1. Introduction

Absolute ballistic gravity meters measure gravity using a free-falling proof mass in a vacuum. A laser interferometer is used to precisely and accurately track its position. Figure 1 shows a simple schematic for the measurement. A laser beam is split into two arms with a beam splitter. One arm (optical path) of the interferometer reflects from a corner-cube retroreflector secured to the free-falling proof mass. This variable length arm is then recombined with a fixed optical path called the “reference arm.” This combined signal creates interference fringes based on relative path length changes that are registered with a high-speed photodetector. Each optical fringe corresponds to a free-fall distance of $\lambda /2$, where $\lambda$ is the wavelength of the laser. Approximately a million fringes are generated when the proof mass falls a distance of about 20 cm. Commercial ballistic absolute gravity meters achieve a typical accuracy of about $10\mathrm{nm}/{\mathrm{s}}^2$, which is equivalent to one part per billion (${10}^{-9}$) of the Earth’s gravity, also called little g. One of the systematic errors in the error budget of this type of instrument [1] is caused by rotation of the proof mass as it falls. Care is taken to minimize the rotation of the proof mass when it is released into free fall, but a small amount of rotation is inevitable.

 

Fig. 1. Simple ballistic gravity interferometer.

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The rotation rate is constant due to the absence of torque on the mass, but the amplitude can vary from drop to drop. A rotation of the proof mass about its center of mass (COM) will cause the optical center (OC) to rotate on an arc with a radius given by the offset between these two points.

Figure 2 shows two extreme cases for the offset between the OC of the corner cube and the COM. In Case I, the OC of the corner cube is displaced laterally from the COM of the pendulum. The OC will appear to move toward or away from the laser as the test mass rotates. Since the rotation rate is constant during free fall, this results in an apparent initial velocity, which does not affect the measured acceleration. For this reason, the horizontal (or lateral) offset between the COM and OC is often ignored in practice.

 

Fig. 2. Two cases for offset between OC and COM of the pendulum proof mass.

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If, on the other hand, the OC of the corner cube is displaced from the COM along the optical axis, in-line with the laser, Case II, the OC will be at an extremum when the COM of the proof mass and the OC are aligned vertically.

When the proof mass rotates about the COM, its OC will cause the OC to rotate upward (or downward if the COM is below the OC) along the arc of a circle with a radius equal to the separation between these two points. This corresponds to Case II in Fig. 2. The vertical displacement from such a rotation will be proportional to the cosine of the rotation angle. Since the proof-mass angular velocity is constant during free fall, the additional vertical displacement of the OC is approximately equal to $\frac{1}{2}\delta {(\omega t)}^2$, where $\delta$ is the inline offset of the corner cube from the COM of the proof mass and the angular velocity is $\omega =\dot{\theta }$. This mimics a spurious acceleration of $\delta {\omega }^2$, impacting the measurement of gravity. In order to achieve a desired accuracy of ${10}^{-8}\mathrm{m}/{\mathrm{s}}^2$, the COM and OC for an FG5 proof mass must be collocated to within 100 μm, and the rotations rate must be kept below 0.01 rad/s. This is a difficult but usually achievable goal.

Rothleitner and Francis [2] provide a thorough analysis of the errors that result from rotation of the corner cube in a ballistic free-fall gravity meter. These errors are reduced by collocating the COM and OC of the proof mass.

The simplest method for balancing of the proof mass requires knowledge of the COM and OC of the corner cube. The procedure used for the commercial FG5 gravity meter involves placing the OC of the proof mass at the center of a balanced pan with two knife edges. One knife edge is placed on a sensitive weighing scale and the other on a precision surface so that the pan is level. The COM of the proof mass is adjusted by rotating (translating) two jam nuts along the optical axis of the mass. The COM is collocated with the OC when the pan can be rotated 180 deg and no mass change is observed on the scale. The accuracy limit of collocation due to machining and measurement errors is about 25 μm. These specifications keep the nominal error for rotation of the proof mass for the FG5 at about $2.5\mathrm{nm}/{\mathrm{s}}^2$ for a rotation rate of 0.01 rad/s.

