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Nanostructured, sub-wavelength anti-reflection layers (NALs) have attracted much attention as a new generation of anti-reflection surfaces. Among different designs, sub-wavelength periodic nanostructures are capable of enhancing transmission of coherent light through an interface without inducing scattering. In this work, we have explored a new profile for periodic NALs capable of transmitting IR light with higher efficiency compared to NALs based on a parabolic profile. To achieve high transmission and low diffraction, the profile and pitch of the nanostructured NALs are calculated using a combination of a multi-layer modeling and Rigorous Coupled Wave (RCWA) analysis.

Traditional anti-reflection coatings (ARCs) are based on destructive interference of light from two or more interfaces [

Nanostructured ARC layers (NALs [

Larger-scale ( ≫ λ ) structured layers are widely used for increasing light transmission in solar cells [_{x}O_{y}N_{z} sputtered layer over the substrate. By altering the composition of each deposited layer, he has gradually decreased the index from high index substrate to a low index SiO_{2} layer on top. The NALs made of SiO_{2} were then generated on top of that layer. By reducing the index of NALs, the jump due to the gap between nanostrutures reduces considerably [

The transmission through NALs can be described based on the concept of “fill factor” [

While different gradient layers over each other [

This work is done at Center for High Technology Materials (CHTM) at University of New Mexico.

NAL adiabatically transforms the otherwise abrupt change of refractive index between the two media. For a single homogeneous ARC layer (which cancels the reflection through destructive interference), the refractive index of antireflection coating should ideally be the geometric mean of the refractive indexes of the two media:

n = n i n t , (1)

where n i and n t are the refractive indexes of the “incidence” and the “transmission” media, respectively. In the next section, we show that this simple relation can be used to design the unit cell of our NAL.

Periodic NALs can be considered as arrays of cone-shaped structures (unit cells) as illustrated in

Thus, by controlling the fill factor one can effectively engineer the refractive index of each slice to any desired value overcoming the difficulty faced in homogeneous antireflection coatings layers. So the NAL can be modeled as stack of slices, which their refractive index is a function of the fill factor according to:

n ( z ) = p ( z ) n S i + ( 1 − p ( z ) ) n A i r , (2)

where p ( z i ) is the fill factor of the i-th slice. Therefore, we will proceed by calculating the unit cell shape based on Equation (1) and Equation (2), and compare the performance of the resulting NAL with the one with parabolic unit cell using in carrier media) using RCWA. While we use Equation (1) to estimate the required fill factor for each slice, we note that the RCWA calculation is based on the geometry in each layer (determined from Equation (2) as described in the next section), and not an effective index approximation, such as the Maxwell-Garnet or Bruggeman models for effective indexes [

Conventionally multilayered anti-reflection coatings are designed to cancel out the reflection for different wavelengths based on destructive interference of reflections. A more recent approach is deposition of thin film layers over substrate, gradually increasing the refractive index from air into the transmission medium. It is well-known that this approach allows broader wavelength range to pass as well as increasing the angular tolerance [

n i = n i − 1 n i + 1 where i = 0 , 1 , 2 , ⋯ , N + 1 . (3)

Calculating the natural logarithm of each term, this nonlinear relation between refractive indexes of subsequent layers is converted to a linear recursive equation:

ln n i = 1 2 ( ln n i − 1 + ln n i + 1 ) where i = 0 , 1 , 2 , ⋯ , N + 1 . (4)

By defining A N = ln ( n i ) Equation (4) can be rewritten as:

( 2 − 1 0 ⋯ 0 0 − 1 2 − 1 ⋯ 0 0 0 − 1 2 ⋯ 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ 2 − 1 0 0 0 ⋯ − 1 2 ) ( A 1 A 2 ⋮ A N − 1 A N ) = ( A 0 0 ⋮ 0 A N + 1 ) , (5)

which can be used to determine the required refractive index of each layer considering A 0 = ln ( n N 0 ) = 0 , A N + 1 = ln ( n N i + 1 ) = ln ( 3.5 ) . As can be seen in the right hand side other than A 0 and A N + 1 the other rows are zero. This is because all the unknowns ( A 1 to A N ) are moved to the left hand side of the linear equations. Solving Equation (4) easily yields ln ( n i ) for each layer.

