Abstract

In this paper, we present 10-pm-order mechanical displacement measurements using heterodyne interferometry. The measuring system includes a single-path heterodyne interferometer and a phase meter based on a phase-locked loop (PLL). It is not easy to measure a mechanical displacement of 10 pm or less owing to electronics and environmental noises in the interferometer. To solve this problem, the improvement of the noise floor is required for the phase meter. A PLL algorithm, which is programmed on a field-programmable gate array module, is used for efficient noise reduction of the phase meter. The interferometer combined with a stiff piezoelectric flexure stage is placed in a vacuum chamber. The measurement comparisons and the noise floor evaluations are performed between air and vacuum to evaluate effects from their environments. The interferometer has two spatially separated beams with different frequencies and two balanced optical arms. The measurement results demonstrate that the system combined with the above components is capable of measuring mechanical displacements of 11 pm in air and vacuum. A noise floor of 0.2 pm/$\sqrt{}{\rm Hz}$ between 50 Hz and 100 Hz can be obtained in vacuum. In this paper, the setup of the interferometer, the signal processing of the PLL, experiments, and results are discussed.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Continuing progress in manufacturing techniques and precision engineering has stimulated demand for the precision measurement of displacements with a resolution of sub-nanometer or 10 pm order. In recent years, many researchers have been competing to realize mechanical displacement measurement with a resolution of 10 pm (or less) using optical interferometers and piezoelectric (PZT) actuators [17]. At present, PZT actuation is one of the most promising ways to create such displacement if the actuator is combined with a stiff flexure stage [8], and optical interferometry is suitable for displacement measurements because of its traceability to the definition of the meter [9,10]. Among the optical interferometers, we consider that heterodyne interferometers [11,12] have the capability to measure picometer-order displacement because they have high noise immunity [13].

Currently, many phase detection methods such as the pulse counting/zero crossing [9,10], lock-in amplifier [1,14], single-bin discrete Fourier transform [15], and two-phase-locked loop (two-PLL) [2] methods are employed for heterodyne interferometry. However, it is not easy to measure a mechanical displacement of 10 pm or less owing to electronics and environmental noises in the interferometer. Such noises are generally from electronics devices, floor vibration, sound, airflow, thermomechanical drift, and other environmental fluctuations. The electronics and environmental noises introduce measurement uncertainties, which should be removed or minimized. We consider that improving the noise reduction algorithms for the phase meter and the utilization of vacuum [1618] can be used to minimize such noises.

In the previous work [6], a PLL phase meter was developed for heterodyne interferometry. An electronics resolution of 0.0174 mrad (${\sim}{1}\;{\rm pm}$ for a single-path interferometer) for the phase meter and 19 pm mechanical step displacement measurement using a four-path commercial interferometer were reported in [6]. However, since the PLL algorithm was programmed on a personal computer (PC) [6], the noise reduction of the PLL program was insufficient owing to the limited calculation speed of the PC. The calculation speed of a field-programmable gate array (FPGA) is generally very high (up to gigahertz or higher [19]. Therefore, we consider that suitable noise reduction of the PLL program can be attained using a high-speed FPGA.

In this paper, we present 10-pm-order mechanical displacement measurements using heterodyne interferometry. The measuring system includes a single-path heterodyne interferometer and a phase meter based on a PLL. The interferometer combined with a PZT flexure stage is placed in a vacuum chamber. The measurement comparisons and the noise floor evaluations between air and vacuum are also discussed to evaluate effects from their environments in this paper. We first describe our new interferometric setup. Second, we show a new PLL signal processing method with noise reduction operations, which is programmed on an FPGA. Finally, we report experiments on displacement measurements and noise floor evaluations in air and vacuum. The results show that the system combined with the above components can measure mechanical step displacements of 11 pm in air and vacuum and can provide a noise level of 0.2 pm/$\sqrt{}{\rm Hz}$ between 50 Hz and 100 Hz in vacuum.

2. HETERODYNE INTERFEROMETER WITH TWO SPATIALLY SEPARATED BEAMS HAVING DIFFERENT FREQUENCIES

Figure 1 shows a single-path heterodyne interferometer with two spatially separated beams having different frequencies. A frequency-stabilized He–Ne laser (Spectra-Physics 117 A, wavelength $\lambda = {633}\;{\rm nm}$, frequency ${f_0}$, frequency-stabilized mode, frequency instability ${\sim}{1}\;{\rm MHz}$ over 1 min [20]) is used as the light source. A beam splitter (BS) separates the incident light from the laser into two beams that are sent to two acousto-optic modulators (AOMs) (Gooch & Housego 3080-125) driven by function generators (FG1 and FG2). The AOMs deflect the incident beams into first-order Bragg diffraction parts with frequencies of ${f_1}$ and ${f_2}$, where the frequency difference is $\Delta f = {f_2} - {f_1}$ (2.1 MHz). Using steering mirrors with holders (M, M1, and M2) (mirror diameter: 25.4 mm, holder: Newport MFM-100), the first-order Bragg diffraction outputs are adjusted and aligned as two parallel incident beams into the interferometer.

 

Fig. 1. Schema of the single-path heterodyne interferometer. ISO, Faraday isolator; BS–BS2, beam splitters; AOM, acousto-optic modulator; FG1 and FG2, function generators; M–M2, mirrors; RR and TR, reference and measurement retroreflectors, respectively; P1 and P2, polarizers; HPF, high-pass filter; LPF, low-pass filter; ADC, analog-to-digital converter; FPGA, field-programmable gate array; PC, personal computer.

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The interferometer, as shown in Fig. 1, consists of BS1 (Agilent 10766 A), BS2 (Agilent 10770 A), and two retroreflectors (Agilent 10767 A): reference RR and target TR. BS1 and BS2 split the incident beams ${f_1}$ and ${f_2}$ into two parts: a reflected beam and a transmitted beam. In the reference arm, the reflected beam ${f_1}$ is reflected off the fixed RR and then combines with the transmitted beam ${f_2}$ at BS1 to generate interference. Similarly, the reflected beam ${f_2}$ from the moving TR combines with the transmitted beam ${f_1}$ at BS2 to create interference in the measurement arm. The interference lights propagate to photodetectors (PD1 and PD2) (Hamamatsu C5460, typical noise power ${\sim}{0.2}\;{\rm pW/\surd Hz}$, typical photoelectric sensitivity ${\sim}{1.5} \times {{10}^6}\;{\rm V}/{\rm W}$) through polarizers (P1 and P2) to form reference and measurement interference signals, respectively. At PD1, the reference signal has a fixed beat frequency $\Delta {f_R} = \Delta f$ (2.1 MHz) because RR is fixed. In contrast, the measurement signal at PD2 has $\Delta {f_M} = \Delta f + \Delta {f_D}$, where $\Delta {f_D}$ is the Doppler frequency, owing to the displacement of TR. At the entrance of each PD, the measured power of the interference laser beam is approximately 2 µW, which produces approximately $\pm {1}\;{\rm V}$ output signals (50 Ω impedance) from PDs. The output signals of the PDs then pass through high-pass filters (HPFs) (Thorlabs EF509, passband ${\gt}{1.8}\;{\rm MHz}$) and low-pass filters (LPFs) (Mini-Circuits, passband ${\lt}{1.9}\;{\rm MHz}$) before reaching the phase meter, which eliminates the residual low-frequency noises of the interference signals and decreases the amplitude of beat-frequency harmonics (4.2 MHz and 6.3 MHz). The signal-to-noise ratio (SNR) of the heterodyne signals measured after the PDs, HPFs, and LPFs is ${\sim}{69.5}\;{\rm dB}$ using a spectrum analyzer (Anritsu M26665C). This SNR is directly associated with the noises of the He–Ne laser head, AOMs, and PDs. The effect of the SNR on the electronics noise will be discussed in Section 4.B.

