High-resolution micro-grating accelerometer based on a gram-scale proof mass

Shan Gao; Shan Gao; Zhen Zhou; Zhen Zhou; Zhen Zhou; Yu Zhang; Yu Zhang; Keke Deng; Keke Deng; Lishuang Feng; Lishuang Feng; Lishuang Feng

doi:10.1364/OE.27.034298

1. Introduction

Diffraction grating interferometry is an attractive method for microscale displacement sensing due to its high sensitivity, and it has been widely used in the Micro-Electro-Mechanical System(MEMS) sensors [1–8]. Micro-grating accelerometer, based on diffraction grating interferometry, has a rapid development over the past two decades. Its advantages in resolution, size, and power consumption make it have a wide range of applications including highly precise inertial navigation, geophysical seismic sensing and oil-fields applications [9–13].

The fundamental limit to the resolution of an accelerometer is the thermal mechanical noise of the proof mass, which can be expressed as [14]

(1)$${a_{tm}} = \sqrt {\frac{{8\pi {k_b}T{f_0}}}{{mQ}}} $$

where ${k_b}$ is Boltzmann’s constant, T is ambient temperature, ${f_0}$ is resonant frequency, m is mass, and Q is the quality factor of resonance. The accelerometer manufactured by the MEMS approach has a mass of milligrams due to the limitation of the process. In order to reduce thermal mechanical noise, accelerometers require a high vacuum to improve the quality factor. N. A. Hall et al. achieved a quality factor of 100 and a mechanical thermal noise of 44.1 ng/√Hz through vacuum packaging [11]. However, vacuum packaging is prone to failure, which reduces the vacuum of the device and results in the performance degradation of the accelerometer [15,16].

We consider another way to reduce mechanical thermal noise. If the mass of the proof mass is improved from milligram-scale to gram-scale, the accelerometer still has a very low level of mechanical thermal noise without vacuum packaging. However, it is difficult to realize a proof mass in gram-scale for accelerometers based on grating interferometry. The accelerometer demonstrated by N. C. Loh et al. has achieved a high mass of 40 mg, but there is still a big gap from gram-scale [10]. If we select a 500-micron-thick silicon wafer to process the proof mass, the size exceeds 29 mm ${\times} $ 29 mm when the proof mass is in gram-scale, and the cantilever beams cannot bear such a heavy proof mass.

We use mechanical processing to process proof mass and MEMS processing to process gratings. Finally, we assemble the two parts. However, the integration tolerance between grating and proof mass is very strict, so it is difficult to form a high contrast ratio optical interference system by mechanical assembly. The contrast ratio of diffraction grating interferometry is a significant characteristic which directly affects the resolution of a micro-grating accelerometer. Previous researches have shown some numerical analyses of contrast ratio [17–19]. However, there has not been a complete analytic model of contrast ratio to provide suggestions of high contrast ratio designs or adjustment methods. The contrast ratio is detrimental to the performance of micro-grating accelerometers, and it is necessary to analyze the influence mechanism of integrated parameters on contrast ratio.

In this paper, an analytic model for the mechanism of the effects of integrated parameters on contrast ratio is established. A prototype with a proof mass of gram-scale is manufactured. A contrast ratio improvement method based on a scanning slit beam profiler is proposed to improve the optical sensitivity of the accelerometer effectively. Experimental results show that the accelerometer with a proof mass of gram-scale has a superior performance in terms of noise floor and bias stability.

2. Sensing principle

The micro-grating accelerometer consists of a proof mass, cantilever beams, a transparent substrate, and a diffraction grating, as shown in Fig. 1. A vertical cavity surface emitting laser (VCSEL) is chosen as the light source. When the light from the VCSEL illuminates the transparent substrate, a portion of the incident light reflects directly by the diffraction grating, while another portion travels through the diffraction grating and reflects at the upper surface of the proof mass. The two reflecting light beams are diffracted by the grating and produce the far field diffraction, where the intensities of diffraction orders are functions of gap change between grating and mirror due to the interference. When there is acceleration along the sensitive direction, the intensity of the diffracted field is modulated by the distance between the mirror and grating with the sensitivity of a Michelson-type interferometer [20].

Fig. 1. Schematic diagram of the micro-grating accelerometer.

