us find a profile characterized by zero derivatives of all orders at the points tex:0 and tex:d: tex:f^{(i)}(0)=f^{(i)}(d)=0, tex:\forall i>0. Without any restriction assume that tex:d=1. Consider first the infinitely differentiable function tex:e^{-z^{-1}(1-z)^{-1}}, that is zero with all its derivatives at tex:z=0 and tex:z=1. After its integration we get a monotone function increasing from 0 to 1 tex:\displaystyle f(z) = C \int_o^ze^{-\zeta^{-1}(1-\zeta)^{-1}}d\zeta, where value of tex:C is chosen to ensure tex:f(1)=1. Using this 'integral' profile leads to the exponential decrease of the reflectance with the growth of tex:d/\lambda due to tex:f^{(i)}(0)=f^{(i)}(d)=0, \forall i>0.


We calculated dependence of the reflectance on tex:d/\lambda for closely packed square pyramids with following profiles:

- tex:k'=1, tex:f(z)=z^2. We calculated flat-sided pyramids where width depends linearly on the height. Since filling fraction is proportional to the width squared, we used tex:f(z)=z^2, but not tex:f(z)=z;

- tex:k'=2, tex:f(z)=3z^2-2z^3;

- tex:k'=3, tex:f(z)=10z^3-15z^4+6z^5.

For these pyramids we obtained tex:R \sim (d/\lambda)^{-2}, tex:R \sim (d/\lambda)^{-4} and tex:R \sim (d/\lambda)^{-6} correspondingly. Note that results obtained by effective medium approximation and FDTD are in good agreement. We used Brauer-Bryngdahl expression for the effective dielectric permittivity of square pyramids.

We calculated the reflectance for gradient index film corresponding to a single-periodic grating with the 'integral' profile. Using this profile leads to the exponential decrease of the reflection with the growth of tex:d/\lambda.

At the incomplete tiling case there are gaps between pyramid bases: tex:0<f(d)<1. It causes a discontinuity of the permittivity tex:\varepsilon(z) at the pyramid bases tex:z=d: tex:\varepsilon(d) \ne \varepsilon_i. Due to this fact, by increasing tex:d/\lambda the reflectance tends to a constant value equal to the reflectance between the media with tex:\varepsilon (d) and tex:\varepsilon_i. To demonstrate this we calculated the reflectance for gradient index film corresponding to cones closely packed in the triangular lattice. We used Maxwell-Garnett expression for the effective dielectric permittivity of cones.


Reflectance for different pyramids. Comparison between results obtained using effective medium approximation (lines) and FDTD (dots).

Geometric optics approximation

1) M.Born and E. Wolf, Principles of Optics, Prgamon, London (1980).
2) G. Franceschetti, “Scattering from plane layered media”, IEEE Trans. Antennas Propag., 12, 754-763 (1964)
 
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