In this section, we determine the refractive index profile and the corresponding shape of the unit cell for a periodic NAL using the framework introduced above. The goal is to enable high (>99%) transmission from air to bulk with a thin NAL. While most reported results indicate that by increasing the NALs thickness the transmission significantly increases [

Each slice of the NAL can be considered a thin homogeneous layer with an index calculated using Equation (5) ( N = 100 slices).

While Equations (2) and (5) determine the fill-factor for each slice, the thickness of the slice has to be determined based on two additional constraints. As light enters the NAL, it experiences a phase lag (compared to air) due to higher index. Instead of selecting a specific layer thickness, we impose a constraint on this phase lag. We determine the thickness of each pillar versus the phase shift it imposes on the incident light. The reason is that, for a same phase shift of π , a longer wavelength requires a taller pillar. This way our comparisons are consistent. The first constraint is imposed on the total phase-change for a wave that

travels from the top to the bottom of the unit-cell; we make the total phase-shift ( ϕ t ) equal to π :

∑ i = 1 N k i t i = ϕ t . (6)

In this equation, t i and k i are the thickness and the wave vector at each of the N slices, respectively. This equation is simply the summation of the phase-changes in each slice. Our numerical simulations have shown that below π phase shift the transmission drops substantially; so a total phase shift of π is the minimum required NAL thickness. Equation (6) can be rewritten as:

∑ i = 1 N n i t i = λ 2 , (7)

where λ is the wavelength of incident light and n i is the refractive index of each slice of NAL. Since Equation (7) can be satisfied using different distributions of t i , the next constraint is imposed on the optical path length (i.e. n i t i ). So the thickness of each layer is related to its index via:

t i = λ 2 N n i . (8)

nanostructure starts as a point at the top (with a fill factor of 0%) and gradually grows, filling the whole unit cell at the base. Thus, each of the designed nanostructures merge with their immediate neighbors as well as with the silicon substrate.

Extensive numerical studies have been reported on the performance of NALs of different shapes and materials. These reports use different methods including Effective Medium Theory (EMT) [

All the logarithmic calculations we did, optimizes the nanopillar for a certain wavelength. That is, for every wavelength there will be a certain NAL with a height dependent on the wavelength. Also, we should note that the X-Y of the base was in unit of area. The XY of NAL at any elevation can be easily calculated by simply multiplying the percentage (X or Y) with pitch value. That is, for creating a NAL we use parameters of wavelength and pitch. Our design maximizes the transmission of light for a certain wavelength. Also, for the specific λ of design (and specific pillar height coupled to it), there is a unique pitch for further maximizing the transmission. This is the main difference of this report with previous works. Previously, researchers have calculated the index of each layer. But, instead of designing a nanopillar, they have just designed continuous films of different indexes, laying on top of each other from silicon surface to air. However, the nanopillar pitch is a parameter that has to be optimized along with the index variation along the nanopillar.

P i t c h ≤ λ i n c 2 n s u b , (9)

where n s u b is the refractive index of the substrate and λ i n c is the wavelength at free space. Our calculations indicate that the transmission is maximum along the p i t c h = 0.27 λ for our design.

minimize reflection, as well as diffraction to higher order spatial frequencies. An extra factor of two is added to n s u b in Equation (9) to avoid backward diffraction at grazing incidence [

An alternative profile previously reported for NALs is the parabolic profile [

A parabolic index profile is one of the most common unit cell profiles previously used for designing NALs [

