Edge preserving super-resolution infrared image reconstruction based on L1- and L2-norms

Abstract

Super-resolution (SR) is a widely used technology that increases image resolution using algorithmic methods. However, preserving the local edge structure and visual quality in infrared (IR) SR images is challenging because of their disadvantages, such as lack of detail, poor contrast, and blurry edges. Traditional and advanced methods maintain the quantitative measures, but they mostly fail to preserve edge and visual quality. This paper proposes an algorithm based on high frequency layer features. This algorithm focuses on the IR image edge texture in the reconstruction process. Experimental results show that the mean gradient of the IR image reconstructed by the proposed algorithm increased by 1.5, 1.4, and 1.2 times than that of the traditional algorithm based on L1-norm, L2-norm, and traditional mixed norm, respectively. The peak signal-to-noise ratio, structural similarity index, and visual effect of the reconstructed image also improved.

Keywords

infrared (IR) super-resolution (SR) image / reconstruction / high frequency layer / edge texture

1 Introduction

Silicon-on-insulator (SOI) is a popular material platform for photonic devices and photonic integrated circuits. Due to its large thermo-optic coefficient and high thermal conductivity, many silicon thermo-optic devices have been designed and fabricated [1-5]. Conventional Si waveguide is wrapped by silica (SiO₂), which has a very low thermal conductivity (k = 1.27 W/(m·K)) [6], thus its thermal response is very slow. The rise time of the conventional Si waveguide is usually several microseconds [3,7], which seriously affects its application in high-speed photonic devices. However, SiN can be fabricated with a complementary-metal-oxide-semiconductor compatible process, it has a relatively low refractive index (n = 1.91 at 1.55 μm) and a relatively large thermal conductivity (k = 30 W/(m·K)) [8]. Therefore, it could be an effective way by replacing SiO₂ claddings with SiN claddings to improve the thermal response of the Si waveguide devices.

Here we propose a waveguide structure based on SiN-Si-SiO₂ system. The high index contrast between a core (Si) and a claddings (SiO₂ [9] and SiN [10]) provides a good confinement to optical field. The SiO₂ bottom-cladding is deeply etched [11] and a trench is formed and filled with SiN. Since the thermal conductivity of SiN is much higher than that of SiO₂, therefore a fast heat dissipation channel is created. Our simulation indicates that the rise time of the proposed waveguide structure is about 110 ns, which is about two orders of magnitude less than that of the conventional Si waveguide.

2 Structure and modeling

Cross-section of the Si waveguide wrapped by SiN is shown in Fig. 1. There are four layers stacked on the Si substrate, which are SiO₂ bottom-cladding, Si core, SiN up-cladding and metal heater (e.g. Cr). It should be noted that the SiN up-cladding is stretched into the SiO₂ bottom-cladding and three sides of the Si core are covered by SiN. That is, the Si waveguide is wrapped by SiN. The SiO₂ bottom-cladding is 1 μm in thickness. The Si core is 400 nm × 220 nm. The metal heater is 400 nm in width and 50 nm in thickness. There are three parameters: etching depth t_s, stretching width w_s and up-cladding thickness t_c. When the Joule heat is generated in the metal heater, it goes through the SiN up-cladding and establishes a stable temperature field in the Si waveguide quickly. The computation domain is indicated by the dashed lines in Fig. 1. We assume a constant temperature of 300 K at the bottom of the computation domain and heat convection boundary condition at the interface of the structure with air. The heat convection coefficient of air is 5.0 W/(m²·K) at 300 K (the same as that in Ref. [7]). The thermal and optical parameters of the materials used in the calculation are listed in Table 1.

Fig.1 Cross-section of Si waveguide wrapped by SiN

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Tab.1 Thermal and optical parameters of materials used in calculation

material	refractive index at 1.55 μm	extinction coefficient at 1.55 μm	thermal conductivity /(W·(m·K)-1)	specific heat /(J·(kg·K)-1)	density /(kg·m-3)
Si	3.476	-	148	710	2.33×103
SiO2	1.445	-	1.27	745	2.3×103
SiN	1.91	4.9×10-6	30	170	2.5×103
Cr	3.674	4.19	93.7	450	7.14×103

In this paper, we use two-dimensional (2D) finite element method (FEM) for thermal analysis and three-dimensional (3D) beam propagation method (BPM) for optical analysis. Noted that rise time refers to the time required for the temperature to change from 300 K to (0.9×(T_max-300)+300) K, where T_max is the value of its maximum.

