Double-grating with multiple diffractions enabled small angle measurement

Jinyao Wang; Chen Liu; Siyi Qin; Guangsong Zhu; Yeding Shao; Songnian Fu; Deming Liu

doi:10.1364/OE.27.005289

1. Introduction

Accurate and precise small angle measurement is essential for the installation of precision mechanical assemblies, optical system and the determination of deflection angle caused by turbulence or vibration [1–5]. Several characterization solutions have been proposed to realize precision angle measurement, such as auto-collimation method [1–3], internal reflection method [5–7], and laser interferometric method [8–14]. The auto-collimation method uses an objective to project the rotation angle of the incident light on the image plane. The deflection angle can be measured with the help of two-dimension image device by analyzing the light spot displacement [1–3]. Such angular resolution is determined by the focal length of the lens and the spatial resolution of the image device. The internal reflection method utilizes the characteristics of the reflectance as the functions of the incident angle under the total reflection condition [5–7]. Its nonlinear response can be reduced with the differential method, and its sensitivity can be enhanced by extending the length of the reflection prisms, in order to increase the number of reflections. However, the optical power after several reflections is detected by a single photodetector. Those factors, including the beam divergence, the installation deviation of the prisms, and the environmental fluctuation, can bring the power penalty during the angle measurement, leading to the measurement error. The interferometric method is based on the interference of two laser beams that traverse with slightly different optical paths caused by the angular displacement. Various configurations, including the Michelson interferometer [8,9], holographic technique [10], heterodyne interferometry [11], polarization interferometry [12], self-mixing interferometry [13,14], and Fabry-Perot etalon [15], have been put forward by characterizing the phase difference of two interference signals. However, the interferometric method is sensitive to the environmental perturbation, the pixel pitch of image device, and phase retrieval algorithms. For all methods mentioned above, the angle to be measured can be transformed to the spot displacement on the image device, optical power variations detected by the power meter, and the optical phase variation derived from the interference pattern on the image device. When the small angle to be measured is at the order of micro-radian level, the characterization setup is bulky, complex, and expensive, because lens with a focal length of meter level and image device with pixel pitch of less than micron are indispensable. Obviously, the resolution of angle measurement is constrained by those key optical components. In order to improve the angular resolution of those methods beyond the limitation of the image device or the power meter, additional optical setups are necessary to amplify the measured parameter before it is received by the conventional angle characterization setup. For example, a telescope module can be inserted into the autocollimator system in order to improve the angular resolution by amplifying the deflection angle [16,17]. Nevertheless, it is still challenging to obtain a compact telescope module.

In this paper, we propose a deflection angle amplification method based on two un-parallel blazed gratings called the double-grating configuration. The configuration of double gratings was previously used in various applications such as the double grating interferometer and the grating compressor [18,19]. In our special double gratings configuration, the optical beam can be diffracted repeatedly and the deflection angle is magnified correspondingly. Such configuration can be inserted before other angular measurement system to improve its angular resolution. According to our experimental results, the double-grating configuration can realize the angle magnification with more than 40 times and micro-radian level angular resolution accordingly.

2. Operation principal

It is well known that diffraction grating can diffract the input light into a number of discrete diffracted orders governed by the following grating equation [20],

\sin θ - \sin i = m \frac{λ}{d}, m = 0, \pm 1, \pm 2, \dots

where m is the order of the diffracted wave, i is the incident angle, θ is the diffraction angle, λ is the operation wavelength, and d is the period of the grating. We define the incident angle as positive when the incidence light and the diffraction light are at the different side with respect to the grating normal, and vice versa.

Due to the polarization independence of the grating diffraction angle, the influence of polarization can be reasonably ignored in the following discussions. When we differentiate the grating equation with respect to the incidence angle i, the angular magnification of the grating diffraction derived from Eq. (1) can be written as,

