A.

The two-mode interference signal

The two-mode [v_opt, v_opt + F] laser beam propagates along two different paths: a reference path of constant length l, and a measurement path of length L, variable. The interference of these two beams provides the signal from which we extract the quantity to be measured, i.e. the length difference between the two paths: ΔL = L - l.

In the diagram above, we assume that the polarization of the initial beam is oriented so that a small fraction of the power, ε² = a few %, propagates along the reference path with S polarization. The remaining power (fraction , close to 100%) propagates along the measurement path with P polarization (S polarization after the half-wave plate) and interferes with the reference beam. The intensity, I(t), of the two-mode interference is detected by photodiode PhD1. The (“reference”) photodiode PhD0 delivers a signal whose amplitude and phase are used to eliminate power and phase fluctuations of the laser sources.

Photodiode PhD1 receives a beam whose intensity has three contributions: beam intensity of the measurement beam (fraction (1−ε²)), intensity of the reference beam (fraction ε²), and the interference term:

where: ω ≡ 2πv_opt =2πc / λ_opt, S ≡ 2πF = 2πc / Λ. Note that this expression is valid only in vacuum. Application to quantitative measurements in air would require that two indices of refraction, at the two wavelengths, are taken into account.

The measurement exploits the phase and amplitude of the cos (δt) terms, modulated at the high frequency F.

We will use the vector representation, wherein a signal A cos(δt + φ) is represented by a vector in the complex plane, and we re-write eq. 1 in the form

where

The ranging procedure has to determine two key parameters of the signal, as shown in Figure 2:

Figure 1.

Schematic diagram of the telemeter

Figure 2.

Addition, in the complex plane, of the three contributions to the PhD1 two-mode interference, signal. The phase , of the PhD0 photodiode, is taken as the phase reference.

B.

Characteristics of the two-mode interference signal

Dependence on the target position

The ΔL dependence is more complex, as two of the three contributions to are affected by a change of the distance:

• i- not only the amplitude of the vector changes, due to the change of the cos ((ω+ δ/2)(L −1)/c) factor, but

• ii- its direction also changes, due to the e^iδ(L+l)/2c factor, and

• iii- finally the direction of the vector also changes, through the e^iδL/c factor.

Finally, the pattern obtained when the target is moved by half a synthetic wavelength displays a large number (ω/δ ≈ 10⁴) of spikes. The spikes are illustrated on Fig. 3, with the choice of ω/δ ≈ 20 for the spikes to be conveniently observable.

Figure 3.

The calculated two-mode interference pattern obtained, at constant v_opt, when the target is moved continuously over Λ/2, with ε = 0.25

Each spike corresponds to one fringe of the optical interference.

C.

The ranging procedure

From a single measurement of this complex interference signal (one single data point in the complex plane) one cannot extract the three contributions separately. However, assuming that a time-of-flight measurement has been done beforehand, with a precision better than 10⁻⁵, one has an estimate of the free spectral range FSR. One can then perform three measurements , in which the optical frequency of the beam is changed from v_opt to v_opt+ FSR / 4 and then to v_opt+ FSR / 2, respectively. Then these three data points of the two-mode interference signal can be processed in a relatively straightforward manner to compute, separately, the interferometric phase Φ_inter =2πΔL/λ_opt (modulo 2π), and synthetic phase Φ_synt =2πΔL/Λ (modulo 2π).

We perform a measurement of every 15 μs and assume that ΔL does not vary during the three-state pattern. With this 3-fold data group , we first calculate the position of the segment centre:

the length of the segment

as well as its direction, given, for instance, by the direction of

From eqns (3) and (4) above, and since ε² << 1, we can use a perturbative treatment to subtract the vector and obtain the vector. Ideally, one can use the direction of to obtain the value of the synthetic phase, that is, the orientation of the signal in the complex plane. In practice, this approach will fail, due to the slow drifts (larger than 10⁻⁴ cycle at the 1minute time scale, see sect. V-A) in the measurement chains. However, the direction of the segment provides another determination of the synthetic phase, and the combination of these two provides (L −1)/Λ in a way which eliminates the slow drifts. This point can be seen by writing the identity

In other words, one can obtain the synthetic phase δΔL/c as twice the angle between the and vectors: in this way, the long term phase drifts between the two detection chains are eliminated. This point, made possible by the choice of recording a two-mode interference signal, is essential in giving access to accuracies far below the optical wavelength.