It is assumed that ${V_i}$ is the constant velocity of TR between the $i$th sampling time (${t_i} = i\Delta t$) and the ($i + 1$)th sampling time (${t_{i + 1}} = {t_i} + \Delta t$), where $\Delta t$ is the sampling time interval. The instantaneous geometrical displacement $\Delta {L_i}$ of TR between ${t_i}$ and ${t_{i + 1}}$ is $\Delta {L_i} = {V_i}\Delta t$. The instantaneous phase difference $\Delta {\varphi _{\textit{mi}}}$ between the reference and measurement signals owing to $\Delta {L_i}$ is expressed as

$$\Delta {\phi _{\textit{mi}}} = 2{\pi}\!\!\int_{{t_i}}^{{t_i} + \Delta t} {\Delta {f_{\textit{Di}}}{\rm d}t} = 2{\pi}\!\!\int_{{t_i}}^{{t_i} + \Delta t} {\frac{{2n{V_i}}}{\lambda}{\rm d}t} = \frac{{4{\pi}\!n}}{\lambda}\Delta {L_i},$$
where $\Delta {f_{\textit{Di}}}$ and $n$ are the instantaneous Doppler frequency and the refractive index of air. The accumulated total phase change $\varphi _m (t_K)$ over an interval from 0 to ${t_K} = K\Delta t$ is written as
$${\varphi _m}\left({{t_K}} \right) = \sum\limits_{i = 0}^K {\Delta {\phi _{\textit{mi}}}} = \frac{{4{\pi}\!n}}{\lambda}\sum\limits_{i = 0}^K {\Delta {L_i}} = \frac{{4{\pi}\!n}}{\lambda}L\!\left({{t_K}} \right)\!,$$
where $L({t_K})$ is the total displacement of TR at ${t_K}$.

In the optical configuration, the two transmitted beams are spatially separated and propagate in opposite directions. The reflected beams are located in the two symmetrically balanced arms. This may help ensure the thermal stability of the interferometer and avoid temperature gradients. It may also suppress the leakage of beams in the optics, leading to reduced periodic nonlinearity in the interferometer [1,2,14,18].

 

Fig. 2. Signal flow and processing of the phase detection: (a) whole layout and (b) phase shifter. PLL, phase-locked loop; LPF, low-pass filter.

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Fig. 3. Overall view of signal processing on the FPGA and PC. ADC, analog-to-digital converter; AI and AO, two channels of ADC; PLL, phase-locked loop; LPF, low-pass filter; FPGA, field-programmable gate array; PC, personal computer.

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3. PLL PHASE METER PROGRAMMED ON THE FPGA

The principle of phase detection with noise reduction is based on a single PLL [6]. Additional noise reduction is based on two moving averaging filters [21,22] combined with the PLL. The software of the PLL and the first moving averaging filter is programmed on an FPGA. The software of the second moving averaging filter is programmed on a host PC. The software is made using LabVIEW (National Instruments, Inc.). Figure 2 shows the signal flow and processing of the phase detection method. The PLL has two input sinusoidal signals (heterodyne signals: reference ${I_r}$ and measurement ${I_m}$) and one output signal (phase shift ${\varphi _m}$ between ${I_r}$ and ${I_m}$). The heterodyne signals are expressed as

$$\begin{split}{I_r} &= A\!\sin(\Delta \omega t), \\ {I_m} &= B\!\sin(\Delta \omega t + {\varphi _m}) = B\!\cos(\Delta \omega t + {\varphi _m} - \pi /{2}) \\&= B\! \cos(\Delta \omega t + {\theta _m}),\end{split}$$
where $A$, $B$, and $\Delta \omega$ are the amplitudes of the heterodyne signal and the angular beat frequency of the heterodyne laser source and ${\varphi _m} = {\theta _m} + \pi /{2}$ is assumed. The PLL structure consists of a phase shifter, a mixer, an LPF, and an integrator. First, ${I_r}$ is $\Phi$-shifted by a phase shifter, and the output signal $I_r^\prime$ becomes
$$I_r^\prime = A\! \sin (\Delta \omega t + \Phi).$$
Then, the mixer makes a product ($I_r^\prime \times {I_m}$) from the input signals. The product is written by
$$\!\!\!I{^\prime _r} \times {I_m} = \frac{{AB}}{2}\big[{\sin ({2\Delta \omega t + \Phi + {\theta _m}} ) + \sin ({\Phi - {\theta _m}} )} \big].\!$$
The product is then low-pass filtered to eliminate beat-frequency components and retain a DC component (i.e., error signal ${O_r}$) that is represented as
$${O_r} = \frac{{AB}}{2}\sin\! \left({\Phi - {\theta _m}} \right) \approx \frac{{AB}}{2}({\Phi - {\theta _m}} ),$$
where we assume $(\Phi - {\theta _m}) \ll {1}$. The integrator accumulates the phase $\Phi [n]$ at the $n$th sampling time so that ${O_r}$ is controlled to zero and $\Phi \; = {\theta _m}$, where ${\theta _m}$ is directly equivalent to the displacement $L({t_K})$ at ${t_K}$. The relationship between the input (${O_r}[n]$) and output ($\Phi [n]$) of the integrator is expressed as
$$\Phi [n] = \Phi [n - {1}]- {O_r}[n].$$
Figure 3 shows an overall view of the program. The total program consists of five loops, four of which are programmed on the FPGA and the other on the host PC. The first loop on the FPGA acquires two signals ${I_r}$ and ${I_m}$ from the analog to digital converter (ADC). The second loop on the FPGA contains a Hilbert transformer [23] to generate the quadrature component of ${I_r}$. The third loop (main processing) on the FPGA implements the PLL algorithm to generate ${\theta _m}$. The first, second, and third loops operate in a parallel working mode at a clock frequency of 120 MHz. Data is transferred among the first, second, and third loops by first-in and first-out (FIFO) access. FIFO is defined as first-in, first-out memory that is used to transfer data between loops on the FPGA, and between the FPGA and the PC [24]. The fourth loop is the moving averaging filter for ${\theta _m}$ [21]. The tap (or averaging) and decimation numbers of 1024 and 96 are selected, respectively, for the moving averaging filter in the fourth loop. The fifth loop is the other moving averaging filter. Its tap and decimation numbers are 2085 and 2085, respectively [22]. In the fourth loop, the clock frequency of 40 MHz is employed to keep the performance of the FPGA. With the decimation number of 96 in the fourth loop, the sampling clock frequency of the output signal ${\theta _m}$ is ${\sim}{417}\;{\rm kHz}$ ($= {40}\;{\rm MHz}/96$). The data ${\theta _m}$ is then streamed from the FPGA to the host PC by direct memory access FIFO. The fifth loop is implemented on the PC, and the final sampling clock frequency of the total system is ${\sim}{200}\;{\rm Hz}$ ($= {417}\;{\rm kHz}/2085$).