M = \frac{d θ}{d i} = \frac{\cos i}{\cos θ}

Under the condition of $| i | \neq | θ |$ , we can obtain $\cos i / \cos θ \neq 1$ , indicating that the angle of the diffraction can be magnified or compressed with respect to the incident angle. Considering that the angular magnification M is equal to 1, and the diffraction order used is not 0, we have i = −θ. Next, when we apply such relation into the grating equation, we can derive i = −arcsin(mλ/2d). Meanwhile, by taking the derivative of Eq. (2) for the diffraction order m>0, we can deduce that the magnification is monotone increasing. So when i > −arcsin(mλ/2d), the magnification M is larger than 1. In addition, from sinθ = sini + mλ/d < 1, we can deduce that i < arcsin(1 − mλ/d). Then, when the incident angle satisfy the condition of −arcsin(mλ/2d) < i < arcsin(1 − mλ/d), the diffraction angle is greater than the incident angle. According to above discussion, we can find that the single grating can be used to magnify the deflection angle of the incident light. However, its magnification is not large enough and varies with the incident angle, as shown in the Eq. (2), leading to the inconvenience for the small angle measurement. Therefore, we propose a double-grating configuration to enable the incident light to be diffracted between two gratings repeatedly. As a result, multi-stage angle amplification of the deflected light can be achieved.

Inspired by the idea of multiple reflections arising in Fabry-Perot etalon [14], we replace the mirror in the Fabry-Perot etalon with a grating to magnify the deflection angle directly, owing to the inequality of incident angle and diffraction angle of grating. We design a double-grating configuration, as shown in Fig. 1. The incident light are expected to be diffracted forth and back between two gratings. Those two gratings with parallel grooves are separated by an included angle of α, and the direction of the grooves are along the x-axis. $\vec{S_{n}}$ is the unit direction vectors of the nth incident light, i_n and θ_n are the incident angle and the diffraction angle at the n^th diffraction, respectively. If we consider a special condition that the beam diffracted back by the grating #2 is parallel to the original incident light on the grating #1, $\vec{S_{n + 2}}$ being equal to $\vec{S_{n}}$ , the cyclical effect can be achieved and the incident light can be confined between those two blazed gratings. When the size of the gratings is large enough, the incident beam with the direction of $\vec{S_{1}}$ will be diffracted between two gratings for many times and the angle magnification effect is substantially enhanced.

Fig. 1 Principle of small angle magnification with the double-grating configuration.

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Now, under the incident condition, the total angle magnification after multiple diffractions can be obtained by multiplying the magnification of each diffraction as,

M_{i_{c}} = \frac{d θ_{N c}}{d i_{1 c}} = \frac{\cos i_{N}}{\cos θ_{N}} \frac{\cos i_{N - 1}}{\cos θ_{N - 1}} \dots \frac{\cos i_{2}}{\cos θ_{2}} \frac{\cos i_{1}}{\cos θ_{1}} = {(\frac{\cos i_{1 c}}{\cos θ_{1 c}} \frac{\cos i_{2 c}}{\cos θ_{2 c}})}^{N / 2}

where M_ic is the angle magnification under the condition of special incident angel of i₁_c. N is the even number of diffraction occurred between the double gratings, which is determined by the width of exit window and the length of gratings. When the incident light is deviated from i₁_c, the total angle magnification will change a little. Meanwhile, we find that the angle magnification increases monotonously with respect to the incident angle. Thus, with an appropriate setting of the double gratings, multi-stage amplification can be realized and the angle magnification can be enhanced significantly. Consequently, by using the double-grating configuration, we can transform the small deflection angle of the incident light into a large deflection angle, achieving the enhancement of angle measurement resolution. When both gratings are identical, we can get i₁_c = i₂_c. According to the geometrical relationship in Fig. 1, we can derive Eq. (4) to describe the relationship among the included angle of the two gratings, the incident angle, and the diffraction angle.

α = θ_{1 c} - i_{2 c} = θ_{1 c} - i_{1 c} =arc \sin (\sin i_{1 c} + m \frac{λ}{d}) - i_{1 c}

For the practical application, the normal of the grating #1 is usually set as the reference of angle measurement. Then the incident angle i₁_c of the first diffraction is correspondingly equal to 0. The included angle of the double gratings α should be equal to arcsin(mλ/d).

In the following experimental verification, we use two identical gratings to construct the proposed double-grating configuration. The incident light is deflected by a small angle relative to the normal of grating #1. By measuring and comparing the incident angle and the emergent angle after the multiple diffractions, the effect of deflection angle amplification can be employed for the small angle measurement. Please note that the proposed configuration can only achieve the deflection angle amplification at the YZ plane. If the amplification of the deflection angle at both directions are requested, we can place another double-grating configuration vertically, in order to amplify the deflection angle at each direction.

