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Lithographic Mask Defects Analysis on an MMI 3 dB Splitter^{ †}

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## Abstract

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## 1. Introduction

- Next follows the Materials and Methods section, where will be described a 1 × 2 multimode interference (MMI) structure consisting of a 3 dB coupler that provides the required optical paths to feed the sensing device and to yield the reference arm signal for further analysis and processing.
- Then, we present the Results section. In this section, we evaluate the imbalance between the outputs of the MMI structure considering an increasing standard deviation of a normal distribution of defects. These perturbations are intended to represent the defects introduced by lithographic mask resolution along the whole structure. Next, these defects’ distribution impacts are also evaluated from a manufacturing perspective where a batch of produced samples with defects is emulated by simulation and the results obtained through statistical analysis are reported.
- Finally, there is the Discussion section where obtained results are interpreted and conclusions are reported. Here, future areas of related research will also be discussed, namely future research/simulation actions that will contribute to mitigating the imbalance introduced by lithographic mask resolution.

## 2. Materials and Methods

_{y}) of the EM field, thus only transverse magnetic (TM) modes fulfil this requirement [10]. For this reason, all subsequent discussion and analysis will be considering a propagating TM mode.

_{m}(m = 0, 1, 2, …) propagating in a high contrast step index multimode device shows an approximate quadratic dependence to the mode number m:

_{0}is the vacuum wavenumber, n

_{eff}the effective refractive index of the structure, λ

_{0}the vacuum wavelength, and W

_{eff}the effective width of the MMI waveguide. The effective refractive index, n

_{eff}, is a characteristic of each propagating mode, representing the “experienced” refractive index, when propagating inside the waveguide structure and the standing-wave condition is matched [12]. Hence, the n

_{eff}of a given propagating mode may be calculated as in Equation (2) below:

_{eff}, is the width when considering the mode field profile penetration depth, due to the Goos-Hähnchen shifts, into the waveguide boundaries. This dimension is polarization dependent and in high refractive index contrast devices, the penetration depth of the EM field beyond the inner walls of the device is practically non-existent, hence W

_{eff}can be approximated by the effective width of the fundamental mode [11]:

_{π}) of the two lowest order modes:

_{π}and 2(3L

_{π}), respectively, while two-fold images form at 1/2(3L

_{π}) and 3/2(3L

_{π}). Single images are, approximately, the same amplitude as the input EM field and each of the two-fold images is affected by a 3 dB attenuation factor, thus offering the ideal conditions for a power splitter device, similar to the structure diagram depicted in Figure 1. This schematic represents an a-SiNx MMI device embedded in SiO

_{2}and associated dimensions, which were used throughout this paper in our simulations and subsequent analysis.

## 3. Results

_{opt}for optimal self-image generation may be expressed by combining Equations (4) and (5), resulting in Equation (6):

_{0}. Hence, MMI section length variation may be expressed by:

_{2}substrate, followed by Plasma Enhanced Chemical Vapor Deposition (PECVD) of the a-SiNx waveguide and an SiO

_{2}cladding which can be obtained by a PECVD process [13] or by plasma gas decomposition [14].

_{2}substrate and same material 50 nm covering layer. Simulations were conducted on a device with the following characteristics and where the MMI section length was determined considering Equation (5) with corresponding parameters for a 3 dB power splitter device (p = 1, N = 2 and a = 4):

- 1 × 0.2 × 10 μm for input and output waveguides width, height and length, respectively;
- 7 × 0.2 × 70.5 μm for MMI section’s width, height and length, respectively;
- MMI device is completely embedded in SiO
_{2}with a 50 nm cover; the superstrate is air.

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Refractive index perturbations due to lithographic mask defects (10 nm standard deviation); (

**b**) Representation of transversal refractive index profile at z = 50 μm (multimode interference section embedded in SiO

_{2}).

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## Share and Cite

**MDPI and ACS Style**

Lourenço, P.; Fantoni, A.; Costa, J.; Vieira, M.
Lithographic Mask Defects Analysis on an MMI 3 dB Splitter. *Photonics* **2019**, *6*, 118.
https://doi.org/10.3390/photonics6040118

**AMA Style**

Lourenço P, Fantoni A, Costa J, Vieira M.
Lithographic Mask Defects Analysis on an MMI 3 dB Splitter. *Photonics*. 2019; 6(4):118.
https://doi.org/10.3390/photonics6040118

**Chicago/Turabian Style**

Lourenço, Paulo, Alessandro Fantoni, João Costa, and Manuela Vieira.
2019. "Lithographic Mask Defects Analysis on an MMI 3 dB Splitter" *Photonics* 6, no. 4: 118.
https://doi.org/10.3390/photonics6040118