The Hilbert transformer, the LPF in the PLL, and the moving average filters are constructed using finite impulse response structures with a distributed arithmetic architecture [25]. In the second loop, ${I_r}$ has two components: in-phase and quadrature, whose phase difference is $\pi /{2}$ rad. If a 12-tap Hilbert filter is chosen for the Hilbert transformer, the in-phase component needs to be delayed by six samples (half a tap) [26]. For the LPF in the PLL, we select an 87-tap triangular window filter that has a cutoff frequency of 12 kHz and a magnitude attenuation of ${\sim}- {41.31}\;{\rm dB}$. Figure 4 shows the Bode diagram of the LPF in the third loop (main PLL). The second and third harmonics (4.2 MHz and 6.3 MHz) of the heterodyne signal are effectively rejected. The moving averaging filters in the fourth and fifth loops act as LPFs. Their cutoff frequencies, which result in ${-}{3}\;{\rm dB}$ attenuation, are 17.3 kHz (fourth loop) and 88.6 Hz (fifth loop) because we selected the tap and decimation numbers for their filters shown before. The PLL and the filters are designed by trial-and-error fashion based on preliminary experiments. Since multiple noise peaks in the band from kHz to 10 kHz are found at the output of the fourth loop, the fifth loop is added to remove them. Compared with the previous PC phase meter [6], the FPGA phase meter can have greater magnitude attenuation owing to many stages of noise filtering and reduction, even though they use almost the same output sampling rate (200 Hz).

 

Fig. 4. Bode diagram of the LPF (87-tap triangular window filter) in the PLL. (a) Magnitude and (b) phase responses.

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4. EXPERIMENTS AND RESULTS

We first test the electronics resolution of the PLL phase meter programmed on the FPGA. After that, we investigate the displacement measurements of the single-path heterodyne interferometer with this phase meter and the PZT flexure stage. The 2.1 MHz electronics and interferometric input signals are fed into an FPGA module (National Instruments PXIe-7972 R) with an ADC (National Instruments 5733, 120 MHz sampling rate, 2 V full-scale range, 16 bits) [27] for all measurements.

A. Electronics Resolution of the PLL Phase Meter Programmed on the FPGA

The configuration schematically shown in Fig. 3 is set up. Two 2.1 MHz purely electronic sinusoidal signals are generated by an FG (Agilent 33522A). The SNR of the measured electronic signals is ${\sim}{81.5}\;{\rm dB}$ using the spectrum analyzer (Anritsu M26665C) [6]. The phase shift between the two signals is measured by the new phase meter programmed on the FPGA. The result is compared with that of the previous PC phase meter in [6]. Both phase meters are implemented with the same instruments and under the same conditions.

Figure 5 shows the measurement results of the two phase meters for a stationary electronics measurement signal over ${\sim}{10}\;{\rm s}$. Figure 5(a) presents the time variation of the phase shifts. A standard deviation ($\sigma$) of ${\sim}{1.7}\;{\unicode{x00B5} \rm rad}$, which is lower than that obtained from the dashed curve (4.2 µrad [6]), is obtained from the solid curve. Figures 5(b) and 5(c) show the phase populations and the noise floors derived from the data in Fig. 5(a), respectively. In Fig. 5(b), the curve fitted to the circular marks has a narrower population than that fitted to the triangular marks, although both populations are almost Gaussian distributions. In Fig. 5(c), the noise floor of the FPGA phase meter is lower than that of the PC one between 0.1 Hz and 100 Hz. The noise floor of the FPGA phase meter is less than ${2}\;{\unicode{x00B5} {\rm rad}/\surd{\rm Hz}}$ below 1 Hz and reaches a limit level of ${\lt}{0.2}\;{\unicode{x00B5} {\rm rad}/\surd{\rm Hz}}$ between 20 Hz and 100 Hz. Moreover, the FPGA phase meter also obtains the same measurable phase step of 0.174 mrad as the PC phase meter [6]. Thus, the phase meter instrumented on the FPGA provides phase measurement with lower noise than that of the previous PC phase meter.

 

Fig. 5. Measurement result of the FPGA phase meter compared with that of the previous PC phase meter [6] for a stationary electronics measurement signal. (a) Phase shifts over time, (b) populations, and (c) phase noise floors of the data in (a).

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B. Displacement Measurements

The PZT flexure stage consists of a high-voltage PZT element (Physik Instrumente P-244.10, 0 to ${-}{1000}\;{\rm V}$, maximum force ${\sim}{2000}\;{\rm N}$, range ${\sim}{10}\;{\unicode{x00B5}{\rm m}}$) and a laboratory-made flexure guide. The flexure guide is made of low-thermal-expansion steel (Nippon Chuo LEX-ZERO, thermal expansion coefficient ${0.19} \times {{10}^{- 6}}/{\rm K}$) and has a monolithic structure. The PZT flexure stage is directly driven by the FG (Agilent 33522A). The PZT flexure stage provides both a sensitivity coefficient of 2.2 nm/V and a mechanical resonant frequency of ${\sim}{1.74}\;{\rm kHz}$.

We carry out measurements under the following four conditions using the setup (= the interferometer optics and the PZT flexure stage) shown in Fig. 6(a): (i) the setup is exposed to normal air; (ii–iv) the setup is placed inside a sealed 192 mm diameter vacuum chamber at atmospheric pressure (ii), ${\sim}{400}\;{\rm Pa}$ (iii), and 7 Pa (iv). TR and RR are mounted on the moving and fixed parts of the flexure stage, respectively. The distance between the retroreflectors and the BSs is ${\sim}{0.5}\;{\rm mm}$. The space between the two arms is less than 5 mm. A thermal insulation board (Yutaka HIB-A, thermal conductivity ${\sim}{0.3}\;{\rm W/mK}$) is inserted between the PZT flexure stage and the bottom of the chamber to impede the heat transmission from the bottom. The whole setup, as shown in Fig. 6(b), is located on an antivibration table within an enclosed room of area ${1.56}\;{{\rm m}^2}$ to suppress vibration, sound, airflow, thermal drift, and environmental effects. A vacuum pump (Osaka Vacuum) is used to decrease the chamber pressure measured by a Pirani gauge (Canon Anelva M-350PG). An overall view of the system is illustrated in Fig. 6(c). The experiments are performed under the conditions shown in Table 1.