Finally, the interferometric phase is obtained from the vector

D.

Conditions for the ranging procedure

For the procedure to be applicable, it has to fulfill two conditions:

Condition i) involves the acquisition rate (the time interval τ between the recordings of and ) and the target speed. With 3-fold data groups , and τ=15μs, the maximum target speed is calculated to be of about 2.6 μm/s which is unacceptable. However, with a modified, yet straightforward procedure with 5 points , a gain by a factor of 100 (at 260 μm/s) can be obtained on the target speed limit. Higher speed limits may still be obtained by a more sophisticated procedure. For a rapidly moving target, for which nanometer accuracy is not required, the bare measurement signal is easily obtained by masking the reference beam and can be used to provide ranging information at high rate and intermediate precision.

Condition ii) involves two different aspects.

The ranging procedure supports the movement of the target and allows to average N 3-fold data groups, yet keeping the benefit of interferometric measurement. The idea is to take advantage of the interferometric measurement, obtained at each cycle i, to correct the value of synthetic measurement ΔL_synt(t_i) for each of N 3-fold data groups to obtain N values of , i.e N values of L at the beginning of the measurement cycle, which lend themselves to averaging.

A.

Optical cross-talk

As well as the measurement and reference beams, characterized by the lengths L and l, and by the power fractions of 1-ε² and ε², a stray optical beam is characterized by the beam path length l_stray and the power fraction ε_stray². Each stray beam adds additional interference terms that tend to distort the segment. Depending on the value of l_stray it can be shown that the distortion of the segment can be an ellipse, or a figure 8, an S shape,… Indeed an analytic treatment (see annex A) shows that it can be described by the corresponding Lissajoux curves. For these contributions to be smaller than 10⁻⁴ one has to achieve Such a rejection factor cannot be expected from an anti-reflection coating. As a consequence all the parallel optics of the setup, such as PBS and BS cubes…, have been replaced by wedged optics. Checking the level of stray contributions can be done by monitoring the phase and amplitude when the beam on the reference arm is blocked, hence preventing the interference: modulations, when the optical wavelength λ_opt is scanned over 1nm, are the signature of the presence of stray beams or multiple reflexions.

Figure 7.

Elliptical distortion of the segment associated with the presence of 1 % sidebands in the laser beat-note spectrum. The optical wavelength of the laser source is scanned by 120 MHz in 12 seconds. a): amplitude, and b) phase of the PhD1 signal. c): complex plane representation, where the transverse coordinate of the segment is magnified to highlight the distortions.

Figure 8.

Segments obtained with 60000 5-fold data groups, when the master laser is slowly scanned (about 200 MHz in 5 minutes).

These recordings are performed with a path length L such that the segment is radial, that is, the situation where the change in the signal amplitude, and hence the AM-to-PM effects are maximum. As in Fig. 7, the transverse coordinate is magnified, to highlight the distortions.

B.

Laser beat-note sidebands

Eq. 1 is valid assuming a pure two-mode laser source. Optical amplifiers can affect the two-mode spectrum: self-modulation in a semi-conductor amplifier can add significant sidebands [12]. Fig. 7 shows data recorded when the beam is amplified in a semiconductor amplifier. The observed ellipticity is prohibitive (nearly 5 10^-2). This is due to the significant fraction (≈10^-2) of optical power in the sidebands, as observed with an optical spectrum analyzer, and has been eliminated by replacing the semiconductor optical amplifier by a fiber amplifier.

By using only wedged optics and a spectrally pure, two-mode beat-note we obtain a signal without distortions related to optical defects.

C.

