# Differences

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en:ar [2012/05/09 04:44] deinega [All texture size to wavelength ratios] |
en:ar [2012/07/20 19:45] (current) valuev |
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Elimination of undesired reflection from optical surfaces is important for many technologies. | 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 photovoltaics reduction of reflection 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. | 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. | 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. | 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. | + | Their work is based on 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. | 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. | 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. | 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. | + | Layers thicknesses 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. | The disadvantage of multi-layer coatings is difficulty to find materials with required refractive indices. | ||

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- | Textured coatings have antireflective properties for wavelengths much smaller then texture size as well. | + | Textured coatings have antireflective properties for wavelengths much smaller than texture size as well. |

In this case reflection reduction can be illustrated geometrically: rays should be reflected many times until being reverted back. | 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. | At the same time transmitted rays deviate from the incident direction that leads to light trapping effect used in solar cells. | ||

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Below we describe how to calculate effective permittivity for following structures: | Below we describe how to calculate effective permittivity for following structures: | ||

- | - structures which are infinite in <tex>z</tex>-direction (dielectric permittivity <tex>\varepsilon</tex> depends only on <tex>x</tex> and <tex>y</tex>). | + | - structures which are infinite in <tex>z</tex>-direction (dielectric permittivity <tex>\varepsilon</tex> depends on <tex>x</tex> and <tex>y</tex> only). |

- structures which are finite <tex>z</tex>-direction. | - structures which are finite <tex>z</tex>-direction. | ||

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If structure possesses central symmetry in <tex>xy</tex>-plane, 2 components of this tensor should be equal <tex>\epsilon_{x}=\epsilon_{y}</tex>. | If structure possesses central symmetry in <tex>xy</tex>-plane, 2 components of this tensor should be equal <tex>\epsilon_{x}=\epsilon_{y}</tex>. | ||

- | As an example, let us consider sequence of parallel plates (<tex>\varepsilon</tex> depends only on one coordinate). | + | As an example, let us consider sequence of parallel plates (<tex>\varepsilon</tex> changes in one direction only). |

- | <tex>t</tex> is width of each plate, <tex>\Lambda</tex> is distance between them, <tex>\varepsilon_s</tex> is plates dielectric permittivity,<tex>\varepsilon_i</tex> is dielectric permittivity of the enviroment. | + | <tex>t</tex> is width of each plate, <tex>\Lambda</tex> is distance between them, <tex>\varepsilon_s</tex> is plates dielectric permittivity,<tex>\varepsilon_i</tex> is dielectric permittivity of the environment. |

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- | If distance between plates <tex>\Lambda</tex> is small compare to wavelength <tex>\lambda</tex>, electric field can be approximated as a constant within a plate and between two closest plates. | + | If distance between plates <tex>\Lambda</tex> is small compare to wavelength <tex>\lambda</tex>, electric field can be assumed to be a constant within a plate and between two closest plates. |

At the plate interface, normal component of vector <tex>\vec D</tex> and tangential component of vector <tex>\vec E</tex> should be continuous. | At the plate interface, normal component of vector <tex>\vec D</tex> and tangential component of vector <tex>\vec E</tex> should be continuous. | ||

- | It leads to the following expressions for components of the tensor <tex>\hat \varepsilon</tex>, corresponding to directions perpendicular and parallel to plates: | + | It leads to the following expressions for components of the tensor <tex>\hat \varepsilon</tex>, corresponding to directions which are perpendicular and parallel to plates: |

<tex> | <tex> | ||

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where <tex>f</tex> is cylinders filling fraction. | where <tex>f</tex> is cylinders filling fraction. | ||

- | For last two cases effective permittivity is tensor <tex>\hat\epsilon</tex> with equal diagonal components <tex>\epsilon_z</tex> and <tex>\epsilon_{x}=\epsilon_{y}</tex> due to the central symmetry in <tex>xy</tex>-plane. | + | In last two cases effective permittivity is tensor <tex>\hat\epsilon</tex> with equal diagonal components <tex>\epsilon_z</tex> and <tex>\epsilon_{x}=\epsilon_{y}</tex> due to the central symmetry in <tex>xy</tex>-plane. |

- | Component <tex>\epsilon_z</tex> is dielectric permittivity averaged value (it follows from continuity of electric field tangential component, as in derivation for <tex>\varepsilon_{\parallel}</tex> in the case of parallel plates). | + | Component <tex>\epsilon_z</tex> is dielectric permittivity average value (it follows from continuity of electric field tangential component, as in derivation for <tex>\varepsilon_{\parallel}</tex> in the case of parallel plates). |

Before we were assuming that structure is infinite along <tex>z</tex>-direction. | Before we were assuming that structure is infinite along <tex>z</tex>-direction. | ||