3. Experiment setup

In order to further investigate the performance of double-grating configuration, we experimentally verify the angle amplification effect, as shown in Fig. 2. The experiment setup can be divided into three parts. The laser and the mirror are used to generate the deflection optical beam. The angle measurement subsystem #2 with a short focal length is used together with the double-grating configuration to detect the spot displacement for the angle measurement. Meanwhile, angle measurement subsystem #1 with a long focal length is used to measure the deflection angle at high angular resolution for the purpose of performance comparison.

Fig. 2 Schematic diagram of the experimental setup with the double-grating configuration.

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In our experiment, we used a DFB laser with operation wavelength of 1550nm as the light source. The beam is firstly expanded and then reflected by a mirror. The mirror is mounted on an optical stage with fine adjustment accuracy of 660µrad per round. The mirror is deflected to generate various deflection angle to be measured. The beam splitter is used to divide the incident light into two beams for the purpose of performance comparison between the conventional auto-collimation method and the proposed double-grating scheme. The auto-collimation angle measurement subsystem #1 with a long focal length is used to measure the deflection angle of the incident light before the double-grating. Meanwhile, the other angle measurement subsystem #2 is used to measure the deflection angle after the double-grating configuration. The infrared image devices (SP928-1550 Ophir Optronics) are used to capture the light spots. Due to the restriction of the lens and image device with a limited angular field of view, the lens #2 and the image device #2 are fixed on a rail to extend the angle measurement range.

The double-grating configuration consists of two identical blazed gratings with parallel grooves. To mitigate the stray light arising from the unused diffraction orders, which may enter the angular measurement system and influence the angular measurement accuracy, we choose an appropriate grating constant for few diffraction orders. Meanwhile, in our experiment, the Littrow gratings do not work under the blazing condition because the measured deflection beam enter the gratings around the grating normal. The power ratio of the desired signal with stray light is more than 20dB.The parameters of the main devices used in the experiment are shown in Table 1. According to our device parameters, the included angle of two gratings is set to 68.43°, which is derived from Eq. (4).

Table 1. Parameter of gratings, lenses, and image devices

View Table

4. Results and discussions

Before the experiment, we carry out a numerical simulation in MATLAB based on the parameters of used components, as shown in Table 1. Each diffraction is calculated using the grating equation. And the iterative formulas are

{\begin{matrix} θ_{n} = arc \sin (\sin i_{n} + m \frac{λ}{d}) \\ i_{n + 1} = θ_{n} - α \end{matrix}

where θ_n and i_n are the diffraction angle and incident angle of the n^th diffraction, respectively. The incident light is diffracted 4 times between the two gratings. The deflection angle is set relative to the normal of grating #1. We calculate the emergent angle of the light beam with the diffraction equation and the angle magnification by taking derivative of the angular curve, as shown in Fig. 3. In accordance with the theoretical analysis, the deflection angle of incident light can be dramatically amplified by the double-grating configuration and the angular magnification is more than 40 for the angle measurement range between −2mrad to 2mrad. Moreover, the simulation curve of the magnification effect also illustrates the characteristic of monotone increasing. Therefore, the double-grating configuration can be used for the deflection angle measurement, and the angular resolution can be theoretically increased by more than 40 times. Furthermore, we can obtain the measurement range of incident angle from −58.32mrad to 3.85mrad for the given grating parameter.

Fig. 3 Simulation and experimental result of the dependence of emergent angle on the incident angle, and the dependence of angle magnification on the incident angle.

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After the simulation, we conduct the experiment. In the double-grating configuration, the incident light is diffracted 4 times and then enters the angle measurement subsystem #2. We can calculate the deflection angle by analyzing the displacement of light spots on the image devices. The experimental results are shown in the inset of Fig. 3, which agree with the simulation excellently. These results verify the feasibility of our proposed solution for small angle measurement. Nevertheless, Fig. 3 also shows that there doesn’t exist a linear relationship between the incident angle and the emergent angle. So, the incident angle cannot be solved with simple treatment. In order to solve the incident angle accurately, we employ the diffraction equation to the emergent angle recursively.

By calculating the displacement of the gravity center of light spot, we can obtain the deflection angle of the light beam. However, in the practical operation, the installation error will inevitably cause a deviation of the parameters from the expected values and subsequently the measurement error occurs. For example, for the auto-collimation angle measurement setup, the distance d₁ between the lens #1 and the image device #1 may not be equal to the focal length of f₁. As for the double-grating structure in Fig. 2, the included angle between two gratings may also not be equal to the ideal value. Moreover, when we set the normal of grating #1 as the reference for the angle measurement, the alignment error between the center of the image devices and the focus lens will definitely bring the systematic measurement error.