Cross-talk in the electronic measurement channels

Electronic crosstalk between the two channels is one of the causes of systematic error. This effect can be corrected for if the four parameters of the cross-talk (two amplitudes and two phases) have been determined beforehand. To this end, we mask the reference path, and move the target by a few Λ to vary the phase difference of the two channels for a few π radians. The crosstalk is of the order of a few 10^-3 in amplitude, and proves to be quite stable over time.

D.

Amplitude-to-phase (AM-to-PM) coupling

Each measurement channel is composed of a photodiode, a HF amplifier, a mixer, an RF amplifier and an analog-to-digital converter. Each of these components may generate amplitude-to-phase coupling. Dedicated experiments [13] made to quantify this effect in different photodiodes have shown that the AM-to-PM coupling includes a significant part which has a transient behavior, with typical time scales in the 10 μs range. Other microwave components (amplifiers for instance) may introduce other time scales in the transient AM-to-PM coupling.

The ranging procedure implies large changes of the signal (up to a factor of 3 in amplitude) every 15 μs. As a consequence, beyond the distortion of the segment due to the stationary AM-to-PM, a hysteresis is expected at the 10-20 μs time scale due to the transient AM-to-PM. This is conspicuous in fig 8, with points . The fact that almost no hysteresis is observed with points and is due to the fact that

Phase of points a2, a3, a4 are measured 8 μs after optical frequency change. The transient effect can be approximately corrected for by the exponential model below:

where δ₁ = 8 μs, δ₂ = 15 μs result from the timing of the measurement procedure; A_i are amplitudes of the five data points; the free parameters are the factor f, and the time constants τ (the value is somewhat different for an increase and a decrease in the signal).

The stationary effect is characterized by measuring the phase as a function of the signal level, when the level is varied slowly. By using a cubic polynomial, the correction can be performed easily on the measured phase and the curvatures on the segment are eliminated.

Figure 13 below shows the two-mode interference signal when scanning the measurement length L, without and with the corrections of the AM-to-PM transfer. The curvature and hysteresis effects on the segments are well corrected and neighboring spikes do not overlap anymore.

Figure 9.

Spikes recorded by scanning the measurement length L, without and with AM/PM effect correction

Figure 10.

Spikes recorded at very high power, for different position of the target: complex plane representation. The optical frequency is kept constant. Blue data: simulation, with ε = 0.22, taking into account the AM-to-PM and saturation effects

Figure 11.

Stabilities of the relative phase and amplitude ratio measurements with an optical signal without interference (the reference path is masked). The average acquisition rate is 27μs. Points “Ampl C1” are almost completely covered by points “Ampl C2”.

Figure 12.

Results of 320 elementary cycles in 43 ms.

The target position and the two-mode optical frequency are fixed. a) In the presence of the measurement/reference interference, the variations in the data result from nm-scale variations in ΔL. Points: 320 measurement values. Blue curve: fit by a sum of harmonics corresponding to the frequency band of the measured acoustic perturbations on the setup. b) By masking the reference path and by converting the observed noise into length data: one measures the instrumental noise.

Figure 13.

The expected resolution of the telemetric measurement. Same data as in Fig. 12 b)

E.

Saturation effect

The electronic components in the measurement channel can give rise to saturation. In the complex plane, saturation effect may add a curvature on the segment and change the segment length. Figure 10 below shows segments recorded at very high power, to highlight this curvature.

The observed curvature results from both saturation and stationary AM-to-PM coupling. It turns out that in one part of the complex plane, the two effects approximately compensate: absence of curvature. The simulation (blue dots) is in agreement with the experimental data.

By working

A.

Statistical noise in the experimental data

Noise on the amplitude and phase of optical data is plotted as Allan deviation curves in Fig. 11. It clearly exceeds the noise recorded when the 20 MHz signal of a synthesizer is split and fed to the two channels of the data acquisition unit.