Unfortunately, the angular rotation of the proof mass can increase, as the contacts on the proof mass are worn or if the bearings on the release mechanism become loose with time. When this occurs, the angular rotation imparted prior to release can increase by an order of magnitude to about $\omega =0.1\mathrm{rad}/\mathrm{s}$ or about the rate of a minute hand on a clock. Since the error is proportional to the square of the rotation rate, this rate requires an improvement of 100 times better balancing for the same rotation error in gravity.

Another difficulty with a direct balancing method is that the location of the OC must be precisely determined prior to balancing. This can be difficult because it requires knowledge of the exact geometry of the corner cube. Quite often, the apex is ground off during the manufacture of the retroreflector, so one must locate a virtual apex from the three sides of the corner cube. This can be done with a coordinate measuring system, but it requires care and an expensive metrological instrument. The OC for a glass corner cube [3] is at a distance $H/n$ below the face of the cube, where $n$ is the index of refraction and $H$ is the height of the prism given by the distance from the face of the cube to the apex. The index for a glass corner cube must be known to 0.1% in order to calculate the OC with an accuracy of 25 μm. It would be difficult to ascertain the index of refraction with better accuracy even if a better direct balancing method were to be developed.

Other methods have been used to accomplish the collocation of the COM of a proof mass with the OC of the imbedded corner cube. Hanada [4] described a method that used a spinning table to locate the COM by finding a location where the proof mass did not slide off the table as it spun rapidly. Once the test mass was spinning around its COM, the authors could observe fringes when the proof mass momentarily aligned with a laser beam emanating from a fixed interferometer on each revolution. This provided a very short observation time of the position of the proof mass in which they had to untangle the combined effects from rotation errors of the corner cube from sliding motion of the proof mass on the table. They were ultimately able to collocate the OC and OC to an accuracy of about 50 μm. This method relied on low friction between the table and the test mass to observe sliding of the object while it was rotating. Germak [5] used a similar method and achieved accuracy of about 14 μm. This paper describes collocation of the COM and OC in all three axes. Rothleitner [6] also developed two different procedures, the best of which located the OC by monitoring fringes and then the COM using a sensitive balance. His estimated uncertainty was about 11 μm. All of these methods have to deal with eccentricity of their rotation tables and are able to measure optical fringes only during a small part of the rotation when the corner cube is aligned with the optical sensor. This limits the ability to integrate their optical signal to gain sensitivity. These methods rely on a two-step procedure, at the end of which there is no single direct measurement of the final error.

Vitouchkine [7] proposed a novel method, which required the use of the entire gravity meter to collocate the COM and OC. The authors intentionally vibrated one foot of the gravity meter in order to induce larger than normal rotation of the object. They iteratively balanced the proof mass in order to minimize the measured DC shift in gravity caused by the vibration. Although the method is elegant because it produces direct measurement of the error with a single measurement, it has several disadvantages. First, the vibration induces rotation and translation of the object (arguably much more translation than rotation). A Coriolis shift in gravity from any east–west motion of the test mass during the vibration measurement could be compensated by an error caused by rotation. Without a separate measurement of translation and rotation of the object, it is unclear what result will be achieved. A practical disadvantage of this method is that the vacuum chamber must be reopened after the measurement in order to rebalance the object. The error resulting from shaking the gravity meter does not produce an estimate for the amount that the COM should be shifted to obtain a properly balanced proof mass.

Our new procedure provides an alternative to all of the other methods to ensure collocation of the COM of the object with the OC of the corner cube. Our approach is to suspend the proof mass from a torsion wire, so that the face of the corner cube is horizontal. The proof mass executes simple harmonic motion as the torsion fiber twists. We interrogate its horizontal translation while the object twists with a laser interferometer. This directly generates the error signal that we want to minimize due to the offset between the COM and OC without requiring knowledge of their absolute position relative to the proof mass.