To mitigate the measurement error induced by those factors, the method of the least mean square fitting is used to fit the experimental results and the theoretical results. Setting the normal of grating #1 as the reference for the angle measurement, the focused light spot of the incident light at the direction of the normal of the lens may deviate from the central point of the image devices, due to the installation deviation. Accordingly, we define an angular deviation in the angle subsystem #1 and #2 as i₀ and θ₀_, and the parameters l₁ and l₂ are the location of focused spot on the image device #1 and the image device #2, respectively. We can describe the incident angle as i = fi (l₁, d₁, i₀ ), where fi is the function to calculate the beam angle for auto-collimation angle approach. Similarly, the emergent angle can be defined as θ = fo (l₂, d₂, θ₀ ), where fo represent the function to calculate the magnified angle with the help of auto-collimation angle approach, and d₂ is the distance between lens #2 and image device #2. Moreover, the simulation results of emergent angle can be defined as θs = fs (i, α), where fs represents the function of the simulation curve. Through adjusting the variables (d₁, i₀, d₂, θ₀, α) to minimize the mean square error between θ and θs, the corrected parameters can be obtained and used in the following angle measurement experiment. Once these parameters are calibrated, it is not required to employ the fitting process for each measurement.

The measured results of small angle deflection are presented in Fig. 4. Meanwhile, the measurement error due to the use of the double-grating configuration is investigated with respect to the measured result from the auto-collimation angle measurement subsystem #1. The maximum error is less than 6.3µrad, which is close to the measurement accuracy of the subsystem #1. Therefore, the experimental results confirm the validity and accuracy of our method for small angle measurement.

Fig. 4 Experimental results and measurement error by using the double-grating configuration.

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According to the pixel pitch of the image device and the focus length of the imaging lens, the theoretical angle resolution of auto-collimation angle measurement subsystem #1 and #2 is about 7.38µrad and 73.8µrad, respectively. Whereas, when the double-grating configuration is inserted before the auto-collimation angle measurement subsystem #2, its angle resolution can reach the order of micro radian, which is better than that of the auto-collimation angle measurement subsystem #1 with ten times longer focal length lens. These results imply that the angular amplification effect of double-grating configuration can improve the resolution of angle measurement, by the use of traditional lens with short focal length. Thus, lens with short focal length together with the double-grating configuration is capable to measure small angle. As for our characterization setup, the length of double-grating configuration is only about 50mm, which is ten times shorter than that of the angle measurement system #1. Therefore, the compact double-grating configuration can be used to replace conventional long focus lens for high angular resolution measurement. Meanwhile, please note that the angle measurement range of the proposed setup is reduced due to the trade-off between the angle measurement range and the angle measurement resolution. In order to extend the dynamic range with the capability of small angle measurement, a simple method may be useful by employing a controllable deflection mirror and a coarse angle measurement system before the precise angle measurement setup with the double-grating.

5. Conclusion

We have proposed a deflection angle amplification method based on the double-grating configuration and applied the proposed scheme to the small angle measurement. The simulation and experiment are implemented in order to comprehensively investigate the performance of angle magnification arising in the double-grating configuration and its feasibility for the small angle measurement. Since the deflection angle can be monotonously amplified for more than 40 times, the double-grating configuration can substantially improve the angular resolution of measurement. The proposed double-grating configuration has comparable sensitivity with respect to the auto-collimation angle approach with long focal lens, while its size is compact.

Funding

National Natural Science Foundation of China (61575071, 61711530043).

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Grating constant	600 lines/mm
Blazed angle of grating	28°41′
Size of grating	50 mm × 50 mm
Focal length of lens #1, f1	500 mm
Focal length of lens #2, f2	50 mm
Pitch of image device #1, p1	3.69 µm × 3.69 µm
Pitch of image device #2, p2	3.69 µm × 3.69 µm

Double-grating with multiple diffractions enabled small angle measurement

Abstract

1. Introduction

2. Operation principal

3. Experiment setup

4. Results and discussions

5. Conclusion

Funding

References

Cited By

Figures (4)

Tables (1)

Equations (5)

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