To verify the displacement resolution, the PZT flexure stage is driven by the FG with two 0.8 Hz square-wave signals with driving peak-to-peak (p-p) amplitudes (mechanical displacements) of 10 mV (22 pm) and 5 mV (11 pm). The measurement results are shown in Figs. 7 and 8, respectively. For all four conditions (i–iv), the measuring system detects mechanical displacements of 22 pm (${\sim}{0.43}\;{\rm mrad}$ corresponds to ${0.43}\;{\rm mrad}/2\pi \times {633/2} \approx {22}\;{\rm pm}$ for a single-path interferometer) and 11 pm (${\sim}{0.22}\;{\rm mrad}$). A standard deviation $\sigma$ of ${\lt}{4}\;{\rm pm}$ on each step plateau is found from Figs. 7 and 8. As can be seen, at 7 Pa, the square-wave line is the most stable, whereas the line is distorted in normal air.

 

Fig. 6. Instrumentation layout of the setup. (a) Photograph of the single-path interferometer with the PZT flexure stage placed inside the chamber, (b) photograph of the whole setup on the antivibration table, and (c) overall view of the system. ISO, Faraday isolator; BS, beam splitter; AOM, acousto-optic modulator; M–M2, mirrors; RR and TR, reference and measurement retroreflectors, respectively; P1 and P2, polarizers; PZT, piezoelectric actuator; PD1 and PD2, photodetectors.

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Tables Icon

Table 1. Instruments and Conditions for Investigating the Picometer-Scale Mechanical Displacement

 

Fig. 7. Results of the measuring system for 0.8 Hz square-wave mechanical displacements with a 22 pm p-p amplitude under the four conditions in the enclosed room. (a) Setup exposed to normal air; setup placed inside a sealed vacuum chamber at (b) atmospheric pressure, (c) 400 Pa, and (d) 7 Pa.

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Fig. 8. Results of the measuring system for 0.8 Hz square-wave mechanical displacements with an 11 pm p-p amplitude under the four conditions in the enclosed room. (a) Setup exposed to normal air; setup placed inside a sealed vacuum chamber at (b) atmospheric pressure, (c) 400 Pa, and (d) 7 Pa.

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Figure 9 shows the phase noise floor and corresponding displacement noise floor analyzed from static measurements when TR is stopped over 30 s under four conditions (A)–(D). The noise floors of the four conditions generally decrease in the following order: ($A$) the experimental setup is not closed and exposed to normal air; ($B$) the experimental setup is placed inside the vacuum chamber at atmospheric pressure; ($C$) the experimental setup is placed inside the vacuum chamber at 8 Pa; and ($D$) there is electronic noise (from the phase meter and the PDs without lights) only. The noise floors of ($A$) and ($B$) have significant components between 0.01 Hz and 1 Hz. In contrast, the noise floor of ($C$) is almost flat, and its magnitude is approximately 40 and 150 times lower than those of ($A$) and ($B$), respectively, between 0.01 Hz and 1 Hz. However, the noise floor of ($C$) has some small peaks between 0.03 Hz and 0.07 Hz. The noise floor of ($D$), which is higher than those of Fig. 5(c) between 0.1 Hz and 100 Hz, is almost lower than ${0.2}\;{\rm pm}/\surd{\rm Hz}$ (${4}\;{\unicode{x00B5} \rm rad}/\surd{\rm Hz}$) between 0.01 Hz and 100 Hz. The combined system, especially the noise floor of ($C$), has a noise level of ${0.2}\;{\rm pm}/\surd{\rm Hz}$ between 50 Hz and 100 Hz. Figure 9 shows that the noise floor in vacuum may be better than that in air. However, a more precise investigation of the noise floors in air and vacuum is subjected to future works.

 

Fig. 9. Phase noise floor and corresponding displacement noise floor of the interferometer under different conditions: setup not enclosed and exposed to normal air ($A$); setup placed inside a vacuum chamber at atmospheric pressure ($B$) and 8 Pa ($C$); and electronic noise only ($D$).

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The electronics noise affected by the SNR of 69.5 dB from the output signals after PDs, HPFs and LPFs (see Section 2) can be estimated from the results shown in Section 4.A. Since the noise floor of the FPGA phase meter is less than ${2}\;{\unicode{x00B5} \rm rad}/\surd{\rm Hz}$ below 1 Hz in the case of the SNR of 81.5 dB, as shown in Fig. 5(c), the noise floor in the case of 69.5 dB can be ${2}\;{\unicode{x00B5} \rm rad}/\surd{\rm Hz} \times {{10}^{(81.5 - 69.5)/20}} = \sim 8\;{{\unicode{x00B5} \rm rad}/\surd{\rm Hz}}({0.4}\;{\rm pm}/\surd{\rm Hz})$ below 1 Hz. This noise floor is less than those shown in (A), (B), and (C) in Figs. 9. In addition, the frequency instability of the He–Ne laser also produces uncertainty in the displacement measurement. According to Eq. (2) the relationship between the uncertainty ($\Delta L$) in $L$ and the frequency variation ($\Delta f$) of the He–Ne laser source is found as $\Delta L/L = \Delta f/f$. With $\Delta f = {1}\;{\rm MHz}$ over 1 min for the 117 A He–Ne laser, $L = {1}\;{\rm mm}$ (dead path), and $f = \sim{474}\;{\rm GHz}$, the uncertainty is $\Delta L = \sim{2}\;{\rm pm}$ (${\sim}{40}\;{\unicode{x00B5} \rm rad}$ phase change) over 1 min. In our evaluation, we have not properly quantified the uncertainties derived from the interferometer (air fluctuations, vibrations, thermal deformations, and the frequency instability of the light source, etc.). It is necessary to investigate these causes in the future.

5. CONCLUSIONS AND FUTURE WORKS

In this study, a PLL phase meter with efficient noise reduction programmed on an FPGA has been developed. This new phase meter attains a lower noise floor than that of the previous PC phase meter [6]. A single-path heterodyne interferometer with two AOMs and a laboratory-made flexure stage combined with a PZT actuator are utilized. The interferometer has two spatially separated incident beams and a balanced optical configuration. Comparisons of mechanical displacements of 11 pm and 22 pm and noise floor evaluations in air and vacuum are performed using the system combined with the above components. The results demonstrate that the combined system can measure mechanical displacements of 10 pm order. The combined system has a noise level of ${0.2}\;{\rm pm}/\surd{\rm Hz}$ between 50 Hz and 100 Hz in vacuum. However, a more precise investigation of the noise floors in air and vacuum is one of future works. The current PLL and filters are designed in a trial-and-error fashion. Therefore, the developed PLL and filters might not be optimized. To optimize the PLL and filters using the control engineering principle, we must investigate the characteristics of electronics and environmental noises in the future. In this experiment, one filter was implemented on the host PC. The development of a phase meter that performs all the functions of the PLL and filters on only the FPGA is also one of the future works.

Funding

Japan Society for the Promotion of Science (20H02042).

Acknowledgment

We would also like to thank Chuo Precision Industrial Co., Ltd., for the use of the equipment.

Disclosures

The authors declare no conflicts of interest.

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12. R. C. Quenelle and L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–9 (1983).