A direct measurement of rotational errors using an interferometer, as in some of the other methods discussed, eliminates the need to find the apex using physical measurements of the geometry of the corner cube using a coordinate measuring machine. Solid glass corner cubes also require knowledge of the index of refraction of the material. Our method is the only one that generates fringes caused by rotational errors continuously as the torsion fiber twists and untwists. The other methods that make a direct measurement of fringes using an interferometer can only measure translation errors when the corner cube is rotated past a fixed laser beam. This allows us to integrate any optical errors for long time periods (up to an hour) to greatly increase the sensitivity of the measurement. This advantage permits direct measurement of the translational error caused by large rotations (of several degrees) of a corner cube to the level of about 1 nm, whereas the other methods are limited to integer fringe measurements with an order of magnitude error of 1 μm.

The very low friction of the single point suspension in our pendulum wire ensures that the COM is hanging below the fiber. Thus, the pendulum mimics the type of rotation that the proof mass will undergo in free fall. If the COM and OC are not collocated, interferometer fringes will be generated as the pendulum executes its twist mode. Alignment of the COM with the twist axis occurs as a natural consequence of the apparatus, whereas the other methods require some sort of rotation table or a spinning bearing surface and a second subsequent measurement to find out if the test mass is creating an out-of-balance condition. The other procedures, therefore, necessarily require an extra measurement compared to our method. They also suffer from uncertainties in the eccentricity of their spinning table and frictional elements that limit their sensitivity.

The method is quite sensitive because it provides direct measurement of the errors generated when the proof mass is rotated by an exaggerated angle of several degrees, whereas the proof mass has much smaller rotations when it is allowed to free fall in the ballistic gravity meter. As we will show later, this method also allows us to characterize the time dependence of the optical errors caused by rotation. Specifically, we can measure the difference between errors that are quadratic or linear with rotation angle because they occur at two different frequencies in the pendulum.

Our method is more general than the other techniques for collocating the COM with the OC because it provides direct measurement of any error generated in an interferometer due to rotations of the proof mass about its COM without any need to reference or to find the COM of the proof mass. In fact, this method does not even rely on the concept of an OC. It will detect and measure any rotational errors independent of their source.

2. Experimental Setup

The apparatus we used consists of a normal FG5 proof mass that contains a 25.4 mm glass corner cube. It is suspended by a single, thin tungsten vertical fiber 76 μm diameter and approximately 64 mm in length. The wire is held fixed at the top of the pendulum and terminated near the test mass with a loop of wire, or hanger, that extends about the center of the proof mass. The pendulum setup is shown in Fig. 3. The COM of the proof mass is free to rotate below the lower termination point of the suspension fiber about the torsion axis. The proof mass is hung so that the optical axis for the corner cube is horizontal. The position of the corner cube is sketched on the proof mass.

 

Fig. 3. Proof mass pendulum setup.

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A quadrature interferometer shown in Fig. 4 was employed in order to increase the resolution of the measurement and to gain information about the direction of motion of the pendulum. A linearly polarized input laser beam (red) is sent through a half-wave ($\lambda /2$) plate to tune the incident angle of polarization. A nonpolarizing beam-splitting cube diverts the beam into two optical paths or interferometer arms. The stationary arm of the interferometer (teal) is transmitted through the beam splitter where it reflects from a stationary corner cube. The variable interferometer arm (orange) reflects from the hanging test mass. Optical interference fringes are created in the recombined beam (purple) as the pendulum moves in the direction of the laser beam. The beam in the variable arm passes through a $\lambda /4$ plate to convert the linearly polarized light into circular polarization. The circularly polarized light has two orthogonal fields that have a relative phase difference of 90 deg. The stationary linearly polarized beam is then broken into two orthogonal linear polarizations with a second polarizing beam splitter. This second beam splitter also breaks the circularly polarized beam from the variable arm into two linearly polarized beams (green and dark blue) that have a 90 deg phase shift relative to one another. This effectively creates two different interferometers with orthogonal polarization. This arrangement is referred to as a quadrature interferometer because the two interferometer signals vary in quadrature (like a sine and cosine) as the path length changes with a constant velocity.

 

Fig. 4. Quadrature interferometer: beam path diagram.