The amplitude ratio (red curve) efficiently eliminates common mode instabilities on the two amplitudes (green and blue curves). At short time scale (< 270 μs), only white noise is observable on the phase difference and amplitude ratio measurements. The humps from 270 μs to 10 ms on amplitudes ratio corresponds to acoustic noise (mainly fans) in the room which disturbs the 7.5 m measurement path. We chose to work with 1600 points (320 5-fold data groups), where the phase and amplitudes ratio have their optimum stability, at 2.5 10^-6 cycles and 2.5 10^-5, respectively. A full cycle of the measurement procedure corresponds to 43 ms of data taking. From 1s to 10s and above, instabilities appear, which can be several 10^-3 cycles at the 15 minute time scale. They are probably related to temperature instabilities of the circuits involved (amplifiers, mixers …). The HF amplifiers, in particular, dissipate significant power (several W). These drifts do not perturb the ranging procedure, performed and completed in a much shorter time (43 ms of data taking).

B.

Resolution of the interferometric measurement

The ranging procedure is tested over a folded ~ 7.5 m path on an optical table (simulating the measurement of a ~ 3.75 m distance). As the ToF measurement unit is not yet integrated on the telemeter setup, we tested the principle of two-mode interference measurement without the ToF measurement, but using a 3D measurement arm and the synthetic phase measurements at 20.04 GHz to obtain a value of L with an accuracy of about 10 μm.

Figure 12 below shows the results of 320 measurements in about 43 ms, corresponding to 320 elementary cycles, with then without interference.

From the noise observed on the data without interference one can expect a resolution of around 100pm on each elementary 5-fold measurement cycle (duration ≈100 μs), and significantly less than 100pm after the full cycle of a telemetric measurement (43 ms).

C.

Convergence of the high-accuracy measurement

For the telemetric measurement to deliver the correct value, it is necessary that the noise on the data be sufficiently low: if the noise is too high, the procedure may converge to a value which is shifted by one or several times λ_opt, even in the absence of systematics. We verified that the noise is low enough that the procedure converges mostly to the same value. Figure 14 below shows 85 successive measurements, during which the target is stationary at <10 nm. Only four errors are observable. This result is satisfactory, as the detection chains have not yet been optimized in terms of noise (HF amplifiers, in particular, are standard components).

Figure 14.

Convergence of the interferometric phase measurement, over 85 results.

D.

The accuracy and the effect of electronic distortions on the result

Here, the range measurement procedure is repeated while the target is moved continuously at about 2 μm/minute. The speed is a few wavelengths per minute. We record 10,000 full measurement cycles in 10 minutes.

The expected, ideal pattern for the two-mode interference signal, shown in Fig. 3, is a regular series of spikes, each spike being an interference fringe. As seen on Fig. 15 below, the recorded spikes are not regular at the 10⁻³ radian level. Due to the slow phase drifts in the two measurement chains at the 10s-1 mn time scale, the spikes are sometimes superposed. However, as mentioned above, the length measurements are completed in a much shorter time, and are not affected.

Figure 15.

The “spikes” of the telemetric signal trajectory in the complex plane when the measurement path length is varied by 25λopt ≈ 39 μm in 10 mn

After the phase and amplitudes measurements of 10,000 full cycles, we proceed to the calculation of 10000 length values , (see end of Sect. II-D), plotted in Fig. 16.

Figure 16.

The result of 10,000 full cycles (a) without and (b) with correction for electronic distortions. For convenience, the length values are shifted by 7, 492717m

Figure 17.

Same as Fig. 16, except for the v_opt frequency update rate, 10 times faster in Fig. 16

Without correcting for the electronic distortions, Figure 16-a, systematic errors are large (~100 μm), and correlated with the interference fringes, as expected from crosstalk and AM-to-PM coupling, since the interference order strongly modulates the signal amplitude. If electronic defects are corrected for, Figure 16-b, the result is much better, the procedure converges with a rate of 74.5%.

The wrong values (shifted by ±λ_opt) do not seem to be randomly scattered. Thus we suspect that more accurate correction of the data would further eliminate systematic effects. To approach 100% convergence, work should focus both on electronic imperfections and on noise.

This operation mode is not acceptable for efficient ranging, since it considerably slows the measurement rate, but it is a benchmark for quantifying the importance of transient effects in fast acquisition mode.