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Based on scalar diffraction theory, the intensities of the 0^th and ± 1^st interferential diffraction orders can be expressed as a function of the distance d between the mirror and grating [21]

(2)$$\begin{array}{l} {I_0} = {I_{in}}{\cos ^2}\left( {\frac{{2\pi d}}{\lambda }} \right),\\ {I_{ {\pm} 1}} = \frac{{4{I_{in}}}}{{{\pi ^2}}}{\sin ^2}\left( {\frac{{2\pi d}}{\lambda }} \right), \end{array}$$

where ${I_0}$ is the intensity of 0^th order, ${I_{ {\pm} 1}}$ is the intensity of ± 1 order, ${I_{in}}$ is the intensity of incident laser intensity, and $\lambda$ is the wavelength of the VCSEL. Either of the diffraction orders can be used to sense the displacement of the proof mass by detecting the intensity change of the corresponding order.

The contrast ratio of any order can be defined by

(3)$$C = \frac{{{I_{\max }} - {I_{\min }}}}{{{I_{\max }} + {I_{\min }}}},$$

where ${I_{\max }}$ and ${I_{\min }}$ are maximum and minimal light intensities of any diffraction order received by the detector during a cyclical fluctuation. The maximum of C can reach 1 in an ideal situation where the minimum light intensities obtained by the detector is zero. Actually, in an actual structure, the contrast ratio cannot reach the maximum due to the unsatisfactory structural parameters, which caused by the limitations of materials and processes. Therefore, it is necessary to establish a theoretical model to analyze the effect of structural parameters on the contrast ratio.

3. Analysis of the contrast ratio

A VCSEL with a center wavelength of 940nm is used as the light source of the sensor. VCSEL has the advantages of small size and low consumption, but the output beam has a large divergence angle. Therefore, an aspherical lens is used to collimate the VCSEL beam. The MFD (mode field diameter) is 1mm after collimation.

The output beam of VCSEL can be regarded as a fundamental mode Gaussian beam, and the amplitude of electric field are given by [22,23]

(4)$$E({x, y, z} )= \frac{A}{{\omega (z)}}\exp \left[ {\frac{{ - ({x^2} + {y^2})}}{{{\omega^2}(z)}}} \right] \times \exp \left\{ { - \textbf{i}k\left[ {\frac{{{x^2} + {y^2}}}{{2R(z)}} + z} \right] + \textbf{i}\varphi (z)} \right\},$$

where A is the amplitude factor, w(z) is the beam diameter and is given by

(5)$$\omega (z) = {\omega _0}\sqrt {1 + {{\left( {\frac{{\lambda z}}{{\pi \omega_0^2}}} \right)}^2}}$$

is the beam diameter,

(6)$$R(z) = z\left[ {1 + {{\left( {\frac{{\pi \omega_0^2}}{{\lambda z}}} \right)}^2}} \right]$$

is the wavefront radius of curvature,

(7)$$\varphi (z) = \arctan \frac{{\lambda z}}{{\pi \omega _0^2}}$$

is a phase factor, and ${\omega _0}$ is the radius of the beam waist.

In practical applications, the beam travel distance is usually within 10cm. Considering ${\omega _0}$ is 0.5mm in this paper, the following approximation can be made:

\omega (z) \approx {\omega _0}.

\frac{{{x^2} + {y^2}}}{{2R(z)}} \approx 0.

\varphi (z) \approx 0.

Therefore, the electric field amplitude of the VCSEL after collimation can be approximated as

(8)$$E({x, y} )= \frac{A}{{{\omega _0}}}\exp \left[ { - \frac{{{x^2} + {y^2}}}{{\omega_0^2}}} \right],$$

that is a plane wave with an amplitude of Gaussian distribution.

The beam path of 0 order can be simplified as shown in Fig. 2. $\varphi$ is the incident angle of the light source, $\theta$ is the angle between the grating and the mirror, d is the distance between the grating and the mirror, h is the thickness of the glass substrate, n is the refractive index of the glass substrate, L is distance between the detector and the upper surface of the glass substrate, $\Delta d$ is the center distance of the spots on the detector plane, ${d_1}$ is the center distance of the spots on the grating, ${d_2}$ is the center distance of the spots on the upper surface of the glass substrate, ${E_1}$ and ${E_2}$ are the electric field amplitude of the two beams at detector plane.

Fig. 2. Beam path diagram of 0 order in diffraction grating interferometry.

