# Design of an Angle Detector for Laser Beams Based on Grating Coupling

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## Abstract

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## 1. Introduction

## 2. Design of the Angle Detector Based on Grating Coupling

_{inc}is incident onto the device from the top. The incident direction is in the x–z plane. The angle between the incident laser and z direction (i.e., surface normal direction) is denoted as θ

_{inc}. When the laser is incident obliquely, θ

_{inc}takes positive values when the incident laser is along +x direction and θ

_{inc}is negative when the incident laser is along –x direction. Our device is designed such that the incident light is efficiently coupled to the “+x traveling mode” and “−x traveling mode” in the slab waveguide. When θ

_{inc}= 0 (that is, normal incidence), “+x mode” and “−x mode” in the slab waveguide are equally strong, due to the geometrical symmetry. When θ

_{inc}≠ 0 (that is, oblique incidence), the “+x mode” and “−x mode” are unbalanced. As a result, θ

_{inc}can be found by comparing the “+x mode” and “−x mode.” In our device, the “+x mode” and “−x mode” are observed by placing two detectors in the slab waveguide. The two detectors are named “+x detector” and “−x detector,” respectively. The “+x detector” is placed to the right of the grating structure and it serves to detect the power of the “+x mode;” similarly, the “−x detector” is placed to the left of the grating structure and it detects the power of the “−x mode.” The photodetectors in the waveguide can be implemented by following available architectures in [16,17,18].

_{inc}values. In our design, the incident light is coupled to the waveguide modes via the second-order Floquet mode predominantly. Consequently, transverse resonance occurs when the following condition is satisfied

_{inc}= 2πf

_{inc}/c, c is the speed of light in free space, and is the wavenumber along x for the fundamental guided mode in the slab waveguide. It is noted that is non-linear with respect to the frequency. The transverse resonant frequency for θ

_{inc}= 0 is denoted as “f

_{0}.” The transverse resonant frequency increases with the increase of θ

_{inc}, as shown in Figure 2. Bandwidth of the resonance is measured by finding the frequencies at which the photodetector’s output drops by 3 dB with respect to the value at the resonant frequency. Further, quality factor is defined as the ratio between the resonant frequency and the bandwidth.

_{0 }is chosen to be close to f

_{inc}and greater than f

_{inc}; and, the offset between f

_{0}and f

_{inc}is characterized by

_{inc}when f

_{inc}is a constant frequency. Meanwhile, output of the “−x detector” exhibits variation with respect to θ

_{inc}as well. It is not a difficult task to sketch the outputs of the “−x detector” based on Figure 2, as negative θ

_{inc}for the “−x detector” is equivalent to positive θ

_{inc}for the “+x detector.” Since f

_{0}and f

_{inc}are close to each other, it is possible to derive the value of θ

_{inc}by observing the outputs of the two detectors if the incident direction is not far off the normal direction. If θ

_{inc}is too large, the outputs of both “+x detector” and “−x detector” would be too weak and hence unreliable. In order to enlarge the detectable range of θ

_{inc}, one feasible way is to reduce the device’s quality factors because lower quality factors lead to wider bandwidths for the curves in Figure 2. Nevertheless, wider bandwidths unavoidably diminishes the detection sensitivity for θ

_{inc}. Other than quality factors, another important design parameter is ∆

_{f}. It is observed that, larger Δ

_{f}results in larger detectable range for θ

_{inc}. However, large ∆

_{f}reduces the coupling efficiencies around θ

_{inc}= 0. In Section 3, the device’s performances with respect to various design parameters are shown by some numerical results.

## 3. Numerical Results

_{g}= 0.26 μm, t

_{wg}= 0.22 μm, and t

_{b}= 2 μm, (please refer to Figure 1 for the definitions of these parameters; thickness of the substrate has negligible impact on the device, according to our observations). The refraction index of box layer is 1.48; and, the other three layers are made of silicon with refraction index 3.48. The incident light is a Gaussian beam with waist radius 23.4 Λ and frequency f

_{inc}= 0.646 c/a, where a = 1 μm; in addition, electric field of the incident light is polarized along y direction. The two detectors are modeled by integrating the Poynting vectors along +x or –x direction within the waveguide. As for each detector, coupling efficiency is defined as P

_{d}/P

_{inc}

_{,}, where P

_{d}is the detector’s output and P

_{inc }is calculated by integrating the incident light’s power density by the detector’s aperture. As discussed in Section 2, our device’s performance is dictated by two major design parameters: quality factors and ∆

_{f}. In the remainder of this section, various values of these two design parameters are employed to adjust the angle detector’s performance. The quality factors are controlled by N, the number of grating elements (apparently, the larger N is, the higher the quality factors are).