13. S. Rahm, C. Loth, A. Dabas, and S. Elouragini, “Improvement of heterodyne detection with optical amplifier and pulsed coherent Doppler Lidar,” J. Mod. Opt. 41, 2145–2151 (1994). [CrossRef]  

14. S. Yokoyama, Y. Hori, T. Yokoyama, and A. Hirai, “A heterodyne interferometer constructed in an integrated optics and its metrological evaluation of a picometre-order periodic error,” Precis. Eng. 54, 206–211 (2018). [CrossRef]  

15. G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004). [CrossRef]  

16. E. C. Teague, “The National Institute of Standards and Technology molecular measuring machine project: metrology and precision engineering design,” J. Vac. Sci. Technol. B 7, 1898–1902 (1989). [CrossRef]  

17. Y. Niwa, K. Arai, A. Ueda, M. Sakagami, N. Gouda, Y. Kobayashi, Y. Yamada, and T. Yano, “Long-term stabilization of a heterodyne metrology interferometer down to a noise level of 20  pm over an hour,” Appl. Opt. 48, 6105–6110 (2009). [CrossRef]  

18. H. Fu, G. Wu, P. Hu, R. Ji, J. Tan, and X. Ding, “Highly thermal-stable heterodyne interferometer with minimized periodic nonlinearity,” Appl. Opt. 57, 1463–1467 (2018). [CrossRef]  

19. National Instruments, “Specifications PXIe-5775, PXI FlexRIO digitizer,” 2020, https://www.ni.com/pdf/manuals/377767b.pdf.

20. Spectra-Physics, “Stabilized Helium-Neon laser—Model 117A,” 2020, https://www.repairfaq.org/sam/brochures/SP117AL/index.html.

21. O. Vainio, “Minimum-phase FIR filters for delay-constrained noise reduction,” IEEE Trans. Instrum. Meas. 48, 1100–1102 (1999). [CrossRef]  

22. U. Hassan and M. S. Anwar, “Reducing noise by repetition: introduction to signal averaging,” Eur. J. Phys. 31, 453–465 (2010). [CrossRef]  

23. S. L. Hahn, Hilbert Transforms in Signal Processing (The Artech House, 1996).

24. K. Coffman, “More digital circuits: counter, RAMs, and FIFOs,” in Real World FPGA Design with Verilog (Prentice-Hall, 1999), pp. 123–155.

25. P. K. Meher, S. Chandrasekaran, and A. Amira, “FPGA realization of FIR filters by efficient and flexible systolization using distributed arithmetic,” IEEE Trans. Signal Process. 56, 3009–3017 (2008). [CrossRef]  

26. N. Kehtarnavaz and S. Mahotra, “Lab 13: FPGA implementation of 4-QAM receiver,” in Digital Signal Processing Laboratory: LabVIEW-Based FPGA Implementation (Brown Walker, 2010), pp. 403–421.

27. National Instruments, “NI 5731/5732/5733/5734R user guide and specifications,” 2020, https://www.ni.com/pdf/manuals/375653a.pdf.

References

  • View by:
  • |
  • |
  • |

  1. J. Lawall and E. Kessler, “Michelson interferometer with 10  pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
    [Crossref]
  2. M. T. L. Hsu, I. C. M. Littler, D. A. Shaddock, J. Herrmann, R. B. Warrington, and M. B. Gray, “Sub-picometer length measurement using heterodyne laser interferometry and all-digital rf phase meters,” Opt. Lett. 35, 4202–4204 (2010).
    [Crossref]
  3. M. Celik, R. Hamid, U. Kuetgens, and A. Yacoot, “Picometre displacement measurements using a differential Fabry–Perot optical interferometer and an x-ray interferometer,” Meas. Sci. Technol. 23, 085901 (2012).
    [Crossref]
  4. M. Pisani and A. Giugni, “A portable picometer reference actuator with 100  µm range, picometer resolution, subnanometer accuracy and submicroradian tip-tilt error for the characterization of measuring instruments at the nanoscale,” Metrologia 55, 541–547 (2018).
    [Crossref]
  5. T. V. Kazieva, K. L. Gubskiy, A. P. Kuznetsov, and V. N. Reshetov, “3D push–pull heterodyne interferometer for SPM metrology,” Appl. Opt. 58, 4000–4006 (2019).
    [Crossref]
  6. T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
    [Crossref]
  7. L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
    [Crossref]
  8. M. Aketagawa, Y. Ikeda, N. Tanyarat, and M. Ishige, “Optical encoder calibration using lattice spacing and optical fringe derived from a scanning tunnelling microscope and optical interferometer,” Meas. Sci. Technol. 18, 503–509 (2007).
    [Crossref]
  9. N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
    [Crossref]
  10. F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9, 1024–1030 (1998).
    [Crossref]
  11. J. A. Dahlquist, D. G. Peterson, and W. Culshaw, “Zeeman laser interferometer,” Appl. Phys. Lett. 9, 181–183 (1966).
    [Crossref]
  12. R. C. Quenelle and L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–9 (1983).
  13. S. Rahm, C. Loth, A. Dabas, and S. Elouragini, “Improvement of heterodyne detection with optical amplifier and pulsed coherent Doppler Lidar,” J. Mod. Opt. 41, 2145–2151 (1994).
    [Crossref]
  14. S. Yokoyama, Y. Hori, T. Yokoyama, and A. Hirai, “A heterodyne interferometer constructed in an integrated optics and its metrological evaluation of a picometre-order periodic error,” Precis. Eng. 54, 206–211 (2018).
    [Crossref]
  15. G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
    [Crossref]
  16. E. C. Teague, “The National Institute of Standards and Technology molecular measuring machine project: metrology and precision engineering design,” J. Vac. Sci. Technol. B 7, 1898–1902 (1989).
    [Crossref]
  17. Y. Niwa, K. Arai, A. Ueda, M. Sakagami, N. Gouda, Y. Kobayashi, Y. Yamada, and T. Yano, “Long-term stabilization of a heterodyne metrology interferometer down to a noise level of 20  pm over an hour,” Appl. Opt. 48, 6105–6110 (2009).
    [Crossref]
  18. H. Fu, G. Wu, P. Hu, R. Ji, J. Tan, and X. Ding, “Highly thermal-stable heterodyne interferometer with minimized periodic nonlinearity,” Appl. Opt. 57, 1463–1467 (2018).
    [Crossref]
  19. National Instruments, “Specifications PXIe-5775, PXI FlexRIO digitizer,” 2020, https://www.ni.com/pdf/manuals/377767b.pdf .
  20. Spectra-Physics, “Stabilized Helium-Neon laser—Model 117A,” 2020, https://www.repairfaq.org/sam/brochures/SP117AL/index.html .
  21. O. Vainio, “Minimum-phase FIR filters for delay-constrained noise reduction,” IEEE Trans. Instrum. Meas. 48, 1100–1102 (1999).
    [Crossref]
  22. U. Hassan and M. S. Anwar, “Reducing noise by repetition: introduction to signal averaging,” Eur. J. Phys. 31, 453–465 (2010).
    [Crossref]
  23. S. L. Hahn, Hilbert Transforms in Signal Processing (The Artech House, 1996).
  24. K. Coffman, “More digital circuits: counter, RAMs, and FIFOs,” in Real World FPGA Design with Verilog (Prentice-Hall, 1999), pp. 123–155.
  25. P. K. Meher, S. Chandrasekaran, and A. Amira, “FPGA realization of FIR filters by efficient and flexible systolization using distributed arithmetic,” IEEE Trans. Signal Process. 56, 3009–3017 (2008).
    [Crossref]
  26. N. Kehtarnavaz and S. Mahotra, “Lab 13: FPGA implementation of 4-QAM receiver,” in Digital Signal Processing Laboratory: LabVIEW-Based FPGA Implementation (Brown Walker, 2010), pp. 403–421.
  27. National Instruments, “NI 5731/5732/5733/5734R user guide and specifications,” 2020, https://www.ni.com/pdf/manuals/375653a.pdf .