- | However, if the thickness of the structure in <tex>z</tex>-direction is more than critical value <tex>\lambda/10</tex>, effective permittivity can be calculated as in "infinite" case. | + | However, if the thickness of the structure in <tex>z</tex>-direction is larger than critical value <tex>\lambda/10</tex>, effective permittivity can be calculated as in "infinite" case. |

- | If structure is multi-layered, than each layer is characterized by its own effective permittivity. | + | In multi-layered structure 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. | + | One can use effective medium approximation for structures with gradually changing profile (like textures coatings) as well. |

- | Optical properties of this structure should be similar to properties of the film gradually changing dielectric permittivity. | + | Optical properties of this structure should be similar to properties of the film with gradually changing dielectric permittivity. |

There are two methods to calculate reflectance from such type of film. | There are two methods to calculate reflectance from such type of film. | ||

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If wavelength is much smaller then texture size, geometric optics approximation can be used. | 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. | 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. | + | Modeled rays propagate in straight lines inside the texture and get reflected or refracted from the texture interface according to Fresnel equations. |

======Geometry optimization====== | ======Geometry optimization====== | ||

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- | 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. | + | We will 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). |

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Textured surface is made from glass (the refractive index <tex>n=1.5</tex>). | Textured surface is made from glass (the refractive index <tex>n=1.5</tex>). | ||

+ | We consider normal light incidence case. | ||

=====Effective medium approximation===== | =====Effective medium approximation===== | ||

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For example, if <tex>f(z)</tex> is polynom of degree <tex>(2k'-1)</tex> with zero derivatives <tex>f^{(i)}(0) = f^{(i)}(d) = 0, 0 < i < k'</tex>, then <tex>R \sim (d/\lambda)^{-2k'}</tex>. | For example, if <tex>f(z)</tex> is polynom of degree <tex>(2k'-1)</tex> with zero derivatives <tex>f^{(i)}(0) = f^{(i)}(d) = 0, 0 < i < k'</tex>, then <tex>R \sim (d/\lambda)^{-2k'}</tex>. | ||

- | In particular, for profiles <tex>f(z)=3z^2-2z^3</tex> and <tex>f(z)=10z^3-15z^4+6z^5</tex> (we assume that <tex>d=1</tex>) have get <tex>R \sim (d/\lambda)^{-4}</tex> and <tex>R \sim (d/\lambda)^{-6}</tex> correspondingly. | + | In particular, for profiles <tex>f(z)=3z^2-2z^3</tex> and <tex>f(z)=10z^3-15z^4+6z^5</tex> (we assume that <tex>d=1</tex>) we have <tex>R \sim (d/\lambda)^{-4}</tex> and <tex>R \sim (d/\lambda)^{-6}</tex> correspondingly. |

Let us find a profile characterized by zero derivatives of all orders at the points <tex>0</tex> and <tex>d</tex>: | Let us find a profile characterized by zero derivatives of all orders at the points <tex>0</tex> and <tex>d</tex>: | ||

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We obtained exponential decrease of the reflection with the growth <tex>d/\Lambda</tex> for the complete tiling case. | We obtained exponential decrease of the reflection with the growth <tex>d/\Lambda</tex> for the complete tiling case. | ||

For the incomplete tiling case the reflection tends to a constant value passing a local minimum while <tex>d/\Lambda</tex> increases. | For the incomplete tiling case the reflection tends to a constant value passing a local minimum while <tex>d/\Lambda</tex> increases. | ||

- | In the following we give our explanation of these results. | + | Below we give our explanation of these results. |

We introduce the following ray classification: | We introduce the following ray classification: | ||

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Only reflected and secondary rays make contribution to the total reflection <tex>R=R_{\rm refl} + R_{\rm sec}</tex>. | Only reflected and secondary rays make contribution to the total reflection <tex>R=R_{\rm refl} + R_{\rm sec}</tex>. | ||

- | It can be shown that (a) number of reflections inside the structure necessary for the propagating rays to obtain the backward direction, growing linearly with the texture height. Since after each reflection ray amplitude is multiplied on reflection coefficient form pyramid surface, <tex>R_{refl}</tex> decreases exponentially with <tex>d/\Lambda</tex>. | + | It can be shown that (a) number of reflections inside the structure, necessary for the propagating rays to obtain the backward direction, grows linearly with the texture height. Since after each reflection ray amplitude is multiplied on reflection coefficient form pyramid surface, <tex>R_{refl}</tex> decreases exponentially with <tex>d/\Lambda</tex>. |

According to our calculations secondary rays make small contribution to the reflection <tex>R_{\rm sec} \approx R_{\rm refl}</tex> which can be explained by the following considerations. | According to our calculations secondary rays make small contribution to the reflection <tex>R_{\rm sec} \approx R_{\rm refl}</tex> which can be explained by the following considerations. | ||