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The optical path inside of a corner cube is insensitive to horizontal motion and rotations of the test mass about the OC. The corner-cube geometry guarantees that the incoming and outgoing laser beams are parallel to within the manufacturing tolerances of the cube, typically $\pm 1-3\mathrm{arc}\mathrm{s}$.

Each detector produces a full optical fringe when the OC of the corner cube inside the proof mass moves by one half of the wavelength of the laser ($\lambda /2$). This is due to the fact that a displacement of the corner cube affects both the path length of the inbound and outbound laser beam. A red He–Ne laser with $\lambda =633\mathrm{nm}$ is typically used in this experiment.

The fringes from each detector are collected using an analog detector and sampled at 5 kHz, which is well above any mechanical resonant frequencies. The two sinusoidal interferometer signals from each detector are then turned into square waves by digitally detecting the zero crossing of each waveform. Figure 5 shows the two square waveforms, A and B, after zero crossing with a 90 deg phase shift. Each full period for either waveform corresponds to a pendulum motion of $\lambda /2$. Every full period of waveform A, or B, can be split into four equal segments delineated by the zero crossings of the two waveforms. Each segment, therefore, corresponds to a displacement of $\lambda /8$ or about 80 nm for a red He–Ne laser.

 

Fig. 5. Quadrature interferometer signals.

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The sequence of zero crossings of each interferometer corresponds to a two-bit Gray code [8] where only one bit changes at a time (00, 10, 11,01 in one direction and 00,01,11,10 in reverse). The progression of the code provides the direction of pendulum motion. The position of the pendulum is incremented or decreased by $\lambda /8$ after each zero crossing. Quadrature decoding is used in many electromechanical devices including shaft encoders.

The wire suspending the proof mass constrains its translation motion in the vertical direction, and the hanger prevents the proof mass from rotation about its long axis (parallel to the laser beam). This permits four modes of oscillation. The proof mass can translate (swing) side to side both in-line with the laser as well as perpendicular to the laser. Additionally, the mass is permitted to “wobble” at a higher frequency about the wire attachment point on horizontal axis perpendicular to the laser. The two swing modes and the wobble mode are driven by the exchange of gravitational potential energy with motion. Finally, the elastic restorative force in the torsion wire drives the twist mode.

Our experiment utilizes the twist mode of the pendulum to produce a rotation of the proof mass about its COM, while its position in the direction of the laser is monitored with a laser interferometer. The frequency of the twist mode is given by ${\omega }_t=\sqrt{{\kappa }_t/\mathrm{I}}$, where ${\kappa }_t=S{A}^2/2\pi \ell$, where $S$ is the shear modulus, $A$ the cross-sectional area, $\ell$ is the length of the fiber, and I is the moment of inertia of the proof mass about the twist axis. The twist frequency can, therefore, be adjusted by changing the diameter or length of the suspension wire or by using a material with a different elastic modulus.

The lowest potential energy position of the test mass in the wobble mode occurs when the COM is directly below the wire on the axis of rotation for the twist mode. Thus, the twist mode effectively rotates the proof mass about an axis that passes through its COM. Once the COM is collocated with the OC of the corner cube, then the twist mode should not generate fringes. This fact allows us to rebalance the object in order to minimize the displacement measured by the interferometer. By performing this measurement, we can then adjust the proof mass balance to collocate the COM with the OC of the attached corner-cube retroreflector. This can be done without the need to determine the position of either fiducial point, the COM or the OC independently.

The existence of the wobble mode ensures that COM of the proof mass naturally seeks its minimum energy position directly underneath the twist axis of the pendulum. This is a very low friction pivot point compared to what is achievable with an air bearing or spinning table.

In general, we can consider an error between the COM and OC expressed as a combination of two orthogonal offsets, $\mathrm{d⃗}={\delta }_x\widehat{x}+{\delta }_z\widehat{z}$, where the $z$ axis is parallel to the laser beam and the $x$ axis is horizontal, or lateral, to the optical setup. The $y$ axis for this reference frame is coincident with the hanging fiber of the pendulum. As the fiber twists, the displacement between the OC and COM rotates about the $y$ axis, which is why the $y$ component of this offset will not contribute to the optical signal at the twist frequency. The $z$ component of this rotated displacement, in line with the laser, is given by $d_z={\delta }_x\sin \theta +{\delta }_z\cos \theta$, where $\theta$ refers to the rotation of the pendulum around the hanging fiber.