2020 (2)

T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
[Crossref]

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

2019 (1)

2018 (3)

S. Yokoyama, Y. Hori, T. Yokoyama, and A. Hirai, “A heterodyne interferometer constructed in an integrated optics and its metrological evaluation of a picometre-order periodic error,” Precis. Eng. 54, 206–211 (2018).
[Crossref]

M. Pisani and A. Giugni, “A portable picometer reference actuator with 100  µm range, picometer resolution, subnanometer accuracy and submicroradian tip-tilt error for the characterization of measuring instruments at the nanoscale,” Metrologia 55, 541–547 (2018).
[Crossref]

H. Fu, G. Wu, P. Hu, R. Ji, J. Tan, and X. Ding, “Highly thermal-stable heterodyne interferometer with minimized periodic nonlinearity,” Appl. Opt. 57, 1463–1467 (2018).
[Crossref]

2012 (1)

M. Celik, R. Hamid, U. Kuetgens, and A. Yacoot, “Picometre displacement measurements using a differential Fabry–Perot optical interferometer and an x-ray interferometer,” Meas. Sci. Technol. 23, 085901 (2012).
[Crossref]

2010 (2)

2009 (1)

2008 (1)

P. K. Meher, S. Chandrasekaran, and A. Amira, “FPGA realization of FIR filters by efficient and flexible systolization using distributed arithmetic,” IEEE Trans. Signal Process. 56, 3009–3017 (2008).
[Crossref]

2007 (1)

M. Aketagawa, Y. Ikeda, N. Tanyarat, and M. Ishige, “Optical encoder calibration using lattice spacing and optical fringe derived from a scanning tunnelling microscope and optical interferometer,” Meas. Sci. Technol. 18, 503–509 (2007).
[Crossref]

2004 (1)

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

2000 (1)

J. Lawall and E. Kessler, “Michelson interferometer with 10  pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[Crossref]

1999 (1)

O. Vainio, “Minimum-phase FIR filters for delay-constrained noise reduction,” IEEE Trans. Instrum. Meas. 48, 1100–1102 (1999).
[Crossref]

1998 (1)

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9, 1024–1030 (1998).
[Crossref]

1994 (1)

S. Rahm, C. Loth, A. Dabas, and S. Elouragini, “Improvement of heterodyne detection with optical amplifier and pulsed coherent Doppler Lidar,” J. Mod. Opt. 41, 2145–2151 (1994).
[Crossref]

1993 (1)

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[Crossref]

1989 (1)

E. C. Teague, “The National Institute of Standards and Technology molecular measuring machine project: metrology and precision engineering design,” J. Vac. Sci. Technol. B 7, 1898–1902 (1989).
[Crossref]

1983 (1)

R. C. Quenelle and L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–9 (1983).

1966 (1)

J. A. Dahlquist, D. G. Peterson, and W. Culshaw, “Zeeman laser interferometer,” Appl. Phys. Lett. 9, 181–183 (1966).
[Crossref]

Aketagawa, M.

T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
[Crossref]

M. Aketagawa, Y. Ikeda, N. Tanyarat, and M. Ishige, “Optical encoder calibration using lattice spacing and optical fringe derived from a scanning tunnelling microscope and optical interferometer,” Meas. Sci. Technol. 18, 503–509 (2007).
[Crossref]

Amira, A.

P. K. Meher, S. Chandrasekaran, and A. Amira, “FPGA realization of FIR filters by efficient and flexible systolization using distributed arithmetic,” IEEE Trans. Signal Process. 56, 3009–3017 (2008).
[Crossref]

Anwar, M. S.

U. Hassan and M. S. Anwar, “Reducing noise by repetition: introduction to signal averaging,” Eur. J. Phys. 31, 453–465 (2010).
[Crossref]

Arai, K.

Braxmaier, C.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Celik, M.

M. Celik, R. Hamid, U. Kuetgens, and A. Yacoot, “Picometre displacement measurements using a differential Fabry–Perot optical interferometer and an x-ray interferometer,” Meas. Sci. Technol. 23, 085901 (2012).
[Crossref]

Chandrasekaran, S.

P. K. Meher, S. Chandrasekaran, and A. Amira, “FPGA realization of FIR filters by efficient and flexible systolization using distributed arithmetic,” IEEE Trans. Signal Process. 56, 3009–3017 (2008).
[Crossref]

Coffman, K.

K. Coffman, “More digital circuits: counter, RAMs, and FIFOs,” in Real World FPGA Design with Verilog (Prentice-Hall, 1999), pp. 123–155.

Culshaw, W.

J. A. Dahlquist, D. G. Peterson, and W. Culshaw, “Zeeman laser interferometer,” Appl. Phys. Lett. 9, 181–183 (1966).
[Crossref]

Dabas, A.

S. Rahm, C. Loth, A. Dabas, and S. Elouragini, “Improvement of heterodyne detection with optical amplifier and pulsed coherent Doppler Lidar,” J. Mod. Opt. 41, 2145–2151 (1994).
[Crossref]

Dahlquist, J. A.

J. A. Dahlquist, D. G. Peterson, and W. Culshaw, “Zeeman laser interferometer,” Appl. Phys. Lett. 9, 181–183 (1966).
[Crossref]

Danzebrink, H.-U.

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

Danzmann, K.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Demarest, F. C.

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9, 1024–1030 (1998).
[Crossref]

Ding, X.

Duong, Q.-A.

T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
[Crossref]

Elouragini, S.

S. Rahm, C. Loth, A. Dabas, and S. Elouragini, “Improvement of heterodyne detection with optical amplifier and pulsed coherent Doppler Lidar,” J. Mod. Opt. 41, 2145–2151 (1994).
[Crossref]

Fluegge, J.

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

Fu, H.

Garcia, A.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Giugni, A.

M. Pisani and A. Giugni, “A portable picometer reference actuator with 100  µm range, picometer resolution, subnanometer accuracy and submicroradian tip-tilt error for the characterization of measuring instruments at the nanoscale,” Metrologia 55, 541–547 (2018).
[Crossref]

Gouda, N.

Gray, M. B.

Gubskiy, K. L.

Hahn, S. L.

S. L. Hahn, Hilbert Transforms in Signal Processing (The Artech House, 1996).

Hamid, R.