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According to Eq. (8), ${E_1}$ and ${E_2}$ can be expressed as

(9)$$\begin{array}{l} {E_1}({x, y} )= \frac{{{A_1}}}{{{\omega _0}}}\exp \left[ { - \frac{{{{({x - {{\Delta d} \mathord{\left/ {\vphantom {{\Delta d} 2}} \right.} 2}} )}^2} + {y^2}}}{{\omega_0^2}}} \right],\\ {E_2}({x, y} )= \frac{{{A_2}}}{{{\omega _0}}}\exp \left[ { - \frac{{{{({x + {{\Delta d} \mathord{\left/ {\vphantom {{\Delta d} 2}} \right.} 2}} )}^2} + {y^2}}}{{\omega_0^2}}} \right]. \end{array}$$

The electric field amplitude of the interference light of the two beams can be expressed as

(10)$${E_3}({x, y} ) = \sqrt {\textbf{2}{E_1}{E_2}} = \frac{{{A_3}}}{{{\omega _0}}}\exp \left[ { - \frac{{{x^2} + {y^2}}}{{\omega_0^2}}} \right],$$

where

(11)$${A_3} = \sqrt {\textbf{2}{A_1}{A_2}} \exp \left( { - \frac{{\Delta {d^2}}}{{4\omega_0^2}}} \right).$$

It can be seen from Eq. (10) that the interference light still is a fundamental mode Gaussian beam and has the same MFD as the incident light, as shown in Fig. 3(b). According to Eq. (11), the amplitude of ${E_3}$ is relevant to the center distance of the spots on the detector plane and the amplitudes of ${E_1}$ and ${E_2}$.

Fig. 3. Schematic diagram of light spots electric field amplitude distribution at detector plane before and after light interference.

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Besides, although ${E_1}$ and ${E_2}$ can be regarded as plane waves in the application range, the two phase planes are not parallel when an angle between the grating and the mirror exists. The non-parallel phase planes stripe the interference spot, which also causes changes in total light intensity of ${E_3}$, as shown in Fig. 3(c).

The electric field amplitudes of the two reflected beams, the separation of the spots on the detector plane, and the angular deviation of beams due to the angle between the grating and the mirror all lead to a decrease in contrast ratio.

3.1 Amplitudes of E₁ and E₂

The interference is incomplete if the amplitudes of ${E_1}$ and ${E_2}$ are not equal, which reduces the contrast ratio. Assuming that D is the duty ratio of grating metal stripe width, ${R_1}$ is the reflectivity of the mirror, ${R_2}$ is the reflectivity of the grating upper surface, and ${I_0}$ is the total light intensity of the 0 diffraction order, the intensity of the reflected light from the mirror can be expressed as ${R_1}D{I_0}$ and the intensity of the reflected light from the grating surface can be expressed as ${R_2}({1 - D} ){I_0}$. According to the relationship between light intensity and light amplitude, there is

{A_1} \propto \sqrt {{R_1}D} {\bf ,}\;{A_2} \propto \sqrt {{R_2}({1 - D} )} .

According to the principle of interference, the proportional coefficient of the influence of the amplitudes of ${E_1}$ and ${E_2}$ on contrast ratio is

(12)$${C_\alpha } = \frac{{2{A_1}{A_2}}}{{A_1^2 + A_2^2}}.$$

${C_\alpha }$ obtains the maximum value of 1 when Eq. (12) satisfies ${R_1}D = {R_2}({1 - D} )$. According to the above formula, there is an optimum duty ratio to reach the maximum contrast ratio when the materials of grating and mirror are selected.

Figure 4 shows the contrast ratio with different duty ratios and reflectivity. For general grating interferometry, whose ratio of ${R_1}$ to ${R_2}$ is between ${1 \mathord{\left/ {\vphantom {1 3}} \right.} 3}$ and 3, and the duty cycle is between 0.3 and 0.7, the value of ${C_\alpha }$ is usually higher than 0.8.

Fig. 4. Numerical evaluation for ${C_\alpha }$ as a function of duty ratio D and the ratio of R1/R2 with a shadow zone in general grating interferometry parameters.

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3.2 The separation of the spots

When the two beams from the grating surface and the mirror are irradiated on the detector, there is a separation between the two spots. The larger the spots separate, the worse spatial correlation of the spots at the detector plane is. The geometric relationship in Fig. 2 can be easily derived:

{d_1} = d\tan \varphi + d\tan ({\varphi + 2\theta } ),

{d_2} = {d_1} + h\left( {\frac{1}{{\sqrt {{{{n^2}} \mathord{\left/ {\vphantom {{{n^2}} {{{\sin }^2}({\varphi + 2\theta } )- 1}}} \right.} {{{\sin }^2}({\varphi + 2\theta } )- 1}}} }} - \frac{1}{{\sqrt {{{{n^2}} \mathord{\left/ {\vphantom {{{n^2}} {{{\sin }^2}\varphi - 1}}} \right.} {{{\sin }^2}\varphi - 1}}} }}} \right),

(13)$$\Delta d = {d_2}\cos \varphi + ({L - {d_2}\sin \varphi } )\tan ({2\theta } ).$$

The first-order Taylor expansion of ${d_2}$ at $\theta = 0$ is

{d_2} = {d_1} + \frac{{2{n^2}\cos \varphi }}{{{{({{n^2} - {{\sin }^2}\varphi } )}^{\frac{3}{2}}}}}\theta h.

Considering $\varphi$ and $\theta$ are both small angles, Eq. (13) can be simplified as

{d_1} = 2({\varphi + \theta } )d,

{d_2} = 2({\varphi + \theta } )d + 2\theta \frac{h}{n},

(14)$$\Delta d = 2d \cdot \varphi + \left( {L + \frac{h}{n} + d} \right) \cdot 2\theta .$$

It can be seen from Eq. (14) that the separation of the spots at the detector plane is caused by two parts: one part is the incident angle, which separates the lights in the air gap between the grating and the mirror; the other part is the angle between the grating and the mirror, which separates the lights in the propagation of the reflecting surface to the detector.

Supposing that the phase planes of ${E_1}$ and ${E_2}$ are parallel at the detector plane, the proportional coefficient of the influence of the separation of the spots on contrast ratio can be expressed as

(15)$${C_\beta } = \frac{{{{\left[ {\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{E_1}(x, y){E_2}(x, y)\textbf{d}x\textbf{d}y} } } \right]}^2}}}{{\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {E_1^2(x, y)\textbf{d}x\textbf{d}y \cdot \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {E_2^2(x, y)\textbf{d}x\textbf{d}y} } } } }}.$$

Figure 5 and Fig. 6 show the light field distribution and contrast at different spot separation distances respectively. When the spots separation distance is less than one-fifth of MFD, the coefficient ${C_\beta }$ is greater than 0.7, which is in excellent condition, while when the spots separation distance is greater than half of MFD, the contrast ratio is lower than 0.15, making it difficult to achieve high-precision detection.

Fig. 5. Schematic diagram of light spots electric field amplitude and interference field distribution at detector plane when the spots separation distances are one-fifth of MFD and one-half of MFD.

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Fig. 6. Numerical evaluation for ${C_\beta }$ as a function of $\Delta d$.

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3.3 Phase modulation

The previous discusses the effects of light spots intensity ratio and separation distance at detector plane on contrast ratio, both of which are considered on the amplitude term of Eq. (4), due to the angle between the grating and the mirror, the phase planes of the two incident spots are not parallel, modulating the interference spot with a sine wave and reducing the contrast ratio.

For any interference spot, the phase difference in the detector plane of the two rays participating in the interference is

(16)$$\phi = \frac{{4\pi d}}{\lambda }.$$

Supposing the coordinates of the detector plane are $({{x_1},{y_1}} )$, interference points at different ${x_1}$ coordinates have different phase differences. As shown in Fig. 7, the relationship of ${x_1}$ and d can be expressed as

(17)$${{\partial d} \mathord{\left/ {\vphantom {{\partial d} {\partial {x_1}}}} \right.} {\partial {x_1}}} = \theta .$$

Fig. 7. Beam path diagram of an interference point of 0 order light at the detector plane.

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When the phase difference changes $2\pi$ at the detector plane, the position of the interference point changes

(18)$$\Delta {x_1} = \frac{\lambda }{{2\theta }}.$$

Interference fringes with an interval of $\Delta {x_1}$ are formed at the detector plane. That is to say, the interference spot is subjected to a sinusoidal modulation with a spatial period of $\Delta {x_1}$. When the acceleration causes the distance d to change, the interference fringes move. The normalized intensity of all interference fringes can be expressed as

(19)$$I = \frac{{\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}[{1 + \sin ({2\pi {x \mathord{\left/ {\vphantom {x {\Delta {x_1}}}} \right.} {\Delta {x_1}}} + {{4\pi d} \mathord{\left/ {\vphantom {{4\pi d} \lambda }} \right.} \lambda }} )} ]\cdot E_3^2(x, y)\textbf{d}x\textbf{d}y} } }}{{\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {E_3^2(x, y)\textbf{d}x\textbf{d}y} } }}.$$

The proportional coefficient of the influence of the phase modulation on contrast ratio can be expressed as

(20)$${C_\gamma } = {I_{\max }} - {I_{\min }}.$$

When the number of interference fringes increases, the contrast ratio decreases rapidly, as shown in Fig. 8 and Fig. 9.

Fig. 8. Schematic diagram of interference fringes and numerical evaluation for light intensity as a function of phase when the number of interference fringes is exactly one to three.

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Fig. 9. Numerical evaluation for ${C_\gamma }$ as a function of $\theta$.

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In summary, the contrast ratio of diffraction grating interferometry can be expressed as

(21)$$C = {C_\alpha } \cdot {C_\beta } \cdot {C_\gamma }.$$

It can be seen that the angle between the grating and the mirror not only affects the spatial coherence of the interference light but also introduces a phase modulation, which is the main factor leading the decrease of the contrast ratio. The system parameters of the actual design are shown in Table 1. If the contrast ratio is above 0.5, the angle between the grating and the mirror needs to be less than 0.028 degrees, which is difficult to achieve such a small angle error by mechanical assembly. Therefore, an angle fine-tuning mechanism should be added to correct the angle. It can be seen from Fig. 9 the optical system has a good contrast ratio when the fringes cannot be observed.

Table 1. The system parameters of the actual design

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4. Experiments and discussions

The experimental setup for testing the 0^th order interference filed distribution and improving the contrast ratio is shown in Fig. 10(a). The sensing probe is integrated by a grating, a moving mass, and a magnetic actuator and the cross-sectional structure of the probe is shown in Fig. 10(b). The double-layer cantilever beam structure, which is made of beryllium bronze sheet with a thickness of 100 µm by laser cutting, supports the copper proof mass with a mass of 1.78 grams. The upper surface of the proof mass is polished to be a mirror. A copper coil is wound on the lower side of the mass, and the coil is placed in the magnetic actuator. When current flows through the coil, the mass is subjected to a force in the axial direction, changing the distance between the grating and the mirror. A VCSEL (940nm wavelength, Model 940Q-B091, Vixar Inc.) is collimated by an aspherical lens and irradiates the grating of the sensing probe. The 0^th order interference field is detected by a scanning slit beam profiler (Model NS2s-GE/3.5/1.8-STD, Ophir-Spiricon LCC).

Fig. 10. (a) Schematic diagram of the experimental setup for testing the 0th order interference filed distribution (b) Cross-sectional structure of the sensing probe (c) Photograph of the fabricated sensing probe.

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The distance between the grating and the mirror was gradually changed 470nm, which is the length of half-wavelength, by changing the coil current. The 0^th order interference filed distribution at different distances was recorded by the beam profiler, as shown in Fig. 11(a). In Fig. 11, the ‘relative phase’ corresponds to the change from 0 to 470 nm, the ‘position’ is the coordinates of the detector plane in the beam profiler, each curve in the waterfall plot is a group of interference filed distribution data after each distance variety, and the ‘normalized intensity’ is the integral of the total intensity of each curve. It can be seen that the interference spot has stripes with an interval of 250 µm. When the relative distance between the grating and the mirror varies, the fringes move, but the total intensity of the spot has no distinct change.

Fig. 11. Interference filed distribution of 0^th order at different distances detected by beam profiler before (a) and after (b) adjustment.

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The spatial angle between the glass substrate and the mirror can be fine-tuned by the adjusting frame on the sensing probe. While adjusting the angle, the spot was observed in real time with beam profiler until no stripes were observed. The 0^th order interference filed distribution at different distances after adjustment is shown in Fig. 11(b). The contrast of the spot is increased to 0.75.

The experimental setup for the static gravity measurement was established to characterize the performance of the accelerometer, as shown in Fig. 12(a). The sensing probe was fixed on a numerical control rotary table, and the input acceleration was generated by revolving the rotary table from 0° to 90°, which means that the input acceleration was variable from 0 g to 1 g. The experimental data were recorded at different values of gravitational acceleration, as shown in Fig. 12(b), and the intensity varies associated with the variation of acceleration with a period of 55.4 mg. According to Eq. (2), the light intensity varies with the distance between the grating and mirror, and its period is half wavelength of the light source, so the mechanical sensitivity of the accelerometer is 8.48µm/g.

Fig. 12. (a) Schematic diagram of the experimental configuration for the static gravity measurement (b) experimental curves for the 0^th order and 1^st order intensity versus acceleration.

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In order to describe the resolution of accelerometer, the noise floor and bias stability are measured through detecting 1^st diffracted order intensity. Figure 13(a) shows the static test results of the accelerometer samples analyzed by power spectral density (PSD) method, and the noise floors are 137 ng/√Hz at low-frequency area from 1 Hz to 10 Hz. Figure 13(b) shows the test results of bias stability by measuring Allan deviation of the output (8 hours’ data), and the minimum of the curve suggests bias stability of 3.1 µg at 1-second measurement interval.

Fig. 13. The noise floor (a) and the Allan deviation (b) of the accelerometer.

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Compared with the micro-grating accelerometer made by our group before [19], the parameters and test results are shown in Table 2. The mechanical resonance frequency decreases as the mass increases to gram-scale, and the mechanical thermal noise is decreased from 211 ng/√Hz to 4.4 ng/√Hz according to Eq. (1). In addition, mechanical sensitivity is greatly improved with the increase of mass. Acceleration noise introduced by the light source and circuit is related to mechanical sensitivity. The greater the mechanical sensitivity, the smaller the acceleration noise introduced by light source and circuit. Greater mechanical sensitivity reduces the noise floor from 0.9 mg/√Hz to 137 ng/√Hz.

Table 2. Comparison of parameters and test result

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5. Conclusion

We demonstrate an accelerometer based on a gram-scale proof mass in this work. The theoretical model for the contrast ratio of a micro-grating accelerometer is established based on Gaussian beam theory, providing theoretical support for design parameters of high-performance micro-grating accelerometers. The adjustment method based on a scanning slit beam profiler effectively improves the contrast ratio of 0^th order to 0.75, which is in excellent agreement with the theoretical analysis. Compared to our former design, the gram-scale proof mass greatly reduces mechanical thermal noise and improves mechanical sensitivity. Experiment results indicate the noise floor (from 1 Hz to 10 Hz) is decreased from 0.9 mg/√Hz to 137 ng/√Hz, and the bias stability (at 1-second measurement interval) is decreased from 0.35 mg to 3.1 µg.

Funding

Defense Industrial Technology Development Program (JCKY201601C006).

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Parameter	symbol	Value
Wavelength of laser	$λ$	940nm
Output power of laser	$I_{i n}$	1.3mW
Radius of beam waist	$ω_{0}$	500 µ m
Grating period	$Λ$	4 µ m
Grating ratio	D	0.5
Reflectivity of mirror	$R_{1}$	0.85
Reflectivity of grating upper surface	$R_{2}$	0.76
Distance from grating to mirror	$d$	100 µ m
Incident angle of laser	$φ$	11.8°
Thickness of glass substrate	$h$	500 µ m
Refractive index of glass substrate	$n$	1.45
Distance from detector to glass substrate	$L$	80mm

Specification	Proposed accelerometer	[19]
Proof mass	1.78g	8.1mg
Resonant frequency	175.8Hz	1676Hz
Quality factor of resonant	6.1	4.9
Mechanical sensitivity	8.48µm/g	89.7nm/g
Thermal mechanical noise	4.4 $ng / \sqrt{Hz}$	211 $ng / \sqrt{Hz}$
Noise floor	137 $ng / \sqrt{Hz}$	0.9 $mg / \sqrt{Hz}$
Bias stability	3.1µg	0.35mg

Parameter	symbol	Value
Wavelength of laser	$λ$	940nm
Output power of laser	$I_{i n}$	1.3mW
Radius of beam waist	$ω_{0}$	500 µ m
Grating period	$Λ$	4 µ m
Grating ratio	D	0.5
Reflectivity of mirror	$R_{1}$	0.85
Reflectivity of grating upper surface	$R_{2}$	0.76
Distance from grating to mirror	$d$	100 µ m
Incident angle of laser	$φ$	11.8°
Thickness of glass substrate	$h$	500 µ m
Refractive index of glass substrate	$n$	1.45
Distance from detector to glass substrate	$L$	80mm

Specification	Proposed accelerometer	[19]
Proof mass	1.78g	8.1mg
Resonant frequency	175.8Hz	1676Hz
Quality factor of resonant	6.1	4.9
Mechanical sensitivity	8.48µm/g	89.7nm/g
Thermal mechanical noise	4.4 $ng / \sqrt{Hz}$	211 $ng / \sqrt{Hz}$
Noise floor	137 $ng / \sqrt{Hz}$	0.9 $mg / \sqrt{Hz}$
Bias stability	3.1µg	0.35mg

High-resolution micro-grating accelerometer based on a gram-scale proof mass

Abstract

1. Introduction

2. Sensing principle