For small angles, the displacement measured by the interferometer is approximately given by

When the pendulum is twisted to a maximum angle of ${\theta }_0$ and then released, a rotational damped periodic angular motion ensues of the form

The result has a single-frequency term, proportional to the horizontal error and another double-frequency term proportional to the in-line error between the COM and OC. This setup allows us to separate the effect of these two errors on the interferometer measurement in the frequency domain. The in-line error between the OC and COM introduces an optical signal at twice the frequency of the pendulum where no mechanical motions of the setup should occur. Thus, the 2f signal frequency can be designed by changing the wire length and diameter so that it occurs at a point in the spectrum where all other real mechanical noise is small. This prevents the smaller 2f signal from being obscured by oscillation of the other modes of motion.

The amplitudes of the single- and double-frequency terms are proportional to the horizontal and inline offsets, but they also contain the initial swing amplitude, ${\theta }_0$, which must be known in order to convert the amplitudes into calibrated offsets. The initial angle of the twist mode was measured by monitoring the displacement of the partial reflection of the laser beam off of the front face of the glass corner cube. Once the initial twist angle is known, the horizontal offset can be calculated. Assuming that this is not changed, the initial twist angle can always be determined by the size of the 1f amplitude and, therefore, only needs to be measured once.

It is important to recognize that there are other modes of the pendulum that cause much larger displacements than the ones associated with rotation. For example, an extremely small swing mode in the direction of the laser beam will generate many more fringes than created when the wire twists. The same is true of other potential modes of the test mass. However, these unintended motions of the pendulum occur at a higher frequency than the slow twist rotational mode and can be removed with a low-pass filter. Separation of optical errors by frequency is one of the elegant features of this method over other spinning or balancing methods that tend to lump all potential error sources together in the same optical measurement in the time domain.

3. Data Analysis

The data from both quadratures of the interferometer are sampled with a standard A/D card from National Instruments at a frequency of about 5 kHz, so that no optical fringes are missed and recorded for several minutes to an hour to increase the averaging time for the signal. The sinusoidal fringes from each detector are turned into square waves with a digital zero-crossing algorithm and then counted to give the position of the pendulum with a precision of $\lambda /8$ or 80 nm, as discussed in the last section. The unique sequence of the quadrature signals provides a means for determining the sign of the proof mass motion as the pendulum oscillates in a superposition of any excited normal modes.

Figure 6 is a spectrum of the motion of the OC of the pendulum in the direction of the laser beam of the interferometer for a nominal twist angle. The data were generated using a nearly balanced test mass. This data represents a 20 min recording time sampled at 5 kHz. For the wire we have chosen, there are three main modes visible at 0.054 Hz (18.5 s), 1.68 Hz(0.6 s), and 3.2 Hz (0.3 s). These are associated with the fringes caused by the twist, swing, and wobble modes of motion, respectively. A smaller peak is visible at two times the twist frequency at 0.11 Hz (9.2 s), resulting from the in-line offset of the COM and OC, as described above.

 

Fig. 6. Spectrum of pendulum motion (20 min record).

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Prior to excitation, the pendulum was magnetically damped in its rest position. The magnet was then moved steadily away from the pendulum mass using a motor drive system in a manner that excited the twist mode but minimized the excitation of the wobble and swing modes. Nonetheless, these higher-frequency modes are visible in the spectrum.

The twist mode shown in Fig. 6 has an amplitude of about 1.9 μm at a frequency of about 0.054 Hz. The swing and wobble modes have amplitudes of about 0.4 and 0.15 μm at 1.7 and 3.2 Hz, respectively. The relative size of these peaks in the spectrum can be misleading. Even though all of these peaks correspond to a similar optical signal, the physical angular motion for swing and wobble modes is vastly smaller than for the twist mode. The twist angle is several degrees, while the swing and wobble modes are damped and very small (corresponding to micrometer displacements). The reason for this apparent discrepancy is that the twist mode is not as effective at generating changes in beam path length especially when the COM and OCs are nearly collocated. The signal component at the double frequency of the twist mode, 0.11 Hz, has an amplitude of only about 5 nanometers with a twist angle of 2.1 deg. The amplitude of the signal is a measure of how well the OC and COM are collocated along the in-line axis, $z$ axis, of the laser beam. It is important to emphasize that this double twist frequency signal is a purely optical signal and is not even associated with any mechanical resonance of the system.