M. Celik, R. Hamid, U. Kuetgens, and A. Yacoot, “Picometre displacement measurements using a differential Fabry–Perot optical interferometer and an x-ray interferometer,” Meas. Sci. Technol. 23, 085901 (2012).
[Crossref]

Hassan, U.

U. Hassan and M. S. Anwar, “Reducing noise by repetition: introduction to signal averaging,” Eur. J. Phys. 31, 453–465 (2010).
[Crossref]

Heinzel, G.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Herrmann, J.

Hetrick, P. S.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[Crossref]

Higuchi, M.

T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
[Crossref]

Hirai, A.

S. Yokoyama, Y. Hori, T. Yokoyama, and A. Hirai, “A heterodyne interferometer constructed in an integrated optics and its metrological evaluation of a picometre-order periodic error,” Precis. Eng. 54, 206–211 (2018).
[Crossref]

Hori, Y.

S. Yokoyama, Y. Hori, T. Yokoyama, and A. Hirai, “A heterodyne interferometer constructed in an integrated optics and its metrological evaluation of a picometre-order periodic error,” Precis. Eng. 54, 206–211 (2018).
[Crossref]

Hoyland, D.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Hsu, M. T. L.

Hu, P.

Ikeda, Y.

M. Aketagawa, Y. Ikeda, N. Tanyarat, and M. Ishige, “Optical encoder calibration using lattice spacing and optical fringe derived from a scanning tunnelling microscope and optical interferometer,” Meas. Sci. Technol. 18, 503–509 (2007).
[Crossref]

Ishige, M.

M. Aketagawa, Y. Ikeda, N. Tanyarat, and M. Ishige, “Optical encoder calibration using lattice spacing and optical fringe derived from a scanning tunnelling microscope and optical interferometer,” Meas. Sci. Technol. 18, 503–509 (2007).
[Crossref]

Jennrich, O.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Ji, R.

Johann, U.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Kazieva, T. V.

Kehtarnavaz, N.

N. Kehtarnavaz and S. Mahotra, “Lab 13: FPGA implementation of 4-QAM receiver,” in Digital Signal Processing Laboratory: LabVIEW-Based FPGA Implementation (Brown Walker, 2010), pp. 403–421.

Kessler, E.

J. Lawall and E. Kessler, “Michelson interferometer with 10  pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[Crossref]

Kobayashi, Y.

Koening, R.

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

Kramar, J. A.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[Crossref]

Kuetgens, U.

M. Celik, R. Hamid, U. Kuetgens, and A. Yacoot, “Picometre displacement measurements using a differential Fabry–Perot optical interferometer and an x-ray interferometer,” Meas. Sci. Technol. 23, 085901 (2012).
[Crossref]

Kuznetsov, A. P.

Lawall, J.

J. Lawall and E. Kessler, “Michelson interferometer with 10  pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[Crossref]

Littler, I. C. M.

Loth, C.

S. Rahm, C. Loth, A. Dabas, and S. Elouragini, “Improvement of heterodyne detection with optical amplifier and pulsed coherent Doppler Lidar,” J. Mod. Opt. 41, 2145–2151 (1994).
[Crossref]

Mahotra, S.

N. Kehtarnavaz and S. Mahotra, “Lab 13: FPGA implementation of 4-QAM receiver,” in Digital Signal Processing Laboratory: LabVIEW-Based FPGA Implementation (Brown Walker, 2010), pp. 403–421.

Meher, P. K.

P. K. Meher, S. Chandrasekaran, and A. Amira, “FPGA realization of FIR filters by efficient and flexible systolization using distributed arithmetic,” IEEE Trans. Signal Process. 56, 3009–3017 (2008).
[Crossref]

Middleton, K.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Molnar, G.

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

Nguyen, T.-D.

T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
[Crossref]

Niwa, Y.

Oldham, N. M.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[Crossref]

Peterson, D. G.

J. A. Dahlquist, D. G. Peterson, and W. Culshaw, “Zeeman laser interferometer,” Appl. Phys. Lett. 9, 181–183 (1966).
[Crossref]

Pisani, M.

M. Pisani and A. Giugni, “A portable picometer reference actuator with 100  µm range, picometer resolution, subnanometer accuracy and submicroradian tip-tilt error for the characterization of measuring instruments at the nanoscale,” Metrologia 55, 541–547 (2018).
[Crossref]

Quenelle, R. C.

R. C. Quenelle and L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–9 (1983).

Rahm, S.

S. Rahm, C. Loth, A. Dabas, and S. Elouragini, “Improvement of heterodyne detection with optical amplifier and pulsed coherent Doppler Lidar,” J. Mod. Opt. 41, 2145–2151 (1994).
[Crossref]

Reshetov, V. N.

Robertson, D.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Rudiger, A.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Sakagami, M.

Schilling, R.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Shaddock, D. A.

Tan, J.

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
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H. Fu, G. Wu, P. Hu, R. Ji, J. Tan, and X. Ding, “Highly thermal-stable heterodyne interferometer with minimized periodic nonlinearity,” Appl. Opt. 57, 1463–1467 (2018).
[Crossref]

Tanyarat, N.

M. Aketagawa, Y. Ikeda, N. Tanyarat, and M. Ishige, “Optical encoder calibration using lattice spacing and optical fringe derived from a scanning tunnelling microscope and optical interferometer,” Meas. Sci. Technol. 18, 503–509 (2007).
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Teague, E. C.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
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E. C. Teague, “The National Institute of Standards and Technology molecular measuring machine project: metrology and precision engineering design,” J. Vac. Sci. Technol. B 7, 1898–1902 (1989).
[Crossref]

Ueda, A.

Vainio, O.

O. Vainio, “Minimum-phase FIR filters for delay-constrained noise reduction,” IEEE Trans. Instrum. Meas. 48, 1100–1102 (1999).
[Crossref]

Vu, T.-T.

T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
[Crossref]

Wand, V.

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Warrington, R. B.

Wei, D.

T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
[Crossref]

Weichert, C.

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

Werner, C. A.

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

Wu, G.

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R. C. Quenelle and L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–9 (1983).

Yacoot, A.

M. Celik, R. Hamid, U. Kuetgens, and A. Yacoot, “Picometre displacement measurements using a differential Fabry–Perot optical interferometer and an x-ray interferometer,” Meas. Sci. Technol. 23, 085901 (2012).
[Crossref]

Yamada, Y.

Yano, T.

Yokoyama, S.

S. Yokoyama, Y. Hori, T. Yokoyama, and A. Hirai, “A heterodyne interferometer constructed in an integrated optics and its metrological evaluation of a picometre-order periodic error,” Precis. Eng. 54, 206–211 (2018).
[Crossref]

Yokoyama, T.

S. Yokoyama, Y. Hori, T. Yokoyama, and A. Hirai, “A heterodyne interferometer constructed in an integrated optics and its metrological evaluation of a picometre-order periodic error,” Precis. Eng. 54, 206–211 (2018).
[Crossref]

Yu, L.