In general, it is well understood that as the index difference between the incidence and substrate medium increases, the transmission decreases. This makes the use of proper NALs profile more crucial. Accordingly, the effectiveness of our design is more critical at higher refractive index media (like silicon with refractive index close to 3.5) as evident from

As the main focus here is to study transmission through silicon-based NALs into the silicon substrate, we have estimated the bandwidth (for transmission above 96%) around the max point at three different values of pitch for silicon substrate (see

Index of refraction | λ (μm) | T_{designed} (μm) | T_{Parabolic} |
---|---|---|---|

n = 1.5 | 4 | 1.00 | 0.9962 |

n = 1.5 | 7 | 0.9999 | 0.9962 |

n = 1.5 | 9 | 0.9998 | 0.9962 |

n = 2.5 | 4 | 0.9995 | 0.9834 |

n = 2.5 | 7 | 0.9995 | 0.9834 |

n = 2.5 | 9 | 0.9949 | 0.9834 |

n = 3.5 | 4 | 0.9911 | 0.9677 |

n = 3.5 | 7 | 0.9911 | 0.9676 |

n = 3.5 | 9 | 0.9911 | 0.9677 |

Index of refraction | Pitch (μm) | Bandwidth_{Parabolic} | Bandwidth_{OurDesign} |
---|---|---|---|

n = 3.5 | 1.04 μm | 2.5 μm | 2.1 μm |

n = 3.5 | 1.23 μm | 2.9 μm | 2.5 μm |

n = 3.5 | 1.42 μm | 3.3 μm | 3 μm |

We have proposed a unit-cell design for a nanostructured antireflection layer (NAL) that results in very high transmission (zeroth-order diffraction) compared to nanostructures based on conventional (parabolic) unit cells. The design is based on dividing the unit cell into multiple slices and selecting the effective refractive index of each slice as the geometrical mean of the adjacent slices. The ultimate shape and height of the structure are determined by the resulting fill-factor subject to two constraints: 1) The optical path length is the same for all the slices. 2) The total optical phase shift along a unit cell is 180˚. The NAL created based on such unit cell provides high transmission with zeroth-order diffraction in spite of its relatively small thickness. We have characterized the resulting NALs in the mid-infrared wavelength range (3 to 9 microns) by calculating the transmission from air into a silicon substrate using the RCWA method. The effectiveness of the proposed design was studied for a range of pitches and wavelengths. Our results show the importance of selecting the optimal pitch for maximum transmission at the desired wavelength.

We have shown that at any wavelength between 3 μm - 9 μm, there exists an optimal pitch that results in almost 100% transmission through the NAL made based on our unit-cell design. In all cases, the pitch is kept sufficiently small that there are no propagating diffraction orders beyond the zero-order. We have also compared the performance of our designed NALs with the ones based on the more common parabolic design and have observed more than 3 times reduction in the reflection with only a small bandwidth reduction. Finally, we found that the optimal pitch-to-wavelength ratio for maximum transmission is 0.27 for our design compared to 0.205 for the parabolic design. This result leaves two questions open: first, what is the optimal structure, and the second is the fabrication process to achieve the promise of the model designs. Optimization is of course somewhat application specific. Some applications may only require a narrow-band design while others may require higher spectral bandwidths. Despite numerous publications on different profiles, this work demonstrates that there remains room for improvement. Fabrication is another difficult issue. In a subsequent publication, we will present our experimental results based on NALs fabricated using interferometric lithography and reactive-ion etching that come close to the promise of these simulations.

This work is done with the funding of Center for High Technology Materials at University of New Mexico.

The authors declare no conflicts of interest regarding the publication of this paper.

Mousavi, B.K., Mousavu, A.K., Busani, T., Zadeh, M.H. and Brueck, S.R.J. (2019) Nanostructured Anti-Reflection Coatings for Enhancing Transmission of Light. Journal of Applied Mathematics and Physics, 7, 3083-3100. https://doi.org/10.4236/jamp.2019.712217