The damping of the twist mode is a measure of the energy lost in one cycle divided by the total energy in that mode. The $Q$ is related to the damping time constant and frequency by $Q=2\pi {(1-e^{-\frac{4\pi }{\omega \tau }})}^{-1}$. The twist mode had a typical damping quality factor of about 1000, which means that the signal could be effectively integrated for about 1 h after the pendulum was excited. The double-frequency signal, however, damps out twice as fast because it is proportional to the square of the twist angle. We found empirically that the best results were obtained for an averaging time of about 20 min.

The background noise of the spectrum near the double frequency for a 20 min average on our optical table is about 1 nm. This background noise limits the minimum detectable signal at the 2f frequency. To reduce noise, a wind shield is used, and the experiment is conducted at night in a quiet, closed room.

Our data acquisition setup included a motor that periodically, every 30 min, moved a magnet near the test mass for a few seconds and then moved it away to induce a rotation in the pendulum without excessively exciting the other pendulum modes. The data collection was initiated 5 min after the motor had stopped. Both quadrature interferometers were sampled at 5 kHz for 20 min. This procedure was automated to allow data collection during quiet periods at night and on the weekend. The distance of closest approach of the magnet was varied to automatically collect data records with different swing amplitudes.

4. Data Interpretation

The interferometer data were digitized by detecting the zero crossing of each waveform: zero when the waveform was negative and unity when positive. These quadrature signals were then decoded to provide a signed digital signal corresponding to the displacement of the OC of the pendulum test mass in the direction of the laser with a resolution of one eighth of the laser wavelength.

These data were then low-pass filtered to remove the swing and wobble modes from the signal. Finally, a nonlinear least-squares fit using Eq. (1) as a model was applied to the low-pass data. This resulted in estimates for the frequencies, phases, and amplitudes for the 1f and 2f signals as well as the damping factor for each excitation of the pendulum.

The noise background of the interferometer as well as the physical ability to adjust the COM accurately place a limit on how well this method can be used to balance the test mass. The relationship between the offset of the COM and the OC of the pendulum test mass, ${\delta }_z$, and the amplitude of the 2f signal, $A_{2f}$, in the interferometer is ${\delta }_z=2A_{2f}/{\theta }^2$. This provides an estimate for the minimum detectable offset between the COM and the OC (in the direction of the laser). We were able to measure a minimum 2f amplitude of about 1 nm using a 20 min record. This translates to an offset of about 1.6 μm between the COM and the OC for a twist amplitude of 2 deg. We could average a few 20 min records to get our error on the balance of the test mass to about 1 μm. Once the offset was measured, we adjusted the balancing weights on the test mass and then remeasured the offset using the pendulum. This procedure was repeated until we could no longer see the 2f signal.

An obvious tactic to increase the sensitivity is to use larger and larger angles. However, a complication arises if the main signal at 1f gets distorted at large angles. This can happen if there is any “clipping” of the signal as the test mass rotates. Since the main 1f signal in our case was about 1000 times larger than the 2f signal of interest, a small distortion of a part per thousand in the amplitude of this signal can create harmonics that will land on top of the 2f signal that we are trying to measure. Fortunately, if the distortions are symmetric, the first harmonic will be at 3f. However, asymmetric distortion of the optical signal can create a harmonic of the 1f signal at 2f. This can happen if the optics are not perfectly aligned, and more light is lost on one side of the twist compared to the other.