_{f}=0.88% and N=31. Since f

_{inc}< f

_{0}in our design, the “−x output” is stronger than the “+x output” for positive θ

_{inc}. The choice of ∆

_{f}makesf

_{inc}coincide with the transverse resonant frequency of θ

_{inc}= 2° for the “−x detector.” As a result, when the ratio between “−x output” and “+x output” is plotted in Figure 3(c), it exhibits a steep increasing slope in range θinc Є [0, 2°]. When θinc is negative, the two detectors’ outputs would be “exchanged:” the ratio between “+x output” and “−x output” exhibits a steep slope in range θinc Є [–2°, 0]. The data for negative θinc are symmetric to those for positive θinc, hence are not shown in Figure 3. Therefore, from the two detectors’ outputs, θ

_{inc}can be reliably derived when it falls in the range −2° < θ

_{inc}< 2°.

**Figure 3.**Numerical results for the angle detector. (

**a**) Coupling efficiency of the “+x detector.” (

**b**) Coupling efficiency of the “−x detector.” (

**c**) Ratio between the two detectors’ outputs.

_{inc}can be enlarged by reducing the quality factors. This analysis is verified by numerical results in Figure 4. There are three curves in Figure 4. One of them is the same as that in Figure 3(c), with N = 31. On the basis of the curve in Figure 3(c), the other two curves in Figure 4 are obtained with N = 25 and N = 21. As expected, the reduction of N diminishes the device’s quality factors, and thus, increases the detectable range. To be specific, the detectable range is [−2°, 2°] when N = 31; it is increased to [−2.5°, 2.5°] with N = 25 and further increased to [−3.5°, 3.5°] with N = 21. Nevertheless, the curves of “N = 21” and “N = 25” have smaller slopes in the range [0, 2°] compared to the slope of “N = 31” curve, which means that lower quality factors result in lower measurement sensitivity for θinc.

_{f}on the device’s performance. One of the two curves in Figure 5 is the same as that in Figure 3(c), where ∆

_{f}=0.88%. The other curve in Figure 5 is generated by increasing ∆

_{f}to 1.03% and with all the other parameters unchanged. It is observed that, the increase of ∆f enlarges the detectable range from [−2°, 2°] to [−4°, 4°]. As a price, the coupling efficiencies around θinc = 0 drop with the increase of ∆f. To be specific, the coupling efficiencies of both detectors at surface normal incidence are 17% when ∆

_{f}= 0.88%; and they drop to 11% when ∆

_{f}increases to 1.03%.

_{f}= 1.03% curve of Figure 5. The device configuration is shown in Figure 6(a). The incident wave is a continuous wave at f

_{inc}= 0.645 c/a. In Figure 6(b–e), electric field E

_{y}at a certain time moment is plotted, with four different incident angles respectively. In Figure 6(b–e), the strongest positive field intensity is represented by dark red color, the strongest negative field intensity is represented by dark blue color, and white color stands for zero field intensity, as specified at the end of Figure 6. Two guided modes (which are traveling toward +x and −x directions, respectively) can be clearly identified in the waveguide. When θ

_{inc}= 0, the two modes are equally strong. With the increase of incident angle, the two modes become more and more unbalanced. When θ

_{inc}= 3°, the −x mode is much stronger than the +x mode.

**Figure 6.**Field distribution plots with four different incident angles. (

**a**) Device configuration. (

**b**) Field distribution plot with θ

_{inc}=0. (

**c**) Field distribution plot with θ

_{inc}=1°. (

**d**) Field distribution plot with θ

_{inc}=2°. (

**e**) Field distribution plot with θ

_{inc}=3°.

## 4. Conclusions

## Acknowledgments

## References and Notes

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**MDPI and ACS Style**

Saha, T.K.; Lu, M.; Ma, Z.; Zhou, W.
Design of an Angle Detector for Laser Beams Based on Grating Coupling. *Micromachines* **2012**, *3*, 36-44.
https://doi.org/10.3390/mi3010036

**AMA Style**

Saha TK, Lu M, Ma Z, Zhou W.
Design of an Angle Detector for Laser Beams Based on Grating Coupling. *Micromachines*. 2012; 3(1):36-44.
https://doi.org/10.3390/mi3010036

**Chicago/Turabian Style**

Saha, Tapas Kumar, Mingyu Lu, Zhenqiang Ma, and Weidong Zhou.
2012. "Design of an Angle Detector for Laser Beams Based on Grating Coupling" *Micromachines* 3, no. 1: 36-44.
https://doi.org/10.3390/mi3010036