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. A. Dahlquist, D. G. Peterson, and W. Culshaw, “Zeeman laser interferometer,” Appl. Phys. Lett. 9, 181–183 (1966).
[Crossref]

Classical Quantum Gravity (1)

G. Heinzel, V. Wand, A. Garcia, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rudiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quantum Gravity 21, S581–S587 (2004).
[Crossref]

Eur. J. Phys. (1)

U. Hassan and M. S. Anwar, “Reducing noise by repetition: introduction to signal averaging,” Eur. J. Phys. 31, 453–465 (2010).
[Crossref]

Hewlett-Packard J. (1)

R. C. Quenelle and L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–9 (1983).

IEEE Trans. Instrum. Meas. (1)

O. Vainio, “Minimum-phase FIR filters for delay-constrained noise reduction,” IEEE Trans. Instrum. Meas. 48, 1100–1102 (1999).
[Crossref]

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P. K. Meher, S. Chandrasekaran, and A. Amira, “FPGA realization of FIR filters by efficient and flexible systolization using distributed arithmetic,” IEEE Trans. Signal Process. 56, 3009–3017 (2008).
[Crossref]

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S. Rahm, C. Loth, A. Dabas, and S. Elouragini, “Improvement of heterodyne detection with optical amplifier and pulsed coherent Doppler Lidar,” J. Mod. Opt. 41, 2145–2151 (1994).
[Crossref]

J. Vac. Sci. Technol. B (1)

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[Crossref]

Meas. Sci. Technol. (4)

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9, 1024–1030 (1998).
[Crossref]

L. Yu, G. Molnar, C. A. Werner, C. Weichert, R. Koening, H.-U. Danzebrink, J. Tan, and J. Fluegge, “Single beam 3DoF homodyne interferometer,” Meas. Sci. Technol. 31, 084006 (2020).
[Crossref]

M. Aketagawa, Y. Ikeda, N. Tanyarat, and M. Ishige, “Optical encoder calibration using lattice spacing and optical fringe derived from a scanning tunnelling microscope and optical interferometer,” Meas. Sci. Technol. 18, 503–509 (2007).
[Crossref]

M. Celik, R. Hamid, U. Kuetgens, and A. Yacoot, “Picometre displacement measurements using a differential Fabry–Perot optical interferometer and an x-ray interferometer,” Meas. Sci. Technol. 23, 085901 (2012).
[Crossref]

Metrologia (1)

M. Pisani and A. Giugni, “A portable picometer reference actuator with 100  µm range, picometer resolution, subnanometer accuracy and submicroradian tip-tilt error for the characterization of measuring instruments at the nanoscale,” Metrologia 55, 541–547 (2018).
[Crossref]

Opt. Lett. (1)

Precis. Eng. (2)

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[Crossref]

S. Yokoyama, Y. Hori, T. Yokoyama, and A. Hirai, “A heterodyne interferometer constructed in an integrated optics and its metrological evaluation of a picometre-order periodic error,” Precis. Eng. 54, 206–211 (2018).
[Crossref]

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J. Lawall and E. Kessler, “Michelson interferometer with 10  pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[Crossref]

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T.-D. Nguyen, Q.-A. Duong, M. Higuchi, T.-T. Vu, D. Wei, and M. Aketagawa, “19-picometer mechanical step displacement measurement using heterodyne interferometer with phase-locked loop and piezoelectric driving flexure-stage,” Sens. Actuators A 304, 111880 (2020).
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Figures (9)

Fig. 1.
Fig. 1. Schema of the single-path heterodyne interferometer. ISO, Faraday isolator; BS–BS2, beam splitters; AOM, acousto-optic modulator; FG1 and FG2, function generators; M–M2, mirrors; RR and TR, reference and measurement retroreflectors, respectively; P1 and P2, polarizers; HPF, high-pass filter; LPF, low-pass filter; ADC, analog-to-digital converter; FPGA, field-programmable gate array; PC, personal computer.
Fig. 2.
Fig. 2. Signal flow and processing of the phase detection: (a) whole layout and (b) phase shifter. PLL, phase-locked loop; LPF, low-pass filter.
Fig. 3.
Fig. 3. Overall view of signal processing on the FPGA and PC. ADC, analog-to-digital converter; AI and AO, two channels of ADC; PLL, phase-locked loop; LPF, low-pass filter; FPGA, field-programmable gate array; PC, personal computer.
Fig. 4.
Fig. 4. Bode diagram of the LPF (87-tap triangular window filter) in the PLL. (a) Magnitude and (b) phase responses.
Fig. 5.
Fig. 5. Measurement result of the FPGA phase meter compared with that of the previous PC phase meter [6] for a stationary electronics measurement signal. (a) Phase shifts over time, (b) populations, and (c) phase noise floors of the data in (a).
Fig. 6.
Fig. 6. Instrumentation layout of the setup. (a) Photograph of the single-path interferometer with the PZT flexure stage placed inside the chamber, (b) photograph of the whole setup on the antivibration table, and (c) overall view of the system. ISO, Faraday isolator; BS, beam splitter; AOM, acousto-optic modulator; M–M2, mirrors; RR and TR, reference and measurement retroreflectors, respectively; P1 and P2, polarizers; PZT, piezoelectric actuator; PD1 and PD2, photodetectors.
Fig. 7.
Fig. 7. Results of the measuring system for 0.8 Hz square-wave mechanical displacements with a 22 pm p-p amplitude under the four conditions in the enclosed room. (a) Setup exposed to normal air; setup placed inside a sealed vacuum chamber at (b) atmospheric pressure, (c) 400 Pa, and (d) 7 Pa.
Fig. 8.
Fig. 8. Results of the measuring system for 0.8 Hz square-wave mechanical displacements with an 11 pm p-p amplitude under the four conditions in the enclosed room. (a) Setup exposed to normal air; setup placed inside a sealed vacuum chamber at (b) atmospheric pressure, (c) 400 Pa, and (d) 7 Pa.
Fig. 9.
Fig. 9. Phase noise floor and corresponding displacement noise floor of the interferometer under different conditions: setup not enclosed and exposed to normal air ( $A$ ); setup placed inside a vacuum chamber at atmospheric pressure ( $B$ ) and 8 Pa ( $C$ ); and electronic noise only ( $D$ ).

Tables (1)

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Table 1. Instruments and Conditions for Investigating the Picometer-Scale Mechanical Displacement

Equations (7)

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Δ ϕ mi = 2 π t i t i + Δ t Δ f Di d t = 2 π t i t i + Δ t 2 n V i λ d t = 4 π n λ Δ L i ,
φ m ( t K ) = i = 0 K Δ ϕ mi = 4 π n λ i = 0 K Δ L i = 4 π n λ L ( t K ) ,
I r = A sin ( Δ ω t ) , I m = B sin ( Δ ω t + φ m ) = B cos ( Δ ω t + φ m π / 2 ) = B cos ( Δ ω t + θ m ) ,
I r = A sin ( Δ ω t + Φ ) .
I r × I m = A B 2 [ sin ( 2 Δ ω t + Φ + θ m ) + sin ( Φ θ m ) ] .
O r = A B 2 sin ( Φ θ m ) A B 2 ( Φ θ m ) ,
Φ [ n ] = Φ [ n 1 ] O r [ n ] .

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