Harmonics caused by distortion are typically linearly dependent on the 1f amplitude. For example, the amplitudes of the harmonics of a square or triangular wave are proportional to the amplitude of the fundamental period. This provides a mechanism to untangle the part of the signal at 2f created by the offset between the OC and COM from harmonic distortion of the 1f signal. The signal of interest depends upon the square of the rotation angle (1f amplitude), whereas asymmetric distortion of 1f signal creates a 2f harmonic that is linearly related to the swing angle. This issue can be dealt with by taking data at different swing amplitudes and fitting the 2f amplitude to a quadratic function (parabola) of the corresponding 1f amplitude. If we assume a constant background noise level of $\mathbb{N}$, a second harmonic caused by distortion of the 1f signal amplitude, given by $\mathscr{H}{A}_{1f}$, and the expected effect caused by a mismatch of the COM and OC, the measured amplitude at 2f, $A_{2f}$, will be given by $A_{2f}=\mathbb{N}+\mathscr{H}{A}_{1f}+\frac{{\delta }_z}{4{\delta }_x^2}{A}_{1f}^2$. This means that all we have to do is fit the 2f amplitude to a quadratic function of the 1f amplitude, and the parabolic coefficient will give us the inline offset, ${\delta }_z$, if we first estimate the horizontal offset using, ${\delta }_x=A_{1f}/{\theta }_0$, with a measured swing angle, ${\theta }_0$.

We have verified that, if the interferometer is not perfectly aligned and a large angular twist angle is excited (above 4 deg), then we can see harmonics of the 1f amplitude at 3f and 4f. It stands to reason that there is also a harmonic at 2f. The physical reason for the distortion of the 1f amplitude is due to the overlap of the two interferometer beams, which changes due to the small horizontal translation of the beam returning from the test mass before it recombines with the stationary arm of the interferometer. This causes the amplitude of the interference to become smaller as the overlap of the beams decreases. This will not create additional optical fringes, but it can mimic subfringe phase variations by delaying or advancing the zero crossings of the interference fringes. If the distortion is symmetric as the wire twists back and forth, this will generate a harmonic at 3f and will not contaminate the 2f amplitude. However, if the overlap is not optimized at zero angle, the distortion of the 1f signal will be asymmetric, as the wire twists clockwise and counterclockwise, and can create a harmonic at 2f (as well as higher harmonics). These harmonics quickly become difficult to observe for small twist angles (below 2 deg), but, even so, we can detect a linear and quadratic correlation between the 1f and 2f signals when the interferometer is not aligned properly. The contamination of a 2f harmonic caused by harmonic distortion of the 1f amplitude actually becomes more problematic as the angle becomes smaller because its amplitude decreases linearly with the twist angle, whereas the contribution that we want to measure due to the mismatch between the OC and COM falls off as the square of the twist angle. This means that one cannot avoid this issue by limiting the measurement to small angles. The harmonic distortion occurring at 2f can be minimized by careful alignment of the interferometer so that the overlap of the interferometer beams is maximized at zero angle. The distortion of the 1f signal will then be symmetric and eliminate the contribution at 2f.

Fortunately, distortion of the 1f signal can effectively be removed by taking data at different swing angles and fitting the 2f amplitudes to a quadratic function of the 1f amplitude. In general, this is the best approach, even when no harmonics are visible in the signal. By varying the optical alignment, we are able to create data sets with small or large harmonic distortion of the 1f signal at 2f verified by the associated linear correlation of the 1f and 2f amplitudes. These experiments are used to verify our model for the source of linear correlation between the 1f and 2f amplitudes. It is important to emphasize that, even if there are unknown causes for the linear correlation of the 1f and 2f amplitudes, it is appropriate to remove the linear correlation because we know that errors linear in the angle will not cause an error for a free-fall measurement of acceleration of gravity.

5. Discussion

Our results show that this method allows us to balance the FG5 test mass, so that the quadratic error in the angle of rotation is below 1 nm for a rotation of about 2 deg (0.03 rad). It is useful to compare this to the original FG5 specification of this error of $2.5\mathrm{nm}/{\mathrm{s}}^2$ for a rotation rate of 0.01 rad/s for a 20 cm free fall (taking 0.2 s). Using this same rotation rate, the test mass rotates only. 002 rad during its free fall, generating an error of only 0.0044 nm with a quadratic dependence on the free-fall time using our new balancing method. Translating this into acceleration, the associated error is $0.22\mathrm{nm}/{\mathrm{s}}^2$. This shows that the new balancing method is about 10 times better than the standard method used to manufacture the FG5.