3 Simulation results

In order to improve the response speed of the proposed structure, the SiN up-cladding should be as thin as possible. However, much too thin SiN up-cladding cannot confine the optical field effectively and results in large optical loss. So the thickness of the SiN up-cladding should be optimized. We use 3D BPM to calculate the effective extinction coefficient of the proposed structure. As shown in Fig. 2, firstly, the effective extinction coefficient decreases rapidly as the up-cladding thickness t_c increases, which is mainly due to the absorption of metal heater to the optical field, and then saturates to about 2×10^-6 when the up-cladding thickness t_c further increases, which is mainly due to the absorption of SiN to the optical field. Therefore, the up-cladding thickness t_c is chosen to be larger than 1 μm. Since optical loss is mainly caused by the absorption of the metal heater to the optical field, the etching depth t_s and stretching width w_s hardly affect the optical loss and are fixed to be 0.5 μm and 1 μm respectively during the above calculation.

Fig.2 Calculated effective extinction coefficient of Si waveguide wrapped by SiN with different up-cladding thicknesses

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Since the thermal conductivity of Si is much larger than those of SiO₂ and SiN, the temperature distribution is almost flat in the Si core. Therefore we use the temperature at the centre point of the Si core to evaluate the whole core. Power consumption is measured via area heat generating rate (AHGR) in the heater with a unit of W/m². We use 2D FEM for the following thermal analysis and the AHGR is chosen to be 5×10¹⁵ W/m².

When we calculate the temperature response of the Si waveguide wrapped by SiN with different etching depths t_s (0.1 to 1.0 μm with an interval of 0.1 μm), the up-cladding thickness t_c is fixed to be 1 μm and the stretching width w_s is fixed to be 1 μm. When we calculate the temperature response of the Si waveguide wrapped by SiN with different up-cladding thicknesses t_c (1.5 to 3 μm with an interval of 0.5 μm), the etching depth t_s is fixed to be 0.1, 0.5 and 0.9 μm respectively, and the stretching width w_s is fixed to be 1 μm. When we calculate the temperature response of the Si waveguide wrapped by SiN with different stretching widths w_s (0 to 1.5 μm with an interval of 0.5 μm), the etching depth t_s is also fixed to be 0.1, 0.5 and 0.9 μm respectively, and the up-cladding thickness t_c is fixed to be 1 μm.

For comparison, the thermal behavior of the conventional Si waveguide wrapped by SiO₂ with the same Si core size and SiO₂ bottom-cladding thickness is also simulated (Fig. 3). The thickness of the SiO₂ up-cladding is also set to be 1 μm and the width of the heater is set to be 12.5 μm, which is the same as the value used in Ref. [1]. As the heater of the conventional Si waveguide wrapped by SiO₂ is wider than that of the Si waveguide wrapped by SiN, the AHGR loading on the heater of the conventional Si waveguide is set to be 1.6×10¹⁴ W/m² in order to get the same power consumption as the Si waveguide wrapped by SiN does.

Fig.3 Cross section of conventional Si waveguide

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Figure 4(a) shows the temperature response of the Si waveguide wrapped by SiN with different etching depths t_s. It is clearly seen that increasing the etching depth t_s impoves the temperature response of the Si waveguide wrapped by SiN greatly (from 2.5 μs for t_s = 0.1 μm to 117 ns for t_s = 1 μm) with the sacrifice of reducing the temperature change greatly (from 318 K for t_s = 0.1 μm to 302 K for t_s = 1 μm). However, the temperature response time of the conventional Si waveguide wrapped by SiO₂ is 7.6 μs (as shown in the red line in Fig. 4(b)), which is about two orders of magnitude larger than that of the Si waveguide wrapped by SiN with t_s = 1 μm.

Figures 4(b) and 4(c) show the rise time and temperature change for different etching depths t_s. It is clearly seen that the rise time and temperature change decrease almost linearly as the etching depth t_s increases, which means that high response speed comes together with high power consumption. When the etching depth t_s increases from 0.1 to 1.0 μm, the rise time of the Si waveguide wrapped by SiN reduces from 1.55 μs to 117 ns. Considering the situation of the etching depth t_s = 1.0 μm, the rise time of the Si waveguide wrapped by SiN is just 1/65 that of the conventional Si waveguide (7.6 μs). The Si waveguide wrapped by SiN has a same temperature change with the conventional Si waveguide when the etching depth t_s is 0.68 μm. The temperature change reduces to about 1.8 K for the etching depth t_s = 1.0 μm, which is only 1/4 that of the conventional Si waveguide. While for the etching depth t_s = 0.1 μm, the temperature change is about 18 K, which is about two times that of the conventional Si waveguide.