- | (b) pyramids deflect secondary rays downward since <tex>n_s>n_i</tex> preventing them to revert back to the incident medium. | + | (b) pyramids deflect secondary rays downward since <tex>n_s>n_i</tex>, preventing them to revert back to the incident medium. |

As a result refracted rays can c) transmit to the substrate directly, d) or move onto the inner pyramid side under the total internal reflection angle. | As a result refracted rays can c) transmit to the substrate directly, d) or move onto the inner pyramid side under the total internal reflection angle. | ||

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In the case of incomplete tiling (cones) the reflectance tends to the constant value with the growth of <tex>d/\Lambda</tex> passing over a local minimum. | In the case of incomplete tiling (cones) the reflectance tends to the constant value with the growth of <tex>d/\Lambda</tex> passing over a local minimum. | ||

It can be explained by the following considerations. | It can be explained by the following considerations. | ||

- | While <tex>d/\Lambda \to \infty</tex> e) normal rays remain almost parallel to the scatterer surface after the first reflection and some of them go to the gap between the bases not reaching the neighbouring scatterer. | + | While <tex>d/\Lambda \to \infty</tex>, e) normal rays remain almost parallel to the scatterer surface after the first reflection and some of them go to the gap between the bases not reaching the neighbouring scatterer. |

Afterwards they are directly reflected back into the incident medium. | Afterwards they are directly reflected back into the incident medium. | ||

For the cones case almost all incident rays behave in this way, therefore while <tex>d/\Lambda \to \infty</tex> the reflectance tends to the substrate reflectance value. | For the cones case almost all incident rays behave in this way, therefore while <tex>d/\Lambda \to \infty</tex> the reflectance tends to the substrate reflectance value. | ||

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It can be explained by the fact that the effective dielectric permittivity does not depend on the <tex>\Lambda/\lambda</tex> at zeroth approximation and the reflectance of corresponding gradient index film decreases while <tex>d/\lambda=(d/\Lambda)(\Lambda/\lambda)</tex> increases. | It can be explained by the fact that the effective dielectric permittivity does not depend on the <tex>\Lambda/\lambda</tex> at zeroth approximation and the reflectance of corresponding gradient index film decreases while <tex>d/\lambda=(d/\Lambda)(\Lambda/\lambda)</tex> increases. | ||

At <tex>\Lambda/\lambda \ge 1</tex> the reflection decreases further passing local minima corresponding to the values of <tex>\Lambda/\lambda</tex> at which next diffraction orders appear. | At <tex>\Lambda/\lambda \ge 1</tex> the reflection decreases further passing local minima corresponding to the values of <tex>\Lambda/\lambda</tex> at which next diffraction orders appear. | ||

- | However these reflectance oscillations become smaller at greater <tex>\Lambda/\lambda<tex> while the curve approaches the geometric optics limit. | + | However these reflectance oscillations become smaller at greater <tex>\Lambda/\lambda</tex> while the curve approaches the geometric optics limit. |

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Using antireflective coatings is one of the way to increase efficiency of solar cells. | Using antireflective coatings is one of the way to increase efficiency of solar cells. | ||

- | For this purpose one can use multi-layered coatings and coatings with porous silicon, which act like film with gradually changing refractive index due to increasing porosity with the depth. | + | For this purpose multi-layered coatings or coatings with porous silicon can be used. Coatings with porous silicon act like a film with gradually changing refractive index if porosity is increasing with the depth. |

However, as we discussed above, these types of coatings are not so easy to produce. | However, as we discussed above, these types of coatings are not so easy to produce. | ||

- | Alternative way to reduce reflectance is using nanotextured coatings. | + | Nanotextured coatings is alternative way for reflection reduction. |

These coatings can be produced using lithographic technique or etching. | These coatings can be produced using lithographic technique or etching. | ||

- | Their use in solar technology leads reducing reflection from the surface of the solar cell by 1 or 2 orders of magnitude. | + | Their use in solar technology reduces reflection from the surface of the solar cell by 1 or 2 orders of magnitude. |

- | Below we present comparison between experimental data and FDTD results for reflectance from chosen textured surface taken from here | + | Below we present comparison between experimental data and FDTD results for reflectance from chosen textured surface taken from |

[[http://www.opticsinfobase.org/abstract.cfm?URI=josaa-28-5-770|http]] | [[http://www.opticsinfobase.org/abstract.cfm?URI=josaa-28-5-770|http]] | ||

{{:deinega_-_minimizing_light_reflection_from_dielectric_textured_surfaces.pdf|PDF}} | {{:deinega_-_minimizing_light_reflection_from_dielectric_textured_surfaces.pdf|PDF}} | ||

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- | In order to model dispersive dielectric permittivity of Silicon in FDTD, one can use technique described in section [[fitting]]. | + | In order to model dispersive dielectric permittivity of silicon in FDTD, one can use technique described in section [[fitting]]. |