As discussed earlier, rotational errors that are proportional to the angle mimic an initial velocity in a free-fall measurement, whereas errors proportional to the square of the angle introduce errors into the measured acceleration. This provides another strong justification for removing any linear correlation of the swing amplitude on our measured 2f amplitudes for estimating the error on a measurement of gravity.

We have not specifically discussed balancing the proof mass in the other two axes because, as we discussed earlier, rotation around these other two axes causes an error that is linearly dependent upon time and, therefore, mimics an initial velocity. The gravity meter application is only sensitive to the acceleration of the freely falling proof mass and is insensitive to the initial velocity.

The test mass can also be balanced in the direction orthogonal to both the laser beam and the twist axis by balancing the test mass to eliminate the errors at the 1f frequency. A full 3D balancing requires rehanging the test mass after rotating it 90 deg around the laser beam axis. All three axes of the test mass can be balanced, so that the resulting optical error in the direction of the laser beam resulting from a rotation of the test mass of a few degrees is reduced to about 1 nm in displacement amplitude.

Our method provides a means to reduce interferometer errors to 1 nm for large rotations of the cube without regard to the mechanism for these errors. However, in order to compare this method to others, it is useful to interpret the result as a minimization of the offset between the COM of the proof mass with the OC of the imbedded corner-cube retroreflector. The relationship between a displacement error, $ϵ$, measured by the interferometer for an inline offset, ${\delta }_{||}$, between the OC and rotation axis (in the same direction as the laser) is $ϵ=\frac{1}{2}{\delta }_{||}{\theta }^2$. The optical error caused by a displacement of the OC, ${\delta }_{\perp }$, from the rotation axis orthogonal to the laser beam is $ϵ={\delta }_{\perp }\theta$.

Our method allows us to balance the object until the optical error is below $ϵ=1\mathrm{nm}$ for a swing angles of the order of several degrees, e.g., $\theta =2°$. This corresponds to a displacement between the OC and COM in the direction of the laser beam of 0.8 μm. This is the usual offset minimized in free-fall gravity meters. Of course, the method is much more sensitive, 0.03 μm, for minimizing the offset between the COM and the OC in the other two axes. In fact, although the noise background shown in our power spectrum is 1 nm for a 20 min average near the pendulum frequencies of interest, it is possible to increase the precision by averaging the results from repeated measurements. One nanometer is, therefore, a conservative error estimate.

6. Conclusion

We have devised a simple technique to provide direct optical measurement to eliminate errors in displacement measurement of a free-falling proof mass due to rotation of the optical retroreflector using a laser interferometer.

The proof mass is suspended by a thin torsion wire and allowed to slowly twist about its COM, while its linear position in a direction perpendicular to the twist is measured with a laser interferometer. This method directly simulates the errors generated when a proof mass is allowed to free fall in a vacuum and its descent is observed using a laser interferometer. The pendulum provides a convenient way to induce rotations around the COM without the need to determine the location of this point on the proof mass. The COM of the suspended mass can be adjusted to minimize the observed translation in the direction of the laser beam. This minimization is equivalent to collocating the OC of the corner-cube retroreflector imbedded in the proof mass with its COM.

The pendulum method constrains the motion of the proof mass to normal modes of the pendulum that can be separated by frequency. This allows different possible optical errors caused by rotation and translation to be untangled in an efficient manner.

Minimizing the rotational errors in an interferometer is important for the measurement of gravity using a freely falling object in a vacuum. Our method lets us balance the proof mass such that the rotational error caused by an angular velocity of 0.01 rad/s is only 0.2 nm or 0.02 μGal. This is 10 times better than the traditional method used for the FG5. This corresponds to a displacement error between the COM of the proof mass and the OC of the test mass in the direction of the laser beam of about 1 μm. This error is significantly smaller than that achieved by other methods.