Fig.4 (a) Temperature response of Si waveguide wrapped by SiN with different etching depths ts; (b) rise time versus etching depths ts; (c) temperature change versus etching depths ts. Corresponding curves of conventional Si waveguide are shown in red lines as references

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It can be seen from Fig. 5(a) that increasing the stretching width w_s is a good way for improving the response speed, but demands for more power consumption. How much the stretching width w_s affecting the rise time and temperature change depends on the etching depth t_s. When the etching depth t_s is 1.0 μm, the Si waveguides wrapped by SiN with different stretching widths w_s have almost same rise time and temperature change, which means that the influence of the stretching width w_s on the rise time and temperature change is weak. When the etching depth t_s is 0.1 μm, the influence of the stretching width w_s on the rise time and temperature change is much stronger. Figure 5(b) shows the rise time of the Si waveguide wrapped by SiN with different stretching depths w_s and etching depths t_s. It is clearly seen that the etching depth t_s plays a dominant role in determining the response speed of the Si waveguide wrapped by SiN. The influence of the stretching width w_s on the response speed of the Si waveguide wrapped by SiN is strong for small etching depth t_s and weak for large etching depth t_s. For example, the rise time change is 0.28 μs for t_s = 0.1 μm, 0.27 μs for t_s = 0.5 μm and 33.5 ns for t_s = 1.0 μm when the stretching width w_s increases from 0 to 1.5 μm. It is clearly seen from Fig. 5(c) that the etching depth t_s also plays a dominant role in determining the temperature change of the Si waveguide wrapped by SiN. The influence of the stretching width w_s on the temperature change of the Si waveguide wrapped by SiN is strong for small etching depth t_s and weak for large etching depth t_s. For example, the temperature change is 7.4 K for t_s = 0.1 μm, 5.4 K for t_s = 0.5 μm and 0.34 K for t_s = 1.0 μm when the stretching width w_s increases from 0 to 1.5 μm.

Fig.5 (a) Temperature response of Si waveguide wrapped by SiN with different stretching widths ws and etching depths ts; (b) rise time versus stretching width ws for different etching depths ts; (c) temperature change versus stretching width ws for different etching depths ts

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Figure 6(a) shows the temperature response of the Si waveguide wrapped with SiN for different up-cladding thicknesses (t_c =1.5, 2.0, 2.5 and 3.0 μm). Here etching depth t_s is chosen to be 0.1, 0.5 and 1.0 μm respectively, and stretching width w_s is chosen to be 1 μm. We can see that for the same etching depth t_s, the temperature change is same for different up-cladding thicknesses. However, the rise time increases almost linearly as the up-cladding thickness increases (see Fig. 6(b)). When up-cladding thickness t_c changes from 1.5 to 3.0 μm, the rise time changes from 1.87 to 2.55 μs for t_s=0.1 μm, from 1.076 to 1.557 μs for t_s = 0.5 μm, and from 147 to 300 ns for t_s = 1.0 μm. It means that when the etching depth t_s is smaller, the up-cladding thickness t_c has a larger influence on the response speed.

Fig.6 (a) Temperature response of Si waveguide wrapped by SiN with different up-cladding thicknesses; (b) rise time versus up-cladding thicknesses tc

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4 Conclusions

A new type of Si waveguide wrapped by SiN and associated optical and thermal analysis are presented. The thickness of SiN up-cladding should be larger than 1 μm in order to prevent the absorption of the metal heater to the optical field. Because of the high thermal conductivity of SiN, the thermal response of the proposed waveguide structure is improved. Moreover, its thermal response can be further improved by creating a fast heat dissipation channel. Our simulation indicates that a rise time of about 110 ns can be achieved for the proposed waveguide structure, which is about two orders of magnitude larger than that of the conventional Si waveguide. The influence of the thickness of up-cladding and the stretching width on the thermal performance are also discussed. Due to the high response speed of the presented waveguide structure, it can be used to improve the response speed of varies thermo-optic devices, such as thermo-optic MZI [1], RODAM [4] and optical logic gate [5] based on tunable micro ring-resonator.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61275099 and 61671094) and the Natural Science foundation of Chongqing Science and Technology Commission (No. CSTC2015JCYJA40032).

RIGHTS & PERMISSIONS

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg