1 Introduction

In last three decades, atom interferometry has been widely used in precision measurements [1,2,3,4,5], and gravity measurement is an important application [1, 6,7,8]. Nowadays, the sensitivity of atom interferometry gravimeter (AIG) has reached 4.2 \(\mu\)Gal/Hz\(^{1/2}\) [6] (1 \(\mu\)Gal=1\(\times 10^{-8}\) m/\(\hbox {s}^2\)), and the uncertainty has reached several \(\mu\)Gal [1, 8,9,10,11], which is mainly limited by the effect of wavefront aberrations [8,9,10,11,12]. The effect of wavefront aberrations in AIG is directly related to the figure of optical elements. To reduce the effect of wavefront aberrations in AIG, high-quality optical elements are required. Furthermore, to accurately evaluate the effect of wavefront aberrations, the figure of the optical elements used in AIG should be accurately measured [13]. Usually, this measurement is performed in atmospheric environment [9, 13]. However, to avoid the largest wavefront aberration uncertainty introduced by the bottom window of vacuum chamber, the retro-reflecting mirror and quarter waveplate need to be installed inside vacuum chamber [9]. Since the measurement of optical elements’ figure in vacuum is rarely reported, one cannot clearly answer the question before this work: does the figure of these optical elements change after being installed inside vacuum chamber?

Usually, the figure of optical element could be determined by measuring the wavefront of laser, which is proportional to the figure of optical element. Shack–Hartmann method [13,14,15] is often used. In this work, a figure measurement device based on Shack–Hartmann wavefront sensor has been built, and this device has been used to measure the figure of optical elements installed inside vacuum. For comparison, the figure of optical elements under pressures ranging from 0.058 Pa to \(1\times 10^5\) Pa has been measured. The root-mean-square (RMS) value of the figure does not change within the error range, and the measured figure distribution of optical elements does not change significantly when the pressure varies from 0.058 Pa to atmospheric pressure.

The article is organized as follows: the principle of the figure measurement in vacuum is described in Sect. 2, and experimental setup is also described in detail in this section. Experimental results are displayed in Sect. 3. Discussions and conclusions are given in Sect. 4.

2 Experimental setup

2.1 Figure measurement device

Fig. 1
figure 1

a Schematic diagram of the experimental setup for figure measurement based on Shack–Hartmann wavefront sensor. WFS wavefront sensor. b Supporting and mobile platform for moving and adjusting the optical elements. ECDP electronically controlled displacement platform, ECOF electronically controlled optical frame

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Schematic diagram of the experimental setup for figure measurement based on Shack–Hartmann wavefront sensor (WFS) (Imagine optics, HASO3 128GE2) used in this work is displayed in Fig. 1a. First, laser beam from a single-mode polarization maintaining fiber is collimated to a diameter of \(D_1\approx 10\) mm (\(1/e^2\)) by a fiber collimator, reflected by a beam splitter, expanded by a beam expander to a diameter of \(D_2\approx 30\) mm, sequentially. Then, the beam is reflected by the reference mirror mounted in the vacuum chamber after passing through the top window of the vacuum chamber. The laser beam sequentially passes through the window, beam expander and beam splitter. Finally, the laser beam is received by the WFS with a diameter of \(D_3\approx D_1\). The measured wavefront of the laser beam in this situation is

$$\begin{aligned} WF_1= & {} WF_{FC}+WF_{BS}+2WF_{BE}+2WF_{M}\nonumber \\&+2WF_{W}+WF_{RM}, \end{aligned}$$
(1)

where \(WF_{FC}\) represents the wavefront contributed by the fiber collimator, \(WF_{BS}\) represents all wavefronts contributed by the beam splitter, \(WF_{BE}\) is the wavefront from the beam expander, \(WF_{M}\) is from the mirror, \(WF_{W}\) is contributed by the window, and \(WF_{RM}\) is from the reference mirror.

Then, to measure the figure of optical element, the test optical element is moved into the optical path. If the test optical element is a transmission optic, the wavefront measured by the WFS is

$$\begin{aligned} WF_2= & {} WF_{FC}+WF_{BS}+2WF_{BE}+2WF_{M}\nonumber \\&+2WF_{W}+WF_{RM}+2WF_{T}, \end{aligned}$$
(2)

where \(WF_{T}\) represents the wavefront contributed by the test optical element. Thus, the wavefront contributed by test optical element is \(WF_2-WF_1\).

When we measure the figure of a reflecting mirror, the reference mirror need to be replaced by it, and the measured wavefront is expressed as

$$\begin{aligned} WF_3= & {} WF_{FC}+WF_{BS}+2WF_{BE}+2WF_{M}\nonumber \\&+2WF_{W}+WF_{TM}. \end{aligned}$$
(3)

Therefore, the wavefront contributed by the test reflecting mirror can be expressed as \(WF_3-WF_1=WF_{TM}-WF_{RM}\approx WF_{TM}\) when the surface quality of the reference mirror is much better than the test reflecting mirror. Actually, a reference mirror with RMS value of \(\lambda /250\) within a diameter of 30 mm is employed in this work, which is measured by a laser interferometer working at \(\lambda\)=633 nm.

2.2 Supporting and mobile platform

When measuring the reference wavefront \(WF_1\), the test optical element is taken out of laser path. Since the optical elements are installed inside the vacuum, an electronic controlled displacement platform (ECDP) is employed to move the optical elements. In addition, when measuring the figure of reflecting mirror, the directions of the mirror need to be adjusted, thus, two electronically controlled optical frames (ECOF) are mounted on ECDP. The designed schematic diagram of this electronic controlled device is displayed in Fig. 1b.

Fig. 2
figure 2

The overall structure of the wavefront measurement device: a the photograph of the system; b the photo of WFS and optical path, WFS wavefront sensor, FC fiber collimator, BE beam expander; c the design diagram of vacuum chamber with supporting and mobile platform

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When the test optical element is a transmission optical element, the reference mirror is mounted on the bottom frame in Fig. 1b, and the test optical element is mounted on right ECOF. When measuring the figure of a reflecting mirror, the reference mirror is mounted on right ECOF, and the test mirror is mounted on left ECOF. The position repeatability of ECDP and angle adjusting resolution of ECOF is better than 0.05 mm and 100 \(\mu\)rad respectively, which meet the experimental requirements.

2.3 Structure of the device

The overall system is shown in Fig. 2a. To reduce the impact of air disturbance for wavefront measurement, the figure measurement device is installed inside a closed chamber. As shown in Fig. 2b, WFS and optical path are installed on a vertical optical platform, which is above the vacuum chamber. As shown in Fig. 2c, the vacuum chamber is designed and used to install the supporting and mobile platform and provides a vacuum environment for measuring optical elements. A mechanical pump and molecular pump are employed to obtain and maintain vacuum environment. To flexibly change the pressure in the chamber, an adjustable valve is used to control the leak rate, shown in Fig. 2a. Therefore, the pressure in this chamber can be modulated from 0.058 Pa to \(1\times 10^5\) Pa. In addition, a vacuum gauge is employed to measure the pressure in the chamber.

3 Experimental results

3.1 Calibration of the figure measuring device

Calibration is a necessary procedure to ensure the precision of the figure measuring device. First, the device is calibrated in atmospheric pressure: (1) measuring the wavefront \(WF_1\) without any test optical element; (2) repeating step 1 to obtain a new wavefront \(WF_1^{'}\); (3) the background wavefront of this device is \(WF_1^{'}-WF_1\). The measured wavefront \(WF_1\) and typical background wavefront are displayed in Fig. 3a and b, respectively. The displayed wavefronts in Fig. 3 are recovered from the measured coefficients of Zernike polynomials [16], and tilts are subtracted in the software of WFS.

Fig. 3
figure 3

Calibration of the figure measuring device. a Measured wavefront \(WF_1\) without any test element in optical path. b Measured background wavefront

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From Fig. 3, the background of wavefront is flat, and the RMS value of the background is only 6 nm, which is as small as \(\lambda /130\), where the working laser wavelength is \(\lambda =780\) nm. The diameter of laser detected by the WFS is about 9.3 mm, thus the measuring range for optical elements is about 28 mm. The same calibration is carried out in vacuum, and similar result is obtained.

3.2 Figure measurement under different pressures

A quarter waveplate with a diameter of 40 mm is firstly employed as the test optical element, which is installed into an adapter and compressed by a pressure ring. The adapter is installed into the right ECOF. The reference mirror is glued on the bottom frame in Fig. 1b by vacuum glue. The test optical element is moved out and into the optical path by moving ECDP left and right. Firstly, wavefront (\(WF_1\)) without test optical element in the optical path is obtained. Then move the test optical element into the optical path by moving ECDP and repeat the measurement to obtain a new wavefront (\(WF_2\)). The wavefront contributed by the test optical element is \(WF_2-WF_1\). To investigate the influence of pressure on the figure of test optical element, measuring it under different pressures is performed, and the results are shown in Fig. 4. The RMS values of the wavefronts within a diameter of 25 mm measured under the pressure of 0.058 Pa, 10 Pa, 1000 Pa, and \(1\times 10^5\) Pa are 34 nm, 32 nm, 33 nm, and 37 nm, respectively.

Fig. 4
figure 4

Measured wavefronts contributed by the quarter waveplate at different pressures: a 0.058 Pa; b 10 Pa; c 1000 Pa; d atmospheric pressure

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All RMS values of the wavefronts measured under different pressures are displayed as black squares in Fig. 5, and each point is averaged 5 times. From the results, the RMS value of the wavefront measured under different pressures varies from 32 nm to 37 nm, which does not change significantly within the error range. Based on these wavefronts, the effect of wavefront aberrations for gravity measurement can be calculated.

In AIG, atoms are cooled and trapped in three-dimensional magneto-optical trap (3D-MOT), then the atomic cloud is launched and further cooled by moving molasses. During the launch, atoms in \(\mid F=1, m_F=0\rangle\) state are selected for gravity measurement. When the atoms enter the interferometry region of the gravimeter, a sequence of \(\pi /2-\pi -\pi /2\) Raman pulses are used to coherently split, reflect, and recombine the atomic wave packet. After the atom interference, the transition probability of atomic state is determined by normalized detection. It can be described as \(P=\frac{1}{2}[1-\cos (\varDelta \varPhi )]\), where \(\varDelta \varPhi\) is the phase shift of the AIG. When the atoms freely fall in an uniform gravity field, the phase shift caused by gravity is \(k_{eff}gT^2\), where \(k_{eff}\) is the effective wave vector of Raman beams, and T is the interval time between two Raman pulses. However, there are many other effects that will introduce systematic errors in gravity measurements, such as the effect of wavefront aberrations.

The effect of wavefront aberrations is directly related to the wavefront distribution of Raman laser, which is used to coherently manipulate atomic cloud. Raman laser is composed of two laser beams propagating in opposite directions, thus, the wavefront felt by atomic cloud is the difference between two lasers. One of the lasers passes through \(\lambda /4\) waveplate and retro-reflecting mirror comparing with another one, therefore, the effect of wavefront aberrations can be determined when the figures of \(\lambda /4\) waveplate and retro-reflecting mirror are measured. The wavefront measured under different pressures can be defined as \(\delta \phi _{wf}^i(r)\), and the distribution of atoms at \(j^{th}\) (\(j=1,2,3\)) Raman pulse is \(n_j(r)\), thus the phase shift contributed by the wavefront measured under \(i^{th}\) pressure is [17]

$$\begin{aligned} \varDelta \phi ^i_{wf}= & {} \int \delta \phi _{wf}^i(r)n_1(r)dr-2\int \delta \phi _{wf}^i(r)n_2(r)dr\nonumber \\&+\int \delta \phi _{wf}^i(r)n_3(r)dr. \end{aligned}$$
(4)

Thus, the contribution of wavefront aberrations to gravity measurement is \(\varDelta g_i=\varDelta \phi ^i_{wf}/k_{eff}T^2\). The biases for gravity measurement contributed by the measured wavefronts under different pressures when \(T=160\) ms are displayed as blue spheres in Fig. 5. The calculated results show that bias does not change with the pressure regularly.

Fig. 5
figure 5

The RMS values of wavefront measured under different pressures and the biases induced by the effect of wavefront aberrations contributed by these wavefronts

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Although the RMS value of wavefronts measured under different pressures does not change within the error range, as can be seen in Fig. 5, distributions of wavefronts measured under different pressures are not exactly the same from Fig. 4. To give the difference of figure measured under different pressures, we subtract the wavefront measured under atmospheric pressure from the wavefronts measured under different pressures, and the results are shown from Fig. 6a to c. It is clear that the wavefronts measured under different pressures are different and the differences are random. The RMS values of the wavefront difference between different pressures and atmospheric pressure are displayed with blue circles in Fig. 7. Furthermore, we also measured the wavefront distributions under atmospheric pressure at different times. Figure 6d–f shows the differences of wavefront measured under atmospheric pressure at different times, and the RMS value of the wavefront difference measured under atmospheric pressure at different times is displayed with red triangle in Fig. 7.

Fig. 6
figure 6

The wavefront differences. The differences between the wavefront measured under the pressure of a 0.058 Pa, b 10 Pa, c 1000 Pa and atmospheric pressure. d–f The differences of the wavefront measured under atmospheric pressure at different times

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The RMS value of the wavefront difference between different pressures and atmospheric pressure varies between 14 nm and 23 nm, and the RMS value of the wavefront difference measured under atmospheric pressure at different times varies between 18.5 nm and 21 nm, which is comparable to the RMS value of the wavefront difference measured under different pressures. That means the change of wavefront measured under different pressures is not completely due to the change of pressure, and there may be some other time-dependent factors contributing to the wavefront measured under different pressures, such as the fluctuation of air density.

The change of wavefront measured under atmospheric pressure at different times may be from air flow, temperature change, vibration of ground and so on. According to the results before, the change of wavefront measured under different pressures cannot be only due to the change of pressure, even the change of wavefront measured under different pressures is mainly from the change of environment. When the wavefront difference measured under atmospheric pressure between different times is regarded as the repeatability of this figure measuring device, the wavefronts measured under different pressures are not changed within the repeatability.

Fig. 7
figure 7

The RMS values of the wavefront difference. Blue circles: the RMS values of wavefront difference between vacuum and atmospheric pressure. Red triangle: the RMS value of wavefront difference between different times under atmospheric pressure

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Furthermore, the figure of reflecting mirror under different pressures is also measured. The test mirror is mounted on an adapter and compressed by a pressure ring, and the adapter is installed into the left ECOF. To reduce the change of the reference mirror’s figure due to the change of air pressure, the reference mirror is quasi-freely glued on another adapter installed into the right ECOF. The RMS value of the wavefront contributed by the reflecting mirror measured under different pressures does not change significantly, which varies between 83 nm and 88 nm. The RMS value of wavefront difference between different pressures and atmospheric pressure varies between 13 nm and 25 nm, which is also comparable to the RMS value of wavefront difference measured under atmospheric pressure at different times. In addition, when the test mirror is glued on the adapter, the wavefront RMS value of which is obviously reduced to 29 nm, which means that this fixing method can significantly reduce the influence of the installation on the figure of the optical element. In addition, to explore the influence of pressure changes on the figure of reference mirror freely glued on the adapter, we employed finite element method [18] to simulate the figure change of reference mirror under different pressures. Simulation result shows that the thickness of reference mirror with a thickness of 10 mm increases about 10 nm, and the peak-to-valley thickness change is about 1 nm when the pressure is reduced from \(1\times 10^5\) Pa to 0.058 Pa, which can be neglected in this measurement.

4 Discussions and conclusions

In this work, a figure measuring device based on Shack–Hartmann wavefront sensor used to measure the figure of optical elements in vacuum has been designed and implemented. This device can measure the figure of optical elements with a precision of better than \(\lambda /100\) in RMS. The figure of optical elements under different pressures has been measured by this device. The RMS value of figure measured under different pressures does not change within the error range. More importantly, the distribution of figure under different pressures does not change significantly, and the variation is mainly from the change of environment after considering the wavefront difference measured under atmospheric pressure at different times.

This work confirms that the figure of optical element measured under atmospheric environment could be still applicable when it is installed into vacuum environment, which is useful for the evaluation of wavefront aberrations effect in AIG. In addition, this device is useful for measuring the figure of optical elements in other applications, such as in inertial confinement nuclear fusion and gravitational wave detection.